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 DOI: 10.1177/1464419314522780published online 24 February 2014

2014 228: 213 originallyProceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body DynamicsHamid M Sedighi

three-mode assumptionsNonlinear free vibrations of quintic inextensional beams lying on Winkler elastic substrate based on

  

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Technical Note

Nonlinear free vibrations of quinticinextensional beams lying on Winklerelastic substrate based on three-modeassumptions

Hamid M Sedighi

Abstract

The vibrational behavior of quintic nonlinear inextensional beam on two-parameter elastic substrate based on the three-

mode assumptions is investigated in this paper. Using the Galerkin decomposition principle by considering three-mode

assumptions, the nonlinear governing equations are derived. An asymptotic approach called parameter expansion

method is employed to obtain the approximate expressions of nonlinear frequency–amplitude relationship for the

first, second, and third modes of vibrations. Afterwards, it is clearly shown that the first term in series expansions

gives excellent results in contrast with numerical calculations. Finally, the effect of the initial configuration of the beam as

well as system parameters on the natural frequencies of quintic nonlinear beams is studied.

Keywords

Quintic nonlinear beam, three-mode assumptions, inextensional beam, parameter expansion method, Winkler elastic

foundation

Date received: 20 August 2013; accepted: 16 January 2014

Introduction

Geometrically, nonlinear vibration of beams hasbecome a subject of numerous research efforts, as evi-denced by many analytical and numerical studiesreported in the literature. From the practical outlook,structures like railroad tracks with soil interaction,1

machine spindles with supporting bearings,2 highwaypavement, buried pipelines, foundation beams,robotic manipulators with elastic tendons and flexiblejoints3,4 are modeled as beams with flexible bound-aries. The dynamic behavior of beam-type structurescan be intensely influenced by the characteristic flex-ible boundaries or elastic foundations. Ayvaz andOzgan1 applied the modified Vlasov model to thefree vibration analysis of beams resting on elasticfoundation as a practical model of soil–beam inter-action. They analysed the effects of the subsoil depth,beam length and the value of the vertical deformationparameter within the subsoil on the frequency param-eters of beams on elastic foundations. Matsubaraet al.2 studied the dynamic modal characteristics ofprecision spindles supported by ball bearings. Theyrepresented the support bearings as piecewise linearsprings with damping provided by concentratedforces external to the bearings at each end of the spin-dle. Lee and Lee3 presented a new methodologyfor the dynamic analysis of tendon-driven robotic

mechanisms with flexible tendons and obtained thejoint reaction forces and the tension in each segmentof tendon. Several researches demonstrated the non-linear frequency of the beams which are very import-ant for the design of many engineering structures. Toinvestigate the dynamics of the vibrational behaviorof these applications, solutions need to be obtained.Emam5 presented approximate analytical solutionsfor the nonlinear free vibrations of symmetrically orasymmetrically laminated composite beams in pre-buckling and postbuckling. He utilised the variationalmethod to derive an approximate analytical solutionfor the nonlinear natural frequency and the nonlinearload–deflection relation by considering first modeassumptions. Post-buckling and nonlinear vibrationanalysis of geometrically imperfect beams with quad-ratic and cubic nonlinearities made of functionallygraded materials resting on nonlinear elastic founda-tion subjected to axial force were studied by Yaghoobi

Proc IMechE Part K:

J Multi-body Dynamics

2014, Vol. 228(2) 213–225

! IMechE 2014

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DOI: 10.1177/1464419314522780

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Mechanical Engineering Department, Engineering Faculty, Shahid

Chamran University, Ahvaz, Iran

Corresponding author:

Hamid M Sedighi, Mechanical Engineering Department, Engineering

Faculty, Shahid Chamran University, PO Box 61357-43337, Ahvaz,

Islamic Republic of Iran.

