Research Article The Effect of High-Frequency Parametric ...downloads.hindawi.com/journals/sv/2014/792673.pdf · perturbation, an approximate equation governing only the low-frequencymotionisderived

Research ArticleThe Effect of High-Frequency Parametric Excitation ona Stochastically Driven Pantograph-Catenary System

R H Huan1 W Q Zhu1 F Ma2 and Z H Liu3

1 Department of Mechanics Zhejiang University Hangzhou 310027 China2Department of Mechanical Engineering University of California Berkeley CA 94720 USA3Department of Civil Engineering Xiamen University Xiamen 361005 China

Correspondence should be addressed to Z H Liu liuzhxmueducn

Received 5 June 2013 Accepted 7 August 2013 Published 11 February 2014

Academic Editor Reza Jazar

Copyright copy 2014 R H Huan 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

In high-speed electric trains a pantograph is mounted on the roof of the train to collect power through contact with an overheadcatenary wire The effect of fast harmonic and parametric excitation on a stochastically driven pantograph-catenary system isstudied in this paper A single-degree-of-freedom model of the pantograph-catenary system is adopted wherein the stiffness ofthe nonlinear spring has a time-varying component characterized by both low and high frequencies Using perturbation andharmonic averaging a Fokker-Planck-Kolmogorov equation governing the stationary response of the pantograph-catenary systemis set up Based on the transition probability density of the stationary response it is found that even small high-frequency parametricexcitation has an appreciable effect on the system response Among other things it shifts the resonant frequency and often changesthe response characteristics markedly

1 Introduction

A number of important structures can be modeled as astochastically driven nonlinear system subjected to both slowand fast harmonic and parametric excitations An exampleis the pantograph-catenary system in railway engineeringHigh-speed electric trains often employ a pantograph tocollect their currents from an overhead catenary systemDuring operation the pantograph is excited by forces dueto train-body vibration ambient air flow contact wireirregularities and other disturbances These disturbancescan be realistically taken as a stochastic excitation to thepantograph Owing to the stiffness variation between thesupport poles and the short-distance droppers the catenarymay be regarded as a nonlinear spring with a time-varyingstiffness component As a result the combined pantograph-catenary system is an example of a stochastically driven non-linear system with both low- and high-frequency parametricexcitations

There is fairly extensive literature on the dynamicsof pantograph-catenary systems [1ndash6] However previousstudies have focused on deterministic excitations It has

been accepted that parametrically induced vibration ofa pantograph-catenary system occurs mainly in the low-frequency region Thus earlier studies usually ignored thehigh-frequency parametric effect generated by the catenary[7 8] Recent theoretical studies on dynamical systemshowever suggest that high-frequency parametric excitationcould shift the resonant frequency or equilibrium states [9ndash12] thus altering the stability [13ndash15] and other responsecharacteristics [16]

In this work the effect of fast harmonic and parametricexcitation on a stochastically driven pantograph-catenarysystem is investigated A nonlinear single-degree-of-freedommodel of the pantograph-catenary system possessing low-and high-frequency time-varying stiffness is adopted Usingperturbation an approximate equation governing only thelow-frequencymotion is derivedThen an averagingmethodbased on harmonic functions [17ndash21] is applied to thelow-frequency equation Subsequently a Fokker-Planck-Kolmogorov equation governing the stationary response ofthe pantograph-catenary system is set up Based on thetransition probability density of the stationary response the

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 792673 8 pageshttpdxdoiorg1011552014792673

2 Shock and Vibration

K(Ωs Ωf t)

m

c

Figure 1 Dynamic model of pantograph-catenary system

effect of fast parametric excitation on the resonant frequencyand the primary resonant response are studied Finally directnumerical simulations of the nonlinear model are performedto validate the analysis presented

2 Model of Pantograph-Catenary System

A commonly used model in railway engineering of thepantograph-catenary system [3 7 8] is the single-degree-of-freedom model shown in Figure 1 Since the stiffnessvariation is repeated in each span of the catenary the catenarystiffness possesses periodic components If we consider thestiffness variation between the vertical droppers the catenaryis usually taken as a spring with time-varying stiffnesscomponents given by [1 7 8]

119870(Ω119904 Ω119891 119905) = 119870

0(119881) (1 + 120572 cos (Ω

119904119905) + 120573 cos (Ω

119891119905))

(1)

where119881 represents the speed of train and1198700(119881) is the speed-

dependent average stiffness of the catenary In the aboveequation 120572 120573 are unspecified coefficients and 120572 cos(Ω

119904119905)

120573 cos(Ω119891119905) account for the stiffness fluctuations between the

support poles and the vertical droppers respectively Let 119871119904

and 119871119889denote respectively the span distance and dropper

distance Then the frequencies of stiffness fluctuation can beexpressed as

Ω119904=2120587119881

119871119904

Ω119891=2120587119881

119871119889

(2)

In general 119871119904≫ 119871119889and therefore Ω

119891≫ Ω

119904 In many

cases the average stiffness 1198700(119881) can be approximated by a

quadratic function of the train speed [8] such that

1198700(119881) = 119870

119904minus 1198701198891198812

(3)

where 119870119904is the static average stiffness of the catenary and

119870119889is a coefficient accounting for dynamic interactions of the

coupled pantograph-catenary system The value of 119870119904can be

calculated by the finite element methodDue to random disturbances (such as train body move-

ment contact wire irregularity and ambient air flow) a weak

random excitation 119888119908120585(119905) is introduced into the equation

of the pantograph-catenary system Incorporating stiffnessnonlinearity the equation of motion of the pantograph-catenary system can be written as

119898 119910 + 119888 119910 + 1198961015840

119910 + 1199103

+ (119870119904minus 1198701198891198812

)

times (1 + 120572 cos (Ω119904119905) + 120573 cos (Ω

119891119905)) 119910 = 119888

119908120585 (119905)

(4)

where119910 is the vertical displacement of the pantograph systemand the fourth item in the left of (4) represents the interactionof pantograph and catenary system

Let1205962119899= 119896119898 and 120591 = 120596

119899119905where 119896 = 119896

1015840

+119870119904minus1198701198891198812 and 120591

is a nondimensional time Equation (4) can be converted intothe nondimensional form

11991010158401015840

+ 21205771199101015840

+ 119910 + 1198871199103

= [minus11986411+ 119864121199032

119904] cos (119903

119904120591) 119910

+ [minus11986421+ 119864221199032

119891] cos (119903

119891120591) 119910 + ℎ

0120585 (120591)

(5)

where 1199101015840 = 119889119910119889120591 11991010158401015840

= 1198892

1199101198891205912 and

2120577 =119888

119898120596119899

119887 =

1198981205962119899

11986411=120572119870119904

1198981205962119899

11986412=120572119870119889

119898(119871119904

2120587)

2

11986421=120573119870119904

1198981205962119899

11986422=120573119870119889

119898(119871119889

2120587)

2

ℎ0=

119888119908

1198981205962119899

119903119904=Ω119904

120596119899

=2120587119881

120596119899119871119904

119903119891=

Ω119891

120596119899

=2120587119881

120596119899119871119889

(6)

Here 119903119904

= 119874(1) is the nondimensional low-frequencyexcitation and 119903

119891≫ 1 is the nondimensional high-frequency

excitation If a train is travelling at a constant speed 119881 then2120577 119864

11 119864121199032

119904 11986421 119864221199032

119891 and ℎ

0are all small constants

Suppose these constants are of the same order in a smallparameter 120576 System (5) is a randomly excited Duffingoscillator with both slow and fast time-varying stiffness Thegoal of the present work is to investigate the influence of thefast parametric excitation on the characteristics of system (5)

3 Approximate Equation Governingthe Slow Motion

Introduce two different time-scales

1198790= 119903119891120591 = 120576minus1

120591 1198791= 120591 (7)

where the slow time 1198791and the fast time 119879

0are considered as

new independent variables in (5) Separate 119910(120591) into a slowpart 119908(119879

1) and a fast part 120576120601(119879

0 1198791) [10] so that

119910 (120591) = 119908 (1198791) + 120576120593 (119879

0 1198791) (8)

Shock and Vibration 3

It has been recognized that the behavior of system (5) ismainly described by the slow part 119908 since 120576120601 is smallcompared to 119908 Let ⟨sdot⟩

