On the homotopy analysis method for non-linear vibration of beams

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On the homotopy analysis method for non-linear vibration of beams

T. Pirbodaghi a, M.T. Ahmadian a,*, M. Fesanghary b

a Center of Excellence in Design, Robotics and Automation, School of Mechanical Engineering, Sharif University of Technology, 11365-9567 Tehran, Iranb Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Avenue, 15875-4413 Tehran, Iran

a r t i c l e i n f o

Article history:

Received 26 August 2007

Received in revised form 15 July 2008

Available online 14 August 2008

Keywords:

Non-linear vibration

Euler-Bernoulli beam

Homotopy analysis method

a b s t r a c t

In this study, the homotopy analysis method (HAM) is used to investigate non-linear vibra-tion behaviour of Euler-Bernoulli beams subjected to axial loads. Analytical expressions for

geometrically non-linear vibration of beams are provided. The effect of vibration amplitude

on the non-linear frequency and buckling load is discussed. Comparison between HAM

results and those available in literature demonstrates the accuracy of this method. This

study shows that a first-order approximation of the HAM leads to highly accurate solutions

which are valid for a wide range of vibration amplitudes.

2008 Elsevier Ltd. All rights reserved.

1. Introduction

Vibration analysis of beams is an important issue in structural engineering applications such as high-rise buildings, long-span bridges, aerospace vehicles and many other industrial usages. As the amplitude of oscillation increases, these structures

are subjected to non-linear vibrations which often lead to material fatigue and structural damage. These effects become

more significant around the natural frequencies of the system. Therefore, it is very important to provide an accurate analysis

towards the understanding of the non-linear vibration characteristics of these structures.

The non-linear vibration of beams is governed by a non-linear partialdifferential equation in space and time. In general,

it is extremely difficult to find an exact or closed-form solution for this equation. Consequently, researchers have been con-

centrated on approximate analytical methods (Evensen, 1968; Pillai and Rao, 1992; Qaisi, 1993; Azrar et al., 1999) and

numerical techniques (Reddy and Singh, 1981; Sarma and Varadan, 1983; Shi and Mei, 1996).

Approximate analytical methods are divided into two main groups: direct techniques and combined techniques. For

weakly non-linear systems, direct techniques, such as perturbation methods, are used to solve directly the non-linear

partialdifferential equation and associated boundary conditions. With the combined approach, the partialdifferential

equation are first discretized into a set of non-linear ordinary-differential equation and then solved analytically in time

domain. One of the most commonly used methods for discretization is the Galerkin procedure. Besides all advantages of

numerical methods, due to convenience for parametric studies and accounting of the physics of the problems, an analytical

solution appears more appealing than the numerical one. Also, Analytical solutions give a reference frame for verification and

validation of other numerical approaches.

The analytical methods have their own limitations. For example perturbation methods, the most extensively used ana-

lytical techniques, are generally restricted to the case of weak non-linearity and are carried out with respect to a small

parameter in the equation. Most of non-linear problems, especially those having strong non-linearity, have no small param-

eters at all. In order to overcome this problem, HAM which does not require small parameters in the equation was developed

0093-6413/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechrescom.2008.08.001

* Corresponding author. Tel.: +98 21 66165503; fax: +98 21 66000021.

E-mail address: [email protected](M.T. Ahmadian).

Mechanics Research Communications 36 (2009) 143148

Contents lists available at ScienceDirect

Mechanics Research Communications

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(Liao and Cheung, 1998; Li and Liao, 2005). The effectiveness and accuracy of the HAM have been demonstrated in the anal-

ysis of various non-linear problems (Liao and Campo, 2002; Liao and Cheung, 2003; Liao, 2007; Hoseini et al., 2008).

The main objective of this study is to obtain analytical expressions for geometrically non-linear vibration of Euler-Ber-

noulli beams. First, the governing non-linear partial differential equation is reduced to a single non-linear ordinary differen-

tial equation. It is assumed that only the fundamental mode is excited. The latter equation is solved analytically in time

domain using HAM.