Emails: hmsedighi@gmail.com; h.msedighi@scu.ac.ir

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and Torabi6 using variational iteration method withfirst mode approximation. Wang et al.7 investigatedthe nonlinear vibration and parametric instability ofthe inextensional beams on the elastic foundation.They also discussed the effects of the boundary con-ditions and foundation models on the nonlinearvibration of the beams. Sedighi et al.8–10 employedmodern asymptotic approaches such as homotopyanalysis method (HAM), homotopy perturbationmethod (HPM) with an auxiliary term, max-minapproach (MMA), and iteration perturbationmethod (IPM) to investigate transverse vibration ofcubic/quantic nonlinear beams using first modeassumption. Bagheri et al.11 studied the nonlinearresponses of clamped–clamped buckled beam. Theyused two efficient mathematical techniques calledHe’s variational approach and Laplace iterationmethod in order to obtain the responses of thebeam vibrations. Wang et al.12 investigated the freevibration nature of beams on the elastic foundationand studied the effects of the rotary inertia on thenatural frequencies. Finally, they discussed the fre-quency differences, due to the different foundationmodels, and the possible modal interaction of thebeam. Wang et al.13 examined the large amplitudevibrations of the inextensional beam resting on theelastic foundation under three-to-one internal reson-ance. They also investigated the nonlinear responseand the associated stability of the system by meansof frequency (force)–response curves. Exact solutionsfor free vibrations and buckling of double taperedcolumns with elastic foundation and tip mass wereinvestigated by Firouz-Abadi et al.14 Vibration ofTimoshenko beam structures resting on a two-parameter elastic foundation using Chebyshev DQmodel was studied by Chen.15 The vibrations ofbeams on a nonlinear elastic foundation were ana-lysed by Mo et al.16 considering the effects of trans-verse shear deformation and the rotational inertia ofbeams. They used a weak form quadrature elementmethod (QEM) for the beam vibration analysis.They presented the fundamental frequencies ofbeams for various slenderness ratios and nonlinearfoundation parameters for both slender and shortbeams. A mixed method for bending and free vibra-tion of beams resting on a Pasternak elastic founda-tion was studied by Chen et al.17 The effects ofPoisson’s ratio and foundation parameters on thenatural frequencies were discussed in their study.A simple finite element method was developed byThambiratnam and Zhuge18 and applied to treatthe free vibration analysis of beams supported onelastic foundations. Thai et al.19 proposed a simplerefined shear deformation theory for bending, buck-ling, and vibration of thick plates resting on elasticfoundation. Al-Qaisia and Hamdan20 presented aninvestigation on the effect of an initial geometricimperfection wavelength, amplitude and degree oflocalisation on the in-plane nonlinear natural

frequencies veering and mode localisation of an elas-tic Euler–Bernoulli beam resting on a Winkler elasticfoundation.

It is very important to provide an accurate ana-lysis towards the understanding of the nonlinearvibration characteristics of beam structures. Elias-Zuniga et al.21,22 proposed a nonlinear transform-ation method to develop equivalent equations ofmotion of nonlinear oscillatory systems with non-linear odd damping terms including quintic non-linear term. Most models dealing with nonlineardynamics of flexible beams include first modemodel terms in the equations of motion. New pre-dictions and understanding require higher order non-linear terms in the equation of motion. This studydemonstrated that the response and natural frequen-cies of quintic beam vibrations highly depend on theinitial conditions of each mode.

In recent times, substantial progresses had beenmade in analytical solutions for nonlinear equationswithout small parameters. There have been severalclassical approaches employed to solve the governingnonlinear differential equations to study the nonlinearvibrations including strong parameters such as energybalance method (EBM),23 frequency–amplitude formu-lation,24 Green quasifunction method,25 Hamiltonianapproach (HA),26 multiscale decomposition method,27

