Research ArticleNonlinear Vibration of an Elastic Soft String Large Amplitudeand Large Curvature

De-Min Zhao Shan-Peng Li Yun Zhang and Jian-Lin Liu

Department of Engineering Mechanics College of Pipeline and Civil Engineering China University of Petroleum (East China)Qingdao 266580 China

Correspondence should be addressed to Jian-Lin Liu liujianlinupceducn

Received 12 April 2018 Accepted 7 June 2018 Published 22 July 2018

Academic Editor Carlos Llopis-Albert

Copyright copy 2018 De-Min 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

Mechanical nonlinear vibration of slender structures such as beams strings rods plates and even shells occurs extensivelyin a variety of areas spanning from aerospace automobile cranes ships offshore platforms and bridges to MEMSNEMS Inthe present study the nonlinear vibration of an elastic string with large amplitude and large curvature has been systematicallyinvestigated Firstly the mechanics model of the string undergoing strong geometric deformation is built based on the Hamiltonprinciple The nonlinear mode shape function was used to discretize the partial differential equation into ordinary differentialequation The modified complex normal form method (CNFM) and the finite difference scheme are used to calculate the criticalparameters of the string vibration including the time history diagram configuration total length and fundamental frequency Itis shown that the calculation results from these two methods are close which are different with those from the linear equationmodel The numerical results are also validated by our experiment and they take excellent agreement These analyses may behelpful to engineer some soft materials and can also provide insight into the design of elementary structures in sensors actuatorsand resonators etc

1 Introduction

Mechanical vibration of slender structures such as beamsstrings rods plates and even shells occurs widely in a vari-ety of areas spanning from aerospace automobile cranesships offshore platforms and bridges to MEMSNEMS(micronanoelectromechanical system) [1ndash5] On one handthe huge vibration amplitude can shorten the lifetime ofthe whole structures with the accumulation of damage orfatigue of materials [6] On the other hand the advantage ofmechanical vibration can be successfully utilized in a lot ofspectrums For instance at the nanoscale Wang et al useda situ transmission electron microscopy (TEM) to measurethe dynamic deflection of a cantilever made of multiwalledcarbon nanotube which was excited to resonance in TEM[7 8] Besides this Zheng and Jiang proposed the concept ofcreation of nanomechanical systems of operating frequencyup to several gigahertz based on the oscillation of a core in amultiwall carbon nanotube [9]

The most intriguing issue on vibration is its nonlineareffect as the strong nonlinearity can lead to many specialphenomena Up to now the nonlinear vibration of structureshas aroused extensive attention in the past decades A centraltask of the nonlinear vibration of beams is to seek theirnonlinear frequencies and much effort has been devotedto this problem For example Hemmatnezhad and Ansari[10] studied the frequency of a functionally graded beam bymeans of a finite element formulation In their model thevonKarman type nonlinear strainndashdisplacement relationshipis employed and the effects of transverse shear deformationare included based upon the Timoshenko beam theorySimilarly to further seek the nonlinear frequencies Gundaand Gupta [11] investigated the vibration of a compositebeam Nikkar and Bagheri [12] explored the cantilever beamwith an intermediate lumped mass and Yu andWu et al [13]studied the beam with immovable spring-hinged ends Morerelated works include that Raju and Rao [14] formulated thenonlinear vibration of the beam using multiterm admissible

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 7909876 11 pageshttpsdoiorg10115520187909876

2 Mathematical Problems in Engineering

functions for the firstmodeThenHe [15] used the variationalapproach to investigate the vibration frequency of a uniformcantilever beam carrying an intermediate lumped massHoseini and Pirbodaghi et al [16] presented the homotopyanalysis method to study the accurate analytical solution forthe nonlinear fundamental natural frequency of a taperedbeam with large amplitude vibration

At the micronanoscale Gheshlaghi [17] developed theEuler-Bernoulli beam model and calculated the nonlinearnatural frequencies of the first two modes of a nanowire Insuccession the harmonic balancemethod and the asymptoticnumerical method were combined to solve the equations of acomprehensivemultiphysicsmodel of cantilevermade of car-bon nanotube [18] Moreover using the different model iethe Eringenrsquos nonlocal elasticity theory Simsek [19] calculatedthe nonlinear vibration frequency of a nanobeamwith axiallyimmovable ends Similarly Nazemnezhad et al [20] got theexact solution for the nonlinear vibration of a nanobeam inuse of the nonlocal Euler-Bernoulli theory For the applica-tions of devices Feng et al [1] studied the nonlinear vibrationof a dielectric elastomer-based microbeam resonator wherethe gas damping and excitation are considered Han andZhang [2] designed a doubly clamped microresonator basedon the large amplitude vibration model Furthermore in theexperiments severalmeasurementmethods for the nonlinearvibration of slender beams were proposed [21 22]

Although much work has been done on the nonlinearvibration of beams and in most of references this behaviorwas termed as ldquolarge amplituderdquo however it should bestressed that only the normal strain with von Karman typewas considered in most of works In practice when a beamvibrates with large displacement its amplitude can evenamount to the value on the same order as its length Thatis to say the large amplitude vibration of the string shouldbe associated with large displacement large rotation andlarge curvature which should be taken into consideration[23] Pai and Nayfeh [24] studied the large deformation bodyespecially considering the large deformation and rotationwhich is a bit complex Babilioa and Lenci [25 26] gavedefinitions on mechanical and geometric curvatures andsimilarly Kopmaz and Gundogdu [27] presented differentconcepts on mathematical and physical curvatures With thesame idea Semler et al [28] developed a beam equation onthe nonlinear vibration of a pipe conveying fluid Zhao etal [29] used the similar model to investigate the nonlinearvibration of a nanobeam with surface effects In additionVlajic and Fitzgerald et al [30] studied the prestressed beamwith large variable curvature and they provided the analyticalformulation for static configurations natural frequenciesand mode shapes which were validated by the experimentand finite element method It is clearly seen that the above-mentioned bibliography mostly focuses on beams and littleliterature on string vibration has been mentioned AlthoughNayfeh and Mook [31] analyzed the nonlinear (free andforced) vibrations of strings and Benedettisi and Rega [32]studied the forced vibration of a suspended cable associatedwith quadratic and cubic nonlinearities only small amplitudeof the string is considered The vibration characteristics of alight axially moving band were investigated by Koivurova in

use of the FourierndashGalerkinndashNewtonmethod [33] where thevon Karman strain was taken into account

In the present study we do not concentrate on thevibration of beams or plates but mainly on an elastic stringor rod made of softmaterials such as rubber materials wherelarge deformation and curvature can often happen when itvibrates In this situation the string will experience a verylarge displacement and especially a large curvatureThis issueis not trivial as the equation is associated with very strongnonlinearity Although the complex normal form method(CNFM) developed by Nayfeh [34] was adopted to analyzethe nonlinear vibration [29] it works very well in weaknonlinear vibration systemThen Leung et al and Zhang et al[35 36]modified the traditional CNFM to predict the naturalfrequency in the strong nonlinear dynamics For the CNFMintroduced by Nayfeh [34] the fundamental frequency isindependent of parameters of the nonlinear terms of theequationwhich equals to the natural frequency of the derivedlinear system However for the modified CNFM [35 36] thefundamental frequency is unknown and needs to be deter-mined by the parameters of the nonlinear terms of the equa-tion

The outline of the paper is organized as follows InSection 2 the dynamics equation of an elastic string withlarge amplitude and large mechanical curvature is derivedbased on the Hamilton principle Next in Section 3 theGalerkinmethod using the nonlinear mode shape function isused to discretize the partial differential equation to ordinarydifferential equation Then the modified complex normalCNFM is adopted to get the semianalytical solution ofthe strong nonlinear vibration of the string in Section 4In Section 5 the semianalytical results including the timehistory diagram string configuration total string length andfundamental frequency are calculated which are comparedwith those from the numerical computation and our self-designed experiment Finally the conclusion is given inSection 6 Although the analysis is aiming to investigate thestring vibration the route of line can be extended to explorethe vibration of some other elastic structures such as rodsbeams plates and shells

2 Model Formulation

We consider an elastic string with an initial configurationof Line 119874119861 as shown in Figure 1 Refer to the Cartesiancoordinate system119874minus119909119910 where the origin is located at point119874 The original length of the string is 119871 and its cross-sectionarea is 119860 Youngrsquos modulus and mass density of the string are119864 and 120588 respectively and point 1198630 is the midpoint of thestring When the string starts to vibrate in the (119909 119910) planeits axis is elongated and the morphology is curvilinear whichis schematized in Figure 1We examine two arbitrary adjacentmaterial points1198620(119909 0) and11986210158400(119909+d119909 0) both in 119909-axis withinfinitesimal distance d119909 in the original configuration thenthey transfer to the positions 1198621(1199091 1199101)and 11986210158401(1199092 1199102) afterdeformation Considering the large deformation of the stringthe curvilinear coordinate is introduced where the arc length119904 is measured from the origin point 119874 along the axis of thestring

Mathematical Problems in Engineering 3

OB

L

L2

D0

C0(x0) C0(x+dx0)

＞s

C1(x1y1) C1(x2y2)

y w

ux

s

＞x

Figure 1 Schematic of the vibrational configuration of an elastic string with large deformation

According to Figure 1 the geometric relations of the stringare given as [28 29 31]

1199091 minus 119909 = 1199061199101 = 1199081199092 minus (119909 + d119909) = 119906 + d1199061199102 = 119908 + d119908

(1)

where 119906 and 119908 are the displacements of point 1198620 along 119909and 119910 directions respectively It can be seen that beforedeformation the microelement between 1198620 and 11986210158400 has thelength d119909 and it becomes d119904 after deformation As a resultthe expression of d119904 is derived as

d119904 = radic(1199092 minus 1199091)2 + (1199102 minus 1199101)2 = radic(1 minus 11990610158402)2 + 11990810158402 (2)

where 1199061015840 = d119906d119909 and 1199081015840 = d119908d119909 Due to the fact that thevalue of 119906 is much smaller than that of 119908 the contributionof 1199061015840 can be negligible in the vibration processTherefore thestrain of an arbitrary point in the string is given by

120576 = d119904 minus d119909d119909 = radic1 + 11990810158402 minus 1 (3)

In order to obtain the governing equation of the string wedeal with this problem by way of energy principle Firstly thekinetic energy of the string can be written as

119879 = int11987101205881198602d119909 (4)

where = d119908d119905 and 119905 is the time variableThe elastic strain energy is decomposed into twoportions

ie the first contribution originates from the pretensionforce of the string 1198730 and the second one comes from theelongation in the vibration process Consequently in a periodfrom 1199051 to 1199052 the strain energy can be expressed as

119880 = 1198641198602 int1199052

1199051

int1198710(1198730119864119860 + 120576)

2

d119909d119905 (5)

Considering there is no work from the nonconservativeforces the application of Hamilton principle yields

120575int11990521199051

(119879 minus 119880) d119905 = 0 (6)

Because we study the soft slender string the mechanicalcurvature is sufficiently accurate and usedmore convenientlyfor the integration [25 26] Therefore in use of the principleof variation the governing equation of the string in vibrationwith large amplitude and large mechanical curvature can bededuced as

120588119860 minus (1198730 minus 119864119860)( 11990810158401015840radic1 + 11990810158402 minus 1199081015840211990810158401015840(1 + 11990810158402)32)

minus 11986411986011990810158401015840 = 0(7)

The boundary conditions at two ends are

119908 (0 119905) = 0119908 (119871 119905) = 0 (8)

where = d2119908d1199052 and 11990810158401015840 = d2119908d1199092It can be seen that when the amplitude of the string is

big enough the term 11990810158402 cannot be ignored in (7) and thusthe governing equation is more complicated as it has a verygreat mechanical curvature As is well known the value of1199081015840 is close to zero when the vibration amplitude is small Ifthis term is omitted (7) can degenerate to the classical stringvibration equation whose amplitude and curvature are bothsmall and the equation is linear

= 119873012058811986011990810158401015840 (9)

For convenience the following nondimensional quantitiesare defined as

119908 = 119908119871 119909 = 119909119871 119903 = 1119871radic1198730120588119860120596 = 120587119903120591 = 120596119905

4 Mathematical Problems in Engineering

1205731 = (1 minus 1198641198601198730)1205872 1205732 = 11986411986011987301205872

(10)

Accordingly (7) can be recast as

= 120573111990810158401015840(1 + 11990810158402)12 minus 12057311199081015840211990810158401015840(1 + 11990810158402)32 + 120573211990810158401015840 (11)

and this equation is termed as the dimensionless largeamplitude vibration (LAV) equation throughout this study

Then the boundary conditions and initial conditions arerewritten as

119908 (0 120591) = 0119908 (1 120591) = 0119908 (119909 0) = 119886 sin (120587119909) (119909 0) = 0

(12)

At the same time the linear equation is nondimensionalizedas

= 1120587211990810158401015840 (13)

In the above equations the related expressions are =d2119908d1205912 1199081015840 = d119908d119909 and 11990810158401015840 = d2119908d1199092 and 119886 is theamplitude of the middle point in the string ie point11986303 Mode Discretization with Exact Mode Shape

The following task is to solve (11) which is an intractableproblem Clearly it seems impossible to directly get the ana-lytical solution of this highly nonlinear equation Thereforein use of the Taylor series (11) can be expanded into thefollowing polynomial expansion until 5th-order terms wherethe terms of higher order are ignored

= 119908101584010158401205872 + 1205731 (minus311990810158402

2 + 15119908101584048 )11990810158401015840 (14)

It is clear that (14) is still a high order and nonlinear partialdifferential equation (PDE) and seeking the closed formsolution is not at hand Herein the Galerkin discretizationmethod is utilized to transform the PDE into the ordinarydifferential equation (ODE) The previous result [18] tells usthat the first-mode analysis of vibration is sufficient to capturethe main nonlinear characteristics and can get the funda-mental frequency accurately enough Hence we take the trialto only consider the first mode of the vibration aiming toget an approximate solution for the first step Moreover themode shape is a critical factor to affect the vibration of thecontinuously beam rod and string For the small amplitudevibration of string with simply supported condition the first-mode shape function is often assumed by linear mode shape

function [31] described by sin(120587119909) However for the largecurvature vibration especially the amplitude of the middlepoint nearly 04-05 times of span of simply supported stringthe mode shape function sin(120587119909) based on the linear themode shape fails The exact mode shape function shouldbe selected considering the physical experiments given inSection 53 Assume that the displacement 119908(119909 120591) can bedescribed as 119908 (119909 120591) = 119902 (120591)119882 (119909) (15)and

119882(119909) = sin (120587119909)radic1 + cos2 (120587119909) (16)

Substituting (15) and (16) into (14) multiplying both sidesof the equation by 119882(119909) and then integrating both sidesfrom 0 to 1 yields the following ODE equation with strongnonlinearities 119902 + 12059620119902 + 11989611199023 + 11989621199025 = 0 (17)

where 12059620 =10745 1198961 = minus818361205731 and 1198962 = minus5351491205731In order to compare the large curvature vibration with

the small amplitude vibration for the linear equation in (13)the classic mode shape function sin(120587119909) is adopted thusthe classic the dimensionless ODE can degenerate to 119902 +119902 = 0 Then one can obtain the theoretical solution easilywhich is named as theoretical solution of the linear equationthroughout the paper

4 Semianalytical Solution UsingCNFM Method

Equation (17) is strong nonlinear system due to the nonlinearterms q3 and q5 which are far more large than the linear term119902 Although Hersquos variational method [6 12 15 19] harmonicbalance method [18] and Homotopy analysis method [16]are used widely in the strong nonlinear system the modifiedCNFMmethod [29 34ndash36] gives more accurate results basedon the experience of the authors Next we use the modifiedCNFM approach to solve (17) We assume that the solution of(17) can be formulated as

119902 = 120585 + 120585119902 = 1198941205961 (120585 minus 120585) (18)

where 120585 is the complex conjugate of 120585 and 1205961 is the unknownfundamental natural frequency to be determined Introducea nonlinear transformation from 120585 to 120578 in the form of120585 = 120578 + ℎ (120578 120578) (19)where 120578 and 120578 are complex conjugates and they are bothcomplex functions

The near identity transformation function ℎ including thevariables 120578 and 120578 is expressed as

ℎ (120578 120578) = Δ 1120578 + Δ 2120578 + Δ 31205783 + Δ 41205782120578 + Δ 51205781205782+ Δ 61205783 + Δ 71205785 + Δ 81205784120578 + Δ 912057831205782+ Δ 1012057821205783 + Δ 111205781205784 + Δ 121205785(20)

where Δ 119894 (119894=1 12) are real numbers

Mathematical Problems in Engineering 5

Substituting (18)ndash(20) into (17) one can get

120578 = 1198941205961120578 + 1198941205961ℎ minus 120597ℎ120597120578 120578 minus 120597ℎ120597120578 + 11989412059612 (1205962012059621 minus 1)

sdot (120578 + 120578 + ℎ + ℎ)+ 11989421205961 [1198961 (120578 + 120578 + ℎ + ℎ)3 + 1198962 (120578 + 120578 + ℎ + ℎ)5]

(21)

where ℎ and ℎ are complex conjugates Substituting (19) and(20) into (18) leads to

119902 = 120578 + 120578 + (Δ 1 + Δ 2) (120578 + 120578) + (Δ 3 + Δ 6) (1205783 + 1205783)+ (Δ 4 + Δ 5) (1205782120578 + 1205781205782) + (Δ 7 + Δ 12) (1205785 + 1205785)+ (Δ 8 + Δ 11) (1205784120578 + 1205781205784)+ (Δ 9 + Δ 10) (12057831205782 + 12057821205783)

(22)

To eliminate the secular terms 120578 1205782120578 and 12057831205782 in (22) thefollowing conditions should be met

Δ 1 + Δ 2 = 0 (23)

Δ 4 + Δ 5 = 0 (24)

Δ 9 + Δ 10 = 0 (25)

Moreover let the coefficients of terms 120578 1205783 1205781205782 1205783 1205785 120578412057812057821205783 1205781205784 and 1205785 in the right side of (22) vanish as they arethe nonresonant terms which are much smaller than term120578 Thus twelve equations are obtained in Appendix Solving(23)ndash(25) and the related equations in Appendix we canacquire the parameters Δ 119894 (119894=1 12) which are given inAppendix

After these operations the resonant terms 120578 1205782120578 and 12057831205782will remain in (21) and the equation is simplified to

120578 = 1198941 + 1205962121205961 120578 + 119894 3119896121205961 1205782120578+ 119894 121205961 (101198962 minus 31198961Δ 6) 12057831205782

(26)

where the real parameters Δ 6 is given in AppendixAssume

120578 = 121198861198901198941205961119905 (27)

where 119886 is the dimensionless vibration amplitude and 1205961is the fundamental frequency which is to be determinedSubstituting (27) into (26) leads to

119886 = 01205961 = 12059620 + 1205962121205961 + 3119896181205961 1198862 + 101198962 minus 31198961Δ 6321205961 1198864 (28)

The first equation in (28) indicates that the amplitude 119886is a constant determined by the initial condition and thesecond equation determines the fundamental frequency 1205961Substituting Δ 119894 (119894=1 12) 119886 and 1205961 into (22) one can getthe final solution of (17) as

119902 = 119886 cos (1205961119905)+ [14 (Δ 3 + Δ 6) 1198863 + 116 (Δ 8 + Δ 11) 1198865]sdot cos (31205961119905) + 116 (Δ 7 + Δ 12) 1198865cos (51205961119905)

(29)

5 Results and Discussion

51 Numerical Scheme To get a more accurate solution thefinite difference method (FDM) is used to directly calculatethe PDE which manifests a robust method The centraldifference is adopted ie 1199081015840 = (119908119899119895+1 minus 119908119899119895minus1)2(Δ119909)11990810158401015840 = (119908119899119895+1 minus 2119908119899119895 + 119908119899119895minus1)(Δ119909)2 and = (119908119899+1119895 minus2119908119899119895 + 119908119899minus1119895 )(Δ120591)2 where 119895 and 119899 are both integers andΔ119909 and Δ120591 are the space and time steps respectively In thesimulation process the string is made of soft material suchas rubberThe physical parameters of the string are measuredas follows mass density 120588=798 kgm3 length 119871=6877 mmcross-section area 119860=2282 mm2 pretension force along theaxial direction 1198730=068 N and Youngrsquos modulus 119864=085MPa The computational program is written in MATLAB 12where the time step Δ120591 is set as 0001 and the space step Δ119909 isset as 001 In the FDMdiscretization when 120591 = 0 and 120591 = Δ120591the displacement120596119894119895 = 119886 sin[120587(Δ119909)(119895minus1)] (119894=1 2) and 119895 takesthe value from 1 to 101

52 Time History Diagram of the Midpoint Evidently themidpoint D0 is a critical point which should be carefullyexamined as it may be associated with the largest amplitudeof the string When the initial amplitude of D0 is biggersuch as 119886=04 and 045 the related curves in the time historydiagram are displayed in Figure 2 It can be seen that withthe evolution of time the difference between the LAV modeland the linear equation becomesmore obvious and the resultfrom the semianalytical solution is in agreement with thenumerical result although there is slight difference as shownin Figure 2 As displayed in Figure 2(a) in the time intervalfrom 0 to 20 there are nearly 4 periods in the LAV modeland only 3 periods in the linear equation model It can alsobe observed when 120591 = 157 that the motion of the pointpredicted by the LAV equation is nearly one-half period fasterthan that by the linear equationThis indicates that the wholemotion of the string predicted by the LAV equation is quickerthan that predicted by the linear equation

53 Vibrational Configurations In what follows we computethe configurations of the string with large amplitude a=04in one period which are shown in Figures 3(a)ndash3(h) Firstlyit is noted that there is great difference between the resultsfrom the LAVmodel and the linear equationmodel as shownin Figure 3 This again stresses that the linear equation is

6 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus08

minus04

00

04

08

w

5 10 15 200

(a)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

50 15 2010

minus08

minus04

00

04

08

w

(b)

Figure 2 Time history diagram of point1198630 with the initial amplitudes (a) 119886 = 04 and (b) 119886 = 045

insufficient to depict the vibration with large amplitude andlarge curvature However in the whole period although thereis difference between the results from the semianalyticalmethod and numerical simulation the configurations pre-dicted by the two methods are close Therefore it should bementioned that the calculated string morphologies by thesemianalytical are more similar to those from the numericalresults than those from the linear equation model

The next phenomenon is that the numerical morpholo-gies of the string are very complicated in the vibrationprocess For instance when 120591=53 60 80 and 105 in Figures3(a) 3(b) 3(d) and 3(h) there are some nearly horizontalsegments appearing in the curves This strange behavior maybe attributed to the complex nonlinear dynamics responsewhere the clamped ends exert strong constraints on thestring and vice versa near the middle part of the stringthe constraint is much weaker Another possible reasonis that the first-mode analysis of the string may be anoversimplified model and the higher-mode behaviors mustbe further considered to illustrate this phenomenon Further-more our semianalytical solution can also have the similarshapes in this figures especially near the middle part of thestring

