African Journal of
Mathematics and Computer Science Research

  • Abbreviation: Afr. J. Math. Comput. Sci. Res.
  • Language: English
  • ISSN: 2006-9731
  • DOI: 10.5897/AJMCSR
  • Start Year: 2008
  • Published Articles: 254

Full Length Research Paper

More accurate approximate analytical solution of pendulum with rotating support

Md. Helal Uddin Molla
  • Md. Helal Uddin Molla
  • Department of Mathematics, Rajshahi University of Engineering and Technology (RUET), Kazla, Rajshahi 6204, Bangladesh.
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M. S. Alam
  • M. S. Alam
  • Department of Mathematics, Rajshahi University of Engineering and Technology (RUET), Kazla, Rajshahi 6204, Bangladesh.
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  •  Received: 08 February 2017
  •  Accepted: 05 April 2017
  •  Published: 30 June 2017

 ABSTRACT

Based on the energy balance method (EBM), a more accurate analytical solution of the pendulum equation with rotating support was presented. The results were compared with those obtained by the differential transformation method (DTM) and He’s improved energy balance method. It was shown that the results are more accurate than the said methods.

Key words: Energy balance method, approximate solutions, nonlinear oscillators, pendulum with rotating support.


 INTRODUCTION

Many scientific problems in natural sciences and engineering are inherently nonlinear, but it is difficult to determine their exact solutions. Many analytical methods are available to find their approximate solution. The perturbation methods (Nayfeh, 1973; He, 2006) were originally developed for handling weak nonlinear problems. Recently, some of them were modified (Cheung et al., 1991) to investigate strong nonlinear problems. Homotopy perturbation (Belendez, 2007; Ganji and Sadighi, 2006; Belendez et al., 2008; Ozis and Yildirim, 2007), iteration method (Haque et al., 2013; Jamshidi and Ganji, 2010; Lim et al., 2006; Baghani et al., 2012; Rafei et al., 2007) are useful for obtaining approximate periodic solution with large amplitude of oscillations; however, they are applicable only for odd nonlinearity problems. Harmonic balance method (Mickens, 1986; Lim et al., 2005; Belendez et al., 2006; Wu et al., 2006; Mickens, 2007; Alam et al., 2007; Belendez et al., 2009; Lai et al., 2009; Hosen et al., 2012) is a powerful method in which truncated Fourier series is used. Iterative homotopy harmonic balance (Guo and Leung, 2010), differential transformation (Ghafoori et al., 2011) and max-min (Yazdi et al., 2012) methods have been developed for solving strongly nonlinear oscillators. Energy balance method (He, 2002; Khan and Mirzabeigy, 2014; Alam et al., 2016; Mehdipour et al., 2010; Ebru et al., 2016; Zhang et al., 2009) is another widely used technique for solving strongly nonlinear oscillators. Though, all these analytical methods have been developed for handling nonlinear oscillator, they provide almost similar results for a particular approximation. Recently, EBM has been modified by truncating some higher order terms of the algebraic equations of related variables to the solution (Alam et al., 2016) and it measures more correct result than the usual method. Moreover, the modification on EBM   used   in   Alam   et al.   (2016)  is   valid  for  some nonlinear oscillators, especially when
 
In this article, the EBM (Alam et al., 2016) was utilized to determine the approximate solution of pendulum equation with rotating support. This type of oscillator was analyzed by Ghafoori et al. (2011) applying differential transformation method (DTM), Belendez et al. (2006) using harmonic balance method and Yazdi et al. (2012) using max-min approach. He (2002) first introduced energy balance method and Khan and Mirzabeigy (2014) was used to improve accuracy of He’s energy balance method to obtain the solution of pendulum equation with rotating support. The present method can be applied to nonlinear oscillatory systems where the nonlinear terms are not small and no perturbation parameter is required. 

 


 THE BASIC IDEA OF HE’S ENERGY BALANCE METHODS

A general form of nonlinear oscillator is 
 
 
The first-order approximate solution of Equation (1) is assumed in the following form:
 
 
 
From Equation (23), the value of  is obtained. But it is a cubical equation. It is noted that the coefficient of is small. Ignoring this term, its solution is obtained as . Then, can be written as . Thus, Equation (23) again becomes a quadratic equation, whose smallest solution is the required value of . Substituting the value of  into Equation (21),  is obtained.


 RESULTS AND DISCUSSION

A more accurate solution of Equation (16) was determined. The solution was compared with those presented by Ghafoori et al. (2011) and Khan and Mirzabeigy (2014). All the results together with numerical solution (obtained by fourth-order Runge–Kutta formula) are presented in Table 1. From the results shown in the table, it is clear that the percentage error of the present solution did not exceed 0.69%. On the contrary, the maximum percentage errors of DTM (Ghafoori et al., 2011) and improved EBM (Khan and Mirzabeigy, 2014) are respectively 29.41 and 2.22%. Thus, the present method provides more accurate solution.
 


 CONCLUSION

Based on EBM, an analytical approximate solution was presented for solving pendulum equation with rotating support. The solution is nicely close to the exact results and are much better than those obtained by differential transformation method (DTM) (Ghafoori et al., 2011) and improved accuracy of He’s energy balance method (Khan and Mirzabeigy, 2014). The relative error of the present method is lower than those obtained by others (Ghafoori et al., 2011; Khan and Mirzabeigy, 2014).


 CONFLICT OF INTERESTS

The authors have not declared any conflict of interests.



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