Science Journal of Applied Mathematics and Statistics
Volume 7, Issue 4, August 2019, Pages: 51-55
Received: Jul. 20, 2019;
Accepted: Sep. 16, 2019;
Published: Oct. 9, 2019
Views 71 Downloads 17
Md. Abul Kashem Mondal, Department of Mathematics, Rajshahi University of Engineering and Technology (RUET), Rajshahi, Bangladesh
Md. Helal Uddin Molla, Department of Mathematics, Rajshahi University of Engineering and Technology (RUET), Rajshahi, Bangladesh
Md. Shamsul Alam, Department of Mathematics, Rajshahi University of Engineering and Technology (RUET), Rajshahi, Bangladesh
The Van der Pol oscillator is a nonlinear damping and non-conservative oscillator. Energy is generated at low amplitude and dissipated at high amplitude. This nonlinear oscillator was first introduced by Dutch electrical engineer and physicist B. Van der Pol and it was originally used to investigate vacuum tubes. Nowadays, it is used in both physical and biological sciences. It is also used in sociology and even in economics. It has a limit cycle and in earlier it was determined by the classical perturbation methods when the nonlinear term is small. Then the harmonic balance method was used to determine the limit cycle for stronger nonlinear case. Moreover, many researchers have been analyzed this oscillator by various numerical approaches. In this article, a new analytical approach based on harmonic balance method is presented to determine the limit cycle as well as approximate solutions of this nonlinear oscillator. The frequency as well as the limit cycle obtained by new approach has been compared with those obtained by other existing methods. The present method gives better result than other existing results and also close to the corresponding numerical result (considered to the exact result). Moreover, the present method is simpler than the existing harmonic balance method.
Md. Abul Kashem Mondal,
Md. Helal Uddin Molla,
Md. Shamsul Alam,
A New Analytical Approach for Solving Van der Pol Oscillator, Science Journal of Applied Mathematics and Statistics.
Vol. 7, No. 4,
2019, pp. 51-55.
Krylov NN, Bogoliubov NN (1947), Introduction to Nonlinear Mechanics. Princeton University Press, New Jersey.
Nayfeh AH (1973), Perturbation Methods. John Wiley and Sons, New York.
Kovacic I and Mickens RE (2012), A generalized van der Pol type oscillator: Investigation of the properties of its limit cycle. J. Math. Comput. Model. 55, 645-653.
Cheung YK, Chen SH and Lau SL (1991), A modified Lindstedt-Poincare method for certain strongly non-linear oscillators. Int. J. Nonlin. Mech. 26 (3), 367-378.
Liu HM (2005), Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt–Poincare method. Chaos, Solit. Fract. 23, 577-579.
Mickens RE (2001), A generalization of the method of harmonic balance. J. Sound Vib. 111, 515-518.
Alam MS, Razzak MA, Hosen MA and Parvez MR (2016), The rapidly convergent solutions of strongly nonlinear oscillators. Spring. Plu. 5, 1258, 16 pages.
Hosen MA and Chowdhury MSH (2015), A new analytical technique based on harmonic balance method to determine approximate periods for Duffing-harmonic oscillator. Alexandria Eng. J. 54, 233-239.
Lim CW and Lai SK (2006), Accurate higher-order analytical approximate solutions to non-conservative nonlinear oscillators and application to Van der Pol damped oscillations. Int. J. Mech. Sci., 48, 483-492.
Rahman MS, Haque ME and Shanta SS (2010), Harmonic balance solution of nonlinear differential equation (Non-conservative). J. Advan. Vib. Engin., 9 (4), 345-356.
GuoZ and Ma X (2014), Residue harmonic balance solution procedure to nonlinear delay differential systems. J. Applied Math. Comp. 237, 20–30.
Ju P and Xue X (2015), Global residue harmonic balance method for large-amplitude oscillations of a nonlinear system. J. Applied Math. Model. 39, 449–454.
Guo Z and Leung AYT (2010), The iterative homotopy harmonic balance method for conservative Helmholtz–Duffing oscillators. J. Applied Math. Comp. 215, 3163–3169.
El-Naggar AM and Ismail GM (2012), Applications of He’s amplitude-frequency formulation to the free vibration of strongly nonlinear oscillators. J. Applied Math. Sci. 6, 2071-2079.
Mishara V, Das S, Jafari H and Ong SH (2016), Study of fractional order van der Pol equation. J. King Saud Univ. Sci. 28, 55-60.
Khan Y and Mirzabeigy A (2014), Improved accuracy of He’s energy balance method for analysis of conservative nonlinear oscillator. Neural Comput. Appli. 25, 889-895.
Molla MHU, Razzak MA and Alam MS (2018), A more accurate solution of nonlinear conservative oscillator by energy balance method. Multidis. Model. Mater. Struct. 14 (4), 634-646.
Mehdipour I, Ganji DD and Mozaffari M (2010), Application of the energy balance method to nonlinear vibrating equations. Current Appli. Phys. 10, 104-112.
Babazadeh H, Ganji DD and Akbarzade M (2008), He’s energy balance method to evaluate the effect of amplitude on the natural frequency in nonlinear vibration systems. Progress In Electromag. Res. M, 4, 143–154.
Molla MHU, Razzak MA and Alam MS (2017), An analytical technique for solving quadratic nonlinear oscillator. Multidis. Model. Mater. Struct. 13 (3), 424-433.
Molla MHU and Alam MS (2017), Higher accuracy analytical approximations to nonlinear oscillators with discontinuity by energy balance method. Result. Phys. 7, 2104–2110.
Belendez A, Arribas E, Ortuno M, Gallego S, Márquez A and Pascual I (2012), Approximate solutions for the nonlinear pendulum equation using a rational harmonic representation. Comput. Math. Appli. 64, 1602-1611.
Alam MS, Haque ME and Hossain MB (2007), A new analytic tecnique to find periodic solutions of nonlinear systems. Int. J. Nonlinear Mech. 42, 1035-1045.
Liping Liu, Earl. H. Dowell, Kenneth C. Hall (2007), A novel harmonic balance analysis for the Van der Pol oscillator, Int. J. Non-Linear Mech. 42, 2-12.
Shen Y, Yang S, Sui C (2014), Analysis on limit cycle of fractional-order van der Pol oscillator, Chaos, Solitons and Fractals, 67, 94-102.
Zhang J, Gu X (2010), A Stability and bifurcation analysis in the delay-coupled van der Pol oscillators, Applie. Math. Model. 34, 2291-2299.
Barro MA (2016), Stability of a ring of coupled van der Pol oscillators with non-uniform distribution of the coupling parameter, J. Appli. Res. Techno. 14, 62–66.
Casaleiroa J, Oliveira LB, Pintoa AC (2014), Van der Pol Approximation Applied to Wien Oscillators, Procedia Techno. 17, 335-342.