research article approximate series solutions for nonlinear ...homotopy analysis method agree well...

Research ArticleApproximate Series Solutions for Nonlinear Free Vibration ofSuspended Cables

Yaobing Zhao1 Ceshi Sun1 Zhiqian Wang2 and Lianhua Wang1

1 College of Civil Engineering Hunan University Changsha Hunan 410082 China2 College of Mechanical and Vehicle Engineering Hunan University Changsha Hunan 410082 China

Correspondence should be addressed to Yaobing Zhao ybzhaohnueducn

Received 16 October 2013 Accepted 24 February 2014 Published 20 March 2014

Academic Editor Didier Remond

Copyright copy 2014 Yaobing Zhao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents approximate series solutions for nonlinear free vibration of suspended cables via the Lindstedt-Poincaremethodand homotopy analysis method respectively Firstly taking into account the geometric nonlinearity of the suspended cable as wellas the quasi-static assumption a mathematical model is presented Secondly two analytical methods are introduced to obtainthe approximate series solutions in the case of nonlinear free vibration Moreover small and large sag-to-span ratios and initialconditions are chosen to study the nonlinear dynamic responses by these two analytical methods The numerical results indicatethat frequency amplitude relationships obtained with different analytical approaches exhibit some quantitative and qualitativedifferences in the cases of motions mode shapes and particular sag-to-span ratios Finally a detailed comparison of the differencesin the displacement fields and cable axial total tensions is made

1 Introduction

As a basic and significant structural element the suspendedcable has been widely applied in many mechanical systemsand engineering fields [1 2] such as civil ocean andaerospace engineering Generally speaking the nonlineardynamics of the suspended cable is very complicated andattracts more andmore attention in recent years for examplereferring to the literature reviews by Rega [3 4]

The suspended cable is a typical weakly nonlinear con-tinuous system which contains the quadratic and cubicnonlinearity terms and the nonlinear free vibration of thesystem has been studied through many analytical methodsin the previous researches Recently Hagedorn and Schafer[5] investigated the nonlinear free vibrations of suspendedcables in the case of small sag via the Lindstedt methodLuongo et al [6 7] applied multiple scales method andLindstedt-Poincare method to study the nonlinear planarfree vibrations of an elastic cable respectively Rega et al[8] examined the nonlinear phenomenon in a large range ofthe cable sag-to-span ratios by the numerical investigationBenedettini et al [9] applied an order-three perturbation

expansion to obtain the solutions of the free nonplanarcoupled equations Srinil et al [10] presented a model toanalyze large amplitude free vibrations of the suspendedcable in three dimensions In these studies the perturbationmethod is the most significant analytical way to study thenonlinear vibration of the suspended cable However it isnoted that the perturbation method should be based on thesmall parameter assumption Because of the limitations inthe perturbationmethod its application is restricted more orless On the other hand due to the limitations of the perturba-tion method some nonperturbation methods are introducedto study the nonlinear free vibrations of suspended cables inthree dimensions for instance the harmonic balancemethod[11] Nevertheless the nonperturbation method could notprovide us with a way to guarantee the convergence of theseries solutions

Hence in order to overcome the limitations in boththe perturbation method and the nonperturbation methodLiao [12] proposed a general analytical method which isthe homotopy analysis method On the one hand it hasbeen successfully applied in many engineering fields withits further development in recent years [13] For example

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 795708 12 pageshttpdxdoiorg1011552014795708

2 Shock and Vibration

Hoseini et al [14] obtained accurate analytical results forthe nonlinear free vibration of a conservative oscillatorwith cubic nonlinearity by using homotopy analysis methodPirbodaghi et al [15] investigated the nonlinear vibrationbehavior of Euler-Bernoulli beams subjected to axial load viahomotopy analysis method Kargarnovin et al [16] appliedhomotopy analysis method to analyze the nonlinear freevibrations of the simple end beams Qian et al [17 18]employed homotopy analysis method to obtain approximatesolutions for an electrostatically actuated microbeam and anelastically restrained beam with a lumped mass respectivelyWu et al [19] contributed to the research of the nonlinearthickness-shear vibrations of a finite crystal plate with homo-topy analysis method

On the other hand there are some comparisons betweenthe homotopy analysis method and perturbation methodSpecifically Yuan and Li [20] found that the solutions ofhomotopy analysis method agree well with the results ofthe modified Linstedt-Poincare method and incrementalharmonic balance method for the primary resonance ofmultiple degree-of-freedom dynamic system with strongnonlinearity Comparisons were made between Adomianrsquosdecomposition method and homotopy analysis method byTan and Abbasbandy [21] You and Xu [22] studied theanalytical approximations for the periodic motion of theduffing system with delayed feedback via the homotopyanalysis method and multiple scales method and pointed outthat the results obtainedwith themultiple scalesmethodwereless accurate

To the best knowledge of the authors no specific studyhas addressed a comparison of homotopy analysis methodand other perturbation method in the case of nonlinearfree vibrations of the suspended cable However the homo-topy analysis method is different from those presented inthe papers published before on the same problem so thisresearch will focus on the comparison of these methodsThe paper consists of four sections firstly the nonlinear freevibration equations of motion are derived by applying theHomiltonrsquos principle and quasi-static assumption and thenthe multimode expansion of the displacement is introducedto obtain a discrete cable model Secondly the approximateseries solutions obtained with Lindstedt-Poincare methodand homotopy analysis method are constructed respectivelyMoreover in the section of the numerical analysis timehistories frequency amplitude curves displacement fieldsand axial tension forces of suspended cables are comparedand illustrated Finally some conclusions are made at the endof the paper

2 Mathematical Model

Figure 1 shows the coordinate system and two configurationsof the suspended cable (the initial deformed configurationof static equilibrium under its own weight and the dynamicconfiguration occupied during the vibration) As described inthis figure the left support 119874 is the origin of the coordinatethe direction 119874119861 is taken as the 119909-coordinate and thedirection perpendicular to119874119861 is the 119910-direction of which thedescending direction is taken as positive The displacements

u(x t)

(x t)

O

X

l

x

yb

B

Dynamic configurationStatic configuration

Figure 1 Two different configurations of the suspended cable

of the point are described by 119906(119909 119905) and V(119909 119905) along thelongitudinal 119909 and vertical 119910 directions respectively

21 Equations of Motion By applying the Hamiltonrsquos prin-ciple and the quasi-static stretching assumption we couldexpress the nonlinear partial differential equation of motionwithout considering the bending torsional and shear rigidi-ties as [3]

119898V minus 119867V10158401015840 minus119864119860

119897(11991010158401015840+ V10158401015840)int

119897

0

(1199101015840V1015840 +

1

2V10158402) d119909 = 0 (1)

where 119898 is the mass per unit length 119860 is the uniform cross-sectional area119864 is themodulus of elasticity119867 is the horizon-tal component of the tension (119867 = 119898119892119897

28119887 119867119864119860 ≪ 1) 119897 is

the span 119887 is the sag at the midspan and 119892 is the accelerationdue to gravity

The corresponding boundary conditions are written as

V (119909 119905) = 0 at 119909 = 0 119909 = 119897 (2)

In this study because the sag-to-span ratio is sufficientlysmall (119891 = 119887119897 lt 18) the static equilibrium configuration isdescribed well through a parabola

119910 (119909) = 4119891 [119909

119897minus (

119909

119897)

2

] (3)

22 Condensed Model In order to make the subsequent sec-tion more general the following nondimensional quantitiesare adopted [23]

Vlowast =V119897 119909

lowast=119909

119897 119910

lowast=119910

119897

119905lowast= radic

119892

8119891119905 120572 =

119864119860

119867

(4)

where the nondimensionalization with respect to the spanwhich affects the range of nondimensionalized responseamplitude is adopted

As a result (1) can be written as follows

119910 (119909) = 4119891 [119909

119897minus (

119909

119897)

2

] (5)

where the asterisks in (5) are omitted for simplicity 119910(119909) =4119891119909(1minus119909) is the nondimensional initial parabolic shape andthe boundary conditions associated with (5) are given as

V (119909 119905) = 0 at 119909 = 0 119909 = 1 (6)

Shock and Vibration 3

23 Mode Shapes and Frequencies Eliminating the nonlinearterms the mode shapes and frequencies can be ascertainedby solving linearized equation of motion Therefore the in-plane 119899th (119899 = odd) symmetric mode shapes are derived by

120593119899(119909) = 120585

119899[1 minus cos (120596

119899119909) minus tan(

120596119899

2) sin (120596

119899119909)]

(119899 = 1 3 5 )

(7)

where the coefficients 120585119899are derived by the normalization

conditions of modes and the mode frequencies 120596119899in (7) are

obtained by solving the following transcendental equation

tan(120596119899

2) =

120596119899

2minus

1

212058221205963

119899 (119899 = 1 3 5 ) (8)

where 1205822 = 119864119860119898119892119871(8119887119871)3 is the Irvine parameter which

is an important factor in the geometrical and mechanicalproperties of the suspended cable

On the other hand the 119899th (119899 = even) in-plane antisym-metric mode shapes and relative frequencies are

120593119899(119909) = radic2 sin (119899120587119909) 120596

119899= 119899120587 (119899 = 2 4 6 )

(9)

24 Discrete Model Assuming that the suspended cable isa multi-degree-of-freedom (MDOF) dynamic system whichis composed of symmetric and antisymmetric modes withrespect to the midspan the Galerkin method is employedto simplify the nonlinear oscillation equation of motionConsidering the boundary conditions the solutions of (5) areexpanded into the following expression

V (119909 119905) =119873

sum

119899=1

119902119899(119905) 120593119899(119909) (10)

where119873 is the number of modes used in the approximation(119873 = 1 2 infin) 119902

119899(119905) is an unknown function of timewhich

is a generalized coordinate of the system response and 120593119899(119909)

is a space coordinate function satisfying the associated linearproblem

Therefore a set of nonlinear ordinary differential equa-tions are yielded by substituting (10) into (5)

119902119899+ 1205962

119899119902119899+

119873

sum

119894=1

119873

sum

119895=1

Γ119899119894119895119902119894119902119895+

119873

sum

119894=1

119873

sum

119895=1

119873

sum

ℎ=1

Λ119899119894119895ℎ119902119894119902119895119902ℎ= 0 (11)

where the dots denote derivatives with respect to 119905 and theexpressions of the coefficients of the quadratic and cubicterms in (11) are as follows

Λ119899119894119895ℎ

= minus1

2120572int

1

0

[12059310158401015840

119895(119909) int

1

0

1205931015840

119894(119909) 1205931015840

ℎ(119909) d119909]120593

119899(119909) d119909

Γ119899119894119895= minus120572int

1

0

[12059310158401015840

119895(119909) int

1

0

1199101015840

(119909) 1205931015840

119894(119909) d119909]120593

119899(119909) d119909

minus1

2120572int

1

0

[11991010158401015840

(119909) int

1

0

1205931015840

119894(119909) 1205931015840

119894(119909) d119909]120593

119899(119909) d119909

(12)

3 Methods of Solution

This section begins with the approximate series solu-tions for the nonlinear free vibrations obtained with theLindstedt-Poincare method followed by the homotopy anal-ysis method For the sake of simplicity only single-modemodel (symmetric mode or antisymmetric mode) is consid-ered

31 Lindstedt-Poincare Method Firstly a new independentvariable is introduced which is

= 120596119899119905 (13)

where 120596119899is the linear frequency of the suspended cable

Therefore (11) is transformed into

119902119899+ 119902119899+Γ119899119899119899

1205962119899

1199022

119899+Λ119899119899119899119899

1205962119899

1199023

119899= 0 (14)

where the coefficient of the linear term is equal to unity sincethe time is nondimensionalized with respect to the linearvibration frequency

By assuming an expansion for 119902119899= 120576119902119899(where 120576 is a small

finite parameter) and omitting the tilde we obtain

119902119899+ 119902119899+ 120576

Γ119899119899119899

1205962119899

1199022

119899+ 1205762Λ119899119899119899119899

1205962119899

1199023

119899= 0 (15)

Following the method of Lindstedt-Poincare [7] we seekthe 4th order approximate solution to (15) by letting

119902119899( 120576) =

4

sum

119898=0

120576119898119902119899119898(119905) (16)

and a strained time coordinate 120591 is introduced

120591 = (1 +

4

sum

119898=1

120576119898120572119898) (17)

Then we could obtain the relation between the nonlinearfrequencyΩ

119899and the linear one 120596

119899as follows

Ω119899minus 120596119899

120596119899

=

4

sum

119898=1

120576119898120572119898 (18)

Substituting (16) and (17) into (15) and equating the coeffi-cients 120576119898 on both sides the nonlinear ordinary equations arereduced to a set of linearized equationsThen the polar formis introduced and the secular terms are set to zero Hence theseries solutions of 119902

119899119898and 120572

119898are obtained based on the fact


that the 4th order series solutions of the frequency amplituderelationship and displacement are as follows [7]

Ω119899= 120596119899[1 + (

3

8

Λ119899119899119899119899

1205962119899

minus5

12

Γ2

119899119899119899

1205964119899

)1198862

+ (minus15

256

Λ2

119899119899119899119899

1205964119899

minus485

1728

Γ4

119899119899119899

1205968119899

+173

192

Λ119899119899119899119899

1205962119899

Γ2

119899119899119899

1205964119899

)1198864]

(19)

119902119899(119905) = 119886 cos (Ω119905 + 120573) + 1198862 [

Γ119899119899119899

61205962119899

cos (2Ω119905 + 2120573) minusΓ119899119899119899

21205962119899

]

+ 1198863(Γ2

119899119899119899

481205964119899

+Λ119899119899119899119899

321205962119899

) cos (3Ω119905 + 3120573)

+ 1198864[(minus

31

96

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

+59

432

Γ3

119899119899119899

1205966119899

)

times cos (2Ω119905 + 2120573)

+ (5

8

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

minus19

72

Γ3

119899119899119899

1205966119899

)

+ (1

432

Γ3

119899119899119899

1205966119899

+1

96

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

)

times cos (4Ω119905 + 4120573) ]

(20)

where 119886 is the actual nondimensional response amplitude120573 is the phase of the oscillation and 119905 is the actual timescale Moreover there is a drift term due to the quadraticnonlinearity in (20) indicating that the equilibrium positionis not at 119902 = 0

32 Homotopy Analysis Method In the following the non-linear free response of the suspended cable is explored byhomotopy analysis method which transforms a nonlinearproblem into an infinite number of linear problems with anembedding parameter 119902 that typically varies from 0 to 1

Introducing a new time scale 120591 = Ω119899119905 (Ω119899is the nonlinear

vibration frequency) and taking into account the quadraticnonlinear term we suppose that

119902119899(119905) = 119906

119899(120591) + 120575

119899 (21)

In (11) the initial conditions are assumed to be

119902119899(0) = 119887

1198990+ 120575119899 119902

119899(0) = 0 (22)

where 1198871198990is the initial condition and 120575

1198990is the nonzero equi-

librium position term due to the quadratic nonlinearity

Under the new time scale transformation the new formof (11) is

Ω2

119899119899(120591) + 120596

2

119899[119906119899(120591) + 120575

119899] + Γ119899119899119899[119906119899(120591) + 120575

119899]2

+ Δ119899119899119899119899

[119906119899(120591) + 120575

119899]3

= 0

(23)

where

119899(120591) =

d2119906119899(120591)

d1205912 (24)

