DynamicAnalysisofRectangularPlateStiffenedbyAny ...downloads.hindawi.com/journals/sv/2019/2364515.pdf · analysis of the rectangular beam-reinforced plate with general boundary restraints,

Research ArticleDynamic Analysis of Rectangular Plate Stiffened by AnyNumber of Beams with Different Lengths and Orientations

Yuan Cao1 Rui Zhong 2 Dong Shao 13 Qingshan Wang 2 and Xianlei Guan2

1Naval Research Academy Beijing 100161 China2State Key Laboratory of High Performance Complex Manufacturing Central South University Changsha 410083 China3Beijing Aerospace Technology Institute China Aerospace Science amp Industry Corp Beijing 100074 China

Correspondence should be addressed to Dong Shao dongshaosatanoutlookcom

Received 4 June 2019 Revised 19 August 2019 Accepted 17 October 2019 Published 7 November 2019

Academic Editor Angelo Marcelo Tusset

Copyright copy 2019 Yuan Cao et al +is 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

+e present work is concerned with dynamic characteristics of beam-stiffened rectangular plate by an improved Fourier seriesmethod (IFSM) including mobility characteristics structural intensity and transient response +e artificial coupling springtechnology is introduced to establish the clamped or elastic connections at the interface between the plate and beams Accordingto IFSM the displacement field of the plate and the stiffening beams are expressed as a combination of the Fourier cosine seriesand its auxiliary functions+en the RayleighndashRitz method is applied to solve the unknown Fourier coefficients which determinesthe dynamic characteristics of the coupled structure +e Newmark method is adopted to obtain the transient response of thecoupled structure where the Rayleigh damping is taken into consideration +e rapid convergence of the current method isshown and good agreement between the predicted results and FEM results is also revealed On this basis the effects of the factorsrelated to the stiffening beam (including the length orientations and arrangement spacing of beams) and elastic parameters aswell as damping coefficients on the dynamic characteristics of the stiffened plate are investigated

1 Introduction

+e stiffened plate component can be regarded as a couplingstructure that is composed of a plate and several beams Inpractical engineering the connection between the plate andbeam does not only involve the classical coupling but theelastic coupling is also frequently encountered In additiondue to the complexity of the actual working conditions thestiffened plates will be often subjected to the elastic boundaryrestrictions A good understanding of the dynamics of thestiffened plate with general boundary restrains can providedirect benefits to the structural design of complex systemsHowever there are only a few literatures on the dynamicanalysis of the rectangular plate with some beams of arbi-trary lengths and orientations+e current work is to presentthe dynamic analysis of the beam-stiffened rectangular platewith general boundary restraints

In the past decades many scholars have made a lot ofefforts on free vibration characteristics of the stiffened plateMukherjee andMukhopadhyay [1] presented the free vibration

of the isoparametric stiffened plate element which could ac-commodate irregular boundaries Holopainen [2] proposed anew finite element model to consider free vibration of ec-centrically stiffened plate In this research a plate with anynumber of arbitrarily oriented stiffeners was presented Basedon the RayleighndashRitzmethod Liew et al [3] and Lee andNg [4]both investigated the free vibration of the stiffened plate witharbitrarily oriented stiffeners +e difference was that Liewstudied the stiffened plate with various shapes while Lee onlyexamined the stiffened rectangular plate A hierarchical finiteelement in conjunction with local trigonometric interpolationfunctions was employed by Barrette et al [5] to investigate thevibration performances of the stiffened plate Employing thedifferential quadrature technique free vibration of plate witheccentric stiffeners was performed by Zeng and Bert [6] Andtheir theoretical solutions were in good agreement with theexperimental results Using the finite element method Sri-vastava et al [7] explored vibration of the stiffened platesubjected to partial edge loading However only the simplysupported and clamped cases were considered in their research

HindawiShock and VibrationVolume 2019 Article ID 2364515 22 pageshttpsdoiorg10115520192364515

On the basis of the first-order shear deformable theory Penget al [8] used mesh-free Galerkin method to carry out the freevibration and stability analyses of the stiffened plate On theframework of the assumed-modes method low-frequency freevibration analysis of the thin rectangular plate with a few lightstiffeners was made by Dozio and Ricciardi [9] Based on theimproved Fourier series method Xu et al [10] examined thefree vibration of the beam-stiffened rectangular plate wherebeams of arbitrary lengths and placement angles were takeninto consideration +ree different modelling approaches werereported by Bhar et al [11] to carry out the component-wisevibration analysis of the stiffened plate and comparativestudies between different models were also given +e staticfree vibration and buckling analysis of the stiffened plate werecarried out byNguyen-+oi et al [12] using CS-FEM-DSG3 InRef [13] the CS-FEM-DSG3 method was also extended toanalyze the static and free vibration of stiffened folded plateswhere three-node triangular element was adopted Four hi-erarchical models were developed by Bhaskar and Pydah [14]to obtain the natural frequencies of the isotropic and ortho-tropic stiffened plate with simply supported boundary con-dition On the basis of the elastic approach Pydah and Bhaskar[15] established an analytical model to investigate the staticsand dynamics of the stiffened plate with simply supported caseBased on the assumed mode method Cho et al [16] in-vestigated natural frequencies and modes of various openingstiffened panels under different boundary conditions wherethe effect of the parameters of the rib on the frequency waschecked+en he and other co-authors [17 18] carried out thefree vibration analysis of the stiffened panel with lumped massand stiffness attachments including rectangular plates withoutopening and with various opening shapes Utilizing theNURBS based on isogeometric analysis approach Qin et al[19] examined the frequency characteristics of curved stiffenedplate under in-plane loading Considering a beam-stiffenedplate Yin et al [20] examined its vibration transmission byapplying the dynamic stiffness method where the transmissionmode was presented Meanwhile Damnjanovic et al [21]adopted the same approach to perform the free vibrationanalysis of the stiffened composite plate based on the high-order shear deformation theory and the results were in goodagreement with the available data and FEM results

+rough the abovementioned review of literatures thepredecessors have conducted in-depth research on the freevibration of the stiffened plate Nevertheless many scholarsare not satisfied with research in the field of free vibrationHarik and Salamoun [22] applied an analytical strip methodto examine bending deflection of the stiffened plate wherethe type of load and the number of ribs were used to de-termine the number of strips Transient response of thesimply supported panel reinforced by stiffeners was pre-dicted by Louca et al [23] using the finite element techniquewhere the effect of boundary condition on transient responsewas considered Sheikh and Mukhopadhyay [24] applied thespline finite strip method to analyze the linear and nonlineartransient vibration of the stiffened plate with arbitraryshapes in which stiffeners had arbitrary orientation andeccentricity Based on finite element analysis Srivastava et al[25] presented the dynamic instability of the stiffened plate

where both in-plane partial and concentrated loadings at theedge were examined Xu et al [26] developed the structuralintensity technique to analyze energy flow for the stiffenedplate applied in the field of marine structures Later the sameauthors [27] investigated the structural intensity of thestiffened plate by means of the finite element method In thisresearch they draw a conclusion that the energy flow of thewhole structure was closely related to the natural frequencyEmploying the harmonic compound strip method Borkovicet al [28] obtained the transient response of the beam-stiffened plate where BernoullindashEuler beam andKirchhoffndashLove thin plate were both applied Combining theMindlin plate theory and Timoshenko beam theory Choet al [29] extended the assumed mode method to predictforced vibration of the stiffened plate and rectangular platewith various boundary restrains Tian and Jie [30] carriedout the research on the dynamic response of the finite ribbedplate under point force moment excitations and he con-cluded that the input mobilities were affected by the distancebetween the excitation point and the beam +en he [31]performed the vibration response analysis and experiment ofthe stiffened plates subjected to the clamped support at edgeIn 2018 he [32] introduced a double cosine integraltransform technique to study the free vibration and forcedresponse of the stiffened plate with free boundary restrainsMeanwhile the energy flow of stiffened panels was reportedby Cho et al [33] where the structural intensity techniquewas applied In order to efficiently solve the dynamic re-sponses of the stiffened plate in wide frequency domain Jiaet al [34] proposed a Composite B-spline Wavelet ElementsMethod (CBWEM) and verified the reliability of the methodby comparison with experimental results Applying theMindlin plate and Timoshenko beam theory Liu et al [35]analyzed the dynamic power transmission characteristics ofa finite stiffened plate with various classical cases where theeffects number and geometric dimensions on the stop bandin the low-frequency domain were shown

+rough the review of the abovementioned literature itcan be found that the existing researches are limited to thedynamic of beam-stiffened plate with classical boundaryconditions (like clamped simply supported and free) Onlythe free vibration analysis related to elastic boundary con-ditions and nonfixed connections between a plate and beamsis carried out in Xursquos et al [10] research However theinvestigations on the mobility characteristic structural in-tensity and transient response analysis of the stiffened platesubjected to the elastic boundary are still blank In an at-tempt to fill this gap the current work presents the dynamicanalysis of the rectangular beam-reinforced plate withgeneral boundary restraints including classical and theelastic restraints as well as elastic coupling of beams andplate IFSM is extended to construct displacement field of thebeam-stiffened plate which has proven to be a very effectivemethod from existing research [10 36ndash38] +e artificialspring technology is introduced to obtain general boundaryconditions and various coupling connections between aplate and beams Based on Xursquos et al [10] research theenergy expressions of the coupling system is obtained whichcan be solved by the RayleighndashRitz method Transient

2 Shock and Vibration

response of the stiffened plate with classical or elasticboundary restrains is also calculated by the Newmarkmethod+e consistency of the current data and FEM resultsis revealed Also some novel numerical results with respectto the structural intensity structural mobility and transientresponse of the stiffened plate are also reported

2 Theoretical Formulations

21Descriptions of theModel +egeometry of a typical beam-stiffened plate and the coordinate system of the couplingstructure are depicted in Figure 1 For the sake of illustrationonly a thin plate reinforced by a beam with any lengths andangles is shown in Figure 1(a) where a b and h denote thelength width and thickness of the plate respectively In ad-dition the primary coordinate system (o-x-y-z) is placed on themiddle surface of the plate where u v and w represent themidsurface displacements of the plate in the primary coordinatesystem respectively +e width and height of the cross sectionfor a stiffening beam are denoted by bb and hb respectivelywhich is in the local coordinate system (orsquo-xrsquo-yrsquo-zrsquo) Four types ofboundary springs are used on each edge of the plate to obtainarbitrary boundary conditions in which three types (kw ku andkv) of linear springs and one type of rotational spring (Kw) areincluded For a thin plate the clamped boundary condition willbe obtained when the spring stiffness value of each side of theplate is infinity Conversely the free boundary is easily achievedwhen the stiffness value is 0 In current work the stiffeningbeam and plate are described in separate systems As shown inFigure 1(a) six sets of coupling springs are used to achieve theconnection between the beam and plate including three linearsprings (kpb1 kpb2 and kpb3) and three types of rotational springs(Kpb1 Kpb2 and Kpb3) By setting stiffness values of couplingsprings different coupling relationships between the beams andplate can be achieved including various elastic coupling andclamped coupling It should be noted that the coupling betweentwo intersecting beams is ignored in this paper FromFigure 1(b) the orientation between the local coordinate system(orsquo-xrsquo-yrsquo-zrsquo) attached to each beam and the primary coordinatesystem (o-x-y-z) is denoted by the symbol of φ when the lengthof the beam is expressed by L And the coordinates of thestarting end of the beam in the primary coordinate system arerepresented by (x0 y0)

22 Energy Functionals In this study both the in-planedisplacement components (u v) and the out-of-plane dis-placement component (w) of the plate are considered+erefore the total strain potential energy (Vp) of the plate iscomposed of the out-of-plane strain potential energy (Vout

p )and the in-plane strain potential energy (Vin

p ) Specifically theexpression of the total potential energy is as follows

Vp Voutp + V

inp (1)

in which

Voutp

DP

2B

S

z2w

zx21113888 1113889

2

+z2w

zy21113888 1113889

2

+ 2μp

z2w

zx2z2w

zy2⎛⎝

+ 2 1 minus μp1113872 1113873z2w

zx zy1113888 1113889

2⎞⎠dS

(2)

Vinp

Gp

2B

S

zu

zx+

zv

zy1113888 1113889

2

minus 2 1 minus μp1113872 1113873zu

zx

zv

zy

⎧⎨

⎩

+1 minus μp

2zv

zx+

zu

zy1113888 1113889

2⎫⎬

⎭dS

(3)

Similarly the kinetic energy (Tp) expression of the platecan be written as

Tp ρph

2B

S

zw

zt1113888 1113889

2

+zu

zt1113888 1113889

2

+zv

zt1113888 1113889

2⎡⎣ ⎤⎦dS

ρphω2

2B

Sw

2+ u

2+ v

21113960 1113961dS

(4)

In equations (2)sim(4) DP Eph3(12(1 minus μp)) is thebending stiffness of the plate and Ep Gp μp ρp and ωrepresent Youngrsquos modulus extensional rigidity Poissonrsquosratio mass density and circular frequency respectively [10]

Since the boundary conditions of the plate are realized bythe artificial virtual spring technology the elastic potentialenergy (Vsp) stored in the boundary springs here should betaken into the potential energy of plate Vsp can be expressedas

