Nonlinear Dynamic Analysis of High-Voltage Overhead ...downloads.hindawi.com/journals/sv/2018/1247523.pdf · ResearchArticle Nonlinear Dynamic Analysis of High-Voltage Overhead Transmission

Research ArticleNonlinear Dynamic Analysis of High-Voltage OverheadTransmission Lines

Meng Zhang 1 Guifeng Zhao 1 and Jie Li 23

1School of Civil Engineering Zhengzhou University Zhengzhou 450001 China2College of Civil Engineering Tongji University Shanghai 200092 China3State Key Laboratory for Disaster Reduction in Civil Engineering Tongji University Shanghai 200092 China

Correspondence should be addressed to Guifeng Zhao gfzhaozzueducn

Received 24 July 2017 Revised 15 February 2018 Accepted 8 March 2018 Published 19 April 2018

Academic Editor Songye Zhu

Copyright copy 2018 Meng Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

According to a generalized Hamiltonrsquos principle three-dimensional (3D) nonlinear vibration equations for overhead transmissionlines that consider geometric nonlinearity are established Based on the characteristics of an actual transmission line the 3Dequations are simplified to two-dimensional equations and the nonlinear vibration behavior of transmission lines is investigatedby combining theoretical analysis with numerical simulation The results show that transmission lines have inherently nonlinearvibration characteristics When in free vibration a transmission line can undergo nonlinear internal resonance even when itsinitial out-of-plane energy is relatively low as its initial out-of-plane energy increases the coupling of in-plane and out-of-planevibration becomes stronger When forced to vibrate by an external excitation due to the combined action of internal and primaryresonance the vibration energy of a transmission line transfers from the out-of-plane direction to the in-plane direction that is notdirectly under the excitation resulting in an increase in the dynamic tension and the displacement amplitude of the transmissionline Increasing damping can consume the vibration energy of a transmission line but cannot prevent the occurrence of internalresonance

1 Introduction

High-voltage overhead transmission lines (HVOTLs) areimportant carriers within power systems and their safetydirectly affects the normal operation of the whole powersystem Currently aluminum conductor steel-reinforced(ACSR) cables are the most widely used transmission lines inengineering applications ACSR cables characterized by theirlight weight small damping and long spans can only bearaxial tension Generally the vibration of a transmission linein the plane where the transmission line is located under theaction of gravity is referred to as in-plane vibration and thevibration in the plane vertical to the aforementioned planeis referred to as out-of-plane vibration A transmission lineis often considered as a single-cable structure in calculationAccording to the suspended cable theory [1] the in-planestiffness of a transmission line is a result of its internalforce and the out-of-plane stiffness of a transmission lineis relatively low Therefore the vibration of a transmission

line exhibits conspicuous ldquolarge displacement small-strainrdquogeometrically nonlinear characteristics In particular wheninduced by wind and rain or strong wind transmission linesare more prone to relatively strong nonlinear vibration Insevere cases such vibration can cause damage to transmissionlines Therefore an in-depth investigation of the nonlinearvibration of HVOTLs is of great significance

Research on cables has progressed from statics to dynam-ics and from the linear theory to the nonlinear theory [1ndash14] In the static theory cables are viewed as two-force barswith an aim to simplify calculation As a result the resultsof calculation based on the static theory differ relatively sig-nificantly from the actual situation [1 2] In comparison thenatural frequencies of the out-of-plane and in-plane vibrationof a suspended cable obtained based on the linear dynamictheory of cables are relatively accurate Moreover it can alsobe derived from the linear dynamic theory of cables thatout-of-plane and in-plane modes of vibration are completelyindependent of one another [3] However cables are flexible

HindawiShock and VibrationVolume 2018 Article ID 1247523 35 pageshttpsdoiorg10115520181247523

2 Shock and Vibration

structures consequently their motion exhibits significantlynonlinear motion characteristics and very complex formsthat are a result of the combined action of external andinternal resonance In addition the geometric and physicalparameters of cables also have an impact on their dynamicresponse [4ndash14] Evidently even the linear dynamic theorycannot accurately describe the forms of motion of cables It isirrefutable that the wind-induced vibration response of long-span lightweight and small-damping transmission lines hasinherently nonlinear characteristics [15ndash24] However thecurrent design specification still requires the use of the linearstatic theory of suspended cables to design transmission lineswhich clearly cannot meet the engineering demand Theabovementioned linear design theory is easily mastered andapplied in engineering but its shortcomings are also obviousFor example it assumes that the relationship between thestatic load and the response of the whole line is a linearfunction in calculations which obviously does not conformto the nonlinearmechanical properties of a transmission lineFurthermore there is a significant difference between the in-plane and out-of-plane vibration of a transmission line andthe former has obvious nonlinearity stiffness that will induceadditional tension of the transmission lineTherefore it is notenough to calculate the response of a transmission line onlyby its first-order out-of-plane vibration In fact as a highlyflexible line system it is inevitable to consider the multiordervibrations of a transmission line and the dynamic tensiongenerated by the vibration of the transmission line under theaction of the pulsating wind is the main load that acts on thetransmission tower

Based on the aforementioned analysis according to ageneralized Hamiltonrsquos principle three-dimensional (3D)nonlinear differential equations of motion for horizontallysuspended transmission lines that consider the initial deflec-tion are established in this work Based on the characteristicsof an actual transmission line the 3D equations are simpli-fied to two-dimensional (2D) equations On this basis themethod of multiple scales is employed to theoretically studythe nonlinear internal resonance behavior of transmissionlines Moreover a higher-order Runge-Kutta method is alsoused to perform a numerical analysis of the nonlinearvibration characteristics of transmission lines to reveal thecharacteristics of the coupled action of the out-of-plane andin-plane vibration of transmission lines when in nonlinearvibration with the aim of providing a basis for the reasonabledesign of HVOTLs

2 3D Nonlinear Equations of Motion forElastic Transmission Lines

21 Establishing 3D Equations of Motion Figure 1 shows anelastic transmission line with its two ends hinge-supportedat the same height The initial static equilibrium geometricconfiguration 1198780 of the transmission line in the plane 119874119909119910 isused as the reference location and is represented by function119909(119910) 119889119888 119871119888 and 119879119888(119910 119905) represent the sag and span of thedynamic tension in the transmission line respectively 119878119863represents the 3D dynamic configuration of the transmission

y

z

x

O

P(z t)

P(x t)

P(y t)

Dynamic state SD

Static state S0

dc

Lc

yc

Figure 1 Spatial configuration of a transmission line

line under an external excitation 119875(119905) 119875(119909 119905) 119875(119910 119905) and119875(119911 119905) represent the components of the external excitation119875(119905) in the 119909- 119910- and 119911-axes respectively 119864119888 and 119860119888represent the elastic modulus and the cross-sectional areaof the transmission line respectively 120574119888 represents the deadweight of the transmission line per unit length (120574119888 = 119898119888119892where 119892 represents the acceleration of gravity) 119906119888(119910 119905)V119888(119910 119905) and 119908119888(119910 119905) represent the displacements of thetransmission line along the longitudinal in-plane vibration(119910-axis direction) the transverse in-plane vibration (119909-axisdirection) and the transverse out-of-plane vibration (119911-axisdirection) respectively

To simplify analysis the following assumptions [12 25]are made (a) the flexural torsional and shear stiffness of thetransmission line is sufficiently low and thus negligible (b)the transmission line only bears tension in the axial directionand its axial strain when vibrating is sufficiently small andthus negligible (c) only geometric nonlinearity is consideredwhereas material nonlinearity is not considered

When the tension in the transmission line is in the initialstatic equilibrium state the length 1198891199040 of the differentialelement at any arbitrary location119910119888 along the 119910-axis directionis

1198891199040 = radic1 + (120597119909120597119910)2119889119910 (1)

According to the definition of Lagrangian strain initiallythe length 119889119904 of the differential element of the transmissionline under the action of gravity is

119889119904 = radic1 + (120597119909120597119910)21 + 1205761198880 119889119910 (2)

where 1205761198880 represents the initial static strain of the transmis-sion line which is calculated using the following equation

1205761198880 = 119879119888 (119910 0)119864119888119860119888 (3)

where 119879119888(119910 0) represents the initial tension in the transmis-sion line

Shock and Vibration 3

The length of the differential element of the transmissionline after it has undergone dynamic deformation under anexternal excitation 119875(119905) is119889119904119863 = radic(1 + 120597119906119888120597119910 )2 + (120597119909120597119910 + 120597V119888120597119910 )2 + (120597119908119888120597119910 )2119889119910 (4)

The total dynamic strain [12] of the transmission line in amotion state under the external excitation is

120576119888 = 119889119904119863 minus 119889119904119889119904 = 1 + 1205761198880radic1 + (120597119909120597119910)2sdot radic(1 + 120597119906119888120597119910 )2 + (120597119909120597119910 + 120597V119888120597119910 )2 + (120597119908119888120597119910 )2 minus 1

(5)

The transient total tension 119879119888(119910 119905) in the differentialelement of the transmission line in a motion state is119879119888 (119910 119905) = 119879119888 (119910 0) + 119864119888119860119888120576119888 = 119879119888 (119910 0)

+ ( 1radic1 + (120597119909120597119910)2sdot radic(1 + 120597119906119888120597119910 )2 + (120597119909120597119910 + 120597V119888120597119910 )2 + (120597119908119888120597119910 )2 minus 1)sdot 119864119888119860119888

(6)

According to a generalized Hamiltonrsquos principle

120575int11990521199051

(119870V minus 119880) 119889119905 + int11990521199051

120575119882119889119905 = 0 (7)

where 119870V 119880 120575119882 represent the kinetic energy and the strainenergy of the transmission line and the virtual work done bynonconservative forces (including the virtual work 120575119882119888 doneby the damping force and the virtual work 120575119882119875 done by theexternal excitation) respectively

The kinetic energy of the transmission line is

119870V = 12 int1198711198880 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) (2119888 + V2119888 + 2119888) 119889119910 (8)

where 120574119888radic1 + (120597119909120597119910)2119892(1 + 1205761198880) represents the mass ofthe transmission line in the static equilibrium state per unitlength and 119888 V119888 and 119888 represent the vibration velocitiesof the transmission line in the 119910- 119909- and 119911-axis directionsrespectively

According to the aforementioned assumptions the strainenergy of the transmission line can only be generated by anaxial tension According to the principle of virtual work thestrain energy is

119880 = int1198711198880119879119888 (119910 0) 120576119888119889119904 + 12 int1198711198880 1198641198881198601198881205762119888119889119904 (9)

The virtual work done by the damping force is

120575119882119888 = minusint1198711198880119888119888 (119888120575119906119888 + V119888120575V119888 + 119888120575119908119888) 119889119910 (10)

where 119888119888 represents the structural damping of the transmis-sion line

The virtual work done by the external excitation 119875(119905) is120575119882119875= int1198711198880(119875 (119910 119905) 120575119906119888 + 119875 (119909 119905) 120575V119888 + 119875 (119911 119905) 120575119908119888) 119889119910 (11)

By substituting (8)ndash(11) into (7) and considering the bound-ary conditions the following is satisfied at the locations119910119888 = 0and 119910119888 = 119871119888 120575119906119888 = 120575V119888 = 120575119908119888 = 0 (12)

Thus the Euler equations for the transmission line alongthe virtual displacement 120575119906119888 120575V119888 and 120575119908119888 directions areobtained

120597120597119910 (119864119888119860119888 (1 + 120597119906119888120597119910)radic1 + (120597119909120597119910)2+ (119879119888 (119910 0) minus 119864119888119860119888) (1 + 120597119906119888120597119910)radic(1 + 120597119906119888120597119910)2 + (120597119909120597119910 + 120597V119888120597119910)2 + (120597119908119888120597119910)2)= 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 119888 + 119888119888119888 minus 119875 (119910 119905)

120597120597119910 (119864119888119860119888 (120597119909120597119910 + 120597V119888120597119910)radic1 + (120597119909120597119910)2+ (119879119888 (119910 0) minus 119864119888119860119888) (120597119909120597119910 + 120597V119888120597119910)radic(1 + 120597119906119888120597119910)2 + (120597119909120597119910 + 120597V119888120597119910)2 + (120597119908119888120597119910)2)= 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) V119888 + 119888119888V119888 minus 119875 (119909 119905)

120597120597119910 (119864119888119860119888 (120597119908119888120597119910)radic1 + (120597119909120597119910)2+ (119879119888 (119910 0) minus 119864119888119860119888) (120597119908119888120597119910)radic(1 + 120597119906119888120597119910)2 + (120597119909120597119910 + 120597V119888120597119910)2 + (120597119908119888120597119910)2)= 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 119888 + 119888119888119888 minus 119875 (119911 119905)

(13)

Because the sag-to-span ratio 119889119888119871119888 of a transmission lineis generally very small 120597119906119888120597119910 ≪ 1 120597V119888120597119910 ≪ 1 120597119908119888120597119910 ≪ 1


and 120597119909120597119910 ≪ 1 under small-strain conditions Thus thesesmall quantities can each be expanded in a Taylor series andtheir higher-order terms are negligible In other words in theabovementioned equations

1radic(1 + 120597119906119888120597119910)2 + (120597119909120597119910 + 120597V119888120597119910)2 + (120597119908119888120597119910)2asymp 1 minus 120597119906119888120597119910 minus (120597119906119888120597119910 )2 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119909120597119910)2+ 12 (120597V119888120597119910 )2 + 12 (120597119908119888120597119910 )2

(14)

Let

120578 = 120597119906119888120597119910 minus (120597119906119888120597119910 )2 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119909120597119910)2+ 12 (120597V119888120597119910 )2 + 12 (120597119908119888120597119910 )2

(15)

By substituting (15) into (13) nonlinear equations of motionfor the elastic transmission line that is horizontally hinge-supported at the two ends are obtained

120597120597119910 [[[

119864119888119860119888radic1 + (120597119909120597119910)2 + (119879119888 (119910 0) minus 119864119888119860119888) (1 minus 120578)]]]sdot (1 + 120597119906119888120597119910 ) = 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 119888 + 119888119888119888minus 119875 (119910 119905)

(16)

120597120597119910 [[[

119864119888119860119888radic1 + (120597119909120597119910)2 + (119879119888 (119910 0) minus 119864119888119860119888) (1 minus 120578)]]]sdot (120597119909120597119910 + 120597V119888120597119910 ) = 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) V119888 + 119888119888V119888minus 119875 (119909 119905)

(17)

120597120597119910 [[[

119864119888119860119888radic1 + (120597119909120597119910)2+ (119879119888 (119910 0) minus 119864119888119860119888) (1 minus 120578)]]]

120597119908119888120597119910 = 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 119888 + 119888119888119888 minus 119875 (119911 119905)

(18)

In (16)ndash(18) 119888 V119888 119888 represent the accelerations of vibra-tion of the transmission line along the 119910- 119909- and 119911-axesrespectively

The above set of partial differential equations contains allthe main vibration characteristics of the initial elastic trans-mission line continuum system that is the quadratic andcubic coupling terms associatedwith the initial deflection theaxial tension and the spatial motion which can be used toanalyze the nonlinear dynamic behavior of the transmissionline

22 Equations of Coupled In-Plane and Out-of-Plane Motionfor a Hinge-Supported Transmission Line Considering thatthe axial stiffness 119864119888119860119888 of a transmission line is generallyrelatively high and its axial elongation due to vibration is farsmaller than its transverse vibration amplitude the axial iner-tia force of a transmission line when in vibration is negligibleUnder normal circumstances the axial damping of a trans-mission line is relatively small thus the axial damping forceof a transmission line when in motion is negligible In addi-tion in Figure 1 since there is no external excitation alongthe axial direction of the transmission line (119910-axis direc-tion) 119875(119910 119905) = 0 Thus (16) can be rewritten as

120597120597119910 [[[

119864119888119860119888radic1 + (120597119909120597119910)2 + (119879119888 (119910 0) minus 119864119888119860119888) (1 minus 120578)]]]sdot (1 + 120597119906119888120597119910 ) = 0

(19)

The following approximation relation is then used

1radic1 + (120597119909120597119910)2 asymp 1 minus 12 (120597119909120597119910)2 (20)

By substituting (20) into (19) integrating the two sides of(19) over the span [0 119871119888] of the transmission line once andignoring the terms unrelated to time 119905 we have119879119888 (119910 0) [1 minus 12 (120597119909120597119910)2] sdot 120597119906119888120597119910 minus (119879119888 (119910 0) minus 119864119888119860119888) 120578

minus (119879119888 (119910 0) minus 119864119888119860119888) 120578 sdot 120597119906119888120597119910 = ℏ (119905) (21)

where ℏ(119905) is an arbitrary function By dividing the twosides of (21) by 119864119888119860119888 and considering that the initial tension119879119888(119910 0) in a transmission line is generally far smaller than itsaxial stiffness 119864119888119860119888 and that the axial deformation of a trans-mission line is a higher-order small quantity compared to itsvertical in-plane deformation and out-of-plane deformationwe have

120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119908119888120597119910 )2 + 12 (120597V119888120597119910 )2 asymp ℏ (119905)119864119888119860119888 (22)


To determine ℏ(119905) the two sides of (22) are integratedover the interval [0 119871119888] and the transmission line boundaryconditions represented by (23) are used After simplificationwe have (24)119906119888 (0 119905) = 0119906119888 (119871119888 119905) = 0 (23)

ℏ (119905)119864119888119860119888 sdot 119871119888 = int1198711198880 [120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119908119888120597119910 )2+ 12 (120597V119888120597119910 )2]119889119910

(24)

Let us set

120582 (119905) = ℏ (119905)119864119888119860119888 = 1119871119888 int1198711198880 [120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910+ 12 (120597119908119888120597119910 )2 + 12 (120597V119888120597119910 )2]119889119910

(25)

By substituting (22) and (25) into (17) and (18) respec-tively we have

120597120597119910 (119879119888 (119910 0) sdot [1 minus 12 (120597119909120597119910)2] sdot 120597V119888120597119910minus (119879119888 (119910 0) minus 119864119888119860119888) sdot 120582 (119905) sdot 120597119909120597119910)minus 120597120597119910 ((119879119888 (119910 0) minus 119864119888119860119888) sdot 120582 (119905) sdot 120597V119888120597119910 )= 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) V119888 + 119888119888V119888 minus 119875 (119909 119905)

(26)

120597120597119910 (119879119888 (119910 0) sdot [1 minus 12 (120597119909120597119910)2] sdot 120597119908119888120597119910minus (119879119888 (119910 0) minus 119864119888119860119888) sdot 120582 (119905) sdot 120597119908119888120597119910 )= 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 119888 + 119888119888119888 minus 119875 (119911 119905)

(27)

For ease of analysis by use of Galerkinrsquos modal trunca-tion method the in-plane and out-of-plane response of thetransmission line is represented by

V119888 (119910 119905) = 119873sum119894=1

120593119894 (119910) sdot 119902119894 (119905) 119908119888 (119910 119905) = 119873sum

119894=1

120593119894 (119910) sdot 119902119894 (119905) (28)