MMA,28 iteration perturbation method (IPM),29

HPM,30 multistage Adomian decomposition method,31

variational iteration method,10 multiple scales method,32

and enhanced differential transform method(EDTM).33 Therefore, many different methods haverecently introduced various ways to eliminate thesmall parameter. The parameter expansion method(PEM) has been shown to solve a large class of non-linear problems efficiently, accurately and easily, withapproximations converging rapidly to solution.Usually, few iterations lead to high accuracy of thesolution. The PEM proposed by He34 is proved to bea very effective and convenient way for handling thenonlinear problems, where one iteration is sufficient toobtain a highly accurate solution. Parameter expan-sion method was successfully applied to variousengineering problems. Sedighi et al.10 revealed theadvantages of recent modern analytical approachessuch as PEM, VIM, HA, MMA, and EBM appliedon the governing equation of transversely vibratingcantilever beams. Liu35 studied approximated periodof nonlinear oscillators with discontinuities by mod-ified PEM. Xu36 applied PEM for strongly nonlinearoscillators. The effectiveness and accuracy of the PEMhave been demonstrated in the analysis of nonlinearproblems such as oscillation of a mass attached to astretched elastic wire,37 nonlinear beam vibrations,38

and pull-in instability and natural frequency of micro-beams vibrations.39

The significant aim of this paper is to achieve ana-lytical expressions for geometrically nonlinear vibra-tion of inextensional Euler–Bernoulli beams with

214 Proc IMechE Part K: J Multi-body Dynamics 228(2)

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quintic nonlinearity using PEM by considering three-mode assumptions. The nonlinear ordinary differen-tial equation of beam vibration is extracted from par-tial differential equation considering three-modeapproximations, based on the Galerkin theory. Theillustrated results demonstrated that the natural fre-quencies of beam vibrations with large deformationshighly depend on the initial configuration of thebeam. The effects of initial conditions as well assystem parameters on the natural frequencies areinvestigated. The results presented in this paper exhi-bit that the analytical method are very effective andconvenient for nonlinear beam vibration for which thehighly nonlinear governing equations exist. The pro-posed analytical method demonstrates that one termin series expansions is sufficient to obtain a highlyaccurate solution of beam vibration.

Mathematical modeling

Consider the simply supported inextensional beam-type structure of length l, moment of inertia I, massper unit length m, and modulus of elasticity E restingon two-parameter elastic foundation, which is axiallycompressed by a loading P as shown in Figure 1.Denoting by w the transverse deflection and k0, k1the Winkler (shear) parameters,40 the differentialequation governing the equilibrium in the deformedsituation is derived as

d2

dx2EIw00 x, tð Þ

1� w02 x, tð Þ½ �1=2

( )þ Pw00 x, tð Þ þ k0w x, tð Þ

� k1w00 x, tð Þ þm €w x, tð Þ ¼ 0 ð1Þ

Neglecting the extension of the beam length, theterm w00 x, tð Þ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� w02 x, tð Þ

pis the expression for

the curvature of inextensional beam,41 using theapproximation

w00 x,tð Þ

1�w02 x,tð Þ½ �1=2ffiw00 x,tð Þ 1þ

1

2w02 x,tð Þþ

3

8w04 x,tð Þ

� �ð2Þ

The nonlinear free vibration of quintic nonlinearbeam in the prebuckling state is governed by

EIw 4ð Þ 1þ1

2w02 þ

3

8w04

� �þ 3EIw000w0w00

þ9

2EIw000w03w00 þ EIw003

þ9

2EIw02w003 þ Pw00 þ k0w� k1w

00 þm €w ¼ 0

ð3Þ

which is subjected to the following boundaryconditions

w 0, tð Þ ¼@2w

@x20, tð Þ ¼ 0, w l, tð Þ ¼

@2w

@x2l, tð Þ ¼ 0 ð4Þ

Assuming three-mode method w x, tð Þ ¼P3i¼1 qi tð Þ�i xð Þ, where �i xð Þ are the ith

eigenmode of the simply supported beam and can beexpressed as

�i xð Þ ¼ sin i�x=lð Þ ð5Þ

Applying the Bubnov–Galerkin method yields

Z l

0

EIw 4ð Þ 1þ1

2w02 þ

3

8w04

� �þ 3EIw000w0w00

�

þ9

2EIw000w03w00 þ EIw003þ

9

2EIw02w003 þ Pw00

þ k0w� k1w00 þm €wÞ�i xð Þdx ¼ 0, i ¼ 1, 2, 3ð Þ

ð6Þ

By introducing the following nondimensional timeand amplitude variables as

� ¼

ffiffiffiffiffiffiffiEI

ml4

rt, �qi ¼

qil, �P ¼

�2Pl2

EI, K0 ¼

k0l4

EI,

K1 ¼�2k1l

2

EIð7Þ

Figure 1. A uniform simply supported beam lying on an elastic substrate.