Vsp 12

1113946b

0kwx0w

2+ Kwx0

zw

zx1113888 1113889

2⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868x0dy +

12

1113946b

0kwxaw

2+ Kwxa

zw

zx1113888 1113889

2⎡⎣ ⎤⎦

xa

dy

+12

1113946a

0kwy0w

2+ Kwy0

zw

zy1113888 1113889

2⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868y0dy +

12

1113946a

0kwybw

2+ Kwyb

zw

zy1113888 1113889

2⎡⎣ ⎤⎦

yb

dy

+12

1113946b

0kux0u

2+ kvx0v

21113960 1113961

11138681113868111386811138681113868x0dy +

12

1113946b

0kuxau

2+ kvxav

21113960 1113961

xady

+12

1113946a

0kvy0v

2+ kuy0u

21113960 1113961

11138681113868111386811138681113868y0dy +

12

1113946a

0kvybv

2+ kuybu

21113960 1113961

ybdy

(5)

Shock and Vibration 3

In this paper four degrees of freedom for a single beamare considered which are two bending displacement com-ponents (wy b and wz b) in the yprime and zprime directions and theaxial displacement component (ub) and torsional displace-ment component (θb) in the xrsquo direction erefore thestrain potential energy (Vbi) of ith beam can be written as

Vbi 12intLi

0Dzbi

z2wzbizxprime

2( )2

+Dybi

z2wybizxprime

2( )2

+ EbiAbizubizxprime( )

2

+ GbiJbizθbizxprime( )

2dxprime

(6)

e kinetic energy of ith beam can be expressed as

Tbi ρbi2intLi

0Abi

zwzbizt

( )2

+ Abizwybizt

( )2

+ Abizubizt

( )2

+ Jbizθbizt

( )2dxprime

ρbiω2

2intLi

0Abi w

zbi( )2 + Abi w

ybi( )

2[

+ Abi ubi( )2 + Jbiθ2bi]dxprime

(7)

In equations (6)sim(7) Dz bi and Dy bi are respectively thebending rigidity in the xrsquo-zrsquo plane and xrsquo-yrsquo planewhen torsionalis denoted by the symbol of Jbi In addition Youngrsquos modulusshear modulus mass density and cross-sectional area of ithbeam are taken place of Ebi Gbi ρbi and Abi respectively

Since the stiening beam and plate is connected by acoupling spring the coupling potential (Vpb Ci) of thecoupling spring needs to be considered in the couplingsystem According to references [10 39] its specic ex-pression can be written as

VpbCi 12intLi

0kpb1 w minus w

zbi( )2 + kpb2 v cosφ minus u sinφ minus w

ybi( )

2[

+ kpb3 v sinφ + u cosφ minus ubi( )2]dxprime

+12intLi

0Kpb1 wpbix minus

zwzbizxprime

( )2

+Kpb212

zv

zxminuszu

zy( ) minus

zwybizxprime

( )2

+Kpb3 wpbiy minus θbi( )

2]dxprime

(8)

It should be noted that six degrees of freedom betweenthe beam and the plate are considered here As shown inequation (8) three sets of linear springs (kpb1 kpb2 and kpb3)are used for linear displacement constraints and three sets ofrotational springs (Kpb1 Kpb2 and Kpb3) are used for angulardisplacement constraints Besides wzbi and w

ybi are shown as

follows

wpbix zw

zxcosφ +

zw

zysinφ

wpbiy minuszw

zxsinφ +

zw

zycosφ

(9)

23 Displacement Function In the present research theadmissible displacement functions of a plate and a beam areestablished by IFSM which has been successfully applied tosolve the vibration of beams [40 41] plates [42 43] andshells [44] e improved Fourier series consists of mainfunctions and auxiliary functions For the auxiliary func-tions theoretically they may have innite forms Con-sidering programming and computational eciency a setof sine functions is used as auxiliary functions Hence thedisplacement functions of the plate can be written asfollows

z

O

b

a

bb

hb

h

kpb2 kpb1

kpb1

kw Kw

ku

kv

Middle surface

kpb3kpb3 kpb2y

w v

u

x

yprime oprime

zprime

xprime

(a)

y0

y1

y

yprime

φ

x0 x1

xprime

x

L

O

(b)

Figure 1 Schematic diagram of the coupling systems (a) Geometry of rectangular plate reinforced by arbitrarily orientated beams(b) Schematic of an arbitrarily placed beam


u(x y t) UAuejωt

v(x y t) VBvejωt

w(x y t) WCwejωt

(10)

in which

U V

cos λ0x( 1113857cos λ0y( 1113857 cos λ0x( 1113857cos λny( 1113857 cos λ0x( 1113857cos λNy( 1113857

cos λmx( 1113857cos λny( 1113857 cos λmx( 1113857cos λNy( 1113857 cos λMx( 1113857cos λNy( 1113857

cos λ0x( 1113857sin λ1y( 1113857 cos λmx( 1113857sin λ1y( 1113857 cos λMx( 1113857sin λ1y( 1113857


sin λ1x( 1113857cos λ0y( 1113857 sin λ1x( 1113857cos λny( 1113857 sin λ1x( 1113857cos λNy( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11a)

W




cos λ0x( 1113857sin λny( 1113857 cos λmx( 1113857sin λny( 1113857 cos λMx( 1113857sin λny( 1113857



sin λmx( 1113857cos λ0y( 1113857 sin λmx( 1113857cos λny( 1113857 sin λmx( 1113857cos λNy( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11b)

Au

Au00 Au

0n Au0N Au

mn AumN Au

MN

a01 am1 aM1 a02 am2 aM2

b10 b1n b1N b20 b2n b2N

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12a)

Bv

Bv00 Bv

0n Bv0N Bv

mn BvmN Bv

MN

c01 cm1 cM1 c02 cm2 cM2

d10 d1n d1N d20 d2n d2N

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12b)

Cw

Cw00 Cw

0n Cw0N Cw

mn CwmN Cw

MN

e01 em1 eM1 e0n emn eMn eM4

f10 f1n f1N fm0 fmn fmN f4N

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12c)

where λm mπa λn nπb andMN represent the numberof truncations Fourier coefficients of displacement func-tions for the plate are expressed by Au mn aml bln Bv mncml dln Cw mn emp and fpn (l 1 2 p 1 2 3 4)respectively

Similarly the displacement field functions of the ithbeam can be obtained by IFSM as follows

wzbi xprime( 1113857 Ψz

biDmiejωt

wy

bi xprime( 1113857 Ψy

biEmiejωt

ubi xprime( 1113857 ΨubiFmie

jωt

θbi xprime( 1113857 ΨθbiGmie

jωt

(13)


where

Ψzbi Ψy

bi sin λminus 4xprime( 1113857 sin λm1xprime( 1113857 sin λminus 1xprime( 1113857

cos λ0xprime( 1113857 cos λm1xprime( 1113857 cos λM1xprime( 1113857⎡⎣ ⎤⎦ (14a)

Ψubi Ψθ

bi sin λminus 2xprime( 1113857 sin λminus 1xprime( 1113857 cos λ0xprime( 1113857 cos λm1xprime( 1113857 cos λM1xprime( 11138571113858 1113859 (14b)

Dmi Dminus 4 Dminus 1 D0 DM11113858 1113859T (15a)

Emi Eminus 4 Eminus 1 E0 EM11113858 1113859T

(15b)

Fmi Fminus 2 Fminus 1 F0 FM11113858 1113859T (15c)

Gmi Gminus 2 Gminus 1 G0 GM11113858 1113859T

(15d)

In the above equations (14a)sim(15d) λm1 m1πLi andM1 indicates the truncated number Dmi Emi Fmi and Gmirepresent the unknown coefficient of ith beamrsquos displace-ment functions

24 Steady-State Vibration Solution For the calculation ofthe steady-state response for the beam-stiffened plate underexternal excitation it is necessary to consider the work doneby the external force In terms of the coupling structureshown in Figure 1(a) the Lagrange function (L) can beexpressed as follows

L Vp + Vsp + 1113944N

i0Vbi + V

pb

Ci1113872 1113873 minus Tp + 1113944N

i0Tbi

⎛⎝ ⎞⎠ minus Wexc

(16)

in which the number of stiffening beams is denoted by Ni 0 means unreinforced bare plate andWexc represents thework done by the external excitation force acting on theplate

Wexc BS

fuu + fvv + fww( 1113857dS (17)

In the above equations (14a) and (14b) the external loaddistribution function on the plate is denoted by the symbolsof fu fv and fw In the current research fw is taken intoconsideration which expresses the lateral load associatedwith out-of-plane displacement (w) of the plate In partic-ular for external point excitation fw is represented by themagnitude (F) of the point excitation force and the 2D deltafunction (δ)

fw Fδ x minus xexc( 1113857δ y minus yexc( 1113857 (18)

Substituting equations (1)sim(8) (10) (13) and (17) intoequation (16) linear equations in the matrix form areavailable by employing the RayleighndashRitz method

K minus ω2M1113872 1113873 H⟶

F⟶

(19)

where the vector H⟶

ATu BT

v CTw DT

mi ETmi FT

mi1113960

GTmi 1113859

Tis composed of the unknown Fourier coefficient

F⟶

0 0 fwCTw 0 0 0 0 1113960 1113961

Tdenotes the exter-

nal force vector and the mass matrix and stiffness matrix ofthe coupled systems are displayed by M and K where theirelements are only related to the material properties geo-metric dimensions and boundary constraints of the beam-stiffened plate structure And their expressions are presentedas follows

M

Muu 0 0 0 0 0

0 Mvv 0 0 0 0

0 0 Mww 0 0 0

0 0 0 ⋱ 0 0

0 0 0 0 Mbi 0

0 0 0 0 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

K

Ku Kuv 0 middot middot middot Kubi middot middot middot

KTuv Kv 0 middot middot middot Kv

bi middot middot middot

0 0 Kw middot middot middot Kwbi middot middot middot

⋮ ⋮ ⋮ ⋱ 0 middot middot middot

Kubi1113858 1113859

T Kvbi1113858 1113859

T Kwbi1113858 1113859

T 0 Kbi 0⋮ ⋮ ⋮ ⋮ 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)

For the external excitation force with any frequency theunknown series expansion coefficient related to displace-ment functions of the coupled structure can be directlycalculated from equation (19) +at is

H⟶

K minus ω2M1113872 1113873minus 1

F⟶

(22)

Substituting equation (22) into equation (10) the dis-placement response of plate reinforced by beams undersome specific excitation will be obtained And vibrationspeed can also be obtained by v jωw Particularly thecomplex Youngrsquos modulus (1113957E) is introduced in the currentnumerical solution process 1113957E Ep(1 + jη) is composed ofYoungrsquos modulus (Ep) and structural loss factor (η) wherej

minus 1

radicis the imaginary unit


+e structure mobility can quantitatively describe thelaw of power flow in the structure +erefore understandingthe admittance characteristics of the stiffened plate is of greatsignificance for structural vibration control +e structuremobility can be calculated by the following equation

Ynk Vk

Fn

(23)

in which Ynk represents the transfer mobility from point n topoint k when Vk is utilized to express the velocity at point kAnd the magnitude of the excitation force on point n ispresented by Fn +e mobilities of the drive point n can beachieved by setting n k

+e vibration energy of the structure can be described bythe structural intensity vector which helps the designersunderstand the energy distribution of the structure +edefinition of structural intensity can be found in references[26 37 45] In this paper the structural intensity vector I (xy) of any point in the stiffened plate can be obtained byvector superposition of the component Ix (x y) along the x-axis and component Iy (x y) along the y-axis +eir re-lationship is seen in the following equation

|I(x y)|

Ix(x y)1113868111386811138681113868

11138681113868111386811138682

+ Iy(x y)11138681113868111386811138681113868

111386811138681113868111386811138682

1113970

(24)

In equation (24) the structural intensity at any pointincludes both bending vibration components and in-planevibration components as shown in equations (25a) and(25b)

Ix(x y) Ixw(x y) + I

xuv(x y) (25a)

Iy(x y) Iyw(x y) + I

yuv(x y) (25b)

In the above equations (25a) and (25b) the structuralintensity components associated with structural bendingvibrations in both directions are

Ixw(x y) minus

12Re Qx

zw

zt1113888 1113889

lowast

minus Mx

z2w

zx zt1113888 1113889

lowast

minus Mxy

z2w

zy zt1113888 1113889

lowast

1113896 1113897

(26a)

Iyw(x y) minus

12Re Qy

zw

zt1113888 1113889

lowast

minus My

z2w

zy zt1113888 1113889

lowast

minus Myx

z2w

zx zt1113888 1113889

lowast

1113896 1113897

(26b)

in which the symbol of (lowast) represents the conjugate of acomplex number And the transverse shear bending mo-ment and torque in the two directions are respectively

Qx minus Dp

z3w

zx3 +z3w

zx zy21113888 1113889

Mx minus Dp

z2w

zx2 + μp

z2w

zy21113888 1113889

Mxy minus 1 minus μp1113872 1113873Dp

z2w

zx zy

(27a)

Qy minus Dp

z3w

zy3 +z3w

zx2zy1113888 1113889

My minus Dp

z2w

zy2 + μp

z2w

zx21113888 1113889

Myx minus 1 minus μp1113872 1113873Dp

z2w

zx zy

(27b)

+e structural intensity associated with the in-planevibration components in equations (25a) and (25b) ispresented as follows

Ixuv(x y) minus

12Re σx

zu

zt1113888 1113889

lowast

+ τxy

zv

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2p

zu

zx+ μ

zv

zy1113888 1113889

zu

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zv

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28a)

Iyuv(x y) minus

12Re σy

zv

zt1113888 1113889

lowast

+ τyx

zu

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2pμ

zu

zx+

zv

zy1113888 1113889

zv

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zu

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28b)

Substituting equations (10) (22) and (26a)sim(28b) into(25a) and (25b) the structural intensity at any point of thestiffened plate can be obtained