Therefore the corresponding initial conditions are

119906119899(0) = 119887

1198990

119899(0) = 0 (25)

Given the fact that the free oscillations of a conservativesystem could be expressed by a series of periodic functionswhich satisfy the initial conditions

cos (119896120591) | 119896 = 1 2 3 (26)

the displacement solution of (23) can be expressed by

119906119899(120591) =

+infin

sum

119896=1

119862119899119896cos (119896120591) (27)

Considering the rule of solution expression and initialconditions in (25) the initial guess of 119906

119899(120591) is chosen as

1199061198990(120591) = 119887

1198990cos 120591 (28)

To construct the homotopy function one may define thelinear auxiliary operator as

L [Φ119899(120591 119902)] = 120596

2

1198990[1205972Φ119899(120591 119902)

1205971205912+ Φ119899(120591 119902)] (29)

which has the property

L (1198621sin 120591 + 119862

2cos 120591) = 0 (30)

for any integration constants 1198621and 119862

2

According to (23) we could define the nonlinear operatoras

N [Φ119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

= Ψ2

119899(119902) [

1205972Φ119899(120591 119902)

1205971205912] + 120596

2

119899[Φ119899(120591 119902) + Δ

119899(119902)]

+ Γ119899119899119899[Φ119899(120591 119902) + Δ

119899(119902)]2

+ Λ119899119899119899119899

[Φ119899(120591 119902) + Δ

119899(119902)]3

(31)

where the unknown function Φ119899(120591 119902) is a mapping of 119906

119899(120591)

and the unknown functions Ψ119899(119902) and Δ

119899(119902) are some kinds

of mapping of the unknown nonlinear frequencyΩ119899and the

equilibrium position 120575119899 respectively In accordance with the

homotopy analysis method we construct the zeroth orderdeformation equation as

(1 minus 119902)L [Φ119899(120591 119902) minus 119906

1198990(120591)]

= 119902ℎ119867 (120591)N [Φ119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

(32)


subjected to the initial conditions

Φ119899(0 119902) = 119887

1198990

120597Φ119899(120591 119902)

120597120591

100381610038161003816100381610038161003816100381610038161003816120591=0

= 0 (33)

where 119902 isin [0 1] is an embedding parameter ℎ = 0 is anauxiliary convergence control parameter119867(120591) = 0 is an aux-iliary function and L(N) is an auxiliary linear (nonlinear)operator

For the sake of simplicity we choose

119867(120591) = 1 (34)

Therefore with the increase of the embedding parameter119902 from 0 to 1 Φ

119899(120591 119902) varies continuously from the initial

guess 1199061198990(120591) to the exact solution 119906

119899(120591) so does Ψ

119899(119902) from

its initial frequency Ω1198990

to the nonlinear physical frequencyΩ119899 SimilarlyΔ

119899(119902) varies from the initial approximation 120575

1198990

to the equilibrium position 120575119899

By using the Taylor series expansion and considering thedeformation derivatives we will obtain

Φ119899(120591 119902) = 119906

1198990(120591) +

+infin

sum

119898=1

119906119899119898(120591) 119902119898

Δ119899(119902) = 120575

1198990+

+infin

sum

119898=1

120575119899119898119902119898 Ψ

119899(119902) = Ω

1198990+

+infin

sum

119898=1

Ω119899119898119902119898

(35)

where

119906119899119898(120591) =

1

119898

120597119898Φ119899(120591 119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

120575119899119898=

1

119898

120597119898Δ119899(119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

Ω119899119898=

1

119898

120597119898Ψ119899(119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

(36)

The ℎ is an important auxiliary parameter that determinesthe convergence for the system Furthermore given that theauxiliary parameter ℎ is properly chosen and all the seriessolutions are converging for 119902 = 1 the series solutions arewritten as

119906119899(120591) = 119906

1198990(120591) +

+infin

sum

119898=1

119906119899119898(120591) 120575

119899= 1205751198990+

+infin

sum

119898=1

120575119899119898

Ω119899= Ω1198990+

+infin

sum

119898=1

Ω119899119898

(37)

For the sake of brevity and simplicity the followingvectors are defined

U119898= 1199061198990(120591) 119906

1198991(120591) 119906

119899119898(120591)

Δ119899119898= 1205751198990 1205751198991 120575

119899119898

Ψ119899119898= Ω1198990 Ω1198991 Ω

119899119898

(38)

Differentiating the zeroth order deformation equation119898 times with respect to the embedding parameters 119902 then

dividing the resulting equations by 119898 and setting 119902 = 0 the119898th-order deformation equations are

L [119906119899119898(120591) minus 120594

119898119906119899119898minus1

(120591)] = ℎR119899119898(U119899119898minus1

Δ119899119898minus1

Ψ119899119898minus1

)

(39)

which is subjected to the initial conditions

119906119899119898(0) = 0

119899119898(0) = 0 (119898 ge 1) (40)

where

120594119898=

0 119898 le 1

1 119898 gt 1

R119899119898=

1

(119898 minus 1)

120597119898minus1N [Φ

119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

120597119902119898minus1

100381610038161003816100381610038161003816100381610038161003816119902=0

=

119898minus1

sum

119896=0

119896

sum

119901=0

Ω119899119901Ω119899119896minus119901

119899119898minus1minus119896

(120591)

+ 1205962

119899[119906119899119898minus1

(120591) + 120575119899119898minus1

]

+ Γ119899119899119899

119898minus1

sum

119896=0

[119906119899119896(120591) 119906119899119898minus1minus119896

(120591) + 120575119899119896120575119899119898minus1minus119896

+2119906119899119896(120591) 120575119899119898minus1minus119896

]

+ Λ119899119899119899119899

119898minus1

sum

119896=0

119896

sum

119901=0

119906119899119901(120591) 119906119899119896minus119901

(120591) 119906119899119898minus1minus119896

(120591)

+

119898minus1

sum

119896=0

119896

sum

119901=0

120575119899119901120575119899119896minus119901

120575119899119898minus1minus119896

+ 3Λ119899119899119899119899

119898minus1

sum

119896=0

119896

sum

119901=0

119906119899119901(120591) 119906119899119896minus119901

(120591) 120575119899119898minus1minus119896

+

119898minus1

sum

119896=0

119896

sum

119901=0

120575119899119901120575119899119896minus119901

119906119899119898minus1minus119896

(120591)

(41)

Moreover the right hand side of the119898th-order deforma-tion equation is expressed as

R119899119898(U119899119898minus1

Δ119899119898minus1

Ψ119899119898minus1

)

= 1198881198991198980

(Δ119899119898minus1

Ψ119899119898minus1

)

+

120583119898

sum

119896=1

119888119899119898119896

(Δ119899119898minus1

Ψ119899119898minus1

) cos (119896120591)

(42)

where 1198881198991198980

is the coefficient of the constant term 119888119899119898119896

isthe coefficient of cos(119896120591) and 120583

119898is the positive integer

dependent on order 119898 According to the property of theauxiliary linear operator L in order to avoid the constant


drift term and the secular terms 120591 cos 120591 their coefficients areset to zero

1198881198991198980

(Δ119899119898minus1

Ψ119899119898minus1

) = 0 1198881198991198981

(Δ119899119898minus1

Ψ119899119898minus1

) = 0

(119898 = 1 2 3 )

(43)

which provide us with two additional algebraic equations forsolvingΩ

119899119898minus1and 120575119899119898minus1

Consequently given the unknownfunctions (1205962

119899 Γ119899119899119899

Λ119899119899119899119899

and 1198871198990) one can calculate the

periodic solutions 119906119899119898(120591) by solving the ordinary differential

equation with the corresponding boundary conditionsTherefore the general periodic solution 119906

119899119898(120591) of (39) is

obtained from

119906119899119898(120591) = 120594

119898119906119899119898minus1

(120591) + 1198621198991sin 120591 + 119862

1198992cos 120591

+ℎ

Ω2

1198990

120583119898

sum

119896=2

119888119899119898119896

(Δ119899119898minus1

Ψ119899119898minus1

)

(1 minus 1198962)cos (119896120591)

(44)

where 1198621198991

must be set to zero to obey the rule of solutionexpression and 119862

1198992is a constant that could be determined

by the initial conditions given by (40) Accordingly the119898th-order analytic approximate solutions of 120575

119899Ω119899 and 119906

119899(120591) are

119906119899(120591) asymp 119906

1198990(120591) +

119898

sum

119898=1

119906119899119898(120591) 120575

119899asymp 1205751198990+

119898

sum

119898=1

120575119899119898

Ω119899asymp Ω1198990+

119898

sum

119898=1

Ω119899119898

(45)

Here we take119898 = 1 for example in order to illustrate thecomputational process of homotopy analysis method In thiscase the right hand side of the 1st order deformation equationcould be expressed as

R1198991[1199061198990(120591) 1205751198990 Ω1198990]

= Ω2

11989901198990(120591) + 120596

2

119899[1199061198990(120591) + 120575

1198990] + Γ119899119899119899[1199061198990(120591) + 120575

1198990]2

+ Λ119899119899119899119899

[1199061198990(120591) + 120575

1198990]3

= [1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990] cos (120591)

+ [1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990] cos (2120591)

+ [1

41198873

1198990Λ119899119899119899119899

] cos (3120591)

+ (1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990

+Γ1198991198991198991205752

1198990+ Λ119899119899119899119899

1205753

1198990)

(46)

In order to satisfy the rule of solution expression the coef-ficients 119888

10(1205751198990 Ω1198990) and 119888

11(1205751198990 Ω1198990) must vanish There-

fore we get two additional algebraic equations aboutΩ1198990and

1205751198990

1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990= 0

1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990+ Γ1198991198991198991205752

1198990

+ Λ119899119899119899119899

1205753

1198990= 0

(47)

The solutions of (47) are

Ω1198990=1

2radic41205962119899+ 31198872

1198990Λ119899119899119899119899

+ 8Γ1198991198991198991205751198990+ 12Λ

1198991198991198991198991205752

1198990

1205751198990= minus

Γ119899119899119899

3Λ119899119899119899119899

minusΥ1198991

3Λ119899119899119899119899

3radic

2

Υ1198992+ radicΥ

2

1198992+ 32Υ

3

1198991

+1

6Λ119899119899119899119899

3

radicΥ1198992+ radicΥ

2

1198992+ 32Υ

3

1198991

2

(48)

whereΥ1198991= minus2Γ

2

119899119899119899+ 61205962

119899Λ119899119899119899119899

+ 91198872

1198990Λ2

119899119899119899119899

Υ1198992= minus16Γ

3

119899119899119899+ 72120596

2

119899Γ119899119899119899Λ119899119899119899119899

(49)

Eliminating the secular term and considering the expres-sion of linear operator the first order deformation equationbecomes

Ω2

1198990[1198991(120591) + 119906

1198991(120591)]

= ℎ [(1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (1

41198873

1198990Λ119899119899119899119899

) cos 3120591]

(50)

It is easy to solve the linear ordinary differential equationwith the initial conditions (119906

1198991(0) = 0

1198991(0) = 0) therefore

the first order approximation is

1199061198991(120591) =

ℎ

96Ω2

1198990

times [(161198872

1198990Γ119899119899119899

+ 31198873

1198990Λ119899119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 120591

minus (161198872

1198990Γ119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (31198873

1198990Λ119899119899119899119899

) cos 3120591] (51)

Following the same procedure the 119898th-order (119898 ge 2)approximation ofΩ

119899119898minus1 120575119899119898minus1

and 119906119899119898(120591) can be obtained

In general the first order approximation of 119902119899(119905) obtained

with the homotopy analysis method is expressed as follows

119902119899(119905) = 119906

1198990[(Ω1198990+ Ω1198991) 119905] + 119906

1198991[(Ω1198990+ Ω1198991) 119905]

+ (1205751198990+ 1205751198991)

(52)


40

30

20

10

0

minus3 minus2 minus1 0 1

Non

linea

r fre

quen

cyΩ

1

Convergent region

Auxiliary parameter ℏ

f = 0002 b10 = 0001f = 0020 b10 = 0010

f = 0040 b10 = 0020

f = 0080 b10 = 0040

Figure 2 Effect of the auxiliary parameter ℎ on the 1st symmetricmode frequency Ω

1obtained with the 5th order homotopy analysis

approximations for four different sag-to-span ratios 119891 and initialconditions 119887

10

Finally it should be pointed out that on the one handonly the zeroth order algebraic equations are nonlinear andall the higher order equations are linear On the other handcompared with the results obtained with Lindstedt-Poincaremethod ((19) and (20)) the zeroth order explicit expressionsof the nonlinear frequency and displacements (48) are muchmore complex Nevertheless with the aid of the computerthe homotopy analysis method still provides us with a veryconvenient way to obtain the higher order approximations

4 Numerical Results and Discussions

The dimensional parameters and material properties of thesuspended cable are chosen as follows [24] the area of thecross-section 119860 = 01257mm2 the mass per unit length119898 =

48655 times 10minus5 kgm the Young modulus 119864 = 134083MPa

and the cable span 119897 = 6005mm Moreover four differentsag-to-span ratios (119891 = 0002 002 004 and 008) arechosen to study the differences between these two analyticalapproaches in the case of the nonlinear free vibrations ofsuspended cables

41 Convergence and Accuracy of Solutions As mentionedin the previous section the auxiliary parameter ℎ playsan important role in the convergence for the approximateseries solutions obtainedwith the homotopy analysismethodFigure 2 shows the effect of the auxiliary parameter ℎ onthe 5th order series solutions for the 1st symmetric modefrequency Ω

1 In order to make the research less complex

as to every sag-to-span ratio 119891 only one initial conditionis selected (119887

10= 1198912) As indicated in Figure 2 there is a

convergent region (ℎ isin [minus20 0]) for the 5th order approx-imations Therefore the auxiliary parameter ℎ is chosen asminus10 in the following study

Nevertheless it should be mentioned that the conver-gence tests or proofs are significant and important for homo-topy analysis method Yet only several sag-to-span ratios

002

001

000

0 2 4 6 8

Time t

Numerical integrationsHomotopy analysis method

minus001

minus002

minus003A f = 0002 b10 = 0001 B f = 0020 b10 = 0010 C f = 0040 b10 = 0020

AB

C

q1(t)

Figure 3 Comparison of the series solution 1199021(119905) obtained with the

1st order homotopy analysis method and numerical integrations inthe case of the 1st symmetric mode

and initial conditions are involved and the convergentregions could not be checked one by one Moreover just asmentioned by Liao [13] it deserves to be further studied inwhich the auxiliary parameter ℎ and function 119867(120591) for anygiven nonlinear problem should be chosen Therefore thehomotopy analysis method needs further improvement anddevelopment in this respect

Once the auxiliary parameter ℎ is chosen appropriatelyapproximate series solutions in the case of nonlinear freevibrations could be obtained Moreover in order to verifythe approximations obtained with the analytical approacheswe could substitute the initial condition 119887

