nonlinear beam vibration with saturation type boundary condition

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online 24 January 2013publishedProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

Hamid M. Sedighi and Kourosh H. ShiraziAsymptotic approach for nonlinear vibrating beams with saturation type boundary condition

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Original Article

Asymptotic approach for nonlinear vibrating beams with saturation typeboundary condition

Hamid M Sedighi and Kourosh H Shirazi

Date received: 1 November 2012; accepted: 20 December 2012

Abstract

This article attempts to analyze the complicated vibrational behavior of the Euler–Bernoulli beam exposed to saturated

nonlinear boundary condition through proposing an innovative precise equivalent function. In this direction, the beamvibrational response is attained by way of a new effective analytical method namely Hamiltonian approach. Despite all theprocedures based on perturbation methods that deadzone or saturation dead-band parameter is omitted during inte-gration, this study indicates that how using Hamiltonian approach, the impact of dead-band parameter is taken intoaccount leading to higher accuracy of the approximated solution. Finally, the precision of the proposed equivalentfunction is evaluated in comparison with the numerical solutions, giving excellent results.

Keywords

Accurate equivalent function, saturation nonlinearity, nonlinear vibration of beam, Hamiltonian approach

Introduction

Most of the beam structures may operate in the non-

linear range during lifetime due to sources of nonli-

nearity such as large deformation effects or boundary

conditions. In the case of non-differentiable nonli-

nearity like saturation type, the analytical solutions

related to the nonlinear problems become very com-

plex. Engineering mechanisms, as electrical power sys-

tems, usually involve saturation type nonlinearities.1

This nonlinearity, due to its inherent difficulty, has

not been modeled exactly by researchers, till present.The approximation of this nonlinear condition in

order to obtain the analytical solution of dynamical

systems behavior has been always the major difficulty

of engineer’s computations. Xin et al.2 analyzed the

impact of saturation nonlinearities and disturbance

rejection on power system small-signal stability

based on the estimated stability region and maximum

endurable disturbance rejection. In the other research,

Xin et al.3 studied the class of linear dynamical sys-

tems subject to saturation nonlinearities and approxi-

mated the considered mechanisms by singular

perturbation dynamical systems based on the notion

of Pade approximation. A frequency-domain criterionfor the elimination of limit cycles in a class of digital

filters using saturation nonlinearity was presented by

Singh.4 The objective of this study is to introduce the

innovative exact equivalent fraction (EF) for satur-

ation nonlinearity as a boundary condition and to

implement the Hamiltonian approach (HA)5 in the

nonlinear beam vibrations.

The introduced function is suitable for analytical

studies of nonlinear dynamical systems using approxi-

mated approaches such as energy balance method,6

variational iteration method,7 modified variational

iteration method,8 Lindstedt–Poincare ´ method,9

Pade ´ technique,10 max-min approach,11 HAM,12,13

parameter expansion method,14–17

amplitude–fre-quency formulation,18 homotopy perturbation trans-

form method,19 Laplace transform method20 and

homotopy perturbation method21–23 as well as numer-

ical studies in direct simulations. A survey of some

recent developments in asymptotic techniques for

strongly nonlinear equations has been investigated

by He.24

Department of Mechanical Engineering, Shahid Chamran University,

Ahvaz, Iran

Corresponding author:Hamid M Sedighi, Department of Mechanical Engineering, Shahid

Chamran University, Ahvaz, Iran.

Email: [email protected]

Proc IMechE Part C:

J Mechanical Engineering Science

0(0) 1–8

! IMechE 2013

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Approximate methods for studying nonlinear

vibrations of distributed parameter systems are

important for investigative and/or designing pur-

poses. Usually, in the analytical procedures in order

to approximate the nonlinear responses, deadzone

and saturation dead-band parameter is deleted

during integration.25–27 This study indicates thatusing HA, the impact of this parameter is included

and so the accuracy of the approximated solution

increases.

The current work attempts to compute nonlinear

frequency–amplitude relation in a cantilever beam in

the presence of saturation type nonlinear boundary

condition. Recently a new powerful method namely

Hamiltonian approach (HA) proposed by He5 has

proven to be a very effective and convenient way of

handling nonlinear problems and has been success-

fully developed and applied to various engineering

problems.