In (28) 120593119894(119910) = sin(119894120587119910119871119888) is the 119894th-order vibrationmode function (the two ends of the conductor line are

considered to be hinge-connected)119873 represents the retainedmode order and 119902119894(119905) represents the generalized coordinatesBy substituting it into (26) and (27) (28) can be transformedto ordinary differential equations of motion which aresubsequently solved Equation (28) is subjected to first-ordermodel truncation usingGalerkinrsquos modal truncationmethodthat is set

V119888 (119910 119905) = 120593V (119910) sdot 119902V (119905) = sin(120587119910119871119888 ) sdot 119902V (119905) 119908119888 (119910 119905) = 120593119908 (119910) sdot 119902119908 (119905) = sin(120587119910119871119888 ) sdot 119902119908 (119905)

(29)

By substituting (29) into (26) and (27) subsequentlymultiplying the two sides of (26) by 120593V(119910) integrating theresulting equation over the interval [0 119871119888] multiplying thetwo sides of (27) by 120593119908(119910) and integrating the resultant equa-tion over the interval [0 119871119888] discrete equations of coupled in-plane and out-of-plane motion for the transmission line areobtained

1198861 119902V (119905) + 1198862119902V (119905) + 11988631199022V (119905) + 11988641199022119908 (119905) + 11988651199023V (119905)+ 11988661199022119908 (119905) 119902V (119905) + 1198867 119902V (119905) = 119875V1198871 119902119908 (119905) + 1198872119902119908 (119905) + 1198873119902V (119905) 119902119908 (119905) + 11988741199022V (119905) 119902119908 (119905)+ 11988751199023119908 (119905) + 1198876 119902119908 (119905) = 119875119908(30)

The factors in (30) are calculated as follows

1198861 = int1198711198880

120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 1205932V (119910) 1198891199101198862 = minusint119871119888

0

120597120597119910 [119879119888 (119910 0) sdot (1 minus 12 (120597119909120597119910)2) sdot 1205931015840V (119910)minus (119879119888 (119910 0) minus 119864119888119860119888119871119888 ) sdot 120597119909120597119910 sdot int1198711198880 120597119909120597119910 sdot 1205931015840V (119910) 119889119910]sdot 120593V (119910) 119889119910

1198863 = int1198711198880

120597120597119910 [(119879119888 (119910 0) minus 1198641198881198601198882119871119888 ) sdot 120597119909120597119910sdot int119871119888012059310158402V (119910) 119889119910 + (119879119888 (119910 0) minus 119864119888119860119888119871119888 )

sdot 1205931015840V (119910) int1198711198880

120597119909120597119910 sdot 1205931015840V (119910) 119889119910]120593V (119910) 1198891199101198864 = int119871119888

0

120597120597119910 [(119879119888 (119910 0) minus 1198641198881198601198882119871119888 ) sdot 120597119909120597119910sdot int119871119888012059310158402119908 (119910) 119889119910]120593V (119910) 119889119910


1198865 = int1198711198880

120597120597119910 [(119879119888 (119910 0) minus 1198641198881198601198882119871119888 ) sdot 1205931015840V (119910)sdot int119871119888012059310158402V (119910) 119889119910]120593V (119910) 119889119910

1198866 = int1198711198880

120597120597119910 [(119879119888 (119910 0) minus 1198641198881198601198882119871119888 ) sdot 1205931015840V (119910)sdot int119871119888012059310158402119908 (119910) 119889119910]120593V (119910) 119889119910

1198867 = int11987111988801198881198881205932V (119910) 119889119910

119875V = int1198711198880119875 (119909 119905) 120593V (119910) 119889119910

1198871 = int1198711198880

120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 1205932119908 (119910) 1198891199101198872 = minusint119871119888

0

120597120597119910 [119879119888 (119910 0) sdot (1 minus 12 (120597119909120597119910)2) sdot 1205931015840119908 (119910)]sdot 120593119908 (119910) 1198891199101198873 = int119871119888

0

120597120597119910 [(119879119888 (119910 0) minus 119864119888119860119888119871119888 ) sdot 1205931015840119908 (119910)sdot int1198711198880

120597119909120597119910 sdot 1205931015840V (119910) 119889119910]120593119908 (119910) 1198891199101198874 = int119871119888

0

120597120597119910 [(119879119888 (119910 0) minus 1198641198881198601198882119871119888 ) sdot 1205931015840119908 (119910)sdot int119871119888012059310158402V (119910) 119889119910]120593119908 (119910) 119889119910

1198875 = int1198711198880

120597120597119910 [(119879119888 (119910 0) minus 1198641198881198601198882119871119888 ) sdot 1205931015840119908 (119910)sdot int119871119888012059310158402119908 (119910) 119889119910]120593119908 (119910) 119889119910

1198876 = int11987111988801198881198881205932119908 (119910) 119889119910

119875119908 = int1198711198880119875 (119911 119905) 120593119908 (119910) 119889119910

(31)

As demonstrated in (30) the in-plane and out-of-planemotion of the transmission line exhibits cubic nonlinear-ity whereas the coupled in-plane and out-of-plane motionexhibits linearity quadratic nonlinearity and cubic nonlin-earity that is the nonlinear coupling is very strong

3 Theoretical Analysis of theCharacteristics of the Nonlinear CoupledIn-Plane and Out-of-Plane InternalResonance of Transmission Lines

To further study the nonlinear coupled in-plane and out-of-plane dynamic characteristics of an overhead transmissionline we conducted the following analysis with the help ofthemultiple-scale (MS)method [26ndash29] which is an effectivemethod for the analysis of nonlinear dynamics

Assume that the external excitation is a harmonic excita-tion that is 119875 (119909 119905) = 119860V cos (Ω119909119905) 119875 (119911 119905) = 119860119908 cos (Ω119911119905) (32)

where 119860V and 119860119908 represent the amplitudes of the in-planeexcitation and the out-of-plane excitation respectively onthe transmission line and Ω119909 and Ω119911 represent the circularfrequencies of the in-plane excitation and the out-of-planeexcitation respectively

Here the in-plane excitation on the transmission line isnot considered that is the generalized in-plane excitationload 119875V is assumed to be zero Now the generalized out-of-plane excitation load on the transmission line is set to 119875119908 =1198750 cosΩ119911119905 In addition the parameters 120576 1205721 1205722 1205831 1205832 12058111205812 1205731 1205732 1205741 and 1198912 are introduced to transform (30) todimensionless equations

119902V (119905) + 12059621119902V (119905) + 12057612057211199022V (119905) + 12057612058111199022119908 (119905) + 12057612057311199023V (119905)+ 12057612057411199022119908 (119905) 119902V (119905) + 21205761205831 119902V (119905) = 0119902119908 (119905) + 12059622119902119908 (119905) + 1205761205722119902V (119905) 119902119908 (119905) + 12057612058121199022V (119905) 119902119908 (119905)+ 12057612057321199023119908 (119905) + 21205761205832 119902119908 (119905) = 21205761198912 cos (Ω119911119905) (33)

where 1205961 and 1205962 represent the circular frequencies of thein-plane and out-of-plane vibrations of the transmissionline respectively The abovementioned parameters have thefollowing conversion relationships

12059621 = 11988621198861 21205761205831 = 11988671198861 1205761205721 = 11988631198861 1205761205811 = 11988641198861 1205761205731 = 11988651198861 1205761205741 = 11988661198861 12059622 = 11988721198871


21205761205832 = 11988761198871 1205761205722 = 11988731198871 1205761205812 = 11988741198871 1205761205732 = 11988751198871 21205761198912 = 11987501198871

(34)

The abovementioned set of nonlinear equations can besolved using the MS method the basic idea of which isto regard the expansion of the response as a function withmultiple independent variables (or multiple scales) [26]

The following independent variables are introduced

119879119899 = 120576119899119905 (119899 = 0 1 ) (35)

that is

1198790 = 1199051198791 = 120576119905 (36)

Thus the derivative with respect to 119905 is transformed to theexpansion of a partial derivative with respect to 119879119899 that is

119889119889119905 = 1198891198790119889119905 1205971205971198790 + 1198891198791119889119905 1205971205971198791 + sdot sdot sdot= 1198630 + 1205761198631 + 12057621198632 + sdot sdot sdot 11988921198891199052 = 1198891198790119889119905 120597212059711987920 + 2120576 120597212059711987901205971198791 + sdot sdot sdot= 11986320 + 212057611986301198631 + 1205762 (11986321 + 211986311198632) + sdot sdot sdot (37)

Hence the solutions to (33) can be expressed as follows

119902V (119905 120576) = 119902V0 (1198790 1198791 1198792 ) + 120576119902V1 (1198790 1198791 1198792 )+ 1205762119902V2 (1198790 1198791 1198792 ) + sdot sdot sdot 119902119908 (119905 120576) = 1199021199080 (1198790 1198791 1198792 ) + 1205761199021199081 (1198790 1198791 1198792 )+ 12057621199021199082 (1198790 1198791 1198792 ) + sdot sdot sdot (38)

By substituting (37) and (38) into (33) we have

(11986321119902V2 + 212058311198631119902V2 + 212058311198632119902V1 + 211986301198631119902V3+ 212058311198630119902V3 + 12057311199023V1 + 21205721119902V1119902V2 + 61205731119902V0119902V1119902V2

+ 21205721119902V0119902V3 + 312057311199022V0119902V3 + 120574111990221199080119902V3+ 21205741119902V211990211990801199021199081 + 120574111990221199081119902V1 + 21205741119902V111990211990801199021199082+ 2120581111990211990811199021199082 + 21205741119902V011990211990811199021199082 + 2120581111990211990801199021199083+ 21205741119902V011990211990801199021199083) 1205764 + (11986320119902V3 + 12059621119902V3 + 212058311198632119902V0+ 11986321119902V1 + 212058311198631119902V1 + 211986301198631119902V2 + 212058311198630119902V2+ 12057211199022V1 + 31205731119902V01199022V1 + 21205721119902V0119902V2 + 312057311199022V0119902V2+ 1205741119902V211990221199080 + 21205741119902V111990211990801199021199081 + 120581111990221199081 + 1205741119902V011990221199081+ 2120581111990211990801199021199082 + 21205741119902V011990211990801199021199082) 1205763 + (11986320119902V2 + 12059621119902V2+ 11986321119902V0 + 212058311198630119902V0 + 211986301198631119902V1 + 212058311198630119902V1+ 312057311199022V0119902V1 + 2120581111990211990801199021199081 + 120574111990221199080119902V1+ 2120574111990211990801199021199081119902V0 + 21205721119902V0119902V1) 1205762 + (11986320119902V1 + 12059621119902V1+ 211986301198631119902V0 + 212058311198630119902V0 + 12057211199022V0 + 120581111990221199080 + 12057311199023V0+ 120574111990221199080119902V0) 120576 + 11986320119902V0 + 12059621119902V0 = 0

(119863211199021199082 + 2120583211986311199021199082 + 2120583211986301199021199083 + 2119863011986311199021199083+ 2120583211986321199021199081 + 120573211990231199081 + 1205722119902V31199021199080 + 21205812119902V0119902V31199021199080+ 21205812119902V1119902V21199021199080 + 12058121199022V11199021199081 + 1205722119902V21199021199081+ 21205812119902V0119902V21199021199081 + 1205722119902V11199021199082 + 21205812119902V0119902V11199021199082+ 61205732119902119908011990211990811199021199082 + 1205722119902V01199021199083 + 12058121199022V01199021199083+ 31205732119902211990801199021199083) 1205764 + (119863201199021199083 + 120596221199021199083 + 2120583211986321199021199080+ 119863211199021199081 + 2120583211986311199021199081 + 2119863011986311199021199082 + 2120583211986301199021199082+ 1205722119902V21199021199080 + 12058121199022V11199021199080 + 31205732119902119908011990221199081 + 1205722119902V11199021199081+ 31205732119902211990801199021199082 + 21205812119902V0119902V21199021199080 + 21205812119902V0119902V11199021199081+ 12058121199022V01199021199080 + 1205722119902V01199021199082) 1205763 + (119863201199021199082 + 120596221199021199082+ 119863211199021199080 + 2120583211986311199021199080 + 2119863011986311199021199082 + 2120583211986301199021199082+ 119863211199021199081 + 2120583211986311199021199081 + 31205732119902211990801199021199082 + 31205732119902119908011990221199081+ 21205812119902V0119902V11199021199081 + 21205812119902V0119902V21199021199080 + 1205722119902V21199021199080+ 1205722119902V11199021199081 + 1205722119902V01199021199082 + 12058121199022V01199021199082) 1205762 + (119863201199021199081+ 120596221199021199081 + 2119863011986311199021199080 + 2120583211986301199021199080 + 1205722119902V01199021199080+ 12058121199022V01199021199080 + 120573211990231199080) 120576 + 119863201199021199080 + 120596221199021199080 = 21205761198912sdot cos (Ω119911119905)

(39)


By setting the sum of the coefficients of the same powerof 120576 to zero we have1205760

11986302119902V0 + 12059612119902V0 = 0119863021199021199080 + 120596221199021199080 = 0 (40)

120576111986302119902V1 + 211986301198631119902V0 + 212058311198630119902V0 + 12059621119902V1 + 12057311199023V0+ 11988611199022V0 + 120574111990221199080119902V0 + 120581111990221199080 = 0119863021199021199081 + 2119863011986311199021199080 + 120573211990231199080 + 120596221199021199081 + 1198862119902V01199021199080+ 2120583211986301199021199080 + 12058121199022V01199021199080 = 21198912 cos (Ω119911119905)

(41)

120576211986320119902V2 + 12059621119902V2 + 11986321119902V0 + 212058311198630119902V0 + 211986301198631119902V1+ 212058311198630119902V1 + 312057311199022V0119902V1 + 2120581111990211990801199021199081 + 120574111990221199080119902V1+ 2120574111990211990801199021199081119902V0 + 21205721119902V0119902V1 = 0119863201199021199082 + 120596221199021199082 + 119863211199021199080 + 2120583211986311199021199080 + 2119863011986311199021199082+ 2120583211986301199021199082 + 119863211199021199081 + 2120583211986311199021199081 + 31205732119902211990801199021199082+ 31205732119902119908011990221199081 + 21205812119902V0119902V11199021199081 + 21205812119902V0119902V21199021199080+ 1205722119902V21199021199080 + 1205722119902V11199021199081 + 1205722119902V01199021199082+ 12058121199022V01199021199082 = 0

(42)

120576311986320119902V3 + 12059621119902V3 + 212058311198632119902V0 + 11986321119902V1 + 212058311198631119902V1+ 211986301198631119902V2 + 212058311198630119902V2 + 12057211199022V1 + 31205731119902V01199022V1+ 21205721119902V0119902V2 + 312057311199022V0119902V2 + 1205741119902V211990221199080+ 21205741119902V111990211990801199021199081 + 120581111990221199081 + 1205741119902V011990221199081 + 2120581111990211990801199021199082+ 21205741119902V011990211990801199021199082 = 0119863201199021199083 + 120596221199021199083 + 2120583211986321199021199080 + 119863211199021199081 + 2120583211986311199021199081+ 2119863011986311199021199082 + 2120583211986301199021199082 + 1205722119902V21199021199080 + 12058121199022V11199021199080+ 31205732119902119908011990221199081 + 1205722119902V11199021199081 + 31205732119902211990801199021199082+ 21205812119902V0119902V21199021199080 + 21205812119902V0119902V11199021199081 + 12058121199022V01199021199080+ 1205722119902V01199021199082 = 0

(43)

By solving the set of equations in (40) we have

119902V0 = 119860V (1198791 1198792) exp (11989412059611198790)+ 119860V (1198791 1198792) exp (minus11989412059611198790)

1199021199080 = 119860119908 (1198791 1198792) exp (11989412059621198790)+ 119860119908 (1198791 1198792) exp (minus11989412059621198790) (44)

where 119860V and 119860119908 are unknown complex functions and119860V and 119860119908 are the conjugates of 119860V and 119860119908 respectivelyFor ease of description conjugate terms are denoted by 119888119888The governing equations for 119860V and 119860119908 can be solved byrequiring 119902V0 119902V1 1199021199080 and 1199021199081 to be functions with a periodof 1198790

By substituting (44) into (41) we have

11986320119902V1 + 12059621119902V1 = minus211989412059611198631119860V119890119894212059611198790minus 211989412059611205831119860V119890119894212059611198790minus 12057411198602119908119860V119890119894(21205962+1205961)1198790minus 12057411198602119908119860V119890119894(1205961minus21205962)1198790minus 312057311198602V119860V11989011989412059611198790minus 21205741119860119908119860119908119860V11989011989412059611198790minus 12057311198603V119890119894312059611198790 minus 12057211198602V119890119894212059611198790minus 21205721119860V119860V minus 21205811119860119908119860119908minus 12058111198602119908119890119894212059621198790 + 119888119888119863201199021199081 + 120596211199021199081 = minus211989412059621198631119860119908119890119894212059621198790minus 211989412059621205832119860119908119890119894212059621198790minus 31205732119860211990811986011990811989011989412059621198790minus 21205812119860V119860V11986011990811989011989412059621198790minus 12057321198603119908119890311989412059621198790minus 1205722119860V119860119908119890119894(1205961+1205962)1198790minus 1205722119860V119860119908119890119894(1205961minus1205962)1198790minus 12058121198602V119860119908119890119894(21205961+1205962)1198790minus 12058121198602V119860119908119890119894(1205962minus21205961)1198790 + 1198912119890119894Ω1199111198790+ 119888119888

(45)

From (45) we can easily find that the terms that contain119890119894212059611198790 and 119890119894212059621198790 can lead to secular terms in the solutionsof the above equation In addition from (30) we know thatthe coupling of the in-plane and out-of-plane motion of thetransmission line exhibits linearity quadratic nonlinearityand cubic nonlinearity Thus if parameters exist that allowthe natural frequencies of the in-plane and out-of-planevibration of the transmission line to satisfy any of the relations1205961 asymp 1205962 1205961 asymp 21205962 and 1205962 asymp 21205961 the solutions to the


abovementioned partial differential equations that describethemotion of the transmission line will contain an additionalterm that links 119902V1 and 1199021199081 which is exactly the secular termcaused by internal resonance Therefore when solving (45)we should consider the cases of 1205961 asymp 1205962 1205961 asymp 21205962 and1205962 asymp 21205961 In view of the complexity of the problem-solvingand the purpose of this study we merely take the case of1205961 asymp 1205962 as an example to illustrate the analysis process

Assume that the natural frequencies of the in-plane andout-of-plane vibrations of the transmission line meet thefollowing relationship

1205962 minus 1205961 = 120576120590 120590 = 119874 (1) (46)

Thus

(1205962 + 1205961) 1198790 = 212059611198790 + 1205761205901198790 = 212059611198790 + 1205901198791= 212059621198790 + 1205901198791212059621198790 = 2 (1205961 + 120576120590) 1198790 = 212059621198790 + 21205901198791(47)

By substituting (47) into (45) we can obtain the require-ments to eliminate the secular terms as follows

21198941205961 (1198631119860V + 1205831119860V) + 12057211198602V + 1205811119860211990811989011989421205901198791 = 021198941205962 (1198631119860119908 + 1205832119860119908) + 1205722119860V1198601199081198901198941205901198791 = 0 (48)