1198990and the equilib-

rium position 120575119899into the initial conditions (22) and then

the numerical integrations are applied to obtain the exactsolutions It is time consuming to obtain the results withregular numerical methods for the undamped periodic freeoscillations though The comparison of the series solution1199021(119905) obtained with the homotopy analysis method and

numerical integrations in the case of nonlinear free vibrationwith the 1st symmetric mode is made in Figure 3 It isnoted that the first order approximations obtained with thehomotopy analysis method are in good agreement with theexact ones obtained through numerical integrations in thesethree different cases

42 Frequency Amplitude Relationships Generally speakingboth the frequency amplitude relationship and the effect ofthe nonlinearities on the law of motion are two importantaspects that need to be examined and analyzed [4] Fur-thermore owing to large flexibility light weight and lowinherent damping of the suspended cable this system is oftensusceptible to exhibit large amplitude vibrations Thereforein the following both of these two aspects are investigatedand illustrated by using the homotopy analysis method andLindstedt-Poincare method In addition in order to clarifythe validity of the obtained results some numerical results ofthe original ordinary differential equation (ODE) are given


Table 1 Three nondimensional parameters of suspended cables

(a) (b) (c) (d)119891 0002 002 004 008120572 9418 94181 188361 3767231205822 (120582120587) 00024 (00156) 24110 (04945) 192882 (13987) 1543060 (39560)

006

004

002

000

000 001 002 003

Response amplitude a

f = 0002

1205822 = 00024a = 9418

A

B

CD

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(a)

000 001 002 003


minus05

15

10

05

00

A

B

CDf = 002

1205822 = 2411a = 94181

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(b)

000 001 002 003


A

B

CD

f = 004

1205822 = 192882a = 188361

Homotopy analysis methodLindstedt-Poincare methodRunge-Kutta method

minus1

3

2

1

0

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(c)

f = 008

1205822 = 15431a = 3767

000 001 002 003



A

B

C

D

Freq

uenc

y ra

tio (Ω

minus120596

)120596

4

2

0

minus2

(d)

Figure 4 Comparisons of frequency amplitude relationships obtained with the 1st order homotopy analysis method and the 4th orderLindstedt-Poincare method for the first four modes (a) 119891 = 0002 (b) 119891 = 002 (c) 119891 = 004 (d) 119891 = 008 A the 1st symmetric mode Bthe 1st antisymmetric mode C the 2nd symmetric mode and D the 2nd antisymmetric mode

At the beginning Table 1 illustrates the following threenondimensional parameters the sag-to-span ratio 119891 theIrvine parameter 1205822 and the nondimensional parameter120572(119864119860119867) As is shown in Table 1 with the rise of the sag-to-span ratio 119891 the Irvine parameter 1205822 increases very quicklyand the coefficients of the quadratic and cubic nonlinearitiesbecome very large tooTherefore in the case of the large sag-to-span ratios there is no small parameter in the equationof motion On the other hand it should be noticed thatthe hardening or softening characteristic of the suspendedcable is largely dependent on the predominance of either thequadratic or the cubic nonlinearity term

Figure 4 shows the frequency amplitude relationships ofsuspended cables under four different sag-to-span ratios (119891 =0002 002 004 and 008) for the first two symmetric andantisymmetric modes Moreover the nondimensionalization

of cable amplitude with respect to cable span is consid-ered (4) and this matter affects the range of the responseamplitude In fact as to the suspended cable no matter thesag-to-span ratio is large or small it is often susceptible toexhibit large response amplitude vibration so the range ofthe dimensional response amplitude is chosen as [0 003]for different sag-to-span ratios and vibration mode shapesNevertheless it should be explained that as to higher ordervibration mode shapes too large amplitude vibrations maynot be easy to exhibit

In Figure 4(a) we show the frequency amplitude rela-tionships of the suspended cable when the sag-to-span ratio119891 = 0002 In this case the suspended cable corresponds toa taut-string and the contribution of the cubic nonlinearityterm is dominant in the nonlinear responses Thereforeas shown in Figure 4(a) the suspended cable exhibits only


hardening behavior for both the symmetric modes (A andC) and antisymmetric modes (B and D) Moreover thefrequency amplitude relationships do not exhibit quantitativeand qualitative differences in the whole range of the responseamplitude and excellent agreements between the homotopyanalysis method and Lindstedt-Poincare method for the firstfour modes are presented in Figure 4(a) The numericalintegrations show that both of these two analytical methodsare appropriate in the case of the taut-string no matter theresponse amplitude is large or small

As the sag-to-span ratio increases (119891 = 002) inFigure 4(b) the contribution of the quadratic nonlinearityterm is still small when compared with that of the cubicone Therefore the suspended cable still exhibits only hard-ening behavior Furthermore according to the conclusionsobtained by Rega et al [8] the dynamic behavior of thesuspended cable when the sag-to-span ratio 119891 = 002 isstrictly hardening Nevertheless as described in Figure 4(b)although for the 1st symmetric and antisymmetric modes(A and B) there are not too many quantitative differencesbetween the results obtained with these two approacheswhereas for the 2nd symmetric and antisymmetric modes(C and D) the differences in the curves obtained by usingthese two analytical approaches increase with the rise of thevibration amplitude To be more specific the curves obtainedwith the Lindstedt-Poincare method start hardening andthey are in good agreement with the curves obtained withthe homotopy analysis method However as the responseamplitude increases the curves obtained with Lindstedt-Poincare method then become softening Hence providedthat the response amplitude of the suspended cable is largethe Lindstedt-Poincare method fails to reflect the character-istic of the suspended cable appropriately However as one ofthe typical perturbation methods too large amplitude valuesare likely considered for the Lindstedt-Poincare solutionshere which is known to hold only for small nonlinearitiesFurthermore as the vibration mode increases the modeshapes become more constrained and the agreement for thefirst mode (A and B) is good up to 0025 while the one forthe second modes (C and D) occurs up to considerably loweramplitude values with also a further decrease when passingfrom the 2nd symmetric (C) to the 2nd antisymmetric mode(D)

Figure 4(c) describes the frequency amplitude relation-ships of the suspended cable when the sag-to-span ratio119891 = 004 and excellent agreements between the solutions ofthe homotopy analysis method and numerical integrationsare presented both for the symmetric modes and anti-symmetric modes However compared with the Lindstedt-Poincare method there are some qualitative and quantitativedifferences between the analytical results Firstly the dynamicbehavior of the suspended cable is initially softening at alow value of the response amplitude of the 1st symmetricmode (A) but at a higher value of it because the cubicnonlinearity term may dominate the nonlinear free oscil-lations the dynamic behavior becomes hardening againProvided that the suspended cable vibrates at low valuesof the response amplitude (eg 119886 le 003) the differencesbetween these two methods could be neglected whereas

if the response amplitude of the suspended cable is largethere are quantitative differences between them Secondlyfor the 2nd symmetric mode (C) when the sag-to-span ratio119891 = 004 although the general trend of frequency amplituderelationship obtained with these two analytical approachesmakes no great differences with the increase of the responseamplitude the Lindstedt-Poincare method predicts morehardening behavior than the homotopy analysis methoddoesThirdly for the 1st and 2nd antisymmetricmodes (B andD) the coefficient of the quadratic nonlinearity term equalsto zero (Γ

119899119899119899equiv 0) Therefore the dynamic behavior of the

suspended cable is definitely hardening in the whole rangeof the vibration amplitude for any antisymmetric modes Asdescribed in Figure 4(c) both of the frequency amplituderelationships exhibit hardening behavior in the case of thesmall amplitude vibration Nevertheless as the responseamplitude increases the dynamic characteristic of the sus-pended cable obtained with the Lindstedt-Poincare methodbecomes softening Therefore as to the antisymmetric mode(119891 = 004) the scope of the response amplitude should beconsidered with care provided that the dynamic behavior ofthe suspended cable needs to be reflected accurately by usingthe Lindstedt-Poincare method Furthermore by comparingthe lower order modes (A and C) with the higher order ones(B and D) the accuracy of the solutions obtained with theLindstedt-Poincare decreases with the increase of the modeorder due to the more constrained mode shapes

Furthermore Figure 4(d) displays the frequency ampli-tude relationships of the suspended cable for the first twosymmetric and antisymmetric modes when the sag-to-spanratio increases continually to 008 In this case for the 1stsymmetric mode (A) the effect of the cubic nonlinearityterm plays a dominant role in the nonlinear vibration andthe characteristic of the results obtained with the homo-topy analysis method is hardening whereas the Lindstedt-Poincare method predicts more hardening behavior Besidesas to the 2nd symmetric mode (C) the suspended cableexhibits hardening or softening characteristic behavior whichis dependent on the vibration amplitude Although the gen-eral trend of the frequency amplitude relationship obtainedwith these two methods shows no great difference thesignificant quantitative difference could be observed for the2nd symmetric mode What is more as to the 1st and 2ndantisymmetric modes just as mentioned in the previousanalysis if the dynamic behavior of the suspended cableobtained with the Lindstedt-Poincare method needs to bereflected correctly the range of the response amplitude islimited especially for the higher order modes As illustratedin Figure 4(d) it is found that the range of agreement isnow larger for the first antisymmetric mode (B) than for thesymmetric one (A) because after the first crossover (120582120587 gt

20) the former is the one with the less constrained modeshape while the latter exhibits now three half waves which isdescribed in Figure 5(d)

43 Displacement Fields In the following a comparisonof displacement fields obtained with these two analyticalapproaches is made Substituting (7) (20) and (52) into(10) the expressions of the displacement field V(119909 119905) are


008

004

000

minus004

minus008

000 025 050 075 100

(xt)

x

t = 0

t = T8t = T4

t = 3T8

t = T2

(a)

010

005

000

minus005

minus010

(xt)

000 025 050 075 100

x

t = 0

t = T8 t = T4

t = 3T8

t = T2

(b)

01

minus02

00

minus01

minus03

(xt)

000 025 050 075 100

x

t = 0t = T8t = T4

t = 3T8t = T2

Homotopy analysis methodLindstedt-Poincare method

(c)

008

004

000

minus004

minus008

(xt)

000 025 050 075 100

x

t = 0t = T8 t = T4

t = 3T8

t = T2


(d)

Figure 5 In-plane displacement fields obtained with homotopy analysis method and Lindstedt-Poincare method in the case of the firstsymmetric mode (a) 119891 = 0002 (Ω

1minus 1205961)1205961= 005 (b) 119891 = 002 (Ω

1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 120596)120596

1= minus010 (d) 119891 = 008

(Ω1minus 1205961)1205961= 20

obtained Figure 5 shows the in-plane displacement fieldsobtained with the homotopy analysis method and Lindstedt-Poincare method in the case of the 1st symmetric mode Forthe sake of convenience only four displacement fields arechosen and illustrated As could be observed in Figure 5(a)the solutions obtained with Lindstedt-Poincare method arein good agreement with the ones obtained with homotopyanalysis method when 119891 = 0002 and (Ω

1minus 1205961)1205961= 005

Nevertheless in some other cases quantitative differencescould be observed (Figures 5(b) 5(c) and 5(d)) and thelocation of the maximum amplitude varies with the changeof the space and time

It should be mentioned that as to the first three casesthe Irvine parameter 120582120587 lt 2 and the first symmetric modefrequency is less than the first antisymmetric one Neverthe-less in the last case in our study the Irvine parameter 120582120587is chosen as 39560 (120582120587 gt 2) As shown in Figure 5(d) thefrequency of the first symmetric mode is larger than the oneof the first antisymmetric mode and the vertical componentof the first symmetric mode has two internal nodes andexhibits now three half waves

44 Axial Tension Forces In the fields of engineering thetension force of the suspended cable plays a very importantroleTherefore the cable total tension obtained with different

analytic methods is studied and analyzed in this sectionAccording to Srinil et al [10] the nondimensional cable totaltension is expressed as

119867119879(119905) = 1 + 120572119902

119899(119905) int

1

0

1199101015840

(119909) 1205931015840

119899(119909) d119909

+120572

21199022

119899(119905) int

1

0

12059310158402

119899(119909) d119909

(53)

where the quasi-static assumption is applied It is interestingto find out that the nondimensional cable total tension isindependent of the coordinate 119909

Figure 6 describes the time histories of the nondimen-sional cable tension force in the case of the first symmetricmode and no negative horizontal tension force is observedA good qualitative or quantitative agreement of the nondi-mensional cable total tension is presented in Figure 6(a)Figure 6(b) displays that the general trends of the timehistories obtained with different methods are the sameexcept the peak value of the cable tension force which isvery important in suspended cables designing to ensuresufficient security coefficient Moreover it is noted that inFigure 6(c) the tiny difference in the vibration amplitudemay lead to significant quantitative differences in cable total


115

110

105

100

0 5 10 15 20

Axi

al te

nsio

n fo

rceH

T

Time t

(a)

0 5 10 15

28

23

18

13

08

Axi

al te

nsio

n fo

rceH

T

Time t

(b)

40

30

20

10

0Axi

al te

nsio

n fo

rceH

T

minus100 5 10 15


Time t

(c)

Axi

al te

nsio

n fo

rceH

T

0

5

10

15

0 1 2 3

Time t


(d)

Figure 6 Timehistories of the nondimensional cable tension force for the case of the first symmetricmode (a)119891 = 0002 (Ω1minus1205961)1205961= 005

(b) 119891 = 002 (Ω1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 1205961)1205961= minus010 (d) 119891 = 008 (Ω

1minus 1205961)1205961= 20

tension Finally in Figure 6(d) one of the evident differencesis the time of the minimum value of the cable total tension

5 Conclusions

In this research the nonlinear free vibrations of the single-mode model of the suspended cable are studied via theLindstedt-Poincare method homotopy analysis method andnumerical integrations and only the first two symmetricand antisymmetric modes are considered Moreover thenumerical results and discussions are extended from a tautstring (119891 = 0002) to a slack cable (119891 = 008)

The homotopy analysis method does not depend onany small parameter assumption and provides us with aconvenient way to ensure the convergence for the seriessolutions It is found that above a certain value of responseamplitude it still continues to agree well with the resultsof the numerical integrations of the ODEs whereas theLindstedt-Poincare solutions partly fail On the one hand inthe case of the taut string (small sag-to-span ratio) these twoanalytical methodsmake no difference On the other hand asto the high order vibrationmodes large response amplitudesand sag-to-span ratios these two approaches may lead tosome quantitative and qualitative differences in the frequencyamplitude relationships However the results obtained with

the homotopy analysis method are in good agreement withthe ones obtained by using numerical integrations in thewhole range of response amplitude Furthermore the homo-topy analysis method has an advantage over the original onein the accuracy of the estimation of the displacement fieldsand second order harmonic to the time history of the cableaxial tension

Conflict of Interests

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

Acknowledgments

The work was supported by the National Natural ScienceFoundation of China (nos 11032004 and 11102063) Theauthors would like to thank the anonymous reviewers fortheir constructive comments and suggestions on the earlierversion of this paper

References

[1] H M Irvine Cable Structures The MIT Press CambridgeMass USA 1981


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

[3] G Rega ldquoNonlinear vibrations of suspended cables Part Imodeling and analysisrdquo Applied Mechanics Reviews vol 57 no1ndash6 pp 443ndash478 2004

[4] G Rega ldquoNonlinear vibrations of suspended cables Part IIdeterministic phenomenardquo Applied Mechanics Reviews vol 57no 1ndash6 pp 479ndash514 2004