Sedighi 215

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The nondimensional nonlinear equations of motion inthe prebuckling state based on the triple-mode meth-ods can be written as

€q1þ�1,1q1þ �1,3q31þ�1,5q

51þ�1,1q1q

22þ�1,2q1q

23

�þ�1,3q

21q3þ�1,4q3q

22þ�1,5q1q

42þ�1,6q1q

43

þ�1,7q41q3þ�1,8q3q

42þ�1,9q

33q

22þ�1,10q1q

22q

23

þ�1,11q21q

22q3þ�1,12q

21q

33þ�1,13q

31q

23þ�1,14q

31q

22Þ¼0

ð8-aÞ

€q2 þ �2,1q2 þ �2,3q32 þ �2,5q

52 þ �2,1q2q

21 þ �2,2q2q

23

�þ �2,3q1q2q3 þ �2,4q

21q

32 þ �2,5q

23q

32

þ �2,6q1q3q32 þ �2,7q

21q

23q2 þ �2,8q1q2q

33

þ �2,9q31q2q3 þ �2,10q2q

43 þ �2,11q

41q2�¼ 0

ð8-bÞ

€q3þ�3,1q3þ �3,3q33þ�3,5q

53þ�3,1q3q

21

�þ�3,2q3q

22þ�3,3q

31þ�3,4q1q

22þ�3,5q

51þ�3,6q

21q

33

þ�3,7q22q

33þ�3,8q

31q

23þ�3,9q1q

22q

23þ�3,10q3q

41

þ�3,11q3q42þ�3,12q

21q

22q3þ�3,13q1q

42þ�3,14q

31q

22

�¼0

ð8-cÞ

where the nondimensional coefficients are described inAppendix 1. In this study, the coupled nonlinear gov-erning equations (8-a) to (8-c) are employed in orderto achieve the first three natural frequencies of quinticbeam vibrations as a function of initial configurationof beam using PEM. The effect of different parameterson the nonlinear behavior of beam structures is alsodiscussed.

Asymptotic solution by parameterexpansion method

Consider the coupled governing equations (8-a) to(8-c) for the vibration of simply supported Euler–Bernoulli beam based on the triple-mode assumptionswith the following general initial conditions

q1 0ð Þ ¼ A1 , _q1 0ð Þ ¼ 0 ð9-aÞ

q2 0ð Þ ¼ A2 , _q2 0ð Þ ¼ 0 ð9-bÞ

q3 0ð Þ ¼ A3 , _q3 0ð Þ ¼ 0 ð9-cÞ

Free oscillation of a system without damping is a peri-odic motion and can be expressed by the followingbase functions

cos m!1�ð Þ, cos m!2�ð Þ, cos m!3�ð Þ, m ¼ 1, 2, 3, . . .

ð10Þ

We denote the natural frequencies of the first, second,and third modes of vibration by !1, !2, !3 and notethat one of our major tasks is to determine

!1 A1, A2, A3ð Þ, !2 A1, A2, A3ð Þ, and !3 A1, A2, A3ð Þ,i.e. the functional behavior of !1, !2, and !3 as afunction of the initial amplitudes A1, A2, and A3. Inthe PEM, an artificial perturbation equation is con-structed by embedding an artificial parameterp 2 0, 1½ � which is used as an expanding parameter.According to PEM the solution of equations (8) isexpanded into a series of p in the form

q1 �ð Þ ¼ q1,0 �ð Þ þ pq1,1 �ð Þ þ p2q1,2 �ð Þ þ � � � ð11-aÞ

q2 �ð Þ ¼ q2,0 �ð Þ þ pq2,1 �ð Þ þ p2q2,2 �ð Þ þ � � � ð11-bÞ

q3 �ð Þ ¼ q3,0 �ð Þ þ pq3,1 �ð Þ þ p2q3,2 �ð Þ þ � � � ð11-cÞ

The coefficients in the equations (8-a) to (8-c) areexpanded in a similar way. Substituting the abovedefinitions into the governing equations of motionand collecting the terms of the same power of p,a series of linear differential equations can beobtained. After some mathematical operations (seeAppendix 2), the first-order solutions of the governingequations can be expressed as