1198790= (int2120587

0

sdot 1198891198790)2120587 be the time-ave-

raging operator over one period of the fast time scale1198790with the slow time 119879

1fixed Assume that 120576120601 and its

derivatives vanish upon 1198790-averaging so that ⟨119910(120591)⟩

1198790

=

119908(1198791) Substitute (8) into (5) to obtain

1198632

1119908 + (120576119863

2

1+ 211986301198631+ 120576minus1

1198632

0) 120593 + 2120577 (119863

1119908 + 119863

0120593 + 120576119863

1120593)

+ (119908 + 120576120593) + 119887(119908 + 120576120593)3

= [minus11986411+ 119864121199032

119904] (119908 + 120576120593) cos (119903

1199041198791)

times [minus11986421+ 120576minus1

11986422119903119891] (119908 + 120576120593) cos (119879

0) + ℎ0120585 (1198791)

(9)

where 119863119895119894= 120597119895

120597119879119895

119894(119894 = 0 1 119895 = 1 2) Average equation

(9) with respect to 1198790and subtract the averaged equation

from (9) an approximate expression for 120601 is obtained byconsidering only the dominant terms of order 120576minus1 as

1198632

0120593 = 119864

22119903119891119908 cos (119879

0) (10)

The stationary solution to first order for 120601 is

120593 = minus11986422119903119891119908 cos (119879

0) (11)

Substitute (11) into (9) and apply 1198790-averaging Retain domi-

nant terms of order 1205760 to obtain

1198632

1119908 + 2120577119863

1119908 + [

[

1 +

(11986422119903119891)2

2

]

]

119908 + 1198871199083

= [minus11986411+ 119864121199032

119904]119908 cos (119903

1199041198791) + ℎ0120585 (1198791)

(12)

Equation (12) governs only the slow motion 119908 of system(5) Note that the fast excitation affects the slow behavior ofsystem (12) by adding (119864

22119903119891)2

2 to the linear stiffness Bynumerical simulations the probability densities 119901(119886) of theamplitude of the original system (5) and of the slow system(12) are plotted in Figure 2 It is observed that the larger thefast excitation parameter 119864

22sdot 119903119891 the bigger the difference

between the two amplitudes

4 Effect of Fast Parametric Excitation

In the last section an approximate equation governing onlythe slow motion of system (5) is obtained by perturbation Inthe following we will discuss the effect of the fast harmonicexcitation on this slow system in greater detail

41 Effect on Resonant Frequency Let 1205962 = [1 + (11986422119903119891)2

2]In order to study the effect of the fast parametric excitation onthe resonant frequency of system (12) we will first considerthe free response of system (12) governed by

1198632

1119908 + 120596

2

119908 + 1198871199083

= 0 (13)

0 100

05

10

15

20

25

30

p(a)

a

12

Figure 2 Probability densities of amplitude of the original system(5) and of the slow system (12) Line 1 no fast excitation line 2 119864

22sdot

119903119891= 08 Solid lines for the original system (5) and dotted lines for

the slow system (12)

The periodic solution of system (13) has the form [17]

119908 (1198791) = 119886 cos 120579 (119879

1) 119863

1119908 (1198791) = minus119886] (119886 120579) sin 120579 (119879

1)

120579 (1198791) = 120601 (119879

1) + 120574

(14)

where 119886 is the amplitude 120574 is the phase angle and

] (119886 120579) =119889120601

1198891198791

= radic(1205962 +31198871198862

4) (1 + 120578 cos (2120579))

120578 =

(1198871198862

4)

(1205962 + 311988711988624)

(15)

The instantaneous frequency ](119886 120579) can be approximated bythe finite sum

] (119886 120579) = 1198870(119886) + 119887

2(119886) cos 2120579 + 119887

4(119886) cos 4120579 + 119887

6(119886) cos 6120579

(16)

where

1198870(119886) = radic(1205962 +

31198871198862

4)(1 minus

1205782

16)

1198872(119886) = radic(1205962 +

31198871198862

4)(

120578

2+31205783

64)

1198874(119886) = radic(1205962 +

31198871198862

4)(minus

1205782

16)

1198876(119886) = radic(1205962 +

31198871198862

4)(

1205783

64)

(17)


0 05 1 15 209

1

11

12

13

14

15

16

17

18

a

3

21

120596

Figure 3 Average resonant frequency for different values of the fastexcitation parameter119864

22sdot 119903119891 Line 1 no fast excitation line 2119864

22sdot119903119891

= 03 line 3 11986422sdot 119903119891= 09

By integrating (16) with respect to 120579 from 0 to 2120587 an averagefrequency

120596 (119886) =1

2120587int

2120587

0

] (119886 120579) 119889120579 = 1198870(119886) (18)

of the oscillator is obtained The approximate relation

120579 (1198791) asymp 120596 (119886) 119879

1+ 120574 (19)

will be used in the averaging process that follows Note thatthe resonant frequency ](119886 120579) depends on both the amplitude119886 and phase 120574 In Figure 3 the average resonant frequency120596(119886) is plotted against the amplitude 119886 for different values of11986422sdot 119903119891 As the fast excitation parameter 119864

22sdot 119903119891increases the

average resonant frequency of the system also increases

42 Effect onResonant Response Wenowproceed to examinethe effect of fast parametric excitation on the resonantresponse of system (12) It is reasonable to assume that neitherlight damping nor a weak random excitation will destabilizesystem (12) In this case the response of system (12) can beregarded as a random spread of the periodic solutions ofsystem (13) As a consequence

119908 (1198791) = 119860 cosΘ(119879

1)

1198631119908 (1198791) = minus119860119881 (119860Θ) sinΘ(119879

1)

Θ (1198791) = Φ (119879

1) + Γ (119879

1)

(20)

where 119860 Θ Φ Γ and 119881 are all random processes Theinstantaneous and average frequencies of system (12) are ofthe same forms given by (15) and (18)

Substitute (20) into (12) and treat (20) as generalized vander Pol transformation from 119908 119863

1119908 to 119860 Γ the following

equations for 119860 and Γ can be obtained

119889119860

1198891198791

= 1198651(119860Θ 119903

1199041198791) + 1198661(119860 Θ) 120585 (119879

1)

119889Γ

1198891198791

= 1198652(119860Θ 119903

1199041198791) + 1198662(119860 Θ) 120585 (119879

1)

(21)

where

1198651(119860Θ 119903

1199041198791)

= minus1

1205962 + 1198871198602(2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904]119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) sinΘ

1198652(119860Θ 119903

1199041198791)

= minus1

1205962119860 + 1198871198603(2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) cosΘ

1198661(119860 Θ) = minus

ℎ0

1205962 + 1198871198602119881 (119860Θ) sinΘ

1198662(119860Θ) = minus

ℎ0

1205962119860 + 1198871198603119881 (119860Θ) cosΘ

(22)

and ℎ0has been specified in (6)

421 The Case with ℎ0= 0 Firstly we consider the pure

parametric harmonic excitation case Neglect the diffusionterms and (21) can be rewritten as

119889119860

1198891198791

= minus1

1205962 + 1198871198602

times (2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) sinΘ

119889Γ

1198891198791

= minus1

1205962119860 + 1198871198603

times (2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) cosΘ

(23)

The nonlinear system (23) is subjected to harmonic para-metric excitations and there is the possibility of parametric


resonance Since large response of the pantograph-catenarysystem may cause malfunctions in power collection we willemphasize the primary parametric resonance case Assumethat in primary parametric resonance there exists

119903119904

120596 (119860)= 2 + 120576120578 (24)

where 120596(119860) is the average frequency of system (12) and 120576120578 ≪1 is the small detuning parameter Multiply (24) by 119879

1and

utilize the approximate relation (19) to obtain

1199031199041198791= 2Θ + 120576120578120596119879

1minus 2Γ (25)

Introduce a new variable Δ = 1205761205781205961198791minus 2Γ so that (25) can

be rewritten as

1199031199041198791= 2Θ + Δ (26)

Substitute (26) into (23) and average (23) with respect tothe rapidly varying process Θ from 0 to 2120587 to generate thefollowing averaged differential equations

119889119860

1198891198791

= (minus119860 [5121205771205962

+ 3201205771198871198602

minus 64 (21198870minus 1198874) (minus11986411+ 119864121199032

119904) sinΔ])

times (512 (1205962

+ 1198871198602

))minus1

119889Δ

1198891198791

= 119903119904minus 21198870+

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032

119904) cosΔ

4 (1205962 + 1198871198602)