[5] P Hagedorn and B Schafer ldquoOn non-linear free vibrations ofan elastic cablerdquo International Journal of Non-Linear Mechanicsvol 15 no 4-5 pp 333ndash340 1980

[6] A Luongo G Rega and F Vestroni ldquoMonofrequent oscillationsof a non-linear model of a suspended cablerdquo Journal of Soundand Vibration vol 82 no 2 pp 247ndash259 1982

[7] A Luongo G Rega and F Vestroni ldquoPlanar non-linear freevibrations of an elastic cablerdquo International Journal of Non-Linear Mechanics vol 19 no 1 pp 39ndash52 1984

[8] G Rega F Vestroni and F Benedettini ldquoParametric analysis oflarge amplitude free vibrations of a suspended cablerdquo Interna-tional Journal of Solids and Structures vol 20 no 2 pp 95ndash1051984

[9] F Benedettini G Rega and F Vestroni ldquoModal coupling in thefree nonplanar finite motion of an elastic cablerdquoMeccanica vol21 no 1 pp 38ndash46 1986

[10] N Srinil G Rega and S Chucheepsakul ldquoThree-dimensionalnon-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cablesrdquo Journalof Sound and Vibration vol 269 no 3ndash5 pp 823ndash852 2004

[11] K Takahashi and Y Konishi ldquoNon-linear vibrations of cables inthree dimensions Part I non-linear free vibrationsrdquo Journal ofSound and Vibration vol 118 no 1 pp 69ndash84 1987

[12] S J Liao The proposed homotopy analysis techniques for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[13] S J Liao Beyond Perturbation Introduction to the homotopyAnalysis Method Chapman and HallCRC Press Boca RatonFla USA 2003

[14] S H Hoseini T Pirbodaghi M Asghari G H Farrahi andM T Ahmadian ldquoNonlinear free vibration of conservativeoscillators with inertia and static type cubic nonlinearities usinghomotopy analysis methodrdquo Journal of Sound and Vibrationvol 316 no 1ndash5 pp 263ndash273 2008

[15] T Pirbodaghi M T Ahmadian and M Fesanghary ldquoOn thehomotopy analysis method for non-linear vibration of beamsrdquoMechanics Research Communications vol 36 no 2 pp 143ndash1482009

[16] M H Kargarnovin and R A Jafari-Talookolaei ldquoApplicationof the homotopy method for the analytic approach of thenonlinear free vibration analysis of the simple end beams usingfour engineering theoriesrdquoActaMechanica vol 212 no 3-4 pp199ndash213 2010

[17] Y H Qian D X Ren S K Lai and S M Chen ldquoAnalyticalapproximations to nonlinear vibration of an electrostaticallyactuated microbeamrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 4 pp 1947ndash1955 2012

[18] Y H Qian W Zhang B W Lin and S K Lai ldquoAnalyti-cal approximate periodic solutions for two-degree-of-freedomcoupled van der Pol-Duffing oscillators by extended homotopyanalysismethodrdquoActaMechanica vol 219 no 1-2 pp 1ndash14 2011

[19] RWu JWang J Du Y Hu andH Hu ldquoSolutions of nonlinearthickness-shear vibrations of an infinite isotropic plate with thehomotopy analysis methodrdquo Numerical Algorithms vol 59 no2 pp 213ndash226 2012

[20] P-X Yuan and Y-Q Li ldquoPrimary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity usingthe homotopy analysis methodrdquo Applied Mathematics andMechanics vol 31 no 10 pp 1293ndash1304 2010

[21] Y Tan and S Abbasbandy ldquoHomotopy analysis method forquadratic Riccati differential equationrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 3 pp539ndash546 2008

[22] X You and H Xu ldquoAnalytical approximations for the periodicmotion of theDuffing systemwith delayed feedbackrdquoNumericalAlgorithms vol 56 no 4 pp 561ndash576 2011

[23] Y Y Zhao L H Wang W C Liu and H B Zhou ldquoDirecttreatment and discretization of nonlinear dynamics of sus-pended cablerdquoActaMechanica Sinica vol 37 pp 329ndash338 2005(Chinese)

[24] G RegaW Lacarbonara A H Nayfeh andCM Chin ldquoMulti-ple resonances in suspended cables direct versus reduced-ordermodelsrdquo International Journal of Non-Linear Mechanics vol 34no 5 pp 901ndash924 1999

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Hoseini et al [14] obtained accurate analytical results forthe nonlinear free vibration of a conservative oscillatorwith cubic nonlinearity by using homotopy analysis methodPirbodaghi et al [15] investigated the nonlinear vibrationbehavior of Euler-Bernoulli beams subjected to axial load viahomotopy analysis method Kargarnovin et al [16] appliedhomotopy analysis method to analyze the nonlinear freevibrations of the simple end beams Qian et al [17 18]employed homotopy analysis method to obtain approximatesolutions for an electrostatically actuated microbeam and anelastically restrained beam with a lumped mass respectivelyWu et al [19] contributed to the research of the nonlinearthickness-shear vibrations of a finite crystal plate with homo-topy analysis method

On the other hand there are some comparisons betweenthe homotopy analysis method and perturbation methodSpecifically Yuan and Li [20] found that the solutions ofhomotopy analysis method agree well with the results ofthe modified Linstedt-Poincare method and incrementalharmonic balance method for the primary resonance ofmultiple degree-of-freedom dynamic system with strongnonlinearity Comparisons were made between Adomianrsquosdecomposition method and homotopy analysis method byTan and Abbasbandy [21] You and Xu [22] studied theanalytical approximations for the periodic motion of theduffing system with delayed feedback via the homotopyanalysis method and multiple scales method and pointed outthat the results obtainedwith themultiple scalesmethodwereless accurate

To the best knowledge of the authors no specific studyhas addressed a comparison of homotopy analysis methodand other perturbation method in the case of nonlinearfree vibrations of the suspended cable However the homo-topy analysis method is different from those presented inthe papers published before on the same problem so thisresearch will focus on the comparison of these methodsThe paper consists of four sections firstly the nonlinear freevibration equations of motion are derived by applying theHomiltonrsquos principle and quasi-static assumption and thenthe multimode expansion of the displacement is introducedto obtain a discrete cable model Secondly the approximateseries solutions obtained with Lindstedt-Poincare methodand homotopy analysis method are constructed respectivelyMoreover in the section of the numerical analysis timehistories frequency amplitude curves displacement fieldsand axial tension forces of suspended cables are comparedand illustrated Finally some conclusions are made at the endof the paper

2 Mathematical Model

Figure 1 shows the coordinate system and two configurationsof the suspended cable (the initial deformed configurationof static equilibrium under its own weight and the dynamicconfiguration occupied during the vibration) As described inthis figure the left support 119874 is the origin of the coordinatethe direction 119874119861 is taken as the 119909-coordinate and thedirection perpendicular to119874119861 is the 119910-direction of which thedescending direction is taken as positive The displacements

u(x t)

(x t)

O

X

l

x

yb

B

Dynamic configurationStatic configuration

Figure 1 Two different configurations of the suspended cable

of the point are described by 119906(119909 119905) and V(119909 119905) along thelongitudinal 119909 and vertical 119910 directions respectively

21 Equations of Motion By applying the Hamiltonrsquos prin-ciple and the quasi-static stretching assumption we couldexpress the nonlinear partial differential equation of motionwithout considering the bending torsional and shear rigidi-ties as [3]

119898V minus 119867V10158401015840 minus119864119860

119897(11991010158401015840+ V10158401015840)int

119897

0

(1199101015840V1015840 +

1

2V10158402) d119909 = 0 (1)

where 119898 is the mass per unit length 119860 is the uniform cross-sectional area119864 is themodulus of elasticity119867 is the horizon-tal component of the tension (119867 = 119898119892119897

28119887 119867119864119860 ≪ 1) 119897 is

the span 119887 is the sag at the midspan and 119892 is the accelerationdue to gravity

The corresponding boundary conditions are written as

V (119909 119905) = 0 at 119909 = 0 119909 = 119897 (2)

In this study because the sag-to-span ratio is sufficientlysmall (119891 = 119887119897 lt 18) the static equilibrium configuration isdescribed well through a parabola

119910 (119909) = 4119891 [119909

119897minus (

119909

119897)

2

] (3)

22 Condensed Model In order to make the subsequent sec-tion more general the following nondimensional quantitiesare adopted [23]

Vlowast =V119897 119909

lowast=119909

119897 119910

lowast=119910

119897

119905lowast= radic

119892

8119891119905 120572 =

119864119860

119867

(4)

where the nondimensionalization with respect to the spanwhich affects the range of nondimensionalized responseamplitude is adopted

As a result (1) can be written as follows

119910 (119909) = 4119891 [119909

119897minus (

119909

119897)

2

] (5)

where the asterisks in (5) are omitted for simplicity 119910(119909) =4119891119909(1minus119909) is the nondimensional initial parabolic shape andthe boundary conditions associated with (5) are given as

V (119909 119905) = 0 at 119909 = 0 119909 = 1 (6)



120593119899(119909) = 120585

119899[1 minus cos (120596

119899119909) minus tan(

120596119899

2) sin (120596

119899119909)]

(119899 = 1 3 5 )

(7)




tan(120596119899

2) =

120596119899

2minus

1

212058221205963

119899 (119899 = 1 3 5 ) (8)




120593119899(119909) = radic2 sin (119899120587119909) 120596

119899= 119899120587 (119899 = 2 4 6 )

(9)


V (119909 119905) =119873

sum

119899=1

119902119899(119905) 120593119899(119909) (10)






119902119899+ 1205962

119899119902119899+

119873

sum

119894=1

119873

sum

119895=1

Γ119899119894119895119902119894119902119895+

119873

sum

119894=1

119873

sum

119895=1

119873

sum

ℎ=1

Λ119899119894119895ℎ119902119894119902119895119902ℎ= 0 (11)


Λ119899119894119895ℎ

= minus1

2120572int

1

0

[12059310158401015840

119895(119909) int

1

0

1205931015840

119894(119909) 1205931015840

ℎ(119909) d119909]120593

119899(119909) d119909

Γ119899119894119895= minus120572int

1

0

[12059310158401015840

119895(119909) int

1

0

1199101015840

(119909) 1205931015840

119894(119909) d119909]120593

119899(119909) d119909

minus1

2120572int

1

0

[11991010158401015840

(119909) int

1

0

1205931015840

119894(119909) 1205931015840

119894(119909) d119909]120593

119899(119909) d119909

(12)




= 120596119899119905 (13)



119902119899+ 119902119899+Γ119899119899119899

1205962119899

1199022

119899+Λ119899119899119899119899

1205962119899

1199023

119899= 0 (14)




119902119899+ 119902119899+ 120576

Γ119899119899119899

1205962119899

1199022

119899+ 1205762Λ119899119899119899119899

1205962119899

1199023

119899= 0 (15)


119902119899( 120576) =

4

sum

119898=0

120576119898119902119899119898(119905) (16)


120591 = (1 +

4

sum

119898=1

120576119898120572119898) (17)



119899as follows

Ω119899minus 120596119899

120596119899

=

4

sum

119898=1

120576119898120572119898 (18)


119899119898and 120572




Ω119899= 120596119899[1 + (

3

8

Λ119899119899119899119899

1205962119899

minus5

12

Γ2

119899119899119899

1205964119899

)1198862

+ (minus15

256

Λ2

119899119899119899119899

1205964119899

minus485

1728

Γ4

119899119899119899

1205968119899

+173

192

Λ119899119899119899119899

1205962119899

Γ2

119899119899119899

1205964119899

)1198864]

(19)

119902119899(119905) = 119886 cos (Ω119905 + 120573) + 1198862 [

Γ119899119899119899

61205962119899

cos (2Ω119905 + 2120573) minusΓ119899119899119899

21205962119899

]

+ 1198863(Γ2

119899119899119899

481205964119899

+Λ119899119899119899119899

321205962119899

) cos (3Ω119905 + 3120573)

+ 1198864[(minus

31

96

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

+59

432

Γ3

119899119899119899

1205966119899

)

times cos (2Ω119905 + 2120573)

+ (5

8

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

minus19

72

Γ3

119899119899119899

1205966119899

)

+ (1

432

Γ3

119899119899119899

1205966119899

+1

96

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

)

times cos (4Ω119905 + 4120573) ]

(20)





119902119899(119905) = 119906

119899(120591) + 120575

119899 (21)


119902119899(0) = 119887

1198990+ 120575119899 119902

119899(0) = 0 (22)





Ω2

119899119899(120591) + 120596

2

119899[119906119899(120591) + 120575

119899] + Γ119899119899119899[119906119899(120591) + 120575

119899]2

+ Δ119899119899119899119899

[119906119899(120591) + 120575

119899]3

= 0

(23)

where

119899(120591) =

d2119906119899(120591)

d1205912 (24)


119906119899(0) = 119887

1198990

119899(0) = 0 (25)


cos (119896120591) | 119896 = 1 2 3 (26)


119906119899(120591) =

+infin

sum

119896=1

119862119899119896cos (119896120591) (27)


119899(120591) is chosen as

1199061198990(120591) = 119887

1198990cos 120591 (28)


L [Φ119899(120591 119902)] = 120596

2

1198990[1205972Φ119899(120591 119902)

1205971205912+ Φ119899(120591 119902)] (29)


L (1198621sin 120591 + 119862

2cos 120591) = 0 (30)


2


N [Φ119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

= Ψ2

119899(119902) [

1205972Φ119899(120591 119902)

1205971205912] + 120596

2

119899[Φ119899(120591 119902) + Δ

119899(119902)]

+ Γ119899119899119899[Φ119899(120591 119902) + Δ

119899(119902)]2

+ Λ119899119899119899119899

[Φ119899(120591 119902) + Δ

119899(119902)]3

(31)


119899(120591)






(1 minus 119902)L [Φ119899(120591 119902) minus 119906

1198990(120591)]

= 119902ℎ119867 (120591)N [Φ119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

(32)



Φ119899(0 119902) = 119887

1198990

120597Φ119899(120591 119902)

120597120591

100381610038161003816100381610038161003816100381610038161003816120591=0

= 0 (33)



119867(120591) = 1 (34)




119899(120591) so does Ψ

119899(119902) from




1198990



Φ119899(120591 119902) = 119906

1198990(120591) +

+infin

sum

119898=1

119906119899119898(120591) 119902119898

Δ119899(119902) = 120575

1198990+

+infin

sum

119898=1

120575119899119898119902119898 Ψ

119899(119902) = Ω

1198990+

+infin

sum

119898=1

Ω119899119898119902119898

(35)

where

119906119899119898(120591) =

1

119898

120597119898Φ119899(120591 119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

120575119899119898=

1

119898

120597119898Δ119899(119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

Ω119899119898=

1

119898

120597119898Ψ119899(119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

(36)


119906119899(120591) = 119906

1198990(120591) +

+infin

sum

119898=1

119906119899119898(120591) 120575

119899= 1205751198990+

+infin

sum

119898=1

120575119899119898

Ω119899= Ω1198990+

+infin

sum

119898=1

Ω119899119898

(37)


U119898= 1199061198990(120591) 119906

1198991(120591) 119906

119899119898(120591)

Δ119899119898= 1205751198990 1205751198991 120575

119899119898

Ψ119899119898= Ω1198990 Ω1198991 Ω

119899119898

(38)