q1,0 �ð Þ ¼ A1 cos !1�ð Þ, ð12-aÞ

q2,0 �ð Þ ¼ A2 cos !2�ð Þ, ð12-bÞ

q3,0 �ð Þ ¼ A3 cos !3�ð Þ, ð12-cÞ

Substitution of these results into the right-hand sideof the second equations gives three differential equa-tions for q1,1 �ð Þ, q2,1 �ð Þ, and q3,1 �ð Þ. Eliminating thesecular terms, the expressions for the first, secondand third natural frequencies !1, !2, and !3 as a func-tion of initial amplitudes A1, A2, and A3 can beobtained as

!1 A1,A2,A3ð Þ¼ �1,1þ3

4�1,3A

21þ

5

8�1,5A

41

�

þ1

2�1,1A

22þ

1

2�1,2A

23þ

3

8�1,5A

42

þ3

8�1,6A

43þ

1

4�1,10A

22A

23þ

3

8�1,13A

21A

23

þ3

8�1,14A

21A

22

�1=2

ð13Þ

!2ðA1,A2,A3Þ¼ �2,1þ3

4�2,3A

22þ5

8�2,5A

42þ1

2�2,1A

21

�

þ1

2�2,2A

23þ3

8�2,4A

21A

22þ3

8�2,5A

22A

23

þ1

4�2,7A

21A

23þ3

8�2,10A

43þ3

8�2,11A

41

�1=2

ð14Þ

216 Proc IMechE Part K: J Multi-body Dynamics 228(2)

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!3 A1,A2,A3ð Þ¼ �3,1þ3

4�3,3A

23þ

5

8�3,5A

43þ

1

2�3,1A

21

�

þ1

2�3,2A

22þ

3

8�3,6A

23A

21

þ3

8�3,7A

22A

23þ

3

8�3,10A

41

þ3

8�3,11A

42þ

1

4�3,12A

22A

21

�1=2

ð15Þ

Replacing !1,!2, and !3 from equations (13) to (15)into equations (12-a) and (12-c) yields

q1 �ð Þ � q1,0 �ð Þ ¼ A1 cos !1 A1, A2,A3ð Þ�ð Þ ð16Þ

q2 �ð Þ � q2,0 �ð Þ ¼ A2 cos !2 A1, A2,A3ð Þ�ð Þ ð17Þ

q3 �ð Þ � q3,0 �ð Þ ¼ A3 cos !3 A1, A2,A3ð Þ�ð Þ ð18Þ

Finally, the transverse deflection w x, �ð Þ as a functionof initial amplitudes A1, A2, A3, location x and non-dimensional time � can be expressed as

w x, �ð Þ ¼ A1 cos !1 A1,A2,A3ð Þ�ð Þ sin �xð Þ

þ A2 cos !2 A1,A2,A3ð Þ�ð Þ sin 2�xð Þ

þ A3 cos !3 A1, A2,A3ð Þ�ð Þ sin 3�xð Þ ð19Þ

Results and discussion

In order to verify the soundness of the asymptoticsolutions by PEM using three-mode models, theauthor plots the analytical solutions together withthe numerical results for different values of initial con-ditions in Figure 2. As can be seen, the first-orderapproximation of mid-point deflection, described inequation (19), from asymptotic approach is in excel-lent agreement with numerical results. The approxi-mate analytical solutions reveal that the first term inseries expansions is sufficient to result in a highlyaccurate solution of the problem. Furthermore,these equations provide excellent approximations tothe period of oscillations by considering the effect ofthe oscillation amplitude. By closely examining thecurves, it is concluded that there are interesting andmeaningful similarities between the trends, peakvalues and frequencies of the third mode assumptionof the approximate solutions by PEM and the numer-ical results.