(27)

Equation (27) involves only slowly varying processes 119860 andΔ By letting 119889119860119889119879

1= 119889Δ119889119879

1= 0 (27) gives the frequency

response relation

[512120577120596

2

119860 + 3201205771198871198603

64119860 (21198870minus 1198874) (minus11986411+ 119864121199032119904)]

2

+ [

4 (119903119904minus 21198870) (1205962

+ 1198871198602

)

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032119904)]

2

= 1

(28)

Numerical results are obtained for 120577 = 01 119887 = 05 11986411= 12

11986412

= 01 and 11986421

= 05 and shown in Figures 4 and5 Figure 4 displays the frequency response under primaryparametric resonance for different values of the fast excitationparameter 119864

22sdot 119903119891 It is observed that fast parametric

excitation shifts the resonant peaks to the right which meansthat a higher frequency of the slow excitation is requiredto produce the resonant response The dependence of theamplitude Δ119886 at 119903

119904= 20 and the width of the resonant region

ΔΩ as a function of the fast excitation parameter 11986422sdot 119903119891

is shown in Figure 5 The amplitude and the width of theresonant region are reduced appreciably by fast parametricexcitation

12 14 16 18 20 22 24 26 2800

02

04

06

08

10

12

14

16

18

a

rs

1

2

3

Figure 4 Frequency response for primary parametric resonanceDashed lines are unstable Line 1 no fast excitation line 2 119864

22sdot 119903119891=

03 line 3 11986422sdot 119903119891= 06

minus01 00 01 02 03 04 05 06 07 08

04

05

06

07

08

09

10

Δa

ΔΩ

ΔaΔ

Ω

Δa

ΔΩ

E22 middot rf

Figure 5The amplitudeΔ119886 at 119903119904= 20 and the width of the resonant

region ΔΩ as functions of the fast excitation parameter 11986422sdot 119903119891

422 The Case with ℎ0

= 0 In practical railway engineeringrandom disturbances are always present and the term 120585(119879

1)

cannot be neglected Suppose that the stochastic excitation120585(1198791) is a weak Gaussian white noise with intensity 2119863 Then

(21) can be modeled as Stratonovich stochastic differentialequations and transformed into the following Ito equationsby adding Wong-Zakai correction terms [18]

119889119860 = 1198981(119860Θ 119903

1199041198791) 1198891198791+ 1205901(119860 Θ) 119889119861 (119879

1)

119889Γ = 1198982(119860Θ 119903

1199041198791) 1198891198791+ 1205902(119860 Θ) 119889119861 (119879

1)

(29)

where 119889119861(1198791) is the unit Wiener process and

1198981(119860Θ 119903

1199041198791) = 1198651(119860Θ 119903

1199041198791) + 119863(119866

1

1205971198661

120597119860+ 1198662

1205971198661

120597Θ)

1198982(119860Θ 119903

1199041198791) = 1198652(119860Θ 119903

1199041198791) + 119863(119866

1

1205971198662

120597119860+ 1198662

1205971198662

120597Θ)


1198871(119860 Θ) = 120590

2

1(119860Θ) = 2119863119866

2

1

1198872(119860 Θ) = 120590

2

2(119860Θ) = 2119863119866

2

2

(30)

and 119865119894and 119866

119894(119894 = 1 2) are given in (22) In primary

parametric resonance utilize (26) and average the rapidlyvarying process Θ from 0 to 2120587 to generate the averaged Itostochastic differential equations

119889119860 = 1198981(119860 Δ) 119889119879

1+ 1205901(119860) 119889119861 (119879

1)

119889Δ = 1198982(119860 Δ) 119889119879

1+ 1205902(119860) 119889119861 (119879

1)

(31)

where the averaged drift and diffusion coefficients are

1198981(119860 Δ) = (minus119860 [512120577120596

2

+ 3201205771198871198602

minus64 (21198870minus 1198874) (minus11986411+ 119864121199032

119904) sinΔ])

times (512 (1205962

+ 1198871198602

))minus1

+

ℎ2

0119863(4119887

2

1198602

+ 121198871205962

1198602

+ 321205964

)

64119860(1205962 + 1198871198602)3

1198982(119860 Δ) = 119903

119904minus 21198870

+

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032

119904) cosΔ

4 (1205962 + 1198871198602)

1198871(119860) = 120590

2

1(119860) =

ℎ2

0119863(16120596

2

+ 101198871198602

)

16(1205962 + 1198871198602)2

1198872(119860) = 120590

2

2(119860) =

ℎ2

0119863(16120596

2

+ 141198871198602

)

161198602(1205962 + 1198871198602)2

(32)

The Fokker-Planck-Kolmogorov (FPK) equation associ-ated with the Ito equations (31) is

120597119901

1205971198791

= minus120597

120597119886(1198981119901) minus

120597

120597Δ(1198982119901)

+1

2

1205972

1205971198862(1198871119901) +

1

2

1205972

120597Δ2(1198872119901)

(33)

where 119901 = 119901(119886 Δ 1198791) is the probability density of amplitude

119886 and phase Δ The initial condition for (33) is

119901 = 120575 (119886 minus 1198860) 120575 (Δ minus Δ

0) 119879

1= 0 (34)

and the boundary conditions for (33) are

119901 = finite 119886 = 0

119901120597119901

120597119886997888rarr 0 119886 997888rarr infin

119901 (119886 Δ + 2119899120587 1198791| 1198860 Δ0 11987910) = 119901 (119886 Δ 119879

1| 1198860 Δ0 11987910)

(35)

00 02 04 06 08 10 12 14 16 18 200

1

212

3

p(a)

a

Figure 6 Stationary probability density 119901(119886) for different values ofthe fast excitation parameter 119864

22sdot 119903119891 Line 1 no fast excitation line

2 11986422sdot 119903119891= 04 line 3 119864

22sdot 119903119891= 08 Dotted lines are from direct

numerical simulations of the nonlinear model (5)

00 01 02 03 04 05 06 07 08

04

06

08

E[a]

E[a]120590lowast10

120590lowast10

E22 middot rf

Figure 7 Mean and variance of the amplitude of the slow system(12)

The nonlinear three-dimensional parabolic problem as givenin (33)ndash(35) does not admit an easy solution analyticallyor numerically Fortunately in practical applications we aremore interested in the stationary solution of the FPK equation(33) In this case (33) can be simplified by letting 120597119901120597120591 =

0 Then the joint stationary probability density 119901(119886 Δ) isobtained readily by using the finite difference method Thestationary probability density of the amplitude 119901(119886) can beobtained from 119901(119886 Δ) by

119901 (119886) = int

2120587

0

119901 (119886 Δ) 119889Δ (36)

Numerical results for 119901(119886) and 119901(119886 Δ) of system (12) inparametric resonance are obtained for 120577 = 01 119887 = 0511986411

= 12 11986412

= 01 11986421

= 05 119903119904= 20 119863 = 10

ℎ0= 01 and shown in Figures 6ndash8 Figure 6 shows the

stationary probability density 119901(119886) for different values of thefast excitation parameter 119864

22sdot 119903119891 Fast parametric excitation


0 04 08 12 16 2

01234560051

152

253

35

0 4 0 8 12 1612345

120575 a

p(a120575)

(a)

005

1 15

01234560020406081

1214

0 51

12345120575

a

p(a120575)

(b)

Figure 8 Joint stationary probability density 119901(119886 Δ) (a) No fast excitation (b) 11986422sdot 119903119891= 08

shifts the probability density curve to the left changing boththe peak height and shape Even when the fast excitationis small the response of the slow system (12) may changedramatically This observation is reinforced in Figure 7 inwhich themean and variance of the amplitude119886of system (12)change significantly upon adding fast parametric excitationThis reflects the increased stiffness of the slow system (12)under fast excitation Finally direct numerical simulations ofthe nonlinear model (5) are performed to generate 119901(119886) Asshown in Figure 6 data from direct numerical simulationsclosely match those generated by (36) thus validating theanalysis presentedThe joint probability density119901(119886 Δ) of theslow system (12) is plotted in Figure 8