L [119906119899119898(120591) minus 120594

119898119906119899119898minus1

(120591)] = ℎR119899119898(U119899119898minus1

Δ119899119898minus1

Ψ119899119898minus1

)

(39)


119906119899119898(0) = 0

119899119898(0) = 0 (119898 ge 1) (40)

where

120594119898=

0 119898 le 1

1 119898 gt 1

R119899119898=

1

(119898 minus 1)

120597119898minus1N [Φ

119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

120597119902119898minus1

100381610038161003816100381610038161003816100381610038161003816119902=0

=

119898minus1

sum

119896=0

119896

sum

119901=0

Ω119899119901Ω119899119896minus119901

119899119898minus1minus119896

(120591)

+ 1205962

119899[119906119899119898minus1

(120591) + 120575119899119898minus1

]

+ Γ119899119899119899

119898minus1

sum

119896=0

[119906119899119896(120591) 119906119899119898minus1minus119896

(120591) + 120575119899119896120575119899119898minus1minus119896

+2119906119899119896(120591) 120575119899119898minus1minus119896

]

+ Λ119899119899119899119899

119898minus1

sum

119896=0

119896

sum

119901=0

119906119899119901(120591) 119906119899119896minus119901

(120591) 119906119899119898minus1minus119896

(120591)

+

119898minus1

sum

119896=0

119896

sum

119901=0

120575119899119901120575119899119896minus119901

120575119899119898minus1minus119896

+ 3Λ119899119899119899119899

119898minus1

sum

119896=0

119896

sum

119901=0

119906119899119901(120591) 119906119899119896minus119901

(120591) 120575119899119898minus1minus119896

+

119898minus1

sum

119896=0

119896

sum

119901=0

120575119899119901120575119899119896minus119901

119906119899119898minus1minus119896

(120591)

(41)


R119899119898(U119899119898minus1

Δ119899119898minus1

Ψ119899119898minus1

)

= 1198881198991198980

(Δ119899119898minus1

Ψ119899119898minus1

)

+

120583119898

sum

119896=1

119888119899119898119896

(Δ119899119898minus1

Ψ119899119898minus1

) cos (119896120591)

(42)

where 1198881198991198980







1198881198991198980

(Δ119899119898minus1

Ψ119899119898minus1

) = 0 1198881198991198981

(Δ119899119898minus1

Ψ119899119898minus1

) = 0

(119898 = 1 2 3 )

(43)


119899119898minus1and 120575119899119898minus1


119899 Γ119899119899119899

Λ119899119899119899119899




119899119898(120591) of (39) is

obtained from

119906119899119898(120591) = 120594

119898119906119899119898minus1

(120591) + 1198621198991sin 120591 + 119862

1198992cos 120591

+ℎ

Ω2

1198990

120583119898

sum

119896=2

119888119899119898119896

(Δ119899119898minus1

Ψ119899119898minus1

)

(1 minus 1198962)cos (119896120591)

(44)

where 1198621198991




119899Ω119899 and 119906

119899(120591) are

119906119899(120591) asymp 119906

1198990(120591) +

119898

sum

119898=1

119906119899119898(120591) 120575

119899asymp 1205751198990+

119898

sum

119898=1

120575119899119898

Ω119899asymp Ω1198990+

119898

sum

119898=1

Ω119899119898

(45)


R1198991[1199061198990(120591) 1205751198990 Ω1198990]

= Ω2

11989901198990(120591) + 120596

2

119899[1199061198990(120591) + 120575

1198990] + Γ119899119899119899[1199061198990(120591) + 120575

1198990]2

+ Λ119899119899119899119899

[1199061198990(120591) + 120575

1198990]3

= [1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990] cos (120591)

+ [1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990] cos (2120591)

+ [1

41198873

1198990Λ119899119899119899119899

] cos (3120591)

+ (1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990

+Γ1198991198991198991205752

1198990+ Λ119899119899119899119899

1205753

1198990)

(46)


10(1205751198990 Ω1198990) and 119888



1205751198990

1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990= 0

1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990+ Γ1198991198991198991205752

1198990

+ Λ119899119899119899119899

1205753

1198990= 0

(47)


Ω1198990=1

2radic41205962119899+ 31198872

1198990Λ119899119899119899119899

+ 8Γ1198991198991198991205751198990+ 12Λ

1198991198991198991198991205752

1198990

1205751198990= minus

Γ119899119899119899

3Λ119899119899119899119899

minusΥ1198991

3Λ119899119899119899119899

3radic

2

Υ1198992+ radicΥ

2

1198992+ 32Υ

3

1198991

+1

6Λ119899119899119899119899

3


2

1198992+ 32Υ

3

1198991

2

(48)


2

119899119899119899+ 61205962

119899Λ119899119899119899119899

+ 91198872

1198990Λ2

119899119899119899119899

Υ1198992= minus16Γ

3

119899119899119899+ 72120596

2

119899Γ119899119899119899Λ119899119899119899119899

(49)


Ω2

1198990[1198991(120591) + 119906

1198991(120591)]

= ℎ [(1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (1

41198873

1198990Λ119899119899119899119899

) cos 3120591]

(50)


1198991(0) = 0

1198991(0) = 0) therefore


1199061198991(120591) =

ℎ

96Ω2

1198990

times [(161198872

1198990Γ119899119899119899

+ 31198873

1198990Λ119899119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 120591

minus (161198872

1198990Γ119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (31198873

1198990Λ119899119899119899119899

) cos 3120591] (51)


119899119898minus1 120575119899119898minus1




119902119899(119905) = 119906

1198990[(Ω1198990+ Ω1198991) 119905] + 119906

1198991[(Ω1198990+ Ω1198991) 119905]

+ (1205751198990+ 1205751198991)

(52)


40

30

20

10

0


Non

linea

r fre

quen

cyΩ

1

Convergent region


f = 0002 b10 = 0001f = 0020 b10 = 0010

f = 0040 b10 = 0020

f = 0080 b10 = 0040




10












002

001

000

0 2 4 6 8

Time t


minus001

minus002

minus003A f = 0002 b10 = 0001 B f = 0020 b10 = 0010 C f = 0040 b10 = 0020

AB

C

q1(t)












(a) (b) (c) (d)119891 0002 002 004 008120572 9418 94181 188361 3767231205822 (120582120587) 00024 (00156) 24110 (04945) 192882 (13987) 1543060 (39560)

006

004

002

000

000 001 002 003


f = 0002

1205822 = 00024a = 9418

A

B

CD

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(a)

000 001 002 003


minus05

15

10

05

00

A

B

CDf = 002

1205822 = 2411a = 94181

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(b)

000 001 002 003


A

B

CD

f = 004

1205822 = 192882a = 188361


minus1

3

2

1

0

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(c)

f = 008

1205822 = 15431a = 3767

000 001 002 003



A

B

C

D

Freq

uenc

y ra

tio (Ω

minus120596

)120596

4

2

0

minus2

(d)

















008

004

000

minus004

minus008

000 025 050 075 100

(xt)

x

t = 0

t = T8t = T4

t = 3T8

t = T2

(a)

010

005

000

minus005

minus010

(xt)

000 025 050 075 100

x

t = 0

t = T8 t = T4

t = 3T8

t = T2

(b)

01

minus02

00

minus01

minus03

(xt)

000 025 050 075 100

x

t = 0t = T8t = T4

t = 3T8t = T2


(c)

008

004

000

minus004

minus008

(xt)

000 025 050 075 100

x

t = 0t = T8 t = T4

t = 3T8

t = T2


(d)


1minus 1205961)1205961= 005 (b) 119891 = 002 (Ω

1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 120596)120596

1= minus010 (d) 119891 = 008

(Ω1minus 1205961)1205961= 20


1minus 1205961)1205961= 005





119867119879(119905) = 1 + 120572119902

119899(119905) int

1

0

1199101015840

(119909) 1205931015840

119899(119909) d119909

+120572

21199022

119899(119905) int

1

0

12059310158402

119899(119909) d119909

(53)




115

110

105

100

0 5 10 15 20

Axi

al te

nsio

n fo

rceH

T

Time t

(a)

0 5 10 15

28

23

18

13

08

Axi

al te

nsio

n fo

rceH

T

Time t

(b)

40

30

20

10

0Axi

al te

nsio

n fo

rceH

T

minus100 5 10 15


Time t

(c)

Axi

al te

nsio

n fo

rceH

T

0

5

10

15

0 1 2 3

Time t


(d)


(b) 119891 = 002 (Ω1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 1205961)1205961= minus010 (d) 119891 = 008 (Ω

1minus 1205961)1205961= 20


5 Conclusions






Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120593119899(119909) = 120585

119899[1 minus cos (120596

119899119909) minus tan(

120596119899

2) sin (120596

119899119909)]

(119899 = 1 3 5 )

(7)




tan(120596119899

2) =

120596119899

2minus

1

212058221205963

119899 (119899 = 1 3 5 ) (8)




120593119899(119909) = radic2 sin (119899120587119909) 120596

119899= 119899120587 (119899 = 2 4 6 )

(9)


V (119909 119905) =119873

sum

119899=1

119902119899(119905) 120593119899(119909) (10)






119902119899+ 1205962

119899119902119899+

119873

sum

119894=1

119873

sum

119895=1

Γ119899119894119895119902119894119902119895+

119873

sum

119894=1

119873

sum

119895=1

119873

sum

ℎ=1

Λ119899119894119895ℎ119902119894119902119895119902ℎ= 0 (11)


Λ119899119894119895ℎ

= minus1

2120572int

1

0

[12059310158401015840

119895(119909) int

1

0

1205931015840

119894(119909) 1205931015840

ℎ(119909) d119909]120593

119899(119909) d119909

Γ119899119894119895= minus120572int

1

0

[12059310158401015840

119895(119909) int

1

0

1199101015840

(119909) 1205931015840

119894(119909) d119909]120593

119899(119909) d119909

minus1

2120572int

1

0

[11991010158401015840

(119909) int

1

0

1205931015840

119894(119909) 1205931015840

119894(119909) d119909]120593

119899(119909) d119909

(12)




= 120596119899119905 (13)



119902119899+ 119902119899+Γ119899119899119899

1205962119899

1199022

119899+Λ119899119899119899119899

1205962119899

1199023

119899= 0 (14)




119902119899+ 119902119899+ 120576

Γ119899119899119899

1205962119899

1199022

119899+ 1205762Λ119899119899119899119899

1205962119899

1199023

119899= 0 (15)


119902119899( 120576) =

4

sum

119898=0

120576119898119902119899119898(119905) (16)


120591 = (1 +

4

sum

119898=1

120576119898120572119898) (17)



119899as follows

Ω119899minus 120596119899

120596119899

=

4

sum

119898=1

120576119898120572119898 (18)


119899119898and 120572




Ω119899= 120596119899[1 + (

3

8

Λ119899119899119899119899

1205962119899

minus5

12

Γ2

119899119899119899

1205964119899

)1198862

+ (minus15

256

Λ2

119899119899119899119899

1205964119899

minus485

1728

Γ4

119899119899119899

1205968119899

+173

192

Λ119899119899119899119899

1205962119899

Γ2

119899119899119899

1205964119899

)1198864]

(19)

119902119899(119905) = 119886 cos (Ω119905 + 120573) + 1198862 [

Γ119899119899119899

61205962119899

cos (2Ω119905 + 2120573) minusΓ119899119899119899

21205962119899

]

+ 1198863(Γ2

119899119899119899

481205964119899

+Λ119899119899119899119899

321205962119899

) cos (3Ω119905 + 3120573)

+ 1198864[(minus

31

96

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

+59

432

Γ3

119899119899119899

1205966119899

)

times cos (2Ω119905 + 2120573)

+ (5

8

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

minus19

72

Γ3

119899119899119899

1205966119899

)

+ (1

432

Γ3

119899119899119899

1205966119899

+1

96

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

)

times cos (4Ω119905 + 4120573) ]

(20)





119902119899(119905) = 119906

119899(120591) + 120575

119899 (21)


119902119899(0) = 119887

1198990+ 120575119899 119902

119899(0) = 0 (22)





Ω2

119899119899(120591) + 120596

2

119899[119906119899(120591) + 120575

119899] + Γ119899119899119899[119906119899(120591) + 120575

119899]2

+ Δ119899119899119899119899

[119906119899(120591) + 120575

119899]3

= 0

(23)

where

119899(120591) =

d2119906119899(120591)

d1205912 (24)


119906119899(0) = 119887

1198990

119899(0) = 0 (25)


cos (119896120591) | 119896 = 1 2 3 (26)


119906119899(120591) =

+infin

sum

119896=1

119862119899119896cos (119896120591) (27)


119899(120591) is chosen as

1199061198990(120591) = 119887

1198990cos 120591 (28)


L [Φ119899(120591 119902)] = 120596

2

1198990[1205972Φ119899(120591 119902)

1205971205912+ Φ119899(120591 119902)] (29)


L (1198621sin 120591 + 119862

2cos 120591) = 0 (30)


2


N [Φ119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

= Ψ2

119899(119902) [

1205972Φ119899(120591 119902)

1205971205912] + 120596

2

119899[Φ119899(120591 119902) + Δ

119899(119902)]

+ Γ119899119899119899[Φ119899(120591 119902) + Δ

119899(119902)]2

+ Λ119899119899119899119899

[Φ119899(120591 119902) + Δ

119899(119902)]3

(31)


119899(120591)






(1 minus 119902)L [Φ119899(120591 119902) minus 119906

1198990(120591)]

= 119902ℎ119867 (120591)N [Φ119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

(32)



Φ119899(0 119902) = 119887

1198990

120597Φ119899(120591 119902)

120597120591

100381610038161003816100381610038161003816100381610038161003816120591=0

= 0 (33)



119867(120591) = 1 (34)




119899(120591) so does Ψ

119899(119902) from




1198990



Φ119899(120591 119902) = 119906

1198990(120591) +

+infin

sum

119898=1

119906119899119898(120591) 119902119898

Δ119899(119902) = 120575

1198990+

+infin

sum

119898=1

120575119899119898119902119898 Ψ

119899(119902) = Ω

1198990+

+infin

sum

119898=1

Ω119899119898119902119898

(35)

where

119906119899119898(120591) =

1

119898

120597119898Φ119899(120591 119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

120575119899119898=

1

119898

120597119898Δ119899(119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

Ω119899119898=

1

119898

120597119898Ψ119899(119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

(36)


119906119899(120591) = 119906

1198990(120591) +

+infin

sum

119898=1

119906119899119898(120591) 120575

119899= 1205751198990+

+infin

sum

119898=1

120575119899119898

Ω119899= Ω1198990+

+infin

sum

119898=1

Ω119899119898

(37)


U119898= 1199061198990(120591) 119906

1198991(120591) 119906

119899119898(120591)

Δ119899119898= 1205751198990 1205751198991 120575

119899119898

Ψ119899119898= Ω1198990 Ω1198991 Ω

119899119898

(38)



L [119906119899119898(120591) minus 120594

119898119906119899119898minus1

(120591)] = ℎR119899119898(U119899119898minus1

Δ119899119898minus1

Ψ119899119898minus1

)

(39)


119906119899119898(0) = 0

119899119898(0) = 0 (119898 ge 1) (40)

where

120594119898=

0 119898 le 1

1 119898 gt 1

R119899119898=

1

(119898 minus 1)