It should be noted that for a vibrating linear Euler–Bernoulli beam on an elastic foundation, the Euler–Lagrange equation is as follows

d2

dx2EIw00 x, tð Þð Þ þ Pw00 x, tð Þ þ k0w x, tð Þ

� k1w00 x, tð Þ þm €w x, tð Þ ¼ 0 ð20Þ

Figures 3 to 5 display the effect of normalised amp-litudes on the nonlinear behavior of a vibrating quin-tic beam. According to equations (13) to (15), thenatural frequencies of beam vibrations are the func-tions of initial amplitudes of vibration; that meanswhen the oscillation amplitudes become larger, theaccuracy of the approximated natural frequencies inthe linear Euler–Bernoulli beam theory decreases.It confirms that the initial configuration of the beamhas a significant effect on nonlinear behavior of thestructure. From these figures it is observed that theresults from linear beam theory are incompatible withquintic nonlinear beam when the initial conditionsbecome larger. In other words, the linear beamresponse is in more agreement with quintic nonlinearbeam, when the amplitudes of vibration approach tozero. From equations (13) to (15), it is clear that thedynamic behavior and natural frequency of linearEuler–Bernoulli beam model are independent ofinitial beam configuration.

Figures 6 to 9 indicate that if the initial conditionsincrease the difference between the predicted resultsby linear beam and quintic nonlinear beam increaseconsiderably. Figure 6 shows the effect of the ampli-tude of first mode on the first natural frequency ofquintic nonlinear beam for some assigned values ofthe second mode amplitude at the side of linearbeam theory results. It is obvious that the first naturalfrequency increases by increasing the amplitude of thefirst mode and the difference between linear beam fre-quency and quintic beam by considering that thethree-mode assumptions get larger as the amplitudeof vibration increases. In addition, the illustratedresults reveal that the amplitude of the second modeof vibration has significant effect on the first naturalfrequency. It is shown that the first natural frequencyincreases as the amplitude of the second eigenmodeincreases. Figure 7 also displays the effect of the thirdmode configuration on the first frequency of quinticnonlinear beam. It appears from the figure that thefirst natural frequency increases by increasing thethird mode initial conditions. Therefore, it is revealedthat a simply supported flexible beam represents richnonlinear dynamics when amplitude of vibrationincreases.

A parametric study that displays the variation ofthe nondimensional natural frequencies with the non-dimensional axial load and the initial configuration ofbeam vibration for simply supported beams is inves-tigated. Figure 8 presents the variation of three non-linear natural frequencies with the nondimensionalaxial load �P in the prebuckling state. As shown inthis figure, it is concluded that the natural frequenciesof quintic beam decrease considerably by increasingthe axial load parameter until the beam buckled andnonlinear frequencies drops to zero. Figure 9 illus-trates the influence of axial load parameter on thenatural frequency of quintic beam for a variety ofamplitude of vibration at the side of linear beam

Sedighi 217

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results. The linear natural frequency is presented as acriterion for comparison, which shows the significanceof the amplitude of vibration on the prediction ofnonlinear natural frequency. The influence of thesecond parameter of elastic foundation K1 onthe first three natural frequencies of quintic beam inthe prebuckling state is indicated in Figure 10. Fromthis figure it is obvious that when the parameter K1

increases, the natural frequencies of quintic beamincrease. Finally, the effect of initial amplitude ofvibration on the vibrational behavior of nonlinearbeam is illustrated in Figure 11. As can be observedfrom Figure 11(a) and (b), at lower values of initialamplitudes, the linear beam theory predicts acceptablesolution of dynamic behavior of the beam. However,as the initial conditions become larger, according to

Figure 2. Comparison between analytical and numerical results. Symbols: analytical solutions, Solid line: numerical solutions;

(a) A1 ¼ 0:3, A2 ¼ 0, A3 ¼ 0, (b) A1 ¼ 0:3, A2 ¼ 0:05, A3 ¼ 0:05, (c) A1¼ 0.15, A2¼ 0.1, A3¼ 0.1.