To verify this numerical result we perform an experimentas a comparison where the string configurations in thevibration are shown in Figures 4(a)ndash4(h) A rubber stringwith 1408times1630mm2 cross-section is fixed horizontally on adesktop at room temperature whose original lengthL is 5085mm To add the pretension force on the string the deformedlength of the string is elongated to 119897=6877 mm and originarea of the cross-section is 2282 mm2 Thus the prestraincan be given as 120576=(lndashL)L=03524 and using the measuredvalue of Youngrsquos modulus 119864=085 MPa the pretension forceis N0=068N At first in the vertical plane the midpoint ofthe string is excited by an original amplitude which is 04times of the string length 119871 Next the string is released and

then its vibration sequences are recorded by the high speedcamera (Phantom v2512 with 10000 frames per second) Asshown in Figures 4(a) 4(b) 4(c) and 4(d) we find that thereare really platform segments appearing in the string and thetotal configurations in these situations are close to the shapeof trapezoid Meanwhile the string is no longer a straightline when it approaches the equilibrium position as shownin Figures 4(e) and 4(f) implying that the length of the stringis bigger than its original length These behaviors are beyondour imagination as they demonstrate the complexity of thenonlinear dynamics

54 Total String Length It can be seen that in both the LAVequation and the linear equation the extensibility of thestring is considered thus its length can be calculated andthe two curves are compared in Figure 5 The nondimen-sional length of the string after deformation is expressed as119897119871 = int119871

0radic1 + 1199082d119909119871 For the linear equation model this

nondimensional length is a periodic function with respectto the time 120591 ie 119897119871(120591 + 1198792) = 119897119871(120591) This means thatthe maximum and minimum values of the string length donot alter in the vibration process Especially when the stringapproaches the 119909-axis it is really a straight line and thus thevalue of 119897119871 is of the smallest value ie one This featurereemphasizes that the small amplitude vibration has linearproperties On the contrary the oscillation behavior of thestring length is more complex in the LAV model whereits minimum value increases with the elapsing time In thissituation the dimensionless string length is always biggerthan one indicating that the string is always in the elongationstate even when it approaches the 119909-axis In the experimentsuch as in Figures 4(e) 4(f) 3(e) and 3(f) the shapes of thestring are notmerely straight lines and its length at this criticaltime must be larger than one Evidently this consistence onthe experimental phenomenon can validate the efficiency ofour numerical method

Mathematical Problems in Engineering 7

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(a)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(b)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(c)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(d)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(e)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(f)

Figure 3 Continued

8 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06w

(g)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(h)

Figure 3 Vibrational configurations of an elastic string when (a) 120591=53 (b) 120591 = 60 (c) 120591 = 70 (d) 120591 = 80 (e) 120591 = 90 (f) 120591 = 93 (g)120591 = 10 (h) 120591 = 105

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4 Vibrational configurations of the string in the experiment

55 Fundamental Frequency Another vibration characteris-tic parameter about vibration is the frequency and in thecurrent study the value of the fundamental frequency isobtained by the solution of (28) and it is also tested by theexperimental measurement When the vibration has a smallamplitude vibration of the string its dynamic behavior canbe formulated by the linear equation 119902 + 119902 = 0 with thenatural frequency always being 1205961 = 1 which is irrelative tothe initial amplitude We observed in Figure 6 the differencebetween the semianalytical method and experiment methodis great The main reason is that for the small amplitudethe mode shape function should be sin(120587119909) rather thansin(120587119909)radic1 + cos2(120587119909) The frequency exhibited in Figure 6demonstrates that our mode shape function should not be

used for the small amplitude vibration However for thelarge amplitude vibration the fundamental frequency is nota constant and its value is greatly affected by the initialamplitude As shown in Figure 6 with the increase of theamplitude the fundamental frequency also increases whichis verified by the experimental results Although there exitsvalue difference between the semianalytical solution and theexperimental data their tendencies are the same and bothof the two results are bigger than one with the increase ofinitial amplitude a This behavior again demonstrates thecomplicated response of the nonlinear dynamics which isvery sensitive to the initial value It should be clarified that ourmode shape function only for the large amplitude vibrationwith the ratio (amplitude of the middle part of the string

Mathematical Problems in Engineering 9

LAV equation numerical solutionLinear equation theoretical solution

1510 200 5

10

11

12

13

14

lL

Figure 5 Total length of the string based on the LAV equation andlinear equation

LAV equation semianalytical solutionLinear equation theoretical solutionExperiment

100

104

108

112

116

120

1

01 02 03 0400a

Figure 6 Relation between the fundamental 1205961 and the initialamplitude a

to the whole span) is nearly 02-05 If the ratio is greaterdifferent mode shape functions should be selected and thesimilar analysis process can be carried out

6 Conclusion

In conclusion the nonlinear vibration of a soft elasticstring with large amplitude and large curvature has beensystematically investigated in this study The mathematicalmodel in the absence of damping is developed based on theHamilton principle The exact mode shape function differentfrom the counterpart derived from the linear equation wasselected based on the experiment For the large amplitudevibration the time history diagram from the semianalyticalis very close to that from the numerical result and thesetwo results are very different from that of the linear modelThe vibration configurations of the elastic string have thesimilar laws as those of the time history diagrams based on

the comparison of the three models We find that in thenumerical calculations there are platform segments with theshape of trapezoid appearing in the configuration curvesand this phenomenon has been verified by our semianalyticalsolution and the experiment The total length of the string inthe large deformation shows that it is not a periodic functionwith respect to the time which is distinct with the result ofthe linear model We also point out that the fundamentalfrequency of the string with large amplitude vibration isgreatly affected by the initial amplitude which has the sametendency as our experiment result

It should be mentioned that this study is only the firsttrial on the nonlinear vibration of elastic structures withlarge amplitude and large curvatures The following work isexpected which will be extended to some other engineeringstructures such as beams rods plates and shells Theseanalyses may be beneficial to engineer some soft materialsand can also shed light on the design of elementary structuresin sensors actuators and resonators etc

Appendix

A Equations to Determine Coefficients ofNonresonant Terms

Besides (23)-(25) the other equations to determine the valueof Δ 119894 (119894=1 12) are

21198941205961Δ 2 + 119894 (12059620 minus 12059621)21205961 = 021198941205961Δ 3 minus 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) minus 119894119896121205961 = 021198941205961Δ 5 + 3119894119896121205961 = 041198941205961Δ 6 + 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) + 119894119896121205961 = 041198941205961Δ 7 minus 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12) minus 3119894119896121205961 (Δ 3 + Δ 6)minus 119894119896221205961 = 0

21198941205961Δ 8 minus 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11) minus 6119894119896121205961 (Δ 3 + Δ 6)minus 5119894119896221205961 = 0

21198941205961Δ 10 + 3119894119896121205961 (Δ 3 + Δ 6) + 10119894119896221205961 = 041198941205961Δ 11 + 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11)+ 6119894119896121205961 (Δ 3 + Δ 6) + 5119894119896221205961 = 0

10 Mathematical Problems in Engineering

61198941205961Δ 12 + 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12)+ 3119894119896121205961 (Δ 3 + Δ 6) + 119894119896221205961 = 0

(A1)

B Formulas of Parameters Δ 119894The parameters Δ 119894 ( i=1 12) are

Δ 1 = minusΔ 2Δ 2 = 12059621 minus 12059620412059621 Δ 3 = minus2Δ 6Δ 6 = 119896112059620 minus 912059621 Δ 4 = minusΔ 5Δ 5 = minus 31198961412059621 Δ 7 = minus32Δ 12Δ 12 = 61198961Δ 6 minus 211989622512059621 minus 12059620 Δ 8 = minus2Δ 11Δ 11 = 61198961Δ 6 minus 51198962912059621 minus 12059620 Δ 9 = minusΔ 10Δ 10 = 31198961Δ 6 minus 101198962412059621

(B1)

Data Availability

No data were used to support this study

Ethical Approval

Ethical approval was obtained for this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study was supported by the National Natural Sci-ence Foundation of China (nos 11672334 11672335 and11611530541) and the Endeavour Australia Cheung KongResearch Fellowship Scholarship from theAustralian govern-ment

References

[1] C Feng L Yu andW Zhang ldquoDynamic analysis of a dielectricelastomer-based microbeam resonator with large vibrationamplituderdquo International Journal of Non-Linear Mechanics vol65 pp 63ndash68 2014

[2] J Han Q Zhang and W Wang ldquoDesign considerations onlarge amplitude vibration of a doubly clamped microresonatorwith two symmetrically located electrodesrdquo Communications inNonlinear Science andNumerical Simulation vol 22 no 1ndash3 pp492ndash510 2015

[3] T Belendez CNeipp andA Belendez ldquoLarge and small deflec-tions of a cantilever beamrdquo European Journal of Physics vol 23no 3 pp 371ndash379 2002

[4] L Wang and Y Zhao ldquoLarge amplitude motion mechanismand non-planar vibration character of stay cables subject to thesupport motionsrdquo Journal of Sound and Vibration vol 327 no1-2 pp 121ndash133 2009

[5] M Spagnuolo and U Andreaus ldquoA targeted review on largedeformations of planar elastic beams extensibility distributedloads buckling and post-bucklingrdquoMathematics andMechanicsof Solids pp 1ndash23 2018

[6] M P Omran A Amani and H G Lemu ldquoAnalytical approxi-mation of nonlinear vibration of string with large amplitudesrdquoJournal of Mechanical Science and Technology vol 27 no 4 pp981ndash986 2013

[7] Z L Wang P Poncharal andW A De Heer ldquoMeasuring phys-ical and mechanical properties of individual carbon nanotubesby in situ TEMrdquo Journal of Physics and Chemistry of Solids vol61 no 7 pp 1025ndash1030 2000

[8] Z LWang R P Gao P PoncharalWADeHeer Z RDai andZ W Pan ldquoMechanical and electrostatic properties of carbonnanotubes and nanowiresrdquo Materials Science and EngineeringC Materials for Biological Applications vol 16 no 1-2 pp 3ndash102001

[9] Q Zheng and Q Jiang ldquoMultiwalled Carbon Nanotubes asGigahertz Oscillatorsrdquo Physical Review Letters vol 88 no 42002

[10] M Hemmatnezhad R Ansari and G H Rahimi ldquoLarge-amplitude free vibrations of functionally graded beams bymeans of a finite element formulationrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 37 no 18-19 pp 8495ndash8504 2013

[11] J B Gunda R K Gupta G Ranga Janardhan and G Venkates-wara Rao ldquoLarge amplitude vibration analysis of compositebeams Simple closed-form solutionsrdquo Composite Structuresvol 93 no 2 pp 870ndash879 2011

[12] A Nikkar S Bagheri and M Saravi ldquoDynamic model of largeamplitude vibration of a uniform cantilever beam carrying anintermediate lumped mass and rotary inertiardquo Latin AmericanJournal of Solids and Structures vol 11 no 2 pp 320ndash329 2014

[13] Y P Yu B S Wu Y H Sun and L Zang ldquoAnalytical approxi-mate solutions to large amplitude vibration of a spring-hingedbeamrdquoMeccanica vol 48 no 10 pp 2569ndash2575 2013

[14] K K Raju andGV Rao ldquoTowards improved evaluation of largeamplitude free-vibration behaviour of uniform beams usingmulti-term admissible functionsrdquo Journal of Sound and Vibra-tion vol 282 no 3-5 pp 1238ndash1246 2005

[15] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons and Fractals vol 34 no 5 pp 1430ndash1439 2007

[16] S H Hoseini T Pirbodaghi M T Ahmadian and G H Far-rahi ldquoOn the large amplitude free vibrations of tapered beams

Mathematical Problems in Engineering 11

an analytical approachrdquo Mechanics Research Communicationsvol 36 no 8 pp 892ndash897 2009

[17] B Gheshlaghi ldquoLarge-amplitude vibrations of nanowires withdissipative surface stress effectsrdquo Acta Mechanica vol 224 no7 pp 1329ndash1334 2013

[18] S Souayeh and N Kacem ldquoComputational models for largeamplitude nonlinear vibrations of electrostatically actuatedcarbon nanotube-basedmass sensorsrdquo Sensors and Actuators APhysical vol 208 pp 10ndash20 2014

[19] M Simsek ldquoLarge amplitude free vibration of nanobeams withvarious boundary conditions based on the nonlocal elasticitytheoryrdquo Composites Part B Engineering vol 56 pp 621ndash6282014

[20] R Nazemnezhad and S Hosseini-Hashemi ldquoExact solution forlarge amplitude flexural vibration of nanobeams using non-local Euler-Bernoulli theoryrdquo Journal ofTheoretical and AppliedMechanics (Poland) vol 55 no 2 pp 649ndash658 2017

[21] K Nakamura K Kakihara M Kawakami and S Ueha ldquoMea-suring vibration characteristics at large amplitude region ofmaterials for high power ultrasonic vibration systemrdquoUltrason-ics vol 38 no 1 pp 122ndash126 2000

[22] AMozuras and E Podzharov ldquoMeasurement of large harmonicvibration amplitudesrdquo Journal of Sound and Vibration vol 271no 3-5 pp 985ndash998 2004

[23] J C Simo ldquoA finite strain beam formulation The three-dimen-sional dynamic problem Part Irdquo Computer Methods AppliedMechanics and Engineering vol 49 no 1 pp 55ndash70 1985

[24] P-F Pai and A-H Nayfeh ldquoA new method for the modeling ofgeometric nonlinearities in structuresrdquoComputersampStrucruresvol 53 no 4 pp 877ndash895 1994

[25] E Babilio and S Lenci ldquoOn the notion of curvature andits mechanical meaning in a geometrically exact plane beamtheoryrdquo International Journal of Mechanical Sciences vol 128-129 pp 277ndash293 2017

[26] S Lenci F Clementi and G Rega ldquoComparing Nonlinear FreeVibrations of Timoshenko Beams with Mechanical or Geo-metric Curvature Definitionrdquo Procedia IUTAM vol 20 pp 34ndash41 2017

[27] O Kopmaz and O Gundogdu ldquoOn the curvature of an Euler-Bernoulli beamrdquo International Journal of Mechanical Engineer-ing Education vol 31 no 2 pp 132ndash142 2003

[28] C Semler G X Li andM P Paıdoussis ldquoThe non-linear equa-tions of motion of pipes conveying fluidrdquo Journal of Sound andVibration vol 169 no 5 pp 577ndash599 1994

[29] D Zhao J Liu and L Wang ldquoNonlinear free vibration of acantilever nanobeam with surface effects Semi-analytical solu-tionsrdquo International Journal of Mechanical Sciences vol 113 pp184ndash195 2016

[30] N Vlajic T Fitzgerald V Nguyen and B BalachandranldquoGeometrically exact planar beams with initial pre-stress andlarge curvature Static configurations natural frequencies andmode shapesrdquo International Journal of Solids and Structures vol51 no 19-20 pp 3361ndash3371 2014

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

[32] F Benedettini and G Rega ldquoNon-linear dynamics of an elasticcable under planar excitationrdquo International Journal of Non-Linear Mechanics vol 22 no 6 pp 497ndash509 1987

[33] H Koivurova ldquoThe numerical study of the nonlinear dynamicsof a light axiallymoving stringrdquo Journal of Sound andVibrationvol 320 no 1-2 pp 373ndash385 2009

[34] A H NayfehTheMethod of Normal Forms Wiley-VCHWein-heim Germany 2nd edition 2011

[35] A Y Leung and Q C Zhang ldquoComplex normal form forstrongly non-linear vibration systems exemplified by Duffing-van der Pol equationrdquo Journal of Sound and Vibration vol 213no 5 pp 907ndash914 1998

[36] W Wang Q C Zhang and X J Wang ldquoApplication of theundetermined fundamental frequency method for analyzingthe critical value of chaosrdquo Acta Physica Sinica vol 58 no 8pp 5162ndash5168 2009

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Mathematical Problems in Engineering 3

OB

L

L2

D0

C0(x0) C0(x+dx0)

＞s

C1(x1y1) C1(x2y2)

y w

ux

s

＞x

Figure 1 Schematic of the vibrational configuration of an elastic string with large deformation

According to Figure 1 the geometric relations of the stringare given as [28 29 31]

1199091 minus 119909 = 1199061199101 = 1199081199092 minus (119909 + d119909) = 119906 + d1199061199102 = 119908 + d119908

(1)

where 119906 and 119908 are the displacements of point 1198620 along 119909and 119910 directions respectively It can be seen that beforedeformation the microelement between 1198620 and 11986210158400 has thelength d119909 and it becomes d119904 after deformation As a resultthe expression of d119904 is derived as

d119904 = radic(1199092 minus 1199091)2 + (1199102 minus 1199101)2 = radic(1 minus 11990610158402)2 + 11990810158402 (2)

where 1199061015840 = d119906d119909 and 1199081015840 = d119908d119909 Due to the fact that thevalue of 119906 is much smaller than that of 119908 the contributionof 1199061015840 can be negligible in the vibration processTherefore thestrain of an arbitrary point in the string is given by

120576 = d119904 minus d119909d119909 = radic1 + 11990810158402 minus 1 (3)

In order to obtain the governing equation of the string wedeal with this problem by way of energy principle Firstly thekinetic energy of the string can be written as

119879 = int11987101205881198602d119909 (4)

where = d119908d119905 and 119905 is the time variableThe elastic strain energy is decomposed into twoportions

ie the first contribution originates from the pretensionforce of the string 1198730 and the second one comes from theelongation in the vibration process Consequently in a periodfrom 1199051 to 1199052 the strain energy can be expressed as

119880 = 1198641198602 int1199052

1199051

int1198710(1198730119864119860 + 120576)

2

d119909d119905 (5)

Considering there is no work from the nonconservativeforces the application of Hamilton principle yields

120575int11990521199051

(119879 minus 119880) d119905 = 0 (6)

Because we study the soft slender string the mechanicalcurvature is sufficiently accurate and usedmore convenientlyfor the integration [25 26] Therefore in use of the principleof variation the governing equation of the string in vibrationwith large amplitude and large mechanical curvature can bededuced as

120588119860 minus (1198730 minus 119864119860)( 11990810158401015840radic1 + 11990810158402 minus 1199081015840211990810158401015840(1 + 11990810158402)32)

minus 11986411986011990810158401015840 = 0(7)

The boundary conditions at two ends are

119908 (0 119905) = 0119908 (119871 119905) = 0 (8)

where = d2119908d1199052 and 11990810158401015840 = d2119908d1199092It can be seen that when the amplitude of the string is

big enough the term 11990810158402 cannot be ignored in (7) and thusthe governing equation is more complicated as it has a verygreat mechanical curvature As is well known the value of1199081015840 is close to zero when the vibration amplitude is small Ifthis term is omitted (7) can degenerate to the classical stringvibration equation whose amplitude and curvature are bothsmall and the equation is linear

= 119873012058811986011990810158401015840 (9)

For convenience the following nondimensional quantitiesare defined as

119908 = 119908119871 119909 = 119909119871 119903 = 1119871radic1198730120588119860120596 = 120587119903120591 = 120596119905

4 Mathematical Problems in Engineering

1205731 = (1 minus 1198641198601198730)1205872 1205732 = 11986411986011987301205872

(10)

Accordingly (7) can be recast as

= 120573111990810158401015840(1 + 11990810158402)12 minus 12057311199081015840211990810158401015840(1 + 11990810158402)32 + 120573211990810158401015840 (11)

and this equation is termed as the dimensionless largeamplitude vibration (LAV) equation throughout this study

Then the boundary conditions and initial conditions arerewritten as

119908 (0 120591) = 0119908 (1 120591) = 0119908 (119909 0) = 119886 sin (120587119909) (119909 0) = 0

(12)

At the same time the linear equation is nondimensionalizedas

= 1120587211990810158401015840 (13)

In the above equations the related expressions are =d2119908d1205912 1199081015840 = d119908d119909 and 11990810158401015840 = d2119908d1199092 and 119886 is theamplitude of the middle point in the string ie point11986303 Mode Discretization with Exact Mode Shape

The following task is to solve (11) which is an intractableproblem Clearly it seems impossible to directly get the ana-lytical solution of this highly nonlinear equation Thereforein use of the Taylor series (11) can be expanded into thefollowing polynomial expansion until 5th-order terms wherethe terms of higher order are ignored

= 119908101584010158401205872 + 1205731 (minus311990810158402

2 + 15119908101584048 )11990810158401015840 (14)

It is clear that (14) is still a high order and nonlinear partialdifferential equation (PDE) and seeking the closed formsolution is not at hand Herein the Galerkin discretizationmethod is utilized to transform the PDE into the ordinarydifferential equation (ODE) The previous result [18] tells usthat the first-mode analysis of vibration is sufficient to capturethe main nonlinear characteristics and can get the funda-mental frequency accurately enough Hence we take the trialto only consider the first mode of the vibration aiming toget an approximate solution for the first step Moreover themode shape is a critical factor to affect the vibration of thecontinuously beam rod and string For the small amplitudevibration of string with simply supported condition the first-mode shape function is often assumed by linear mode shape

function [31] described by sin(120587119909) However for the largecurvature vibration especially the amplitude of the middlepoint nearly 04-05 times of span of simply supported stringthe mode shape function sin(120587119909) based on the linear themode shape fails The exact mode shape function shouldbe selected considering the physical experiments given inSection 53 Assume that the displacement 119908(119909 120591) can bedescribed as 119908 (119909 120591) = 119902 (120591)119882 (119909) (15)and

119882(119909) = sin (120587119909)radic1 + cos2 (120587119909) (16)

Substituting (15) and (16) into (14) multiplying both sidesof the equation by 119882(119909) and then integrating both sidesfrom 0 to 1 yields the following ODE equation with strongnonlinearities 119902 + 12059620119902 + 11989611199023 + 11989621199025 = 0 (17)

where 12059620 =10745 1198961 = minus818361205731 and 1198962 = minus5351491205731In order to compare the large curvature vibration with

the small amplitude vibration for the linear equation in (13)the classic mode shape function sin(120587119909) is adopted thusthe classic the dimensionless ODE can degenerate to 119902 +119902 = 0 Then one can obtain the theoretical solution easilywhich is named as theoretical solution of the linear equationthroughout the paper

4 Semianalytical Solution UsingCNFM Method

Equation (17) is strong nonlinear system due to the nonlinearterms q3 and q5 which are far more large than the linear term119902 Although Hersquos variational method [6 12 15 19] harmonicbalance method [18] and Homotopy analysis method [16]are used widely in the strong nonlinear system the modifiedCNFMmethod [29 34ndash36] gives more accurate results basedon the experience of the authors Next we use the modifiedCNFM approach to solve (17) We assume that the solution of(17) can be formulated as

119902 = 120585 + 120585119902 = 1198941205961 (120585 minus 120585) (18)

where 120585 is the complex conjugate of 120585 and 1205961 is the unknownfundamental natural frequency to be determined Introducea nonlinear transformation from 120585 to 120578 in the form of120585 = 120578 + ℎ (120578 120578) (19)where 120578 and 120578 are complex conjugates and they are bothcomplex functions

The near identity transformation function ℎ including thevariables 120578 and 120578 is expressed as

ℎ (120578 120578) = Δ 1120578 + Δ 2120578 + Δ 31205783 + Δ 41205782120578 + Δ 51205781205782+ Δ 61205783 + Δ 71205785 + Δ 81205784120578 + Δ 912057831205782+ Δ 1012057821205783 + Δ 111205781205784 + Δ 121205785(20)

where Δ 119894 (119894=1 12) are real numbers

Mathematical Problems in Engineering 5

Substituting (18)ndash(20) into (17) one can get

120578 = 1198941205961120578 + 1198941205961ℎ minus 120597ℎ120597120578 120578 minus 120597ℎ120597120578 + 11989412059612 (1205962012059621 minus 1)

sdot (120578 + 120578 + ℎ + ℎ)+ 11989421205961 [1198961 (120578 + 120578 + ℎ + ℎ)3 + 1198962 (120578 + 120578 + ℎ + ℎ)5]