28–32

Previously, He

6

introduced the energybalance method based on collocation and the

Hamiltonian. This approach is very simple but

strongly depends on the chosen location point.

Recently, He5 proposed the Hamiltonian approach.

This approach is a kind of energy method with a

vast application in conservative oscillatory systems.

He et al.28 used HA to establish frequency–amplitude

relationship of Duffing-harmonic oscillator. Some

new asymptotic methods for the solitary solutions of

nonlinear differential equations, nonlinear differen-

tial-difference equations, and nonlinear fractional dif-

ferential equations were studied by He.29 Sedighi

et al.30 have presented the advantages of some effect-ive analytical approaches such as min-max approach,

parameter expansion method, Hamiltonian approach,

variational iteration method and energy balance

method on the asymptotic solutions of governing

equation of transversely vibrating cantilever beams.

The objective of this article is to substantiate the

acceptable ability of HA in predicting the analytical

response of nonlinear systems dealing with nondiffer-

entiable nonlinearities. The innovative EF for nondif-

ferentiable saturated nonlinearity has been employed

in the analytical procedures, here. The effects of vibra-

tion amplitude besides saturation dead-band param-eter on natural frequency are taken into

consideration. The results presented in this article

exhibit that the analytical method is very effective

and high-accuracy for nonlinear vibration for which

the highly nonlinear governing equations exist.

Mathematical formulation

The cantilever beam studied in this work has length L,

mass per unit length of the beam m, moment of inertia

I and modulus of elasticity E as shown in Figure 1.

Assume that the Euler–Bernoulli theorem can be

adopted. Crespo da Silva and Glenn33 derived theequations of motion governing the nonlinear nonpla-

nar vibrations of Euler–Bernoulli beams. The integral

partial-differential equations are simplified to the case

of planar motion of cantilever beam under transverse

excitation. The governing partial differential equation

for the nonlinear flexural vibration of the beam is, asfollows

m €v þ EIviv þ EI v0 v0v00ð Þ0Â Ã0þ 1

2m v0

Z xL

@2

@t2

Z x0

v02dx

24

35dx

8<:

9=;

0

¼ 0 ð1Þ

Here x is the axial coordinate which is measured from

the origin, v denotes the lateral vibration in y direc-

tion. The boundary conditions for the beam including

saturation nonlinear type can be expressed as

v 0, tð Þ ¼ @v

@x0, tð Þ ¼ 0,

@2v

@x2L, tð Þ ¼ 0,

EI @3v

@x3L, tð Þ ¼ F sat L, tð Þ ð2Þ

where F sat L, tð Þ is nonlinear boundary condition at its

end as shown in Figure 2 and is described by the fol-

lowing nonlinear saturation formula

F sat vð Þ ¼ f 1 v

ð Þv5

À

k pv À 4v4

f 2 vð Þ v4

8<: ð3Þ

Figure 1. Configuration of cantilever beam with saturation

type boundary condition.

2 Proc IMechE Part C: J Mechanical Engineering Science 0(0)








where k p is the constant of primary spring. Assuming

v x, tð Þ ¼ q tð Þ’ xð Þ, where ’ xð Þ is the first eigenmode of

the clamped–free beam and can be expressed as

’ xð Þ ¼ cosh xð Þ À cos xð Þ

Àcosh Lð Þ þ cos Lð Þsinh Lð Þ þ sin Lð Þ

sinh xð Þ À

sin xð Þð Þ

ð4Þ

where ¼ 1:875 is the root of characteristic equation

for first eigenmode. Applying the weighted residual

Bubnov–Galerkin method yields

Z L0

m €v þ EIviv þ EI v0 v0v00ð Þ0Â Ã0

þ 1

2m v0

Z x

L

@2

@t2 Z x

0

v02dx

24

35dx

8<:

9=;