The following formulas are introduced

119860119903 = 1198861199032 119890119895120579119903 119903 = V 119908 (49)

By substituting (49) into (48) and separating the real partfrom the imaginary part we have

1198631119886V + 1205831119886V + 12057211198862V21205961 sin 120579V+ 1205811119886211990841205961 sin 2 (120579119908 + 1205901198791) sec 120579V = 0119886V1198631120579Vminus csc 120579V41205961 [12057211198862V cos 2120579V + 12058111198862119908 cos 2 (120579119908 + 1205901198791)]= 0

1198631119886119908 + 1205832119886119908 + 1205722119886V11988611990841205962 cos 120579119908 sin (120579V + 120579119908 + 1205901198791) = 01198861199081198631120579119908 + 1205722119886V11988611990841205962 cos 120579119908 cos (120579V + 120579119908 + 1205901198791) = 0

(50)

Let us introduce the new phase angle as follows

1205931 = 2 (120579119908 + 1205901198791) 1205932 = 120579V + 120579119908 + 1205901198791 (51)


1198631119886V + 1205831119886V + 12057211198862V21205961 sin 120579V + 1205811119886211990841205961 sin1205931 sec 120579V = 0

119886V1198631120579V minus csc 120579V41205961 [12057211198862V cos 2120579V + 12058111198862119908 cos1205931] = 01198631119886119908 + 1205832119886119908 + 1205722119886V11988611990841205962 cos 120579119908 sin1205932 = 0

1198861199081198631120579119908 + 1205722119886V11988611990841205962 cos 120579119908 cos1205932 = 0(52)

When solving (52) wemust discuss whether the dampingof the system is considered In addition even if we can obtainthe one-order approximate steady-state solution of the equa-tion we must investigate its asymptotic stability to determinewhether it actually exists Hence we can see that the aboveissue is so complicated that it should be specifically studiedDue to the limitation of the abovementioned approximationmethod it is difficult to obtain the theoretical solutions to theequations of coupled in-plane and out-of-plane motion fortransmission lines Therefore in the subsequent section weintend to perform the quantitative analysis of (30) by usingnumerical methods

Because the main purpose of this article is to reveal theinherently nonlinear vibration characteristics of transmissionlines the above MS analysis is merely used to determine therequirements for eliminating secular terms in the solutionsof the partial differential equations of transmission linesBecause the second-order approximation solutions can pro-vide enough information for most practical problems it isusually necessary to give only the second-order approxima-tion solutions of a nonlinear system by using the MS method[27ndash29] For this reason we consider only the power of 120576up to 2 in this paper From the requirements for eliminatingsecular terms it is clear that internal resonance is consideredto exist in the transmission line if parameters exist thatallow the natural frequencies of the in-plane and out-of-plane vibration of the transmission line to satisfy any of thefollowing relations 1205961 asymp 1205962 1205961 asymp 21205962 and 1205962 asymp 212059614 Numerical Analysis of theCharacteristics of the Nonlinear CoupledIn-Plane and Out-of-Plane InternalResonance of Transmission Lines

In numerical analysis the Runge-Kutta method is widelyused to obtain the approximate solutions of differentialequations This method is so accurate that most computerpackages designed to find numerical solutions for differentialequations will use it by defaultmdashthe fourth-order Runge-Kutta method Therefore we employ a higher-order Runge-Kutta method to obtain the numerical solutions to theseequations with the aim of revealing the nonlinear vibrationcharacteristics of transmission lines


Figure 2 A high-voltage transmission line

The following state variables are introduced

1198841 = 119902V1198842 = 119902V1198843 = 1199021199084 = 119902119908(53)

Thus (30) can be transformed to state functions

1 = 11988422= (119875V minus 11988621198841 minus 119886311988421 minus 119886411988423 minus 119886511988431 minus 1198866119884231198841 minus 11988671198842)1198861 3 = 11988444 = (119875119908 minus 11988721198843 minus 119887311988411198843 minus 1198874119884211198843 minus 119887511988433 minus 11988761198844)1198871

(54)

41 Calculation of the Parameters A relatively large single-span transmission line in a high-voltage transmission linesystem in East China (Figure 2) is selected as an example foranalysis [30]The conductors of this span of the transmissionline are composed of LGJ-63045 ASCR cables Table 1 liststhe design parameters of the selected transmission line Themaximum design wind speed at a height of 10m above theground for safe operation of the selected transmission lineis 253ms In the actual transmission line the conductorsof each phase are quad-bundled conductors To simplifycalculation in the finite element (FE) modeling processthe quad-bundled conductors of each phase are equivalentlysimplified to one conductor based on the following principlesthe bundled conductors have the same windward area thesame total operational tension and the same linear densityTable 2 lists the parameters of the transmission line aftersimplification

The sag-to-span ratio 119889119888119871119888 of the selected transmissionline is calculated based on the parameters listed in Table 1 tobe approximately 129which is far smaller than 18Thus (30)can be used to analyze the nonlinear vibration characteristicsof this transmission line

42 Modal Analysis of the Transmission Line Based on theactual characteristics of the selected transmission line incombination with the design parameters listed in Tables 1and 2 LINK10 elements in ANSYS are used to construct anFE model for the selected transmission line First a modalanalysis is performed to determine the natural frequency ofvibration of each order of the transmission line model thisprovides a basis for the subsequent numerical simulation Tovalidate the FE model constructed for a single transmissionline in this section the FE simulation results with respect tothe natural vibration frequencies of the transmission line arealso compared with the theoretical solutions

A transmission line is a typical type of suspended cablestructure and the analytical solutions for the natural frequen-cies of its vibration can be obtained based on the single-cable theory A transmission line resists external loadingprimarily by self-stretching and consequently will undergorelatively large displacement under loading Therefore thegeometric nonlinearity of a transmission line must be takeninto consideration in the calculation When constructing anFE model of a transmission line one crucial point is todetermine the initial configuration of the cable Specificallyon the one hand this means to determine the spatial locationof the transmission line by determining the equilibrium anddeformation compatibility equations for the transmission linebased on the existing suspended cable theory on the otherhand this means to transition the transmission line from itsinitial stress-free state to an initial loaded state based on acertain configuration-seeking approach

The catenary and parabolic methods are the main meth-ods used to determine the equilibrium equation for a sus-pended cable The former is an accurate method whereasthe latter is an approximate method However when thesag-to-span ratio 119889119888119871119888 of a cable is less than 18 theparabolic method can also yield a relatively accurate solutionfor the initial configuration of the transmission line Hereassuming that the vertical load on a transmission line isevenly distributed along the span as shown in Figure 3 theequation for the spatial configuration of the transmission lineis

119909 (119910) = 11990211988821198791198880119910 (119871119888 minus 119910) + Δ 119888119871119888 119910 (55)

where 119902119888 represents the vertical load on the transmission linewhich in this case is the dead weight of the transmissionline 1198791198880 represents the initial horizontal tension in thetransmission line (1198791198880 = (1199021198881198712119888)(8119889119888)) 119871119888 represents thehorizontal span of the transmission line Δ 119888 represents thedifference in height between the two ends of the transmissionline and 119889119888 represents the mid-span sag of the transmissionline

Considering that a transmission line generally has a smallsag the first-order natural frequency corresponding to theout-of-plane vibration of the transmission line is the smallestfrequency of all the frequencies of its vibration Accordingto the linear vibration theory of suspended cables for ahorizontal suspended cable with two ends hinged at the sameheight and whose mass is evenly distributed the natural


Table1Orig

inaldesig

nparameterso

fthe

selected

transm

issionlin

e

Items

Diameter119889

(mm)

Cross-sectionalarea119860 119888

(mm2)

Linear

density119898 119888

(kgm)

Elastic

mod

ulus

119864 119888(MPa)

Averageo

peratio

nal

tension

(N)

Tensionatdesig

nwind

speed

(N)

Breaking

force

(N)

Span119871 119888

(m)

Sag119889 119888

(m)

Con

ductors

336

66655

206

63000

3531

643688

1412

65480

166


Table 2 Design parameters of the selected transmission line after simplification

Items Diameter 119889(mm)

Density 120588119888(kgm3)

Linear density119898119888(kgm)

Linear load 119902119888(Nm)

Initial strain1205761198880Conductors 1344 30905 824 80752 000084

frequencies of vibration and the corresponding modes underthe action of gravity alone are as follows

(a) The out-of-plane swing and the in-plane vibrationof the suspended cable are not coupled Thus the naturalcircular frequencies of the out-of-plane swing and the cor-responding modes can be expressed as follows

120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)

where 120596119908 represents the natural circular frequency of theout-of-plane vibration of the suspended cable 119899 representsthe modal number 119898119888 represents the mass of the suspendedcable per unit length Φ119908 represents the out-of-plane modeof vibration of the suspended cable and 119860119899 represents themodal amplitude (each of the other symbols has the samemeaning as previously described)

(b) The in-plane antisymmetric vibration of the sus-pended cable will not generate an increment in its horizontaldynamic tensionThe natural circular frequencies of its vibra-tion and the corresponding modes can be expressed as fol-lows

120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)

where 120596V represents the natural circular frequency of thein-plane antisymmetric vibration of the suspended cable 119899represents the modal number ΦV represents the mode of thein-plane antisymmetric vibration of the suspended cable and119861119899 represents themodal amplitude (each of the other symbolshas the same meaning as previously described)

(c) The in-plane symmetric vibration of the suspendedcable will generate an additional increment in its dynamictension The natural circular frequencies of its vibration andthe corresponding modes can be determined by solving thefollowing transcendental equations

tan1206032 = 1206032 minus 41205822 (1206032 )3 120603 = 120596vant119871119888radic1198791198880119898119888

O

B

y

x

qc

Lc

Δ c

Figure 3 A schematic diagram of the calculation for a single cablewhen the load is evenly distributed along the span

1205822 = (1198981198881198921198711198881198791198880)2 1198711198881198791198880119871119888 (119864119888119860119888) Φvant = ℎ11987911988801206032 (1 minus tan 1206032 sin

120603119910119871119888 minus cos120603119910119871119888 )

(58)

where 120596vant represents the natural circular frequency of thein-plane symmetric vibration of the suspended cable Φvantrepresents the in-plane symmetric mode of vibration of thesuspended cable 119864119888 represents the elastic modulus of thesuspended cable119860119888 represents the cross-sectional area of thesuspended cable and ℎ represents the additional tension inthe suspended cable (each of the other symbols has the samemeaning as previously described)

By substituting the parameters listed in Tables 1 and 2 into(55)ndash(58) the theoretical solutions for the natural vibrationfrequencies of the first eight orders of the transmission linein the three directions can be obtained (they are shownin Table 3) Table 4 summarizes the linear natural vibrationfrequencies of the transmission line obtained from FE simu-lation

A comparison of Tables 3 and 4 shows that the naturalvibration frequencies of the transmission linemodel obtainedfrom FE simulation are in relatively good agreement withthe theoretical values with the largest difference beingapproximately 1 This indicates that the transmission linemodel constructed in this work is relatively reliable More-over Tables 3 and 4 also demonstrate that the followingrelationships between the natural frequencies of the in-planeand out-of-plane modes of vibration of the transmission line


Table 3 Theoretical linear natural vibration frequencies of the transmission line model

Calculationequation

Firstorder

Secondorder

Thirdorder

Fourthorder Fifth order Sixth order Seventh

orderEighthorder

Out-of-plane vibrationfrequency 119891119908 (Hz)

119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091

In-plane symmetric vibrationfrequency 119891V (Hz)

119891V = 120596V(2120587) 0344 046 ndash ndash ndash ndash ndash ndash

In-plane antisymmetricvibration frequency 119891vant (Hz)

119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182

Table 4 Natural vibration frequencies of the transmission line model obtained from FE simulation

Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218

always exist 120596V asymp 120596119908 120596V asymp 2120596119908 and 120596V asymp 3120596119908 Inother words within the scope of the linear theory the naturalvibration frequencies of the transmission line are alwaysmultiples of one another

Figure 4 shows the modes of vibration of the first nineorders obtained from FE simulation As demonstrated inFigure 4 of all the modes of vibration of the transmissionline the first-order out-of-plane vibration is the most easilyexcitable The higher-order frequencies of in-plane vibrationare approximately integer multiples of the fundamental fre-quency and this relationship also exists between the frequen-cies of in-plane antisymmetric vibration and the fundamentalfrequency In addition to the relationship to the fundamentalfrequency the higher-order frequencies of the out-of-planevibration are also approximately equal to those of the in-plane vibration and the in-plane and out-of-plane modesof vibration corresponding to the same frequency are alsosimilar to one another

43 Numerical Analysis of the Nonlinear Internal Resonanceof the Transmission Line When in Free Vibration When onlythe free vibration of the transmission line under the actionof the initial displacement is considered in (54) 119875V = 0 and119875119908 = 0

The structural damping of a transmission line is often rel-atively small and it can generally be set to 00005 In additionthe effects of the structural damping of a transmission lineon its vibration particularly under strong wind conditionsare relatively insignificant whereas aerodynamic dampingrelatively significantly affects the vibration of a transmissionline By comprehensively considering the aforementionedfactors in this study when analyzing the free vibration of thetransmission line the damping 119888119888 of its in-plane and out-of-plane vibration is set to 0001 In the following sections theeffects of the initial potential energy of the transmission lineon its internal resonance are analyzed in two scenarios

431 When the Initial Potential Energy of the TransmissionLine Is Relatively Low (ie When the Initial Mid-Span Dis-placement of the Transmission Line Is Relatively Small) Letus set the initial out-of-plane mid-span displacement and theinitial velocity of out-of-planemotion of the transmission lineto 1m and 000005ms respectively and both its initial in-planemid-span displacement and the initial velocity of its in-plane motion to 0 that is the initial conditions described in(53) are set as follows 119902V = 0 119902V = 0 119902119908 = 1 119902119908 = 000005The displacements of the transmission line obtained fromnumerical simulation are then divided by the horizontal span119871119888 to obtain dimensionless displacementsThe dimensionlessvelocities of in-plane and out-of-plane can also be obtainedfrom the numerical simulation results of the in-plane andout-of-plane velocities multiplied by the first-order out-of-plane vibration period 1198790 and then divided by the horizontalspan 119871119888 The results are presented in Figures 5ndash7

As demonstrated in Figure 5 when the initial energyof the transmission line is relatively low its in-plane vibra-tion displacement is relatively small with the maximumamplitude being approximately 110 that of its out-of-planevibration displacement in addition the ldquobeat vibrationrdquophenomenon is inconspicuous in the response time historyAs demonstrated in the Fourier spectra of the displacementresponse of the transmission line (Figure 6) the first-ordermode of vibration is the primary out-of-plane mode ofvibrationThe out-of-plane vibration of the transmission linegives rise to the first-order in-plane antisymmetric mode ofvibration and the first-order symmetric mode of vibration Inaddition the first-order in-plane symmetric vibration energyis also greater than the first-order in-plane antisymmetricvibration energy and the in-plane vibration energy amplitudeis smaller than the out-of-plane vibration energy amplitudeby one order of magnitude This indicates that the trans-mission line is inherently capable of generating nonlinearinternal resonance and can undergo internal resonance even


X

Y

Z

First out-of-plane mode ofvibration (symmetric)

(a) First order 0136Hz

X

Y

Z

First in-plane mode ofvibration (antisymmetric)

(b) Second order 027Hz

X

Y

Z

Second out-of-plane mode ofvibration (antisymmetric)

(c) Third order 0272Hz

X

Y

Z

First in-plane mode ofvibration (symmetric)

(d) Fourth order 0346Hz

X

Y

Z

Third out-of-plane mode ofvibration (symmetric)

(e) Fifth order 0408Hz

X

Y

Z

Second in-plane mode ofvibration (symmetric)

(f) Sixth order 046Hz

X

Y

Z

Second in-plane mode ofvibration (antisymmetric)

(g) Seventh order 0544Hz

X

Y

Z

Fourth out-of-plane mode ofvibration (antisymmetric)

(h) Eighth order 0544Hz

X

Y

Z

Fifth out-of-plane mode ofvibration (symmetric)

(i) Ninth order 068Hz

Figure 4 Modes of vibration of the transmission line of the first nine orders obtained from FE simulation

when its initial energy is relatively low but energy transferand mode coupling are quite weak under this conditionAs demonstrated in Figure 7 the out-of-plane and in-planemodes of vibration of the transmission line both exhibitperiodic motion characteristics

432 When the Initial Potential Energy of the TransmissionLine Is Relatively High (ie When the Initial Mid-Span Dis-placement of the Transmission Line Is Relatively Large) Letus set the initial out-of-plane mid-span displacement of thetransmission line to a relatively large value (10m) the initialvelocity of its out-of-plane mid-span motion to 000005msand both its initial in-plane mid-span displacement and theinitial velocity of its in-plane motion to 0 that is the initialconditions described in (53) are set as follows 119902V = 0 119902V = 0

119902119908 = 10 119902119908 = 000005 Figures 8ndash10 show the numericalsimulation results

As demonstrated in Figure 8 when the initial out-of-plane displacement of the transmission line is relatively largeas a result of the internal resonance the system energyconstantly transfers between the modes that are in the 120596V asymp120596119908 120596V asymp 2120596119908 and 120596V asymp 3120596119908 relationships in addition thesystem energy also transfers between degrees of freedom thatare similarly related to multiples of the frequency On the onehand the out-of-plane vibration energy transfers to the in-plane direction resulting in vibration with a relatively largeamplitude Figure 8 also demonstrates that the order of mag-nitude of the in-plane vibration displacement amplitude iscommensurate with the out-of-plane vibration displacementamplitude a phenomenon similar to ldquobeat vibrationrdquo is alsoobserved in the displacement response time history that is


minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)

(a) In-plane mid-span displacement time history

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3

(b) Out-of-plane mid-span displacement time history

Figure 5 In-plane and out-of-planemid-span displacement time histories of the transmission linewhen its initial potential energy is relativelylow

0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ

outminusofminusplane

f = 01361（Ｔ

(b) Out-of-plane

Figure 6 Fourier spectra of the in-plane and out-of-plane mid-span displacements of the transmission line when its initial potential energyis relatively low

as the out-of-plane vibration response gradually decreasesthe in-plane vibration response at the corresponding timegradually increases and vice versa Furthermore from Fig-ures 5 and 8 we can also see that both of the maximum in-plane and out-of-plane vibration amplitudes decrease withtime This is principally because the damping dissipates thefree vibration energy On the other hand when Galerkinrsquosmethod is employed in numerical simulation to performfirst-order modal truncation on the vibration response ofthe transmission line to simplify the analytical process the

simulation results show that the second-order out-of-planemode of vibration of the transmission line as well as its first-order symmetric mode of vibration and its first- to fourth-order antisymmetric modes of vibration are still excitedAs a result of the internal resonance the first-order out-of-plane mode of vibration which has relatively high energygives rise to the second-order out-of-planemode of vibrationwhose frequency is twice its own frequency Because thenatural frequencies of the second-order out-of-plane modeof vibration and the first-order in-plane antisymmetric mode


minus4minus2

02

40

05

1

equilibrium configurationinitial statusoutminusofminusplane displaced state

004

003

002

001

0

times10minus3

xL

c

yLc

wcL c

(a) Configuration of out-of-plane motion at various times

times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c

minus15 minus1 minus05 0 05 1 15 2minus2wcLc

(b) In-plane and out-of-plane displacement responses in the phasespace

minus1 minus05 0 05 1 15minus15minus2

minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c

(c) In-plane mid-span phase trajectory

minus1 0 1 2minus2wcLc

minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3

(d) Out-of-plane mid-span phase trajectory

Figure 7 Configuration of the out-of-plane motion and the phase-space response of the transmission line when its initial potential energyis relatively low

of vibration are closely spaced the out-of-plane vibrationenergy gradually transfers to the in-plane direction In addi-tion because the natural frequency of the in-plane vibrationof each order is two or three times the in-plane fundamentalfrequency the energy also transfers between different in-plane modes of vibration The Fourier spectra of the dis-placement response shown in Figure 9 clearly demonstratethat the in-plane vibration also contains the first-order out-of-plane mode of vibration Similarly as demonstrated inFigure 10 the in-plane motion of the transmission line isstrongly coupled to its out-of-plane motion