2. Theoretical formulation

Consider a straight Euler-Bernoulli beam of length L, a cross-sectional area A, the mass per unit length of the beam m, a

moment of inertiaI, and a modulus of elasticityEthat is subjected to an axial force of magnitude Pas shown inFig. 1. The

equation of motion including the effects of mid-plane stretching is given by:

mo2w0

ot02 EI

o4w0

ox04 P

o2w0

ox02 EA

2L

o2w0

ox02

Z L0

ow0

ox0

2dx0 0 1

For convenience, the following non-dimensional variables are used:

x x0=L;w w0=q; t t0EI=mL41=2; P PL2=EI

whereq = (I/A)1/2 is the radius of gyration of the cross-section. As a result Eq. (1)can be written as follows:

o2

wot2

o4

wox4

Po2

wox2

1

2

o2

wox2

Z 10

ow

ox

2dx 0 2

assuming w(x,t) = W(t)/(x) where/(x) is the first eigenmode of the beam(Tse et al., 1978) and applying the Galerkin method,

the equation of motion is obtained as follows:

d2Wt

dt2 a1Pa2Wt a3W

3t 0 3

the Eq.(3)is the differential equation of motion governing the non-linear vibration of Euler-Bernoulli beams. The center of

the beam is subjected to the following initial conditions:

W0 Wmax;dW0

dt 0 4

whereWmax denotes the non-dimensional maximum amplitude of oscillation.

Under the transformations = xt, the Eq.(3)can be written as:

x2 W a1Pa2W a3W3 0 5

where x is the non-linear frequency and double-dot denotes differentiation with respect to s anda1,a2 anda3 are asfollows:

a1 Z 10

o4/x

ox4

!/xdx

!,Z 10

/2xdx 6a

a2 Z 10

o2/x

ox2

!/xdx

!,Z 10

/2xdx 6b

a3 1

2 Z 1

0

o2/x

ox2 Z

1

0

o/x

ox

2

dx !/xdx !,Z 1

0

/2xdx 6c

Fig. 1. A schematic of an Euler-Bernoulli beam subjected to an axial load.

144 T. Pirbodaghi et al. / Mechanics Research Communications 36 (2009) 143148


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Post-buckling loaddeflection relation for the problem can be obtained from Eq. (5)by substitutingx = 0 as:

P a1 a3W2=a2 7

neglecting the contribution ofWin Eq.(7), the buckling load can be determined as:

Pc a1=a2 8

3. Homotopy analysis method

3.1. An overview

Homotopy analysis is a general analytic method for solving the non-linear differential equations (Liao and Cheung, 1998;

Li and Liao, 2005; Hoseini et al., 2008). The HAM transforms a non-linear problem into an infinite number of linear problems

with embedding an auxiliary parameter (q) that typically ranges from zero to one. As q increases from 0 to 1, the solution

varies from the initial guess to the exact solution. To illustrate the basic ideas of theHAM, consider the following non-linear

differential equation:

Nxt 0 9

where Nis a general non-linear operator,t denotes the time, andx(t) is an unknown variable. The homotopy function is con-

structed as follows:

H/;q;h;ht 1 q/t;q x0t qhhtN/t;q 10

where h is a non-zero auxiliary parameter, (t) is a non-zero auxiliary function and denotes an auxiliary linear operator. As q

increases from 0 to 1, the /(t;q) varies from the initial approximation to the exact solution. In the other words,/ (t;0) =x0(t)

is the solution of the H(/;q,h,(t))jq=0 = 0 and /(t;1) =x(t) is the solution of the H(/;q,h,(t))jq=1= 0./(t;q) can be expanded in

a power series of theq using Taylors theorem as:

/t;q /t;0 X1m1

1

m!

om/t;q

oqm

q0

qm x0t X1m1

xmtqm 11

where xm(t) is called the m-order deformation derivative. Setting H(/;q,h,(t)) = 0, the zero-order deformation equation is

constructed:

1 q/t;q x0t qhhtN/t;q 12

with the following initial conditions:

/0;q const:;d/0;q

dt 0 13

differentiating zero-order deformation equation with respect to q and then settingq = 0, yields the first-order deformation

equation which give the first-order approximation of the x(t):

x1t hhtN/t;qjq0 14

x10 0;dx10

dt 0 15

The higher order approximations of the solution can be obtained by calculating the m-order (m> 1) deformation equationwhich can be calculated by differentiating Eqs.(12) and (13)m times with respect to the q (Li and Liao, 2005).

