variational iteration method for nonlinear vibration

IJRRAS 5 (3) ● December 2010

Fereidoon & al. ● Variational Iteration Method

VARIATIONAL ITERATION METHOD FOR NONLINEAR VIBRATION OF SYSTEMS WITH LINEAR AND NONLINEAR STIFFNESS A. Fereidoon1, M. Ghadimi2,*, H.D. Kaliji3, M. Eftari4 & S. Alinia5 1

Department of Mechanical Engineering, Faculty of Engineering, Semnan University, Iran. 2,3 Department of Mechanical Engineering Islamic Azad University, Semnan, Iran. 4 Department of Mechanical Engineering Islamic Azad University, South Tehran Branch, Iran. 5 Department of Mechanical Engineering, Babol University of Technology, Babol, Iran. * Email: [email protected] ABSTRACT In this study, accurate analytical solution for the nonlinear free vibration of an oscillator with inertia and static type cubic nonlinearities is derived. This solution is called He’s Variational Iteration Method (VIM). Comparing this method with numerical integration solutions using a built-in ODE-solver in MATLAB and He’s Homotopy Analysis Method (HAM) is shown its drastic approximation. By using VIM, only a few iterations lead us to high accuracy of the solutions and it is valid for whole solution domain. Keywords: Variational Iteration Method (VIM), Nonlinear vibration, Approximate frequency 1.

INTRODUCTION

Oscillation systems have been widely used in many areas of physics and engineering. These systems have significant importance in engineering particularly in mechanical and structural dynamic because many practical engineering components consist of vibrating systems that can be modeled using oscillator systems such as elastic beams supported by two springs or mass-on-moving belt or nonlinear pendulum and vibration of a milling machine and a mass with serial linear and nonlinear stiffness on a frictionless contact surface [1-4]. However, many physical phenomena have nonlinear terms in its differential equation. Researchers modeled these phenomena with linear differential equation for its simplicity solutions. Recently, by improving approximate methods which are easy to use, they can model the physical problems realistically with better precisian. So, the study on various methods that use for solving nonlinear differential equations is an important topic for the analysis of engineering practical problems. Some of these methods are, He’s Homotopy Perturbation Method (HPM) [5-7], Homotopy Analysis Method (HAM) [8-10], He’s Parameter-Expanding Method [11,12], He’s Energy Balance Method (EBM) [13-15] and He’s Variational Iteration Method (VIM) [16-18]. The main propose of this study is to obtain highly accurate analytical solution for the equation of elastic cantilever beam carrying a concentrated mass at an intermediate point along its span [19,20]. The VIM solution has been compared with HAM and numerical integration solutions using a built-in ODE-solver in MATLAB. Also the amount of frequencies achieved with VIM has been compared with those obtained from former studies. The results will show its effective and convenient approximate solution. 2.

SYSTEM DESCRIPTION AND ASSUMPTIONS

Consider free vibration of a conservative, single-degree-of-freedom system with a mass attached to linear and nonlinear springs in series as shown in figure 1. After transformation, the motion is governed by a nonlinear differential equation of motion [21] as: 2

(1  3 zu 2 )

d 2u  du   6 zu    eu  eu 3  0, dt 2  dt 

(1)

With the initial conditions: u (0)  A,

du

dt

(0)  0

(2) y2

Linear spring

k1

y1

Frictionless

Nonlinear spring

k2 , 

mass

Figure 1. a mass with serial linear and nonlinear stiffnesses on a frictionless contact surface.

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Where


u(t )  y2 (t )  y1 (t ).

 k

(3) (4)

2

  k2 k

(5)

Z

(6)

1

1 

e  k2 m(1   )

(7)

k1 and k 2 are the linear and nonlinear spring constant, respectively. Parameters  ,  , v , e , m and  are perturbation parameter, coefficient of the nonlinear spring force, deflection In which

of the nonlinear spring, natural frequency, mass and the ratio of the springs constant. 3.