5 Conclusions

In the present paper the effect of fast parametric excitationon a stochastically excited pantograph-catenary system hasbeen investigated A nonlinear model of the pantograph-catenary system has been adopted wherein the stiffness ofthe nonlinear spring has a time-varying component char-acterized by both low and high frequencies The overallparametrically inducedmotion of the system is separated intotwo parts a dominant low-frequency vibration which is themain motion and a small high-frequency vibration whichaffects the low-frequency motion by altering the stiffnessUsing perturbation an approximate equation governing onlythe low-frequency motion has been derived An averagingmethod for harmonic functions has been applied to obtainthe primary resonant response of the low-frequency motion

Analytical results show that the effect of fast parametricexcitation is not negligible The addition of even a smallamount of high-frequency parametric excitation may dra-matically increase the resonant frequency and change theprimary resonant response of a system From a theoreticalviewpoint an investigation of a Duffing oscillator subjectedto both stochastic and parametric forces has been conductedto study the surprising effect of high-frequency input Practi-cally speaking many structures outside railway engineeringcan be modeled as a stochastically driven nonlinear systemexcited by both slow and fast parametric excitations Hencethe results of this investigation could be useful in otherapplications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial sup-blackport provided by the Natural Science Foundation ofChina (nos 10932009 and 11372271) 973 Program (no2011CB711105) National Key Technology Support Program(no 2009BAG12A01) and Natural Science Foundation ofZhejiang Province (no LY12A02004) Opinions findingsand conclusions expressed in this paper are those of theauthors and do not necessarily reflect the views of thesponsors

References

[1] G Galeotti and P Toni ldquoNonlinear modeling of a railwaypantograph for high speed runningrdquo Transactions on Modellingand Simulation vol 5 pp 421ndash436 1993

[2] P H Poznic J Jerrelind and L Drugge ldquoExperimentalevaluation of nonlinear dynamics and coupled motions in apantographrdquo in Proceedings of the ASME International DesignEngineering Technical Conferences and Computers and Informa-tion in Engineering Conference (DETC rsquo09) pp 619ndash626 SanDiego Calif USA September 2009

[3] L Drugge T Larsson A Berghuvud and A Stensson ldquoThenonlinear behavior of a pantograph current collector suspen-sionrdquo in Proceedings of the ASME Design Engineering TechnicalConferences pp 1ndash7 Las Vegas Nev USA 1999

[4] S Levy J A Bain and E J Leclerc ldquoRailway overhead contactsystems catenary-pantograph dynamics for power collection athigh speedsrdquo Journal of Engineering for Industry vol 90 no 4pp 692ndash699 1968

[5] J-W Kim H-C Chae B-S Park S-Y Lee C-S Han and J-H Jang ldquoState sensitivity analysis of the pantograph system for ahigh-speed rail vehicle considering span length and static upliftforcerdquo Journal of Sound andVibration vol 303 no 3ndash5 pp 405ndash427 2007

[6] G Poetsch J Evans R Meisinger et al ldquoPantographcatenarydynamics and controlrdquo Vehicle System Dynamics vol 28 no 2-3 pp 159ndash195 1997


[7] T X Wu and M J Brennan ldquoBasic analytical study ofpantograph-catenary systemdynamicsrdquoVehicle SystemDynam-ics vol 30 no 6 pp 443ndash456 1998

[8] T X Wu and M J Brennan ldquoDynamic stiffness of a railwayoverhead wire system and its effect on pantograph-catenarysystem dynamicsrdquo Journal of Sound and Vibration vol 219 no3 pp 483ndash502 1999

[9] D Tcherniak and J J Thomsen ldquoSlow effects of fast harmonicexcitation for elastic structuresrdquoNonlinearDynamics vol 17 no3 pp 227ndash246 1998

[10] R Bourkha and M Belhaq ldquoEffect of fast harmonic excitationon a self-excited motion in van der Pol oscillatorrdquo ChaosSolitons amp Fractals vol 34 no 2 pp 621ndash627 2007

[11] J J Thomsen ldquoSome general effects of strong high-frequencyexcitation stiffening biasing and smootheningrdquo Journal ofSound and Vibration vol 253 no 4 pp 807ndash831 2002

[12] J JThomsen ldquoUsing fast vibrations to quench friction-inducedoscillationsrdquo Journal of Sound and Vibration vol 228 no 5 pp1079ndash1102 1999

[13] J S Jensen ldquoNon-linear dynamics of the follower-loadeddoublependulumwith added support-excitationrdquo Journal of Sound andVibration vol 215 no 1 pp 125ndash142 1998

[14] J S Jensen ldquoQuasi-static equilibria of a buckled beam withadded high-frequency excitationrdquo DCAMM Report TechnicalUniversity of Denmark Copenhagen Denmark 1998

[15] J S Jensen ldquoPipes conveying fluid pulsating with high fre-quencyrdquo DCAMM Report no 563 Technical University ofDenmark Copenhagen Denmark 1998

[16] E P Popov and I P Paltov Approximate Methods for AnalyzingNonlinear Automatic Systems Fizmatgiz Moscow Russia 1960

[17] Z Xu and Y K Cheung ldquoAveraging method using generalizedharmonic functions for strongly non-linear oscillatorsrdquo Journalof Sound and Vibration vol 174 no 4 pp 563ndash576 1994

[18] Z L Huang and W Q Zhu ldquoAveraging method for quasi-integrable Hamiltonian systemsrdquo Journal of Sound and Vibra-tion vol 284 no 1-2 pp 325ndash341 2005

[19] Z L Huang W Q Zhu and Y Suzuki ldquoStochastic averaging ofstrongly non-linear oscillators under combined harmonic andwhite-noise excitationsrdquo Journal of Sound and Vibration vol238 no 2 pp 233ndash256 2000

[20] G O Cai and Y K Lin ldquoNonlinearly damped systems undersimultaneous broad-band and harmonic excitationsrdquoNonlinearDynamics vol 6 no 2 pp 163ndash177 1994

[21] R Haiwu X Wei M Guang and F Tong ldquoResponse of aDuffing oscillator to combined deterministic harmonic andrandom excitationrdquo Journal of Sound and Vibration vol 242no 2 pp 362ndash368 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



K(Ωs Ωf t)

m

c

Figure 1 Dynamic model of pantograph-catenary system

effect of fast parametric excitation on the resonant frequencyand the primary resonant response are studied Finally directnumerical simulations of the nonlinear model are performedto validate the analysis presented

2 Model of Pantograph-Catenary System

A commonly used model in railway engineering of thepantograph-catenary system [3 7 8] is the single-degree-of-freedom model shown in Figure 1 Since the stiffnessvariation is repeated in each span of the catenary the catenarystiffness possesses periodic components If we consider thestiffness variation between the vertical droppers the catenaryis usually taken as a spring with time-varying stiffnesscomponents given by [1 7 8]

119870(Ω119904 Ω119891 119905) = 119870

0(119881) (1 + 120572 cos (Ω

119904119905) + 120573 cos (Ω

119891119905))

(1)

where119881 represents the speed of train and1198700(119881) is the speed-

dependent average stiffness of the catenary In the aboveequation 120572 120573 are unspecified coefficients and 120572 cos(Ω

119904119905)

120573 cos(Ω119891119905) account for the stiffness fluctuations between the

support poles and the vertical droppers respectively Let 119871119904

and 119871119889denote respectively the span distance and dropper

distance Then the frequencies of stiffness fluctuation can beexpressed as

Ω119904=2120587119881

119871119904

Ω119891=2120587119881

119871119889

(2)

In general 119871119904≫ 119871119889and therefore Ω

119891≫ Ω

119904 In many

cases the average stiffness 1198700(119881) can be approximated by a

quadratic function of the train speed [8] such that

1198700(119881) = 119870

119904minus 1198701198891198812

(3)

where 119870119904is the static average stiffness of the catenary and

119870119889is a coefficient accounting for dynamic interactions of the

coupled pantograph-catenary system The value of 119870119904can be

calculated by the finite element methodDue to random disturbances (such as train body move-

ment contact wire irregularity and ambient air flow) a weak

random excitation 119888119908120585(119905) is introduced into the equation

of the pantograph-catenary system Incorporating stiffnessnonlinearity the equation of motion of the pantograph-catenary system can be written as

119898 119910 + 119888 119910 + 1198961015840

119910 + 1199103

+ (119870119904minus 1198701198891198812

)

times (1 + 120572 cos (Ω119904119905) + 120573 cos (Ω

119891119905)) 119910 = 119888

119908120585 (119905)