3.2. Application of the HAM

Consider the Eqs.(3) and (4)for the vibration of an Euler-Bernoulli beam. Free oscillation of a system without damping is

a periodic motion and can be expressed by the following base functions:

cosmxt; m 1;2;3;. . . 16

under the transformation W(t) = V(s), Eqs.(3) and (4)becomes as follows:

x2d2Vsds2

a1Pa2Vs a3Vs3

0 17

V0 Wmax;

dV0

ds 0 18

T. Pirbodaghi et al. / Mechanics Research Communications 36 (2009) 143148 145


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in order to satisfy the initial conditions, the initial guess ofV(s) is chosen as follows:

V0s Wmax coss 19

to construct the homotopy function, the auxiliary linear operator is selected as:

Vs;q x2 o

2Vs;qos2

Vs;q

! 20

from Eq.(17), the non-linear operator is defined as:

NVs;q;x x2o2Vs;q

os2 a1Pa2Vs; q a3Vs; q

321

the first-order deformation equation which give the first-order approximation of theV(s) can be written as:

V1s hhsNVs;q;xjq0 22

V10 0;dV10

ds 0 23

the auxiliary function ((s)) and the auxiliary parameter (h) which adjust convergence region and rate of approximate solu-tion must be chosen in such a way that the solution of Eq.(22)could be expressed by a set of base functions. Assumingh = 1

and(s) = 1 can satisfy this constraint. As a result, the Eqs.(22) and (23)are become as:

x2 o2

V1sos2

V1s !

x2o2

V0sos2

a1Pa2V0s a3V0s3 24

V10 0;dV10

ds 0 25

solving Eqs.(24) and (25), the V1(s) is obtained as follows:

V1s Wmax8x2NL

4a1Pa2 3a3W2

max 4x2

NL

s sins

a332x2NL

W3maxcos s cos3s 26

The solution should obey the general form of the base function. Therefore, the coefficient of the secular term (ssins) mustbe zero. This provides an algebraic equation which yields the non-linear frequency related to axial force and maximum non-

dimensional amplitude of the beam center. The non-linear natural frequency and the deflection of the beam center become

as follows:

xNL1

23a3W

2

max 4a1Pa21=2

27

V1s 1

32x2NLa3W

3

maxcoss cos 3s 28

thus the first-order approximation of theV(s) is become as:

Vs V0s V1s Wmax32x2NL

a3W2

max 32x2NL

coss

a332x2NL

W3max

cos3s 29

and

Wt Wmax32x2NL

a3W2

max 32x2NL cosxNLt

a332x2NL

W3max

cos3xNLt 30

the higher order approximations of the V(s) can be obtained by calculating the m-order (m> 1) deformation equation andperforming the same procedure as done above.

4. Results and discussions

In order to demonstrate the accuracy and effectiveness of the HAM, The procedure explained in previous section is ap-

plied to simply supported and clamped beams. Table 1 shows the comparison of non-linear to linear frequency ratio

(xNL/xL) with those reported in the literature.It can be observed that there is an excellent agreement between the results obtained from the HAM and those reported by

Azrar et al. (1999) and Qaisi (1993). By increasing the amplitude of vibration, the difference between the non-linear fre-

quency and linear frequency increases. In general, large vibration amplitude will yield a higher frequency ratio. It can be seen

that for high-amplitude ratios the present method overestimates the frequencies of clamped beams but gives close agree-

ment with published results for simply supported beams. This is because of using the trigonometric base functions in the

application of HAM, which means that we assumed the general form of solution is a combination of trigonometric functions.



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Since the eigenmodes for simply supported beams involve only the sinusoidal component, the HAM gives more accurate re-

sults in comparison with clamped beams which have hyperbolic component in their eigenmodes.