(21)

where ℎ and ℎ are complex conjugates Substituting (19) and(20) into (18) leads to

119902 = 120578 + 120578 + (Δ 1 + Δ 2) (120578 + 120578) + (Δ 3 + Δ 6) (1205783 + 1205783)+ (Δ 4 + Δ 5) (1205782120578 + 1205781205782) + (Δ 7 + Δ 12) (1205785 + 1205785)+ (Δ 8 + Δ 11) (1205784120578 + 1205781205784)+ (Δ 9 + Δ 10) (12057831205782 + 12057821205783)

(22)

To eliminate the secular terms 120578 1205782120578 and 12057831205782 in (22) thefollowing conditions should be met

Δ 1 + Δ 2 = 0 (23)

Δ 4 + Δ 5 = 0 (24)

Δ 9 + Δ 10 = 0 (25)

Moreover let the coefficients of terms 120578 1205783 1205781205782 1205783 1205785 120578412057812057821205783 1205781205784 and 1205785 in the right side of (22) vanish as they arethe nonresonant terms which are much smaller than term120578 Thus twelve equations are obtained in Appendix Solving(23)ndash(25) and the related equations in Appendix we canacquire the parameters Δ 119894 (119894=1 12) which are given inAppendix

After these operations the resonant terms 120578 1205782120578 and 12057831205782will remain in (21) and the equation is simplified to

120578 = 1198941 + 1205962121205961 120578 + 119894 3119896121205961 1205782120578+ 119894 121205961 (101198962 minus 31198961Δ 6) 12057831205782

(26)

where the real parameters Δ 6 is given in AppendixAssume

120578 = 121198861198901198941205961119905 (27)

where 119886 is the dimensionless vibration amplitude and 1205961is the fundamental frequency which is to be determinedSubstituting (27) into (26) leads to

119886 = 01205961 = 12059620 + 1205962121205961 + 3119896181205961 1198862 + 101198962 minus 31198961Δ 6321205961 1198864 (28)

The first equation in (28) indicates that the amplitude 119886is a constant determined by the initial condition and thesecond equation determines the fundamental frequency 1205961Substituting Δ 119894 (119894=1 12) 119886 and 1205961 into (22) one can getthe final solution of (17) as

119902 = 119886 cos (1205961119905)+ [14 (Δ 3 + Δ 6) 1198863 + 116 (Δ 8 + Δ 11) 1198865]sdot cos (31205961119905) + 116 (Δ 7 + Δ 12) 1198865cos (51205961119905)

(29)

5 Results and Discussion

51 Numerical Scheme To get a more accurate solution thefinite difference method (FDM) is used to directly calculatethe PDE which manifests a robust method The centraldifference is adopted ie 1199081015840 = (119908119899119895+1 minus 119908119899119895minus1)2(Δ119909)11990810158401015840 = (119908119899119895+1 minus 2119908119899119895 + 119908119899119895minus1)(Δ119909)2 and = (119908119899+1119895 minus2119908119899119895 + 119908119899minus1119895 )(Δ120591)2 where 119895 and 119899 are both integers andΔ119909 and Δ120591 are the space and time steps respectively In thesimulation process the string is made of soft material suchas rubberThe physical parameters of the string are measuredas follows mass density 120588=798 kgm3 length 119871=6877 mmcross-section area 119860=2282 mm2 pretension force along theaxial direction 1198730=068 N and Youngrsquos modulus 119864=085MPa The computational program is written in MATLAB 12where the time step Δ120591 is set as 0001 and the space step Δ119909 isset as 001 In the FDMdiscretization when 120591 = 0 and 120591 = Δ120591the displacement120596119894119895 = 119886 sin[120587(Δ119909)(119895minus1)] (119894=1 2) and 119895 takesthe value from 1 to 101

52 Time History Diagram of the Midpoint Evidently themidpoint D0 is a critical point which should be carefullyexamined as it may be associated with the largest amplitudeof the string When the initial amplitude of D0 is biggersuch as 119886=04 and 045 the related curves in the time historydiagram are displayed in Figure 2 It can be seen that withthe evolution of time the difference between the LAV modeland the linear equation becomesmore obvious and the resultfrom the semianalytical solution is in agreement with thenumerical result although there is slight difference as shownin Figure 2 As displayed in Figure 2(a) in the time intervalfrom 0 to 20 there are nearly 4 periods in the LAV modeland only 3 periods in the linear equation model It can alsobe observed when 120591 = 157 that the motion of the pointpredicted by the LAV equation is nearly one-half period fasterthan that by the linear equationThis indicates that the wholemotion of the string predicted by the LAV equation is quickerthan that predicted by the linear equation

53 Vibrational Configurations In what follows we computethe configurations of the string with large amplitude a=04in one period which are shown in Figures 3(a)ndash3(h) Firstlyit is noted that there is great difference between the resultsfrom the LAVmodel and the linear equationmodel as shownin Figure 3 This again stresses that the linear equation is

6 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus08

minus04

00

04

08

w

5 10 15 200

(a)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

50 15 2010

minus08

minus04

00

04

08

w

(b)

Figure 2 Time history diagram of point1198630 with the initial amplitudes (a) 119886 = 04 and (b) 119886 = 045

insufficient to depict the vibration with large amplitude andlarge curvature However in the whole period although thereis difference between the results from the semianalyticalmethod and numerical simulation the configurations pre-dicted by the two methods are close Therefore it should bementioned that the calculated string morphologies by thesemianalytical are more similar to those from the numericalresults than those from the linear equation model

The next phenomenon is that the numerical morpholo-gies of the string are very complicated in the vibrationprocess For instance when 120591=53 60 80 and 105 in Figures3(a) 3(b) 3(d) and 3(h) there are some nearly horizontalsegments appearing in the curves This strange behavior maybe attributed to the complex nonlinear dynamics responsewhere the clamped ends exert strong constraints on thestring and vice versa near the middle part of the stringthe constraint is much weaker Another possible reasonis that the first-mode analysis of the string may be anoversimplified model and the higher-mode behaviors mustbe further considered to illustrate this phenomenon Further-more our semianalytical solution can also have the similarshapes in this figures especially near the middle part of thestring

To verify this numerical result we perform an experimentas a comparison where the string configurations in thevibration are shown in Figures 4(a)ndash4(h) A rubber stringwith 1408times1630mm2 cross-section is fixed horizontally on adesktop at room temperature whose original lengthL is 5085mm To add the pretension force on the string the deformedlength of the string is elongated to 119897=6877 mm and originarea of the cross-section is 2282 mm2 Thus the prestraincan be given as 120576=(lndashL)L=03524 and using the measuredvalue of Youngrsquos modulus 119864=085 MPa the pretension forceis N0=068N At first in the vertical plane the midpoint ofthe string is excited by an original amplitude which is 04times of the string length 119871 Next the string is released and

then its vibration sequences are recorded by the high speedcamera (Phantom v2512 with 10000 frames per second) Asshown in Figures 4(a) 4(b) 4(c) and 4(d) we find that thereare really platform segments appearing in the string and thetotal configurations in these situations are close to the shapeof trapezoid Meanwhile the string is no longer a straightline when it approaches the equilibrium position as shownin Figures 4(e) and 4(f) implying that the length of the stringis bigger than its original length These behaviors are beyondour imagination as they demonstrate the complexity of thenonlinear dynamics

54 Total String Length It can be seen that in both the LAVequation and the linear equation the extensibility of thestring is considered thus its length can be calculated andthe two curves are compared in Figure 5 The nondimen-sional length of the string after deformation is expressed as119897119871 = int119871

0radic1 + 1199082d119909119871 For the linear equation model this

nondimensional length is a periodic function with respectto the time 120591 ie 119897119871(120591 + 1198792) = 119897119871(120591) This means thatthe maximum and minimum values of the string length donot alter in the vibration process Especially when the stringapproaches the 119909-axis it is really a straight line and thus thevalue of 119897119871 is of the smallest value ie one This featurereemphasizes that the small amplitude vibration has linearproperties On the contrary the oscillation behavior of thestring length is more complex in the LAV model whereits minimum value increases with the elapsing time In thissituation the dimensionless string length is always biggerthan one indicating that the string is always in the elongationstate even when it approaches the 119909-axis In the experimentsuch as in Figures 4(e) 4(f) 3(e) and 3(f) the shapes of thestring are notmerely straight lines and its length at this criticaltime must be larger than one Evidently this consistence onthe experimental phenomenon can validate the efficiency ofour numerical method

Mathematical Problems in Engineering 7

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(a)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(b)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(c)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(d)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(e)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(f)

Figure 3 Continued

8 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06w

(g)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(h)

Figure 3 Vibrational configurations of an elastic string when (a) 120591=53 (b) 120591 = 60 (c) 120591 = 70 (d) 120591 = 80 (e) 120591 = 90 (f) 120591 = 93 (g)120591 = 10 (h) 120591 = 105

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4 Vibrational configurations of the string in the experiment

55 Fundamental Frequency Another vibration characteris-tic parameter about vibration is the frequency and in thecurrent study the value of the fundamental frequency isobtained by the solution of (28) and it is also tested by theexperimental measurement When the vibration has a smallamplitude vibration of the string its dynamic behavior canbe formulated by the linear equation 119902 + 119902 = 0 with thenatural frequency always being 1205961 = 1 which is irrelative tothe initial amplitude We observed in Figure 6 the differencebetween the semianalytical method and experiment methodis great The main reason is that for the small amplitudethe mode shape function should be sin(120587119909) rather thansin(120587119909)radic1 + cos2(120587119909) The frequency exhibited in Figure 6demonstrates that our mode shape function should not be

used for the small amplitude vibration However for thelarge amplitude vibration the fundamental frequency is nota constant and its value is greatly affected by the initialamplitude As shown in Figure 6 with the increase of theamplitude the fundamental frequency also increases whichis verified by the experimental results Although there exitsvalue difference between the semianalytical solution and theexperimental data their tendencies are the same and bothof the two results are bigger than one with the increase ofinitial amplitude a This behavior again demonstrates thecomplicated response of the nonlinear dynamics which isvery sensitive to the initial value It should be clarified that ourmode shape function only for the large amplitude vibrationwith the ratio (amplitude of the middle part of the string

Mathematical Problems in Engineering 9

LAV equation numerical solutionLinear equation theoretical solution

1510 200 5

10

11

12

13

14

lL

Figure 5 Total length of the string based on the LAV equation andlinear equation

LAV equation semianalytical solutionLinear equation theoretical solutionExperiment

100

104

108

112

116

120

1

01 02 03 0400a

Figure 6 Relation between the fundamental 1205961 and the initialamplitude a

to the whole span) is nearly 02-05 If the ratio is greaterdifferent mode shape functions should be selected and thesimilar analysis process can be carried out

6 Conclusion

In conclusion the nonlinear vibration of a soft elasticstring with large amplitude and large curvature has beensystematically investigated in this study The mathematicalmodel in the absence of damping is developed based on theHamilton principle The exact mode shape function differentfrom the counterpart derived from the linear equation wasselected based on the experiment For the large amplitudevibration the time history diagram from the semianalyticalis very close to that from the numerical result and thesetwo results are very different from that of the linear modelThe vibration configurations of the elastic string have thesimilar laws as those of the time history diagrams based on

the comparison of the three models We find that in thenumerical calculations there are platform segments with theshape of trapezoid appearing in the configuration curvesand this phenomenon has been verified by our semianalyticalsolution and the experiment The total length of the string inthe large deformation shows that it is not a periodic functionwith respect to the time which is distinct with the result ofthe linear model We also point out that the fundamentalfrequency of the string with large amplitude vibration isgreatly affected by the initial amplitude which has the sametendency as our experiment result

It should be mentioned that this study is only the firsttrial on the nonlinear vibration of elastic structures withlarge amplitude and large curvatures The following work isexpected which will be extended to some other engineeringstructures such as beams rods plates and shells Theseanalyses may be beneficial to engineer some soft materialsand can also shed light on the design of elementary structuresin sensors actuators and resonators etc

Appendix

A Equations to Determine Coefficients ofNonresonant Terms

Besides (23)-(25) the other equations to determine the valueof Δ 119894 (119894=1 12) are

21198941205961Δ 2 + 119894 (12059620 minus 12059621)21205961 = 021198941205961Δ 3 minus 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) minus 119894119896121205961 = 021198941205961Δ 5 + 3119894119896121205961 = 041198941205961Δ 6 + 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) + 119894119896121205961 = 041198941205961Δ 7 minus 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12) minus 3119894119896121205961 (Δ 3 + Δ 6)minus 119894119896221205961 = 0

21198941205961Δ 8 minus 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11) minus 6119894119896121205961 (Δ 3 + Δ 6)minus 5119894119896221205961 = 0

21198941205961Δ 10 + 3119894119896121205961 (Δ 3 + Δ 6) + 10119894119896221205961 = 041198941205961Δ 11 + 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11)+ 6119894119896121205961 (Δ 3 + Δ 6) + 5119894119896221205961 = 0

10 Mathematical Problems in Engineering

61198941205961Δ 12 + 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12)+ 3119894119896121205961 (Δ 3 + Δ 6) + 119894119896221205961 = 0

(A1)

B Formulas of Parameters Δ 119894The parameters Δ 119894 ( i=1 12) are

Δ 1 = minusΔ 2Δ 2 = 12059621 minus 12059620412059621 Δ 3 = minus2Δ 6Δ 6 = 119896112059620 minus 912059621 Δ 4 = minusΔ 5Δ 5 = minus 31198961412059621 Δ 7 = minus32Δ 12Δ 12 = 61198961Δ 6 minus 211989622512059621 minus 12059620 Δ 8 = minus2Δ 11Δ 11 = 61198961Δ 6 minus 51198962912059621 minus 12059620 Δ 9 = minusΔ 10Δ 10 = 31198961Δ 6 minus 101198962412059621

(B1)

Data Availability

No data were used to support this study

Ethical Approval

Ethical approval was obtained for this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study was supported by the National Natural Sci-ence Foundation of China (nos 11672334 11672335 and11611530541) and the Endeavour Australia Cheung KongResearch Fellowship Scholarship from theAustralian govern-ment

References

[1] C Feng L Yu andW Zhang ldquoDynamic analysis of a dielectricelastomer-based microbeam resonator with large vibrationamplituderdquo International Journal of Non-Linear Mechanics vol65 pp 63ndash68 2014

[2] J Han Q Zhang and W Wang ldquoDesign considerations onlarge amplitude vibration of a doubly clamped microresonatorwith two symmetrically located electrodesrdquo Communications inNonlinear Science andNumerical Simulation vol 22 no 1ndash3 pp492ndash510 2015

[3] T Belendez CNeipp andA Belendez ldquoLarge and small deflec-tions of a cantilever beamrdquo European Journal of Physics vol 23no 3 pp 371ndash379 2002

[4] L Wang and Y Zhao ldquoLarge amplitude motion mechanismand non-planar vibration character of stay cables subject to thesupport motionsrdquo Journal of Sound and Vibration vol 327 no1-2 pp 121ndash133 2009

[5] M Spagnuolo and U Andreaus ldquoA targeted review on largedeformations of planar elastic beams extensibility distributedloads buckling and post-bucklingrdquoMathematics andMechanicsof Solids pp 1ndash23 2018

[6] M P Omran A Amani and H G Lemu ldquoAnalytical approxi-mation of nonlinear vibration of string with large amplitudesrdquoJournal of Mechanical Science and Technology vol 27 no 4 pp981ndash986 2013

[7] Z L Wang P Poncharal andW A De Heer ldquoMeasuring phys-ical and mechanical properties of individual carbon nanotubesby in situ TEMrdquo Journal of Physics and Chemistry of Solids vol61 no 7 pp 1025ndash1030 2000

[8] Z LWang R P Gao P PoncharalWADeHeer Z RDai andZ W Pan ldquoMechanical and electrostatic properties of carbonnanotubes and nanowiresrdquo Materials Science and EngineeringC Materials for Biological Applications vol 16 no 1-2 pp 3ndash102001

[9] Q Zheng and Q Jiang ldquoMultiwalled Carbon Nanotubes asGigahertz Oscillatorsrdquo Physical Review Letters vol 88 no 42002

[10] M Hemmatnezhad R Ansari and G H Rahimi ldquoLarge-amplitude free vibrations of functionally graded beams bymeans of a finite element formulationrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 37 no 18-19 pp 8495ndash8504 2013

[11] J B Gunda R K Gupta G Ranga Janardhan and G Venkates-wara Rao ldquoLarge amplitude vibration analysis of compositebeams Simple closed-form solutionsrdquo Composite Structuresvol 93 no 2 pp 870ndash879 2011

[12] A Nikkar S Bagheri and M Saravi ldquoDynamic model of largeamplitude vibration of a uniform cantilever beam carrying anintermediate lumped mass and rotary inertiardquo Latin AmericanJournal of Solids and Structures vol 11 no 2 pp 320ndash329 2014

[13] Y P Yu B S Wu Y H Sun and L Zang ldquoAnalytical approxi-mate solutions to large amplitude vibration of a spring-hingedbeamrdquoMeccanica vol 48 no 10 pp 2569ndash2575 2013

[14] K K Raju andGV Rao ldquoTowards improved evaluation of largeamplitude free-vibration behaviour of uniform beams usingmulti-term admissible functionsrdquo Journal of Sound and Vibra-tion vol 282 no 3-5 pp 1238ndash1246 2005

[15] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons and Fractals vol 34 no 5 pp 1430ndash1439 2007

[16] S H Hoseini T Pirbodaghi M T Ahmadian and G H Far-rahi ldquoOn the large amplitude free vibrations of tapered beams

Mathematical Problems in Engineering 11

an analytical approachrdquo Mechanics Research Communicationsvol 36 no 8 pp 892ndash897 2009

[17] B Gheshlaghi ldquoLarge-amplitude vibrations of nanowires withdissipative surface stress effectsrdquo Acta Mechanica vol 224 no7 pp 1329ndash1334 2013

[18] S Souayeh and N Kacem ldquoComputational models for largeamplitude nonlinear vibrations of electrostatically actuatedcarbon nanotube-basedmass sensorsrdquo Sensors and Actuators APhysical vol 208 pp 10ndash20 2014

[19] M Simsek ldquoLarge amplitude free vibration of nanobeams withvarious boundary conditions based on the nonlocal elasticitytheoryrdquo Composites Part B Engineering vol 56 pp 621ndash6282014

[20] R Nazemnezhad and S Hosseini-Hashemi ldquoExact solution forlarge amplitude flexural vibration of nanobeams using non-local Euler-Bernoulli theoryrdquo Journal ofTheoretical and AppliedMechanics (Poland) vol 55 no 2 pp 649ndash658 2017

[21] K Nakamura K Kakihara M Kawakami and S Ueha ldquoMea-suring vibration characteristics at large amplitude region ofmaterials for high power ultrasonic vibration systemrdquoUltrason-ics vol 38 no 1 pp 122ndash126 2000

[22] AMozuras and E Podzharov ldquoMeasurement of large harmonicvibration amplitudesrdquo Journal of Sound and Vibration vol 271no 3-5 pp 985ndash998 2004

[23] J C Simo ldquoA finite strain beam formulation The three-dimen-sional dynamic problem Part Irdquo Computer Methods AppliedMechanics and Engineering vol 49 no 1 pp 55ndash70 1985

[24] P-F Pai and A-H Nayfeh ldquoA new method for the modeling ofgeometric nonlinearities in structuresrdquoComputersampStrucruresvol 53 no 4 pp 877ndash895 1994

[25] E Babilio and S Lenci ldquoOn the notion of curvature andits mechanical meaning in a geometrically exact plane beamtheoryrdquo International Journal of Mechanical Sciences vol 128-129 pp 277ndash293 2017

[26] S Lenci F Clementi and G Rega ldquoComparing Nonlinear FreeVibrations of Timoshenko Beams with Mechanical or Geo-metric Curvature Definitionrdquo Procedia IUTAM vol 20 pp 34ndash41 2017

[27] O Kopmaz and O Gundogdu ldquoOn the curvature of an Euler-Bernoulli beamrdquo International Journal of Mechanical Engineer-ing Education vol 31 no 2 pp 132ndash142 2003

[28] C Semler G X Li andM P Paıdoussis ldquoThe non-linear equa-tions of motion of pipes conveying fluidrdquo Journal of Sound andVibration vol 169 no 5 pp 577ndash599 1994

[29] D Zhao J Liu and L Wang ldquoNonlinear free vibration of acantilever nanobeam with surface effects Semi-analytical solu-tionsrdquo International Journal of Mechanical Sciences vol 113 pp184ndash195 2016

[30] N Vlajic T Fitzgerald V Nguyen and B BalachandranldquoGeometrically exact planar beams with initial pre-stress andlarge curvature Static configurations natural frequencies andmode shapesrdquo International Journal of Solids and Structures vol51 no 19-20 pp 3361ndash3371 2014

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

[32] F Benedettini and G Rega ldquoNon-linear dynamics of an elasticcable under planar excitationrdquo International Journal of Non-Linear Mechanics vol 22 no 6 pp 497ndash509 1987

[33] H Koivurova ldquoThe numerical study of the nonlinear dynamicsof a light axiallymoving stringrdquo Journal of Sound andVibrationvol 320 no 1-2 pp 373ndash385 2009

[34] A H NayfehTheMethod of Normal Forms Wiley-VCHWein-heim Germany 2nd edition 2011

[35] A Y Leung and Q C Zhang ldquoComplex normal form forstrongly non-linear vibration systems exemplified by Duffing-van der Pol equationrdquo Journal of Sound and Vibration vol 213no 5 pp 907ndash914 1998

[36] W Wang Q C Zhang and X J Wang ldquoApplication of theundetermined fundamental frequency method for analyzingthe critical value of chaosrdquo Acta Physica Sinica vol 58 no 8pp 5162ndash5168 2009

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Mathematical Problems in Engineering

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Submit your manuscripts atwwwhindawicom

4 Mathematical Problems in Engineering

1205731 = (1 minus 1198641198601198730)1205872 1205732 = 11986411986011987301205872

(10)

Accordingly (7) can be recast as

= 120573111990810158401015840(1 + 11990810158402)12 minus 12057311199081015840211990810158401015840(1 + 11990810158402)32 + 120573211990810158401015840 (11)

and this equation is termed as the dimensionless largeamplitude vibration (LAV) equation throughout this study

Then the boundary conditions and initial conditions arerewritten as

119908 (0 120591) = 0119908 (1 120591) = 0119908 (119909 0) = 119886 sin (120587119909) (119909 0) = 0

(12)

At the same time the linear equation is nondimensionalizedas

= 1120587211990810158401015840 (13)

In the above equations the related expressions are =d2119908d1205912 1199081015840 = d119908d119909 and 11990810158401015840 = d2119908d1199092 and 119886 is theamplitude of the middle point in the string ie point11986303 Mode Discretization with Exact Mode Shape

The following task is to solve (11) which is an intractableproblem Clearly it seems impossible to directly get the ana-lytical solution of this highly nonlinear equation Thereforein use of the Taylor series (11) can be expanded into thefollowing polynomial expansion until 5th-order terms wherethe terms of higher order are ignored