(4)

where119910 is the vertical displacement of the pantograph systemand the fourth item in the left of (4) represents the interactionof pantograph and catenary system

Let1205962119899= 119896119898 and 120591 = 120596

119899119905where 119896 = 119896

1015840

+119870119904minus1198701198891198812 and 120591

is a nondimensional time Equation (4) can be converted intothe nondimensional form

11991010158401015840

+ 21205771199101015840

+ 119910 + 1198871199103

= [minus11986411+ 119864121199032

119904] cos (119903

119904120591) 119910

+ [minus11986421+ 119864221199032

119891] cos (119903

119891120591) 119910 + ℎ

0120585 (120591)

(5)

where 1199101015840 = 119889119910119889120591 11991010158401015840

= 1198892

1199101198891205912 and

2120577 =119888

119898120596119899

119887 =

1198981205962119899

11986411=120572119870119904

1198981205962119899

11986412=120572119870119889

119898(119871119904

2120587)

2

11986421=120573119870119904

1198981205962119899

11986422=120573119870119889

119898(119871119889

2120587)

2

ℎ0=

119888119908

1198981205962119899

119903119904=Ω119904

120596119899

=2120587119881

120596119899119871119904

119903119891=

Ω119891

120596119899

=2120587119881

120596119899119871119889

(6)

Here 119903119904

= 119874(1) is the nondimensional low-frequencyexcitation and 119903

119891≫ 1 is the nondimensional high-frequency

excitation If a train is travelling at a constant speed 119881 then2120577 119864

11 119864121199032

119904 11986421 119864221199032

119891 and ℎ

0are all small constants

Suppose these constants are of the same order in a smallparameter 120576 System (5) is a randomly excited Duffingoscillator with both slow and fast time-varying stiffness Thegoal of the present work is to investigate the influence of thefast parametric excitation on the characteristics of system (5)

3 Approximate Equation Governingthe Slow Motion

Introduce two different time-scales

1198790= 119903119891120591 = 120576minus1

120591 1198791= 120591 (7)

where the slow time 1198791and the fast time 119879

0are considered as

new independent variables in (5) Separate 119910(120591) into a slowpart 119908(119879

1) and a fast part 120576120601(119879

0 1198791) [10] so that

119910 (120591) = 119908 (1198791) + 120576120593 (119879

0 1198791) (8)



1198790= (int2120587

0





1198790

=


1198632

1119908 + (120576119863

2

1+ 211986301198631+ 120576minus1

1198632

0) 120593 + 2120577 (119863

1119908 + 119863

0120593 + 120576119863

1120593)

+ (119908 + 120576120593) + 119887(119908 + 120576120593)3

= [minus11986411+ 119864121199032

119904] (119908 + 120576120593) cos (119903

1199041198791)


11986422119903119891] (119908 + 120576120593) cos (119879

0) + ℎ0120585 (1198791)

(9)

where 119863119895119894= 120597119895

120597119879119895




1198632

0120593 = 119864

22119903119891119908 cos (119879

0) (10)


120593 = minus11986422119903119891119908 cos (119879

0) (11)



1198632

1119908 + 2120577119863

1119908 + [

[

1 +

(11986422119903119891)2

2

]

]

119908 + 1198871199083

= [minus11986411+ 119864121199032

119904]119908 cos (119903

1199041198791) + ℎ0120585 (1198791)

(12)


22119903119891)2








1198632

1119908 + 120596

2

119908 + 1198871199083

= 0 (13)

0 100

05

10

15

20

25

30

p(a)

a

12


22sdot




119908 (1198791) = 119886 cos 120579 (119879

1) 119863

1119908 (1198791) = minus119886] (119886 120579) sin 120579 (119879

1)

120579 (1198791) = 120601 (119879

1) + 120574

(14)


] (119886 120579) =119889120601

1198891198791

= radic(1205962 +31198871198862

4) (1 + 120578 cos (2120579))

120578 =

(1198871198862

4)

(1205962 + 311988711988624)

(15)


] (119886 120579) = 1198870(119886) + 119887

2(119886) cos 2120579 + 119887

4(119886) cos 4120579 + 119887

6(119886) cos 6120579

(16)

where

1198870(119886) = radic(1205962 +

31198871198862

4)(1 minus

1205782

16)

1198872(119886) = radic(1205962 +

31198871198862

4)(

120578

2+31205783

64)

1198874(119886) = radic(1205962 +

31198871198862

4)(minus

1205782

16)

1198876(119886) = radic(1205962 +

31198871198862

4)(

1205783

64)

(17)


0 05 1 15 209

1

11

12

13

14

15

16

17

18

a

3

21

120596



22sdot119903119891

= 03 line 3 11986422sdot 119903119891= 09


120596 (119886) =1

2120587int

2120587

0

] (119886 120579) 119889120579 = 1198870(119886) (18)


120579 (1198791) asymp 120596 (119886) 119879

1+ 120574 (19)





119908 (1198791) = 119860 cosΘ(119879

1)

1198631119908 (1198791) = minus119860119881 (119860Θ) sinΘ(119879

1)

Θ (1198791) = Φ (119879

1) + Γ (119879

1)

(20)





119889119860

1198891198791

= 1198651(119860Θ 119903

1199041198791) + 1198661(119860 Θ) 120585 (119879

1)

119889Γ

1198891198791

= 1198652(119860Θ 119903

1199041198791) + 1198662(119860 Θ) 120585 (119879

1)

(21)

where

1198651(119860Θ 119903

1199041198791)

= minus1

1205962 + 1198871198602(2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904]119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) sinΘ

1198652(119860Θ 119903

1199041198791)

= minus1

1205962119860 + 1198871198603(2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) cosΘ

1198661(119860 Θ) = minus

ℎ0

1205962 + 1198871198602119881 (119860Θ) sinΘ

1198662(119860Θ) = minus

ℎ0

1205962119860 + 1198871198603119881 (119860Θ) cosΘ

(22)




119889119860

1198891198791

= minus1

1205962 + 1198871198602

times (2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) sinΘ

119889Γ

1198891198791

= minus1

1205962119860 + 1198871198603

times (2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) cosΘ

(23)




119903119904

120596 (119860)= 2 + 120576120578 (24)


1and


1199031199041198791= 2Θ + 120576120578120596119879

1minus 2Γ (25)


be rewritten as

1199031199041198791= 2Θ + Δ (26)


119889119860

1198891198791

= (minus119860 [5121205771205962

+ 3201205771198871198602


119904) sinΔ])

times (512 (1205962

+ 1198871198602

))minus1

119889Δ

1198891198791

= 119903119904minus 21198870+

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032

119904) cosΔ

4 (1205962 + 1198871198602)

(27)


1= 119889Δ119889119879


response relation

[512120577120596

2

119860 + 3201205771198871198603

64119860 (21198870minus 1198874) (minus11986411+ 119864121199032119904)]

2

+ [

4 (119903119904minus 21198870) (1205962

+ 1198871198602

)

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032119904)]

2

= 1

(28)


11986412

= 01 and 11986421







12 14 16 18 20 22 24 26 2800

02

04

06

08

10

12

14

16

18

a

rs

1

2

3


22sdot 119903119891=

03 line 3 11986422sdot 119903119891= 06

minus01 00 01 02 03 04 05 06 07 08

04

05

06

07

08

09

10

Δa

ΔΩ

ΔaΔ

Ω

Δa

ΔΩ

E22 middot rf





1)



119889119860 = 1198981(119860Θ 119903

1199041198791) 1198891198791+ 1205901(119860 Θ) 119889119861 (119879

1)

119889Γ = 1198982(119860Θ 119903

1199041198791) 1198891198791+ 1205902(119860 Θ) 119889119861 (119879

1)

(29)


1198981(119860Θ 119903

1199041198791) = 1198651(119860Θ 119903

1199041198791) + 119863(119866

1

1205971198661

120597119860+ 1198662

1205971198661

120597Θ)

1198982(119860Θ 119903

1199041198791) = 1198652(119860Θ 119903

1199041198791) + 119863(119866

1

1205971198662

120597119860+ 1198662

1205971198662

120597Θ)