The aforementioned analysis shows that the out-of-planevibration energy of the transmission line can give rise notonly to in-plane antisymmetric vibration but also to first-or-der in-plane symmetric vibration whose energy is the highest

of all the in-plane modes of vibration According to thesuspended cable theory in-plane symmetric vibration is theprimary factor that causes changes in the dynamic tension ina suspended cable To further illustrate the effects of internalresonance on the dynamic tension in the transmission linethe dynamic tension in the transmission line is calculatedunder the aforementioned two initial potential energy condi-tions (see Figure 11 for the results) In Figure 11 the tensionis a dimensionless value obtained by dividing the dynamictension 119879119888 obtained from simulation by the initial tension1198791198880 As demonstrated in Figure 11 when the transmissionline vibrates with relatively low initial potential energy thedynamic tension in the line is always positive indicating thatthe transmission line is under tension Meanwhile when theline is in vibration the dynamic tension in the transmission


minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3

(a) In-plane mid-span displacement

minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)

(b) Out-of-plane mid-span displacement

Figure 8 In-plane and out-of-planemid-span displacement time histories of the transmission linewhen its initial potential energy is relativelyhigh

02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane

Figure 9 Fourier spectra of the in-plane and out-of-plane mid-span displacements of the transmission line when its initial potential energyis relatively high

line is approximately the same as the initial tension Thissuggests that when the transmission line vibrates freelywith relatively low initial potential energy the effects ofinternal resonance on the tension in the transmission line arenegligible The spectral distribution of the dynamic tensionshown in Figure 11(c) also demonstrates that when the initialpotential energy of the transmission line is relatively lowthe tension in the transmission line is almost unaffectedby its vibration When the initial potential energy of thetransmission line is relatively high the dynamic tension inthe transmission line may reach twice the initial tensionThe spectral distribution of the dynamic tension shown in

Figure 11(d) also demonstrates that the increase in the tensionin the line is related to its in-plane vibration This indicatesthat when the transmission line is in motion with a largeamplitude its in-plane vibration is coupled to its out-of-planevibration that is when the initial out-of-plane energy ofthe transmission line is relatively high significant internalresonance occurs in the transmission line when it startsvibrating resulting in the transfer of out-of-plane vibrationenergy to the in-plane direction This in turn not only givesrise to in-plane antisymmetric vibration but also causes in-plane symmetric vibration ultimately leading to a significantincrease in the tension in the transmission line


minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c


(b) In-plane and out-of-plane displacement response in the phase space

minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c

minus0005 0 0005 001minus001vcLc



minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c


Figure 10 Configuration of the out-of-plane motion and the phase-space response of the transmission line when its initial potential energyis relatively high

433 Effects of Damping on the Nonlinear Internal Resonanceof the Transmission Line It was noted in the previoussections that the structural damping of a transmission lineis very small To effectively reduce the vibration responsesof transmission lines control devices (eg dampers spiraldampers damping spacers and damper lines) are ofteninstalled on transmission lines in engineering practice thesecontrol devices can generate relatively large dampingThus inthis section the effects of damping on the internal resonanceof the transmission line are analyzed

Here the same initial potential energy conditions as thoseused in Section 432 are used but the initial damping isvaried Figures 12 and 13 show the free vibration responseof the transmission line obtained in the simulation undervarious damping conditions A comparison of Figures 12 and13 with Figures 8ndash10 shows that under the same conditionsincreasing the damping of the transmission line results

in a significant decrease in the displacement of and thetension in the transmission line as well as a decrease in theduration of vibration of the transmission line This indicatesthat increasing the damping of the transmission line caneffectively consume its vibration energy control its vibrationresponse amplitude and prevent an increase in its dynamictension but cannot hamper the occurrence of its nonlinearinternal resonance

44 Numerical Analysis of the Nonlinear Internal Resonanceof the Transmission Line When in Forced Vibration Basedon the aforementioned analysis we know that the naturalfrequencies of the lower-order modes of vibration of theoverhead transmission line are relatively low However in anactual environment excitation energy (primarily wind loadenergy) is mainly distributed in the 0-1Hz band particularlyin the 0ndash02Hz bandTherefore under an external excitation


0

05

1

15

200 400 600 800 10000t (s)

dynamic tensioninitial tension

nonminus

dim

ensio

nal t

ensio

nT

cT

c0

(a) Dynamic tension time history of the transmission line when its initialenergy is relatively low

minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0

(b) Dynamic tension time history of the transmission line when its initialenergy is relatively high

02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum

(c) Fourier spectrum of the dynamic tension in the transmission linewhen its initial energy is relatively low

0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ

(d) Fourier spectrum of the dynamic tension in the transmission linewhen its initial energy is relatively high

Figure 11 Dynamic tension in the transmission line and its Fourier spectra under two initial energy conditions

(eg a wind load) there is a relatively high probability thatthe transmission line will resonate with it To further studythe effects of the internal resonance of the transmission lineon its wind-induced vibration in this section the effects of theinternal resonance of the transmission line under a harmonicexcitation on the primary resonance are first analyzed basedon the results the forced vibration characteristics of thetransmission line under a wind load are investigated

441 Analysis of the Internal Resonance of the TransmissionLine under a Harmonic Excitation Here it is assumed thatthe transmission line is only subjected to an out-of-planeharmonic excitation 119875119908 = 1198750 cosΩ119911119905 The relationship be-tween the circular frequency Ω119911 of the excitation and thenatural circular frequency 120596119908 of the out-of-plane vibration

of the transmission line is characterized by introducing adetuning parameter 1205901 a nonnegative real number 119896 and asmall parameter 120576

Ω119911 = 119896120596119908 + 1205761205901 (59)

To study the primary resonance response of the system119896 120576 and 1205901 are set to 1 0001 and 10 respectively Bysubstituting these values into (51) the frequency119891119911 of the out-of-plane excitation is determined 119891119911 = Ω119911(2120587) = 0146HzThe natural frequency 119891119908 of the first-order out-of-planevibration of the transmission line that is directly inducedby the excitation is as follows 119891119908 = 120596119908(2120587) = 0136Hz119891119911 approximately equals 119891119908 Thus it can be considered thatΩ119911 asymp 120596119908


minus12minus10

minus8minus6minus4minus2

02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)

(b) Out-of-plane mid-span displacement time his-tory

02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ

(c) Fourier spectrum of the in-plane mid-spandisplacement

0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ

(d) Fourier spectrum of the out-of-plane mid-span displacement


0001002003004

v∙ cT0L

c

minus1 0 1 2minus2vcLc times10minus3

(e) In-plane mid-span phase trajectory

times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c

minus2 0 2 4 6minus4wcLc

(f) Out-of-plane mid-span phase trajectory

0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


(g) Dynamic tension time history

0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ

(h) Fourier spectrum of the dynamic tension

Figure 12 Free vibration response of the transmission line when its damping 119888119888 is set to 001


100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ

(c) Fourier spectrumof the in-planemid-spandis-placement

0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3

times10minus4minus3 minus2 minus1 0 1 2 3 4minus4

vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ




Let us set the amplitude 1198750 of the generalized excitationload to 3305N and the initial conditions for simulation to119902V = 0 119902V = 0 119902119908 = 01 119902119908 = 000005 By substitutingthese values into (54) the nonlinear forced vibration responseof the transmission line under various damping conditionsis determined Figures 14ndash17 show the numerical simulationresults obtainedwhen the damping 119888119888 is set to 0001 001 005and 01 respectively

As demonstrated in Figure 14 the out-of-plane vibrationof the transmission line directly induced by the excitation hasa very large amplitude suggesting that the transmission lineis forced to resonate when Ω119911 asymp 120596119908 Due to the internalresonance the vibration energy of the transmission linetransfers from the out-of-plane direction to the in-planedirection which is not directly under excitation In additionthe in-plane vibration of the transmission line also exhibitsresonance characteristics In other words due to the internalresonance the forced resonance of the transmission line caninvolve several modes of resonance When only consideringthe small structural damping the resonance amplitude of thetransmission line will jump that is within a certain periodof time the response occurs at a relatively small amplitudeand vibration with a relatively large amplitude occurs at acertain time and can continue steadily for an extended periodof time This differs from the resonance of a linear systemDue to the nonlinear coupling and internal resonance theforced resonance modulates the vibration frequency of thesystem and the frequencies of the out-of-plane and in-plane vibration of the transmission line are no longer linearnatural frequencies and are also not the same as the excitationfrequency but instead are manifested as a relatively widedistribution of vibration energy (Figures 14(c) and 14(d)) Asdemonstrated in the phase diagrams of the in-plane and out-of-plane response (Figures 14(e) and 14(f)) the in-plane andout-of-plane response is aperiodic with an amplitude thatjumps Considering that the external excitation has a verylarge amplitude due to the nonlinear internal resonance theforced resonance excites the transmission line to resonateinternally resulting in a sharp increase in the dynamic ten-sion in the transmission line In addition because of the in-plane vibration amplitude jumps a corresponding jumpingphenomenon is also observed in the dynamic tension timehistory curve (Figure 14(g)) As shown in the Fourier spec-trum of the dynamic tension (Figure 14(h)) the dynamic ten-sion in the transmission line is mainly significantly affectedby the in-plane vibration Because its amplitude is far greaterthan the design-breaking load the dynamic tension maycause the transmission to break

Figure 15 shows the simulation results obtained by in-creasing the damping 119888119888 to 001 while retaining the other con-ditions As demonstrated in Figure 15 amplitude jumps areeliminated due to the damping effects but internal resonancestill exists The in-plane vibration of the transmission linenot directly induced by the excitation has a large amplitudewhich are of the same order of magnitude of the out-of-plane vibration In addition the frequency modulation bythe forced resonance becomes more complicated comparedto the situation in which the damping is small (Figures 15(c)and 15(d)) As demonstrated in the phase trajectory of the

steady-state response (Figures 15(e) and 15(f)) there are stilljumps in the in-plane vibration amplitude of the transmissionline and the corresponding dynamic tension amplitude andthe maximum value of these jumps can be as high as 3 timesthe initial tensionThis indicates that it is still possible for thetransmission line to be broken under these conditions

Figures 16 and 17 show the simulation results that areobtained when the damping 119888119888 is further increased to 005and 01 respectively while retaining the other conditions Asdemonstrated in Figures 16 and 17 due to the large dampingeffects the vibration amplitude of the transmission linereaches a stable state within a short period of time howeverthe large damping is unable to prevent the occurrence ofinternal resonance and the in-plane vibration still has a verylarge amplitude the numerical value of which is commensu-rate with that of the out-of-plane vibration The dynamictension in the transmission line when it is in steady-statevibration can still reach approximately three times the ini-tial tension This suggests that even under relatively largedamping effects the forced resonance of the transmission linecannot be maintained as the steady-state mode of motiondirectly induced by the excitation but is instead manifestedas the nonlinear vibration of coupled modes Due to the fre-quency-modulating effect of the forced resonance the naturalfrequencies of the modes of vibration of the transmissionline directly induced by the excitation more accuratelyapproach the excitation frequency and correspondingly themodulated frequencies inherit the relationships between theoriginal linear natural frequencies they are either closelyspaced with or multiples of one another In addition thevibration still contains higher-order modes of vibration

As demonstrated in the phase trajectories of the response(Figures 16(e) 16(f) 17(e) and 17(f)) when its damping isrelatively high the resonance response of the transmissionline reaches saturation In other words because the non-linear vibration equations for the transmission line exhibitquadratic nonlinearity and the linear natural vibration fre-quencies of the transmission line satisfy the 120596119908 asymp 120596V and120596V asymp 2120596119908 conditions when the frequency Ω119911 of the externalexcitation is approximately equal to 120596119908 and the excitationamplitude meets a certain condition the response energyof the mode of vibration directly induced by the excitationreaches saturation and all the input excitation energy entersanother mode Thus while its damping can be increasedby installing energy dissipating equipment in engineeringpractice the motion of a transmission line exhibits coupledin-plane and out-of-plane vibration characteristics due to thenonlinear internal resonance in addition a transmission linecan still continue to vibrate with a relatively large amplitudeunder an external excitation with a relatively large amplitudeand the vibration energy continuously transfers between rele-vant modes without being attenuated Due to the presence ofthis response saturation phenomenon the dynamic tensionin a transmission line when in vibration can still possiblyreach a higher level evenwhen its damping is set to a relativelylarge value which is disadvantageous to practical engineer-ing

To compare the vibration response of the transmissionline when not in resonance with the vibration response of


200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c

(b) Out-of-planemid-span displacement time history

02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c

minus008 minus006 minus004 minus002 0 002 004minus01vcLc


minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c

minus004 minus002 0 002 004 006minus006wcLc


200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ


Figure 14 Primary resonance response of the transmission line when its damping 119888119888 is set to 0001


minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c

minus002 minus0015 minus001 minus0005 0 0005 001minus0025

vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ


minus0015 minus001 minus0005 0 0005 001minus002vcLc

minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ




the transmission line when in resonance the following setof parameters is selected frequency 119891119911 of the external exci-tation 068Hz damping 119888119888 0001 all the other parametersare the same as those used to simulate the transmissionline when in resonance Figure 18 shows the simulationresults A comparison of Figures 14 and 18 shows that whenthe external excitation amplitude and the damping of thetransmission line remain unchanged the vibration amplitudeof the transmission line is far smaller when not in reso-nance than when in resonance While an internal resonancephenomenon can still be observed numerically when thetransmission line is not in resonance the internal resonance-induced in-plane vibration amplitude is very small and isnegligible When the transmission line is not in resonancethe vibration of the system is primarily composed of theout-of-plane forced vibration induced by the out-of-planeexcitation (Figure 18(d)) in addition the dynamic tensionin the transmission line changes to a relatively small extent(Figure 18(g)) It can be seen that internal resonance has agreat influence on themain resonance of transmission line sothe design of transmission line needs to consider its influencereasonably

442 Analysis of the Internal Resonance of the TransmissionLine When in Wind-Induced Vibration Under actual oper-ating conditions overhead transmission lines will inevitablyundergo wind-induced vibration Therefore in this sectionthe nonlinear vibration characteristics of the transmissionline under a wind load are studied Considering that thevertical in-plane wind load on a transmission line is generallyvery small to simplify the analytical process it is assumedthat the wind load acts in the out-of-plane direction of thetransmission line The quasi-steady hypothesis is used in cal-culation when calculating the wind load on the transmissionline only the effects of the vibration of the transmission lineon the wind speed are considered whereas the aerodynamiccoupling effects are not considered that is the load terms119875(119909 119905) and119875(119911 119905) in (26) and (27) can be expressed as follows

119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)

where 120588119886 represents the air density 119863119888 represents the wind-ward width of the transmission line 119862119863 represents the shapefactor of the transmission line V119886-119909 and V119886-119911 represent thewind speeds along the 119909- and 119911-axis directions respectivelyV119888 119888 represent the in-plane and out-of-plane vibrationvelocities of the transmission line respectively

By substituting (60) into (26) and (27) and finding thefirst-order modal solutions for the in-plane and out-of-planevibration using Galerkinrsquos modal truncation method thecontinuous partial differential equations can be transformedto discrete ordinary partial equations as shown in (30) inwhich the factors 1198867 1198876 119875V and 119875119908 are calculated using thefollowing equations (all the other factors are calculated usingthe same equations as introduced in Section 22)

1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)

Wind speeds used in this section are selected fromthe database of wind speeds recorded by the field windmeasurement platform installed on the transmission line(Figure 3) Three wind speed time histories with differentaverage winds are selected which are three wind speedrecords with an average value 1198800 of 52 956 and 201msrespectively These three wind speeds are equivalent to theBeaufort Scales 3 5 and 8 respectively The in-plane verticalwind load on the transmission line is converted from thevertical fluctuating wind speed recorded by the wind speedmeasurement device and its value is generally small and neg-ligible Figure 19 shows the frequency spectra correspondingto the aforementioned three wind speed time histories Asdemonstrated in Figure 19 the wind speed energy is mainlydistributed in the 0-1Hz band

Equation (30) is solved using a higher-order Runge-KuttamethodThe damping 119888119888 of the transmission line is set to 001and the initial conditions for calculation are set as follows119902V = 0119902V = 0119902119908 = 05119902119908 = 000005

(62)

Based on Tables 3 and 4 we know that the naturalfrequencies of the first-order in-plane and out-of-plane vibra-tion of the transmission satisfy the following condition 120596V asymp3120596119908 Figures 20ndash22 show the numerical simulation resultswith respect to the vibration response of the transmission lineunder a wind load

As demonstrated in Figure 20 when 1198800 = 52ms theout-of-plane mid-span vibration amplitude of the transmis-sion line can exceed 1m and its in-plane mid-span vibrationamplitude is far greater than its cross-sectional diameter inaddition the in-plane vibration is associatedwith a noticeablejumping phenomenon (Figure 20(a)) This indicates that thetransmission line also undergoes internal resonance whenin wind-induced vibration and that the vibration energytransfers between the out-of-plane and the in-plane modesof vibration The overall vibration of the transmission lineis manifested as coupled in-plane and out-of-plane vibration(Figure 20(b)) In-plane vibration amplitude jumps can alsobe found by observing the phase trajectories of the transmis-sion line (Figures 20(c) and 20(d)) The out-of-plane vibra-tion is a complex aperiodic motion Due to the excitationfrom the wind load and the frequency-modulating effect of


200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c

minus4 minus3 minus2 minus1 0 1 2 3minus5vcLc


times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)


Figure 18 Nonresonance response of the transmission line when its damping 119888119888 is set to 0001


U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um

10minus2 10minus1 100 10110minus4 10minus3

f (Hz)

(a) Fourier spectrum of the wind speed (1198800 = 52ms)

U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)

(b) Fourier spectrum of the wind speed (1198800 = 956ms)

U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)

(c) Fourier spectrum of the wind speed (1198800 = 201ms)

Figure 19 Fourier spectra of the wind speed

the internal resonance the natural characteristics of the in-plane and out-of-plane vibration undergo relatively signifi-cant changes and its energy is distributed in a relatively widefrequency band (Figures 20(e) and 20(f)) In addition dueto the in-plane vibration there are significant jumps in thedynamic tension amplitude of the transmission line (Figures20(g) and 20(h))