25 Transient Vibration Solution +e mass matrix M andstiffness matrix K of the coupled structure have been pre-sented in equations (20)sim(21) in Section 24 Hence thedynamic equation of the structural system in the time do-main can be described as [46]

Meurox + C _x + Kx Fp (29)

where Fp is the external excitation vector which isthe function of time and x denotes the displacement attime t

Generally damping constants have frequency-varyingcharacteristics +erefore it is difficult to accurately definethe damping matrix in numerical simulation For the sake ofsimplicity only Rayleigh damping of the coupling structureis considered in this study namely C α0M + β0K And α0


and β0 are the Rayleigh damping coefficients independent offrequency

In order to solve the transient response of the system theNewmark method is adopted which is an extensive implicitalgorithm Its integration method is as follows

_xn+1 _xn +(1 minus c)Δt euroxn + cΔt euroxn+1 (30)

xn+1 xn + Δt _xn +12

minus β1113874 1113875Δt2 euroxn + βΔt2 euroxn+1 (31)

In equations (30)sim(31) when the velocity function andthe displacement function are expanded by Taylor series theexpansion is retained to the second derivative and equations(32a)sim(33) can be calculated as follows

euroxn+1 1

βΔt2xn+1 minus xn( 1113857 minus

1βΔt

_xn minus12β

minus 11113888 1113889 euroxn (32a)

_xn+1 c

βΔtxn+1 minus xn( 1113857 + 1 minus

c

β1113888 1113889 _xn + 1 minus

c

2β1113888 1113889Δt2 euroxn

(32b)

Combining equations (29) (32a) and (32b) the fol-lowing matrix equation (33) can be obtained

Kxn+1 Fn+1p (33)

in which

K K +1

βΔt2M

Fn+1p fn+1 + M

1βΔt2

xn +1

βΔt_xn +

12β

minus 11113888 1113889euroxn1113890 1113891

(34)

+e accuracy and stability of the Newmark methoddepends on β and c In this study β 025 and c 05 areused which is known as the average acceleration Newmarkmethod In addition the force in the above equation (29) canbe either point force or pressure

3 Results and Discussions

In this section a series of numerical results for dynamicbehaviors of the stiffened plate with general boundary re-straints are carried out based on the theoretical modelestablished in Section 2 Further analysis will be given in thefollowing sections Particularly unless otherwise stated thebeam and the plate are assumed to have the same materialproperties ie Ep Ebi 207GPa ρp ρbi 7800 kgm3μp μbi 03 and η 001 It should also be noted that forthe convenience of research two reinforced plates of dif-ferent geometric parameters are included in the currentresearch which are rectangular plate and square plate re-spectively For a rectangular plate its geometric parametersare b 1m ab 2 and h 001m where the measuredpoints can be expressed as follows according to the o-x-ycoordinate in Figure 1 point A (1m 05m) and point B(05m 05m) For the square plate except for ab 1 theother parameters are the same as the rectangular plate of

which the measured points are point C (05m 05m) pointD (025m 025m) and point E (04m 04m) In thefollowing analysis the geometric and material parameters ofthe two plates are not repeatedly described It is worthemphasizing that the response at the above-measured pointsis the lateral vibration response of the stiffened plate

In addition for the sake of simplicity the symbols of C SF and E are utilized to represent the clamped simply sup-ported free and elastic boundary conditions respectively allof which can be achieved by changing the spring stiffness+espring stiffness values for different boundary conditions usedin this paper are presented as follows

At x constant or y constant

C ku 1012 Nm kv 1012 Nm kw 1012 Nm andKw 1012 NmiddotmradS ku 1012 Nm kv 1012 Nm kw 1012 Nm andKw 0NmiddotmradF ku 0Nm kv 0Nm kw 0Nm andKw 0NmiddotmradE2 ku 1012 Nm kv 1012 Nm kw 1012 Nm andKw 102 NmiddotmradE5 ku 1012 Nm kv 1012 Nm kw 1012 Nm andKw 105 NmiddotmradE7 ku 1012 Nm kv 1012 Nm kw 1012 Nm andKw 107 NmiddotmradE55 ku 105 Nm kv 105 Nm kw 105 Nm andKw 105 NmiddotmradE77 ku 107 Nm kv 107 Nm kw 107 Nm andKw 107 Nmiddotmrad

Another special note is the type and magnitude of theexternal load used in this paper+e excitation forces appliedin the current research are point excitation and uniformpressure in the normal direction of the stiffened plate whichare denoted by the symbol of qw and pw respectively Andthe location of the concentrated point force is at point Q inFigure 2(a) when locally acting area of uniform pressure isshown by Ω in Figure 2(b) What needs special mention isqw 1 and pw 1 in the following calculation

31 Steady-State Vibration Analysis

311 Convergence and Correctness Analysis In this sectionthe stiffened rectangular plate with CCCC case is first se-lected to check the convergence of the current method +ecross-section parameters of the beam are bbh 1 and hbh 1 Figure 3 shows convergence of velocity response forthe rectangular plate with one central x-wise beam and onecentral y-wise beam under qw 1 when the similar curvesfor the same coupling structure under pw 1 are presentedin Figure 4 It can be observed from Figures 3sim4 that whenthe truncated numbers M and N are equal to 12 or 14 theresponse curves of the coupling structure under the action ofpoint force or surface force are basically coincident andmatch well with the results obtained by FEM softwareAnsys150 +e above analysis fully proves that the currentmethod has good convergence In order to balance both


calculation efficiency and calculation accuracy the dis-placement series expressions of beams and plates aretruncated by MNM1 12 in the following calculationsBesides it can be seen by comparing Figures 3 and 4 that thetrend of the response of drive point A or transfer point Bdoes not change under different forms of force and only themagnitude of the response is different

Next a square plate with one beam located at its diagonalline is utilized to implement a comparative study between thecurrent method and FEM result (Ansys150) It is clarified that theconfigurations of the reinforcing component are L1

2

radicm bb

h 1 hbh 1 x0 y0 0 and φ 45∘ and two points (point Cand point D) on the square plate supported by CCCC areselected as analysis points +e comparison examples aredepicted in Figures 5sim6 which present the comparison ofvelocity response for the stiffened square plate under qw 1and under pw 1 respectively It is easy to be observed fromthe two figures that the current calculation method and thefinite element result are in good agreement which proves thatthe construction of the steady-state response dynamic analysis

model by IFSM is correct Simultaneously compared with thefinite element method element mesh is not required in thecurrent calculation model which greatly saves the memoryresources and improves the solving efficiency

312 Structural Mobility Analysis In this part the effect ofthe stiffening beam with different lengths on mobilitycharacteristics of the reinforced plate subjected to SSSS willbe investigated at first where qw 1 is adopted As shown inFigure 7 four different lengths of beams are selected for theanalysis of the mobility characteristics of the coupledstructure including L1 0m 02m 05m and 1m Par-ticularly except for the length of the stiffening beam and thecross-section parameter hbh 15 the remaining geometricconfigurations are consistent with Figure 5 in this exampleFrom Figure 7 the following points can be found Firstlyregardless of the drive point or the transfer point thenumber of peak points of mobility curves for the plate withdifferent lengths of stiffening beam compared with the bare

0 100 200 300 400

Vel

ocity

(dB

ref =

10ndash9

ms

)

Frequency (Hz)

M = N = 3M = N = 9M = N = 14

M = N = 6M = N = 12FEM result

60

90

120

150

(a)

Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


0 100 200 300 400

Frequency (Hz)

60

90

120

150

(b)

Figure 3 Convergence of velocity response for rectangular plate with one central x-wise beam and one central y-wise beam under unit force(qw 1) (a) Point A (1m 05m) (b) Point B (05m 05m)

b

Q

a

(a)

02b

a

b

02a

Ω

(b)

F0

t0O t

F

F(t) = F0

(c)

Figure 2 External load types for the stiffened plates (a) point force (b) surface force and (c) rectangular pluse


plate will increase which may be because the beam changesthe local stiffness and mass distribution of the plate Sec-ondly when the reinforcing beam passes through thetransfer point D or the drive point C corresponding toL1 05m or L1 1m the peak of their mobility curves willshift to the high frequency andmagnitude of some peaks willappear to decrease in the range of 200ndash800Hz especiallyL1 1m which shows that the rib increases the ratio ofstiffness to mass of the plate in high frequency +e abovephenomena also reveals that in structural vibration control

reinforcement helps to reduce the response peak of the platein the midhigh-frequency range and changes the structuralresonance frequency but has a little effect on the response inlow-frequency domain

Figure 8 studies the mobility characteristics of the di-agonally stiffened square plate subjected to local uniformpressure with various elastic boundary conditions (ieE2SE2S E2E5E2E5 E5E5E5E5 and CE5CE5) +e points C andD are selected as the observed points and a local uniformpressure pw acts on the coupling system containing two

Present methodFEM result

Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(b)

Figure 5 Comparison of velocity response for square plate with one beam located at its diagonal line under unit force (qw 1) (a) Point C(05m 05m) (b) Point D (025m 025m)

Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(a)V

eloc

ity (d

B re

f = 1

0ndash9 m

s)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(b)

Figure 4 Convergence of velocity response for rectangular plate with one central x-wise beam and one central y-wise beam under localuniform pressure (pw 1) (a) Point A (1m 05m) (b) Point B (05m 05m)


reinforcing beams For convenience of explanation (x0 y0φ) in Figure 1(b) is used to indicate the arrangement of theribs Hence the arrangement of the beams used in Figure 8are expressed as (0m 0m 45∘) and (1m 0m 135∘) +egeometric configurations of the stiffening beams areL1 L2

2

radic bb1h 1 bb2h 1 hb1h 1 and hb2h 1 It

can be seen from Figure 8 that as the boundary restrains varyfrom E2SE2S to CE5CE5 the resonant peak of the structuralmobility response moves to the high-frequency directionand the peak value of the response curves will show a

downward trend especially under CE5CE5 +erefore it canbe known that the frequency position corresponding to thevibration formant can be adjusted by modifying theboundary constraint spring stiffness to achieve structuralvibration control Another interesting finding is that thenumber of formants of the mobility responses increaseswhen the stiffened plate is subjected to E2E5E2E5 and CE5CE5boundary conditions

In order to investigate the effect of coupling springstiffnesses on the structural mobility characteristics a

0 200 400 600 800

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

Frequency (Hz)

None05

021

(a)

None05

021

0 200 400 600 800ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

Frequency (Hz)

(b)

Figure 7 Mobility characteristics for square plate with one beam of different lengths located at its diagonal line under unit force (qw 1)(a) Point C (05m 05m) (b) Point D (025m 025m)

0 100 200 300 400 500 600

Frequency (Hz)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

100 200 300 400 500 6000Frequency (Hz)

(b)

Figure 6 Comparison of velocity response for square plate with one beam located at its diagonal line under local uniform pressure (pw 1)(a) Point C (05m 05m) (b) Point D (025m 025m)


stiffened rectangular plate with E2E2E2E2 case is presented inFigure 9 +e analytical model used in Figure 9 is a rect-angular plate with one central x-wise beam and three y-wiseevenly distributed beams +e geometric parameters of thereinforced beam used are L1 2m in the x directionL2 L3 L4 1m in the y direction bb1h 1bb2 bb3 bb4 bb1 hb1h 15 and hb2 hb3 hb4 hb1And point A and point B are selected as the drive point andtransfer point respectively In addition it should be notedthat the coupling springs (including kpb1 kpb2 kpb3 Kpb1Kpb2 and Kpb3) shown in Section 2 are replaced by thesymbol of kb +e curves in Figure 9 show that with theincrease of the coupled spring stiffness the resonant peaks ofthe structural velocity mobility responses move faster andfaster in the high-frequency direction And the resonantpeaks of the structural mobility move faster in the range of150Hz to 500Hz than the range of 0 to 100Hz whichindicates that the structural mobility responses in the highfrequency are more sensitive to the coupled springs Inaddition we can also see that an increase in coupling springstiffness will result in an increase in the peak values of themobility response in the range of 0 to 150Hz which in-dicates that the reinforcement of the beam and the plate willhave a negative effect on the low-frequency vibrationreduction

In the analytical model constructed in this paperstructural damping is introduced by means of complexYoungrsquos modulus Damping effects on the mobility char-acteristics of a rectangular plate stiffened by three y-wiseevenly distributed beams are illustrated in Figure 10 Exceptfor removing a beam in the x direction the remaininggeometric parameters of the current model with E5E5E5E5boundary restrains and measured points follow the samevalues of the cases studied in Figure 9 It can be seen from

Figure 10 that due to the existence of structural damping thepeak values of the vibration mobility responses are reducedIt is found after further observation that the magnitude ofresonance peaks for mobility responses shows a decreasingtrend as the structural damping increases and the peakreduction in the range of 150sim500Hz is more obvious thanthat in the range of 0sim150Hz Besides the structuraldamping of the stiffened plate can only effectively reduce themobility response amplitude of the resonance peak whilethe effect of structural damping on the mobility response inthe nonresonant region is not obvious

313 Structural Intensity Analysis In the previous sectionthe mobility characteristics of the stiffened plates are clearlydescribed Undoubtedly these numerical results providereference data for further research However for the specificstructural design it is extremely necessary to understand thedistribution and transmission of vibration energy in thestructural system To this end the following analysis ofstructural intensity will be performed to show the flowstrength and direction of the vibrational energy in thestructure