1198871(119860 Θ) = 120590

2

1(119860Θ) = 2119863119866

2

1

1198872(119860 Θ) = 120590

2

2(119860Θ) = 2119863119866

2

2

(30)

and 119865119894and 119866



119889119860 = 1198981(119860 Δ) 119889119879

1+ 1205901(119860) 119889119861 (119879

1)

119889Δ = 1198982(119860 Δ) 119889119879

1+ 1205902(119860) 119889119861 (119879

1)

(31)


1198981(119860 Δ) = (minus119860 [512120577120596

2

+ 3201205771198871198602


119904) sinΔ])

times (512 (1205962

+ 1198871198602

))minus1

+

ℎ2

0119863(4119887

2

1198602

+ 121198871205962

1198602

+ 321205964

)

64119860(1205962 + 1198871198602)3

1198982(119860 Δ) = 119903

119904minus 21198870

+

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032

119904) cosΔ

4 (1205962 + 1198871198602)

1198871(119860) = 120590

2

1(119860) =

ℎ2

0119863(16120596

2

+ 101198871198602

)

16(1205962 + 1198871198602)2

1198872(119860) = 120590

2

2(119860) =

ℎ2

0119863(16120596

2

+ 141198871198602

)

161198602(1205962 + 1198871198602)2

(32)


120597119901

1205971198791

= minus120597

120597119886(1198981119901) minus

120597

120597Δ(1198982119901)

+1

2

1205972

1205971198862(1198871119901) +

1

2

1205972

120597Δ2(1198872119901)

(33)



119901 = 120575 (119886 minus 1198860) 120575 (Δ minus Δ

0) 119879

1= 0 (34)


119901 = finite 119886 = 0

119901120597119901

120597119886997888rarr 0 119886 997888rarr infin

119901 (119886 Δ + 2119899120587 1198791| 1198860 Δ0 11987910) = 119901 (119886 Δ 119879

1| 1198860 Δ0 11987910)

(35)

00 02 04 06 08 10 12 14 16 18 200

1

212

3

p(a)

a



2 11986422sdot 119903119891= 04 line 3 119864



00 01 02 03 04 05 06 07 08

04

06

08

E[a]

E[a]120590lowast10

120590lowast10

E22 middot rf




119901 (119886) = int

2120587

0

119901 (119886 Δ) 119889Δ (36)


= 12 11986412

= 01 11986421

= 05 119903119904= 20 119863 = 10





0 04 08 12 16 2

01234560051

152

253

35

0 4 0 8 12 1612345

120575 a

p(a120575)

(a)

005

1 15

01234560020406081

1214

0 51

12345120575

a

p(a120575)

(b)



5 Conclusions





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198790= (int2120587

0





1198790

=


1198632

1119908 + (120576119863

2

1+ 211986301198631+ 120576minus1

1198632

0) 120593 + 2120577 (119863

1119908 + 119863

0120593 + 120576119863

1120593)

+ (119908 + 120576120593) + 119887(119908 + 120576120593)3

= [minus11986411+ 119864121199032

119904] (119908 + 120576120593) cos (119903

1199041198791)


11986422119903119891] (119908 + 120576120593) cos (119879

0) + ℎ0120585 (1198791)

(9)

where 119863119895119894= 120597119895

120597119879119895




1198632

0120593 = 119864

22119903119891119908 cos (119879

0) (10)


120593 = minus11986422119903119891119908 cos (119879

0) (11)



1198632

1119908 + 2120577119863

1119908 + [

[

1 +

(11986422119903119891)2

2

]

]

119908 + 1198871199083

= [minus11986411+ 119864121199032

119904]119908 cos (119903

1199041198791) + ℎ0120585 (1198791)

(12)


22119903119891)2








1198632

1119908 + 120596

2

119908 + 1198871199083

= 0 (13)

0 100

05

10

15

20

25

30

p(a)

a

12


22sdot




119908 (1198791) = 119886 cos 120579 (119879

1) 119863

1119908 (1198791) = minus119886] (119886 120579) sin 120579 (119879

1)

120579 (1198791) = 120601 (119879

1) + 120574

(14)


] (119886 120579) =119889120601

1198891198791

= radic(1205962 +31198871198862

4) (1 + 120578 cos (2120579))

120578 =

(1198871198862

4)

(1205962 + 311988711988624)

(15)


] (119886 120579) = 1198870(119886) + 119887

2(119886) cos 2120579 + 119887

4(119886) cos 4120579 + 119887

6(119886) cos 6120579

(16)

where

1198870(119886) = radic(1205962 +

31198871198862

4)(1 minus

1205782

16)

1198872(119886) = radic(1205962 +

31198871198862

4)(

120578

2+31205783

64)

1198874(119886) = radic(1205962 +

31198871198862

4)(minus

1205782

16)

1198876(119886) = radic(1205962 +

31198871198862

4)(

1205783

64)

(17)


0 05 1 15 209

1

11

12

13

14

15

16

17

18

a

3

21

120596



22sdot119903119891

= 03 line 3 11986422sdot 119903119891= 09


120596 (119886) =1

2120587int

2120587

0

] (119886 120579) 119889120579 = 1198870(119886) (18)


120579 (1198791) asymp 120596 (119886) 119879

1+ 120574 (19)





119908 (1198791) = 119860 cosΘ(119879

1)

1198631119908 (1198791) = minus119860119881 (119860Θ) sinΘ(119879

1)

Θ (1198791) = Φ (119879

1) + Γ (119879

1)

(20)





119889119860

1198891198791

= 1198651(119860Θ 119903

1199041198791) + 1198661(119860 Θ) 120585 (119879

1)

119889Γ

1198891198791

= 1198652(119860Θ 119903

1199041198791) + 1198662(119860 Θ) 120585 (119879

1)

(21)

where

1198651(119860Θ 119903

1199041198791)

= minus1

1205962 + 1198871198602(2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904]119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) sinΘ

1198652(119860Θ 119903

1199041198791)

= minus1

1205962119860 + 1198871198603(2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) cosΘ

1198661(119860 Θ) = minus

ℎ0

1205962 + 1198871198602119881 (119860Θ) sinΘ

1198662(119860Θ) = minus

ℎ0

1205962119860 + 1198871198603119881 (119860Θ) cosΘ

(22)




119889119860

1198891198791

= minus1

1205962 + 1198871198602

times (2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) sinΘ

119889Γ

1198891198791

= minus1

1205962119860 + 1198871198603

times (2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) cosΘ

(23)




119903119904

120596 (119860)= 2 + 120576120578 (24)


1and


1199031199041198791= 2Θ + 120576120578120596119879

1minus 2Γ (25)


be rewritten as

1199031199041198791= 2Θ + Δ (26)


119889119860

1198891198791

= (minus119860 [5121205771205962

+ 3201205771198871198602


119904) sinΔ])

times (512 (1205962

+ 1198871198602

))minus1

119889Δ

1198891198791

= 119903119904minus 21198870+

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032

119904) cosΔ

4 (1205962 + 1198871198602)

(27)


1= 119889Δ119889119879


response relation

[512120577120596

2

119860 + 3201205771198871198603

64119860 (21198870minus 1198874) (minus11986411+ 119864121199032119904)]

2

+ [

4 (119903119904minus 21198870) (1205962

+ 1198871198602

)

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032119904)]

2

= 1

(28)


11986412

= 01 and 11986421







12 14 16 18 20 22 24 26 2800

02

04

06

08

10

12

14

16

18

a

rs

1

2

3


22sdot 119903119891=

03 line 3 11986422sdot 119903119891= 06

minus01 00 01 02 03 04 05 06 07 08

04

05

06

07

08

09

10

Δa

ΔΩ

ΔaΔ

Ω

Δa

ΔΩ

E22 middot rf





1)



119889119860 = 1198981(119860Θ 119903

1199041198791) 1198891198791+ 1205901(119860 Θ) 119889119861 (119879

1)

119889Γ = 1198982(119860Θ 119903

1199041198791) 1198891198791+ 1205902(119860 Θ) 119889119861 (119879

1)

(29)


1198981(119860Θ 119903

1199041198791) = 1198651(119860Θ 119903

1199041198791) + 119863(119866

1

1205971198661

120597119860+ 1198662

1205971198661

120597Θ)

1198982(119860Θ 119903

1199041198791) = 1198652(119860Θ 119903

1199041198791) + 119863(119866

1

1205971198662

120597119860+ 1198662

1205971198662

120597Θ)