Figure 21 shows the simulation results obtained when theaverage wind speed 1198800 is increased to 956ms As demon-strated in Figure 21 the transmission line vibrates with a largeamplitude in addition the out-of-plane mid-span vibrationamplitude of the transmission line can reach half of its maxi-mum sag and its in-plane mid-span vibration amplitude canalso reach half of its maximum out-of-plane vibration ampli-tude (Figure 21(a)) moreover there are also jumps in the

in-plane and out-of-plane vibration amplitudes of the trans-mission line However due to the inherent randomness ofa wind load the internal resonance of the transmission linewhen in wind-induced vibration is not as conspicuous as theinternal resonance that occurs when the line is in free vibra-tion or under a harmonic excitation However as demon-strated by the response frequency spectra changes in the nat-ural vibration characteristics of the system are more signifi-cant than those that occur in a linear system Higher-orderin-plane and out-of-planemodes of vibration of the transmis-sion line can be excited by a wind load with a relatively highwind speed (Figures 21(e) and 21(f)) under this condition thephase trajectory of the response has an oval shape and thereare jumps in the displacement amplitude (the outer and innertrajectories represent lower-order and higher-order modes of


minus2minus1

0123

54

12Lc inminusplaneoutminusofminusplaneu0 = 520 ＧＭ

times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c

(a) Dimensionless mid-span displacement

times10minus3

times10minus3minus1 0 1 2 3 4 5minus2

wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c

(b) Relationship between the out-of-plane and in-plane mid-span displacements

minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c

minus1 minus05 0 05 1 15minus15vcLc times10minus3


minus2 minus1 0 1 2 3 4 5minus0015

minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ

(e) Fourier spectrum of the in-plane mid-span dis-placement


0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ

(f) Fourier spectrum of the out-of-plane mid-span displacement

0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ


Figure 20 Wind-induced vibration response of the transmission line when 1198800 = 52ms


0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004

12Lc inminusplaneoutminusofminusplaneu0 = 956ＧＭ

cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c

0 2 4 6 8 10 12minus2wcLc times10minus3


02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ

(e) Fourier spectrum of the in-plane mid-spandisplacement

0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c

(a) Dimensionless in-plane mid-span displacement

100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c

(b) Dimensionless out-of-plane mid-span displace-ment

minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c


(c) In-plane mid-span phase-plane response

minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c


(d) Out-of-plane mid-span phase-plane response

inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ

(e) Fourier spectrum of the in-plane mid-span displacement response

0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ

(f) Fourier spectrum of the out-of-plane mid-span displacement response

0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ




vibration resp) (Figures 21(c) and 21(d)) A comparison ofFigures 20 and 21 shows that as the wind speed increases thedynamic tension in the transmission line increases Becausethe dynamic tension is mainly affected by the in-planevibration the jumps in the dynamic tension amplitude arequite noticeable If the external wind excitation lasts for asufficiently long period of time the dynamic tension in thetransmission line can even surpass its breaking load limit atsome moments during the vibration process (Figures 21(g)and 21(h)) resulting in damage to the transmission line Thisshould warrant attention in practical engineering

Figure 22 shows the simulation results obtained when theaverage wind speed 1198800 is increased to 201ms As demon-strated in Figure 22 the vibration amplitude of the transmis-sion line further increases under a strong wind in additionthe in-plane vibration amplitude of the transmission lineis even greater than its out-of-plane vibration amplitudeand the out-of-plane and in-plane vibration amplitudes ofthe transmission line are of the same order of magnitudeas its maximum arc sag (Figures 22(a) and 22(b)) and caneven surpass its maximum arc sag which will result indamage to the transmission line under practical conditionsAs demonstrated in the in-plane and out-of-plane displace-ment response time histories and phase-plane response(Figures 22(c) and 22(d)) the wind-induced vibration of thetransmission line exhibits significant nonlinear vibrationcharacteristics in addition the out-of-plane vibration is alsocoupled to the in-plane vibration and the vibration energy ofthe structure transfers from the out-of-plane modes that areunder the wind load to the in-plane modes

A comparison of Figures 22(e) and 22(f) shows that theenergy-containing range of the out-of-plane mid-span dis-placement response of the transmission line is 009ndash0294Hzand the energy-containing range of its in-plane mid-spandisplacement response is 012ndash025Hz whereas the mainenergy-containing range of the wind load is 001ndash1Hz Thusunder a strong wind the wind-induced vibration of thetransmission line modulates not only the linersquos vibrationamplitude but also its vibration frequency and can cause thetransmission line to undergo high-frequency vibration inaddition the dynamic tension in the transmission line in-creases significantly and can even reach or surpass its break-ing load in a short time which is very disadvantageous to thetransmission line under practical conditions A comparisonof Figures 14 and 22 shows that themotion of the transmissionline under a strong wind is similar to that of a primaryresonance under harmonic excitation that is the amplitudeof vibration is very large and the in-plane vibration isstrongly coupled to its out-of-plane vibration

5 Conclusions

In this study 3D nonlinear vibration equations for trans-mission lines that consider geometric nonlinearity are estab-lished Based on the boundary conditions and the assump-tions used formodeling the 3D equations are simplified to 2Dequations Galerkinrsquos modal truncation method is employedto simplify the continuous partial differential equations todiscrete partial differential equations that are analyzed using

the method of multiple scales and solved by a higher-orderRunge-Kutta method On this basis the nonlinear dynamiccharacteristics of transmission lines when in free and forcedvibration respectively are analyzed The main conclusionsdrawn from this study are summarized as follows(1) A transmission line has inherent nonlinear vibrationcharacteristics When in vibration the in-plane motion of atransmission line is coupled to its out-of-plane motion andlinear quadratic nonlinear and cubic nonlinear modes ofinternal resonance can easily occur(2) When a transmission line vibrates freely nonlinearinternal resonance can occur even when the initial out-of-plane energy of the transmission line is relatively lowhowever under this condition energy transfer and modecoupling phenomena are not noticeable As its initial out-of-plane energy increases the internal resonance of a trans-mission line becomes prominent and the coupling of its in-plane and out-of-plane vibration becomes stronger resultingin a transfer of out-of-plane vibration energy to the in-plane direction and an increase in the in-plane vibrationamplitude and the dynamic tension in the transmission lineDue to the transmission linersquos mode coupling and nonlinearinternal resonance the out-of-plane and in-plane vibrationfrequencies of the line differ relatively significantly from thenatural vibration frequencies of a linear system(3) Under a harmonic excitation a transmission lineundergoes forced resonance Under the combined action ofinternal and primary resonance the in-plane vibration of thetransmission line is coupled to its out-of-plane vibration andthe vibration energy transfers from the out-of-plane directionto the in-plane direction that is not directly under the excita-tion resulting in a significant increase in the dynamic tensionand the displacement amplitude of the transmission line Inaddition as the external excitation amplitude increases theforced resonance effects become more significant and theforced resonance has a modulating effect on the vibrationfrequency of the transmission line As a result of the forcedresonance the in-plane and out-of-plane vibration energyof the transmission line is distributed over a relatively widerange(4) When a transmission line is in wind-induced vibra-tion as the wind speed increases the nonlinear internalresonance of the transmission line increases the frequency-modulating effect of its nonlinear internal resonance alsoincreases which may cause the transmission line to undergohigh-frequency vibration Because the in-plane vibration am-plitude of a transmission line and its dynamic tension in-crease significantly as the wind speed increases a noticeableamplitude jumping phenomenon occurs consequently whenin vibration for a long period of time the dynamic tensionamplitude of the transmission line may exceed the breakingload limit resulting in the breaking of the transmission line(5) Regardless of whether a transmission line is in freeor forced vibration increasing its damping can effectivelyconsume its vibration energy thereby allowing the vibrationamplitude of the transmission line to reach a stable statewithin a short period of time However increasing thedamping cannot prevent the occurrence of nonlinear internalresonance in a transmission line When a transmission line


is in forced vibration and the external excitation has a relative-ly large amplitude even if the damping of the transmissionline is increased to a relatively large value the transmissionline can still maintain its nonlinear vibration of coupledmodes with a large amplitude due to the nonlinear internalresonance in addition the vibration energy also constantlytransfers between the coupled modes and does not decreasesignificantly When a transmission line is in vibration foran extended period of time this response saturation phe-nomenon can still cause the dynamic tension in the transmis-sion line to surpass the design-breaking load thereby causingdamage to the transmission line

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research has been funded by the National NaturalScience Foundation of China (Grants nos 51578512 and51108425) Outstanding Young Talent Research Fund ofZhengzhou University (Grant no 1521322004) and Founda-tion for University Young Key Teacher by Henan Province(Grant no 2015GGJS-151)

References

[1] H M Irvine Cable Structure MIT Press 1981[2] H M Irvine and J H Griffin ldquoOn the dynamic response of

a suspended cablerdquo Earthquake Engineering amp Structural Dy-namics vol 4 no 4 pp 389ndash402 1976

[3] H Yamaguchi and M Ito ldquoLinear theory of free vibrations ofan inclined cable in three dimensionsrdquo Proceedings of the JapanSociety of Civil Engineers vol 1979 no 286 pp 29ndash36 1979

[4] H Yamaguchi T Miyata and M Ito ldquoTime response analysisof a cable under harmonic excitationrdquo Proceedings of the JapanSociety of Civil Engineers vol 1981 no 308 pp 424ndash427 1981

[5] N C Perkins and C D Mote Jr ldquoThree-dimensional vibrationof travelling elastic cablesrdquo Journal of Sound and Vibration vol114 no 2 pp 325ndash340 1987

[6] G Visweswara Rao and R N Iyengar ldquoInternal resonanceand non-linear response of a cable under periodic excitationrdquoJournal of Sound and Vibration vol 149 no 1 pp 25ndash41 1991

[7] F Benedettini G Rega and R Alaggio ldquoNon-linear oscillationsof a four-degree-of-freedom model of a suspended cable undermultiple internal resonance conditionsrdquo Journal of Sound andVibration vol 182 no 5 pp 775ndash798 1995

[8] M Aufaure ldquoThree-node cable element ensuring the continuityof the horizontal tension A clamp-cable elementrdquoComputers ampStructures vol 74 no 2 pp 243ndash251 2000

[9] WKChang andRA Ibrahim ldquoMultiple Internal Resonance inSuspendedCables under Random In-Plane LoadingrdquoNonlinearDynamics vol 12 no 3 pp 275ndash303 1997

[10] A S K Kwan ldquoA new approach to geometric nonlinearity ofcable structuresrdquoComputersamp Structures vol 67 no 4 pp 243ndash252 1998

[11] H N Arafat and A H Nayfeh ldquoNon-linear responses of sus-pended cables to primary resonance excitationsrdquo Journal ofSound and Vibration vol 266 no 2 pp 325ndash354 2003

[12] N Srinil G Rega and S Chucheepsakul ldquoThree-dimensionalnon-linear coupling and dynamic tension in the large-ampli-tude free vibrations of arbitrarily sagged cablesrdquo Sound andVibration vol 269 no 3ndash5 852 pages 2004

[13] N Srinil G Rega and S Chucheepsakul ldquoTwo-to-one resonantmulti-modal dynamics of horizontalinclined cables I Theo-retical formulation andmodel validationrdquoNonlinear Dynamicsvol 48 no 3 pp 231ndash252 2007

[14] X Xiao J Yang J Druez and Y Xu ldquoNonlinear oscillationsof a suspended cablerdquo Zhendong Ceshi Yu ZhenduanJournalof Vibration Measurement and Diagnosis vol 23 no 2 p 1102003

[15] D Dempsey ldquoWinds wreak havoc on linesrdquo Transmission Dis-tribution World vol 48 no 6 pp 32ndash40 1996

[16] KWang andG Tang ldquoResponse analysis of nonlinear vibrationof overhead power line under suspension chain staterdquoVibrationand Shock vol 22 no 2 72 pages 2003

[17] N Barbieri O H de Souza Junior and R Barbieri ldquoDynamicalanalysis of transmission line cables Part 1mdashLinear theoryrdquoMechanical Systems and Signal Processing vol 18 no 3 pp 659ndash669 2004

[18] R Barbieri N Barbieri and O H de Souza Junior ldquoDynamicalanalysis of transmission line cables Part 3-Nonlinear theoryrdquoMechanical Systems and Signal Processing vol 22 no 4 pp 992ndash1007 2008

[19] A Berlioz and C-H Lamarque ldquoA non-linear model for thedynamics of an inclined cablerdquo Journal of Sound and Vibrationvol 279 no 3-5 pp 619ndash639 2005

[20] S W Rienstra ldquoNonlinear free vibrations of coupled spans ofoverhead transmission linesrdquo Journal of EngineeringMathemat-ics vol 53 no 3-4 pp 337ndash348 2005

[21] K Tucker and A Haldar ldquoNumerical model validation andsensitivity study of a transmission-line insulator failure usingfull-scale test datardquo IEEE Transactions on Power Delivery vol22 no 4 pp 2439ndash2444 2007

[22] G F Zhao Q Xie and J Li ldquoWind-induced nonlinear interac-tions in the out-of-plane and in-plane dynamics of transmissionline conductorsrdquo in Proceedings of the in Proceedings of SecondInternational Conference on Modelling and Simulation pp 80ndash85 UK 2009

[23] L Wang and B Fang ldquoSimulations on transient characteristicsof 500 kV capacitor voltage transformerrdquoGaodianya JishuHighVoltage Engineering vol 38 no 9 pp 2389ndash2396 2012

[24] X Xie X Hu J Peng and Z Wang ldquoRefined modeling andfree vibration of two-span suspended transmission linesrdquo ActaMechanica vol 228 no 2 pp 673ndash681 2017

[25] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979

[26] B Wen Y Li P Xu and Q Han Engineering NonlinearVibration Science Press Beijing China 2007

[27] G Veronis ldquoA note on themethod of multiple scalesrdquoQuarterlyof Applied Mathematics vol 38 no 3 pp 363ndash368 198081

[28] H Wang and H Hu ldquoRemarks on the perturbation methods insolving the second-order delay differential equationsrdquo Nonlin-ear Dynamics vol 33 no 4 pp 379ndash398 2003


[29] Hu Huailei WangHaiyan ldquoA note on the method of multiplescalesrdquo Journal of Dynamics and Control vol 2 no 3 pp 10ndash132004

[30] M Zhang G Zhao L Wang and J Li ldquoWind-induced cou-pling vibration effects of high-voltage transmission tower-linesystemsrdquo Shock and Vibration vol 2017 34 pages 2017

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structures consequently their motion exhibits significantlynonlinear motion characteristics and very complex formsthat are a result of the combined action of external andinternal resonance In addition the geometric and physicalparameters of cables also have an impact on their dynamicresponse [4ndash14] Evidently even the linear dynamic theorycannot accurately describe the forms of motion of cables It isirrefutable that the wind-induced vibration response of long-span lightweight and small-damping transmission lines hasinherently nonlinear characteristics [15ndash24] However thecurrent design specification still requires the use of the linearstatic theory of suspended cables to design transmission lineswhich clearly cannot meet the engineering demand Theabovementioned linear design theory is easily mastered andapplied in engineering but its shortcomings are also obviousFor example it assumes that the relationship between thestatic load and the response of the whole line is a linearfunction in calculations which obviously does not conformto the nonlinearmechanical properties of a transmission lineFurthermore there is a significant difference between the in-plane and out-of-plane vibration of a transmission line andthe former has obvious nonlinearity stiffness that will induceadditional tension of the transmission lineTherefore it is notenough to calculate the response of a transmission line onlyby its first-order out-of-plane vibration In fact as a highlyflexible line system it is inevitable to consider the multiordervibrations of a transmission line and the dynamic tensiongenerated by the vibration of the transmission line under theaction of the pulsating wind is the main load that acts on thetransmission tower

Based on the aforementioned analysis according to ageneralized Hamiltonrsquos principle three-dimensional (3D)nonlinear differential equations of motion for horizontallysuspended transmission lines that consider the initial deflec-tion are established in this work Based on the characteristicsof an actual transmission line the 3D equations are simpli-fied to two-dimensional (2D) equations On this basis themethod of multiple scales is employed to theoretically studythe nonlinear internal resonance behavior of transmissionlines Moreover a higher-order Runge-Kutta method is alsoused to perform a numerical analysis of the nonlinearvibration characteristics of transmission lines to reveal thecharacteristics of the coupled action of the out-of-plane andin-plane vibration of transmission lines when in nonlinearvibration with the aim of providing a basis for the reasonabledesign of HVOTLs

2 3D Nonlinear Equations of Motion forElastic Transmission Lines

21 Establishing 3D Equations of Motion Figure 1 shows anelastic transmission line with its two ends hinge-supportedat the same height The initial static equilibrium geometricconfiguration 1198780 of the transmission line in the plane 119874119909119910 isused as the reference location and is represented by function119909(119910) 119889119888 119871119888 and 119879119888(119910 119905) represent the sag and span of thedynamic tension in the transmission line respectively 119878119863represents the 3D dynamic configuration of the transmission

y

z

x

O

P(z t)

P(x t)

P(y t)

Dynamic state SD

Static state S0

dc

Lc

yc

Figure 1 Spatial configuration of a transmission line

line under an external excitation 119875(119905) 119875(119909 119905) 119875(119910 119905) and119875(119911 119905) represent the components of the external excitation119875(119905) in the 119909- 119910- and 119911-axes respectively 119864119888 and 119860119888represent the elastic modulus and the cross-sectional areaof the transmission line respectively 120574119888 represents the deadweight of the transmission line per unit length (120574119888 = 119898119888119892where 119892 represents the acceleration of gravity) 119906119888(119910 119905)V119888(119910 119905) and 119908119888(119910 119905) represent the displacements of thetransmission line along the longitudinal in-plane vibration(119910-axis direction) the transverse in-plane vibration (119909-axisdirection) and the transverse out-of-plane vibration (119911-axisdirection) respectively

To simplify analysis the following assumptions [12 25]are made (a) the flexural torsional and shear stiffness of thetransmission line is sufficiently low and thus negligible (b)the transmission line only bears tension in the axial directionand its axial strain when vibrating is sufficiently small andthus negligible (c) only geometric nonlinearity is consideredwhereas material nonlinearity is not considered

When the tension in the transmission line is in the initialstatic equilibrium state the length 1198891199040 of the differentialelement at any arbitrary location119910119888 along the 119910-axis directionis

1198891199040 = radic1 + (120597119909120597119910)2119889119910 (1)

According to the definition of Lagrangian strain initiallythe length 119889119904 of the differential element of the transmissionline under the action of gravity is

119889119904 = radic1 + (120597119909120597119910)21 + 1205761198880 119889119910 (2)

where 1205761198880 represents the initial static strain of the transmis-sion line which is calculated using the following equation

1205761198880 = 119879119888 (119910 0)119864119888119860119888 (3)

where 119879119888(119910 0) represents the initial tension in the transmis-sion line





(5)



(6)


120575int11990521199051

(119870V minus 119880) 119889119905 + int11990521199051

120575119882119889119905 = 0 (7)



