HARMONIC BALANCE APPROACHES TO A NONLINEAR SINGULAR OSCILLATOR

HU Hui;GUO Yuan-jun;ZHENG Min-yi

Journal of Vibration and Shock ›› 2009, Vol. 28 ›› Issue (2) : 121-123.

PDF(567 KB)
PDF(567 KB)
Journal of Vibration and Shock ›› 2009, Vol. 28 ›› Issue (2) : 121-123.
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HARMONIC BALANCE APPROACHES TO A NONLINEAR SINGULAR OSCILLATOR

  • HU Hui,GUO Yuan-jun,ZHENG Min-yi
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Abstract

The method of harmonic balance is used to calculate analytical approximations to the periodic solutions of the nonlinear oscillator in which the restoring force is inversely proportional to the dependent variable. Unlike Mickens method, the second order nonlinear singular differential equation is directly solved. The Fourier series expansion coefficients of the nonlinear restoring forces corresponding to the first and second order harmonic balance solutions are easily calculated using corresponding integrals. The nonlinear algebraic equations for the second order approximate solution are solved by using symbolic computation software. The percentage error of the first approximate frequency in relation to the exact one is 12.8%, and the percentage error for the second approximate frequency is lower than 1.28%. A comparison of the first and second analytical approximate periodic solutions with the numerically exact solutions shows that the second analytical approximate periodic solution is much more accurate than the first analytical approximate periodic solution. It is usually rather difficult to use the harmonic balance method to produce higher order analytical approximations because it requires solutions of sets of complicated nonlinear algebraic equations. This difficulty may be overcome to some extent by using symbolic computation software such as Matlab or Mathematica. The derivation in this paper can be considered as a typical example.

Key words

angular frequency / nonlinear singular oscillator / symbolic computation software / the method of harmonic balance

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HU Hui;GUO Yuan-jun;ZHENG Min-yi. HARMONIC BALANCE APPROACHES TO A NONLINEAR SINGULAR OSCILLATOR[J]. Journal of Vibration and Shock, 2009, 28(2): 121-123
PDF(567 KB)

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