01A’ xð Þdx ¼ 0 ð5Þ

to implement the end nonlinear boundary condition,

applying integration by parts on equation (5), it is

converted to the following

Z L0

m €vþEI v0 v0v00ð Þ0Â Ã0

þ1

2m v0

Z xL

@2

@t2

Z x0

v02dx

2

4

3

5dx

8<:

9=;

1

A’ xð Þdx

þZ L0

EIviv’ xð Þdx¼0 ð6Þ

Z L0

m €v þ EI v0 v0v00ð Þ0Â Ã0

þ 1

2m v0

Z xL

@2

@t2

Z x0

v02dx

24

35dx

8<:

9=;1A’ xð Þdx

þ EIv000’ xð Þ

L

0ÀZ

L

0

EIv000d ’ xð Þð Þ ¼ 0 ð7Þ

In the above equation the boundary condition term

EIv000 L, tð Þ is replaced by F sat L, tð Þ. By introducing the

following nondimensional variables

¼ ffiffiffiffiffiffiffiffiffi

EI

mL4

r t, q ¼ q

Lð8Þ

and applying the weighted residual Bubnov–Galerkin

method, the nondimensional nonlinear equation of

motion can be expressed as

€q þ 1q þ 2q3 þ 3q _q2 þ 4q2€q þ F sat L, tð Þ ¼ 0

ð9Þ

where

1 ¼ 12:3624,2 ¼ 40:44, 3 ¼ 4 ¼ 4:6 ð10Þ

To solve nonlinear ordinary equation (9) analytically,

the saturation condition F sat, must be formulated,

properly. The EF for the shifted Heaviside function

is expressed in the following relation

H v À að Þ ¼ 1

2þ 1

2

v À aj jv À a

ð11Þ

In this article, we introduce novel exact equivalentfunction for this nonlinearity as

F sat vð Þ ¼ k pv þ Àk pv þ f 2 vð ÞÀ ÁH v À að Þ

þ Àk pv þ f 1 vð ÞÀ ÁH Àv À að Þ ð12Þ

Using this new definition of F sat, and setting

f 1 vð Þ ¼ f 2 vð Þ ¼ ksv equation (9) can be rewritten as

follows

€q þ 01q þ 2q3 þ 3q _q2 þ 4q2

€q

þ5 q

À0 À q

þ0 À Á ¼

0

ð13-a

Þwhere

01 ¼ 1 þ p þ s, 5 ¼ s

2, 0 ¼

2L,

p ¼ 4k pL3

EI , s ¼ 4ksL3

EI ð13-bÞ

Overview of the Hamiltonian approach

Consider a general form of nonlinear differential

equation

€q þ f q, _q, €qð Þ ¼ 0, q 0ð Þ ¼ A, _q 0ð Þ ¼ 0, ð14Þ

Figure 2. Plot of generalized saturation nonlinearity.

Sedighi and Shirazi 3








Establishing variational principle for equation (14)

proposed by He5 yields

J qð Þ ¼Z T =

4

0

À _q2

2þ F q, _q, €qð Þ

dt, ð15Þ

where T is the period of the nonlinear oscillator.

In the functional (15), _q2=2 is the kinetic energy and

F q, _q, €qð Þ satisfies @F =@q ¼ f . Throughout the oscilla-

tion since the system is conservative, the total energy

remains unchanged during the motion. Hamiltonian

of the oscillator becomes a constant value H ¼ K :E :þP:E : ¼ H 0, where the terms K :E : and P:E : are kinetic

and potential energies of the system, respectively. The

new function H qð Þ can be written as5

^H qð Þ ¼ Z

T =4

0

_q2

2 þ F q, _q, €qð Þ dt ¼T

4 H 0 ð16Þ

It is obvious that

@H

@T ¼ H 0

4ð17Þ

and then the frequency–amplitude relation is obtained

by setting

@

@Ai

@H

@T ¼0 or

@

@Ai

@H

@ 1=!ð Þ ¼

0

ð18

Þ

Approximation by Hamiltonian

technique

Assuming q ¼ Pni ¼1 Ai cos i ! ð Þ as an approximate

solution. Using first term approximation, the solution

can be assumed as

q ¼ A cos ! ð Þ ð19ÞLet us consider the nonlinear equation (13-a) the

function F q, _q, €qð Þ can be expressed as

F q, _q, €qð Þ ¼ _q2

21 þ 4q2À Áþ 0

1

2q2 þ 2

4q4

þ Z 5 q À 0 À q þ 0 À Á dq |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}I :P:

ð20Þ

The last part of equation (20) yields

I :P: ¼ 5

Àq2=2 þ 0q q40

q2=2 À 0q þ 02 q4 0

&

À

Àq2=2 À 0q q4À 0

q2

=2 þ 0q þ 02

q4À 0& ! ð21

ÞAssuming q ¼ A cos ! ð Þ and A4 0, in the one-

fourth of oscillation period, equation (21) can be sim-

plifies as

I :P:¼À5

q2

2þ0qþ02

þ Àq2=2þ0q q40

q2=2À0qþ02 q40

(

ð22Þ

Therefore, the Hamiltonian of the system is easily

established as

H qð Þ ¼Z T =4

0

_q2

2þ 0

1

2q2 þ 2

4q4 þ 4

2q2

_q2

À5

1

2q2 þ 0q þ 02

dt

þZ 1=!ÂcosÀ1 0=Að Þ

0

51

2q2 À 0q þ 02

dt

þ Z T =4

1=!ÂcosÀ1 0=Að Þ

5 À 12

q2 þ 0q

dt

ð23Þ

Substituting equation (19) into (23) leads to

setting

@

@A

@H

@ 1=!ð Þ

¼ 0:39233!

2A3 À 0:5892A3

þ 0:7854 !2 À 01 þ 25À ÁA À 5A cos À1 0

A À 5

03

A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2 À 02p þ 5

0A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 À 02

p ¼ 0 ð25Þ

H ¼ 1

!0:14732A4 þ 0:39270

1A2 þ 0:3927A2!2 À 0:39275A2 þ 0:09823A4!2 À 1:5708502

ÀÀ1:55A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi1 À 0

A

2s þ 0:55A2 cos À1 0

A

þ 5

2 cos À1 0A

À 0:1255A2

1A ð24Þ









From equation (25) we can easily find that the solution ! is

! Að Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:52A2 þ 2 0

1 À 25

À Áþ 2:545 cos À1 0A

À Áþ 03

A2 ffiffiffiffiffiffiffiffiffiffi

A2À02p À 0 ffiffiffiffiffiffiffiffiffiffi

A2À02p

3A2 þ 2

vuut ð26Þ

Replacing ! from equation (26) into equation (19) yields

q ð Þ ¼ A cos

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:52A2 þ 2 0

1 À 25

À Áþ 2:545 cosÀ1 0A

À Áþ 03

A2 ffiffiffiffiffiffiffiffiffiffi

A2À02p À 0 ffiffiffiffiffiffiffiffiffiffi

A2À02p

3A2 þ 2

vuut

0B@

1CA ð27Þ

Figure 3. Comparison of the results of analytical solutions with the numerical simulations for 0¼ 0:5 A , p¼s¼ 0:1. (a) Time

history; (b) Phase portrait. Symbols: numerical solution; Solid line: analytical solutions.









Results and discussion

In order to demonstrate the integrity of proposed EF

and solutions by HA, the analytical results together

with the corresponding numerical results have been

presented, graphically. First-order approximation of

q

ð Þusing HA analytic method and introduced EF,

exhibits a good agreement with numerical resultsfrom fourth-order Runge–Kutta method, as depicted

in Figure 3. In addition, the phase portrait curves for

different amplitudes A ¼ 0.01, 0.05, 0.08, 0.1 and 1.2 is

plotted to reveal the high accuracy of asymptotic solu-

tions. Consequently, the phase-space curves generated

from HA are thoroughly acceptable as verified by

numerical results.

To explain the effect of saturation dead-band par-

ameter 0 on the nonlinear response of beam vibration,

the normalized frequency as a function of amplitude A

is shown in Figure 4 for different values of 0. For the

same normalized amplitudes, the frequency of beam

vibration shifts downward, when the saturation par-

ameter 0 moves upward. It is evident that the analyticsolution converges rapidly and is valid for a wide

range of saturation parameter and initial conditions.