120597119898minus1N [Φ

119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

120597119902119898minus1

100381610038161003816100381610038161003816100381610038161003816119902=0

=

119898minus1

sum

119896=0

119896

sum

119901=0

Ω119899119901Ω119899119896minus119901

119899119898minus1minus119896

(120591)

+ 1205962

119899[119906119899119898minus1

(120591) + 120575119899119898minus1

]

+ Γ119899119899119899

119898minus1

sum

119896=0

[119906119899119896(120591) 119906119899119898minus1minus119896

(120591) + 120575119899119896120575119899119898minus1minus119896

+2119906119899119896(120591) 120575119899119898minus1minus119896

]

+ Λ119899119899119899119899

119898minus1

sum

119896=0

119896

sum

119901=0

119906119899119901(120591) 119906119899119896minus119901

(120591) 119906119899119898minus1minus119896

(120591)

+

119898minus1

sum

119896=0

119896

sum

119901=0

120575119899119901120575119899119896minus119901

120575119899119898minus1minus119896

+ 3Λ119899119899119899119899

119898minus1

sum

119896=0

119896

sum

119901=0

119906119899119901(120591) 119906119899119896minus119901

(120591) 120575119899119898minus1minus119896

+

119898minus1

sum

119896=0

119896

sum

119901=0

120575119899119901120575119899119896minus119901

119906119899119898minus1minus119896

(120591)

(41)


R119899119898(U119899119898minus1

Δ119899119898minus1

Ψ119899119898minus1

)

= 1198881198991198980

(Δ119899119898minus1

Ψ119899119898minus1

)

+

120583119898

sum

119896=1

119888119899119898119896

(Δ119899119898minus1

Ψ119899119898minus1

) cos (119896120591)

(42)

where 1198881198991198980







1198881198991198980

(Δ119899119898minus1

Ψ119899119898minus1

) = 0 1198881198991198981

(Δ119899119898minus1

Ψ119899119898minus1

) = 0

(119898 = 1 2 3 )

(43)


119899119898minus1and 120575119899119898minus1


119899 Γ119899119899119899

Λ119899119899119899119899




119899119898(120591) of (39) is

obtained from

119906119899119898(120591) = 120594

119898119906119899119898minus1

(120591) + 1198621198991sin 120591 + 119862

1198992cos 120591

+ℎ

Ω2

1198990

120583119898

sum

119896=2

119888119899119898119896

(Δ119899119898minus1

Ψ119899119898minus1

)

(1 minus 1198962)cos (119896120591)

(44)

where 1198621198991




119899Ω119899 and 119906

119899(120591) are

119906119899(120591) asymp 119906

1198990(120591) +

119898

sum

119898=1

119906119899119898(120591) 120575

119899asymp 1205751198990+

119898

sum

119898=1

120575119899119898

Ω119899asymp Ω1198990+

119898

sum

119898=1

Ω119899119898

(45)


R1198991[1199061198990(120591) 1205751198990 Ω1198990]

= Ω2

11989901198990(120591) + 120596

2

119899[1199061198990(120591) + 120575

1198990] + Γ119899119899119899[1199061198990(120591) + 120575

1198990]2

+ Λ119899119899119899119899

[1199061198990(120591) + 120575

1198990]3

= [1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990] cos (120591)

+ [1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990] cos (2120591)

+ [1

41198873

1198990Λ119899119899119899119899

] cos (3120591)

+ (1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990

+Γ1198991198991198991205752

1198990+ Λ119899119899119899119899

1205753

1198990)

(46)


10(1205751198990 Ω1198990) and 119888



1205751198990

1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990= 0

1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990+ Γ1198991198991198991205752

1198990

+ Λ119899119899119899119899

1205753

1198990= 0

(47)


Ω1198990=1

2radic41205962119899+ 31198872

1198990Λ119899119899119899119899

+ 8Γ1198991198991198991205751198990+ 12Λ

1198991198991198991198991205752

1198990

1205751198990= minus

Γ119899119899119899

3Λ119899119899119899119899

minusΥ1198991

3Λ119899119899119899119899

3radic

2

Υ1198992+ radicΥ

2

1198992+ 32Υ

3

1198991

+1

6Λ119899119899119899119899

3


2

1198992+ 32Υ

3

1198991

2

(48)


2

119899119899119899+ 61205962

119899Λ119899119899119899119899

+ 91198872

1198990Λ2

119899119899119899119899

Υ1198992= minus16Γ

3

119899119899119899+ 72120596

2

119899Γ119899119899119899Λ119899119899119899119899

(49)


Ω2

1198990[1198991(120591) + 119906

1198991(120591)]

= ℎ [(1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (1

41198873

1198990Λ119899119899119899119899

) cos 3120591]

(50)


1198991(0) = 0

1198991(0) = 0) therefore


1199061198991(120591) =

ℎ

96Ω2

1198990

times [(161198872

1198990Γ119899119899119899

+ 31198873

1198990Λ119899119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 120591

minus (161198872

1198990Γ119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (31198873

1198990Λ119899119899119899119899

) cos 3120591] (51)


119899119898minus1 120575119899119898minus1




119902119899(119905) = 119906

1198990[(Ω1198990+ Ω1198991) 119905] + 119906

1198991[(Ω1198990+ Ω1198991) 119905]

+ (1205751198990+ 1205751198991)

(52)


40

30

20

10

0


Non

linea

r fre

quen

cyΩ

1

Convergent region


f = 0002 b10 = 0001f = 0020 b10 = 0010

f = 0040 b10 = 0020

f = 0080 b10 = 0040




10












002

001

000

0 2 4 6 8

Time t


minus001

minus002

minus003A f = 0002 b10 = 0001 B f = 0020 b10 = 0010 C f = 0040 b10 = 0020

AB

C

q1(t)












(a) (b) (c) (d)119891 0002 002 004 008120572 9418 94181 188361 3767231205822 (120582120587) 00024 (00156) 24110 (04945) 192882 (13987) 1543060 (39560)

006

004

002

000

000 001 002 003


f = 0002

1205822 = 00024a = 9418

A

B

CD

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(a)

000 001 002 003


minus05

15

10

05

00

A

B

CDf = 002

1205822 = 2411a = 94181

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(b)

000 001 002 003


A

B

CD

f = 004

1205822 = 192882a = 188361


minus1

3

2

1

0

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(c)

f = 008

1205822 = 15431a = 3767

000 001 002 003



A

B

C

D

Freq

uenc

y ra

tio (Ω

minus120596

)120596

4

2

0

minus2

(d)

















008

004

000

minus004

minus008

000 025 050 075 100

(xt)

x

t = 0

t = T8t = T4

t = 3T8

t = T2

(a)

010

005

000

minus005

minus010

(xt)

000 025 050 075 100

x

t = 0

t = T8 t = T4

t = 3T8

t = T2

(b)

01

minus02

00

minus01

minus03

(xt)

000 025 050 075 100

x

t = 0t = T8t = T4

t = 3T8t = T2


(c)

008

004

000

minus004

minus008

(xt)

000 025 050 075 100

x

t = 0t = T8 t = T4

t = 3T8

t = T2


(d)


1minus 1205961)1205961= 005 (b) 119891 = 002 (Ω

1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 120596)120596

1= minus010 (d) 119891 = 008

(Ω1minus 1205961)1205961= 20


1minus 1205961)1205961= 005





119867119879(119905) = 1 + 120572119902

119899(119905) int

1

0

1199101015840

(119909) 1205931015840

119899(119909) d119909

+120572

21199022

119899(119905) int

1

0

12059310158402

119899(119909) d119909

(53)




115

110

105

100

0 5 10 15 20

Axi

al te

nsio

n fo

rceH

T

Time t

(a)

0 5 10 15

28

23

18

13

08

Axi

al te

nsio

n fo

rceH

T

Time t

(b)

40

30

20

10

0Axi

al te

nsio

n fo

rceH

T

minus100 5 10 15


Time t

(c)

Axi

al te

nsio

n fo

rceH

T

0

5

10

15

0 1 2 3

Time t


(d)


(b) 119891 = 002 (Ω1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 1205961)1205961= minus010 (d) 119891 = 008 (Ω

1minus 1205961)1205961= 20


5 Conclusions






Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Ω119899= 120596119899[1 + (

3

8

Λ119899119899119899119899

1205962119899

minus5

12

Γ2

119899119899119899

1205964119899

)1198862

+ (minus15

256

Λ2

119899119899119899119899

1205964119899

minus485

1728

Γ4

119899119899119899

1205968119899

+173

192

Λ119899119899119899119899

1205962119899

Γ2

119899119899119899

1205964119899

)1198864]

(19)

119902119899(119905) = 119886 cos (Ω119905 + 120573) + 1198862 [

Γ119899119899119899

61205962119899

cos (2Ω119905 + 2120573) minusΓ119899119899119899

21205962119899

]

+ 1198863(Γ2

119899119899119899

481205964119899

+Λ119899119899119899119899

321205962119899

) cos (3Ω119905 + 3120573)

+ 1198864[(minus

31

96

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

+59

432

Γ3

119899119899119899

1205966119899

)

times cos (2Ω119905 + 2120573)

+ (5

8

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

minus19

72

Γ3

119899119899119899

1205966119899

)

+ (1

432

Γ3

119899119899119899

1205966119899

+1

96

Γ119899119899119899

1205962119899

Λ119899119899119899119899

1205962119899

)

times cos (4Ω119905 + 4120573) ]

(20)





119902119899(119905) = 119906

119899(120591) + 120575

119899 (21)


119902119899(0) = 119887

1198990+ 120575119899 119902

119899(0) = 0 (22)





Ω2

119899119899(120591) + 120596

2

119899[119906119899(120591) + 120575

119899] + Γ119899119899119899[119906119899(120591) + 120575

119899]2

+ Δ119899119899119899119899

[119906119899(120591) + 120575

119899]3

= 0

(23)

where

119899(120591) =

d2119906119899(120591)

d1205912 (24)


119906119899(0) = 119887

1198990

119899(0) = 0 (25)


cos (119896120591) | 119896 = 1 2 3 (26)


119906119899(120591) =

+infin

sum

119896=1

119862119899119896cos (119896120591) (27)


119899(120591) is chosen as

1199061198990(120591) = 119887

1198990cos 120591 (28)


L [Φ119899(120591 119902)] = 120596

2

1198990[1205972Φ119899(120591 119902)

1205971205912+ Φ119899(120591 119902)] (29)


L (1198621sin 120591 + 119862

2cos 120591) = 0 (30)


2


N [Φ119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

= Ψ2

119899(119902) [

1205972Φ119899(120591 119902)

1205971205912] + 120596

2

119899[Φ119899(120591 119902) + Δ

119899(119902)]

+ Γ119899119899119899[Φ119899(120591 119902) + Δ

119899(119902)]2

+ Λ119899119899119899119899

[Φ119899(120591 119902) + Δ

119899(119902)]3

(31)


119899(120591)






(1 minus 119902)L [Φ119899(120591 119902) minus 119906

1198990(120591)]

= 119902ℎ119867 (120591)N [Φ119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

(32)



Φ119899(0 119902) = 119887

1198990

120597Φ119899(120591 119902)

120597120591

100381610038161003816100381610038161003816100381610038161003816120591=0

= 0 (33)



119867(120591) = 1 (34)




119899(120591) so does Ψ

119899(119902) from




1198990



Φ119899(120591 119902) = 119906

1198990(120591) +

+infin

sum

119898=1

119906119899119898(120591) 119902119898

Δ119899(119902) = 120575

1198990+

+infin

sum

119898=1

120575119899119898119902119898 Ψ

119899(119902) = Ω

1198990+

+infin

sum

119898=1

Ω119899119898119902119898

(35)

where

119906119899119898(120591) =

1

119898

120597119898Φ119899(120591 119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

120575119899119898=

1

119898

120597119898Δ119899(119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

Ω119899119898=

1

119898

120597119898Ψ119899(119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

(36)


119906119899(120591) = 119906

1198990(120591) +

+infin

sum

119898=1

119906119899119898(120591) 120575

119899= 1205751198990+

+infin

sum

119898=1

120575119899119898

Ω119899= Ω1198990+

+infin

sum

119898=1

Ω119899119898

(37)


U119898= 1199061198990(120591) 119906

1198991(120591) 119906

119899119898(120591)

Δ119899119898= 1205751198990 1205751198991 120575

119899119898

Ψ119899119898= Ω1198990 Ω1198991 Ω

119899119898

(38)



L [119906119899119898(120591) minus 120594

119898119906119899119898minus1

(120591)] = ℎR119899119898(U119899119898minus1

Δ119899119898minus1

Ψ119899119898minus1

)

(39)


119906119899119898(0) = 0

119899119898(0) = 0 (119898 ge 1) (40)

where

120594119898=

0 119898 le 1

1 119898 gt 1

R119899119898=

1

(119898 minus 1)

120597119898minus1N [Φ

119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

120597119902119898minus1

100381610038161003816100381610038161003816100381610038161003816119902=0

=

119898minus1

sum

119896=0

119896

sum

119901=0

Ω119899119901Ω119899119896minus119901

119899119898minus1minus119896

(120591)

+ 1205962

119899[119906119899119898minus1

(120591) + 120575119899119898minus1

]

+ Γ119899119899119899

119898minus1

sum

119896=0

[119906119899119896(120591) 119906119899119898minus1minus119896

(120591) + 120575119899119896120575119899119898minus1minus119896

+2119906119899119896(120591) 120575119899119898minus1minus119896

]

+ Λ119899119899119899119899

119898minus1

sum

119896=0

119896

sum

119901=0

119906119899119901(120591) 119906119899119896minus119901

(120591) 119906119899119898minus1minus119896

(120591)

+

119898minus1

sum

119896=0

119896

sum

119901=0

120575119899119901120575119899119896minus119901

120575119899119898minus1minus119896

+ 3Λ119899119899119899119899

119898minus1

sum

119896=0

119896

sum

119901=0

119906119899119901(120591) 119906119899119896minus119901

(120591) 120575119899119898minus1minus119896

+

119898minus1

sum

119896=0

119896

sum

119901=0

120575119899119901120575119899119896minus119901

119906119899119898minus1minus119896

(120591)

(41)


R119899119898(U119899119898minus1

Δ119899119898minus1

Ψ119899119898minus1

)

= 1198881198991198980

(Δ119899119898minus1

Ψ119899119898minus1

)

+

120583119898

sum

119896=1

119888119899119898119896

(Δ119899119898minus1

Ψ119899119898minus1

) cos (119896120591)

(42)

where 1198881198991198980







1198881198991198980

(Δ119899119898minus1

Ψ119899119898minus1

) = 0 1198881198991198981

(Δ119899119898minus1

Ψ119899119898minus1

) = 0

(119898 = 1 2 3 )

(43)


119899119898minus1and 120575119899119898minus1


119899 Γ119899119899119899

Λ119899119899119899119899




119899119898(120591) of (39) is

obtained from

119906119899119898(120591) = 120594

119898119906119899119898minus1

(120591) + 1198621198991sin 120591 + 119862

1198992cos 120591

+ℎ

Ω2

1198990

120583119898

sum

119896=2

119888119899119898119896

(Δ119899119898minus1

Ψ119899119898minus1

)