119870V = 12 int1198711198880 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) (2119888 + V2119888 + 2119888) 119889119910 (8)



119880 = int1198711198880119879119888 (119910 0) 120576119888119889119904 + 12 int1198711198880 1198641198881198601198881205762119888119889119904 (9)


120575119882119888 = minusint1198711198880119888119888 (119888120575119906119888 + V119888120575V119888 + 119888120575119908119888) 119889119910 (10)








(13)





(14)

Let

120578 = 120597119906119888120597119910 minus (120597119906119888120597119910 )2 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119909120597119910)2+ 12 (120597V119888120597119910 )2 + 12 (120597119908119888120597119910 )2

(15)


120597120597119910 [[[


(16)

120597120597119910 [[[


(17)

120597120597119910 [[[


120597119908119888120597119910 = 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 119888 + 119888119888119888 minus 119875 (119911 119905)

(18)




120597120597119910 [[[


(19)






120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119908119888120597119910 )2 + 12 (120597V119888120597119910 )2 asymp ℏ (119905)119864119888119860119888 (22)



ℏ (119905)119864119888119860119888 sdot 119871119888 = int1198711198880 [120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119908119888120597119910 )2+ 12 (120597V119888120597119910 )2]119889119910

(24)

Let us set

120582 (119905) = ℏ (119905)119864119888119860119888 = 1119871119888 int1198711198880 [120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910+ 12 (120597119908119888120597119910 )2 + 12 (120597V119888120597119910 )2]119889119910

(25)



(26)


(27)


V119888 (119910 119905) = 119873sum119894=1

120593119894 (119910) sdot 119902119894 (119905) 119908119888 (119910 119905) = 119873sum

119894=1

120593119894 (119910) sdot 119902119894 (119905) (28)




(29)


1198861 119902V (119905) + 1198862119902V (119905) + 11988631199022V (119905) + 11988641199022119908 (119905) + 11988651199023V (119905)+ 11988661199022119908 (119905) 119902V (119905) + 1198867 119902V (119905) = 119875V1198871 119902119908 (119905) + 1198872119902119908 (119905) + 1198873119902V (119905) 119902119908 (119905) + 11988741199022V (119905) 119902119908 (119905)+ 11988751199023119908 (119905) + 1198876 119902119908 (119905) = 119875119908(30)


1198861 = int1198711198880

120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 1205932V (119910) 1198891199101198862 = minusint119871119888

0


1198863 = int1198711198880


sdot 1205931015840V (119910) int1198711198880

120597119909120597119910 sdot 1205931015840V (119910) 119889119910]120593V (119910) 1198891199101198864 = int119871119888

0



1198865 = int1198711198880


1198866 = int1198711198880


1198867 = int11987111988801198881198881205932V (119910) 119889119910

119875V = int1198711198880119875 (119909 119905) 120593V (119910) 119889119910

1198871 = int1198711198880

120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 1205932119908 (119910) 1198891199101198872 = minusint119871119888

0


0


120597119909120597119910 sdot 1205931015840V (119910) 119889119910]120593119908 (119910) 1198891199101198874 = int119871119888

0


1198875 = int1198711198880


1198876 = int11987111988801198881198881205932119908 (119910) 119889119910

119875119908 = int1198711198880119875 (119911 119905) 120593119908 (119910) 119889119910

(31)







119902V (119905) + 12059621119902V (119905) + 12057612057211199022V (119905) + 12057612058111199022119908 (119905) + 12057612057311199023V (119905)+ 12057612057411199022119908 (119905) 119902V (119905) + 21205761205831 119902V (119905) = 0119902119908 (119905) + 12059622119902119908 (119905) + 1205761205722119902V (119905) 119902119908 (119905) + 12057612058121199022V (119905) 119902119908 (119905)+ 12057612057321199023119908 (119905) + 21205761205832 119902119908 (119905) = 21205761198912 cos (Ω119911119905) (33)


12059621 = 11988621198861 21205761205831 = 11988671198861 1205761205721 = 11988631198861 1205761205811 = 11988641198861 1205761205731 = 11988651198861 1205761205741 = 11988661198861 12059622 = 11988721198871


21205761205832 = 11988761198871 1205761205722 = 11988731198871 1205761205812 = 11988741198871 1205761205732 = 11988751198871 21205761198912 = 11987501198871

(34)



119879119899 = 120576119899119905 (119899 = 0 1 ) (35)

that is

1198790 = 1199051198791 = 120576119905 (36)






(11986321119902V2 + 212058311198631119902V2 + 212058311198632119902V1 + 211986301198631119902V3+ 212058311198630119902V3 + 12057311199023V1 + 21205721119902V1119902V2 + 61205731119902V0119902V1119902V2



(39)



11986302119902V0 + 12059612119902V0 = 0119863021199021199080 + 120596221199021199080 = 0 (40)

120576111986302119902V1 + 211986301198631119902V0 + 212058311198630119902V0 + 12059621119902V1 + 12057311199023V0+ 11988611199022V0 + 120574111990221199080119902V0 + 120581111990221199080 = 0119863021199021199081 + 2119863011986311199021199080 + 120573211990231199080 + 120596221199021199081 + 1198862119902V01199021199080+ 2120583211986301199021199080 + 12058121199022V01199021199080 = 21198912 cos (Ω119911119905)

(41)


(42)


(43)


119902V0 = 119860V (1198791 1198792) exp (11989412059611198790)+ 119860V (1198791 1198792) exp (minus11989412059611198790)

1199021199080 = 119860119908 (1198791 1198792) exp (11989412059621198790)+ 119860119908 (1198791 1198792) exp (minus11989412059621198790) (44)




(45)





1205962 minus 1205961 = 120576120590 120590 = 119874 (1) (46)

Thus

(1205962 + 1205961) 1198790 = 212059611198790 + 1205761205901198790 = 212059611198790 + 1205901198791= 212059621198790 + 1205901198791212059621198790 = 2 (1205961 + 120576120590) 1198790 = 212059621198790 + 21205901198791(47)


21198941205961 (1198631119860V + 1205831119860V) + 12057211198602V + 1205811119860211990811989011989421205901198791 = 021198941205962 (1198631119860119908 + 1205832119860119908) + 1205722119860V1198601199081198901198941205901198791 = 0 (48)


119860119903 = 1198861199032 119890119895120579119903 119903 = V 119908 (49)




(50)


1205931 = 2 (120579119908 + 1205901198791) 1205932 = 120579V + 120579119908 + 1205901198791 (51)


1198631119886V + 1205831119886V + 12057211198862V21205961 sin 120579V + 1205811119886211990841205961 sin1205931 sec 120579V = 0


1198861199081198631120579119908 + 1205722119886V11988611990841205962 cos 120579119908 cos1205932 = 0(52)







1198841 = 119902V1198842 = 119902V1198843 = 1199021199084 = 119902119908(53)



(54)






119909 (119910) = 11990211988821198791198880119910 (119871119888 minus 119910) + Δ 119888119871119888 119910 (55)




Table1Orig

inaldesig

nparameterso

fthe

selected

transm

issionlin

e

Items

Diameter119889

(mm)


(mm2)

Linear

density119898 119888

(kgm)

Elastic

mod

ulus

119864 119888(MPa)

Averageo

peratio

nal

tension

(N)

Tensionatdesig

nwind

speed

(N)

Breaking

force

(N)

Span119871 119888

(m)

Sag119889 119888

(m)

Con

ductors

336

66655

206

63000

3531

643688

1412

65480

166




Density 120588119888(kgm3)






120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)



120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)




O

B

y

x

qc

Lc

Δ c



120603119910119871119888 minus cos120603119910119871119888 )

(58)






Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






(5)



(6)


120575int11990521199051

(119870V minus 119880) 119889119905 + int11990521199051

120575119882119889119905 = 0 (7)



119870V = 12 int1198711198880 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) (2119888 + V2119888 + 2119888) 119889119910 (8)



119880 = int1198711198880119879119888 (119910 0) 120576119888119889119904 + 12 int1198711198880 1198641198881198601198881205762119888119889119904 (9)


120575119882119888 = minusint1198711198880119888119888 (119888120575119906119888 + V119888120575V119888 + 119888120575119908119888) 119889119910 (10)








(13)





(14)

Let

120578 = 120597119906119888120597119910 minus (120597119906119888120597119910 )2 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119909120597119910)2+ 12 (120597V119888120597119910 )2 + 12 (120597119908119888120597119910 )2

(15)


120597120597119910 [[[


(16)

120597120597119910 [[[


(17)

120597120597119910 [[[


120597119908119888120597119910 = 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 119888 + 119888119888119888 minus 119875 (119911 119905)

(18)




120597120597119910 [[[


(19)






120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119908119888120597119910 )2 + 12 (120597V119888120597119910 )2 asymp ℏ (119905)119864119888119860119888 (22)



ℏ (119905)119864119888119860119888 sdot 119871119888 = int1198711198880 [120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119908119888120597119910 )2+ 12 (120597V119888120597119910 )2]119889119910

(24)

Let us set

120582 (119905) = ℏ (119905)119864119888119860119888 = 1119871119888 int1198711198880 [120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910+ 12 (120597119908119888120597119910 )2 + 12 (120597V119888120597119910 )2]119889119910

(25)



(26)


(27)


V119888 (119910 119905) = 119873sum119894=1

120593119894 (119910) sdot 119902119894 (119905) 119908119888 (119910 119905) = 119873sum

119894=1

120593119894 (119910) sdot 119902119894 (119905) (28)




(29)


1198861 119902V (119905) + 1198862119902V (119905) + 11988631199022V (119905) + 11988641199022119908 (119905) + 11988651199023V (119905)+ 11988661199022119908 (119905) 119902V (119905) + 1198867 119902V (119905) = 119875V1198871 119902119908 (119905) + 1198872119902119908 (119905) + 1198873119902V (119905) 119902119908 (119905) + 11988741199022V (119905) 119902119908 (119905)+ 11988751199023119908 (119905) + 1198876 119902119908 (119905) = 119875119908(30)


1198861 = int1198711198880

120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 1205932V (119910) 1198891199101198862 = minusint119871119888

0


1198863 = int1198711198880


sdot 1205931015840V (119910) int1198711198880

120597119909120597119910 sdot 1205931015840V (119910) 119889119910]120593V (119910) 1198891199101198864 = int119871119888

0



1198865 = int1198711198880


1198866 = int1198711198880


1198867 = int11987111988801198881198881205932V (119910) 119889119910

119875V = int1198711198880119875 (119909 119905) 120593V (119910) 119889119910

1198871 = int1198711198880

120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 1205932119908 (119910) 1198891199101198872 = minusint119871119888

0


0


120597119909120597119910 sdot 1205931015840V (119910) 119889119910]120593119908 (119910) 1198891199101198874 = int119871119888

0


1198875 = int1198711198880


1198876 = int11987111988801198881198881205932119908 (119910) 119889119910

119875119908 = int1198711198880119875 (119911 119905) 120593119908 (119910) 119889119910

(31)







119902V (119905) + 12059621119902V (119905) + 12057612057211199022V (119905) + 12057612058111199022119908 (119905) + 12057612057311199023V (119905)+ 12057612057411199022119908 (119905) 119902V (119905) + 21205761205831 119902V (119905) = 0119902119908 (119905) + 12059622119902119908 (119905) + 1205761205722119902V (119905) 119902119908 (119905) + 12057612058121199022V (119905) 119902119908 (119905)+ 12057612057321199023119908 (119905) + 21205761205832 119902119908 (119905) = 21205761198912 cos (Ω119911119905) (33)


12059621 = 11988621198861 21205761205831 = 11988671198861 1205761205721 = 11988631198861 1205761205811 = 11988641198861 1205761205731 = 11988651198861 1205761205741 = 11988661198861 12059622 = 11988721198871


21205761205832 = 11988761198871 1205761205722 = 11988731198871 1205761205812 = 11988741198871 1205761205732 = 11988751198871 21205761198912 = 11987501198871

(34)



119879119899 = 120576119899119905 (119899 = 0 1 ) (35)

that is

1198790 = 1199051198791 = 120576119905 (36)






(11986321119902V2 + 212058311198631119902V2 + 212058311198632119902V1 + 211986301198631119902V3+ 212058311198630119902V3 + 12057311199023V1 + 21205721119902V1119902V2 + 61205731119902V0119902V1119902V2



(39)



11986302119902V0 + 12059612119902V0 = 0119863021199021199080 + 120596221199021199080 = 0 (40)

120576111986302119902V1 + 211986301198631119902V0 + 212058311198630119902V0 + 12059621119902V1 + 12057311199023V0+ 11988611199022V0 + 120574111990221199080119902V0 + 120581111990221199080 = 0119863021199021199081 + 2119863011986311199021199080 + 120573211990231199080 + 120596221199021199081 + 1198862119902V01199021199080+ 2120583211986301199021199080 + 12058121199022V01199021199080 = 21198912 cos (Ω119911119905)

(41)


(42)


(43)


119902V0 = 119860V (1198791 1198792) exp (11989412059611198790)+ 119860V (1198791 1198792) exp (minus11989412059611198790)

1199021199080 = 119860119908 (1198791 1198792) exp (11989412059621198790)+ 119860119908 (1198791 1198792) exp (minus11989412059621198790) (44)




(45)





1205962 minus 1205961 = 120576120590 120590 = 119874 (1) (46)

Thus

(1205962 + 1205961) 1198790 = 212059611198790 + 1205761205901198790 = 212059611198790 + 1205901198791= 212059621198790 + 1205901198791212059621198790 = 2 (1205961 + 120576120590) 1198790 = 212059621198790 + 21205901198791(47)


21198941205961 (1198631119860V + 1205831119860V) + 12057211198602V + 1205811119860211990811989011989421205901198791 = 021198941205962 (1198631119860119908 + 1205832119860119908) + 1205722119860V1198601199081198901198941205901198791 = 0 (48)


119860119903 = 1198861199032 119890119895120579119903 119903 = V 119908 (49)




(50)


1205931 = 2 (120579119908 + 1205901198791) 1205932 = 120579V + 120579119908 + 1205901198791 (51)


1198631119886V + 1205831119886V + 12057211198862V21205961 sin 120579V + 1205811119886211990841205961 sin1205931 sec 120579V = 0


1198861199081198631120579119908 + 1205722119886V11988611990841205962 cos 120579119908 cos1205932 = 0(52)







1198841 = 119902V1198842 = 119902V1198843 = 1199021199084 = 119902119908(53)



(54)






119909 (119910) = 11990211988821198791198880119910 (119871119888 minus 119910) + Δ 119888119871119888 119910 (55)




Table1Orig

inaldesig

nparameterso

fthe

selected

transm

issionlin

e

Items

Diameter119889

(mm)


(mm2)

Linear

density119898 119888

(kgm)

Elastic

mod

ulus

119864 119888(MPa)

Averageo

peratio

nal

tension

(N)

Tensionatdesig

nwind

speed

(N)

Breaking

force

(N)

Span119871 119888

(m)

Sag119889 119888

(m)

Con

ductors

336

66655

206

63000

3531

643688

1412

65480

166




Density 120588119888(kgm3)






120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)



120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)




O

B

y

x

qc

Lc

Δ c



120603119910119871119888 minus cos120603119910119871119888 )

(58)






Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





(14)

Let

120578 = 120597119906119888120597119910 minus (120597119906119888120597119910 )2 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119909120597119910)2+ 12 (120597V119888120597119910 )2 + 12 (120597119908119888120597119910 )2

(15)


120597120597119910 [[[


(16)

120597120597119910 [[[


(17)

120597120597119910 [[[


120597119908119888120597119910 = 120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 119888 + 119888119888119888 minus 119875 (119911 119905)

(18)




120597120597119910 [[[


(19)






120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119908119888120597119910 )2 + 12 (120597V119888120597119910 )2 asymp ℏ (119905)119864119888119860119888 (22)



ℏ (119905)119864119888119860119888 sdot 119871119888 = int1198711198880 [120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119908119888120597119910 )2+ 12 (120597V119888120597119910 )2]119889119910

(24)

Let us set

120582 (119905) = ℏ (119905)119864119888119860119888 = 1119871119888 int1198711198880 [120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910+ 12 (120597119908119888120597119910 )2 + 12 (120597V119888120597119910 )2]119889119910

(25)



(26)


(27)


V119888 (119910 119905) = 119873sum119894=1

120593119894 (119910) sdot 119902119894 (119905) 119908119888 (119910 119905) = 119873sum

119894=1

120593119894 (119910) sdot 119902119894 (119905) (28)




(29)


1198861 119902V (119905) + 1198862119902V (119905) + 11988631199022V (119905) + 11988641199022119908 (119905) + 11988651199023V (119905)+ 11988661199022119908 (119905) 119902V (119905) + 1198867 119902V (119905) = 119875V1198871 119902119908 (119905) + 1198872119902119908 (119905) + 1198873119902V (119905) 119902119908 (119905) + 11988741199022V (119905) 119902119908 (119905)+ 11988751199023119908 (119905) + 1198876 119902119908 (119905) = 119875119908(30)


1198861 = int1198711198880

120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 1205932V (119910) 1198891199101198862 = minusint119871119888

0


1198863 = int1198711198880


sdot 1205931015840V (119910) int1198711198880

120597119909120597119910 sdot 1205931015840V (119910) 119889119910]120593V (119910) 1198891199101198864 = int119871119888

0



1198865 = int1198711198880


1198866 = int1198711198880


1198867 = int11987111988801198881198881205932V (119910) 119889119910

119875V = int1198711198880119875 (119909 119905) 120593V (119910) 119889119910

1198871 = int1198711198880

120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 1205932119908 (119910) 1198891199101198872 = minusint119871119888

0


0


120597119909120597119910 sdot 1205931015840V (119910) 119889119910]120593119908 (119910) 1198891199101198874 = int119871119888

0


1198875 = int1198711198880


1198876 = int11987111988801198881198881205932119908 (119910) 119889119910

119875119908 = int1198711198880119875 (119911 119905) 120593119908 (119910) 119889119910

(31)







119902V (119905) + 12059621119902V (119905) + 12057612057211199022V (119905) + 12057612058111199022119908 (119905) + 12057612057311199023V (119905)+ 12057612057411199022119908 (119905) 119902V (119905) + 21205761205831 119902V (119905) = 0119902119908 (119905) + 12059622119902119908 (119905) + 1205761205722119902V (119905) 119902119908 (119905) + 12057612058121199022V (119905) 119902119908 (119905)+ 12057612057321199023119908 (119905) + 21205761205832 119902119908 (119905) = 21205761198912 cos (Ω119911119905) (33)


12059621 = 11988621198861 21205761205831 = 11988671198861 1205761205721 = 11988631198861 1205761205811 = 11988641198861 1205761205731 = 11988651198861 1205761205741 = 11988661198861 12059622 = 11988721198871


21205761205832 = 11988761198871 1205761205722 = 11988731198871 1205761205812 = 11988741198871 1205761205732 = 11988751198871 21205761198912 = 11987501198871

(34)



119879119899 = 120576119899119905 (119899 = 0 1 ) (35)

that is

1198790 = 1199051198791 = 120576119905 (36)