The influence of primary and secondary spring par-

ameters p, s on the natural frequency has been illu-

strated in Figure 5 as a function of normalized

amplitude of cantilever beam vibration. As can be

observed, the greater spring parameters p, s

Figure 4. Fundamental normalized frequency vs. normalized amplitude: effect of parameter 0.

Figure 5. Fundamental normalized frequency vs. normalized amplitude: effect of parameters p, s.









produce larger limit cycle frequency. It is inferred

from Figure 4 and 5 that the fundamental frequency

increases as the vibration amplitude gets larger,

Regardless of dead-band and spring parameters.

Furthermore, the effect of different boundary con-

ditions (free, saturated and dead-zone) on the asymp-

totic solution of cantilever beam vibration has been

studied through Figure 6. Free end cantilever beam

has the less nonlinear frequency. However, the fre-

quency of beam under saturated boundary condition

is larger than similar beam under dead-zone boundary

condition. It should be pointed out that dead-zoneboundary condition appears via substituting p ¼ 0

in the equation of motion.

The impact of saturation parameter is also investi-

gated in the following. As illustrated in Figure 7,

increasing saturation parameter causes the vibrating

frequency to increase and consequently makes the

time period of cantilever beam oscillation decrease.

Conclusion

An advanced effective asymptotic method namely HA

was employed to establish the frequency–amplitude

relationship of vibrating cantilever beam under satur-

ation type boundary condition. In this direction, theinnovative EF for nondifferentiable saturated nonli-

nearity has been engaged to predict the analytic

Figure 6. Comparison between solutions of vibrating cantilever beam with free, saturated and dead-zone boundary conditions.

Figure 7. The impact of saturated boundary condition on the asymptotic solution.









response of nonlinear beam vibrations. The saturated

nonlinearity was rewritten precisely using continuous

functions. The accuracy of the obtained results using

introduced EF, confirms the strength of the presented

modeling.

FundingThis research received no specific grant from any funding

agency in the public, commercial, or not-for-profit sectors.

References

1. Hu T and Lin Z. Control systems with actuator satur-

ation: analysis and design. Boston: Birkha ¨ user, 2001.

2. Xin H, Gan D, Qu Z, et al. Impact of saturation non-

linearities/disturbances on the small-signal stability of

power systems: an analytical approach. Electric Power

Syst Res 2008; 78(5): 849–860.

3. Xin H, Gan D and Qiu J. Stability analysis of linear

dynamical systems with saturation nonlinearities and a

short time delay. Phys Lett A 2008; 372(22): 3999–4009.4. Singh V. Novel frequency-domain criterion for elimin-

ation of limit cycles in a class of digital filters with single

saturation nonlinearity. Chaos Solit Fract 2008; 38(1):

178–183.

5. He JH. Hamiltonian approach to nonlinear oscillators.

Phys Lett A 2010; 374(23): 2312–2314.

6. He JH. Preliminary report on the energy balance for

nonlinear oscillations. Mech Res Commun 2002; 29(2–

3): 107–111.

7. He JH. Variational iteration method for autonomous

ordinary differential systems. Appl Math Comput

2000; 114(2–3): 115–123.

8. Yang QW, Chen YM, Liu JK, et al. A modified vari-ational iteration method for nonlinear oscillators. Int J

Nonlinear Sci Numer Simul DOI: 10.1515/

IJNSNS.2011.045.

9. He JH. Modified Lindsted-Poincare methods for some

strongly nonlinear oscillations Part III: double series

expansion. Int J Nonlinear Sci. Numer Simul 2001;

2(4): 317–320.

10. Baker GA. Essentials of Pade approximants. New York:

Academic Press, 1975.

11. He JH. Max-Min approach to nonlinear oscillators. Int

J Nonlinear Sci Numer Simul 2008; 9: 207–210.

12. Liao SJ. Beyond perturbation. Boca Raton: CRC Press,

2003.

13. Sedighi HM, Shirazi KH and Zare J. An analytic solu-tion of transversal oscillation of quintic nonlinear beam

with homotopy analysis method. Int J Nonlinear Mech

2012; 47: 777–784.