(1 minus 1198962)cos (119896120591)

(44)

where 1198621198991




119899Ω119899 and 119906

119899(120591) are

119906119899(120591) asymp 119906

1198990(120591) +

119898

sum

119898=1

119906119899119898(120591) 120575

119899asymp 1205751198990+

119898

sum

119898=1

120575119899119898

Ω119899asymp Ω1198990+

119898

sum

119898=1

Ω119899119898

(45)


R1198991[1199061198990(120591) 1205751198990 Ω1198990]

= Ω2

11989901198990(120591) + 120596

2

119899[1199061198990(120591) + 120575

1198990] + Γ119899119899119899[1199061198990(120591) + 120575

1198990]2

+ Λ119899119899119899119899

[1199061198990(120591) + 120575

1198990]3

= [1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990] cos (120591)

+ [1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990] cos (2120591)

+ [1

41198873

1198990Λ119899119899119899119899

] cos (3120591)

+ (1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990

+Γ1198991198991198991205752

1198990+ Λ119899119899119899119899

1205753

1198990)

(46)


10(1205751198990 Ω1198990) and 119888



1205751198990

1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990= 0

1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990+ Γ1198991198991198991205752

1198990

+ Λ119899119899119899119899

1205753

1198990= 0

(47)


Ω1198990=1

2radic41205962119899+ 31198872

1198990Λ119899119899119899119899

+ 8Γ1198991198991198991205751198990+ 12Λ

1198991198991198991198991205752

1198990

1205751198990= minus

Γ119899119899119899

3Λ119899119899119899119899

minusΥ1198991

3Λ119899119899119899119899

3radic

2

Υ1198992+ radicΥ

2

1198992+ 32Υ

3

1198991

+1

6Λ119899119899119899119899

3


2

1198992+ 32Υ

3

1198991

2

(48)


2

119899119899119899+ 61205962

119899Λ119899119899119899119899

+ 91198872

1198990Λ2

119899119899119899119899

Υ1198992= minus16Γ

3

119899119899119899+ 72120596

2

119899Γ119899119899119899Λ119899119899119899119899

(49)


Ω2

1198990[1198991(120591) + 119906

1198991(120591)]

= ℎ [(1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (1

41198873

1198990Λ119899119899119899119899

) cos 3120591]

(50)


1198991(0) = 0

1198991(0) = 0) therefore


1199061198991(120591) =

ℎ

96Ω2

1198990

times [(161198872

1198990Γ119899119899119899

+ 31198873

1198990Λ119899119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 120591

minus (161198872

1198990Γ119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (31198873

1198990Λ119899119899119899119899

) cos 3120591] (51)


119899119898minus1 120575119899119898minus1




119902119899(119905) = 119906

1198990[(Ω1198990+ Ω1198991) 119905] + 119906

1198991[(Ω1198990+ Ω1198991) 119905]

+ (1205751198990+ 1205751198991)

(52)


40

30

20

10

0


Non

linea

r fre

quen

cyΩ

1

Convergent region


f = 0002 b10 = 0001f = 0020 b10 = 0010

f = 0040 b10 = 0020

f = 0080 b10 = 0040




10












002

001

000

0 2 4 6 8

Time t


minus001

minus002

minus003A f = 0002 b10 = 0001 B f = 0020 b10 = 0010 C f = 0040 b10 = 0020

AB

C

q1(t)












(a) (b) (c) (d)119891 0002 002 004 008120572 9418 94181 188361 3767231205822 (120582120587) 00024 (00156) 24110 (04945) 192882 (13987) 1543060 (39560)

006

004

002

000

000 001 002 003


f = 0002

1205822 = 00024a = 9418

A

B

CD

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(a)

000 001 002 003


minus05

15

10

05

00

A

B

CDf = 002

1205822 = 2411a = 94181

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(b)

000 001 002 003


A

B

CD

f = 004

1205822 = 192882a = 188361


minus1

3

2

1

0

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(c)

f = 008

1205822 = 15431a = 3767

000 001 002 003



A

B

C

D

Freq

uenc

y ra

tio (Ω

minus120596

)120596

4

2

0

minus2

(d)

















008

004

000

minus004

minus008

000 025 050 075 100

(xt)

x

t = 0

t = T8t = T4

t = 3T8

t = T2

(a)

010

005

000

minus005

minus010

(xt)

000 025 050 075 100

x

t = 0

t = T8 t = T4

t = 3T8

t = T2

(b)

01

minus02

00

minus01

minus03

(xt)

000 025 050 075 100

x

t = 0t = T8t = T4

t = 3T8t = T2


(c)

008

004

000

minus004

minus008

(xt)

000 025 050 075 100

x

t = 0t = T8 t = T4

t = 3T8

t = T2


(d)


1minus 1205961)1205961= 005 (b) 119891 = 002 (Ω

1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 120596)120596

1= minus010 (d) 119891 = 008

(Ω1minus 1205961)1205961= 20


1minus 1205961)1205961= 005





119867119879(119905) = 1 + 120572119902

119899(119905) int

1

0

1199101015840

(119909) 1205931015840

119899(119909) d119909

+120572

21199022

119899(119905) int

1

0

12059310158402

119899(119909) d119909

(53)




115

110

105

100

0 5 10 15 20

Axi

al te

nsio

n fo

rceH

T

Time t

(a)

0 5 10 15

28

23

18

13

08

Axi

al te

nsio

n fo

rceH

T

Time t

(b)

40

30

20

10

0Axi

al te

nsio

n fo

rceH

T

minus100 5 10 15


Time t

(c)

Axi

al te

nsio

n fo

rceH

T

0

5

10

15

0 1 2 3

Time t


(d)


(b) 119891 = 002 (Ω1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 1205961)1205961= minus010 (d) 119891 = 008 (Ω

1minus 1205961)1205961= 20


5 Conclusions






Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Φ119899(0 119902) = 119887

1198990

120597Φ119899(120591 119902)

120597120591

100381610038161003816100381610038161003816100381610038161003816120591=0

= 0 (33)



119867(120591) = 1 (34)




119899(120591) so does Ψ

119899(119902) from




1198990



Φ119899(120591 119902) = 119906

1198990(120591) +

+infin

sum

119898=1

119906119899119898(120591) 119902119898

Δ119899(119902) = 120575

1198990+

+infin

sum

119898=1

120575119899119898119902119898 Ψ

119899(119902) = Ω

1198990+

+infin

sum

119898=1

Ω119899119898119902119898

(35)

where

119906119899119898(120591) =

1

119898

120597119898Φ119899(120591 119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

120575119899119898=

1

119898

120597119898Δ119899(119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

Ω119899119898=

1

119898

120597119898Ψ119899(119902)

120597119902119898

100381610038161003816100381610038161003816100381610038161003816119902=0

(36)


119906119899(120591) = 119906

1198990(120591) +

+infin

sum

119898=1

119906119899119898(120591) 120575

119899= 1205751198990+

+infin

sum

119898=1

120575119899119898

Ω119899= Ω1198990+

+infin

sum

119898=1

Ω119899119898

(37)


U119898= 1199061198990(120591) 119906

1198991(120591) 119906

119899119898(120591)

Δ119899119898= 1205751198990 1205751198991 120575

119899119898

Ψ119899119898= Ω1198990 Ω1198991 Ω

119899119898

(38)



L [119906119899119898(120591) minus 120594

119898119906119899119898minus1

(120591)] = ℎR119899119898(U119899119898minus1

Δ119899119898minus1

Ψ119899119898minus1

)

(39)


119906119899119898(0) = 0

119899119898(0) = 0 (119898 ge 1) (40)

where

120594119898=

0 119898 le 1

1 119898 gt 1

R119899119898=

1

(119898 minus 1)

120597119898minus1N [Φ

119899(120591 119902) Δ

119899(119902) Ψ

119899(119902)]

120597119902119898minus1

100381610038161003816100381610038161003816100381610038161003816119902=0

=

119898minus1

sum

119896=0

119896

sum

119901=0

Ω119899119901Ω119899119896minus119901

119899119898minus1minus119896

(120591)

+ 1205962

119899[119906119899119898minus1

(120591) + 120575119899119898minus1

]

+ Γ119899119899119899

119898minus1

sum

119896=0

[119906119899119896(120591) 119906119899119898minus1minus119896

(120591) + 120575119899119896120575119899119898minus1minus119896

+2119906119899119896(120591) 120575119899119898minus1minus119896

]

+ Λ119899119899119899119899

119898minus1

sum

119896=0

119896

sum

119901=0

119906119899119901(120591) 119906119899119896minus119901

(120591) 119906119899119898minus1minus119896

(120591)

+

119898minus1

sum

119896=0

119896

sum

119901=0

120575119899119901120575119899119896minus119901

120575119899119898minus1minus119896

+ 3Λ119899119899119899119899

119898minus1

sum

119896=0

119896

sum

119901=0

119906119899119901(120591) 119906119899119896minus119901

(120591) 120575119899119898minus1minus119896

+

119898minus1

sum

119896=0

119896

sum

119901=0

120575119899119901120575119899119896minus119901

119906119899119898minus1minus119896

(120591)

(41)


R119899119898(U119899119898minus1

Δ119899119898minus1

Ψ119899119898minus1

)

= 1198881198991198980

(Δ119899119898minus1

Ψ119899119898minus1

)

+

120583119898

sum

119896=1

119888119899119898119896

(Δ119899119898minus1

Ψ119899119898minus1

) cos (119896120591)

(42)

where 1198881198991198980







1198881198991198980

(Δ119899119898minus1

Ψ119899119898minus1

) = 0 1198881198991198981

(Δ119899119898minus1

Ψ119899119898minus1

) = 0

(119898 = 1 2 3 )

(43)


119899119898minus1and 120575119899119898minus1


119899 Γ119899119899119899

Λ119899119899119899119899




119899119898(120591) of (39) is

obtained from

119906119899119898(120591) = 120594

119898119906119899119898minus1

(120591) + 1198621198991sin 120591 + 119862

1198992cos 120591

+ℎ

Ω2

1198990

120583119898

sum

119896=2

119888119899119898119896

(Δ119899119898minus1

Ψ119899119898minus1

)

(1 minus 1198962)cos (119896120591)

(44)

where 1198621198991




119899Ω119899 and 119906

119899(120591) are

119906119899(120591) asymp 119906

1198990(120591) +

119898

sum

119898=1

119906119899119898(120591) 120575

119899asymp 1205751198990+

119898

sum

119898=1

120575119899119898

Ω119899asymp Ω1198990+

119898

sum

119898=1

Ω119899119898

(45)


R1198991[1199061198990(120591) 1205751198990 Ω1198990]

= Ω2

11989901198990(120591) + 120596

2

119899[1199061198990(120591) + 120575

1198990] + Γ119899119899119899[1199061198990(120591) + 120575

1198990]2

+ Λ119899119899119899119899

[1199061198990(120591) + 120575

1198990]3

= [1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990] cos (120591)

+ [1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990] cos (2120591)

+ [1

41198873

1198990Λ119899119899119899119899

] cos (3120591)

+ (1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990

+Γ1198991198991198991205752

1198990+ Λ119899119899119899119899

1205753

1198990)

(46)


10(1205751198990 Ω1198990) and 119888



1205751198990

1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990= 0

1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990+ Γ1198991198991198991205752

1198990

+ Λ119899119899119899119899

1205753

1198990= 0

(47)


Ω1198990=1

2radic41205962119899+ 31198872

1198990Λ119899119899119899119899

+ 8Γ1198991198991198991205751198990+ 12Λ

1198991198991198991198991205752

1198990

1205751198990= minus

Γ119899119899119899

3Λ119899119899119899119899

minusΥ1198991

3Λ119899119899119899119899

3radic

2

Υ1198992+ radicΥ

2

1198992+ 32Υ

3

1198991

+1

6Λ119899119899119899119899

3


2

1198992+ 32Υ

3

1198991

2

(48)


2

119899119899119899+ 61205962

119899Λ119899119899119899119899

+ 91198872

1198990Λ2

119899119899119899119899

Υ1198992= minus16Γ

3

119899119899119899+ 72120596

2

119899Γ119899119899119899Λ119899119899119899119899

(49)


Ω2

1198990[1198991(120591) + 119906

1198991(120591)]

= ℎ [(1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (1

41198873

1198990Λ119899119899119899119899

) cos 3120591]

(50)


1198991(0) = 0

1198991(0) = 0) therefore


1199061198991(120591) =

ℎ

96Ω2

1198990

times [(161198872

1198990Γ119899119899119899

+ 31198873

1198990Λ119899119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 120591

minus (161198872

1198990Γ119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (31198873

1198990Λ119899119899119899119899

) cos 3120591] (51)


119899119898minus1 120575119899119898minus1




119902119899(119905) = 119906

1198990[(Ω1198990+ Ω1198991) 119905] + 119906

1198991[(Ω1198990+ Ω1198991) 119905]

+ (1205751198990+ 1205751198991)

(52)


40

30

20

10

0


Non

linea

r fre

quen

cyΩ

1

Convergent region


f = 0002 b10 = 0001f = 0020 b10 = 0010

f = 0040 b10 = 0020

f = 0080 b10 = 0040




10












002

001

000

0 2 4 6 8

Time t


minus001

minus002

minus003A f = 0002 b10 = 0001 B f = 0020 b10 = 0010 C f = 0040 b10 = 0020

AB

C

q1(t)












(a) (b) (c) (d)119891 0002 002 004 008120572 9418 94181 188361 3767231205822 (120582120587) 00024 (00156) 24110 (04945) 192882 (13987) 1543060 (39560)

006

004

002

000

000 001 002 003


f = 0002

1205822 = 00024a = 9418

A

B

CD

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(a)

000 001 002 003


minus05

15

10

05

00

A

B

CDf = 002

1205822 = 2411a = 94181

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(b)

000 001 002 003


A

B

CD

f = 004

1205822 = 192882a = 188361


minus1

3

2

1

0

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(c)

f = 008

1205822 = 15431a = 3767

000 001 002 003



A

B

C

D

Freq

uenc

y ra

tio (Ω

minus120596

)120596

4

2

0

minus2

(d)

















008

004

000

minus004

minus008

000 025 050 075 100

(xt)

x

t = 0

t = T8t = T4

t = 3T8

t = T2

(a)

010

005

000

minus005

minus010

(xt)

000 025 050 075 100

x

t = 0

t = T8 t = T4

t = 3T8

t = T2

(b)

01

minus02

00

minus01

minus03

(xt)

000 025 050 075 100

x

t = 0t = T8t = T4

t = 3T8t = T2


(c)

008

004

000

minus004

minus008

(xt)

000 025 050 075 100

x

t = 0t = T8 t = T4

t = 3T8

t = T2


(d)


1minus 1205961)1205961= 005 (b) 119891 = 002 (Ω

1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 120596)120596

1= minus010 (d) 119891 = 008

(Ω1minus 1205961)1205961= 20


1minus 1205961)1205961= 005





119867119879(119905) = 1 + 120572119902

119899(119905) int

1

0

1199101015840

(119909) 1205931015840

119899(119909) d119909

+120572

21199022

119899(119905) int

1

0

12059310158402

119899(119909) d119909

(53)




115

110

105

100

0 5 10 15 20

Axi

al te

nsio

n fo

rceH

T

Time t

(a)