218 Proc IMechE Part K: J Multi-body Dynamics 228(2)

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Figure 5. The impact of nonlinear terms on the dynamic behavior of beam vibration for A1 ¼ 0:4, A2 ¼ 0:1, A3 ¼ 0:05.

Figure 3. The impact of nonlinear terms on the dynamic behavior of beam vibration for A1 ¼ 0:3, A2 ¼ 0, A3 ¼ 0.

Figure 4. The impact of nonlinear terms on the dynamic behavior of beam vibration for A1 ¼ 0:2, A2 ¼ 0:1, A3 ¼ 0:1.

Sedighi 219

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Figure 7. The effect of the first and third mode initial conditions on the first natural frequency of quintic beam.

Figure 6. The effect of the first and second mode initial conditions on the first natural frequency of quintic beam.

Figure 8. The influence of axial load parameter on the first three natural frequencies of quintic beam in the prebuckling state.

220 Proc IMechE Part K: J Multi-body Dynamics 228(2)

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Figure 9. The influence of axial load parameter and first mode initial condition on the first natural frequency of quintic beam in the

prebuckling state.

Figure 10. The influence of the second parameter of elastic foundation on the first three natural frequencies of quintic beam in the

prebuckling state.

Figure 11. Comparison between of linear beam and quintic nonlinear beam: (a) and (b) small deformation; (c) and (d) large

deformation.

Sedighi 221

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Figure 11(c) and (d), the difference between predictedresults gets larger. As can be seen, in this situation, thelinear beam model predicts the quasiperiodic motionby turning around the torus whereas the quintic non-linear beam shows period-3 motion in the phase planeof the system.

Conclusion

In this research, nonlinear behavior of quintic inex-tensional beams resting on elastic foundation wasinvestigated. Approximate expressions for the firstthree natural frequencies of beam vibration as func-tions of initial conditions were presented usingmodern asymptotic approach. It demonstrated that

the fundamental frequency based upon linear theorycan be different from the natural frequency of quinticnonlinear beam at large vibration amplitudes. Thecontribution of each eigenmode configuration on thefirst natural frequency was examined. The integrity ofthe obtained asymptotic approximate solutions byPEM is verified by numerical results.

Funding

This research received no specific grant from any fundingagency in the public, commercial, or not-for-profit sectors.

Conflict of interest

None declared.

Figure 11. Continued.

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Appendix 1

�1,1 ¼ �4 � �Pþ K0 þ K1, �1,3 ¼

�6

8,

�1,5 ¼3�8

64, �1,1 ¼ �

6, �1,2 ¼9�6

4,

�1,3 ¼3�6

8, �1,4 ¼

3�6

2, �1,5 ¼

9�8

4,

�1,6 ¼729�8

64, �1,7 ¼

4�8

128, �1,8 ¼

9�8

2,

�1,9 ¼243�8

16, �1,10 ¼

405�8

16, �1,11 ¼

135�8

16,

�1,12 ¼243�8

64, �1,13 ¼

81�8

32, �1,14 ¼

21�8

16

�2,1 ¼ 16�4 � 4 �Pþ K0 þ 4K1, �2,3 ¼ 8�6,

�2,5 ¼ 12�8, �2,1 ¼ 4�6, �2,2 ¼ 36�6,

�2,3 ¼ 12�6, �2,4 ¼ 18�8, �2,5 ¼ 162�8

�2,6 ¼ 72�8, �2,7 ¼405�8

4, �2,8 ¼

243�8

2,

�2,9 ¼45�8

2, �2,10 ¼

729�8

4, �2,11 ¼

21�8

8

�3,1 ¼ 81�4 � 9 �Pþ K0 þ 9K1, �3,3 ¼729�6

8,

�3,5 ¼19683�8

64, �3,1 ¼

81�6

4, �3,2 ¼ 81�6,

�3,3 ¼9�6

8, �3,4 ¼

27�6

2, �3,5 ¼

81�8

128,

�3,6 ¼6561�8

32, �3,7 ¼

6561�8

8, �3,8 ¼

2187�8

64,

�3,9 ¼6561�8

16, �3,10 ¼

729�8

64, �3,11 ¼

729�8

4,

�3,12 ¼3645�8

16, �3,13 ¼

81�8

32, �3,14 ¼

405�8

16

Appendix 2

The coefficients of equations (8-a) to (8-c) can beexpanded as follows

in equation (8-a)

