Improvement of a KrylovBogoliubov method that uses Jacobi elliptic functions
Abstract
An improved version of a KrylovBogoliubov method that gives the approximate solution of the nonlinear cubic oscillator ẍ + c _{1}x + c _{3}x ^{3} + ∊f(x, xdot) = 0 in terms of Jacobi elliptic functions is described. Compact general expressions are given for the time derivatives of the amplitude and phase similar to those obtained by the usual KrylovBogoliubov method (which gives the approximate solution in terms of circular functions). These expressions are especially simple for quasilinear ( c_{3} = 0) and quasipurecubic ( c_{1} = 0) oscillators. Two types of cubic oscillators have been used as examples: the linear damped oscillator f(x, xdot) = xdot, and the van der Pol oscillator f(x, xdot) = (α  βx ^{2}) xdot. The approximate solutions of these quasilinear and quasipurecubic oscillators are simple and accurate. The influence of the nonlinearity on the rate of variation of the amplitude of these two types of cubic oscillators was also studied.
 Publication:

Journal of Sound Vibration
 Pub Date:
 May 1990
 DOI:
 10.1016/0022460X(90)90781T
 Bibcode:
 1990JSV...139..151B