14. He JH. Modified Lindstedt-Poincare methods for some

strongly non-linear oscillations Part I: expansion of a

constant. Int J Nonlinear Mech 2002; 37(2): 309–314.

15. Sedighi HM, Shirazi KH, Noghrehabadi AR, et al.

Asymptotic investigation of buckled beam nonlinear

vibration. Iranian J Sci Technol Trans Mech Eng

2012; 36(M2): 107–116.

16. Sedighi HM, Shirazi KH, Reza A, et al. Accurate mod-

eling of preload discontinuity in the analytical approach

of the nonlinear free vibration of beams. Proc IMechE,

Part C: J Mechanical Engineering Science 2012; 226(10):

2474–2484.

17. Sedighi HM and Shirazi KH. Vibrations of micro-

beams actuated by an electric field via parameter expan-

sion method. Acta Astronaut 2013; 85: 19–24.

18. He JH. An improved amplitude-frequency formulation

for nonlinear oscillators. Int J Nonlinear Sci Numer

Simul 2008; 9(2): 211–212.

19. Khan Y, Va ´ zquez-Leal H and Faraz N. An efficient

new iterative method for oscillator differential equation.

Scientia Iranica A DOI:10.1016/j.scient.2012.10.018.

20. Rafieipour H, Lotfavar A and Mansoori MH. New

analytical approach to nonlinear behavior study of

asymmetrically LCBs on nonlinear elastic foundation

under steady axial and thermal loading. Latin Am J

Solids Struct 2012; 9: 531–545.

21. He JH. Homotopy perturbation method for bifurcation

of nonlinear problems. Int J Nonlinear Sci Numer Simul

2005; 6(2): 207–208.

22. He JH. Comparison of homotopy perturbation method

and homotopy analysis method. Appl Math Comput

2004; 156(2): 527–539.

23. He JH. The homotopy perturbation method for non-linear oscillators with discontinuities. Appl Math

Comput 2004; 151(1): 287–292.

24. He JH. Some asymptotic methods for strongly non-

linear equations. Int J Mod Phys B 2006; 20: 1141–1199.

25. Sedighi HM and Shirazi KH. A new approach to ana-

lytical solution of cantilever beam vibration with non-

linear boundary condition. ASME J Comput Nonlinear

Dyn 2012; 7(3): 034502.

26. Sedighi HM, Shirazi KH and Zare J. Novel equivalent

function for deadzone nonlinearity: applied to analyt-

ical solution of beam vibration using He’s parameter

expanding method. Latin Am J Solids Struct 2012; 9:

443–451.

27. Sedighi HM and Shirazi KH. Bifurcation analysis in

hunting dynamical behavior in a railway bogie: using

novel exact equivalent functions for discontinuous non-

linearities. Scientia Iranica 2012; 19(6): 1493–1501.

28. He JH, Zhong T and Tang L. Hamiltonian approach to

Duffing-harmonic equation. Int J Nonlinear Sci Numer

Simul 2010; 11: S43–S46.

29. He JH. Asymptotic methods for solitary solutions and

compactons. Abstract and Applied Analysis 2012;

916793.

30. Sedighi HM, Shirazi KH and Noghrehabadi A.

Application of recent powerful analytical approaches

on the non-linear vibration of cantilever beams. Int J

Nonlinear Sci Numer Simul 2012; 13(7–8): 487–494.31. Bayat M, Pakar I and Ganji DD. Recent developments

of some asymptotic methods and their applications for

nonlinear vibration equations in engineering problems:

a review. Latin Am J Solids Struct 2012; 9: 145–234.

32. Akbarzade M and Kargar A. Application of the

Hamiltonian approach to nonlinear vibrating equa-

tions. Math Comput Model 2011; 54: 2504–2514.

33. Crespo da Silva MRM and Glynn CC. Nonlinear flex-

ural-flexural-torsional dynamics of inextensional

beams. I. Equations of motion. J Struct Mech 1978; 6:

437–448.







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nonlinear beam vibration with saturation type boundary condition