1 ¼ 1þ pa1,1 þ p2a1,2 þ � � �

�1,1 ¼ !21 � pb1,1 � p2b1,2 þ � � �

1 ¼ pc1,1 þ p2c1,2 þ � � �

ð21Þ

in equation (8-b)

1 ¼ 1þ pa2,1 þ p2a2,2 þ � � �

�2,1 ¼ !22 � pb2,1 � p2b2,2 þ � � �

1 ¼ pc2,1 þ p2c2,2 þ � � �

ð22Þ

and for equation (8-c)

1 ¼ 1þ pa3,1 þ p2a3,2 þ � � �

�3,1 ¼ !23 � pb3,1 � p2b3,2 þ � � �

1 ¼ pc3,1 þ p2c3,2 þ � � �

ð23Þ

where aj,i, bj,i, cj,i j ¼ 1, 2, 3ð Þ and i ¼ 1, 2, 3, . . . ,ð Þ

are unknown parameter which should be determined.When p ¼ 0, the governing equations become thelinear differential equations for which an exact solu-tion can be calculated for p ¼ 1. Substituting equa-tions (21), (22), and (23) into equations (8-a) to (8-c)and collecting the terms of the same power of p, weobtain two series of linear equations which the firstequations are

€q1,0 �ð Þ þ !21q1,0 �ð Þ ¼ 0, q1,0 0ð Þ ¼ A1, _q1,0 0ð Þ ¼ 0

ð24Þ

€q2,0 �ð Þ þ !22q2,0 �ð Þ ¼ 0, q2,0 0ð Þ ¼ A2, _q2,0 0ð Þ ¼ 0

ð25Þ

€q3,0 �ð Þ þ !23q3,0 �ð Þ ¼ 0, q3,0 0ð Þ ¼ A3, _q3,0 0ð Þ ¼ 0

ð26Þ

By assuming p ¼ 1, equations (21) to (23) yields

a1,1 ¼ 0, b1,1 ¼ !21 � �1,1, c1,1 ¼ 1 ð27Þ

a2,1 ¼ 0, b2,1 ¼ !22 � �2,1, c2,1 ¼ 1 ð28Þ

a3,1 ¼ 0, b3,1 ¼ !23 � �3,1, c3,1 ¼ 1 ð29Þ

224 Proc IMechE Part K: J Multi-body Dynamics 228(2)

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The secular terms in the right-hand side of the secondseries of governing equations result in

� !21 � �1,1

� �A1 þ

3

4�1,3A

31 þ

5

8�1,5A

51 þ

1

2�1,1A1A

22

þ1

2�1,2A1A

23 þ

3

8�1,5A1A

42 þ

3

8�1,6A1A

43

þ1

4�1,10A1A

22A

23 þ

3

8�1,13A

31A

23 þ

3

8�1,14A

31A

22 ¼ 0

ð30Þ

� !22 � �2,1

� �A2 þ

3

4�2,3A

32 þ

5

8�2,5A

52 þ

1

2�2,1A2A

21

þ1

2�2,2A2A

23 þ

3

8�2,4A

21A

32 þ

3

8�2,5A

32A

23

þ1

4�2,7A

21A2A

23 þ

3

8�2,10A2A

43 þ

3

8�2,11A

41A2 ¼ 0

ð31Þ

� !23 � �3,1

� �A3 þ

3

4�3,3A

33 þ

5

8�3,5A

53 þ

1

2�3,1A3A

21

þ1

2�3,2A3A

22 þ

3

8�3,6A

33A

21 þ

3

8�3,7A

22A

33

þ3

8�3,10A3A

41 þ

3

8�3,11A

42A3 þ

1

4�3,12A

22A3A

21 ¼ 0

ð32Þ

Sedighi 225

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