1198871(119860 Θ) = 120590

2

1(119860Θ) = 2119863119866

2

1

1198872(119860 Θ) = 120590

2

2(119860Θ) = 2119863119866

2

2

(30)

and 119865119894and 119866



119889119860 = 1198981(119860 Δ) 119889119879

1+ 1205901(119860) 119889119861 (119879

1)

119889Δ = 1198982(119860 Δ) 119889119879

1+ 1205902(119860) 119889119861 (119879

1)

(31)


1198981(119860 Δ) = (minus119860 [512120577120596

2

+ 3201205771198871198602


119904) sinΔ])

times (512 (1205962

+ 1198871198602

))minus1

+

ℎ2

0119863(4119887

2

1198602

+ 121198871205962

1198602

+ 321205964

)

64119860(1205962 + 1198871198602)3

1198982(119860 Δ) = 119903

119904minus 21198870

+

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032

119904) cosΔ

4 (1205962 + 1198871198602)

1198871(119860) = 120590

2

1(119860) =

ℎ2

0119863(16120596

2

+ 101198871198602

)

16(1205962 + 1198871198602)2

1198872(119860) = 120590

2

2(119860) =

ℎ2

0119863(16120596

2

+ 141198871198602

)

161198602(1205962 + 1198871198602)2

(32)


120597119901

1205971198791

= minus120597

120597119886(1198981119901) minus

120597

120597Δ(1198982119901)

+1

2

1205972

1205971198862(1198871119901) +

1

2

1205972

120597Δ2(1198872119901)

(33)



119901 = 120575 (119886 minus 1198860) 120575 (Δ minus Δ

0) 119879

1= 0 (34)


119901 = finite 119886 = 0

119901120597119901

120597119886997888rarr 0 119886 997888rarr infin

119901 (119886 Δ + 2119899120587 1198791| 1198860 Δ0 11987910) = 119901 (119886 Δ 119879

1| 1198860 Δ0 11987910)

(35)

00 02 04 06 08 10 12 14 16 18 200

1

212

3

p(a)

a



2 11986422sdot 119903119891= 04 line 3 119864



00 01 02 03 04 05 06 07 08

04

06

08

E[a]

E[a]120590lowast10

120590lowast10

E22 middot rf




119901 (119886) = int

2120587

0

119901 (119886 Δ) 119889Δ (36)


= 12 11986412

= 01 11986421

= 05 119903119904= 20 119863 = 10





0 04 08 12 16 2

01234560051

152

253

35

0 4 0 8 12 1612345

120575 a

p(a120575)

(a)

005

1 15

01234560020406081

1214

0 51

12345120575

a

p(a120575)

(b)



5 Conclusions





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 1 15 209

1

11

12

13

14

15

16

17

18

a

3

21

120596



22sdot119903119891

= 03 line 3 11986422sdot 119903119891= 09


120596 (119886) =1

2120587int

2120587

0

] (119886 120579) 119889120579 = 1198870(119886) (18)


120579 (1198791) asymp 120596 (119886) 119879

1+ 120574 (19)





119908 (1198791) = 119860 cosΘ(119879

1)

1198631119908 (1198791) = minus119860119881 (119860Θ) sinΘ(119879

1)

Θ (1198791) = Φ (119879

1) + Γ (119879

1)

(20)





119889119860

1198891198791

= 1198651(119860Θ 119903

1199041198791) + 1198661(119860 Θ) 120585 (119879

1)

119889Γ

1198891198791

= 1198652(119860Θ 119903

1199041198791) + 1198662(119860 Θ) 120585 (119879

1)

(21)

where

1198651(119860Θ 119903

1199041198791)

= minus1

1205962 + 1198871198602(2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904]119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) sinΘ

1198652(119860Θ 119903

1199041198791)

= minus1

1205962119860 + 1198871198603(2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) cosΘ

1198661(119860 Θ) = minus

ℎ0

1205962 + 1198871198602119881 (119860Θ) sinΘ

1198662(119860Θ) = minus

ℎ0

1205962119860 + 1198871198603119881 (119860Θ) cosΘ

(22)




119889119860

1198891198791

= minus1

1205962 + 1198871198602

times (2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) sinΘ

119889Γ

1198891198791

= minus1

1205962119860 + 1198871198603

times (2120577119860119881 (119860Θ) sinΘ

+ [minus11986411+ 119864121199032

119904] 119860 cosΘ cos (119903

1199041198791))

times 119881 (119860Θ) cosΘ

(23)




119903119904

120596 (119860)= 2 + 120576120578 (24)


1and


1199031199041198791= 2Θ + 120576120578120596119879

1minus 2Γ (25)


be rewritten as

1199031199041198791= 2Θ + Δ (26)


119889119860

1198891198791

= (minus119860 [5121205771205962

+ 3201205771198871198602


119904) sinΔ])

times (512 (1205962

+ 1198871198602

))minus1

119889Δ

1198891198791

= 119903119904minus 21198870+

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032

119904) cosΔ

4 (1205962 + 1198871198602)

(27)


1= 119889Δ119889119879


response relation

[512120577120596

2

119860 + 3201205771198871198603

64119860 (21198870minus 1198874) (minus11986411+ 119864121199032119904)]

2

+ [

4 (119903119904minus 21198870) (1205962

+ 1198871198602

)

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032119904)]

2

= 1

(28)


11986412

= 01 and 11986421







12 14 16 18 20 22 24 26 2800

02

04

06

08

10

12

14

16

18

a

rs

1

2

3


22sdot 119903119891=

03 line 3 11986422sdot 119903119891= 06

minus01 00 01 02 03 04 05 06 07 08

04

05

06

07

08

09

10

Δa

ΔΩ

ΔaΔ

Ω

Δa

ΔΩ

E22 middot rf





1)



119889119860 = 1198981(119860Θ 119903

1199041198791) 1198891198791+ 1205901(119860 Θ) 119889119861 (119879

1)

119889Γ = 1198982(119860Θ 119903

1199041198791) 1198891198791+ 1205902(119860 Θ) 119889119861 (119879

1)

(29)


1198981(119860Θ 119903

1199041198791) = 1198651(119860Θ 119903

1199041198791) + 119863(119866

1

1205971198661

120597119860+ 1198662

1205971198661

120597Θ)

1198982(119860Θ 119903

1199041198791) = 1198652(119860Θ 119903

1199041198791) + 119863(119866

1

1205971198662

120597119860+ 1198662

1205971198662

120597Θ)


1198871(119860 Θ) = 120590

2

1(119860Θ) = 2119863119866

2

1

1198872(119860 Θ) = 120590

2

2(119860Θ) = 2119863119866

2

2

(30)

and 119865119894and 119866



119889119860 = 1198981(119860 Δ) 119889119879

1+ 1205901(119860) 119889119861 (119879

1)

119889Δ = 1198982(119860 Δ) 119889119879

1+ 1205902(119860) 119889119861 (119879

1)

(31)


1198981(119860 Δ) = (minus119860 [512120577120596

2

+ 3201205771198871198602


119904) sinΔ])

times (512 (1205962

+ 1198871198602

))minus1

+

ℎ2

0119863(4119887

2

1198602

+ 121198871205962

1198602

+ 321205964

)

64119860(1205962 + 1198871198602)3

1198982(119860 Δ) = 119903

119904minus 21198870

+

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032

119904) cosΔ

4 (1205962 + 1198871198602)

1198871(119860) = 120590

2

1(119860) =

ℎ2

0119863(16120596

2

+ 101198871198602

)

16(1205962 + 1198871198602)2

1198872(119860) = 120590

2

2(119860) =

ℎ2

0119863(16120596

2

+ 141198871198602

)

161198602(1205962 + 1198871198602)2

(32)


120597119901

1205971198791

= minus120597

120597119886(1198981119901) minus

120597

120597Δ(1198982119901)

+1

2

1205972

1205971198862(1198871119901) +

1

2

1205972

120597Δ2(1198872119901)

(33)



119901 = 120575 (119886 minus 1198860) 120575 (Δ minus Δ

0) 119879

1= 0 (34)


119901 = finite 119886 = 0

119901120597119901

120597119886997888rarr 0 119886 997888rarr infin

119901 (119886 Δ + 2119899120587 1198791| 1198860 Δ0 11987910) = 119901 (119886 Δ 119879

1| 1198860 Δ0 11987910)