ANALYSIS AND SOLUTION OF VARIATIONAL ITERATION METHOD

To illustrate the basic concept of the Variational Iteration Method, we consider the following general nonlinear system: (8) Lu  Nu  g (t ), where L is a linear operator and N is a nonlinear operator and g(t) an inhomogeneous term. According to the Variational Iteration Method, we can construct following iteration formulation: t (9) u n 1 (t )  u n (t )   (Lu n ( )  Nu n ( )  g ( )) d  ,



0

Here the subscript n denotes the nth-order approximation and λ is called a general Lagrange multiplier [16-18 ,22], which can be identified optimally via the variational theory, u n is considered as a restricted variation [23], i.e.,

u n  0 . Supposing that the angular frequency of the equation (1) is ω, we have the following linearized equation: (10) u   2u  0. So we can rewrite Eq. (1) in the form (11) u   2u  g (u )  0. Where (12) g (u )  (e 2   2 )u  3 zu 2u  6 zuu 2  e 2u 3 Applying the Variational Iteration Method, we can construct the following functional equation: t

u n 1 (t )  u n (t )    (u ( )   2u ( )  g (u ( ))) d  ,

(13)

 ( )   2 ( )  0,  ( )  t  0,

(14)

0

g is considered as a restricted variation, i.e,  g  0. Calculating variation with respect to u n , and noting that  g (u n )  0, we have the following stationary Where

conditions:

1   ( )  t  0. Therefore, the multiplier, can be identified as



1



(15)

sin  (  t ).

Substituting the identified multiplier into Eq. (13) results in the following iteration formula: t

1

0



u n 1 (t )  u n (t )  

sin  (  t )  u  e 2u  3 zu 2u  6 zuu 2  e 2u 3  d  .

(16)

We can determine the angular frequency [17]: 2

  A cos 0

2

t  2  e 2  3 z  2 A 2 cos2 t  6 zA 2 2 sin 2 t  e 2 A 2 cos2 t   0.

From the above equation, one can easily conclude that

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(17)


  e


0.75 A 2  1 . 0.75 zA 2  1

(18)

And period of oscillation for this system by variational iteration method is 2

T 

e

0.75 A  1 0.75 zA 2  1 2

(19)

.

Now, assuming its initial approximate solution has the form u 0 (t )  A cos(t ). Appling Eq. (20) and substituting into Eq. (16), we can obtain A ((9 z  2  e 2 )A 2 cos t  (e 2  9 z  2 )A 2 cos 3t  32 2 (16 3  16e 2  12 3 zA 3  12e 2 A 2 )t sin t )

u1 (t )  A cos t 

(20) (21)

Here the angular frequency ω is defined as Eq. (18). 4.

RESULTS

Figure 2 shows the displacement of the mass ( y2 (t ) ) with VIM, HAM [4] and ODE-solver in MATLAB. Also, comparison between frequencies obtained by VIM and HAM are illustrated in figure 3. The results are obtained for the following amounts. m  1,

k1  4,

k2  16,

z  0.8,

h  1.408.

Figure 2. The comparison between VIM solution with HAM and ODE-solver in MATLAB for A   / 9 and   0.5 .

Figure 3.The results of HAM and VIM for frequency versus amplitude associated with influence of  . 262


5.


OVERALL CONCLUTIONS

The Variational Iteration Method has been used to obtain an analytical solution for the nonlinear free vibration of a conservative oscillator with inertia and static type cubic nonlinearities. The results show that there is suitable precision between this method with Homotopy Analysis method and ODE-solver in MATLAB. Moreover, VIM is suitable not only for weak nonlinear problems, but also for strongly nonlinear problems. The most significant features of this method are its excellent accuracy for the whole range of oscillation amplitude values. Also, it can be used to solve other conservative truly nonlinear oscillators with complex nonlinearities. 6.

REFERENCES

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