Firstly structural intensity of a rectangular plate at-tached by beams with different configurations under two-point excitation is carried out where four symmetricalboundary conditions are employed namely CSCS SFSFE77E77E77E77 and E55E55E55E55 +e geometric parametersof the beams (see Figure 11) are L1 L2 L3 1 hb1h 2hb2 hb3 hb1 and bb1 bb2 bb3 h In this example re-gardless of the arrangement spacing of the beams theorientation φ of all beams and their coordinates y0 in the ydirection (see Figure 1(b)) are the same φ 90∘ and y0 0+erefore the coordinates x0 in the x direction will be used

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)

Figure 8 Mobility characteristics of diagonally stiffened square plate subjected to local uniform pressure with various elastic boundaryconditions (pw 1) (a) Point C (05m 05m) (b) Point D (025m 025m)


to show the arrangements of the three beams at the spacingof each beam For example (0 03 1) means that the ar-rangements of the three beams attached to the rectangularplate are (0m 0m 90∘) (03m 0m 90∘) and (1m 0m 90∘)According to the above description the arrangements of thethree-pitch reinforcing beams are (06m 08m 1m) (06m09m 12m) and (06m 1m 14m) which are representedby Type I Type II and Type III respectively In addition theexcitation frequency of the two-point forces is 7254Hz(denoting 2nd frequency of the rectangular plate with CSCS

case) the amplitude of which are 1N For symmetrical two-point forces (see Figure 11) the locations of two-points forceare point G (05m 05m) and point H (15m 05m)respectively +e results in Figure 11 indicate that as thestiffness of the boundary spring weakens the presence of thereinforcement will intensify the effect of the vibrationalenergy gathering toward the edges of the plate and thereflection and backflow of vibrational energy occur at theedges of the plate Moreover regardless of the boundaryconditions the increased spacing of the ribs will have a

kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)

Figure 9 Mobility characteristics of beam-reinforced rectangular plate with different coupling spring stiffnesses (pw 1) (a) Point A (1m05m) (b) Point B (05m 05m)

η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)

Figure 10 Damping effects on the mobility characteristics of a rectangular plates stiffened by three y-wise evenly distributed beams (pw 1)(a) Point A (1m 05m) (b) Point B (05m 05m)


lowast

lowastlowast

lowast

lowast

lowastlowast

lowast

D Type IIIC Type IIB Type IA None stiffener

(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast

E None stiffener F Type I G Type II H Type III

(b)

Figure 11 Continued


positive increasing effect on the structural intensity of theplate between the reinforcements especially for elasticboundary cases (E5E5E5E5 and E7E7E7E7) +e above phe-nomenon may be because reinforcements hinder energyflow on the plate resulting in energy accumulation in someareas By further observing it is found that in the case ofsymmetrical boundary cases and symmetrical point forcesthere is always at least a section where the vibrational energycancels each other out which is called the power-insulationsection Furthermore the number of such power-insulation

section can be increased by setting an appropriate re-inforcement spacing Also it can be intuitively found thatthe vibration energy is transmitted from the excitation to theperiphery of the plate but the force source is not alwaysoutput source of the vibration energy (see Figure 11(d))

Next structural intensity of an SSSS square plate stiff-ened by one beam with various orientations is shown inFigure 12 where two-point forces at point S (075m 025m)and point T (025m 075m) are adopted +e geometricparameters of the attaching beam are L1 1 bb1h 1 and

lowast lowast lowast lowast


I None stiffener J Type I K Type II L Type III

(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast

M None stiffener N Type I O Type II P Type III

(d)

Figure 11 Structural intensity of a rectangular plate attached by beams with different configurations under two-point excitation (a) CSCS(b) SFSF (c) E77E77E77E77 (d) E55E55E55E55


hb1h 2 and four arrangements of the beams are taken intoconsideration including (0 0 0∘) (0 0 20∘) (0 0 30∘) and(0 0 45∘) according to the beamrsquos arrangement methodadopted in Figure 8 Moreover the excitation frequency is12244Hz (see Figure 12(AsimD)) and 19590Hz (seeFigure 12(EsimH)) respectively which represent the 2ndfrequency and 4th frequency for the square plate FromFigure 12 it is no difficult to find that as the azimuth angle ofthe reinforcement varies from 0∘ to 45∘ the energy flow nearthe excitation source S first decreases and then increases andthe structural intensity reaches a minimum when the re-inforcement passes through the excitation source (that isφ 20∘) Besides there are significant differences in theinfluence of the azimuth variation of the reinforcement onthe structural density distribution of the plate at differentexcitation frequencies which reflects that the effect of thereinforcement on the structural density of the plate dependsdramatically on the excitation frequency

32 Transient Response Analysis +is section is concernedwith the transient vibration analysis of the reinforced plateunder point force and local uniform pressure +e con-vergence and accuracy of transient responses for the platewith beams attached are first validated For a validation casetransient responses of a coupling structure identical to thereinforced rectangular plate used in Figures 3sim4 are ex-amined in Figures 13sim14 and the external excitation loads intime domain are the point excitation force qw and localuniform pressure pw respectively the time of which is

t0 02 s shown in Figure 2(c) It is well revealed fromFigures 13sim14 that the results calculated by the currentmethod converge very rapidly as M N and M1 are in-creased Convergence for transient responses of the stiffenedplate is achieved by MNM1 12 In addition it can beseen through further observation that the present results arehighly consistent with FEM results which verifies the cor-rectness of the transient response model of the stiffenedplates established in this paper In order to further verify thecorrectness of the damping treatment method in this papera comparison between the current method and FEM resultsis implemented in Figure 15 where only the Rayleighdamping is added and the other parameters are consistentwith these from Figure 13 And the selected Rayleighdamping coefficients are set as α0 0999 and β0 0001 Ascan be seen from Figure 15 a good agreement between thecurrent results and those obtained by FEM is achievedwhich indicates that the presented damping treatment ofstiffened plates is reasonable

Next transient responses of an SSSS square plate stiff-ened by a beam with various orientations are presented inFigure 16 For the present analysis the time increment istaken to be 033ms and normal point load qw located atpoint C is applied +e geometric parameters and ar-rangement of the beam are the same as those in Figure 12but the arrangement of the beam with (0 0 20∘) is replacedby (0 0 15∘) here +e curves in Figure 16 reveal that re-gardless of the measured point A or B when the orientationof the reinforcing beam changes from 0∘ to 45∘ the oscil-lation periods of the time response curve are shortened and

lowast

lowast

lowast

lowast

lowast

lowast

lowast

A φ = 0deg B φ = 20deg C φ = 30deg D φ = 45deg

(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast

E φ = 0deg F φ = 20deg G φ = 30deg H φ = 45deg

(b)

Figure 12 Structural intensity of an SSSS square plate stiffened by one beam with various orientations (qw 1) (a) 2nd frequency(12244Hz) (b) 4th frequency (19590Hz)


the amplitude of the displacement responses shows a de-creasing trend +is is undoubtedly expected because thecloser the stiffening beam is to the measured points thegreater the bending stiffness at the measured points is

As mentioned in Section 25 the Rayleigh damping isselected to implement the transient vibration analysis of thebeam-stiffened plate and it can be obtained by setting the

values of α0 and β0 +erefore the influence of the Rayleighdamping coefficients on transient performances of thestiffened plate will receive attention in the following ex-ample As shown in Figure 17 a SCSC square plate rein-forced by two identical beams is considered to performcurrent analysis of which the configurations are (0m 0m45∘) and (0m 0m 135∘) respectively And the geometric

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)

Figure 14 Convergence of transient responses for rectangular plate with one central x-wise beam and one central y-wise beam under localuniform pressure (pw 1) (a) Point A (1m 05m) (b) Point B (05m 05m)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)

Figure 13 Convergence of transient responses for rectangular plate with one central x-wise beam and one central y-wise beam under unitforce (qw 1) (a) Point A (1m 05m) (b) Point B (05m 05m)


parameters of the used beams are set as L1 L2 2

radic hb1

h 1 hb2 hb1 and bb1 bb2 h In this investigation thesame form and incentive position of normal load as Fig-ure 17 is utilized where the observed point is point C FromFigure 17 it can be intuitively seen that due to the existenceof damping the vibration energy of the coupling structurewill be continuously consumed and the transient vibration ofthe coupled structure exhibits a completely different be-havior than the undamped transient vibration Secondly thechange of the mass damping coefficient α0 has a little effecton the transient vibration of the stiffened plate (see

Figure 17(a)) However a small change in the stiffnessdamping coefficient β0 will cause a significant change in thetransient response (see Figure 17(b)) Specifically when theparameter β0 is smaller than a certain value denoted by thesymbol β10 the transient amplitude of the coupled structurewill gradually decrease and finally stabilize When β0 exceedsthe value β10 the vibration response of the reinforcedstructure will gradually increase and finally stabilize at acertain value Also as the value of β0 increases the time forthe transient response to reach the steady state will beextended

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)

Figure 16 Transient response of an SSSS square plate stiffened by one beam with various orientations (qw 1) (a) Point C (05m 05m)(b) Point E (04m 04m)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)

Figure 15 Comparison of transient responses for rectangular plate with one central x-wise beam and one central y-wise beam under localuniform pressure (pw minus 1) (a) Point A (1m 05m) (b) Point B (05m 05m)


In the last transient study transient responses of arectangular plate with three x-wise and one y-wise evenlydistributed beams are examined in Figure 18 in whichnormal local uniform pressure (pw 1) is consideredExcept for the difference of the arrangement of the beamsthe parameters of the beams employed in Figure 3 areused here and the measured point is point B What needsto be explained here is that when the same type of springsat edges of the stiffened plate is studied the stiffnessvalues of the remaining types of springs are set to infinityreplaced by 1012 for instance while the elastic parameterskw varies from 0 to 1012 the stiffness values of ku kv andKw are designated as 1012 Figure 18 shows that as thestiffness value of transverse springs kw varies from 105 to1011 the amplitude and phase of the lateral displacementresponse will change significantly Specifically as thestiffness value increases the oscillation period of theresponse curve will be shortened and the vibration am-plitude will decrease When the stiffness value of springkw is greater than 1011 the transient displacement re-sponse remains unchanged and is close to the responseunder CCCC case For a rotating spring Kw the lateraldisplacement response will only change significantly if itsstiffness value varies from 103 to 108 but its effect is not asobvious as that of the transverse spring kw In addition itcan be seen from Figures 18(a) and 18(b) that the lateraldisplacement response is not affected by the variation ofthe in-plane linear spring ku and kv which is consistentwith the current classical thin plate theory In additionthe results in Figure 18 indicate that the definition of theboundary conditions at the beginning of this section isreasonable

4 Conclusions

An analysis model is established for investigating the lineardynamic analysis of rectangular plates reinforced by beams ofany lengths and orientations Employing IFSM displacementfunctions of the plate and reinforcements are obtained re-spectively Based on RayleighndashRitz method dynamic responseof the coupling structure is obtained inwhich various boundaryrestrains and beam-plate coupling relationships are achievedusing the artificial spring technology Several numerical analysesverify the effectiveness of the current method and a series ofnovel numerical results are also given Besides it should also benoted that even though only the linear dynamic performancesof the beam-stiffened plate are reported here the presentedmethods can also be extended to perform other dynamicsanalysis related to the stiffened plate such as the vibrationcontrol of the stiffened plate in electrothermal-magnetic fieldand vibration analysis of reinforced coupling plates

In addition there are some conclusions that are stated asfollows (1) +e mobility characteristics of the beam-stiff-ened plate are not only related to the length of the beams butalso to coupling spring stiffness and structural dampingespecially in the high-frequency range which provides adesign basis for structural vibration control (2) +e ar-rangement spacing and orientation of stiffening beams havea significant effect on the structural intensity distribution ofthe plate Moreover the structural intensity distribution ofthe stiffened plates is sensitive to the frequency of the ex-citation force and boundary conditions which give a ref-erence for discriminating the position of the load and thetransmission path of the vibration energy (3) Reducing thedistance between the measured point and the reinforcing

Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)

Figure 17 Effect of Rayleigh damping coefficient on transient response of stiffened square plate with SCSC case (qw 1) (a) Mass dampingcoefficient α0 (b) Stiffness damping coefficient β0


beams or increasing the spring stiffness (kw and Kw) willhave a negative effect on the amplitude and oscillationperiod of the transverse transient response of the measuredpoint However the in-plane springs (ku and kv) have noeffect on the lateral transient vibration of the stiffened plate(4) When the Rayleigh damping is considered in the tran-sient analysis of the stiffened plate the transient response isheavily affected by the stiffness damping coefficient ratherthan the mass damping coefficient

Data Availability

All the underlying data related to this article are availableupon request

Conflicts of Interest

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

Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)

Figure 18 Effect of the boundary spring stiffness on transient response of stiffened plate under local uniform pressure (pw 1) (a) Linearspring ku (b) Linear spring kv (c) Linear spring kw (d) Rotational spring Kw


Acknowledgments

+e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(grant no 51905511)

References

[1] A Mukherjee and M Mukhopadhyay ldquoFinite element freevibration of eccentrically stiffened platesrdquo Computers ampStructures vol 30 no 6 pp 1303ndash1317 1988

[2] T P Holopainen ldquoFinite element free vibration analysis ofeccentrically stiffened platesrdquo Computers amp Structures vol 56no 6 pp 993ndash1007 1995

[3] K M Liew Y Xiang S Kitipornchai and J L MeekldquoFormulation of Mindlin-Engesser model for stiffened platevibrationrdquo Computer Methods in Applied Mechanics andEngineering vol 120 no 3-4 pp 339ndash353 1995