= 119908101584010158401205872 + 1205731 (minus311990810158402

2 + 15119908101584048 )11990810158401015840 (14)

It is clear that (14) is still a high order and nonlinear partialdifferential equation (PDE) and seeking the closed formsolution is not at hand Herein the Galerkin discretizationmethod is utilized to transform the PDE into the ordinarydifferential equation (ODE) The previous result [18] tells usthat the first-mode analysis of vibration is sufficient to capturethe main nonlinear characteristics and can get the funda-mental frequency accurately enough Hence we take the trialto only consider the first mode of the vibration aiming toget an approximate solution for the first step Moreover themode shape is a critical factor to affect the vibration of thecontinuously beam rod and string For the small amplitudevibration of string with simply supported condition the first-mode shape function is often assumed by linear mode shape

function [31] described by sin(120587119909) However for the largecurvature vibration especially the amplitude of the middlepoint nearly 04-05 times of span of simply supported stringthe mode shape function sin(120587119909) based on the linear themode shape fails The exact mode shape function shouldbe selected considering the physical experiments given inSection 53 Assume that the displacement 119908(119909 120591) can bedescribed as 119908 (119909 120591) = 119902 (120591)119882 (119909) (15)and

119882(119909) = sin (120587119909)radic1 + cos2 (120587119909) (16)

Substituting (15) and (16) into (14) multiplying both sidesof the equation by 119882(119909) and then integrating both sidesfrom 0 to 1 yields the following ODE equation with strongnonlinearities 119902 + 12059620119902 + 11989611199023 + 11989621199025 = 0 (17)

where 12059620 =10745 1198961 = minus818361205731 and 1198962 = minus5351491205731In order to compare the large curvature vibration with

the small amplitude vibration for the linear equation in (13)the classic mode shape function sin(120587119909) is adopted thusthe classic the dimensionless ODE can degenerate to 119902 +119902 = 0 Then one can obtain the theoretical solution easilywhich is named as theoretical solution of the linear equationthroughout the paper

4 Semianalytical Solution UsingCNFM Method

Equation (17) is strong nonlinear system due to the nonlinearterms q3 and q5 which are far more large than the linear term119902 Although Hersquos variational method [6 12 15 19] harmonicbalance method [18] and Homotopy analysis method [16]are used widely in the strong nonlinear system the modifiedCNFMmethod [29 34ndash36] gives more accurate results basedon the experience of the authors Next we use the modifiedCNFM approach to solve (17) We assume that the solution of(17) can be formulated as

119902 = 120585 + 120585119902 = 1198941205961 (120585 minus 120585) (18)

where 120585 is the complex conjugate of 120585 and 1205961 is the unknownfundamental natural frequency to be determined Introducea nonlinear transformation from 120585 to 120578 in the form of120585 = 120578 + ℎ (120578 120578) (19)where 120578 and 120578 are complex conjugates and they are bothcomplex functions

The near identity transformation function ℎ including thevariables 120578 and 120578 is expressed as

ℎ (120578 120578) = Δ 1120578 + Δ 2120578 + Δ 31205783 + Δ 41205782120578 + Δ 51205781205782+ Δ 61205783 + Δ 71205785 + Δ 81205784120578 + Δ 912057831205782+ Δ 1012057821205783 + Δ 111205781205784 + Δ 121205785(20)

where Δ 119894 (119894=1 12) are real numbers

Mathematical Problems in Engineering 5

Substituting (18)ndash(20) into (17) one can get

120578 = 1198941205961120578 + 1198941205961ℎ minus 120597ℎ120597120578 120578 minus 120597ℎ120597120578 + 11989412059612 (1205962012059621 minus 1)

sdot (120578 + 120578 + ℎ + ℎ)+ 11989421205961 [1198961 (120578 + 120578 + ℎ + ℎ)3 + 1198962 (120578 + 120578 + ℎ + ℎ)5]

(21)

where ℎ and ℎ are complex conjugates Substituting (19) and(20) into (18) leads to

119902 = 120578 + 120578 + (Δ 1 + Δ 2) (120578 + 120578) + (Δ 3 + Δ 6) (1205783 + 1205783)+ (Δ 4 + Δ 5) (1205782120578 + 1205781205782) + (Δ 7 + Δ 12) (1205785 + 1205785)+ (Δ 8 + Δ 11) (1205784120578 + 1205781205784)+ (Δ 9 + Δ 10) (12057831205782 + 12057821205783)

(22)

To eliminate the secular terms 120578 1205782120578 and 12057831205782 in (22) thefollowing conditions should be met

Δ 1 + Δ 2 = 0 (23)

Δ 4 + Δ 5 = 0 (24)

Δ 9 + Δ 10 = 0 (25)

Moreover let the coefficients of terms 120578 1205783 1205781205782 1205783 1205785 120578412057812057821205783 1205781205784 and 1205785 in the right side of (22) vanish as they arethe nonresonant terms which are much smaller than term120578 Thus twelve equations are obtained in Appendix Solving(23)ndash(25) and the related equations in Appendix we canacquire the parameters Δ 119894 (119894=1 12) which are given inAppendix

After these operations the resonant terms 120578 1205782120578 and 12057831205782will remain in (21) and the equation is simplified to

120578 = 1198941 + 1205962121205961 120578 + 119894 3119896121205961 1205782120578+ 119894 121205961 (101198962 minus 31198961Δ 6) 12057831205782

(26)

where the real parameters Δ 6 is given in AppendixAssume

120578 = 121198861198901198941205961119905 (27)

where 119886 is the dimensionless vibration amplitude and 1205961is the fundamental frequency which is to be determinedSubstituting (27) into (26) leads to

119886 = 01205961 = 12059620 + 1205962121205961 + 3119896181205961 1198862 + 101198962 minus 31198961Δ 6321205961 1198864 (28)

The first equation in (28) indicates that the amplitude 119886is a constant determined by the initial condition and thesecond equation determines the fundamental frequency 1205961Substituting Δ 119894 (119894=1 12) 119886 and 1205961 into (22) one can getthe final solution of (17) as

119902 = 119886 cos (1205961119905)+ [14 (Δ 3 + Δ 6) 1198863 + 116 (Δ 8 + Δ 11) 1198865]sdot cos (31205961119905) + 116 (Δ 7 + Δ 12) 1198865cos (51205961119905)

(29)

5 Results and Discussion

51 Numerical Scheme To get a more accurate solution thefinite difference method (FDM) is used to directly calculatethe PDE which manifests a robust method The centraldifference is adopted ie 1199081015840 = (119908119899119895+1 minus 119908119899119895minus1)2(Δ119909)11990810158401015840 = (119908119899119895+1 minus 2119908119899119895 + 119908119899119895minus1)(Δ119909)2 and = (119908119899+1119895 minus2119908119899119895 + 119908119899minus1119895 )(Δ120591)2 where 119895 and 119899 are both integers andΔ119909 and Δ120591 are the space and time steps respectively In thesimulation process the string is made of soft material suchas rubberThe physical parameters of the string are measuredas follows mass density 120588=798 kgm3 length 119871=6877 mmcross-section area 119860=2282 mm2 pretension force along theaxial direction 1198730=068 N and Youngrsquos modulus 119864=085MPa The computational program is written in MATLAB 12where the time step Δ120591 is set as 0001 and the space step Δ119909 isset as 001 In the FDMdiscretization when 120591 = 0 and 120591 = Δ120591the displacement120596119894119895 = 119886 sin[120587(Δ119909)(119895minus1)] (119894=1 2) and 119895 takesthe value from 1 to 101

52 Time History Diagram of the Midpoint Evidently themidpoint D0 is a critical point which should be carefullyexamined as it may be associated with the largest amplitudeof the string When the initial amplitude of D0 is biggersuch as 119886=04 and 045 the related curves in the time historydiagram are displayed in Figure 2 It can be seen that withthe evolution of time the difference between the LAV modeland the linear equation becomesmore obvious and the resultfrom the semianalytical solution is in agreement with thenumerical result although there is slight difference as shownin Figure 2 As displayed in Figure 2(a) in the time intervalfrom 0 to 20 there are nearly 4 periods in the LAV modeland only 3 periods in the linear equation model It can alsobe observed when 120591 = 157 that the motion of the pointpredicted by the LAV equation is nearly one-half period fasterthan that by the linear equationThis indicates that the wholemotion of the string predicted by the LAV equation is quickerthan that predicted by the linear equation

53 Vibrational Configurations In what follows we computethe configurations of the string with large amplitude a=04in one period which are shown in Figures 3(a)ndash3(h) Firstlyit is noted that there is great difference between the resultsfrom the LAVmodel and the linear equationmodel as shownin Figure 3 This again stresses that the linear equation is

6 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus08

minus04

00

04

08

w

5 10 15 200

(a)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

50 15 2010

minus08

minus04

00

04

08

w

(b)

Figure 2 Time history diagram of point1198630 with the initial amplitudes (a) 119886 = 04 and (b) 119886 = 045

insufficient to depict the vibration with large amplitude andlarge curvature However in the whole period although thereis difference between the results from the semianalyticalmethod and numerical simulation the configurations pre-dicted by the two methods are close Therefore it should bementioned that the calculated string morphologies by thesemianalytical are more similar to those from the numericalresults than those from the linear equation model

The next phenomenon is that the numerical morpholo-gies of the string are very complicated in the vibrationprocess For instance when 120591=53 60 80 and 105 in Figures3(a) 3(b) 3(d) and 3(h) there are some nearly horizontalsegments appearing in the curves This strange behavior maybe attributed to the complex nonlinear dynamics responsewhere the clamped ends exert strong constraints on thestring and vice versa near the middle part of the stringthe constraint is much weaker Another possible reasonis that the first-mode analysis of the string may be anoversimplified model and the higher-mode behaviors mustbe further considered to illustrate this phenomenon Further-more our semianalytical solution can also have the similarshapes in this figures especially near the middle part of thestring

To verify this numerical result we perform an experimentas a comparison where the string configurations in thevibration are shown in Figures 4(a)ndash4(h) A rubber stringwith 1408times1630mm2 cross-section is fixed horizontally on adesktop at room temperature whose original lengthL is 5085mm To add the pretension force on the string the deformedlength of the string is elongated to 119897=6877 mm and originarea of the cross-section is 2282 mm2 Thus the prestraincan be given as 120576=(lndashL)L=03524 and using the measuredvalue of Youngrsquos modulus 119864=085 MPa the pretension forceis N0=068N At first in the vertical plane the midpoint ofthe string is excited by an original amplitude which is 04times of the string length 119871 Next the string is released and

then its vibration sequences are recorded by the high speedcamera (Phantom v2512 with 10000 frames per second) Asshown in Figures 4(a) 4(b) 4(c) and 4(d) we find that thereare really platform segments appearing in the string and thetotal configurations in these situations are close to the shapeof trapezoid Meanwhile the string is no longer a straightline when it approaches the equilibrium position as shownin Figures 4(e) and 4(f) implying that the length of the stringis bigger than its original length These behaviors are beyondour imagination as they demonstrate the complexity of thenonlinear dynamics

54 Total String Length It can be seen that in both the LAVequation and the linear equation the extensibility of thestring is considered thus its length can be calculated andthe two curves are compared in Figure 5 The nondimen-sional length of the string after deformation is expressed as119897119871 = int119871

0radic1 + 1199082d119909119871 For the linear equation model this

nondimensional length is a periodic function with respectto the time 120591 ie 119897119871(120591 + 1198792) = 119897119871(120591) This means thatthe maximum and minimum values of the string length donot alter in the vibration process Especially when the stringapproaches the 119909-axis it is really a straight line and thus thevalue of 119897119871 is of the smallest value ie one This featurereemphasizes that the small amplitude vibration has linearproperties On the contrary the oscillation behavior of thestring length is more complex in the LAV model whereits minimum value increases with the elapsing time In thissituation the dimensionless string length is always biggerthan one indicating that the string is always in the elongationstate even when it approaches the 119909-axis In the experimentsuch as in Figures 4(e) 4(f) 3(e) and 3(f) the shapes of thestring are notmerely straight lines and its length at this criticaltime must be larger than one Evidently this consistence onthe experimental phenomenon can validate the efficiency ofour numerical method

Mathematical Problems in Engineering 7

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(a)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(b)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(c)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(d)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(e)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(f)

Figure 3 Continued

8 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06w

(g)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(h)

Figure 3 Vibrational configurations of an elastic string when (a) 120591=53 (b) 120591 = 60 (c) 120591 = 70 (d) 120591 = 80 (e) 120591 = 90 (f) 120591 = 93 (g)120591 = 10 (h) 120591 = 105

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4 Vibrational configurations of the string in the experiment

55 Fundamental Frequency Another vibration characteris-tic parameter about vibration is the frequency and in thecurrent study the value of the fundamental frequency isobtained by the solution of (28) and it is also tested by theexperimental measurement When the vibration has a smallamplitude vibration of the string its dynamic behavior canbe formulated by the linear equation 119902 + 119902 = 0 with thenatural frequency always being 1205961 = 1 which is irrelative tothe initial amplitude We observed in Figure 6 the differencebetween the semianalytical method and experiment methodis great The main reason is that for the small amplitudethe mode shape function should be sin(120587119909) rather thansin(120587119909)radic1 + cos2(120587119909) The frequency exhibited in Figure 6demonstrates that our mode shape function should not be

used for the small amplitude vibration However for thelarge amplitude vibration the fundamental frequency is nota constant and its value is greatly affected by the initialamplitude As shown in Figure 6 with the increase of theamplitude the fundamental frequency also increases whichis verified by the experimental results Although there exitsvalue difference between the semianalytical solution and theexperimental data their tendencies are the same and bothof the two results are bigger than one with the increase ofinitial amplitude a This behavior again demonstrates thecomplicated response of the nonlinear dynamics which isvery sensitive to the initial value It should be clarified that ourmode shape function only for the large amplitude vibrationwith the ratio (amplitude of the middle part of the string

Mathematical Problems in Engineering 9

LAV equation numerical solutionLinear equation theoretical solution

1510 200 5

10

11

12

13

14

lL

Figure 5 Total length of the string based on the LAV equation andlinear equation

LAV equation semianalytical solutionLinear equation theoretical solutionExperiment

100

104

108

112

116

120

1

01 02 03 0400a

Figure 6 Relation between the fundamental 1205961 and the initialamplitude a

to the whole span) is nearly 02-05 If the ratio is greaterdifferent mode shape functions should be selected and thesimilar analysis process can be carried out

6 Conclusion

In conclusion the nonlinear vibration of a soft elasticstring with large amplitude and large curvature has beensystematically investigated in this study The mathematicalmodel in the absence of damping is developed based on theHamilton principle The exact mode shape function differentfrom the counterpart derived from the linear equation wasselected based on the experiment For the large amplitudevibration the time history diagram from the semianalyticalis very close to that from the numerical result and thesetwo results are very different from that of the linear modelThe vibration configurations of the elastic string have thesimilar laws as those of the time history diagrams based on

the comparison of the three models We find that in thenumerical calculations there are platform segments with theshape of trapezoid appearing in the configuration curvesand this phenomenon has been verified by our semianalyticalsolution and the experiment The total length of the string inthe large deformation shows that it is not a periodic functionwith respect to the time which is distinct with the result ofthe linear model We also point out that the fundamentalfrequency of the string with large amplitude vibration isgreatly affected by the initial amplitude which has the sametendency as our experiment result

It should be mentioned that this study is only the firsttrial on the nonlinear vibration of elastic structures withlarge amplitude and large curvatures The following work isexpected which will be extended to some other engineeringstructures such as beams rods plates and shells Theseanalyses may be beneficial to engineer some soft materialsand can also shed light on the design of elementary structuresin sensors actuators and resonators etc

Appendix

A Equations to Determine Coefficients ofNonresonant Terms

Besides (23)-(25) the other equations to determine the valueof Δ 119894 (119894=1 12) are

21198941205961Δ 2 + 119894 (12059620 minus 12059621)21205961 = 021198941205961Δ 3 minus 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) minus 119894119896121205961 = 021198941205961Δ 5 + 3119894119896121205961 = 041198941205961Δ 6 + 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) + 119894119896121205961 = 041198941205961Δ 7 minus 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12) minus 3119894119896121205961 (Δ 3 + Δ 6)minus 119894119896221205961 = 0

21198941205961Δ 8 minus 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11) minus 6119894119896121205961 (Δ 3 + Δ 6)minus 5119894119896221205961 = 0

21198941205961Δ 10 + 3119894119896121205961 (Δ 3 + Δ 6) + 10119894119896221205961 = 041198941205961Δ 11 + 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11)+ 6119894119896121205961 (Δ 3 + Δ 6) + 5119894119896221205961 = 0

10 Mathematical Problems in Engineering

61198941205961Δ 12 + 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12)+ 3119894119896121205961 (Δ 3 + Δ 6) + 119894119896221205961 = 0

(A1)

B Formulas of Parameters Δ 119894The parameters Δ 119894 ( i=1 12) are

Δ 1 = minusΔ 2Δ 2 = 12059621 minus 12059620412059621 Δ 3 = minus2Δ 6Δ 6 = 119896112059620 minus 912059621 Δ 4 = minusΔ 5Δ 5 = minus 31198961412059621 Δ 7 = minus32Δ 12Δ 12 = 61198961Δ 6 minus 211989622512059621 minus 12059620 Δ 8 = minus2Δ 11Δ 11 = 61198961Δ 6 minus 51198962912059621 minus 12059620 Δ 9 = minusΔ 10Δ 10 = 31198961Δ 6 minus 101198962412059621

(B1)

Data Availability

No data were used to support this study

Ethical Approval

Ethical approval was obtained for this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study was supported by the National Natural Sci-ence Foundation of China (nos 11672334 11672335 and11611530541) and the Endeavour Australia Cheung KongResearch Fellowship Scholarship from theAustralian govern-ment

References

[1] C Feng L Yu andW Zhang ldquoDynamic analysis of a dielectricelastomer-based microbeam resonator with large vibrationamplituderdquo International Journal of Non-Linear Mechanics vol65 pp 63ndash68 2014

[2] J Han Q Zhang and W Wang ldquoDesign considerations onlarge amplitude vibration of a doubly clamped microresonatorwith two symmetrically located electrodesrdquo Communications inNonlinear Science andNumerical Simulation vol 22 no 1ndash3 pp492ndash510 2015

[3] T Belendez CNeipp andA Belendez ldquoLarge and small deflec-tions of a cantilever beamrdquo European Journal of Physics vol 23no 3 pp 371ndash379 2002

[4] L Wang and Y Zhao ldquoLarge amplitude motion mechanismand non-planar vibration character of stay cables subject to thesupport motionsrdquo Journal of Sound and Vibration vol 327 no1-2 pp 121ndash133 2009

[5] M Spagnuolo and U Andreaus ldquoA targeted review on largedeformations of planar elastic beams extensibility distributedloads buckling and post-bucklingrdquoMathematics andMechanicsof Solids pp 1ndash23 2018

[6] M P Omran A Amani and H G Lemu ldquoAnalytical approxi-mation of nonlinear vibration of string with large amplitudesrdquoJournal of Mechanical Science and Technology vol 27 no 4 pp981ndash986 2013

[7] Z L Wang P Poncharal andW A De Heer ldquoMeasuring phys-ical and mechanical properties of individual carbon nanotubesby in situ TEMrdquo Journal of Physics and Chemistry of Solids vol61 no 7 pp 1025ndash1030 2000

[8] Z LWang R P Gao P PoncharalWADeHeer Z RDai andZ W Pan ldquoMechanical and electrostatic properties of carbonnanotubes and nanowiresrdquo Materials Science and EngineeringC Materials for Biological Applications vol 16 no 1-2 pp 3ndash102001

[9] Q Zheng and Q Jiang ldquoMultiwalled Carbon Nanotubes asGigahertz Oscillatorsrdquo Physical Review Letters vol 88 no 42002

[10] M Hemmatnezhad R Ansari and G H Rahimi ldquoLarge-amplitude free vibrations of functionally graded beams bymeans of a finite element formulationrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 37 no 18-19 pp 8495ndash8504 2013

[11] J B Gunda R K Gupta G Ranga Janardhan and G Venkates-wara Rao ldquoLarge amplitude vibration analysis of compositebeams Simple closed-form solutionsrdquo Composite Structuresvol 93 no 2 pp 870ndash879 2011

[12] A Nikkar S Bagheri and M Saravi ldquoDynamic model of largeamplitude vibration of a uniform cantilever beam carrying anintermediate lumped mass and rotary inertiardquo Latin AmericanJournal of Solids and Structures vol 11 no 2 pp 320ndash329 2014

[13] Y P Yu B S Wu Y H Sun and L Zang ldquoAnalytical approxi-mate solutions to large amplitude vibration of a spring-hingedbeamrdquoMeccanica vol 48 no 10 pp 2569ndash2575 2013

[14] K K Raju andGV Rao ldquoTowards improved evaluation of largeamplitude free-vibration behaviour of uniform beams usingmulti-term admissible functionsrdquo Journal of Sound and Vibra-tion vol 282 no 3-5 pp 1238ndash1246 2005

[15] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons and Fractals vol 34 no 5 pp 1430ndash1439 2007

[16] S H Hoseini T Pirbodaghi M T Ahmadian and G H Far-rahi ldquoOn the large amplitude free vibrations of tapered beams

Mathematical Problems in Engineering 11

an analytical approachrdquo Mechanics Research Communicationsvol 36 no 8 pp 892ndash897 2009

[17] B Gheshlaghi ldquoLarge-amplitude vibrations of nanowires withdissipative surface stress effectsrdquo Acta Mechanica vol 224 no7 pp 1329ndash1334 2013

[18] S Souayeh and N Kacem ldquoComputational models for largeamplitude nonlinear vibrations of electrostatically actuatedcarbon nanotube-basedmass sensorsrdquo Sensors and Actuators APhysical vol 208 pp 10ndash20 2014

[19] M Simsek ldquoLarge amplitude free vibration of nanobeams withvarious boundary conditions based on the nonlocal elasticitytheoryrdquo Composites Part B Engineering vol 56 pp 621ndash6282014

[20] R Nazemnezhad and S Hosseini-Hashemi ldquoExact solution forlarge amplitude flexural vibration of nanobeams using non-local Euler-Bernoulli theoryrdquo Journal ofTheoretical and AppliedMechanics (Poland) vol 55 no 2 pp 649ndash658 2017

[21] K Nakamura K Kakihara M Kawakami and S Ueha ldquoMea-suring vibration characteristics at large amplitude region ofmaterials for high power ultrasonic vibration systemrdquoUltrason-ics vol 38 no 1 pp 122ndash126 2000

[22] AMozuras and E Podzharov ldquoMeasurement of large harmonicvibration amplitudesrdquo Journal of Sound and Vibration vol 271no 3-5 pp 985ndash998 2004

[23] J C Simo ldquoA finite strain beam formulation The three-dimen-sional dynamic problem Part Irdquo Computer Methods AppliedMechanics and Engineering vol 49 no 1 pp 55ndash70 1985

[24] P-F Pai and A-H Nayfeh ldquoA new method for the modeling ofgeometric nonlinearities in structuresrdquoComputersampStrucruresvol 53 no 4 pp 877ndash895 1994

[25] E Babilio and S Lenci ldquoOn the notion of curvature andits mechanical meaning in a geometrically exact plane beamtheoryrdquo International Journal of Mechanical Sciences vol 128-129 pp 277ndash293 2017

