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Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator



Mohammadreza Azimi and Alireza Azimi
 
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ABSTRACT

In this study, we illustrate the nonlinear oscillator with a discontinuous term which is arising from the motion of rigid rod on the circular surface without slipping. At first we explain the problem. Two powerful methods, Parameter Expansion Method (PEM) and Variational Iteration Method (VIM), which has good accuracy and efficiency applied to solution. The results compared with the forth order Runge-Kutta. The advantages of using these methods are high accuracy and simple procedure in comparison to exact solution.

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  How to cite this article:

Mohammadreza Azimi and Alireza Azimi, 2012. Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator. Trends in Applied Sciences Research, 7: 514-522.

DOI: 10.3923/tasr.2012.514.522

URL: https://scialert.net/abstract/?doi=tasr.2012.514.522
 
Received: February 07, 2012; Accepted: March 08, 2012; Published: May 25, 2012



INTRODUCTION

Nonlinear phenomena play important roles in engineering. Moreover, obtaining exact solutions for these problems has many difficulties. To overcome the shortcomings, many new analytical methods have purposed nowadays. Some of these methods are: Parameter Expansion Method (PEM) (Ghasempour et al., 2009), Energy Balance Method (EBM) (Bayat and Pakar, 2011; Ganji et al., 2008; Mehdipour et al., 2010), Variational Iteration Method (VIM) (Choobbasti et al., 2008; Ganji et al., 2007; Khatami et al., 2008; Shakeri et al., 2009; He, 1999, 2007), Homotopy Perturbation Method (HPM) (He, 2008a; Fazeli et al., 2008; Mirgolbabaei and Ganji, 2009; Sharma and Methi, 2011; Ghotbi et al., 2008; Chowdhury, 2011), Amplitude Frequency Formulation (AFF) (He, 2008b), The Max-Min Approach (MMA) (Zeng, 2009; He, 2008c) and Homotopy Analysis Method (HAM) (Zahedi et al., 2008).

In this study, we clarify the rigid rod rocks on the circular surface problem by using Parameter Expansion Method (PEM), Variational Iteration Method (VIM). And numerical Runge-Kutta method of order 4 will be compare with the analytical results.

PROBLEM DESCRIPTIONS

The motion's equation of the rigid rod which rocks on the circular surface without slipping is:
(1) Where l is rigid rod's length, r is radius of circular surface and u is function of angle in each time:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(1)

SOLUTION PROCEDURE

Parameter expansion method (PEM): We can rewrite Eq. 1 as following form:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(2)

where, Substitution of approximation:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(3)

into Eq. 2, yields:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(4)

To apply parameter expansion method to Eq. 3, we rewrite it as follows:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(5)

According to the parameter expansion method, the solution and coefficients of Eq. 4 can be expanded as following terms:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(6)

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(7)

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(8)

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(9)

where, p is a book keeping parameter. Inserting Eq. 5-8 into Eq. 4, we have:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(10)

Equating terms with the identical powers of p yields:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(11)

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(12)

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(13)

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(14)

Considering the initial conditions u0(0) = A, Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator (0) = 0, the solution of Eq. 10 is u0 = Acos(ωt). Substituting the result into Eq. 7, yields:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(15)

To find secular term, we use Fourier expansion as follows:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(16)

Avoiding secular term, needs:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(17)

Let p = 1 into Eq. 5-7, gives:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator

Frequency can be yield:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(18)

where, g is gravitational acceleration:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator

Variational iteration method (VIM): To clarify the basic ideas of VIM, we consider the following differential equation:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(19)

where, L is a linear operator and N is a nonlinear operator.

According to the Variational Iteration Method we can construct the following iteration formulation:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(20)

where, λ is general Lagrange multiplier which can be identified optimally via the variational theory. The subscript indicates the nth approximation and Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator is considered as restricted variation, i.e., Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator.

Assuming that the angular frequency is ω, we can rewrite Eq. 2 as following form:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(21)

We can write following equation:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(22)

where, Lu = u”+ω2u, Nu = a(u2u”)+bu cos(u)-ω2u and ũ is considered as restricted variation, i.e., δũ = 0. Making the above correction functional stationary, we obtain the following stationary conditions:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(23)

The Lagrange multipliers, so, can be identified as:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(24)

Substituting Eq. 25 into Eq. 23 obtained as follows:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(25)

Substituting u0(t) = A cos(ωt) as trial function into Eq. 2, yields the residual as follows:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(26)

Using power Fourier series, we can obtain:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(27)

Now we apply Fourier expansion series on to achieve secular term:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(28)

Substituting R0 (t) into Eq. 26, we have:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(29)

Avoiding secular term needs avoid resonance, so relationship between amplitude A and frequency ω can be yield:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
(30)

Like previous part, g is gravitational acceleration:

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator

RESULTS AND DISCUSSION

Now, we want to investigate the approximate solutions in some numerical cases. Table 1-2 give information about the values of approximate solutions and numerical forth order Runge-Kutta results and also, analytical error between these quantities for two numerical cases in each 0.5 sec.

As can be seen in Fig. 1 and 2 the approximate solutions have a good adjustment with numerical method Runge-Kutta. High accuracy and validity reveal that both methods are powerful and effective to use Solution gives us possibility to obtain frequency in different cases.

Table 1: Comparison between PEM, VIM and Runge-Kutta for g = 9.8 m sec-1, r = 1 m, l = 2 m, a = 3, b = 29.4 and Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator

Table 2: Comparison between PEM, VIM and Runge-Kutta for g = 9.8 m sec-2, r = 1 m, l = 2 m, a = 3, b = 29.4 and Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
Fig. 1: Comparison between PEM, VIM and Runge-Kutta for g = 9.8 m sec-1, r = 1 m, l = 2 m, a = 3, b = 29.4 and Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator

Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator
Fig. 2: Comparison between PEM, VIM and Runge-Kutta for g = 9.8 m sec-1, r = 1 m, l = 2 m, a = 3, b = 29.4 and Image for - Application of Parameter Expansion Method and Variational Iteration Method to Strongly Nonlinear Oscillator

CONCLUSION

In this study, we applied the parameter-expansion method and VIM which are two powerful and efficient methods, to obtain the relationship between amplitude and frequency for the nonlinear equation which comes from the motion of rigid rod on the circular surface without slipping. Also, the solution compared with the numerical method forth order Runge-Kutta. The high accuracy and validity of approximate solutions assure us about the solution and reveal these methods can be used for nonlinear oscillators even with high order of nonlinearity.

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