(35)

00 02 04 06 08 10 12 14 16 18 200

1

212

3

p(a)

a



2 11986422sdot 119903119891= 04 line 3 119864



00 01 02 03 04 05 06 07 08

04

06

08

E[a]

E[a]120590lowast10

120590lowast10

E22 middot rf




119901 (119886) = int

2120587

0

119901 (119886 Δ) 119889Δ (36)


= 12 11986412

= 01 11986421

= 05 119903119904= 20 119863 = 10





0 04 08 12 16 2

01234560051

152

253

35

0 4 0 8 12 1612345

120575 a

p(a120575)

(a)

005

1 15

01234560020406081

1214

0 51

12345120575

a

p(a120575)

(b)



5 Conclusions





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119903119904

120596 (119860)= 2 + 120576120578 (24)


1and


1199031199041198791= 2Θ + 120576120578120596119879

1minus 2Γ (25)


be rewritten as

1199031199041198791= 2Θ + Δ (26)


119889119860

1198891198791

= (minus119860 [5121205771205962

+ 3201205771198871198602


119904) sinΔ])

times (512 (1205962

+ 1198871198602

))minus1

119889Δ

1198891198791

= 119903119904minus 21198870+

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032

119904) cosΔ

4 (1205962 + 1198871198602)

(27)


1= 119889Δ119889119879


response relation

[512120577120596

2

119860 + 3201205771198871198603

64119860 (21198870minus 1198874) (minus11986411+ 119864121199032119904)]

2

+ [

4 (119903119904minus 21198870) (1205962

+ 1198871198602

)

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032119904)]

2

= 1

(28)


11986412

= 01 and 11986421







12 14 16 18 20 22 24 26 2800

02

04

06

08

10

12

14

16

18

a

rs

1

2

3


22sdot 119903119891=

03 line 3 11986422sdot 119903119891= 06

minus01 00 01 02 03 04 05 06 07 08

04

05

06

07

08

09

10

Δa

ΔΩ

ΔaΔ

Ω

Δa

ΔΩ

E22 middot rf





1)



119889119860 = 1198981(119860Θ 119903

1199041198791) 1198891198791+ 1205901(119860 Θ) 119889119861 (119879

1)

119889Γ = 1198982(119860Θ 119903

1199041198791) 1198891198791+ 1205902(119860 Θ) 119889119861 (119879

1)

(29)


1198981(119860Θ 119903

1199041198791) = 1198651(119860Θ 119903

1199041198791) + 119863(119866

1

1205971198661

120597119860+ 1198662

1205971198661

120597Θ)

1198982(119860Θ 119903

1199041198791) = 1198652(119860Θ 119903

1199041198791) + 119863(119866

1

1205971198662

120597119860+ 1198662

1205971198662

120597Θ)


1198871(119860 Θ) = 120590

2

1(119860Θ) = 2119863119866

2

1

1198872(119860 Θ) = 120590

2

2(119860Θ) = 2119863119866

2

2

(30)

and 119865119894and 119866



119889119860 = 1198981(119860 Δ) 119889119879

1+ 1205901(119860) 119889119861 (119879

1)

119889Δ = 1198982(119860 Δ) 119889119879

1+ 1205902(119860) 119889119861 (119879

1)

(31)


1198981(119860 Δ) = (minus119860 [512120577120596

2

+ 3201205771198871198602


119904) sinΔ])

times (512 (1205962

+ 1198871198602

))minus1

+

ℎ2

0119863(4119887

2

1198602

+ 121198871205962

1198602

+ 321205964

)

64119860(1205962 + 1198871198602)3

1198982(119860 Δ) = 119903

119904minus 21198870

+

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032

119904) cosΔ

4 (1205962 + 1198871198602)

1198871(119860) = 120590

2

1(119860) =

ℎ2

0119863(16120596

2

+ 101198871198602

)

16(1205962 + 1198871198602)2

1198872(119860) = 120590

2

2(119860) =

ℎ2

0119863(16120596

2

+ 141198871198602

)

161198602(1205962 + 1198871198602)2

(32)


120597119901

1205971198791

= minus120597

120597119886(1198981119901) minus

120597

120597Δ(1198982119901)

+1

2

1205972

1205971198862(1198871119901) +

1

2

1205972

120597Δ2(1198872119901)

(33)



119901 = 120575 (119886 minus 1198860) 120575 (Δ minus Δ

0) 119879

1= 0 (34)


119901 = finite 119886 = 0

119901120597119901

120597119886997888rarr 0 119886 997888rarr infin

119901 (119886 Δ + 2119899120587 1198791| 1198860 Δ0 11987910) = 119901 (119886 Δ 119879

1| 1198860 Δ0 11987910)

(35)

00 02 04 06 08 10 12 14 16 18 200

1

212

3

p(a)

a



2 11986422sdot 119903119891= 04 line 3 119864



00 01 02 03 04 05 06 07 08

04

06

08

E[a]

E[a]120590lowast10

120590lowast10

E22 middot rf




119901 (119886) = int

2120587

0

119901 (119886 Δ) 119889Δ (36)


= 12 11986412

= 01 11986421

= 05 119903119904= 20 119863 = 10





0 04 08 12 16 2

01234560051

152

253

35

0 4 0 8 12 1612345

120575 a

p(a120575)

(a)

005

1 15

01234560020406081

1214

0 51

12345120575

a

p(a120575)

(b)



5 Conclusions





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1198871(119860 Θ) = 120590

2

1(119860Θ) = 2119863119866

2

1

1198872(119860 Θ) = 120590

2

2(119860Θ) = 2119863119866

2

2

(30)

and 119865119894and 119866



119889119860 = 1198981(119860 Δ) 119889119879

1+ 1205901(119860) 119889119861 (119879

1)

119889Δ = 1198982(119860 Δ) 119889119879

1+ 1205902(119860) 119889119861 (119879

1)

(31)


1198981(119860 Δ) = (minus119860 [512120577120596

2

+ 3201205771198871198602


119904) sinΔ])

times (512 (1205962

+ 1198871198602

))minus1

+

ℎ2

0119863(4119887

2

1198602

+ 121198871205962

1198602

+ 321205964

)

64119860(1205962 + 1198871198602)3

1198982(119860 Δ) = 119903

119904minus 21198870

+

(21198870+ 21198872+ 1198874) (minus11986411+ 119864121199032

119904) cosΔ

4 (1205962 + 1198871198602)

1198871(119860) = 120590

2

1(119860) =

ℎ2

0119863(16120596

2

+ 101198871198602

)

16(1205962 + 1198871198602)2

1198872(119860) = 120590

2

2(119860) =

ℎ2

0119863(16120596

2

+ 141198871198602

)

161198602(1205962 + 1198871198602)2

(32)


120597119901

1205971198791

= minus120597

120597119886(1198981119901) minus

120597

120597Δ(1198982119901)

+1

2

1205972

1205971198862(1198871119901) +

1

2

1205972

120597Δ2(1198872119901)

(33)



119901 = 120575 (119886 minus 1198860) 120575 (Δ minus Δ

0) 119879

1= 0 (34)


119901 = finite 119886 = 0

119901120597119901

120597119886997888rarr 0 119886 997888rarr infin

119901 (119886 Δ + 2119899120587 1198791| 1198860 Δ0 11987910) = 119901 (119886 Δ 119879

1| 1198860 Δ0 11987910)

(35)

00 02 04 06 08 10 12 14 16 18 200

1

212

3

p(a)

a



2 11986422sdot 119903119891= 04 line 3 119864



00 01 02 03 04 05 06 07 08

04

06

08

E[a]

E[a]120590lowast10

120590lowast10

E22 middot rf




119901 (119886) = int

2120587

0

119901 (119886 Δ) 119889Δ (36)


= 12 11986412

= 01 11986421

= 05 119903119904= 20 119863 = 10





0 04 08 12 16 2

01234560051

152

253

35

0 4 0 8 12 1612345

120575 a

p(a120575)

(a)

005

1 15

01234560020406081

1214

0 51

12345120575

a

p(a120575)

(b)



5 Conclusions





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 04 08 12 16 2

01234560051

152

253

35

0 4 0 8 12 1612345

120575 a

p(a120575)

(a)

005

1 15

01234560020406081

1214

0 51

12345120575

a

p(a120575)

(b)



5 Conclusions





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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