[26] S Lenci F Clementi and G Rega ldquoComparing Nonlinear FreeVibrations of Timoshenko Beams with Mechanical or Geo-metric Curvature Definitionrdquo Procedia IUTAM vol 20 pp 34ndash41 2017

[27] O Kopmaz and O Gundogdu ldquoOn the curvature of an Euler-Bernoulli beamrdquo International Journal of Mechanical Engineer-ing Education vol 31 no 2 pp 132ndash142 2003

[28] C Semler G X Li andM P Paıdoussis ldquoThe non-linear equa-tions of motion of pipes conveying fluidrdquo Journal of Sound andVibration vol 169 no 5 pp 577ndash599 1994

[29] D Zhao J Liu and L Wang ldquoNonlinear free vibration of acantilever nanobeam with surface effects Semi-analytical solu-tionsrdquo International Journal of Mechanical Sciences vol 113 pp184ndash195 2016

[30] N Vlajic T Fitzgerald V Nguyen and B BalachandranldquoGeometrically exact planar beams with initial pre-stress andlarge curvature Static configurations natural frequencies andmode shapesrdquo International Journal of Solids and Structures vol51 no 19-20 pp 3361ndash3371 2014

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

[32] F Benedettini and G Rega ldquoNon-linear dynamics of an elasticcable under planar excitationrdquo International Journal of Non-Linear Mechanics vol 22 no 6 pp 497ndash509 1987

[33] H Koivurova ldquoThe numerical study of the nonlinear dynamicsof a light axiallymoving stringrdquo Journal of Sound andVibrationvol 320 no 1-2 pp 373ndash385 2009

[34] A H NayfehTheMethod of Normal Forms Wiley-VCHWein-heim Germany 2nd edition 2011

[35] A Y Leung and Q C Zhang ldquoComplex normal form forstrongly non-linear vibration systems exemplified by Duffing-van der Pol equationrdquo Journal of Sound and Vibration vol 213no 5 pp 907ndash914 1998

[36] W Wang Q C Zhang and X J Wang ldquoApplication of theundetermined fundamental frequency method for analyzingthe critical value of chaosrdquo Acta Physica Sinica vol 58 no 8pp 5162ndash5168 2009

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Mathematical Problems in Engineering 5

Substituting (18)ndash(20) into (17) one can get

120578 = 1198941205961120578 + 1198941205961ℎ minus 120597ℎ120597120578 120578 minus 120597ℎ120597120578 + 11989412059612 (1205962012059621 minus 1)

sdot (120578 + 120578 + ℎ + ℎ)+ 11989421205961 [1198961 (120578 + 120578 + ℎ + ℎ)3 + 1198962 (120578 + 120578 + ℎ + ℎ)5]

(21)

where ℎ and ℎ are complex conjugates Substituting (19) and(20) into (18) leads to

119902 = 120578 + 120578 + (Δ 1 + Δ 2) (120578 + 120578) + (Δ 3 + Δ 6) (1205783 + 1205783)+ (Δ 4 + Δ 5) (1205782120578 + 1205781205782) + (Δ 7 + Δ 12) (1205785 + 1205785)+ (Δ 8 + Δ 11) (1205784120578 + 1205781205784)+ (Δ 9 + Δ 10) (12057831205782 + 12057821205783)

(22)

To eliminate the secular terms 120578 1205782120578 and 12057831205782 in (22) thefollowing conditions should be met

Δ 1 + Δ 2 = 0 (23)

Δ 4 + Δ 5 = 0 (24)

Δ 9 + Δ 10 = 0 (25)

Moreover let the coefficients of terms 120578 1205783 1205781205782 1205783 1205785 120578412057812057821205783 1205781205784 and 1205785 in the right side of (22) vanish as they arethe nonresonant terms which are much smaller than term120578 Thus twelve equations are obtained in Appendix Solving(23)ndash(25) and the related equations in Appendix we canacquire the parameters Δ 119894 (119894=1 12) which are given inAppendix

After these operations the resonant terms 120578 1205782120578 and 12057831205782will remain in (21) and the equation is simplified to

120578 = 1198941 + 1205962121205961 120578 + 119894 3119896121205961 1205782120578+ 119894 121205961 (101198962 minus 31198961Δ 6) 12057831205782

(26)

where the real parameters Δ 6 is given in AppendixAssume

120578 = 121198861198901198941205961119905 (27)

where 119886 is the dimensionless vibration amplitude and 1205961is the fundamental frequency which is to be determinedSubstituting (27) into (26) leads to

119886 = 01205961 = 12059620 + 1205962121205961 + 3119896181205961 1198862 + 101198962 minus 31198961Δ 6321205961 1198864 (28)

The first equation in (28) indicates that the amplitude 119886is a constant determined by the initial condition and thesecond equation determines the fundamental frequency 1205961Substituting Δ 119894 (119894=1 12) 119886 and 1205961 into (22) one can getthe final solution of (17) as

119902 = 119886 cos (1205961119905)+ [14 (Δ 3 + Δ 6) 1198863 + 116 (Δ 8 + Δ 11) 1198865]sdot cos (31205961119905) + 116 (Δ 7 + Δ 12) 1198865cos (51205961119905)

(29)

5 Results and Discussion

51 Numerical Scheme To get a more accurate solution thefinite difference method (FDM) is used to directly calculatethe PDE which manifests a robust method The centraldifference is adopted ie 1199081015840 = (119908119899119895+1 minus 119908119899119895minus1)2(Δ119909)11990810158401015840 = (119908119899119895+1 minus 2119908119899119895 + 119908119899119895minus1)(Δ119909)2 and = (119908119899+1119895 minus2119908119899119895 + 119908119899minus1119895 )(Δ120591)2 where 119895 and 119899 are both integers andΔ119909 and Δ120591 are the space and time steps respectively In thesimulation process the string is made of soft material suchas rubberThe physical parameters of the string are measuredas follows mass density 120588=798 kgm3 length 119871=6877 mmcross-section area 119860=2282 mm2 pretension force along theaxial direction 1198730=068 N and Youngrsquos modulus 119864=085MPa The computational program is written in MATLAB 12where the time step Δ120591 is set as 0001 and the space step Δ119909 isset as 001 In the FDMdiscretization when 120591 = 0 and 120591 = Δ120591the displacement120596119894119895 = 119886 sin[120587(Δ119909)(119895minus1)] (119894=1 2) and 119895 takesthe value from 1 to 101

52 Time History Diagram of the Midpoint Evidently themidpoint D0 is a critical point which should be carefullyexamined as it may be associated with the largest amplitudeof the string When the initial amplitude of D0 is biggersuch as 119886=04 and 045 the related curves in the time historydiagram are displayed in Figure 2 It can be seen that withthe evolution of time the difference between the LAV modeland the linear equation becomesmore obvious and the resultfrom the semianalytical solution is in agreement with thenumerical result although there is slight difference as shownin Figure 2 As displayed in Figure 2(a) in the time intervalfrom 0 to 20 there are nearly 4 periods in the LAV modeland only 3 periods in the linear equation model It can alsobe observed when 120591 = 157 that the motion of the pointpredicted by the LAV equation is nearly one-half period fasterthan that by the linear equationThis indicates that the wholemotion of the string predicted by the LAV equation is quickerthan that predicted by the linear equation

53 Vibrational Configurations In what follows we computethe configurations of the string with large amplitude a=04in one period which are shown in Figures 3(a)ndash3(h) Firstlyit is noted that there is great difference between the resultsfrom the LAVmodel and the linear equationmodel as shownin Figure 3 This again stresses that the linear equation is

6 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus08

minus04

00

04

08

w

5 10 15 200

(a)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

50 15 2010

minus08

minus04

00

04

08

w

(b)

Figure 2 Time history diagram of point1198630 with the initial amplitudes (a) 119886 = 04 and (b) 119886 = 045

insufficient to depict the vibration with large amplitude andlarge curvature However in the whole period although thereis difference between the results from the semianalyticalmethod and numerical simulation the configurations pre-dicted by the two methods are close Therefore it should bementioned that the calculated string morphologies by thesemianalytical are more similar to those from the numericalresults than those from the linear equation model

The next phenomenon is that the numerical morpholo-gies of the string are very complicated in the vibrationprocess For instance when 120591=53 60 80 and 105 in Figures3(a) 3(b) 3(d) and 3(h) there are some nearly horizontalsegments appearing in the curves This strange behavior maybe attributed to the complex nonlinear dynamics responsewhere the clamped ends exert strong constraints on thestring and vice versa near the middle part of the stringthe constraint is much weaker Another possible reasonis that the first-mode analysis of the string may be anoversimplified model and the higher-mode behaviors mustbe further considered to illustrate this phenomenon Further-more our semianalytical solution can also have the similarshapes in this figures especially near the middle part of thestring

To verify this numerical result we perform an experimentas a comparison where the string configurations in thevibration are shown in Figures 4(a)ndash4(h) A rubber stringwith 1408times1630mm2 cross-section is fixed horizontally on adesktop at room temperature whose original lengthL is 5085mm To add the pretension force on the string the deformedlength of the string is elongated to 119897=6877 mm and originarea of the cross-section is 2282 mm2 Thus the prestraincan be given as 120576=(lndashL)L=03524 and using the measuredvalue of Youngrsquos modulus 119864=085 MPa the pretension forceis N0=068N At first in the vertical plane the midpoint ofthe string is excited by an original amplitude which is 04times of the string length 119871 Next the string is released and

then its vibration sequences are recorded by the high speedcamera (Phantom v2512 with 10000 frames per second) Asshown in Figures 4(a) 4(b) 4(c) and 4(d) we find that thereare really platform segments appearing in the string and thetotal configurations in these situations are close to the shapeof trapezoid Meanwhile the string is no longer a straightline when it approaches the equilibrium position as shownin Figures 4(e) and 4(f) implying that the length of the stringis bigger than its original length These behaviors are beyondour imagination as they demonstrate the complexity of thenonlinear dynamics

54 Total String Length It can be seen that in both the LAVequation and the linear equation the extensibility of thestring is considered thus its length can be calculated andthe two curves are compared in Figure 5 The nondimen-sional length of the string after deformation is expressed as119897119871 = int119871

0radic1 + 1199082d119909119871 For the linear equation model this

nondimensional length is a periodic function with respectto the time 120591 ie 119897119871(120591 + 1198792) = 119897119871(120591) This means thatthe maximum and minimum values of the string length donot alter in the vibration process Especially when the stringapproaches the 119909-axis it is really a straight line and thus thevalue of 119897119871 is of the smallest value ie one This featurereemphasizes that the small amplitude vibration has linearproperties On the contrary the oscillation behavior of thestring length is more complex in the LAV model whereits minimum value increases with the elapsing time In thissituation the dimensionless string length is always biggerthan one indicating that the string is always in the elongationstate even when it approaches the 119909-axis In the experimentsuch as in Figures 4(e) 4(f) 3(e) and 3(f) the shapes of thestring are notmerely straight lines and its length at this criticaltime must be larger than one Evidently this consistence onthe experimental phenomenon can validate the efficiency ofour numerical method

Mathematical Problems in Engineering 7

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(a)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(b)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(c)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(d)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(e)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(f)

Figure 3 Continued

8 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06w

(g)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(h)

Figure 3 Vibrational configurations of an elastic string when (a) 120591=53 (b) 120591 = 60 (c) 120591 = 70 (d) 120591 = 80 (e) 120591 = 90 (f) 120591 = 93 (g)120591 = 10 (h) 120591 = 105

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4 Vibrational configurations of the string in the experiment

55 Fundamental Frequency Another vibration characteris-tic parameter about vibration is the frequency and in thecurrent study the value of the fundamental frequency isobtained by the solution of (28) and it is also tested by theexperimental measurement When the vibration has a smallamplitude vibration of the string its dynamic behavior canbe formulated by the linear equation 119902 + 119902 = 0 with thenatural frequency always being 1205961 = 1 which is irrelative tothe initial amplitude We observed in Figure 6 the differencebetween the semianalytical method and experiment methodis great The main reason is that for the small amplitudethe mode shape function should be sin(120587119909) rather thansin(120587119909)radic1 + cos2(120587119909) The frequency exhibited in Figure 6demonstrates that our mode shape function should not be

used for the small amplitude vibration However for thelarge amplitude vibration the fundamental frequency is nota constant and its value is greatly affected by the initialamplitude As shown in Figure 6 with the increase of theamplitude the fundamental frequency also increases whichis verified by the experimental results Although there exitsvalue difference between the semianalytical solution and theexperimental data their tendencies are the same and bothof the two results are bigger than one with the increase ofinitial amplitude a This behavior again demonstrates thecomplicated response of the nonlinear dynamics which isvery sensitive to the initial value It should be clarified that ourmode shape function only for the large amplitude vibrationwith the ratio (amplitude of the middle part of the string

Mathematical Problems in Engineering 9

LAV equation numerical solutionLinear equation theoretical solution

1510 200 5

10

11

12

13

14

lL

Figure 5 Total length of the string based on the LAV equation andlinear equation

LAV equation semianalytical solutionLinear equation theoretical solutionExperiment

100

104

108

112

116

120

1

01 02 03 0400a

Figure 6 Relation between the fundamental 1205961 and the initialamplitude a

to the whole span) is nearly 02-05 If the ratio is greaterdifferent mode shape functions should be selected and thesimilar analysis process can be carried out

6 Conclusion

In conclusion the nonlinear vibration of a soft elasticstring with large amplitude and large curvature has beensystematically investigated in this study The mathematicalmodel in the absence of damping is developed based on theHamilton principle The exact mode shape function differentfrom the counterpart derived from the linear equation wasselected based on the experiment For the large amplitudevibration the time history diagram from the semianalyticalis very close to that from the numerical result and thesetwo results are very different from that of the linear modelThe vibration configurations of the elastic string have thesimilar laws as those of the time history diagrams based on

the comparison of the three models We find that in thenumerical calculations there are platform segments with theshape of trapezoid appearing in the configuration curvesand this phenomenon has been verified by our semianalyticalsolution and the experiment The total length of the string inthe large deformation shows that it is not a periodic functionwith respect to the time which is distinct with the result ofthe linear model We also point out that the fundamentalfrequency of the string with large amplitude vibration isgreatly affected by the initial amplitude which has the sametendency as our experiment result

It should be mentioned that this study is only the firsttrial on the nonlinear vibration of elastic structures withlarge amplitude and large curvatures The following work isexpected which will be extended to some other engineeringstructures such as beams rods plates and shells Theseanalyses may be beneficial to engineer some soft materialsand can also shed light on the design of elementary structuresin sensors actuators and resonators etc

Appendix

A Equations to Determine Coefficients ofNonresonant Terms

Besides (23)-(25) the other equations to determine the valueof Δ 119894 (119894=1 12) are

21198941205961Δ 2 + 119894 (12059620 minus 12059621)21205961 = 021198941205961Δ 3 minus 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) minus 119894119896121205961 = 021198941205961Δ 5 + 3119894119896121205961 = 041198941205961Δ 6 + 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) + 119894119896121205961 = 041198941205961Δ 7 minus 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12) minus 3119894119896121205961 (Δ 3 + Δ 6)minus 119894119896221205961 = 0

21198941205961Δ 8 minus 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11) minus 6119894119896121205961 (Δ 3 + Δ 6)minus 5119894119896221205961 = 0

21198941205961Δ 10 + 3119894119896121205961 (Δ 3 + Δ 6) + 10119894119896221205961 = 041198941205961Δ 11 + 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11)+ 6119894119896121205961 (Δ 3 + Δ 6) + 5119894119896221205961 = 0

10 Mathematical Problems in Engineering

61198941205961Δ 12 + 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12)+ 3119894119896121205961 (Δ 3 + Δ 6) + 119894119896221205961 = 0

(A1)

B Formulas of Parameters Δ 119894The parameters Δ 119894 ( i=1 12) are

Δ 1 = minusΔ 2Δ 2 = 12059621 minus 12059620412059621 Δ 3 = minus2Δ 6Δ 6 = 119896112059620 minus 912059621 Δ 4 = minusΔ 5Δ 5 = minus 31198961412059621 Δ 7 = minus32Δ 12Δ 12 = 61198961Δ 6 minus 211989622512059621 minus 12059620 Δ 8 = minus2Δ 11Δ 11 = 61198961Δ 6 minus 51198962912059621 minus 12059620 Δ 9 = minusΔ 10Δ 10 = 31198961Δ 6 minus 101198962412059621

(B1)

Data Availability

No data were used to support this study

Ethical Approval

Ethical approval was obtained for this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study was supported by the National Natural Sci-ence Foundation of China (nos 11672334 11672335 and11611530541) and the Endeavour Australia Cheung KongResearch Fellowship Scholarship from theAustralian govern-ment

References

[1] C Feng L Yu andW Zhang ldquoDynamic analysis of a dielectricelastomer-based microbeam resonator with large vibrationamplituderdquo International Journal of Non-Linear Mechanics vol65 pp 63ndash68 2014

[2] J Han Q Zhang and W Wang ldquoDesign considerations onlarge amplitude vibration of a doubly clamped microresonatorwith two symmetrically located electrodesrdquo Communications inNonlinear Science andNumerical Simulation vol 22 no 1ndash3 pp492ndash510 2015

[3] T Belendez CNeipp andA Belendez ldquoLarge and small deflec-tions of a cantilever beamrdquo European Journal of Physics vol 23no 3 pp 371ndash379 2002

[4] L Wang and Y Zhao ldquoLarge amplitude motion mechanismand non-planar vibration character of stay cables subject to thesupport motionsrdquo Journal of Sound and Vibration vol 327 no1-2 pp 121ndash133 2009

[5] M Spagnuolo and U Andreaus ldquoA targeted review on largedeformations of planar elastic beams extensibility distributedloads buckling and post-bucklingrdquoMathematics andMechanicsof Solids pp 1ndash23 2018

[6] M P Omran A Amani and H G Lemu ldquoAnalytical approxi-mation of nonlinear vibration of string with large amplitudesrdquoJournal of Mechanical Science and Technology vol 27 no 4 pp981ndash986 2013

[7] Z L Wang P Poncharal andW A De Heer ldquoMeasuring phys-ical and mechanical properties of individual carbon nanotubesby in situ TEMrdquo Journal of Physics and Chemistry of Solids vol61 no 7 pp 1025ndash1030 2000

[8] Z LWang R P Gao P PoncharalWADeHeer Z RDai andZ W Pan ldquoMechanical and electrostatic properties of carbonnanotubes and nanowiresrdquo Materials Science and EngineeringC Materials for Biological Applications vol 16 no 1-2 pp 3ndash102001

[9] Q Zheng and Q Jiang ldquoMultiwalled Carbon Nanotubes asGigahertz Oscillatorsrdquo Physical Review Letters vol 88 no 42002

[10] M Hemmatnezhad R Ansari and G H Rahimi ldquoLarge-amplitude free vibrations of functionally graded beams bymeans of a finite element formulationrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 37 no 18-19 pp 8495ndash8504 2013

[11] J B Gunda R K Gupta G Ranga Janardhan and G Venkates-wara Rao ldquoLarge amplitude vibration analysis of compositebeams Simple closed-form solutionsrdquo Composite Structuresvol 93 no 2 pp 870ndash879 2011

[12] A Nikkar S Bagheri and M Saravi ldquoDynamic model of largeamplitude vibration of a uniform cantilever beam carrying anintermediate lumped mass and rotary inertiardquo Latin AmericanJournal of Solids and Structures vol 11 no 2 pp 320ndash329 2014

[13] Y P Yu B S Wu Y H Sun and L Zang ldquoAnalytical approxi-mate solutions to large amplitude vibration of a spring-hingedbeamrdquoMeccanica vol 48 no 10 pp 2569ndash2575 2013

[14] K K Raju andGV Rao ldquoTowards improved evaluation of largeamplitude free-vibration behaviour of uniform beams usingmulti-term admissible functionsrdquo Journal of Sound and Vibra-tion vol 282 no 3-5 pp 1238ndash1246 2005

[15] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons and Fractals vol 34 no 5 pp 1430ndash1439 2007

[16] S H Hoseini T Pirbodaghi M T Ahmadian and G H Far-rahi ldquoOn the large amplitude free vibrations of tapered beams

Mathematical Problems in Engineering 11

an analytical approachrdquo Mechanics Research Communicationsvol 36 no 8 pp 892ndash897 2009

[17] B Gheshlaghi ldquoLarge-amplitude vibrations of nanowires withdissipative surface stress effectsrdquo Acta Mechanica vol 224 no7 pp 1329ndash1334 2013

[18] S Souayeh and N Kacem ldquoComputational models for largeamplitude nonlinear vibrations of electrostatically actuatedcarbon nanotube-basedmass sensorsrdquo Sensors and Actuators APhysical vol 208 pp 10ndash20 2014

[19] M Simsek ldquoLarge amplitude free vibration of nanobeams withvarious boundary conditions based on the nonlocal elasticitytheoryrdquo Composites Part B Engineering vol 56 pp 621ndash6282014

[20] R Nazemnezhad and S Hosseini-Hashemi ldquoExact solution forlarge amplitude flexural vibration of nanobeams using non-local Euler-Bernoulli theoryrdquo Journal ofTheoretical and AppliedMechanics (Poland) vol 55 no 2 pp 649ndash658 2017

[21] K Nakamura K Kakihara M Kawakami and S Ueha ldquoMea-suring vibration characteristics at large amplitude region ofmaterials for high power ultrasonic vibration systemrdquoUltrason-ics vol 38 no 1 pp 122ndash126 2000

[22] AMozuras and E Podzharov ldquoMeasurement of large harmonicvibration amplitudesrdquo Journal of Sound and Vibration vol 271no 3-5 pp 985ndash998 2004

[23] J C Simo ldquoA finite strain beam formulation The three-dimen-sional dynamic problem Part Irdquo Computer Methods AppliedMechanics and Engineering vol 49 no 1 pp 55ndash70 1985

[24] P-F Pai and A-H Nayfeh ldquoA new method for the modeling ofgeometric nonlinearities in structuresrdquoComputersampStrucruresvol 53 no 4 pp 877ndash895 1994

[25] E Babilio and S Lenci ldquoOn the notion of curvature andits mechanical meaning in a geometrically exact plane beamtheoryrdquo International Journal of Mechanical Sciences vol 128-129 pp 277ndash293 2017

[26] S Lenci F Clementi and G Rega ldquoComparing Nonlinear FreeVibrations of Timoshenko Beams with Mechanical or Geo-metric Curvature Definitionrdquo Procedia IUTAM vol 20 pp 34ndash41 2017

[27] O Kopmaz and O Gundogdu ldquoOn the curvature of an Euler-Bernoulli beamrdquo International Journal of Mechanical Engineer-ing Education vol 31 no 2 pp 132ndash142 2003

[28] C Semler G X Li andM P Paıdoussis ldquoThe non-linear equa-tions of motion of pipes conveying fluidrdquo Journal of Sound andVibration vol 169 no 5 pp 577ndash599 1994

[29] D Zhao J Liu and L Wang ldquoNonlinear free vibration of acantilever nanobeam with surface effects Semi-analytical solu-tionsrdquo International Journal of Mechanical Sciences vol 113 pp184ndash195 2016

[30] N Vlajic T Fitzgerald V Nguyen and B BalachandranldquoGeometrically exact planar beams with initial pre-stress andlarge curvature Static configurations natural frequencies andmode shapesrdquo International Journal of Solids and Structures vol51 no 19-20 pp 3361ndash3371 2014