Non-linear to linear frequency ratio versus non-dimensional amplitude ratio is illustrated in Fig. 2. It can be seen that the

rate of increase in non-linear fundamental frequency with increasing displacement is very low at small amplitudes. This can

lead to the conclusion that a very good agreement can be noticed between data from a linear model and non-linear model at

small deflections. However, as the maximum amplitude increases, the non-linearity effect becomes significant.

Table 1

Comparison of non-linear to linear frequency ratio (xNL/xL)

Wmax Simply supported Clamped

Azrar et al. (1999) Qaisi (1993) Present study Azrar et al. (1999) Qaisi (1993) Present study

1 1.0891 1.0897 1.0897 1.0221 1.0628 1.0572

2 1.3177 1.3229 1.3228 1.0856 1.2140 1.2125

3 1.6256 1.6394 1.6393 1.1831 1.3904 1.4344

4 1.9999 1.3064 1.5635 1.6171

Fig. 2. Non-linear to linear frequency ratio versus non-dimensional amplitude ratio.

Fig. 3. Variation of the buckling load parameter (P/Pc) versus the non-dimensional amplitude ratio.

T. Pirbodaghi et al. / Mechanics Research Communications 36 (2009) 143148 147


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Fig. 3shows the variation of the buckling load parameter (P/Pc) versus the non-dimensional amplitude ratio. In large

amplitude ratios, the difference between the present study results and those obtained by Emam (2002) increases. This is be-

cause of ignoring the effect of mid-plane stretching by Emam (2002). In large amplitude ratios, the effect of the non-linearity

due to mid-plane stretching is dominant and neglecting it introduces error in the results.

To demonstrate the accuracy of the obtained analytical results, the authors also calculate the variation of non-dimen-

sional amplitude ratio versuss for the beam center using fourth-order RungeKutta method.Fig. 4illustrates comparisonbetween these results.

As can be seen in the figure, the results obtained using the HAM have a good agreement with numerical results.

5. Conclusions

In this study, the HAM was employed to obtain analytical expressions for the non-linear fundamental frequency and

deflection of Euler-Bernoulli beams. Unlike the solutions obtained by the other analytical techniques such as perturbation

methods, these expressions are valid for a wide range of vibration amplitudes. The HAM solution converges quickly and

its components can be simply calculated. Also, compared to other analytical methods, it can be observed that the results

of HAM require smaller computational effort and only a first-order approximation leads to accurate solutions. Beside all

the advantages of the HAM, there are no rigorous theories to direct us to choose the initial approximations, auxiliary linear

operators, auxiliary functions, and auxiliary parameter. However, further research is needed to better understand the effect

of these parameters on the solution quality.

References

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amplitudes part I: general theory and application to the single mode approach to free and forced vibration analysis. J. Sound Vib. 224, 183207.

Emam, A.S., 2002. A theoretical and experimental study of nonlinear dynamics of buckled beams. PhD Thesis, Virginia Polytechnic Institute and State

University.

Evensen, D.A., 1968. Nonlinear vibrations of beams with various boundary conditions. AIAA J. 6, 370372.

Hoseini, S.H., Pirbodaghi, T., Asghari, M., Farrahi, G.H., Ahmadian, M.T., 2008. Nonlinear free vibration of conservative oscillators with inertia and static type

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Li, S., Liao, S.J., 2005. An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl. Math. Comput. 169, 854865.

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Liao, S.J., Campo, A., 2002. Analytic solutions of the temperature distribution in Blasius viscous flow problems. J. Fluid Mech. 453, 411425.

Liao, S.J., Cheung, A.T., 1998. Application of homotopy analysis method in nonlinear oscillations. ASME J. Appl. Mech. 65, 914922.

Liao, S.J., Cheung, K.F., 2003. Homotopy analysis of nonlinear progressive waves in deep water. J. Eng. Math. 45, 105116.

Pillai, S.R.R., Rao, B.N., 1992. On nonlinear free vibrations of simply supported uniform beams. J. Sound Vib. 159, 527531.

Qaisi, M.I., 1993. Application of the harmonic balance principle to the nonlinear free vibration of beams. Appl. Acoust. 40, 141151.

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Fig. 4. Variation of the non-dimensional amplitude ratio versus s.

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On the homotopy analysis method for non-linear vibration of beams