[4] H P Lee and T Y Ng ldquoEffects of torsional and bendingrestraints of intermediate stiffeners on the free vibration ofrectangular platesrdquo Mechanics of Structures and Machinesvol 23 no 3 pp 309ndash320 1995

[5] M Barrette A Berry and O Beslin ldquoVibration of stiffenedplates using hierarchical trigonometric functionsrdquo Journal ofSound and Vibration vol 235 no 5 pp 727ndash747 2000

[6] H Zeng and C W Bert ldquoA differential quadrature analysis ofvibration for rectangular stiffened platesrdquo Journal of Soundand Vibration vol 241 no 2 pp 247ndash252 2001

[7] A K L Srivastava P K Datta and A H Sheikh ldquoBucklingand vibration of stiffened plates subjected to partial edgeloadingrdquo International Journal of Mechanical Sciences vol 45no 1 pp 73ndash93 2003

[8] L X Peng KM Liew and S Kitipornchai ldquoBuckling and freevibration analyses of stiffened plates using the FSDT mesh-free methodrdquo Journal of Sound and Vibration vol 289 no 3pp 421ndash449 2006

[9] L Dozio and M Ricciardi ldquoFree vibration analysis of ribbedplates by a combined analytical-numerical methodrdquo Journalof Sound and Vibration vol 319 no 1-2 pp 681ndash697 2009

[10] H Xu J Du and W L Li ldquoVibrations of rectangular platesreinforced by any number of beams of arbitrary lengths andplacement anglesrdquo Journal of Sound and Vibration vol 329no 18 pp 3759ndash3779 2010

[11] A Bhar S S Phoenix and S K Satsangi ldquoFinite elementanalysis of laminated composite stiffened plates using FSDTand HSDT a comparative perspectiverdquo Composite Structuresvol 92 no 2 pp 312ndash321 2010

[12] T Nguyen-+oi T Bui-Xuan P Phung-Van H Nguyen-Xuan and P Ngo-+anh ldquoStatic free vibration and bucklinganalyses of stiffened plates by CS-FEM-DSG3 using triangularelementsrdquo Computers amp Structures vol 125 no 9pp 100ndash113 2013

[13] N Nguyen-Minh T Nguyen-+oi T Bui-Xuan and T Vo-Duy ldquoStatic and free vibration analyses of stiffened foldedplates using a cell-based smoothed discrete shear gap method(CS-FEM-DSG3)rdquo Applied Mathematics and Computationvol 266 pp 212ndash234 2015

[14] K Bhaskar and A Pydah ldquoAn elasticity approach for simply-supported isotropic and orthotropic stiffened platesrdquo In-ternational Journal of Mechanical Sciences vol 89 pp 21ndash302014

[15] A Pydah and K Bhaskar ldquoAccurate discrete modelling ofstiffened isotropic and orthotropic rectangular platesrdquo Lin-Walled Structures vol 97 pp 266ndash278 2015

[16] D S Cho N Vladimir and T M Choi ldquoNumerical pro-cedure for the vibration analysis of arbitrarily constrainedstiffened panels with openingsrdquo International Journal ofNaval Architecture and Ocean Engineering vol 6 no 4pp 763ndash774 2014

[17] D S Cho B H Kim J-H Kim T M Choi and N VladimirldquoFree vibration analysis of stiffened panels with lumped massand stiffness attachmentsrdquo Ocean Engineering vol 124pp 84ndash93 2016

[18] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supportedrectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[19] X C Qin C Y Dong F Wang and X Y Qu ldquoStatic anddynamic analyses of isogeometric curvilinearly stiffenedplatesrdquoAppliedMathematical Modelling vol 45 pp 336ndash3642017

[20] X Yin W Wu H Li and K Zhong ldquoVibration transmissionwithin beam-stiffened plate structures using dynamic stiffnessmethodrdquo Procedia Engineering vol 199 pp 411ndash416 2017

[21] E Damnjanovic M Nefovska-Danilovic M Petronijevic andM Marjanovic ldquoApplication of the dynamic stiffness methodin the vibration analysis of stiffened composite platesrdquo Pro-cedia Engineering vol 199 pp 224ndash229 2017

[22] I E Harik andG L Salamoun ldquo+e analytical stripmethod ofsolution for stiffened rectangular platesrdquo Computers ampStructures vol 29 no 2 pp 283ndash291 1988

[23] L A Louca Y G Pan and J E Harding ldquoResponse ofstiffened and unstiffened plates subjected to blast loadingrdquoEngineering Structures vol 20 no 12 pp 1079ndash1086 1998

[24] A H Sheikh and M Mukhopadhyay ldquoLinear and nonlineartransient vibration analysis of stiffened plate structuresrdquo Fi-nite Elements in Analysis and Design vol 38 no 6pp 477ndash502 2002

[25] A K L Srivastava P K Datta and A H Sheikh ldquoDynamicinstability of stiffened plates subjected to non-uniform har-monic in-plane edge loadingrdquo Journal of Sound and Vibra-tion vol 262 no 5 pp 1171ndash1189 2003

[26] X D Xu H P Lee Y Y Wang and C Lu ldquo+e energy flowanalysis in stiffened plates of marine structuresrdquoLin-WalledStructures vol 42 no 7 pp 979ndash994 2004

[27] X D Xu H P Lee and C Lu ldquoPower flow paths in stiffenedplatesrdquo Journal of Sound and Vibration vol 282 no 3ndash5pp 1264ndash1272 2005

[28] A Borkovic N MrCa and S Kovacevic ldquoDynamical analysisof stiffened plates using the compound strip methodrdquo En-gineering Structures vol 50 no 3 pp 56ndash67 2013

[29] D S Cho B H Kim J-H Kim N Vladimir and T M ChoildquoForced vibration analysis of arbitrarily constrained rectan-gular plates and stiffened panels using the assumed modemethodrdquo Lin-Walled Structures vol 90 pp 182ndash190 2015

[30] R L Tian and P Jie ldquoA closed form solution for the dynamicresponse of finite ribbed platesrdquo Le Journal of the AcousticalSociety of America vol 119 no 2 pp 917ndash925 2006

[31] R L Tian ldquoAn analytical and experimental study of the vi-bration response of a clamped ribbed platerdquo Journal of Soundand Vibration vol 331 no 4 pp 902ndash913 2012

[32] T R Lin and K Zhang ldquoAn analytical study of the free andforced vibration response of a ribbed plate with free boundaryconditionsrdquo Journal of Sound and Vibration vol 422pp 15ndash33 2018

[33] D-S Cho T-M Choi J-H Kim and N Vladimir ldquoDom-inant components of vibrational energy flow in stiffenedpanels analysed by the structural intensity techniquerdquo


International Journal of Naval Architecture and Ocean En-gineering vol 10 no 5 pp 583ndash595 2018

[34] G Jia X Zhang C Wang Y He and X Chen ldquoPredictingdynamic response of stiffened-plate composite structures in awide-frequency domain based on composite B-spline waveletelements method (CBWEM)rdquo International Journal of Me-chanical Sciences vol 144 pp 708ndash722 2018

[35] C Liu J Zhang and F Li ldquoPower transmission and sup-pression characteristics of stiffened Mindlin plate underdifferent boundary constraintsrdquo Archive of Applied Me-chanics vol 89 no 9 pp 1705ndash1721 2019

[36] D Shi H Zhang Q Wang and S Zha ldquoFree and forcedvibration of the moderately thick laminated compositerectangular plate on various elastic winkler and pasternakfoundationsrdquo Shock and Vibration vol 2017 Article ID7820130 23 pages 2017

[37] Y Chen G Jin M Zhu Z Liu J Du andW L Li ldquoVibrationbehaviors of a box-type structure built up by plates and energytransmission through the structurerdquo Journal of Sound andVibration vol 331 no 4 pp 849ndash867 2012

[38] Y Chen G Jin J Du and Z Liu ldquoPower transmissionanalysis of coupled rectangular plates with elastically re-strained coupling edge including in-plane vibrationrdquo inProceedings of the 20th International Congress on Acoustics(ICA) Sydney Australia August 2010

[39] S Jiang W L Li T Yang and J Du ldquoFree vibration analysisof doubly curved shallow shells reinforced by any number ofbeams with arbitrary lengthsrdquo Journal of Vibration andControl vol 22 no 2 pp 570ndash584 2016

[40] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4pp 709ndash725 2000

[41] S Zhu G Jin Y Wang and X Ye ldquoA general Fourier for-mulation for vibration analysis of functionally graded sand-wich beams with arbitrary boundary condition and resting onelastic foundationsrdquo Acta Mechanica vol 227 no 5pp 1493ndash1514 2016

[42] X Shi D Shi W L Li and Q Wang ldquoA unified method forfree vibration analysis of circular annular and sector plateswith arbitrary boundary conditionsrdquo Journal of Vibration andControl vol 22 no 2 pp 442ndash456 2014

[43] J Du W L Li G Jin T Yang and Z Liu ldquoAn analyticalmethod for the in-plane vibration analysis of rectangularplates with elastically restrained edgesrdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 908ndash927 2007

[44] D Shi S Zha H Zhang and Q Wang ldquoFree vibrationanalysis of the unified functionally graded shallow shell withgeneral boundary conditionsrdquo Shock and Vibration vol 2017Article ID 7025190 19 pages 2017

[45] D-S Cho T-M Choi J-H Kim and N Vladimir ldquoStruc-tural intensity analysis of stepped thickness rectangular platesutilizing the finite element methodrdquo Lin-Walled Structuresvol 109 pp 1ndash12 2016

[46] M A Neto A Amaro L Roseiro J Cirne and R LealEngineering Computation of Structures Le Finite ElementMethod Springer Berlin Germany 2015


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On the basis of the first-order shear deformable theory Penget al [8] used mesh-free Galerkin method to carry out the freevibration and stability analyses of the stiffened plate On theframework of the assumed-modes method low-frequency freevibration analysis of the thin rectangular plate with a few lightstiffeners was made by Dozio and Ricciardi [9] Based on theimproved Fourier series method Xu et al [10] examined thefree vibration of the beam-stiffened rectangular plate wherebeams of arbitrary lengths and placement angles were takeninto consideration +ree different modelling approaches werereported by Bhar et al [11] to carry out the component-wisevibration analysis of the stiffened plate and comparativestudies between different models were also given +e staticfree vibration and buckling analysis of the stiffened plate werecarried out byNguyen-+oi et al [12] using CS-FEM-DSG3 InRef [13] the CS-FEM-DSG3 method was also extended toanalyze the static and free vibration of stiffened folded plateswhere three-node triangular element was adopted Four hi-erarchical models were developed by Bhaskar and Pydah [14]to obtain the natural frequencies of the isotropic and ortho-tropic stiffened plate with simply supported boundary con-dition On the basis of the elastic approach Pydah and Bhaskar[15] established an analytical model to investigate the staticsand dynamics of the stiffened plate with simply supported caseBased on the assumed mode method Cho et al [16] in-vestigated natural frequencies and modes of various openingstiffened panels under different boundary conditions wherethe effect of the parameters of the rib on the frequency waschecked+en he and other co-authors [17 18] carried out thefree vibration analysis of the stiffened panel with lumped massand stiffness attachments including rectangular plates withoutopening and with various opening shapes Utilizing theNURBS based on isogeometric analysis approach Qin et al[19] examined the frequency characteristics of curved stiffenedplate under in-plane loading Considering a beam-stiffenedplate Yin et al [20] examined its vibration transmission byapplying the dynamic stiffness method where the transmissionmode was presented Meanwhile Damnjanovic et al [21]adopted the same approach to perform the free vibrationanalysis of the stiffened composite plate based on the high-order shear deformation theory and the results were in goodagreement with the available data and FEM results

+rough the abovementioned review of literatures thepredecessors have conducted in-depth research on the freevibration of the stiffened plate Nevertheless many scholarsare not satisfied with research in the field of free vibrationHarik and Salamoun [22] applied an analytical strip methodto examine bending deflection of the stiffened plate wherethe type of load and the number of ribs were used to de-termine the number of strips Transient response of thesimply supported panel reinforced by stiffeners was pre-dicted by Louca et al [23] using the finite element techniquewhere the effect of boundary condition on transient responsewas considered Sheikh and Mukhopadhyay [24] applied thespline finite strip method to analyze the linear and nonlineartransient vibration of the stiffened plate with arbitraryshapes in which stiffeners had arbitrary orientation andeccentricity Based on finite element analysis Srivastava et al[25] presented the dynamic instability of the stiffened plate

where both in-plane partial and concentrated loadings at theedge were examined Xu et al [26] developed the structuralintensity technique to analyze energy flow for the stiffenedplate applied in the field of marine structures Later the sameauthors [27] investigated the structural intensity of thestiffened plate by means of the finite element method In thisresearch they draw a conclusion that the energy flow of thewhole structure was closely related to the natural frequencyEmploying the harmonic compound strip method Borkovicet al [28] obtained the transient response of the beam-stiffened plate where BernoullindashEuler beam andKirchhoffndashLove thin plate were both applied Combining theMindlin plate theory and Timoshenko beam theory Choet al [29] extended the assumed mode method to predictforced vibration of the stiffened plate and rectangular platewith various boundary restrains Tian and Jie [30] carriedout the research on the dynamic response of the finite ribbedplate under point force moment excitations and he con-cluded that the input mobilities were affected by the distancebetween the excitation point and the beam +en he [31]performed the vibration response analysis and experiment ofthe stiffened plates subjected to the clamped support at edgeIn 2018 he [32] introduced a double cosine integraltransform technique to study the free vibration and forcedresponse of the stiffened plate with free boundary restrainsMeanwhile the energy flow of stiffened panels was reportedby Cho et al [33] where the structural intensity techniquewas applied In order to efficiently solve the dynamic re-sponses of the stiffened plate in wide frequency domain Jiaet al [34] proposed a Composite B-spline Wavelet ElementsMethod (CBWEM) and verified the reliability of the methodby comparison with experimental results Applying theMindlin plate and Timoshenko beam theory Liu et al [35]analyzed the dynamic power transmission characteristics ofa finite stiffened plate with various classical cases where theeffects number and geometric dimensions on the stop bandin the low-frequency domain were shown