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

[32] F Benedettini and G Rega ldquoNon-linear dynamics of an elasticcable under planar excitationrdquo International Journal of Non-Linear Mechanics vol 22 no 6 pp 497ndash509 1987

[33] H Koivurova ldquoThe numerical study of the nonlinear dynamicsof a light axiallymoving stringrdquo Journal of Sound andVibrationvol 320 no 1-2 pp 373ndash385 2009

[34] A H NayfehTheMethod of Normal Forms Wiley-VCHWein-heim Germany 2nd edition 2011

[35] A Y Leung and Q C Zhang ldquoComplex normal form forstrongly non-linear vibration systems exemplified by Duffing-van der Pol equationrdquo Journal of Sound and Vibration vol 213no 5 pp 907ndash914 1998

[36] W Wang Q C Zhang and X J Wang ldquoApplication of theundetermined fundamental frequency method for analyzingthe critical value of chaosrdquo Acta Physica Sinica vol 58 no 8pp 5162ndash5168 2009

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

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AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus08

minus04

00

04

08

w

5 10 15 200

(a)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

50 15 2010

minus08

minus04

00

04

08

w

(b)

Figure 2 Time history diagram of point1198630 with the initial amplitudes (a) 119886 = 04 and (b) 119886 = 045

insufficient to depict the vibration with large amplitude andlarge curvature However in the whole period although thereis difference between the results from the semianalyticalmethod and numerical simulation the configurations pre-dicted by the two methods are close Therefore it should bementioned that the calculated string morphologies by thesemianalytical are more similar to those from the numericalresults than those from the linear equation model

The next phenomenon is that the numerical morpholo-gies of the string are very complicated in the vibrationprocess For instance when 120591=53 60 80 and 105 in Figures3(a) 3(b) 3(d) and 3(h) there are some nearly horizontalsegments appearing in the curves This strange behavior maybe attributed to the complex nonlinear dynamics responsewhere the clamped ends exert strong constraints on thestring and vice versa near the middle part of the stringthe constraint is much weaker Another possible reasonis that the first-mode analysis of the string may be anoversimplified model and the higher-mode behaviors mustbe further considered to illustrate this phenomenon Further-more our semianalytical solution can also have the similarshapes in this figures especially near the middle part of thestring

To verify this numerical result we perform an experimentas a comparison where the string configurations in thevibration are shown in Figures 4(a)ndash4(h) A rubber stringwith 1408times1630mm2 cross-section is fixed horizontally on adesktop at room temperature whose original lengthL is 5085mm To add the pretension force on the string the deformedlength of the string is elongated to 119897=6877 mm and originarea of the cross-section is 2282 mm2 Thus the prestraincan be given as 120576=(lndashL)L=03524 and using the measuredvalue of Youngrsquos modulus 119864=085 MPa the pretension forceis N0=068N At first in the vertical plane the midpoint ofthe string is excited by an original amplitude which is 04times of the string length 119871 Next the string is released and

then its vibration sequences are recorded by the high speedcamera (Phantom v2512 with 10000 frames per second) Asshown in Figures 4(a) 4(b) 4(c) and 4(d) we find that thereare really platform segments appearing in the string and thetotal configurations in these situations are close to the shapeof trapezoid Meanwhile the string is no longer a straightline when it approaches the equilibrium position as shownin Figures 4(e) and 4(f) implying that the length of the stringis bigger than its original length These behaviors are beyondour imagination as they demonstrate the complexity of thenonlinear dynamics

54 Total String Length It can be seen that in both the LAVequation and the linear equation the extensibility of thestring is considered thus its length can be calculated andthe two curves are compared in Figure 5 The nondimen-sional length of the string after deformation is expressed as119897119871 = int119871

0radic1 + 1199082d119909119871 For the linear equation model this

nondimensional length is a periodic function with respectto the time 120591 ie 119897119871(120591 + 1198792) = 119897119871(120591) This means thatthe maximum and minimum values of the string length donot alter in the vibration process Especially when the stringapproaches the 119909-axis it is really a straight line and thus thevalue of 119897119871 is of the smallest value ie one This featurereemphasizes that the small amplitude vibration has linearproperties On the contrary the oscillation behavior of thestring length is more complex in the LAV model whereits minimum value increases with the elapsing time In thissituation the dimensionless string length is always biggerthan one indicating that the string is always in the elongationstate even when it approaches the 119909-axis In the experimentsuch as in Figures 4(e) 4(f) 3(e) and 3(f) the shapes of thestring are notmerely straight lines and its length at this criticaltime must be larger than one Evidently this consistence onthe experimental phenomenon can validate the efficiency ofour numerical method

Mathematical Problems in Engineering 7

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(a)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(b)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(c)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(d)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(e)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(f)

Figure 3 Continued

8 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06w

(g)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(h)

Figure 3 Vibrational configurations of an elastic string when (a) 120591=53 (b) 120591 = 60 (c) 120591 = 70 (d) 120591 = 80 (e) 120591 = 90 (f) 120591 = 93 (g)120591 = 10 (h) 120591 = 105

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4 Vibrational configurations of the string in the experiment

55 Fundamental Frequency Another vibration characteris-tic parameter about vibration is the frequency and in thecurrent study the value of the fundamental frequency isobtained by the solution of (28) and it is also tested by theexperimental measurement When the vibration has a smallamplitude vibration of the string its dynamic behavior canbe formulated by the linear equation 119902 + 119902 = 0 with thenatural frequency always being 1205961 = 1 which is irrelative tothe initial amplitude We observed in Figure 6 the differencebetween the semianalytical method and experiment methodis great The main reason is that for the small amplitudethe mode shape function should be sin(120587119909) rather thansin(120587119909)radic1 + cos2(120587119909) The frequency exhibited in Figure 6demonstrates that our mode shape function should not be

used for the small amplitude vibration However for thelarge amplitude vibration the fundamental frequency is nota constant and its value is greatly affected by the initialamplitude As shown in Figure 6 with the increase of theamplitude the fundamental frequency also increases whichis verified by the experimental results Although there exitsvalue difference between the semianalytical solution and theexperimental data their tendencies are the same and bothof the two results are bigger than one with the increase ofinitial amplitude a This behavior again demonstrates thecomplicated response of the nonlinear dynamics which isvery sensitive to the initial value It should be clarified that ourmode shape function only for the large amplitude vibrationwith the ratio (amplitude of the middle part of the string

Mathematical Problems in Engineering 9

LAV equation numerical solutionLinear equation theoretical solution

1510 200 5

10

11

12

13

14

lL

Figure 5 Total length of the string based on the LAV equation andlinear equation

LAV equation semianalytical solutionLinear equation theoretical solutionExperiment

100

104

108

112

116

120

1

01 02 03 0400a

Figure 6 Relation between the fundamental 1205961 and the initialamplitude a

to the whole span) is nearly 02-05 If the ratio is greaterdifferent mode shape functions should be selected and thesimilar analysis process can be carried out

6 Conclusion

In conclusion the nonlinear vibration of a soft elasticstring with large amplitude and large curvature has beensystematically investigated in this study The mathematicalmodel in the absence of damping is developed based on theHamilton principle The exact mode shape function differentfrom the counterpart derived from the linear equation wasselected based on the experiment For the large amplitudevibration the time history diagram from the semianalyticalis very close to that from the numerical result and thesetwo results are very different from that of the linear modelThe vibration configurations of the elastic string have thesimilar laws as those of the time history diagrams based on

the comparison of the three models We find that in thenumerical calculations there are platform segments with theshape of trapezoid appearing in the configuration curvesand this phenomenon has been verified by our semianalyticalsolution and the experiment The total length of the string inthe large deformation shows that it is not a periodic functionwith respect to the time which is distinct with the result ofthe linear model We also point out that the fundamentalfrequency of the string with large amplitude vibration isgreatly affected by the initial amplitude which has the sametendency as our experiment result

It should be mentioned that this study is only the firsttrial on the nonlinear vibration of elastic structures withlarge amplitude and large curvatures The following work isexpected which will be extended to some other engineeringstructures such as beams rods plates and shells Theseanalyses may be beneficial to engineer some soft materialsand can also shed light on the design of elementary structuresin sensors actuators and resonators etc

Appendix

A Equations to Determine Coefficients ofNonresonant Terms

Besides (23)-(25) the other equations to determine the valueof Δ 119894 (119894=1 12) are

21198941205961Δ 2 + 119894 (12059620 minus 12059621)21205961 = 021198941205961Δ 3 minus 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) minus 119894119896121205961 = 021198941205961Δ 5 + 3119894119896121205961 = 041198941205961Δ 6 + 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) + 119894119896121205961 = 041198941205961Δ 7 minus 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12) minus 3119894119896121205961 (Δ 3 + Δ 6)minus 119894119896221205961 = 0

21198941205961Δ 8 minus 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11) minus 6119894119896121205961 (Δ 3 + Δ 6)minus 5119894119896221205961 = 0

21198941205961Δ 10 + 3119894119896121205961 (Δ 3 + Δ 6) + 10119894119896221205961 = 041198941205961Δ 11 + 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11)+ 6119894119896121205961 (Δ 3 + Δ 6) + 5119894119896221205961 = 0

10 Mathematical Problems in Engineering

61198941205961Δ 12 + 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12)+ 3119894119896121205961 (Δ 3 + Δ 6) + 119894119896221205961 = 0

(A1)

B Formulas of Parameters Δ 119894The parameters Δ 119894 ( i=1 12) are

Δ 1 = minusΔ 2Δ 2 = 12059621 minus 12059620412059621 Δ 3 = minus2Δ 6Δ 6 = 119896112059620 minus 912059621 Δ 4 = minusΔ 5Δ 5 = minus 31198961412059621 Δ 7 = minus32Δ 12Δ 12 = 61198961Δ 6 minus 211989622512059621 minus 12059620 Δ 8 = minus2Δ 11Δ 11 = 61198961Δ 6 minus 51198962912059621 minus 12059620 Δ 9 = minusΔ 10Δ 10 = 31198961Δ 6 minus 101198962412059621

(B1)

Data Availability

No data were used to support this study

Ethical Approval

Ethical approval was obtained for this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study was supported by the National Natural Sci-ence Foundation of China (nos 11672334 11672335 and11611530541) and the Endeavour Australia Cheung KongResearch Fellowship Scholarship from theAustralian govern-ment

References

[1] C Feng L Yu andW Zhang ldquoDynamic analysis of a dielectricelastomer-based microbeam resonator with large vibrationamplituderdquo International Journal of Non-Linear Mechanics vol65 pp 63ndash68 2014

[2] J Han Q Zhang and W Wang ldquoDesign considerations onlarge amplitude vibration of a doubly clamped microresonatorwith two symmetrically located electrodesrdquo Communications inNonlinear Science andNumerical Simulation vol 22 no 1ndash3 pp492ndash510 2015

[3] T Belendez CNeipp andA Belendez ldquoLarge and small deflec-tions of a cantilever beamrdquo European Journal of Physics vol 23no 3 pp 371ndash379 2002

[4] L Wang and Y Zhao ldquoLarge amplitude motion mechanismand non-planar vibration character of stay cables subject to thesupport motionsrdquo Journal of Sound and Vibration vol 327 no1-2 pp 121ndash133 2009

[5] M Spagnuolo and U Andreaus ldquoA targeted review on largedeformations of planar elastic beams extensibility distributedloads buckling and post-bucklingrdquoMathematics andMechanicsof Solids pp 1ndash23 2018

[6] M P Omran A Amani and H G Lemu ldquoAnalytical approxi-mation of nonlinear vibration of string with large amplitudesrdquoJournal of Mechanical Science and Technology vol 27 no 4 pp981ndash986 2013

[7] Z L Wang P Poncharal andW A De Heer ldquoMeasuring phys-ical and mechanical properties of individual carbon nanotubesby in situ TEMrdquo Journal of Physics and Chemistry of Solids vol61 no 7 pp 1025ndash1030 2000

[8] Z LWang R P Gao P PoncharalWADeHeer Z RDai andZ W Pan ldquoMechanical and electrostatic properties of carbonnanotubes and nanowiresrdquo Materials Science and EngineeringC Materials for Biological Applications vol 16 no 1-2 pp 3ndash102001

[9] Q Zheng and Q Jiang ldquoMultiwalled Carbon Nanotubes asGigahertz Oscillatorsrdquo Physical Review Letters vol 88 no 42002

[10] M Hemmatnezhad R Ansari and G H Rahimi ldquoLarge-amplitude free vibrations of functionally graded beams bymeans of a finite element formulationrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 37 no 18-19 pp 8495ndash8504 2013

[11] J B Gunda R K Gupta G Ranga Janardhan and G Venkates-wara Rao ldquoLarge amplitude vibration analysis of compositebeams Simple closed-form solutionsrdquo Composite Structuresvol 93 no 2 pp 870ndash879 2011

[12] A Nikkar S Bagheri and M Saravi ldquoDynamic model of largeamplitude vibration of a uniform cantilever beam carrying anintermediate lumped mass and rotary inertiardquo Latin AmericanJournal of Solids and Structures vol 11 no 2 pp 320ndash329 2014

[13] Y P Yu B S Wu Y H Sun and L Zang ldquoAnalytical approxi-mate solutions to large amplitude vibration of a spring-hingedbeamrdquoMeccanica vol 48 no 10 pp 2569ndash2575 2013

[14] K K Raju andGV Rao ldquoTowards improved evaluation of largeamplitude free-vibration behaviour of uniform beams usingmulti-term admissible functionsrdquo Journal of Sound and Vibra-tion vol 282 no 3-5 pp 1238ndash1246 2005

[15] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons and Fractals vol 34 no 5 pp 1430ndash1439 2007

[16] S H Hoseini T Pirbodaghi M T Ahmadian and G H Far-rahi ldquoOn the large amplitude free vibrations of tapered beams

Mathematical Problems in Engineering 11

an analytical approachrdquo Mechanics Research Communicationsvol 36 no 8 pp 892ndash897 2009

[17] B Gheshlaghi ldquoLarge-amplitude vibrations of nanowires withdissipative surface stress effectsrdquo Acta Mechanica vol 224 no7 pp 1329ndash1334 2013

[18] S Souayeh and N Kacem ldquoComputational models for largeamplitude nonlinear vibrations of electrostatically actuatedcarbon nanotube-basedmass sensorsrdquo Sensors and Actuators APhysical vol 208 pp 10ndash20 2014

[19] M Simsek ldquoLarge amplitude free vibration of nanobeams withvarious boundary conditions based on the nonlocal elasticitytheoryrdquo Composites Part B Engineering vol 56 pp 621ndash6282014

[20] R Nazemnezhad and S Hosseini-Hashemi ldquoExact solution forlarge amplitude flexural vibration of nanobeams using non-local Euler-Bernoulli theoryrdquo Journal ofTheoretical and AppliedMechanics (Poland) vol 55 no 2 pp 649ndash658 2017

[21] K Nakamura K Kakihara M Kawakami and S Ueha ldquoMea-suring vibration characteristics at large amplitude region ofmaterials for high power ultrasonic vibration systemrdquoUltrason-ics vol 38 no 1 pp 122ndash126 2000

[22] AMozuras and E Podzharov ldquoMeasurement of large harmonicvibration amplitudesrdquo Journal of Sound and Vibration vol 271no 3-5 pp 985ndash998 2004

[23] J C Simo ldquoA finite strain beam formulation The three-dimen-sional dynamic problem Part Irdquo Computer Methods AppliedMechanics and Engineering vol 49 no 1 pp 55ndash70 1985

[24] P-F Pai and A-H Nayfeh ldquoA new method for the modeling ofgeometric nonlinearities in structuresrdquoComputersampStrucruresvol 53 no 4 pp 877ndash895 1994

[25] E Babilio and S Lenci ldquoOn the notion of curvature andits mechanical meaning in a geometrically exact plane beamtheoryrdquo International Journal of Mechanical Sciences vol 128-129 pp 277ndash293 2017

[26] S Lenci F Clementi and G Rega ldquoComparing Nonlinear FreeVibrations of Timoshenko Beams with Mechanical or Geo-metric Curvature Definitionrdquo Procedia IUTAM vol 20 pp 34ndash41 2017

[27] O Kopmaz and O Gundogdu ldquoOn the curvature of an Euler-Bernoulli beamrdquo International Journal of Mechanical Engineer-ing Education vol 31 no 2 pp 132ndash142 2003

[28] C Semler G X Li andM P Paıdoussis ldquoThe non-linear equa-tions of motion of pipes conveying fluidrdquo Journal of Sound andVibration vol 169 no 5 pp 577ndash599 1994

[29] D Zhao J Liu and L Wang ldquoNonlinear free vibration of acantilever nanobeam with surface effects Semi-analytical solu-tionsrdquo International Journal of Mechanical Sciences vol 113 pp184ndash195 2016

[30] N Vlajic T Fitzgerald V Nguyen and B BalachandranldquoGeometrically exact planar beams with initial pre-stress andlarge curvature Static configurations natural frequencies andmode shapesrdquo International Journal of Solids and Structures vol51 no 19-20 pp 3361ndash3371 2014

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

[32] F Benedettini and G Rega ldquoNon-linear dynamics of an elasticcable under planar excitationrdquo International Journal of Non-Linear Mechanics vol 22 no 6 pp 497ndash509 1987

[33] H Koivurova ldquoThe numerical study of the nonlinear dynamicsof a light axiallymoving stringrdquo Journal of Sound andVibrationvol 320 no 1-2 pp 373ndash385 2009

[34] A H NayfehTheMethod of Normal Forms Wiley-VCHWein-heim Germany 2nd edition 2011

[35] A Y Leung and Q C Zhang ldquoComplex normal form forstrongly non-linear vibration systems exemplified by Duffing-van der Pol equationrdquo Journal of Sound and Vibration vol 213no 5 pp 907ndash914 1998

[36] W Wang Q C Zhang and X J Wang ldquoApplication of theundetermined fundamental frequency method for analyzingthe critical value of chaosrdquo Acta Physica Sinica vol 58 no 8pp 5162ndash5168 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(a)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(b)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(c)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06w

02 04 06 08 1000

x

(d)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(e)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

minus06

minus04

minus02

00

02

04

06

w

02 04 06 08 1000

x

(f)

Figure 3 Continued

8 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06w

(g)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(h)

Figure 3 Vibrational configurations of an elastic string when (a) 120591=53 (b) 120591 = 60 (c) 120591 = 70 (d) 120591 = 80 (e) 120591 = 90 (f) 120591 = 93 (g)120591 = 10 (h) 120591 = 105

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4 Vibrational configurations of the string in the experiment

55 Fundamental Frequency Another vibration characteris-tic parameter about vibration is the frequency and in thecurrent study the value of the fundamental frequency isobtained by the solution of (28) and it is also tested by theexperimental measurement When the vibration has a smallamplitude vibration of the string its dynamic behavior canbe formulated by the linear equation 119902 + 119902 = 0 with thenatural frequency always being 1205961 = 1 which is irrelative tothe initial amplitude We observed in Figure 6 the differencebetween the semianalytical method and experiment methodis great The main reason is that for the small amplitudethe mode shape function should be sin(120587119909) rather thansin(120587119909)radic1 + cos2(120587119909) The frequency exhibited in Figure 6demonstrates that our mode shape function should not be

used for the small amplitude vibration However for thelarge amplitude vibration the fundamental frequency is nota constant and its value is greatly affected by the initialamplitude As shown in Figure 6 with the increase of theamplitude the fundamental frequency also increases whichis verified by the experimental results Although there exitsvalue difference between the semianalytical solution and theexperimental data their tendencies are the same and bothof the two results are bigger than one with the increase ofinitial amplitude a This behavior again demonstrates thecomplicated response of the nonlinear dynamics which isvery sensitive to the initial value It should be clarified that ourmode shape function only for the large amplitude vibrationwith the ratio (amplitude of the middle part of the string

Mathematical Problems in Engineering 9

LAV equation numerical solutionLinear equation theoretical solution

1510 200 5

10

11

12

13

14

lL

Figure 5 Total length of the string based on the LAV equation andlinear equation

LAV equation semianalytical solutionLinear equation theoretical solutionExperiment

100

104

108

112

116

120

1

01 02 03 0400a

Figure 6 Relation between the fundamental 1205961 and the initialamplitude a

to the whole span) is nearly 02-05 If the ratio is greaterdifferent mode shape functions should be selected and thesimilar analysis process can be carried out

6 Conclusion

In conclusion the nonlinear vibration of a soft elasticstring with large amplitude and large curvature has beensystematically investigated in this study The mathematicalmodel in the absence of damping is developed based on theHamilton principle The exact mode shape function differentfrom the counterpart derived from the linear equation wasselected based on the experiment For the large amplitudevibration the time history diagram from the semianalyticalis very close to that from the numerical result and thesetwo results are very different from that of the linear modelThe vibration configurations of the elastic string have thesimilar laws as those of the time history diagrams based on

the comparison of the three models We find that in thenumerical calculations there are platform segments with theshape of trapezoid appearing in the configuration curvesand this phenomenon has been verified by our semianalyticalsolution and the experiment The total length of the string inthe large deformation shows that it is not a periodic functionwith respect to the time which is distinct with the result ofthe linear model We also point out that the fundamentalfrequency of the string with large amplitude vibration isgreatly affected by the initial amplitude which has the sametendency as our experiment result

It should be mentioned that this study is only the firsttrial on the nonlinear vibration of elastic structures withlarge amplitude and large curvatures The following work isexpected which will be extended to some other engineeringstructures such as beams rods plates and shells Theseanalyses may be beneficial to engineer some soft materialsand can also shed light on the design of elementary structuresin sensors actuators and resonators etc

Appendix

A Equations to Determine Coefficients ofNonresonant Terms

Besides (23)-(25) the other equations to determine the valueof Δ 119894 (119894=1 12) are

21198941205961Δ 2 + 119894 (12059620 minus 12059621)21205961 = 021198941205961Δ 3 minus 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) minus 119894119896121205961 = 021198941205961Δ 5 + 3119894119896121205961 = 041198941205961Δ 6 + 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) + 119894119896121205961 = 041198941205961Δ 7 minus 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12) minus 3119894119896121205961 (Δ 3 + Δ 6)minus 119894119896221205961 = 0

21198941205961Δ 8 minus 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11) minus 6119894119896121205961 (Δ 3 + Δ 6)minus 5119894119896221205961 = 0

21198941205961Δ 10 + 3119894119896121205961 (Δ 3 + Δ 6) + 10119894119896221205961 = 041198941205961Δ 11 + 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11)+ 6119894119896121205961 (Δ 3 + Δ 6) + 5119894119896221205961 = 0

10 Mathematical Problems in Engineering

61198941205961Δ 12 + 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12)+ 3119894119896121205961 (Δ 3 + Δ 6) + 119894119896221205961 = 0

(A1)

B Formulas of Parameters Δ 119894The parameters Δ 119894 ( i=1 12) are

Δ 1 = minusΔ 2Δ 2 = 12059621 minus 12059620412059621 Δ 3 = minus2Δ 6Δ 6 = 119896112059620 minus 912059621 Δ 4 = minusΔ 5Δ 5 = minus 31198961412059621 Δ 7 = minus32Δ 12Δ 12 = 61198961Δ 6 minus 211989622512059621 minus 12059620 Δ 8 = minus2Δ 11Δ 11 = 61198961Δ 6 minus 51198962912059621 minus 12059620 Δ 9 = minusΔ 10Δ 10 = 31198961Δ 6 minus 101198962412059621