(11986321119902V2 + 212058311198631119902V2 + 212058311198632119902V1 + 211986301198631119902V3+ 212058311198630119902V3 + 12057311199023V1 + 21205721119902V1119902V2 + 61205731119902V0119902V1119902V2



(39)



11986302119902V0 + 12059612119902V0 = 0119863021199021199080 + 120596221199021199080 = 0 (40)

120576111986302119902V1 + 211986301198631119902V0 + 212058311198630119902V0 + 12059621119902V1 + 12057311199023V0+ 11988611199022V0 + 120574111990221199080119902V0 + 120581111990221199080 = 0119863021199021199081 + 2119863011986311199021199080 + 120573211990231199080 + 120596221199021199081 + 1198862119902V01199021199080+ 2120583211986301199021199080 + 12058121199022V01199021199080 = 21198912 cos (Ω119911119905)

(41)


(42)


(43)


119902V0 = 119860V (1198791 1198792) exp (11989412059611198790)+ 119860V (1198791 1198792) exp (minus11989412059611198790)

1199021199080 = 119860119908 (1198791 1198792) exp (11989412059621198790)+ 119860119908 (1198791 1198792) exp (minus11989412059621198790) (44)




(45)





1205962 minus 1205961 = 120576120590 120590 = 119874 (1) (46)

Thus

(1205962 + 1205961) 1198790 = 212059611198790 + 1205761205901198790 = 212059611198790 + 1205901198791= 212059621198790 + 1205901198791212059621198790 = 2 (1205961 + 120576120590) 1198790 = 212059621198790 + 21205901198791(47)


21198941205961 (1198631119860V + 1205831119860V) + 12057211198602V + 1205811119860211990811989011989421205901198791 = 021198941205962 (1198631119860119908 + 1205832119860119908) + 1205722119860V1198601199081198901198941205901198791 = 0 (48)


119860119903 = 1198861199032 119890119895120579119903 119903 = V 119908 (49)




(50)


1205931 = 2 (120579119908 + 1205901198791) 1205932 = 120579V + 120579119908 + 1205901198791 (51)


1198631119886V + 1205831119886V + 12057211198862V21205961 sin 120579V + 1205811119886211990841205961 sin1205931 sec 120579V = 0


1198861199081198631120579119908 + 1205722119886V11988611990841205962 cos 120579119908 cos1205932 = 0(52)







1198841 = 119902V1198842 = 119902V1198843 = 1199021199084 = 119902119908(53)



(54)






119909 (119910) = 11990211988821198791198880119910 (119871119888 minus 119910) + Δ 119888119871119888 119910 (55)




Table1Orig

inaldesig

nparameterso

fthe

selected

transm

issionlin

e

Items

Diameter119889

(mm)


(mm2)

Linear

density119898 119888

(kgm)

Elastic

mod

ulus

119864 119888(MPa)

Averageo

peratio

nal

tension

(N)

Tensionatdesig

nwind

speed

(N)

Breaking

force

(N)

Span119871 119888

(m)

Sag119889 119888

(m)

Con

ductors

336

66655

206

63000

3531

643688

1412

65480

166




Density 120588119888(kgm3)






120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)



120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)




O

B

y

x

qc

Lc

Δ c



120603119910119871119888 minus cos120603119910119871119888 )

(58)






Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




ℏ (119905)119864119888119860119888 sdot 119871119888 = int1198711198880 [120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910 + 12 (120597119908119888120597119910 )2+ 12 (120597V119888120597119910 )2]119889119910

(24)

Let us set

120582 (119905) = ℏ (119905)119864119888119860119888 = 1119871119888 int1198711198880 [120597119906119888120597119910 + 120597119909120597119910 sdot 120597V119888120597119910+ 12 (120597119908119888120597119910 )2 + 12 (120597V119888120597119910 )2]119889119910

(25)



(26)


(27)


V119888 (119910 119905) = 119873sum119894=1

120593119894 (119910) sdot 119902119894 (119905) 119908119888 (119910 119905) = 119873sum

119894=1

120593119894 (119910) sdot 119902119894 (119905) (28)




(29)


1198861 119902V (119905) + 1198862119902V (119905) + 11988631199022V (119905) + 11988641199022119908 (119905) + 11988651199023V (119905)+ 11988661199022119908 (119905) 119902V (119905) + 1198867 119902V (119905) = 119875V1198871 119902119908 (119905) + 1198872119902119908 (119905) + 1198873119902V (119905) 119902119908 (119905) + 11988741199022V (119905) 119902119908 (119905)+ 11988751199023119908 (119905) + 1198876 119902119908 (119905) = 119875119908(30)


1198861 = int1198711198880

120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 1205932V (119910) 1198891199101198862 = minusint119871119888

0


1198863 = int1198711198880


sdot 1205931015840V (119910) int1198711198880

120597119909120597119910 sdot 1205931015840V (119910) 119889119910]120593V (119910) 1198891199101198864 = int119871119888

0



1198865 = int1198711198880


1198866 = int1198711198880


1198867 = int11987111988801198881198881205932V (119910) 119889119910

119875V = int1198711198880119875 (119909 119905) 120593V (119910) 119889119910

1198871 = int1198711198880

120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 1205932119908 (119910) 1198891199101198872 = minusint119871119888

0


0


120597119909120597119910 sdot 1205931015840V (119910) 119889119910]120593119908 (119910) 1198891199101198874 = int119871119888

0


1198875 = int1198711198880


1198876 = int11987111988801198881198881205932119908 (119910) 119889119910

119875119908 = int1198711198880119875 (119911 119905) 120593119908 (119910) 119889119910

(31)







119902V (119905) + 12059621119902V (119905) + 12057612057211199022V (119905) + 12057612058111199022119908 (119905) + 12057612057311199023V (119905)+ 12057612057411199022119908 (119905) 119902V (119905) + 21205761205831 119902V (119905) = 0119902119908 (119905) + 12059622119902119908 (119905) + 1205761205722119902V (119905) 119902119908 (119905) + 12057612058121199022V (119905) 119902119908 (119905)+ 12057612057321199023119908 (119905) + 21205761205832 119902119908 (119905) = 21205761198912 cos (Ω119911119905) (33)


12059621 = 11988621198861 21205761205831 = 11988671198861 1205761205721 = 11988631198861 1205761205811 = 11988641198861 1205761205731 = 11988651198861 1205761205741 = 11988661198861 12059622 = 11988721198871


21205761205832 = 11988761198871 1205761205722 = 11988731198871 1205761205812 = 11988741198871 1205761205732 = 11988751198871 21205761198912 = 11987501198871

(34)



119879119899 = 120576119899119905 (119899 = 0 1 ) (35)

that is

1198790 = 1199051198791 = 120576119905 (36)






(11986321119902V2 + 212058311198631119902V2 + 212058311198632119902V1 + 211986301198631119902V3+ 212058311198630119902V3 + 12057311199023V1 + 21205721119902V1119902V2 + 61205731119902V0119902V1119902V2



(39)



11986302119902V0 + 12059612119902V0 = 0119863021199021199080 + 120596221199021199080 = 0 (40)

120576111986302119902V1 + 211986301198631119902V0 + 212058311198630119902V0 + 12059621119902V1 + 12057311199023V0+ 11988611199022V0 + 120574111990221199080119902V0 + 120581111990221199080 = 0119863021199021199081 + 2119863011986311199021199080 + 120573211990231199080 + 120596221199021199081 + 1198862119902V01199021199080+ 2120583211986301199021199080 + 12058121199022V01199021199080 = 21198912 cos (Ω119911119905)

(41)


(42)


(43)


119902V0 = 119860V (1198791 1198792) exp (11989412059611198790)+ 119860V (1198791 1198792) exp (minus11989412059611198790)

1199021199080 = 119860119908 (1198791 1198792) exp (11989412059621198790)+ 119860119908 (1198791 1198792) exp (minus11989412059621198790) (44)




(45)





1205962 minus 1205961 = 120576120590 120590 = 119874 (1) (46)

Thus

(1205962 + 1205961) 1198790 = 212059611198790 + 1205761205901198790 = 212059611198790 + 1205901198791= 212059621198790 + 1205901198791212059621198790 = 2 (1205961 + 120576120590) 1198790 = 212059621198790 + 21205901198791(47)


21198941205961 (1198631119860V + 1205831119860V) + 12057211198602V + 1205811119860211990811989011989421205901198791 = 021198941205962 (1198631119860119908 + 1205832119860119908) + 1205722119860V1198601199081198901198941205901198791 = 0 (48)


119860119903 = 1198861199032 119890119895120579119903 119903 = V 119908 (49)




(50)


1205931 = 2 (120579119908 + 1205901198791) 1205932 = 120579V + 120579119908 + 1205901198791 (51)


1198631119886V + 1205831119886V + 12057211198862V21205961 sin 120579V + 1205811119886211990841205961 sin1205931 sec 120579V = 0


1198861199081198631120579119908 + 1205722119886V11988611990841205962 cos 120579119908 cos1205932 = 0(52)







1198841 = 119902V1198842 = 119902V1198843 = 1199021199084 = 119902119908(53)



(54)






119909 (119910) = 11990211988821198791198880119910 (119871119888 minus 119910) + Δ 119888119871119888 119910 (55)




Table1Orig

inaldesig

nparameterso

fthe

selected

transm

issionlin

e

Items

Diameter119889

(mm)


(mm2)

Linear

density119898 119888

(kgm)

Elastic

mod

ulus

119864 119888(MPa)

Averageo

peratio

nal

tension

(N)

Tensionatdesig

nwind

speed

(N)

Breaking

force

(N)

Span119871 119888

(m)

Sag119889 119888

(m)

Con

ductors

336

66655

206

63000

3531

643688

1412

65480

166




Density 120588119888(kgm3)






120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)



120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)




O

B

y

x

qc

Lc

Δ c



120603119910119871119888 minus cos120603119910119871119888 )

(58)






Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

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1


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003

002

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0

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times10minus3

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0

1

2

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4

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times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

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200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

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times10minus3

minus12

minus10

minus8

minus6

minus4

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0

2

4

6

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minus01

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0

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minus02

minus015

minus01

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c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

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0001002003004

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02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



1198865 = int1198711198880


1198866 = int1198711198880


1198867 = int11987111988801198881198881205932V (119910) 119889119910

119875V = int1198711198880119875 (119909 119905) 120593V (119910) 119889119910

1198871 = int1198711198880

120574119888radic1 + (120597119909120597119910)2119892 (1 + 1205761198880) 1205932119908 (119910) 1198891199101198872 = minusint119871119888

0


0


120597119909120597119910 sdot 1205931015840V (119910) 119889119910]120593119908 (119910) 1198891199101198874 = int119871119888

0


1198875 = int1198711198880


1198876 = int11987111988801198881198881205932119908 (119910) 119889119910

119875119908 = int1198711198880119875 (119911 119905) 120593119908 (119910) 119889119910

(31)







119902V (119905) + 12059621119902V (119905) + 12057612057211199022V (119905) + 12057612058111199022119908 (119905) + 12057612057311199023V (119905)+ 12057612057411199022119908 (119905) 119902V (119905) + 21205761205831 119902V (119905) = 0119902119908 (119905) + 12059622119902119908 (119905) + 1205761205722119902V (119905) 119902119908 (119905) + 12057612058121199022V (119905) 119902119908 (119905)+ 12057612057321199023119908 (119905) + 21205761205832 119902119908 (119905) = 21205761198912 cos (Ω119911119905) (33)


12059621 = 11988621198861 21205761205831 = 11988671198861 1205761205721 = 11988631198861 1205761205811 = 11988641198861 1205761205731 = 11988651198861 1205761205741 = 11988661198861 12059622 = 11988721198871


21205761205832 = 11988761198871 1205761205722 = 11988731198871 1205761205812 = 11988741198871 1205761205732 = 11988751198871 21205761198912 = 11987501198871

(34)



119879119899 = 120576119899119905 (119899 = 0 1 ) (35)

that is

1198790 = 1199051198791 = 120576119905 (36)






(11986321119902V2 + 212058311198631119902V2 + 212058311198632119902V1 + 211986301198631119902V3+ 212058311198630119902V3 + 12057311199023V1 + 21205721119902V1119902V2 + 61205731119902V0119902V1119902V2



(39)



11986302119902V0 + 12059612119902V0 = 0119863021199021199080 + 120596221199021199080 = 0 (40)

120576111986302119902V1 + 211986301198631119902V0 + 212058311198630119902V0 + 12059621119902V1 + 12057311199023V0+ 11988611199022V0 + 120574111990221199080119902V0 + 120581111990221199080 = 0119863021199021199081 + 2119863011986311199021199080 + 120573211990231199080 + 120596221199021199081 + 1198862119902V01199021199080+ 2120583211986301199021199080 + 12058121199022V01199021199080 = 21198912 cos (Ω119911119905)

(41)


(42)


(43)


119902V0 = 119860V (1198791 1198792) exp (11989412059611198790)+ 119860V (1198791 1198792) exp (minus11989412059611198790)

1199021199080 = 119860119908 (1198791 1198792) exp (11989412059621198790)+ 119860119908 (1198791 1198792) exp (minus11989412059621198790) (44)




(45)





1205962 minus 1205961 = 120576120590 120590 = 119874 (1) (46)

Thus

(1205962 + 1205961) 1198790 = 212059611198790 + 1205761205901198790 = 212059611198790 + 1205901198791= 212059621198790 + 1205901198791212059621198790 = 2 (1205961 + 120576120590) 1198790 = 212059621198790 + 21205901198791(47)


21198941205961 (1198631119860V + 1205831119860V) + 12057211198602V + 1205811119860211990811989011989421205901198791 = 021198941205962 (1198631119860119908 + 1205832119860119908) + 1205722119860V1198601199081198901198941205901198791 = 0 (48)


119860119903 = 1198861199032 119890119895120579119903 119903 = V 119908 (49)




(50)


1205931 = 2 (120579119908 + 1205901198791) 1205932 = 120579V + 120579119908 + 1205901198791 (51)


1198631119886V + 1205831119886V + 12057211198862V21205961 sin 120579V + 1205811119886211990841205961 sin1205931 sec 120579V = 0


1198861199081198631120579119908 + 1205722119886V11988611990841205962 cos 120579119908 cos1205932 = 0(52)







1198841 = 119902V1198842 = 119902V1198843 = 1199021199084 = 119902119908(53)



(54)






119909 (119910) = 11990211988821198791198880119910 (119871119888 minus 119910) + Δ 119888119871119888 119910 (55)




Table1Orig

inaldesig

nparameterso

fthe

selected

transm

issionlin

e

Items

Diameter119889

(mm)


(mm2)

Linear

density119898 119888

(kgm)

Elastic

mod

ulus

119864 119888(MPa)

Averageo

peratio

nal

tension

(N)

Tensionatdesig

nwind

speed

(N)

Breaking

force

(N)

Span119871 119888

(m)

Sag119889 119888

(m)

Con

ductors

336

66655

206

63000

3531

643688

1412

65480

166




Density 120588119888(kgm3)






120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)



120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)




O

B

y

x

qc

Lc

Δ c



120603119910119871119888 minus cos120603119910119871119888 )

(58)






Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

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times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

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minus15

minus1

minus05

0

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1

15

2

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minus0015

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minus0005

0

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times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

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mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

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yLc

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times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

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minus02

minus015

minus01

minus005

0

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01

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minus02

minus015

minus01

minus005

0

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0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

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nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

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nT

cT

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02 04 06 08 10f (Hz)

0

005

01

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02

025

03

035

04

Four

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spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

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02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

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02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



21205761205832 = 11988761198871 1205761205722 = 11988731198871 1205761205812 = 11988741198871 1205761205732 = 11988751198871 21205761198912 = 11987501198871

(34)



119879119899 = 120576119899119905 (119899 = 0 1 ) (35)

that is

1198790 = 1199051198791 = 120576119905 (36)






(11986321119902V2 + 212058311198631119902V2 + 212058311198632119902V1 + 211986301198631119902V3+ 212058311198630119902V3 + 12057311199023V1 + 21205721119902V1119902V2 + 61205731119902V0119902V1119902V2



(39)



11986302119902V0 + 12059612119902V0 = 0119863021199021199080 + 120596221199021199080 = 0 (40)

120576111986302119902V1 + 211986301198631119902V0 + 212058311198630119902V0 + 12059621119902V1 + 12057311199023V0+ 11988611199022V0 + 120574111990221199080119902V0 + 120581111990221199080 = 0119863021199021199081 + 2119863011986311199021199080 + 120573211990231199080 + 120596221199021199081 + 1198862119902V01199021199080+ 2120583211986301199021199080 + 12058121199022V01199021199080 = 21198912 cos (Ω119911119905)

(41)


(42)


(43)


119902V0 = 119860V (1198791 1198792) exp (11989412059611198790)+ 119860V (1198791 1198792) exp (minus11989412059611198790)

1199021199080 = 119860119908 (1198791 1198792) exp (11989412059621198790)+ 119860119908 (1198791 1198792) exp (minus11989412059621198790) (44)




(45)





1205962 minus 1205961 = 120576120590 120590 = 119874 (1) (46)

Thus

(1205962 + 1205961) 1198790 = 212059611198790 + 1205761205901198790 = 212059611198790 + 1205901198791= 212059621198790 + 1205901198791212059621198790 = 2 (1205961 + 120576120590) 1198790 = 212059621198790 + 21205901198791(47)


21198941205961 (1198631119860V + 1205831119860V) + 12057211198602V + 1205811119860211990811989011989421205901198791 = 021198941205962 (1198631119860119908 + 1205832119860119908) + 1205722119860V1198601199081198901198941205901198791 = 0 (48)


119860119903 = 1198861199032 119890119895120579119903 119903 = V 119908 (49)




(50)


1205931 = 2 (120579119908 + 1205901198791) 1205932 = 120579V + 120579119908 + 1205901198791 (51)


1198631119886V + 1205831119886V + 12057211198862V21205961 sin 120579V + 1205811119886211990841205961 sin1205931 sec 120579V = 0


1198861199081198631120579119908 + 1205722119886V11988611990841205962 cos 120579119908 cos1205932 = 0(52)







1198841 = 119902V1198842 = 119902V1198843 = 1199021199084 = 119902119908(53)



(54)






119909 (119910) = 11990211988821198791198880119910 (119871119888 minus 119910) + Δ 119888119871119888 119910 (55)




Table1Orig

inaldesig

nparameterso

fthe

selected

transm

issionlin

e

Items

Diameter119889

(mm)


(mm2)

Linear

density119898 119888

(kgm)

Elastic

mod

ulus

119864 119888(MPa)

Averageo

peratio

nal

tension

(N)

Tensionatdesig

nwind

speed

(N)

Breaking

force

(N)

Span119871 119888

(m)

Sag119889 119888

(m)

Con

ductors

336

66655

206

63000

3531

643688

1412

65480

166




Density 120588119888(kgm3)






120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)



120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)




O

B

y

x

qc

Lc

Δ c



120603119910119871119888 minus cos120603119910119871119888 )

(58)






Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

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200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

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xL

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times10minus4

times10minus3

minus5

minus4

minus3

minus2

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0

1

2

3

4

5

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c




minus15

minus1

minus05

0

05

1

15

2

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times10minus3

vcLc

v∙ cT0L

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minus0015

minus001

minus0005

0

0005

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0015w

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c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