+rough the review of the abovementioned literature itcan be found that the existing researches are limited to thedynamic of beam-stiffened plate with classical boundaryconditions (like clamped simply supported and free) Onlythe free vibration analysis related to elastic boundary con-ditions and nonfixed connections between a plate and beamsis carried out in Xursquos et al [10] research However theinvestigations on the mobility characteristic structural in-tensity and transient response analysis of the stiffened platesubjected to the elastic boundary are still blank In an at-tempt to fill this gap the current work presents the dynamicanalysis of the rectangular beam-reinforced plate withgeneral boundary restraints including classical and theelastic restraints as well as elastic coupling of beams andplate IFSM is extended to construct displacement field of thebeam-stiffened plate which has proven to be a very effectivemethod from existing research [10 36ndash38] +e artificialspring technology is introduced to obtain general boundaryconditions and various coupling connections between aplate and beams Based on Xursquos et al [10] research theenergy expressions of the coupling system is obtained whichcan be solved by the RayleighndashRitz method Transient








Vp Voutp + V

inp (1)

in which

Voutp

DP

2B

S

z2w

zx21113888 1113889

2

+z2w

zy21113888 1113889

2

+ 2μp

z2w

zx2z2w

zy2⎛⎝

+ 2 1 minus μp1113872 1113873z2w

zx zy1113888 1113889

2⎞⎠dS

(2)

Vinp

Gp

2B

S

zu

zx+

zv

zy1113888 1113889

2


zx

zv

zy

⎧⎨

⎩

+1 minus μp

2zv

zx+

zu

zy1113888 1113889

2⎫⎬

⎭dS

(3)


Tp ρph

2B

S

zw

zt1113888 1113889

2

+zu

zt1113888 1113889

2

+zv

zt1113888 1113889

2⎡⎣ ⎤⎦dS

ρphω2

2B

Sw

2+ u

2+ v

21113960 1113961dS

(4)



Vsp 12

1113946b

0kwx0w

2+ Kwx0

zw

zx1113888 1113889

2⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868x0dy +

12

1113946b

0kwxaw

2+ Kwxa

zw

zx1113888 1113889

2⎡⎣ ⎤⎦

xa

dy

+12

1113946a

0kwy0w

2+ Kwy0

zw

zy1113888 1113889

2⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868y0dy +

12

1113946a

0kwybw

2+ Kwyb

zw

zy1113888 1113889

2⎡⎣ ⎤⎦

yb

dy

+12

1113946b

0kux0u

2+ kvx0v

21113960 1113961

11138681113868111386811138681113868x0dy +

12

1113946b

0kuxau

2+ kvxav

21113960 1113961

xady

+12

1113946a

0kvy0v

2+ kuy0u

21113960 1113961

11138681113868111386811138681113868y0dy +

12

1113946a

0kvybv

2+ kuybu

21113960 1113961

ybdy

(5)



Vbi 12intLi

0Dzbi

z2wzbizxprime

2( )2

+Dybi

z2wybizxprime

2( )2


2


2dxprime

(6)


Tbi ρbi2intLi

0Abi

zwzbizt

( )2

+ Abizwybizt

( )2

+ Abizubizt

( )2

+ Jbizθbizt

( )2dxprime

ρbiω2

2intLi

0Abi w

zbi( )2 + Abi w

ybi( )

2[


(7)



VpbCi 12intLi

0kpb1 w minus w


ybi( )

2[


+12intLi

0Kpb1 wpbix minus

zwzbizxprime

( )2

+Kpb212

zv

zxminuszu

zy( ) minus

zwybizxprime

( )2


2]dxprime

(8)


ybi are shown as

follows

wpbix zw

zxcosφ +

zw

zysinφ

wpbiy minuszw

zxsinφ +

zw

zycosφ

(9)


z

O

b

a

bb

hb

h

kpb2 kpb1

kpb1

kw Kw

ku

kv

Middle surface

kpb3kpb3 kpb2y

w v

u

x

yprime oprime

zprime

xprime

(a)

y0

y1

y

yprime

φ

x0 x1

xprime

x

L

O

(b)



u(x y t) UAuejωt

v(x y t) VBvejωt

w(x y t) WCwejωt

(10)

in which

U V







⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11a)

W









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11b)

Au

Au00 Au

0n Au0N Au

mn AumN Au

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12a)

Bv

Bv00 Bv

0n Bv0N Bv

mn BvmN Bv

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12b)

Cw

Cw00 Cw

0n Cw0N Cw

mn CwmN Cw

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12c)




biDmiejωt

wy


biEmiejωt


jωt


jωt

(13)


where

Ψzbi Ψy



Ψubi Ψθ




(15b)



(15d)



L Vp + Vsp + 1113944N

i0Vbi + V

pb

Ci1113872 1113873 minus Tp + 1113944N

i0Tbi


(16)


Wexc BS

fuu + fvv + fww( 1113857dS (17)




K minus ω2M1113872 1113873 H⟶

F⟶

(19)


ATu BT

v CTw DT

mi ETmi FT

mi1113960

GTmi 1113859


F⟶

0 0 fwCTw 0 0 0 0 1113960 1113961

Tdenotes the exter-


M

Muu 0 0 0 0 0

0 Mvv 0 0 0 0

0 0 Mww 0 0 0

0 0 0 ⋱ 0 0

0 0 0 0 Mbi 0

0 0 0 0 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

K






Kubi1113858 1113859

T Kvbi1113858 1113859

T Kwbi1113858 1113859

T 0 Kbi 0⋮ ⋮ ⋮ ⋮ 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


H⟶

K minus ω2M1113872 1113873minus 1

F⟶

(22)


minus 1




Ynk Vk

Fn

(23)



|I(x y)|

Ix(x y)1113868111386811138681113868

11138681113868111386811138682

+ Iy(x y)11138681113868111386811138681113868

111386811138681113868111386811138682

1113970

(24)



xuv(x y) (25a)


yuv(x y) (25b)


Ixw(x y) minus

12Re Qx

zw

zt1113888 1113889

lowast

minus Mx

z2w

zx zt1113888 1113889

lowast

minus Mxy

z2w

zy zt1113888 1113889

lowast

1113896 1113897

(26a)

Iyw(x y) minus

12Re Qy

zw

zt1113888 1113889

lowast

minus My

z2w

zy zt1113888 1113889

lowast

minus Myx

z2w

zx zt1113888 1113889

lowast

1113896 1113897

(26b)


Qx minus Dp

z3w

zx3 +z3w

zx zy21113888 1113889

Mx minus Dp

z2w

zx2 + μp

z2w

zy21113888 1113889


z2w

zx zy

(27a)

Qy minus Dp

z3w

zy3 +z3w

zx2zy1113888 1113889

My minus Dp

z2w

zy2 + μp

z2w

zx21113888 1113889


z2w

zx zy

(27b)


Ixuv(x y) minus

12Re σx

zu

zt1113888 1113889

lowast

+ τxy

zv

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2p

zu

zx+ μ

zv

zy1113888 1113889

zu

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zv

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28a)

Iyuv(x y) minus

12Re σy

zv

zt1113888 1113889

lowast

+ τyx

zu

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2pμ

zu

zx+

zv

zy1113888 1113889

zv

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zu

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28b)













euroxn+1 1


1βΔt

_xn minus12β

minus 11113888 1113889 euroxn (32a)

_xn+1 c


c

β1113888 1113889 _xn + 1 minus

c

2β1113888 1113889Δt2 euroxn

(32b)


Kxn+1 Fn+1p (33)

in which

K K +1

βΔt2M

Fn+1p fn+1 + M

1βΔt2

xn +1

βΔt_xn +

12β

minus 11113888 1113889euroxn1113890 1113891

(34)














2

radicm bb




0 100 200 300 400

Vel

ocity

(dB

ref =

10ndash9

ms

)

Frequency (Hz)

M = N = 3M = N = 9M = N = 14


60

90

120

150

(a)

Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


0 100 200 300 400

Frequency (Hz)

60

90

120

150

(b)


b

Q

a

(a)

02b

a

b

02a

Ω

(b)

F0

t0O t

F

F(t) = F0

(c)







Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(b)


Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(a)V

eloc

ity (d

B re

f = 1

0ndash9 m

s)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(b)




2





0 200 400 600 800

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

Frequency (Hz)

None05

021

(a)

None05

021

0 200 400 600 800ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

Frequency (Hz)

(b)


0 100 200 300 400 500 600

Frequency (Hz)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

100 200 300 400 500 6000Frequency (Hz)

(b)








ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)





kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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








Vp Voutp + V

inp (1)

in which

Voutp

DP

2B

S

z2w

zx21113888 1113889

2

+z2w

zy21113888 1113889

2

+ 2μp

z2w

zx2z2w

zy2⎛⎝

+ 2 1 minus μp1113872 1113873z2w

zx zy1113888 1113889

2⎞⎠dS

(2)

Vinp

Gp

2B

S

zu

zx+

zv

zy1113888 1113889

2


zx

zv

zy

⎧⎨

⎩

+1 minus μp

2zv

zx+

zu

zy1113888 1113889

2⎫⎬

⎭dS

(3)


Tp ρph

2B

S

zw

zt1113888 1113889

2

+zu

zt1113888 1113889

2

+zv

zt1113888 1113889

2⎡⎣ ⎤⎦dS

ρphω2

2B

Sw

2+ u

2+ v

21113960 1113961dS

(4)



Vsp 12

1113946b

0kwx0w

2+ Kwx0

zw

zx1113888 1113889

2⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868x0dy +

12

1113946b

0kwxaw

2+ Kwxa

zw

zx1113888 1113889

2⎡⎣ ⎤⎦

xa

dy

+12

1113946a

0kwy0w

2+ Kwy0

zw

zy1113888 1113889

2⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868y0dy +

12

1113946a

0kwybw

2+ Kwyb

zw

zy1113888 1113889

2⎡⎣ ⎤⎦

yb

dy

+12

1113946b

0kux0u

2+ kvx0v

21113960 1113961

11138681113868111386811138681113868x0dy +

12

1113946b

0kuxau

2+ kvxav

21113960 1113961

xady

+12

1113946a

0kvy0v

2+ kuy0u

21113960 1113961

11138681113868111386811138681113868y0dy +

12

1113946a

0kvybv

2+ kuybu

21113960 1113961

ybdy

(5)



Vbi 12intLi

0Dzbi

z2wzbizxprime

2( )2

+Dybi

z2wybizxprime

2( )2


2


2dxprime

(6)


Tbi ρbi2intLi

0Abi

zwzbizt

( )2

+ Abizwybizt

( )2

+ Abizubizt

( )2

+ Jbizθbizt

( )2dxprime

ρbiω2

2intLi

0Abi w

zbi( )2 + Abi w

ybi( )

2[


(7)



VpbCi 12intLi

0kpb1 w minus w


ybi( )

2[


+12intLi

0Kpb1 wpbix minus

zwzbizxprime

( )2

+Kpb212

zv

zxminuszu

zy( ) minus

zwybizxprime

( )2


2]dxprime

(8)


ybi are shown as

follows

wpbix zw

zxcosφ +

zw

zysinφ

wpbiy minuszw

zxsinφ +

zw

zycosφ

(9)


z

O

b

a

bb

hb

h

kpb2 kpb1

kpb1

kw Kw

ku

kv

Middle surface

kpb3kpb3 kpb2y

w v

u

x

yprime oprime

zprime

xprime

(a)

y0

y1

y

yprime

φ

x0 x1

xprime

x

L

O

(b)



u(x y t) UAuejωt

v(x y t) VBvejωt

w(x y t) WCwejωt

(10)

in which

U V







⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11a)

W









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11b)

Au

Au00 Au

0n Au0N Au

mn AumN Au

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12a)

Bv

Bv00 Bv

0n Bv0N Bv

mn BvmN Bv

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12b)

Cw

Cw00 Cw

0n Cw0N Cw

mn CwmN Cw

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12c)




biDmiejωt

wy


biEmiejωt


jωt


jωt

(13)


where

Ψzbi Ψy



Ψubi Ψθ




(15b)



(15d)



L Vp + Vsp + 1113944N

i0Vbi + V

pb

Ci1113872 1113873 minus Tp + 1113944N

i0Tbi


(16)


Wexc BS

fuu + fvv + fww( 1113857dS (17)




K minus ω2M1113872 1113873 H⟶

F⟶

(19)


ATu BT

v CTw DT

mi ETmi FT

mi1113960

GTmi 1113859


F⟶

0 0 fwCTw 0 0 0 0 1113960 1113961

Tdenotes the exter-


M

Muu 0 0 0 0 0

0 Mvv 0 0 0 0

0 0 Mww 0 0 0

0 0 0 ⋱ 0 0

0 0 0 0 Mbi 0

0 0 0 0 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

K






Kubi1113858 1113859

T Kvbi1113858 1113859

T Kwbi1113858 1113859

T 0 Kbi 0⋮ ⋮ ⋮ ⋮ 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


H⟶

K minus ω2M1113872 1113873minus 1

F⟶

(22)


minus 1




Ynk Vk

Fn

(23)



|I(x y)|

Ix(x y)1113868111386811138681113868

11138681113868111386811138682

+ Iy(x y)11138681113868111386811138681113868

111386811138681113868111386811138682

1113970

(24)



xuv(x y) (25a)


yuv(x y) (25b)


Ixw(x y) minus

12Re Qx

zw

zt1113888 1113889

lowast

minus Mx

z2w

zx zt1113888 1113889

lowast

minus Mxy

z2w

zy zt1113888 1113889

lowast

1113896 1113897

(26a)

Iyw(x y) minus

12Re Qy

zw

zt1113888 1113889

lowast

minus My

z2w

zy zt1113888 1113889

lowast

minus Myx

z2w

zx zt1113888 1113889

lowast

1113896 1113897

(26b)


Qx minus Dp

z3w

zx3 +z3w

zx zy21113888 1113889

Mx minus Dp

z2w

zx2 + μp

z2w

zy21113888 1113889


z2w

zx zy

(27a)

Qy minus Dp

z3w

zy3 +z3w

zx2zy1113888 1113889

My minus Dp

z2w

zy2 + μp

z2w

zx21113888 1113889


z2w

zx zy

(27b)


Ixuv(x y) minus

12Re σx

zu

zt1113888 1113889

lowast

+ τxy

zv

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2p

zu

zx+ μ

zv

zy1113888 1113889

zu

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zv

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28a)

Iyuv(x y) minus

12Re σy

zv

zt1113888 1113889

lowast

+ τyx

zu

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2pμ

zu

zx+

zv

zy1113888 1113889

zv

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zu

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28b)













euroxn+1 1


1βΔt

_xn minus12β

minus 11113888 1113889 euroxn (32a)

_xn+1 c


c

β1113888 1113889 _xn + 1 minus

c

2β1113888 1113889Δt2 euroxn

(32b)


Kxn+1 Fn+1p (33)

in which

K K +1

βΔt2M

Fn+1p fn+1 + M

1βΔt2

xn +1

βΔt_xn +

12β

minus 11113888 1113889euroxn1113890 1113891

(34)














2

radicm bb




0 100 200 300 400

Vel

ocity

(dB

ref =

10ndash9

ms

)

Frequency (Hz)

M = N = 3M = N = 9M = N = 14


60

90

120

150

(a)

Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


0 100 200 300 400

Frequency (Hz)

60

90

120

150

(b)


b

Q

a

(a)

02b

a

b

02a

Ω

(b)

F0

t0O t

F

F(t) = F0

(c)







Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(b)


Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(a)V

eloc

ity (d

B re

f = 1

0ndash9 m

s)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(b)




2





0 200 400 600 800

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

Frequency (Hz)

None05

021

(a)

None05

021

0 200 400 600 800ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

Frequency (Hz)

(b)


0 100 200 300 400 500 600

Frequency (Hz)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

100 200 300 400 500 6000Frequency (Hz)

(b)








ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)





kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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



Vbi 12intLi

0Dzbi

z2wzbizxprime

2( )2

+Dybi

z2wybizxprime

2( )2


2


2dxprime

(6)


Tbi ρbi2intLi

0Abi

zwzbizt

( )2

+ Abizwybizt

( )2

+ Abizubizt

( )2

+ Jbizθbizt

( )2dxprime

ρbiω2

2intLi

0Abi w

zbi( )2 + Abi w

ybi( )

2[


(7)



VpbCi 12intLi

0kpb1 w minus w


ybi( )

2[


+12intLi

0Kpb1 wpbix minus

zwzbizxprime

( )2

+Kpb212

zv

zxminuszu

zy( ) minus

zwybizxprime

( )2


2]dxprime

(8)


ybi are shown as

follows

wpbix zw

zxcosφ +

zw

zysinφ

wpbiy minuszw

zxsinφ +

zw

zycosφ

(9)


z

O

b

a

bb

hb

h

kpb2 kpb1

kpb1

kw Kw

ku

kv

Middle surface

kpb3kpb3 kpb2y

w v

u

x

yprime oprime

zprime

xprime

(a)

y0

y1

y

yprime

φ

x0 x1

xprime

x

L

O

(b)



u(x y t) UAuejωt

v(x y t) VBvejωt

w(x y t) WCwejωt

(10)

in which

U V







⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11a)

W









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11b)

Au

Au00 Au

0n Au0N Au

mn AumN Au

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12a)

Bv

Bv00 Bv

0n Bv0N Bv

mn BvmN Bv

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12b)

Cw

Cw00 Cw

0n Cw0N Cw

mn CwmN Cw

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12c)




biDmiejωt

wy


biEmiejωt


jωt


jωt

(13)


where

Ψzbi Ψy



Ψubi Ψθ




(15b)



(15d)



L Vp + Vsp + 1113944N

i0Vbi + V

pb

Ci1113872 1113873 minus Tp + 1113944N

i0Tbi


(16)


Wexc BS

fuu + fvv + fww( 1113857dS (17)




K minus ω2M1113872 1113873 H⟶

F⟶

(19)


ATu BT

v CTw DT

mi ETmi FT

mi1113960

GTmi 1113859


F⟶

0 0 fwCTw 0 0 0 0 1113960 1113961

Tdenotes the exter-


M

Muu 0 0 0 0 0

0 Mvv 0 0 0 0

0 0 Mww 0 0 0

0 0 0 ⋱ 0 0

0 0 0 0 Mbi 0

0 0 0 0 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

K






Kubi1113858 1113859

T Kvbi1113858 1113859

T Kwbi1113858 1113859

T 0 Kbi 0⋮ ⋮ ⋮ ⋮ 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


H⟶

K minus ω2M1113872 1113873minus 1

F⟶

(22)


minus 1




Ynk Vk

Fn

(23)



|I(x y)|

Ix(x y)1113868111386811138681113868

11138681113868111386811138682

+ Iy(x y)11138681113868111386811138681113868

111386811138681113868111386811138682

1113970

(24)



xuv(x y) (25a)


yuv(x y) (25b)


Ixw(x y) minus

12Re Qx

zw

zt1113888 1113889

lowast

minus Mx

z2w

zx zt1113888 1113889

lowast

minus Mxy

z2w

zy zt1113888 1113889

lowast

1113896 1113897

(26a)

Iyw(x y) minus

12Re Qy

zw

zt1113888 1113889

lowast

minus My

z2w

zy zt1113888 1113889

lowast

minus Myx

z2w

zx zt1113888 1113889

lowast

1113896 1113897

(26b)


Qx minus Dp

z3w

zx3 +z3w

zx zy21113888 1113889

Mx minus Dp

z2w

zx2 + μp

z2w

zy21113888 1113889


z2w

zx zy

(27a)

Qy minus Dp

z3w

zy3 +z3w

zx2zy1113888 1113889

My minus Dp

z2w

zy2 + μp

z2w

zx21113888 1113889


z2w

zx zy

(27b)


Ixuv(x y) minus

12Re σx

zu

zt1113888 1113889

lowast

+ τxy

zv

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2p

zu

zx+ μ

zv

zy1113888 1113889

zu

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zv

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28a)

Iyuv(x y) minus

12Re σy

zv

zt1113888 1113889

lowast

+ τyx

zu

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2pμ

zu

zx+

zv

zy1113888 1113889

zv

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zu

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28b)













euroxn+1 1


1βΔt

_xn minus12β

minus 11113888 1113889 euroxn (32a)

_xn+1 c


c

β1113888 1113889 _xn + 1 minus

c

2β1113888 1113889Δt2 euroxn

(32b)


Kxn+1 Fn+1p (33)

in which

K K +1

βΔt2M

Fn+1p fn+1 + M

1βΔt2

xn +1

βΔt_xn +

12β

minus 11113888 1113889euroxn1113890 1113891

(34)














2

radicm bb




0 100 200 300 400

Vel

ocity

(dB

ref =

10ndash9

ms

)

Frequency (Hz)

M = N = 3M = N = 9M = N = 14


60

90

120

150

(a)

Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


0 100 200 300 400

Frequency (Hz)

60

90

120

150

(b)


b

Q

a

(a)

02b

a

b

02a

Ω

(b)

F0

t0O t

F

F(t) = F0

(c)







Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(b)


Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(a)V

eloc

ity (d

B re

f = 1

0ndash9 m

s)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(b)




2





0 200 400 600 800

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

Frequency (Hz)

None05

021

(a)

None05

021

0 200 400 600 800ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

Frequency (Hz)

(b)


0 100 200 300 400 500 600

Frequency (Hz)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

100 200 300 400 500 6000Frequency (Hz)

(b)








ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)





kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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


u(x y t) UAuejωt

v(x y t) VBvejωt

w(x y t) WCwejωt

(10)

in which

U V







⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11a)

W









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11b)

Au

Au00 Au

0n Au0N Au

mn AumN Au

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12a)

Bv

Bv00 Bv

0n Bv0N Bv

mn BvmN Bv

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12b)

Cw

Cw00 Cw

0n Cw0N Cw

mn CwmN Cw

MN



⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

T

(12c)




biDmiejωt

wy


biEmiejωt


jωt


jωt

(13)


where

Ψzbi Ψy



Ψubi Ψθ




(15b)



(15d)



L Vp + Vsp + 1113944N

i0Vbi + V

pb

Ci1113872 1113873 minus Tp + 1113944N

i0Tbi


(16)


Wexc BS

fuu + fvv + fww( 1113857dS (17)




K minus ω2M1113872 1113873 H⟶

F⟶

(19)


ATu BT

v CTw DT

mi ETmi FT

mi1113960

GTmi 1113859


F⟶

0 0 fwCTw 0 0 0 0 1113960 1113961

Tdenotes the exter-


M

Muu 0 0 0 0 0

0 Mvv 0 0 0 0

0 0 Mww 0 0 0

0 0 0 ⋱ 0 0

0 0 0 0 Mbi 0

0 0 0 0 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

K






Kubi1113858 1113859

T Kvbi1113858 1113859

T Kwbi1113858 1113859

T 0 Kbi 0⋮ ⋮ ⋮ ⋮ 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


H⟶

K minus ω2M1113872 1113873minus 1

F⟶

(22)


minus 1




Ynk Vk

Fn

(23)



|I(x y)|

Ix(x y)1113868111386811138681113868

11138681113868111386811138682

+ Iy(x y)11138681113868111386811138681113868

111386811138681113868111386811138682

1113970

(24)



xuv(x y) (25a)


yuv(x y) (25b)


Ixw(x y) minus

12Re Qx

zw

zt1113888 1113889

lowast

minus Mx

z2w

zx zt1113888 1113889

lowast

minus Mxy

z2w

zy zt1113888 1113889

lowast

1113896 1113897

(26a)

Iyw(x y) minus

12Re Qy

zw

zt1113888 1113889

lowast

minus My

z2w

zy zt1113888 1113889

lowast

minus Myx

z2w

zx zt1113888 1113889

lowast

1113896 1113897

(26b)


Qx minus Dp

z3w

zx3 +z3w

zx zy21113888 1113889

Mx minus Dp

z2w

zx2 + μp

z2w

zy21113888 1113889


z2w

zx zy

(27a)

Qy minus Dp

z3w

zy3 +z3w

zx2zy1113888 1113889

My minus Dp

z2w

zy2 + μp

z2w

zx21113888 1113889


z2w

zx zy

(27b)


Ixuv(x y) minus

12Re σx

zu

zt1113888 1113889

lowast

+ τxy

zv

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2p

zu

zx+ μ

zv

zy1113888 1113889

zu

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zv

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28a)

Iyuv(x y) minus

12Re σy

zv

zt1113888 1113889

lowast

+ τyx

zu

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2pμ

zu

zx+

zv

zy1113888 1113889

zv

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zu

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28b)













euroxn+1 1


1βΔt

_xn minus12β

minus 11113888 1113889 euroxn (32a)

_xn+1 c


c

β1113888 1113889 _xn + 1 minus

c

2β1113888 1113889Δt2 euroxn

(32b)


Kxn+1 Fn+1p (33)

in which

K K +1

βΔt2M

Fn+1p fn+1 + M

1βΔt2

xn +1

βΔt_xn +

12β

minus 11113888 1113889euroxn1113890 1113891

(34)














2

radicm bb




0 100 200 300 400

Vel

ocity

(dB

ref =

10ndash9

ms

)

Frequency (Hz)

M = N = 3M = N = 9M = N = 14


60

90

120

150

(a)

Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


0 100 200 300 400

Frequency (Hz)

60

90

120

150

(b)


b

Q

a

(a)

02b

a

b

02a

Ω

(b)

F0

t0O t

F

F(t) = F0

(c)







Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(b)


Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(a)V

eloc

ity (d

B re

f = 1

0ndash9 m

s)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(b)




2





0 200 400 600 800

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

Frequency (Hz)

None05

021

(a)

None05

021

0 200 400 600 800ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

Frequency (Hz)

(b)


0 100 200 300 400 500 600

Frequency (Hz)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

100 200 300 400 500 6000Frequency (Hz)

(b)








ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)





kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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


where

Ψzbi Ψy



Ψubi Ψθ




(15b)



(15d)



L Vp + Vsp + 1113944N

i0Vbi + V

pb

Ci1113872 1113873 minus Tp + 1113944N

i0Tbi


(16)


Wexc BS

fuu + fvv + fww( 1113857dS (17)




K minus ω2M1113872 1113873 H⟶

F⟶

(19)


ATu BT

v CTw DT

mi ETmi FT

mi1113960

GTmi 1113859


F⟶

0 0 fwCTw 0 0 0 0 1113960 1113961

Tdenotes the exter-


M

Muu 0 0 0 0 0

0 Mvv 0 0 0 0

0 0 Mww 0 0 0

0 0 0 ⋱ 0 0

0 0 0 0 Mbi 0

0 0 0 0 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

K






Kubi1113858 1113859

T Kvbi1113858 1113859

T Kwbi1113858 1113859

T 0 Kbi 0⋮ ⋮ ⋮ ⋮ 0 ⋱

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


H⟶

K minus ω2M1113872 1113873minus 1

F⟶

(22)


minus 1




Ynk Vk

Fn

(23)



|I(x y)|

Ix(x y)1113868111386811138681113868

11138681113868111386811138682

+ Iy(x y)11138681113868111386811138681113868

111386811138681113868111386811138682

1113970

(24)



xuv(x y) (25a)


yuv(x y) (25b)


Ixw(x y) minus

12Re Qx

zw

zt1113888 1113889

lowast

minus Mx

z2w

zx zt1113888 1113889

lowast

minus Mxy

z2w

zy zt1113888 1113889

lowast

1113896 1113897

(26a)

Iyw(x y) minus

12Re Qy

zw

zt1113888 1113889

lowast

minus My

z2w

zy zt1113888 1113889

lowast

minus Myx

z2w

zx zt1113888 1113889

lowast

1113896 1113897

(26b)


Qx minus Dp

z3w

zx3 +z3w

zx zy21113888 1113889

Mx minus Dp

z2w

zx2 + μp

z2w

zy21113888 1113889


z2w

zx zy

(27a)

Qy minus Dp

z3w

zy3 +z3w

zx2zy1113888 1113889

My minus Dp

z2w

zy2 + μp

z2w

zx21113888 1113889


z2w

zx zy

(27b)


Ixuv(x y) minus

12Re σx

zu

zt1113888 1113889

lowast

+ τxy

zv

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2p

zu

zx+ μ

zv

zy1113888 1113889

zu

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zv

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28a)

Iyuv(x y) minus

12Re σy

zv

zt1113888 1113889

lowast

+ τyx

zu

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2pμ

zu

zx+

zv

zy1113888 1113889

zv

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zu

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28b)













euroxn+1 1


1βΔt

_xn minus12β

minus 11113888 1113889 euroxn (32a)

_xn+1 c


c

β1113888 1113889 _xn + 1 minus

c

2β1113888 1113889Δt2 euroxn

(32b)


Kxn+1 Fn+1p (33)

in which

K K +1

βΔt2M

Fn+1p fn+1 + M

1βΔt2

xn +1

βΔt_xn +

12β

minus 11113888 1113889euroxn1113890 1113891

(34)














2

radicm bb




0 100 200 300 400

Vel

ocity

(dB

ref =

10ndash9

ms

)

Frequency (Hz)

M = N = 3M = N = 9M = N = 14


60

90

120

150

(a)

Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


0 100 200 300 400

Frequency (Hz)

60

90

120

150

(b)


b

Q

a

(a)

02b

a

b

02a

Ω

(b)

F0

t0O t

F

F(t) = F0

(c)







Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(b)


Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(a)V

eloc

ity (d

B re

f = 1

0ndash9 m

s)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(b)




2





0 200 400 600 800

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

Frequency (Hz)

None05

021

(a)

None05

021

0 200 400 600 800ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

Frequency (Hz)

(b)


0 100 200 300 400 500 600

Frequency (Hz)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

100 200 300 400 500 6000Frequency (Hz)

(b)








ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)





kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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



Ynk Vk

Fn

(23)



|I(x y)|

Ix(x y)1113868111386811138681113868

11138681113868111386811138682

+ Iy(x y)11138681113868111386811138681113868

111386811138681113868111386811138682

1113970

(24)



xuv(x y) (25a)


yuv(x y) (25b)


Ixw(x y) minus

12Re Qx

zw

zt1113888 1113889

lowast

minus Mx

z2w

zx zt1113888 1113889

lowast

minus Mxy

z2w

zy zt1113888 1113889

lowast

1113896 1113897

(26a)

Iyw(x y) minus

12Re Qy

zw

zt1113888 1113889

lowast

minus My

z2w

zy zt1113888 1113889

lowast

minus Myx

z2w

zx zt1113888 1113889

lowast

1113896 1113897

(26b)


Qx minus Dp

z3w

zx3 +z3w

zx zy21113888 1113889

Mx minus Dp

z2w

zx2 + μp

z2w

zy21113888 1113889


z2w

zx zy

(27a)

Qy minus Dp

z3w

zy3 +z3w

zx2zy1113888 1113889

My minus Dp

z2w

zy2 + μp

z2w

zx21113888 1113889


z2w

zx zy

(27b)


Ixuv(x y) minus

12Re σx

zu

zt1113888 1113889

lowast

+ τxy

zv

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2p

zu

zx+ μ

zv

zy1113888 1113889

zu

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zv

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28a)

Iyuv(x y) minus

12Re σy

zv

zt1113888 1113889

lowast

+ τyx

zu

zt1113888 1113889

lowast

1113896 1113897 minus12Re

Eph

1 minus μ2pμ

zu

zx+

zv

zy1113888 1113889

zv

zt1113888 1113889

lowast

+Eph

2 1 + μp1113872 1113873

zu

zy+

zv

zx1113888 1113889

zu

zt1113888 1113889

lowast⎧⎨

⎩

⎫⎬

⎭ (28b)













euroxn+1 1


1βΔt

_xn minus12β

minus 11113888 1113889 euroxn (32a)

_xn+1 c


c

β1113888 1113889 _xn + 1 minus

c

2β1113888 1113889Δt2 euroxn

(32b)


Kxn+1 Fn+1p (33)

in which

K K +1

βΔt2M

Fn+1p fn+1 + M

1βΔt2

xn +1

βΔt_xn +

12β

minus 11113888 1113889euroxn1113890 1113891

(34)














2

radicm bb




0 100 200 300 400

Vel

ocity

(dB

ref =

10ndash9

ms

)

Frequency (Hz)

M = N = 3M = N = 9M = N = 14


60

90

120

150

(a)

Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


0 100 200 300 400

Frequency (Hz)

60

90

120

150

(b)


b

Q

a

(a)

02b

a

b

02a

Ω

(b)

F0

t0O t

F

F(t) = F0

(c)







Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(b)


Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(a)V

eloc

ity (d

B re

f = 1

0ndash9 m

s)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(b)




2





0 200 400 600 800

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

Frequency (Hz)

None05

021

(a)

None05

021

0 200 400 600 800ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

Frequency (Hz)

(b)


0 100 200 300 400 500 600

Frequency (Hz)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

100 200 300 400 500 6000Frequency (Hz)

(b)








ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)





kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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








euroxn+1 1


1βΔt

_xn minus12β

minus 11113888 1113889 euroxn (32a)

_xn+1 c


c

β1113888 1113889 _xn + 1 minus

c

2β1113888 1113889Δt2 euroxn

(32b)


Kxn+1 Fn+1p (33)

in which

K K +1

βΔt2M

Fn+1p fn+1 + M

1βΔt2

xn +1

βΔt_xn +

12β

minus 11113888 1113889euroxn1113890 1113891

(34)














2

radicm bb




0 100 200 300 400

Vel

ocity

(dB

ref =

10ndash9

ms

)

Frequency (Hz)

M = N = 3M = N = 9M = N = 14


60

90

120

150

(a)

Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


0 100 200 300 400

Frequency (Hz)

60

90

120

150

(b)


b

Q

a

(a)

02b

a

b

02a

Ω

(b)

F0

t0O t

F

F(t) = F0

(c)







Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(b)


Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(a)V

eloc

ity (d

B re

f = 1

0ndash9 m

s)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(b)




2





0 200 400 600 800

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

Frequency (Hz)

None05

021

(a)

None05

021

0 200 400 600 800ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

Frequency (Hz)

(b)


0 100 200 300 400 500 600

Frequency (Hz)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

100 200 300 400 500 6000Frequency (Hz)

(b)








ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)





kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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




2

radicm bb




0 100 200 300 400

Vel

ocity

(dB

ref =

10ndash9

ms

)

Frequency (Hz)

M = N = 3M = N = 9M = N = 14


60

90

120

150

(a)

Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


0 100 200 300 400

Frequency (Hz)

60

90

120

150

(b)


b

Q

a

(a)

02b

a

b

02a

Ω

(b)

F0

t0O t

F

F(t) = F0

(c)







Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(b)


Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(a)V

eloc

ity (d

B re

f = 1

0ndash9 m

s)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(b)




2





0 200 400 600 800

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

Frequency (Hz)

None05

021

(a)

None05

021

0 200 400 600 800ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

Frequency (Hz)

(b)


0 100 200 300 400 500 600

Frequency (Hz)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

100 200 300 400 500 6000Frequency (Hz)

(b)








ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)





kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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






Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

60

90

120

150

100 200 300 400 500 6000Frequency (Hz)

(b)


Vel

ocity

(dB

ref =

10ndash9

ms

)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(a)V

eloc

ity (d

B re

f = 1

0ndash9 m

s)

M = N = 3M = N = 9M = N = 14


30

60

90

120

100 200 300 400 5000Frequency (Hz)

(b)




2





0 200 400 600 800

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

Frequency (Hz)

None05

021

(a)

None05

021

0 200 400 600 800ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

Frequency (Hz)

(b)


0 100 200 300 400 500 600

Frequency (Hz)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

100 200 300 400 500 6000Frequency (Hz)

(b)








ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)





kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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



2





0 200 400 600 800

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

Frequency (Hz)

None05

021

(a)

None05

021

0 200 400 600 800ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

Frequency (Hz)

(b)


0 100 200 300 400 500 600

Frequency (Hz)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

(a)


Vel

ocity

(dB

ref =

10ndash9

ms

)

30

60

90

120

100 200 300 400 500 6000Frequency (Hz)

(b)








ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)





kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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







ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20M

obili

ty (d

B)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

200 400 600 8000Frequency (Hz)

(a)

Mob

ility

(dB)

E2SE2SE5E5E5E5

E2E5E2E5

CE5CE5

ndash120

ndash100

ndash80

ndash60

ndash40

200 400 600 8000Frequency (Hz)

(b)





kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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




kb = 103Nm kb = 105Nm kb = 1012Nm

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

kb = 103Nm kb = 105Nm kb = 1012Nm

100 200 300 400 5000Frequency (Hz)

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)


η = 0η = 004

η = 002η = 006

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

Mob

ility

(dB)

100 200 300 400 5000Frequency (Hz)

(a)

η = 0η = 004

η = 002η = 006

100 200 300 400 5000Frequency (Hz)

ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

Mob

ility

(dB)

(b)



lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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


lowast

lowastlowast

lowast

lowast

lowastlowast

lowast


(a)

lowast

lowast

lowast

lowast lowast

lowast lowast

lowast


(b)

Figure 11 Continued








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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








(c)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(d)







lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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






lowast

lowast

lowast

lowast

lowast

lowast

lowast


(a)

lowast

lowast

lowast

lowast

lowast

lowast

lowast

lowast


(b)






M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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





M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

3

6

9

12

15

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash8m

)

005 010 015 020000Time (s)

0

2

4

6

8

10

(b)


M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

ndash12

ndash06

00

06

12

18

24

005 010 015 020000Time (s)

(a)

M = N = 6M = N = 10M = N = 14


Disp

lace

men

t (times1

0ndash6m

)

005 010 015 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

(b)




radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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



radic hb1



0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

14

002 004 006 008 010000Time (s)

(a)

0deg30deg

15deg45deg

Disp

lace

men

t (times1

0ndash6m

)

00

02

04

06

08

10

12

002 004 006 008 010000Time (s)

(b)


FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash45

ndash36

ndash27

ndash18

ndash09

00

(a)

FEM resultsPresent

Disp

lace

men

t (times1

0ndash7m

)

005 010 015 020000Time (s)

ndash180

ndash135

ndash090

ndash045

000

(b)




4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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



4 Conclusions



Disp

lace

men

t (times1

0ndash8m

)

ndash10

0

10

20

30

40

50

60

005 010 015 020000Time (s)

α0 = 0 β0 = 0α0 = 01 β0 = 0001

α0 = 001 β0 = 0001α0 = 04 β0 = 0001

(a)D

ispla

cem

ent (

times10ndash8

m)

ndash20

0

20

40

60

004 008 012 016 020000Time (s)

α0 = 0 β0 = 0α0 = 001 β0 = 0002α0 = 001 β0 = 001

α0 = 001 β0 = 0001α0 = 001 β0 = 0004α0 = 001 β0 = 004

(b)




Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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



Data Availability




Disp

lace

men

t (times1

0ndash8m

)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(a)D

ispla

cem

ent (

times10ndash8

m)

kt = 102kt = 108

kt = 0

kt = 105kt = 1012

ndash2

ndash1

0

1

2

004 008 012 016 020000Time (s)

(b)

Disp

lace

men

t (times1

0ndash8m

)

kt = 108

kt = 105

kt = 1012kt = 1011

004 008 012 016 020000Time (s)

0

5

10

15

20

25

30

(c)

Disp

lace

men

t (times1

0ndash8m

)

kt = 105kt = 1012kt = 0

kt = 108

kt = 103

004 008 012 016 020000Time (s)

ndash3

ndash2

ndash1

0

1

2

3

4

5

6

7

(d)



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

DynamicAnalysisofRectangularPlateStiffenedbyAny ...downloads.hindawi.com/journals/sv/2019/2364515.pdf · analysis of the rectangular beam-reinforced plate with general boundary restraints,