(B1)

Data Availability

No data were used to support this study

Ethical Approval

Ethical approval was obtained for this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study was supported by the National Natural Sci-ence Foundation of China (nos 11672334 11672335 and11611530541) and the Endeavour Australia Cheung KongResearch Fellowship Scholarship from theAustralian govern-ment

References

[1] C Feng L Yu andW Zhang ldquoDynamic analysis of a dielectricelastomer-based microbeam resonator with large vibrationamplituderdquo International Journal of Non-Linear Mechanics vol65 pp 63ndash68 2014

[2] J Han Q Zhang and W Wang ldquoDesign considerations onlarge amplitude vibration of a doubly clamped microresonatorwith two symmetrically located electrodesrdquo Communications inNonlinear Science andNumerical Simulation vol 22 no 1ndash3 pp492ndash510 2015

[3] T Belendez CNeipp andA Belendez ldquoLarge and small deflec-tions of a cantilever beamrdquo European Journal of Physics vol 23no 3 pp 371ndash379 2002

[4] L Wang and Y Zhao ldquoLarge amplitude motion mechanismand non-planar vibration character of stay cables subject to thesupport motionsrdquo Journal of Sound and Vibration vol 327 no1-2 pp 121ndash133 2009

[5] M Spagnuolo and U Andreaus ldquoA targeted review on largedeformations of planar elastic beams extensibility distributedloads buckling and post-bucklingrdquoMathematics andMechanicsof Solids pp 1ndash23 2018

[6] M P Omran A Amani and H G Lemu ldquoAnalytical approxi-mation of nonlinear vibration of string with large amplitudesrdquoJournal of Mechanical Science and Technology vol 27 no 4 pp981ndash986 2013

[7] Z L Wang P Poncharal andW A De Heer ldquoMeasuring phys-ical and mechanical properties of individual carbon nanotubesby in situ TEMrdquo Journal of Physics and Chemistry of Solids vol61 no 7 pp 1025ndash1030 2000

[8] Z LWang R P Gao P PoncharalWADeHeer Z RDai andZ W Pan ldquoMechanical and electrostatic properties of carbonnanotubes and nanowiresrdquo Materials Science and EngineeringC Materials for Biological Applications vol 16 no 1-2 pp 3ndash102001

[9] Q Zheng and Q Jiang ldquoMultiwalled Carbon Nanotubes asGigahertz Oscillatorsrdquo Physical Review Letters vol 88 no 42002

[10] M Hemmatnezhad R Ansari and G H Rahimi ldquoLarge-amplitude free vibrations of functionally graded beams bymeans of a finite element formulationrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 37 no 18-19 pp 8495ndash8504 2013

[11] J B Gunda R K Gupta G Ranga Janardhan and G Venkates-wara Rao ldquoLarge amplitude vibration analysis of compositebeams Simple closed-form solutionsrdquo Composite Structuresvol 93 no 2 pp 870ndash879 2011

[12] A Nikkar S Bagheri and M Saravi ldquoDynamic model of largeamplitude vibration of a uniform cantilever beam carrying anintermediate lumped mass and rotary inertiardquo Latin AmericanJournal of Solids and Structures vol 11 no 2 pp 320ndash329 2014

[13] Y P Yu B S Wu Y H Sun and L Zang ldquoAnalytical approxi-mate solutions to large amplitude vibration of a spring-hingedbeamrdquoMeccanica vol 48 no 10 pp 2569ndash2575 2013

[14] K K Raju andGV Rao ldquoTowards improved evaluation of largeamplitude free-vibration behaviour of uniform beams usingmulti-term admissible functionsrdquo Journal of Sound and Vibra-tion vol 282 no 3-5 pp 1238ndash1246 2005

[15] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons and Fractals vol 34 no 5 pp 1430ndash1439 2007

[16] S H Hoseini T Pirbodaghi M T Ahmadian and G H Far-rahi ldquoOn the large amplitude free vibrations of tapered beams

Mathematical Problems in Engineering 11

an analytical approachrdquo Mechanics Research Communicationsvol 36 no 8 pp 892ndash897 2009

[17] B Gheshlaghi ldquoLarge-amplitude vibrations of nanowires withdissipative surface stress effectsrdquo Acta Mechanica vol 224 no7 pp 1329ndash1334 2013

[18] S Souayeh and N Kacem ldquoComputational models for largeamplitude nonlinear vibrations of electrostatically actuatedcarbon nanotube-basedmass sensorsrdquo Sensors and Actuators APhysical vol 208 pp 10ndash20 2014

[19] M Simsek ldquoLarge amplitude free vibration of nanobeams withvarious boundary conditions based on the nonlocal elasticitytheoryrdquo Composites Part B Engineering vol 56 pp 621ndash6282014

[20] R Nazemnezhad and S Hosseini-Hashemi ldquoExact solution forlarge amplitude flexural vibration of nanobeams using non-local Euler-Bernoulli theoryrdquo Journal ofTheoretical and AppliedMechanics (Poland) vol 55 no 2 pp 649ndash658 2017

[21] K Nakamura K Kakihara M Kawakami and S Ueha ldquoMea-suring vibration characteristics at large amplitude region ofmaterials for high power ultrasonic vibration systemrdquoUltrason-ics vol 38 no 1 pp 122ndash126 2000

[22] AMozuras and E Podzharov ldquoMeasurement of large harmonicvibration amplitudesrdquo Journal of Sound and Vibration vol 271no 3-5 pp 985ndash998 2004

[23] J C Simo ldquoA finite strain beam formulation The three-dimen-sional dynamic problem Part Irdquo Computer Methods AppliedMechanics and Engineering vol 49 no 1 pp 55ndash70 1985

[24] P-F Pai and A-H Nayfeh ldquoA new method for the modeling ofgeometric nonlinearities in structuresrdquoComputersampStrucruresvol 53 no 4 pp 877ndash895 1994

[25] E Babilio and S Lenci ldquoOn the notion of curvature andits mechanical meaning in a geometrically exact plane beamtheoryrdquo International Journal of Mechanical Sciences vol 128-129 pp 277ndash293 2017

[26] S Lenci F Clementi and G Rega ldquoComparing Nonlinear FreeVibrations of Timoshenko Beams with Mechanical or Geo-metric Curvature Definitionrdquo Procedia IUTAM vol 20 pp 34ndash41 2017

[27] O Kopmaz and O Gundogdu ldquoOn the curvature of an Euler-Bernoulli beamrdquo International Journal of Mechanical Engineer-ing Education vol 31 no 2 pp 132ndash142 2003

[28] C Semler G X Li andM P Paıdoussis ldquoThe non-linear equa-tions of motion of pipes conveying fluidrdquo Journal of Sound andVibration vol 169 no 5 pp 577ndash599 1994

[29] D Zhao J Liu and L Wang ldquoNonlinear free vibration of acantilever nanobeam with surface effects Semi-analytical solu-tionsrdquo International Journal of Mechanical Sciences vol 113 pp184ndash195 2016

[30] N Vlajic T Fitzgerald V Nguyen and B BalachandranldquoGeometrically exact planar beams with initial pre-stress andlarge curvature Static configurations natural frequencies andmode shapesrdquo International Journal of Solids and Structures vol51 no 19-20 pp 3361ndash3371 2014

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

[32] F Benedettini and G Rega ldquoNon-linear dynamics of an elasticcable under planar excitationrdquo International Journal of Non-Linear Mechanics vol 22 no 6 pp 497ndash509 1987

[33] H Koivurova ldquoThe numerical study of the nonlinear dynamicsof a light axiallymoving stringrdquo Journal of Sound andVibrationvol 320 no 1-2 pp 373ndash385 2009

[34] A H NayfehTheMethod of Normal Forms Wiley-VCHWein-heim Germany 2nd edition 2011

[35] A Y Leung and Q C Zhang ldquoComplex normal form forstrongly non-linear vibration systems exemplified by Duffing-van der Pol equationrdquo Journal of Sound and Vibration vol 213no 5 pp 907ndash914 1998

[36] W Wang Q C Zhang and X J Wang ldquoApplication of theundetermined fundamental frequency method for analyzingthe critical value of chaosrdquo Acta Physica Sinica vol 58 no 8pp 5162ndash5168 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06w

(g)

LAV equation numerical solutionLAV equation semianalytical solutionLinear equation theoretical solution

02 04 06 08 1000

x

minus06

minus04

minus02

00

02

04

06

w

(h)

Figure 3 Vibrational configurations of an elastic string when (a) 120591=53 (b) 120591 = 60 (c) 120591 = 70 (d) 120591 = 80 (e) 120591 = 90 (f) 120591 = 93 (g)120591 = 10 (h) 120591 = 105

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4 Vibrational configurations of the string in the experiment

55 Fundamental Frequency Another vibration characteris-tic parameter about vibration is the frequency and in thecurrent study the value of the fundamental frequency isobtained by the solution of (28) and it is also tested by theexperimental measurement When the vibration has a smallamplitude vibration of the string its dynamic behavior canbe formulated by the linear equation 119902 + 119902 = 0 with thenatural frequency always being 1205961 = 1 which is irrelative tothe initial amplitude We observed in Figure 6 the differencebetween the semianalytical method and experiment methodis great The main reason is that for the small amplitudethe mode shape function should be sin(120587119909) rather thansin(120587119909)radic1 + cos2(120587119909) The frequency exhibited in Figure 6demonstrates that our mode shape function should not be

used for the small amplitude vibration However for thelarge amplitude vibration the fundamental frequency is nota constant and its value is greatly affected by the initialamplitude As shown in Figure 6 with the increase of theamplitude the fundamental frequency also increases whichis verified by the experimental results Although there exitsvalue difference between the semianalytical solution and theexperimental data their tendencies are the same and bothof the two results are bigger than one with the increase ofinitial amplitude a This behavior again demonstrates thecomplicated response of the nonlinear dynamics which isvery sensitive to the initial value It should be clarified that ourmode shape function only for the large amplitude vibrationwith the ratio (amplitude of the middle part of the string

Mathematical Problems in Engineering 9

LAV equation numerical solutionLinear equation theoretical solution

1510 200 5

10

11

12

13

14

lL

Figure 5 Total length of the string based on the LAV equation andlinear equation

LAV equation semianalytical solutionLinear equation theoretical solutionExperiment

100

104

108

112

116

120

1

01 02 03 0400a

Figure 6 Relation between the fundamental 1205961 and the initialamplitude a

to the whole span) is nearly 02-05 If the ratio is greaterdifferent mode shape functions should be selected and thesimilar analysis process can be carried out

6 Conclusion

In conclusion the nonlinear vibration of a soft elasticstring with large amplitude and large curvature has beensystematically investigated in this study The mathematicalmodel in the absence of damping is developed based on theHamilton principle The exact mode shape function differentfrom the counterpart derived from the linear equation wasselected based on the experiment For the large amplitudevibration the time history diagram from the semianalyticalis very close to that from the numerical result and thesetwo results are very different from that of the linear modelThe vibration configurations of the elastic string have thesimilar laws as those of the time history diagrams based on

the comparison of the three models We find that in thenumerical calculations there are platform segments with theshape of trapezoid appearing in the configuration curvesand this phenomenon has been verified by our semianalyticalsolution and the experiment The total length of the string inthe large deformation shows that it is not a periodic functionwith respect to the time which is distinct with the result ofthe linear model We also point out that the fundamentalfrequency of the string with large amplitude vibration isgreatly affected by the initial amplitude which has the sametendency as our experiment result

It should be mentioned that this study is only the firsttrial on the nonlinear vibration of elastic structures withlarge amplitude and large curvatures The following work isexpected which will be extended to some other engineeringstructures such as beams rods plates and shells Theseanalyses may be beneficial to engineer some soft materialsand can also shed light on the design of elementary structuresin sensors actuators and resonators etc

Appendix

A Equations to Determine Coefficients ofNonresonant Terms

Besides (23)-(25) the other equations to determine the valueof Δ 119894 (119894=1 12) are

21198941205961Δ 2 + 119894 (12059620 minus 12059621)21205961 = 021198941205961Δ 3 minus 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) minus 119894119896121205961 = 021198941205961Δ 5 + 3119894119896121205961 = 041198941205961Δ 6 + 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) + 119894119896121205961 = 041198941205961Δ 7 minus 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12) minus 3119894119896121205961 (Δ 3 + Δ 6)minus 119894119896221205961 = 0

21198941205961Δ 8 minus 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11) minus 6119894119896121205961 (Δ 3 + Δ 6)minus 5119894119896221205961 = 0

21198941205961Δ 10 + 3119894119896121205961 (Δ 3 + Δ 6) + 10119894119896221205961 = 041198941205961Δ 11 + 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11)+ 6119894119896121205961 (Δ 3 + Δ 6) + 5119894119896221205961 = 0

10 Mathematical Problems in Engineering

61198941205961Δ 12 + 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12)+ 3119894119896121205961 (Δ 3 + Δ 6) + 119894119896221205961 = 0

(A1)

B Formulas of Parameters Δ 119894The parameters Δ 119894 ( i=1 12) are

Δ 1 = minusΔ 2Δ 2 = 12059621 minus 12059620412059621 Δ 3 = minus2Δ 6Δ 6 = 119896112059620 minus 912059621 Δ 4 = minusΔ 5Δ 5 = minus 31198961412059621 Δ 7 = minus32Δ 12Δ 12 = 61198961Δ 6 minus 211989622512059621 minus 12059620 Δ 8 = minus2Δ 11Δ 11 = 61198961Δ 6 minus 51198962912059621 minus 12059620 Δ 9 = minusΔ 10Δ 10 = 31198961Δ 6 minus 101198962412059621

(B1)

Data Availability

No data were used to support this study

Ethical Approval

Ethical approval was obtained for this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study was supported by the National Natural Sci-ence Foundation of China (nos 11672334 11672335 and11611530541) and the Endeavour Australia Cheung KongResearch Fellowship Scholarship from theAustralian govern-ment

References

[1] C Feng L Yu andW Zhang ldquoDynamic analysis of a dielectricelastomer-based microbeam resonator with large vibrationamplituderdquo International Journal of Non-Linear Mechanics vol65 pp 63ndash68 2014

[2] J Han Q Zhang and W Wang ldquoDesign considerations onlarge amplitude vibration of a doubly clamped microresonatorwith two symmetrically located electrodesrdquo Communications inNonlinear Science andNumerical Simulation vol 22 no 1ndash3 pp492ndash510 2015

[3] T Belendez CNeipp andA Belendez ldquoLarge and small deflec-tions of a cantilever beamrdquo European Journal of Physics vol 23no 3 pp 371ndash379 2002

[4] L Wang and Y Zhao ldquoLarge amplitude motion mechanismand non-planar vibration character of stay cables subject to thesupport motionsrdquo Journal of Sound and Vibration vol 327 no1-2 pp 121ndash133 2009

[5] M Spagnuolo and U Andreaus ldquoA targeted review on largedeformations of planar elastic beams extensibility distributedloads buckling and post-bucklingrdquoMathematics andMechanicsof Solids pp 1ndash23 2018

[6] M P Omran A Amani and H G Lemu ldquoAnalytical approxi-mation of nonlinear vibration of string with large amplitudesrdquoJournal of Mechanical Science and Technology vol 27 no 4 pp981ndash986 2013

[7] Z L Wang P Poncharal andW A De Heer ldquoMeasuring phys-ical and mechanical properties of individual carbon nanotubesby in situ TEMrdquo Journal of Physics and Chemistry of Solids vol61 no 7 pp 1025ndash1030 2000

[8] Z LWang R P Gao P PoncharalWADeHeer Z RDai andZ W Pan ldquoMechanical and electrostatic properties of carbonnanotubes and nanowiresrdquo Materials Science and EngineeringC Materials for Biological Applications vol 16 no 1-2 pp 3ndash102001

[9] Q Zheng and Q Jiang ldquoMultiwalled Carbon Nanotubes asGigahertz Oscillatorsrdquo Physical Review Letters vol 88 no 42002

[10] M Hemmatnezhad R Ansari and G H Rahimi ldquoLarge-amplitude free vibrations of functionally graded beams bymeans of a finite element formulationrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 37 no 18-19 pp 8495ndash8504 2013

[11] J B Gunda R K Gupta G Ranga Janardhan and G Venkates-wara Rao ldquoLarge amplitude vibration analysis of compositebeams Simple closed-form solutionsrdquo Composite Structuresvol 93 no 2 pp 870ndash879 2011

[12] A Nikkar S Bagheri and M Saravi ldquoDynamic model of largeamplitude vibration of a uniform cantilever beam carrying anintermediate lumped mass and rotary inertiardquo Latin AmericanJournal of Solids and Structures vol 11 no 2 pp 320ndash329 2014

[13] Y P Yu B S Wu Y H Sun and L Zang ldquoAnalytical approxi-mate solutions to large amplitude vibration of a spring-hingedbeamrdquoMeccanica vol 48 no 10 pp 2569ndash2575 2013

[14] K K Raju andGV Rao ldquoTowards improved evaluation of largeamplitude free-vibration behaviour of uniform beams usingmulti-term admissible functionsrdquo Journal of Sound and Vibra-tion vol 282 no 3-5 pp 1238ndash1246 2005

[15] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons and Fractals vol 34 no 5 pp 1430ndash1439 2007

[16] S H Hoseini T Pirbodaghi M T Ahmadian and G H Far-rahi ldquoOn the large amplitude free vibrations of tapered beams

Mathematical Problems in Engineering 11

an analytical approachrdquo Mechanics Research Communicationsvol 36 no 8 pp 892ndash897 2009

[17] B Gheshlaghi ldquoLarge-amplitude vibrations of nanowires withdissipative surface stress effectsrdquo Acta Mechanica vol 224 no7 pp 1329ndash1334 2013

[18] S Souayeh and N Kacem ldquoComputational models for largeamplitude nonlinear vibrations of electrostatically actuatedcarbon nanotube-basedmass sensorsrdquo Sensors and Actuators APhysical vol 208 pp 10ndash20 2014

[19] M Simsek ldquoLarge amplitude free vibration of nanobeams withvarious boundary conditions based on the nonlocal elasticitytheoryrdquo Composites Part B Engineering vol 56 pp 621ndash6282014

[20] R Nazemnezhad and S Hosseini-Hashemi ldquoExact solution forlarge amplitude flexural vibration of nanobeams using non-local Euler-Bernoulli theoryrdquo Journal ofTheoretical and AppliedMechanics (Poland) vol 55 no 2 pp 649ndash658 2017

[21] K Nakamura K Kakihara M Kawakami and S Ueha ldquoMea-suring vibration characteristics at large amplitude region ofmaterials for high power ultrasonic vibration systemrdquoUltrason-ics vol 38 no 1 pp 122ndash126 2000

[22] AMozuras and E Podzharov ldquoMeasurement of large harmonicvibration amplitudesrdquo Journal of Sound and Vibration vol 271no 3-5 pp 985ndash998 2004

[23] J C Simo ldquoA finite strain beam formulation The three-dimen-sional dynamic problem Part Irdquo Computer Methods AppliedMechanics and Engineering vol 49 no 1 pp 55ndash70 1985

[24] P-F Pai and A-H Nayfeh ldquoA new method for the modeling ofgeometric nonlinearities in structuresrdquoComputersampStrucruresvol 53 no 4 pp 877ndash895 1994

[25] E Babilio and S Lenci ldquoOn the notion of curvature andits mechanical meaning in a geometrically exact plane beamtheoryrdquo International Journal of Mechanical Sciences vol 128-129 pp 277ndash293 2017

[26] S Lenci F Clementi and G Rega ldquoComparing Nonlinear FreeVibrations of Timoshenko Beams with Mechanical or Geo-metric Curvature Definitionrdquo Procedia IUTAM vol 20 pp 34ndash41 2017

[27] O Kopmaz and O Gundogdu ldquoOn the curvature of an Euler-Bernoulli beamrdquo International Journal of Mechanical Engineer-ing Education vol 31 no 2 pp 132ndash142 2003

[28] C Semler G X Li andM P Paıdoussis ldquoThe non-linear equa-tions of motion of pipes conveying fluidrdquo Journal of Sound andVibration vol 169 no 5 pp 577ndash599 1994

[29] D Zhao J Liu and L Wang ldquoNonlinear free vibration of acantilever nanobeam with surface effects Semi-analytical solu-tionsrdquo International Journal of Mechanical Sciences vol 113 pp184ndash195 2016

[30] N Vlajic T Fitzgerald V Nguyen and B BalachandranldquoGeometrically exact planar beams with initial pre-stress andlarge curvature Static configurations natural frequencies andmode shapesrdquo International Journal of Solids and Structures vol51 no 19-20 pp 3361ndash3371 2014

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

[32] F Benedettini and G Rega ldquoNon-linear dynamics of an elasticcable under planar excitationrdquo International Journal of Non-Linear Mechanics vol 22 no 6 pp 497ndash509 1987

[33] H Koivurova ldquoThe numerical study of the nonlinear dynamicsof a light axiallymoving stringrdquo Journal of Sound andVibrationvol 320 no 1-2 pp 373ndash385 2009

[34] A H NayfehTheMethod of Normal Forms Wiley-VCHWein-heim Germany 2nd edition 2011

[35] A Y Leung and Q C Zhang ldquoComplex normal form forstrongly non-linear vibration systems exemplified by Duffing-van der Pol equationrdquo Journal of Sound and Vibration vol 213no 5 pp 907ndash914 1998

[36] W Wang Q C Zhang and X J Wang ldquoApplication of theundetermined fundamental frequency method for analyzingthe critical value of chaosrdquo Acta Physica Sinica vol 58 no 8pp 5162ndash5168 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

LAV equation numerical solutionLinear equation theoretical solution

1510 200 5

10

11

12

13

14

lL

Figure 5 Total length of the string based on the LAV equation andlinear equation

LAV equation semianalytical solutionLinear equation theoretical solutionExperiment

100

104

108

112

116

120

1

01 02 03 0400a

Figure 6 Relation between the fundamental 1205961 and the initialamplitude a

to the whole span) is nearly 02-05 If the ratio is greaterdifferent mode shape functions should be selected and thesimilar analysis process can be carried out

6 Conclusion

In conclusion the nonlinear vibration of a soft elasticstring with large amplitude and large curvature has beensystematically investigated in this study The mathematicalmodel in the absence of damping is developed based on theHamilton principle The exact mode shape function differentfrom the counterpart derived from the linear equation wasselected based on the experiment For the large amplitudevibration the time history diagram from the semianalyticalis very close to that from the numerical result and thesetwo results are very different from that of the linear modelThe vibration configurations of the elastic string have thesimilar laws as those of the time history diagrams based on

the comparison of the three models We find that in thenumerical calculations there are platform segments with theshape of trapezoid appearing in the configuration curvesand this phenomenon has been verified by our semianalyticalsolution and the experiment The total length of the string inthe large deformation shows that it is not a periodic functionwith respect to the time which is distinct with the result ofthe linear model We also point out that the fundamentalfrequency of the string with large amplitude vibration isgreatly affected by the initial amplitude which has the sametendency as our experiment result

It should be mentioned that this study is only the firsttrial on the nonlinear vibration of elastic structures withlarge amplitude and large curvatures The following work isexpected which will be extended to some other engineeringstructures such as beams rods plates and shells Theseanalyses may be beneficial to engineer some soft materialsand can also shed light on the design of elementary structuresin sensors actuators and resonators etc