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yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

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minus02

minus015

minus01

minus005

0

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01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

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cT

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02 04 06 08 10f (Hz)

0

005

01

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02

025

03

035

04

Four

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itude

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trum


0

01

02

03

04

05

06

07

08

09

1Fo

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02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




11986302119902V0 + 12059612119902V0 = 0119863021199021199080 + 120596221199021199080 = 0 (40)

120576111986302119902V1 + 211986301198631119902V0 + 212058311198630119902V0 + 12059621119902V1 + 12057311199023V0+ 11988611199022V0 + 120574111990221199080119902V0 + 120581111990221199080 = 0119863021199021199081 + 2119863011986311199021199080 + 120573211990231199080 + 120596221199021199081 + 1198862119902V01199021199080+ 2120583211986301199021199080 + 12058121199022V01199021199080 = 21198912 cos (Ω119911119905)

(41)


(42)


(43)


119902V0 = 119860V (1198791 1198792) exp (11989412059611198790)+ 119860V (1198791 1198792) exp (minus11989412059611198790)

1199021199080 = 119860119908 (1198791 1198792) exp (11989412059621198790)+ 119860119908 (1198791 1198792) exp (minus11989412059621198790) (44)




(45)





1205962 minus 1205961 = 120576120590 120590 = 119874 (1) (46)

Thus

(1205962 + 1205961) 1198790 = 212059611198790 + 1205761205901198790 = 212059611198790 + 1205901198791= 212059621198790 + 1205901198791212059621198790 = 2 (1205961 + 120576120590) 1198790 = 212059621198790 + 21205901198791(47)


21198941205961 (1198631119860V + 1205831119860V) + 12057211198602V + 1205811119860211990811989011989421205901198791 = 021198941205962 (1198631119860119908 + 1205832119860119908) + 1205722119860V1198601199081198901198941205901198791 = 0 (48)


119860119903 = 1198861199032 119890119895120579119903 119903 = V 119908 (49)




(50)


1205931 = 2 (120579119908 + 1205901198791) 1205932 = 120579V + 120579119908 + 1205901198791 (51)


1198631119886V + 1205831119886V + 12057211198862V21205961 sin 120579V + 1205811119886211990841205961 sin1205931 sec 120579V = 0


1198861199081198631120579119908 + 1205722119886V11988611990841205962 cos 120579119908 cos1205932 = 0(52)







1198841 = 119902V1198842 = 119902V1198843 = 1199021199084 = 119902119908(53)



(54)






119909 (119910) = 11990211988821198791198880119910 (119871119888 minus 119910) + Δ 119888119871119888 119910 (55)




Table1Orig

inaldesig

nparameterso

fthe

selected

transm

issionlin

e

Items

Diameter119889

(mm)


(mm2)

Linear

density119898 119888

(kgm)

Elastic

mod

ulus

119864 119888(MPa)

Averageo

peratio

nal

tension

(N)

Tensionatdesig

nwind

speed

(N)

Breaking

force

(N)

Span119871 119888

(m)

Sag119889 119888

(m)

Con

ductors

336

66655

206

63000

3531

643688

1412

65480

166




Density 120588119888(kgm3)






120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)



120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)




O

B

y

x

qc

Lc

Δ c



120603119910119871119888 minus cos120603119910119871119888 )

(58)






Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

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c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





1205962 minus 1205961 = 120576120590 120590 = 119874 (1) (46)

Thus

(1205962 + 1205961) 1198790 = 212059611198790 + 1205761205901198790 = 212059611198790 + 1205901198791= 212059621198790 + 1205901198791212059621198790 = 2 (1205961 + 120576120590) 1198790 = 212059621198790 + 21205901198791(47)


21198941205961 (1198631119860V + 1205831119860V) + 12057211198602V + 1205811119860211990811989011989421205901198791 = 021198941205962 (1198631119860119908 + 1205832119860119908) + 1205722119860V1198601199081198901198941205901198791 = 0 (48)


119860119903 = 1198861199032 119890119895120579119903 119903 = V 119908 (49)




(50)


1205931 = 2 (120579119908 + 1205901198791) 1205932 = 120579V + 120579119908 + 1205901198791 (51)


1198631119886V + 1205831119886V + 12057211198862V21205961 sin 120579V + 1205811119886211990841205961 sin1205931 sec 120579V = 0


1198861199081198631120579119908 + 1205722119886V11988611990841205962 cos 120579119908 cos1205932 = 0(52)







1198841 = 119902V1198842 = 119902V1198843 = 1199021199084 = 119902119908(53)



(54)






119909 (119910) = 11990211988821198791198880119910 (119871119888 minus 119910) + Δ 119888119871119888 119910 (55)




Table1Orig

inaldesig

nparameterso

fthe

selected

transm

issionlin

e

Items

Diameter119889

(mm)


(mm2)

Linear

density119898 119888

(kgm)

Elastic

mod

ulus

119864 119888(MPa)

Averageo

peratio

nal

tension

(N)

Tensionatdesig

nwind

speed

(N)

Breaking

force

(N)

Span119871 119888

(m)

Sag119889 119888

(m)

Con

ductors

336

66655

206

63000

3531

643688

1412

65480

166




Density 120588119888(kgm3)






120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)



120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)




O

B

y

x

qc

Lc

Δ c



120603119910119871119888 minus cos120603119910119871119888 )

(58)






Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





1198841 = 119902V1198842 = 119902V1198843 = 1199021199084 = 119902119908(53)



(54)






119909 (119910) = 11990211988821198791198880119910 (119871119888 minus 119910) + Δ 119888119871119888 119910 (55)




Table1Orig

inaldesig

nparameterso

fthe

selected

transm

issionlin

e

Items

Diameter119889

(mm)


(mm2)

Linear

density119898 119888

(kgm)

Elastic

mod

ulus

119864 119888(MPa)

Averageo

peratio

nal

tension

(N)

Tensionatdesig

nwind

speed

(N)

Breaking

force

(N)

Span119871 119888

(m)

Sag119889 119888

(m)

Con

ductors

336

66655

206

63000

3531

643688

1412

65480

166




Density 120588119888(kgm3)






120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)



120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)




O

B

y

x

qc

Lc

Δ c



120603119910119871119888 minus cos120603119910119871119888 )

(58)






Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Table1Orig

inaldesig

nparameterso

fthe

selected

transm

issionlin

e

Items

Diameter119889

(mm)


(mm2)

Linear

density119898 119888

(kgm)

Elastic

mod

ulus

119864 119888(MPa)

Averageo

peratio

nal

tension

(N)

Tensionatdesig

nwind

speed

(N)

Breaking

force

(N)

Span119871 119888

(m)

Sag119889 119888

(m)

Con

ductors

336

66655

206

63000

3531

643688

1412

65480

166




Density 120588119888(kgm3)






120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)



120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)




O

B

y

x

qc

Lc

Δ c



120603119910119871119888 minus cos120603119910119871119888 )

(58)






Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Density 120588119888(kgm3)






120596119908 = 119899120587119871119888radic1198791198880119898119888 119899 = 1 2 3 Φ119908 = 119860119899 sin 119899120587119910119871119888 119899 = 1 2 3 (56)



120596V = 2119899120587119871119888 radic1198791198880119898119888 119899 = 1 2 3 ΦV = 119861119899 sin2119899120587119910119871119888 119899 = 1 2 3 (57)




O

B

y

x

qc

Lc

Δ c



120603119910119871119888 minus cos120603119910119871119888 )

(58)






Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0273 0409 0546 0682 0818 0955 1091




119891vant = 120596vant(2120587) 0273 0546 0818 1091 1364 1637 191 2182


Calculationequation

Firstorder

Secondorder

Thirdorder


orderEighthorder


119891119908 = 120596119908(2120587) 0136 0272 0408 0544 068 0816 0952 1088


119891V = 120596V(2120587) 0346 046 0686 0954 1225 1497 1769 2042


119891vant = 120596vant(2120587) 027 0544 0816 1088 136 1633 191 218








X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z



X

Y

Z









minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



minus2

minus15

minus1

minus05

0

05

1

15times10minus4

v cL

c

200 400 600 800 10000t (s)


minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

wcL

c

200 400 600 800 10000t (s)

times10minus3



0

0005

001

0015

002

0025

003

0035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 054（Ｔ

f = 027（Ｔ

inminusplane

(a) In-plane

01 02 03 04 05 060f (Hz)

0

01

02

03

04

05

06

07

Four

ier a

mpl

itude

spec

trum

f = 027（Ｔ


f = 01361（Ｔ

(b) Out-of-plane





minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



minus4minus2

02

40

05

1


004

003

002

001

0

times10minus3

xL

c

yLc

wcL c


times10minus4

times10minus3

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

v cL

c




minus15

minus1

minus05

0

05

1

15

2

times10minus4

times10minus3

vcLc

v∙ cT0L

c



minus0015

minus001

minus0005

0

0005

001

0015w

∙ cT0L

c

times10minus3







minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus0025

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

0025

wcL

c

200 400 600 800 10000t (s)



02 04 06 08 10f (Hz)

0

05

1

15

2

25

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 04022（Ｔ

f = 027（Ｔ

f = 013（Ｔ

(a) In-plane

0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01371（Ｔ

f = 027（Ｔ

(b) Out-of-plane





minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



minus004minus002

0002

0040

05

1


004

003

002

001

0

xL

c

yLc

wcL c


times10minus3

minus12

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

v cL

c



minus02

minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c




minus02

minus015

minus01

minus005

0

005

01

015w

∙ cT0L

c








0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

05

1

15

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


minus1

minus05

0

05

1

15

2

25

3

200 400 600 800 10000t (s)


nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0

005

01

015

02

025

03

035

04

Four

ier a

mpl

itude

spec

trum


0

01

02

03

04

05

06

07

08

09

1Fo

urie

r am

plitu

de sp

ectr

um

02 04 06 08 10f (Hz)

f = 040（Ｔ

f = 013 （Ｔ

f = 027 （Ｔ






Ω119911 = 119896120596119908 + 1205761205901 (59)



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



minus12minus10


02468

v cL

c

200 400 600 800 10000t (s)

times10minus3


minus002minus0015

minus001minus0005

00005

0010015

0020025

wcL

c

200 400 600 800 10000t (s)


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 026 （Ｔ

f = 038 （Ｔ

f = 013 （Ｔ


0

05

1

15

2

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 026 （Ｔ

f = 01310 （Ｔ



0001002003004

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



0 200 400 600 800 1000minus1

minus050

051

152

253

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 10

005

01

015

02

025

03

f (Hz)

Four

ier a

mpl

itude

spec

trum

f = 038 （Ｔ

f = 026 （Ｔ




100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



100 200 300 400 5000t (s)

minus12minus10


02468

v cL

c

times10minus3


100 200 300 400 5000t (s)

minus0015minus001

minus00050

0005001

0015002

0025

wcL

c


0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 060 （Ｔ

f = 020 （Ｔ

f = 03024 （Ｔ


0

02

04

06

08

1

12

14

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 020 （Ｔ

f = 01050 （Ｔ



02468

v∙ cT0L

c

times10minus3


vcLc



times10minus3

times10minus3

minus6

minus4

minus2

0

2

4

6

w∙ cT0L

c


100 200 300 400 5000t (s)

minus1minus05

005

115

225

3

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04045

Four

ier a

mpl

itude

spec

trum

f = 030 （Ｔ












200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



200 400 600 800 10000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

v cL

c


200 400 600 800 10000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


02 04 06 08 10f (Hz)

inminusplane

005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

f = 034 （Ｔ

f = 023 （Ｔ

f = 005 （Ｔ

f = 02 （Ｔ


005

115

225

335

445

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 034 （Ｔf = 008 （Ｔ

f = 01361 （Ｔ

f = 01965 （Ｔ


minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



200 400 600 800 10000t (s)

minus6minus4minus2

02468

10

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 005 （Ｔ

f = 020 （Ｔ

f = 047 （Ｔ




minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



minus004

minus003

minus002

minus001

0

001

002

v cL

c

200 400 600 800 10000t (s)



0001002003004

wcL

c

200 400 600 800 10000t (s)


0

05

1

15

2

25

3

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 004 （Ｔ

f = 028 （Ｔ

f = 023 （Ｔ


005

115

225

335

445

555

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 01382 （Ｔ

f = 0094 （Ｔ


minus03

minus02

minus01

0

01

02

03

v∙ cT0L

c


vcLc


minus03

minus02

minus01

0

01

02

03

w∙ cT0L

c



200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 028 （Ｔ

f = 023 （Ｔ




200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



200 400 600 800 10000t (s)

minus004

minus003

minus002

minus001

0

001

002

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


02 04 06 08 10f (Hz)

005

115

225

335

445

5

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 028 （Ｔ

f = 055 （Ｔf = 023 （Ｔ


0123456789

10

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01389 （Ｔ



minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c


200 400 600 800 10000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


02 04 06 08 10f (Hz)

0010203040506070809

1

Four

ier a

mpl

itude

spec

trum

f = 023 （Ｔ f = 056 （Ｔ

f = 028 （Ｔ




200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



200 400 600 800 10000t (s)

minus0035minus003

minus0025minus002

minus0015minus001

minus00050

0005001

0015

v cL

c


200 400 600 800 10000t (s)


0001002003004

wcL

c


0

1

2

3

4

5

6

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

inminusplane

f = 055 （Ｔf = 023 （Ｔ

f = 02771 （Ｔ


0

2

4

6

8

10

12

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 009 （Ｔ f = 042 （Ｔ

f = 01386 （Ｔ


minus02minus015

minus01minus005

0005

01015

02

v∙ cT0L

c



minus025minus02

minus015minus01

minus0050

00501

01502

025

w∙ cT0L

c



0 200 400 600 800 1000t (s)


minus6

minus4

minus2

0

2

4

6

8

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

14

12

Four

ier a

mpl

itude

spec

trum

f = 028 （Ｔ

f = 023 （Ｔ

f = 056 （Ｔ






119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





119875 (119909 119905) = 05120588119886119863119888119862119863 (V119886-119909 minus V119888)2 119875 (119911 119905) = 05120588119886119863119888119862119863 (V119886-119911 minus 119888)2 (60)



1198867 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119909) 1205932V (119910) 119889119910

1198876 = int1198711198880(119888119888 + 120588119886119863119888119862119863V119886-119911) 1205932119908 (119910) 119889119910

119875V = int119871119888005120588119886119863119888119862119863 (V119886-119909)2 120593V (119910) 119889119910

119875119908 = int119871119888005120588119886119863119888119862119863 (V119886-119911)2 120593119908 (119910) 119889119910

(61)



(62)




200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



200 400 600 800 10000t (s)

times10minus5

minus8

minus6

minus4

minus2

0

2

4

v cL

c


200 400 600 800 10000t (s)

times10minus3

minus2minus15

minus1minus05

005

115

2

wcL

c


02 04 06 08 10f (Hz)

00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 055 （Ｔ

f = 040 （Ｔf = 027 （Ｔ


02 04 06 08 10f (Hz)

0

01

02

03

04

05

06

Four

ier a

mpl

itude

spec

trum


f = 0136 （Ｔf = 068 （Ｔ


times10minus5

times10minus4


02468

v∙ cT0L

c



times10minus3

minus003

minus002

minus001

0

001

002

003

w∙ cT0L

c



200 400 600 800 10000t (s)


0506070809

111121314

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


0

005

01

015

02

025

03

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)




U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



U0 = 520 ＧＭ

10minus5

10minus4

10minus3

10minus1

10minus2

100Fo

urie

r am

plitu

de sp

ectr

um


f (Hz)


U0 = 956 ＧＭ

10minus3

10minus2

10minus1

101

100

Four

ier a

mpl

itude

spec

trum


f (Hz)


U0 = 201 ＧＭ

10minus6

10minus4

10minus2

100

102

Four

ier a

mpl

itude

spec

trum


f (Hz)







minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



minus2minus1

0123

54


times10minus3

200 400 600 800 1000 12000t (s)

cL

camp

wcL

c


times10minus3


wcLc

minus15

minus1

minus05

0

05

1

15

v cL

c


minus002minus0015

minus0005minus001

00005

0010015

002

v∙ cT0L

c




minus001

minus0005

0

0005

001

0015

002

w∙ cT0L

c

wcLc times10minus3


02 04 06 08 10f (Hz)

inminusplane

0

0005

001

0015

002

0025

Four

ier a

mpl

itude

spec

trum f = 03260 （Ｔ



0

005

01

015

02

Four

ier a

mpl

itude

spec

trum

0201 04030 0605f (Hz)

f = 01362 （Ｔ


0 200 400 600 800 1000 1200t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0


002040608

11214


00005

0010015

0020025

0030035

004

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 0326 （Ｔ




0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0 200 400 600 800 1000 1200t (s)

minus002

minus001

0

001

002

003

004


cL

camp

wcL

c


minus001 0 001 002 003 004minus002wcLc

minus002

minus0015

minus001

minus0005

0

0005

001

0015

v cL

c



minus015

minus01

minus005

0

005

01

015

v∙ cT0L

c



0001002003004

w∙ cT0L

c



02 04 06 08 10f (Hz)

0010203040506070809

Four

ier a

mpl

itude

spec

trum

inminusplane

f = 03788 （Ｔ


0

01

02

03

04

05

Four

ier a

mpl

itude

spec

trum

01 02 03 04 05 060f (Hz)


f = 03788 （Ｔ

f = 018 （Ｔ


0 200 400 600 800 1000 1200minus2

minus1

0

1

2

3

4

5

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



02 04 06 08 10f (Hz)

0005

01015

02025

03035

04

Four

ier a

mpl

itude

spec

trum f = 03788 （Ｔ




100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



100 200 300 400 500 6000t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

cL

c


100 200 300 400 500 6000t (s)

minus006

minus004

minus002

0

002

004

006

wcL

c


minus08

minus06

minus04

minus02

0

02

04

06

v∙ cT0L

c



minus08

minus06

minus04

minus02

0

02

04

06

w∙ cT0L

c



inminusplane

0

1

2

3

4

5

Four

ier a

mpl

itude

spec

trum

02 04 06 08 10f (Hz)

f = 01931 （Ｔ

f = 02336 （Ｔ

f = 012 （Ｔ


0 01 02 03 04 05 06f (Hz)


0

1

2

3

4

5

6

7

Four

ier a

mpl

itude

spec

trum

f = 0294 （Ｔ

f = 00981 （Ｔ

f = 02056 （Ｔ


0 100 200 300 400 500 600minus4

minus2

0

2

4

6

8

10

t (s)

nonminus

dim

ensio

nal t

ensio

nT

cT

c0



0 02 04 06 08 1f (Hz)

0

02

04

06

08

1

Four

ier a

mpl

itude

spec

trum

f = 02336 （Ｔ

f = 01931 （Ｔ

f = 0475 （Ｔ







5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






5 Conclusions







Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






Acknowledgments


References


































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

Nonlinear Dynamic Analysis of High-Voltage Overhead ...downloads.hindawi.com/journals/sv/2018/1247523.pdf · ResearchArticle Nonlinear Dynamic Analysis of High-Voltage Overhead Transmission