0 5 10 15

28

23

18

13

08

Axi

al te

nsio

n fo

rceH

T

Time t

(b)

40

30

20

10

0Axi

al te

nsio

n fo

rceH

T

minus100 5 10 15


Time t

(c)

Axi

al te

nsio

n fo

rceH

T

0

5

10

15

0 1 2 3

Time t


(d)


(b) 119891 = 002 (Ω1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 1205961)1205961= minus010 (d) 119891 = 008 (Ω

1minus 1205961)1205961= 20


5 Conclusions






Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198881198991198980

(Δ119899119898minus1

Ψ119899119898minus1

) = 0 1198881198991198981

(Δ119899119898minus1

Ψ119899119898minus1

) = 0

(119898 = 1 2 3 )

(43)


119899119898minus1and 120575119899119898minus1


119899 Γ119899119899119899

Λ119899119899119899119899




119899119898(120591) of (39) is

obtained from

119906119899119898(120591) = 120594

119898119906119899119898minus1

(120591) + 1198621198991sin 120591 + 119862

1198992cos 120591

+ℎ

Ω2

1198990

120583119898

sum

119896=2

119888119899119898119896

(Δ119899119898minus1

Ψ119899119898minus1

)

(1 minus 1198962)cos (119896120591)

(44)

where 1198621198991




119899Ω119899 and 119906

119899(120591) are

119906119899(120591) asymp 119906

1198990(120591) +

119898

sum

119898=1

119906119899119898(120591) 120575

119899asymp 1205751198990+

119898

sum

119898=1

120575119899119898

Ω119899asymp Ω1198990+

119898

sum

119898=1

Ω119899119898

(45)


R1198991[1199061198990(120591) 1205751198990 Ω1198990]

= Ω2

11989901198990(120591) + 120596

2

119899[1199061198990(120591) + 120575

1198990] + Γ119899119899119899[1199061198990(120591) + 120575

1198990]2

+ Λ119899119899119899119899

[1199061198990(120591) + 120575

1198990]3

= [1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990] cos (120591)

+ [1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990] cos (2120591)

+ [1

41198873

1198990Λ119899119899119899119899

] cos (3120591)

+ (1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990

+Γ1198991198991198991205752

1198990+ Λ119899119899119899119899

1205753

1198990)

(46)


10(1205751198990 Ω1198990) and 119888



1205751198990

1205962

119899+3

41198872

1198990Λ119899119899119899119899

+ 2Γ1198991198991198991205751198990+ 3Λ119899119899119899119899

1205752

1198990minus Ω2

1198990= 0

1

21198872

1198990Γ119899119899119899

+ 1205962

1198991205751198990+3

21198872

1198990Λ119899119899119899119899

1205751198990+ Γ1198991198991198991205752

1198990

+ Λ119899119899119899119899

1205753

1198990= 0

(47)


Ω1198990=1

2radic41205962119899+ 31198872

1198990Λ119899119899119899119899

+ 8Γ1198991198991198991205751198990+ 12Λ

1198991198991198991198991205752

1198990

1205751198990= minus

Γ119899119899119899

3Λ119899119899119899119899

minusΥ1198991

3Λ119899119899119899119899

3radic

2

Υ1198992+ radicΥ

2

1198992+ 32Υ

3

1198991

+1

6Λ119899119899119899119899

3


2

1198992+ 32Υ

3

1198991

2

(48)


2

119899119899119899+ 61205962

119899Λ119899119899119899119899

+ 91198872

1198990Λ2

119899119899119899119899

Υ1198992= minus16Γ

3

119899119899119899+ 72120596

2

119899Γ119899119899119899Λ119899119899119899119899

(49)


Ω2

1198990[1198991(120591) + 119906

1198991(120591)]

= ℎ [(1

21198872

1198990Γ119899119899119899

+2

31198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (1

41198873

1198990Λ119899119899119899119899

) cos 3120591]

(50)


1198991(0) = 0

1198991(0) = 0) therefore


1199061198991(120591) =

ℎ

96Ω2

1198990

times [(161198872

1198990Γ119899119899119899

+ 31198873

1198990Λ119899119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 120591

minus (161198872

1198990Γ119899119899119899

+ 481198872

1198990Λ119899119899119899119899

1205751198990) cos 2120591

+ (31198873

1198990Λ119899119899119899119899

) cos 3120591] (51)


119899119898minus1 120575119899119898minus1




119902119899(119905) = 119906

1198990[(Ω1198990+ Ω1198991) 119905] + 119906

1198991[(Ω1198990+ Ω1198991) 119905]

+ (1205751198990+ 1205751198991)

(52)


40

30

20

10

0


Non

linea

r fre

quen

cyΩ

1

Convergent region


f = 0002 b10 = 0001f = 0020 b10 = 0010

f = 0040 b10 = 0020

f = 0080 b10 = 0040




10












002

001

000

0 2 4 6 8

Time t


minus001

minus002

minus003A f = 0002 b10 = 0001 B f = 0020 b10 = 0010 C f = 0040 b10 = 0020

AB

C

q1(t)












(a) (b) (c) (d)119891 0002 002 004 008120572 9418 94181 188361 3767231205822 (120582120587) 00024 (00156) 24110 (04945) 192882 (13987) 1543060 (39560)

006

004

002

000

000 001 002 003


f = 0002

1205822 = 00024a = 9418

A

B

CD

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(a)

000 001 002 003


minus05

15

10

05

00

A

B

CDf = 002

1205822 = 2411a = 94181

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(b)

000 001 002 003


A

B

CD

f = 004

1205822 = 192882a = 188361


minus1

3

2

1

0

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(c)

f = 008

1205822 = 15431a = 3767

000 001 002 003



A

B

C

D

Freq

uenc

y ra

tio (Ω

minus120596

)120596

4

2

0

minus2

(d)

















008

004

000

minus004

minus008

000 025 050 075 100

(xt)

x

t = 0

t = T8t = T4

t = 3T8

t = T2

(a)

010

005

000

minus005

minus010

(xt)

000 025 050 075 100

x

t = 0

t = T8 t = T4

t = 3T8

t = T2

(b)

01

minus02

00

minus01

minus03

(xt)

000 025 050 075 100

x

t = 0t = T8t = T4

t = 3T8t = T2


(c)

008

004

000

minus004

minus008

(xt)

000 025 050 075 100

x

t = 0t = T8 t = T4

t = 3T8

t = T2


(d)


1minus 1205961)1205961= 005 (b) 119891 = 002 (Ω

1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 120596)120596

1= minus010 (d) 119891 = 008

(Ω1minus 1205961)1205961= 20


1minus 1205961)1205961= 005





119867119879(119905) = 1 + 120572119902

119899(119905) int

1

0

1199101015840

(119909) 1205931015840

119899(119909) d119909

+120572

21199022

119899(119905) int

1

0

12059310158402

119899(119909) d119909

(53)




115

110

105

100

0 5 10 15 20

Axi

al te

nsio

n fo

rceH

T

Time t

(a)

0 5 10 15

28

23

18

13

08

Axi

al te

nsio

n fo

rceH

T

Time t

(b)

40

30

20

10

0Axi

al te

nsio

n fo

rceH

T

minus100 5 10 15


Time t

(c)

Axi

al te

nsio

n fo

rceH

T

0

5

10

15

0 1 2 3

Time t


(d)


(b) 119891 = 002 (Ω1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 1205961)1205961= minus010 (d) 119891 = 008 (Ω

1minus 1205961)1205961= 20


5 Conclusions






Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










40

30

20

10

0


Non

linea

r fre

quen

cyΩ

1

Convergent region


f = 0002 b10 = 0001f = 0020 b10 = 0010

f = 0040 b10 = 0020

f = 0080 b10 = 0040




10












002

001

000

0 2 4 6 8

Time t


minus001

minus002

minus003A f = 0002 b10 = 0001 B f = 0020 b10 = 0010 C f = 0040 b10 = 0020

AB

C

q1(t)












(a) (b) (c) (d)119891 0002 002 004 008120572 9418 94181 188361 3767231205822 (120582120587) 00024 (00156) 24110 (04945) 192882 (13987) 1543060 (39560)

006

004

002

000

000 001 002 003


f = 0002

1205822 = 00024a = 9418

A

B

CD

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(a)

000 001 002 003


minus05

15

10

05

00

A

B

CDf = 002

1205822 = 2411a = 94181

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(b)

000 001 002 003


A

B

CD

f = 004

1205822 = 192882a = 188361


minus1

3

2

1

0

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(c)

f = 008

1205822 = 15431a = 3767

000 001 002 003



A

B

C

D

Freq

uenc

y ra

tio (Ω

minus120596

)120596

4

2

0

minus2

(d)

















008

004

000

minus004

minus008

000 025 050 075 100

(xt)

x

t = 0

t = T8t = T4

t = 3T8

t = T2

(a)

010

005

000

minus005

minus010

(xt)

000 025 050 075 100

x

t = 0

t = T8 t = T4

t = 3T8

t = T2

(b)

01

minus02

00

minus01

minus03

(xt)

000 025 050 075 100

x

t = 0t = T8t = T4

t = 3T8t = T2


(c)

008

004

000

minus004

minus008

(xt)

000 025 050 075 100

x

t = 0t = T8 t = T4

t = 3T8

t = T2


(d)


1minus 1205961)1205961= 005 (b) 119891 = 002 (Ω

1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 120596)120596

1= minus010 (d) 119891 = 008

(Ω1minus 1205961)1205961= 20


1minus 1205961)1205961= 005





119867119879(119905) = 1 + 120572119902

119899(119905) int

1

0

1199101015840

(119909) 1205931015840

119899(119909) d119909

+120572

21199022

119899(119905) int

1

0

12059310158402

119899(119909) d119909

(53)




115

110

105

100

0 5 10 15 20

Axi

al te

nsio

n fo

rceH

T

Time t

(a)

0 5 10 15

28

23

18

13

08

Axi

al te

nsio

n fo

rceH

T

Time t

(b)

40

30

20

10

0Axi

al te

nsio

n fo

rceH

T

minus100 5 10 15


Time t

(c)

Axi

al te

nsio

n fo

rceH

T

0

5

10

15

0 1 2 3

Time t


(d)


(b) 119891 = 002 (Ω1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 1205961)1205961= minus010 (d) 119891 = 008 (Ω

1minus 1205961)1205961= 20


5 Conclusions






Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(a) (b) (c) (d)119891 0002 002 004 008120572 9418 94181 188361 3767231205822 (120582120587) 00024 (00156) 24110 (04945) 192882 (13987) 1543060 (39560)

006

004

002

000

000 001 002 003


f = 0002

1205822 = 00024a = 9418

A

B

CD

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(a)

000 001 002 003


minus05

15

10

05

00

A

B

CDf = 002

1205822 = 2411a = 94181

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(b)

000 001 002 003


A

B

CD

f = 004

1205822 = 192882a = 188361


minus1

3

2

1

0

Freq

uenc

y ra

tio (Ω

minus120596

)120596

(c)

f = 008

1205822 = 15431a = 3767

000 001 002 003



A

B

C

D

Freq

uenc

y ra

tio (Ω

minus120596

)120596

4

2

0

minus2

(d)

















008

004

000

minus004

minus008

000 025 050 075 100

(xt)

x

t = 0

t = T8t = T4

t = 3T8

t = T2

(a)

010

005

000

minus005

minus010

(xt)

000 025 050 075 100

x

t = 0

t = T8 t = T4

t = 3T8

t = T2

(b)

01

minus02

00

minus01

minus03

(xt)

000 025 050 075 100

x

t = 0t = T8t = T4

t = 3T8t = T2


(c)

008

004

000

minus004

minus008

(xt)

000 025 050 075 100

x

t = 0t = T8 t = T4

t = 3T8

t = T2


(d)


1minus 1205961)1205961= 005 (b) 119891 = 002 (Ω

1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 120596)120596

1= minus010 (d) 119891 = 008

(Ω1minus 1205961)1205961= 20


1minus 1205961)1205961= 005





119867119879(119905) = 1 + 120572119902

119899(119905) int

1

0

1199101015840

(119909) 1205931015840

119899(119909) d119909

+120572

21199022

119899(119905) int

1

0

12059310158402

119899(119909) d119909

(53)




115

110

105

100

0 5 10 15 20

Axi

al te

nsio

n fo

rceH

T

Time t

(a)

0 5 10 15

28

23

18

13

08

Axi

al te

nsio

n fo

rceH

T

Time t

(b)

40

30

20

10

0Axi

al te

nsio

n fo

rceH

T

minus100 5 10 15


Time t

(c)

Axi

al te

nsio

n fo

rceH

T

0

5

10

15

0 1 2 3

Time t


(d)


(b) 119891 = 002 (Ω1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 1205961)1205961= minus010 (d) 119891 = 008 (Ω

1minus 1205961)1205961= 20


5 Conclusions






Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















008

004

000

minus004

minus008

000 025 050 075 100

(xt)

x

t = 0

t = T8t = T4

t = 3T8

t = T2

(a)

010

005

000

minus005

minus010

(xt)

000 025 050 075 100

x

t = 0

t = T8 t = T4

t = 3T8

t = T2

(b)

01

minus02

00

minus01

minus03

(xt)

000 025 050 075 100

x

t = 0t = T8t = T4

t = 3T8t = T2


(c)

008

004

000

minus004

minus008

(xt)

000 025 050 075 100

x

t = 0t = T8 t = T4

t = 3T8

t = T2


(d)


1minus 1205961)1205961= 005 (b) 119891 = 002 (Ω

1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 120596)120596

1= minus010 (d) 119891 = 008

(Ω1minus 1205961)1205961= 20


1minus 1205961)1205961= 005





119867119879(119905) = 1 + 120572119902

119899(119905) int

1

0

1199101015840

(119909) 1205931015840

119899(119909) d119909

+120572

21199022

119899(119905) int

1

0

12059310158402

119899(119909) d119909

(53)




115

110

105

100

0 5 10 15 20

Axi

al te

nsio

n fo

rceH

T

Time t

(a)

0 5 10 15

28

23

18

13

08

Axi

al te

nsio

n fo

rceH

T

Time t

(b)

40

30

20

10

0Axi

al te

nsio

n fo

rceH

T

minus100 5 10 15


Time t

(c)

Axi

al te

nsio

n fo

rceH

T

0

5

10

15

0 1 2 3

Time t


(d)


(b) 119891 = 002 (Ω1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 1205961)1205961= minus010 (d) 119891 = 008 (Ω

1minus 1205961)1205961= 20


5 Conclusions






Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










115

110

105

100

0 5 10 15 20

Axi

al te

nsio

n fo

rceH

T

Time t

(a)

0 5 10 15

28

23

18

13

08

Axi

al te

nsio

n fo

rceH

T

Time t

(b)

40

30

20

10

0Axi

al te

nsio

n fo

rceH

T

minus100 5 10 15


Time t

(c)

Axi

al te

nsio

n fo

rceH

T

0

5

10

15

0 1 2 3

Time t


(d)


(b) 119891 = 002 (Ω1minus 1205961)1205961= 025 (c) 119891 = 004 (Ω

1minus 1205961)1205961= minus010 (d) 119891 = 008 (Ω

1minus 1205961)1205961= 20


5 Conclusions






Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article approximate series solutions for nonlinear ...homotopy analysis method agree well...

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