Appendix

A Equations to Determine Coefficients ofNonresonant Terms

Besides (23)-(25) the other equations to determine the valueof Δ 119894 (119894=1 12) are

21198941205961Δ 2 + 119894 (12059620 minus 12059621)21205961 = 021198941205961Δ 3 minus 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) minus 119894119896121205961 = 021198941205961Δ 5 + 3119894119896121205961 = 041198941205961Δ 6 + 119894 (12059620 minus 12059621)21205961 (Δ 3 + Δ 6) + 119894119896121205961 = 041198941205961Δ 7 minus 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12) minus 3119894119896121205961 (Δ 3 + Δ 6)minus 119894119896221205961 = 0

21198941205961Δ 8 minus 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11) minus 6119894119896121205961 (Δ 3 + Δ 6)minus 5119894119896221205961 = 0

21198941205961Δ 10 + 3119894119896121205961 (Δ 3 + Δ 6) + 10119894119896221205961 = 041198941205961Δ 11 + 119894 (12059620 minus 12059621)21205961 (Δ 8 + Δ 11)+ 6119894119896121205961 (Δ 3 + Δ 6) + 5119894119896221205961 = 0

10 Mathematical Problems in Engineering

61198941205961Δ 12 + 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12)+ 3119894119896121205961 (Δ 3 + Δ 6) + 119894119896221205961 = 0

(A1)

B Formulas of Parameters Δ 119894The parameters Δ 119894 ( i=1 12) are

Δ 1 = minusΔ 2Δ 2 = 12059621 minus 12059620412059621 Δ 3 = minus2Δ 6Δ 6 = 119896112059620 minus 912059621 Δ 4 = minusΔ 5Δ 5 = minus 31198961412059621 Δ 7 = minus32Δ 12Δ 12 = 61198961Δ 6 minus 211989622512059621 minus 12059620 Δ 8 = minus2Δ 11Δ 11 = 61198961Δ 6 minus 51198962912059621 minus 12059620 Δ 9 = minusΔ 10Δ 10 = 31198961Δ 6 minus 101198962412059621

(B1)

Data Availability

No data were used to support this study

Ethical Approval

Ethical approval was obtained for this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study was supported by the National Natural Sci-ence Foundation of China (nos 11672334 11672335 and11611530541) and the Endeavour Australia Cheung KongResearch Fellowship Scholarship from theAustralian govern-ment

References

[1] C Feng L Yu andW Zhang ldquoDynamic analysis of a dielectricelastomer-based microbeam resonator with large vibrationamplituderdquo International Journal of Non-Linear Mechanics vol65 pp 63ndash68 2014

[2] J Han Q Zhang and W Wang ldquoDesign considerations onlarge amplitude vibration of a doubly clamped microresonatorwith two symmetrically located electrodesrdquo Communications inNonlinear Science andNumerical Simulation vol 22 no 1ndash3 pp492ndash510 2015

[3] T Belendez CNeipp andA Belendez ldquoLarge and small deflec-tions of a cantilever beamrdquo European Journal of Physics vol 23no 3 pp 371ndash379 2002

[4] L Wang and Y Zhao ldquoLarge amplitude motion mechanismand non-planar vibration character of stay cables subject to thesupport motionsrdquo Journal of Sound and Vibration vol 327 no1-2 pp 121ndash133 2009

[5] M Spagnuolo and U Andreaus ldquoA targeted review on largedeformations of planar elastic beams extensibility distributedloads buckling and post-bucklingrdquoMathematics andMechanicsof Solids pp 1ndash23 2018

[6] M P Omran A Amani and H G Lemu ldquoAnalytical approxi-mation of nonlinear vibration of string with large amplitudesrdquoJournal of Mechanical Science and Technology vol 27 no 4 pp981ndash986 2013

[7] Z L Wang P Poncharal andW A De Heer ldquoMeasuring phys-ical and mechanical properties of individual carbon nanotubesby in situ TEMrdquo Journal of Physics and Chemistry of Solids vol61 no 7 pp 1025ndash1030 2000

[8] Z LWang R P Gao P PoncharalWADeHeer Z RDai andZ W Pan ldquoMechanical and electrostatic properties of carbonnanotubes and nanowiresrdquo Materials Science and EngineeringC Materials for Biological Applications vol 16 no 1-2 pp 3ndash102001

[9] Q Zheng and Q Jiang ldquoMultiwalled Carbon Nanotubes asGigahertz Oscillatorsrdquo Physical Review Letters vol 88 no 42002

[10] M Hemmatnezhad R Ansari and G H Rahimi ldquoLarge-amplitude free vibrations of functionally graded beams bymeans of a finite element formulationrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 37 no 18-19 pp 8495ndash8504 2013

[11] J B Gunda R K Gupta G Ranga Janardhan and G Venkates-wara Rao ldquoLarge amplitude vibration analysis of compositebeams Simple closed-form solutionsrdquo Composite Structuresvol 93 no 2 pp 870ndash879 2011

[12] A Nikkar S Bagheri and M Saravi ldquoDynamic model of largeamplitude vibration of a uniform cantilever beam carrying anintermediate lumped mass and rotary inertiardquo Latin AmericanJournal of Solids and Structures vol 11 no 2 pp 320ndash329 2014

[13] Y P Yu B S Wu Y H Sun and L Zang ldquoAnalytical approxi-mate solutions to large amplitude vibration of a spring-hingedbeamrdquoMeccanica vol 48 no 10 pp 2569ndash2575 2013

[14] K K Raju andGV Rao ldquoTowards improved evaluation of largeamplitude free-vibration behaviour of uniform beams usingmulti-term admissible functionsrdquo Journal of Sound and Vibra-tion vol 282 no 3-5 pp 1238ndash1246 2005

[15] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons and Fractals vol 34 no 5 pp 1430ndash1439 2007

[16] S H Hoseini T Pirbodaghi M T Ahmadian and G H Far-rahi ldquoOn the large amplitude free vibrations of tapered beams

Mathematical Problems in Engineering 11

an analytical approachrdquo Mechanics Research Communicationsvol 36 no 8 pp 892ndash897 2009

[17] B Gheshlaghi ldquoLarge-amplitude vibrations of nanowires withdissipative surface stress effectsrdquo Acta Mechanica vol 224 no7 pp 1329ndash1334 2013

[18] S Souayeh and N Kacem ldquoComputational models for largeamplitude nonlinear vibrations of electrostatically actuatedcarbon nanotube-basedmass sensorsrdquo Sensors and Actuators APhysical vol 208 pp 10ndash20 2014

[19] M Simsek ldquoLarge amplitude free vibration of nanobeams withvarious boundary conditions based on the nonlocal elasticitytheoryrdquo Composites Part B Engineering vol 56 pp 621ndash6282014

[20] R Nazemnezhad and S Hosseini-Hashemi ldquoExact solution forlarge amplitude flexural vibration of nanobeams using non-local Euler-Bernoulli theoryrdquo Journal ofTheoretical and AppliedMechanics (Poland) vol 55 no 2 pp 649ndash658 2017

[21] K Nakamura K Kakihara M Kawakami and S Ueha ldquoMea-suring vibration characteristics at large amplitude region ofmaterials for high power ultrasonic vibration systemrdquoUltrason-ics vol 38 no 1 pp 122ndash126 2000

[22] AMozuras and E Podzharov ldquoMeasurement of large harmonicvibration amplitudesrdquo Journal of Sound and Vibration vol 271no 3-5 pp 985ndash998 2004

[23] J C Simo ldquoA finite strain beam formulation The three-dimen-sional dynamic problem Part Irdquo Computer Methods AppliedMechanics and Engineering vol 49 no 1 pp 55ndash70 1985

[24] P-F Pai and A-H Nayfeh ldquoA new method for the modeling ofgeometric nonlinearities in structuresrdquoComputersampStrucruresvol 53 no 4 pp 877ndash895 1994

[25] E Babilio and S Lenci ldquoOn the notion of curvature andits mechanical meaning in a geometrically exact plane beamtheoryrdquo International Journal of Mechanical Sciences vol 128-129 pp 277ndash293 2017

[26] S Lenci F Clementi and G Rega ldquoComparing Nonlinear FreeVibrations of Timoshenko Beams with Mechanical or Geo-metric Curvature Definitionrdquo Procedia IUTAM vol 20 pp 34ndash41 2017

[27] O Kopmaz and O Gundogdu ldquoOn the curvature of an Euler-Bernoulli beamrdquo International Journal of Mechanical Engineer-ing Education vol 31 no 2 pp 132ndash142 2003

[28] C Semler G X Li andM P Paıdoussis ldquoThe non-linear equa-tions of motion of pipes conveying fluidrdquo Journal of Sound andVibration vol 169 no 5 pp 577ndash599 1994

[29] D Zhao J Liu and L Wang ldquoNonlinear free vibration of acantilever nanobeam with surface effects Semi-analytical solu-tionsrdquo International Journal of Mechanical Sciences vol 113 pp184ndash195 2016

[30] N Vlajic T Fitzgerald V Nguyen and B BalachandranldquoGeometrically exact planar beams with initial pre-stress andlarge curvature Static configurations natural frequencies andmode shapesrdquo International Journal of Solids and Structures vol51 no 19-20 pp 3361ndash3371 2014

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

[32] F Benedettini and G Rega ldquoNon-linear dynamics of an elasticcable under planar excitationrdquo International Journal of Non-Linear Mechanics vol 22 no 6 pp 497ndash509 1987

[33] H Koivurova ldquoThe numerical study of the nonlinear dynamicsof a light axiallymoving stringrdquo Journal of Sound andVibrationvol 320 no 1-2 pp 373ndash385 2009

[34] A H NayfehTheMethod of Normal Forms Wiley-VCHWein-heim Germany 2nd edition 2011

[35] A Y Leung and Q C Zhang ldquoComplex normal form forstrongly non-linear vibration systems exemplified by Duffing-van der Pol equationrdquo Journal of Sound and Vibration vol 213no 5 pp 907ndash914 1998

[36] W Wang Q C Zhang and X J Wang ldquoApplication of theundetermined fundamental frequency method for analyzingthe critical value of chaosrdquo Acta Physica Sinica vol 58 no 8pp 5162ndash5168 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

61198941205961Δ 12 + 119894 (12059620 minus 12059621)21205961 (Δ 7 + Δ 12)+ 3119894119896121205961 (Δ 3 + Δ 6) + 119894119896221205961 = 0

(A1)

B Formulas of Parameters Δ 119894The parameters Δ 119894 ( i=1 12) are

Δ 1 = minusΔ 2Δ 2 = 12059621 minus 12059620412059621 Δ 3 = minus2Δ 6Δ 6 = 119896112059620 minus 912059621 Δ 4 = minusΔ 5Δ 5 = minus 31198961412059621 Δ 7 = minus32Δ 12Δ 12 = 61198961Δ 6 minus 211989622512059621 minus 12059620 Δ 8 = minus2Δ 11Δ 11 = 61198961Δ 6 minus 51198962912059621 minus 12059620 Δ 9 = minusΔ 10Δ 10 = 31198961Δ 6 minus 101198962412059621

(B1)

Data Availability

No data were used to support this study

Ethical Approval

Ethical approval was obtained for this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study was supported by the National Natural Sci-ence Foundation of China (nos 11672334 11672335 and11611530541) and the Endeavour Australia Cheung KongResearch Fellowship Scholarship from theAustralian govern-ment

References

[1] C Feng L Yu andW Zhang ldquoDynamic analysis of a dielectricelastomer-based microbeam resonator with large vibrationamplituderdquo International Journal of Non-Linear Mechanics vol65 pp 63ndash68 2014

[2] J Han Q Zhang and W Wang ldquoDesign considerations onlarge amplitude vibration of a doubly clamped microresonatorwith two symmetrically located electrodesrdquo Communications inNonlinear Science andNumerical Simulation vol 22 no 1ndash3 pp492ndash510 2015

[3] T Belendez CNeipp andA Belendez ldquoLarge and small deflec-tions of a cantilever beamrdquo European Journal of Physics vol 23no 3 pp 371ndash379 2002

[4] L Wang and Y Zhao ldquoLarge amplitude motion mechanismand non-planar vibration character of stay cables subject to thesupport motionsrdquo Journal of Sound and Vibration vol 327 no1-2 pp 121ndash133 2009

[5] M Spagnuolo and U Andreaus ldquoA targeted review on largedeformations of planar elastic beams extensibility distributedloads buckling and post-bucklingrdquoMathematics andMechanicsof Solids pp 1ndash23 2018

[6] M P Omran A Amani and H G Lemu ldquoAnalytical approxi-mation of nonlinear vibration of string with large amplitudesrdquoJournal of Mechanical Science and Technology vol 27 no 4 pp981ndash986 2013

[7] Z L Wang P Poncharal andW A De Heer ldquoMeasuring phys-ical and mechanical properties of individual carbon nanotubesby in situ TEMrdquo Journal of Physics and Chemistry of Solids vol61 no 7 pp 1025ndash1030 2000

[8] Z LWang R P Gao P PoncharalWADeHeer Z RDai andZ W Pan ldquoMechanical and electrostatic properties of carbonnanotubes and nanowiresrdquo Materials Science and EngineeringC Materials for Biological Applications vol 16 no 1-2 pp 3ndash102001

[9] Q Zheng and Q Jiang ldquoMultiwalled Carbon Nanotubes asGigahertz Oscillatorsrdquo Physical Review Letters vol 88 no 42002

[10] M Hemmatnezhad R Ansari and G H Rahimi ldquoLarge-amplitude free vibrations of functionally graded beams bymeans of a finite element formulationrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 37 no 18-19 pp 8495ndash8504 2013

[11] J B Gunda R K Gupta G Ranga Janardhan and G Venkates-wara Rao ldquoLarge amplitude vibration analysis of compositebeams Simple closed-form solutionsrdquo Composite Structuresvol 93 no 2 pp 870ndash879 2011

[12] A Nikkar S Bagheri and M Saravi ldquoDynamic model of largeamplitude vibration of a uniform cantilever beam carrying anintermediate lumped mass and rotary inertiardquo Latin AmericanJournal of Solids and Structures vol 11 no 2 pp 320ndash329 2014

[13] Y P Yu B S Wu Y H Sun and L Zang ldquoAnalytical approxi-mate solutions to large amplitude vibration of a spring-hingedbeamrdquoMeccanica vol 48 no 10 pp 2569ndash2575 2013

[14] K K Raju andGV Rao ldquoTowards improved evaluation of largeamplitude free-vibration behaviour of uniform beams usingmulti-term admissible functionsrdquo Journal of Sound and Vibra-tion vol 282 no 3-5 pp 1238ndash1246 2005

[15] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons and Fractals vol 34 no 5 pp 1430ndash1439 2007

[16] S H Hoseini T Pirbodaghi M T Ahmadian and G H Far-rahi ldquoOn the large amplitude free vibrations of tapered beams

Mathematical Problems in Engineering 11

an analytical approachrdquo Mechanics Research Communicationsvol 36 no 8 pp 892ndash897 2009

[17] B Gheshlaghi ldquoLarge-amplitude vibrations of nanowires withdissipative surface stress effectsrdquo Acta Mechanica vol 224 no7 pp 1329ndash1334 2013

[18] S Souayeh and N Kacem ldquoComputational models for largeamplitude nonlinear vibrations of electrostatically actuatedcarbon nanotube-basedmass sensorsrdquo Sensors and Actuators APhysical vol 208 pp 10ndash20 2014

[19] M Simsek ldquoLarge amplitude free vibration of nanobeams withvarious boundary conditions based on the nonlocal elasticitytheoryrdquo Composites Part B Engineering vol 56 pp 621ndash6282014

[20] R Nazemnezhad and S Hosseini-Hashemi ldquoExact solution forlarge amplitude flexural vibration of nanobeams using non-local Euler-Bernoulli theoryrdquo Journal ofTheoretical and AppliedMechanics (Poland) vol 55 no 2 pp 649ndash658 2017

[21] K Nakamura K Kakihara M Kawakami and S Ueha ldquoMea-suring vibration characteristics at large amplitude region ofmaterials for high power ultrasonic vibration systemrdquoUltrason-ics vol 38 no 1 pp 122ndash126 2000

[22] AMozuras and E Podzharov ldquoMeasurement of large harmonicvibration amplitudesrdquo Journal of Sound and Vibration vol 271no 3-5 pp 985ndash998 2004

[23] J C Simo ldquoA finite strain beam formulation The three-dimen-sional dynamic problem Part Irdquo Computer Methods AppliedMechanics and Engineering vol 49 no 1 pp 55ndash70 1985

[24] P-F Pai and A-H Nayfeh ldquoA new method for the modeling ofgeometric nonlinearities in structuresrdquoComputersampStrucruresvol 53 no 4 pp 877ndash895 1994

[25] E Babilio and S Lenci ldquoOn the notion of curvature andits mechanical meaning in a geometrically exact plane beamtheoryrdquo International Journal of Mechanical Sciences vol 128-129 pp 277ndash293 2017

[26] S Lenci F Clementi and G Rega ldquoComparing Nonlinear FreeVibrations of Timoshenko Beams with Mechanical or Geo-metric Curvature Definitionrdquo Procedia IUTAM vol 20 pp 34ndash41 2017

[27] O Kopmaz and O Gundogdu ldquoOn the curvature of an Euler-Bernoulli beamrdquo International Journal of Mechanical Engineer-ing Education vol 31 no 2 pp 132ndash142 2003

[28] C Semler G X Li andM P Paıdoussis ldquoThe non-linear equa-tions of motion of pipes conveying fluidrdquo Journal of Sound andVibration vol 169 no 5 pp 577ndash599 1994

[29] D Zhao J Liu and L Wang ldquoNonlinear free vibration of acantilever nanobeam with surface effects Semi-analytical solu-tionsrdquo International Journal of Mechanical Sciences vol 113 pp184ndash195 2016

[30] N Vlajic T Fitzgerald V Nguyen and B BalachandranldquoGeometrically exact planar beams with initial pre-stress andlarge curvature Static configurations natural frequencies andmode shapesrdquo International Journal of Solids and Structures vol51 no 19-20 pp 3361ndash3371 2014

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

[32] F Benedettini and G Rega ldquoNon-linear dynamics of an elasticcable under planar excitationrdquo International Journal of Non-Linear Mechanics vol 22 no 6 pp 497ndash509 1987

[33] H Koivurova ldquoThe numerical study of the nonlinear dynamicsof a light axiallymoving stringrdquo Journal of Sound andVibrationvol 320 no 1-2 pp 373ndash385 2009

[34] A H NayfehTheMethod of Normal Forms Wiley-VCHWein-heim Germany 2nd edition 2011

[35] A Y Leung and Q C Zhang ldquoComplex normal form forstrongly non-linear vibration systems exemplified by Duffing-van der Pol equationrdquo Journal of Sound and Vibration vol 213no 5 pp 907ndash914 1998

[36] W Wang Q C Zhang and X J Wang ldquoApplication of theundetermined fundamental frequency method for analyzingthe critical value of chaosrdquo Acta Physica Sinica vol 58 no 8pp 5162ndash5168 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

an analytical approachrdquo Mechanics Research Communicationsvol 36 no 8 pp 892ndash897 2009

[17] B Gheshlaghi ldquoLarge-amplitude vibrations of nanowires withdissipative surface stress effectsrdquo Acta Mechanica vol 224 no7 pp 1329ndash1334 2013

[18] S Souayeh and N Kacem ldquoComputational models for largeamplitude nonlinear vibrations of electrostatically actuatedcarbon nanotube-basedmass sensorsrdquo Sensors and Actuators APhysical vol 208 pp 10ndash20 2014

[19] M Simsek ldquoLarge amplitude free vibration of nanobeams withvarious boundary conditions based on the nonlocal elasticitytheoryrdquo Composites Part B Engineering vol 56 pp 621ndash6282014

[20] R Nazemnezhad and S Hosseini-Hashemi ldquoExact solution forlarge amplitude flexural vibration of nanobeams using non-local Euler-Bernoulli theoryrdquo Journal ofTheoretical and AppliedMechanics (Poland) vol 55 no 2 pp 649ndash658 2017

[21] K Nakamura K Kakihara M Kawakami and S Ueha ldquoMea-suring vibration characteristics at large amplitude region ofmaterials for high power ultrasonic vibration systemrdquoUltrason-ics vol 38 no 1 pp 122ndash126 2000

[22] AMozuras and E Podzharov ldquoMeasurement of large harmonicvibration amplitudesrdquo Journal of Sound and Vibration vol 271no 3-5 pp 985ndash998 2004

[23] J C Simo ldquoA finite strain beam formulation The three-dimen-sional dynamic problem Part Irdquo Computer Methods AppliedMechanics and Engineering vol 49 no 1 pp 55ndash70 1985

[24] P-F Pai and A-H Nayfeh ldquoA new method for the modeling ofgeometric nonlinearities in structuresrdquoComputersampStrucruresvol 53 no 4 pp 877ndash895 1994

[25] E Babilio and S Lenci ldquoOn the notion of curvature andits mechanical meaning in a geometrically exact plane beamtheoryrdquo International Journal of Mechanical Sciences vol 128-129 pp 277ndash293 2017

[26] S Lenci F Clementi and G Rega ldquoComparing Nonlinear FreeVibrations of Timoshenko Beams with Mechanical or Geo-metric Curvature Definitionrdquo Procedia IUTAM vol 20 pp 34ndash41 2017

[27] O Kopmaz and O Gundogdu ldquoOn the curvature of an Euler-Bernoulli beamrdquo International Journal of Mechanical Engineer-ing Education vol 31 no 2 pp 132ndash142 2003

[28] C Semler G X Li andM P Paıdoussis ldquoThe non-linear equa-tions of motion of pipes conveying fluidrdquo Journal of Sound andVibration vol 169 no 5 pp 577ndash599 1994

[29] D Zhao J Liu and L Wang ldquoNonlinear free vibration of acantilever nanobeam with surface effects Semi-analytical solu-tionsrdquo International Journal of Mechanical Sciences vol 113 pp184ndash195 2016

[30] N Vlajic T Fitzgerald V Nguyen and B BalachandranldquoGeometrically exact planar beams with initial pre-stress andlarge curvature Static configurations natural frequencies andmode shapesrdquo International Journal of Solids and Structures vol51 no 19-20 pp 3361ndash3371 2014

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

[32] F Benedettini and G Rega ldquoNon-linear dynamics of an elasticcable under planar excitationrdquo International Journal of Non-Linear Mechanics vol 22 no 6 pp 497ndash509 1987

[33] H Koivurova ldquoThe numerical study of the nonlinear dynamicsof a light axiallymoving stringrdquo Journal of Sound andVibrationvol 320 no 1-2 pp 373ndash385 2009

[34] A H NayfehTheMethod of Normal Forms Wiley-VCHWein-heim Germany 2nd edition 2011

[35] A Y Leung and Q C Zhang ldquoComplex normal form forstrongly non-linear vibration systems exemplified by Duffing-van der Pol equationrdquo Journal of Sound and Vibration vol 213no 5 pp 907ndash914 1998

[36] W Wang Q C Zhang and X J Wang ldquoApplication of theundetermined fundamental frequency method for analyzingthe critical value of chaosrdquo Acta Physica Sinica vol 58 no 8pp 5162ndash5168 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom