tailieunhanh - A modified asymptotic method for a van-der-pol oscillator with large cubic restoring nonlinearity

A modified asymptotic method is proposed and applied to evaluate the frequency of self-excited oscillation in a Van-Der-Pol oscillator with large cubic restoring nonlinearity. The result obtained can be used for large enough nonlinearity. | Vietnam Journal of Mechanics, VAST, Vol. 26, 2004, No 2 (122 - 128) A MODIFIED ASYMPTOTIC METHOD FOR A VAN-DER-POL OSCILLATOR WITH LARGE CUBIC RESTORING NONLINEARITY NGUYEN VAN DINH Institute of Mechanics ABSTRACT. A modified asymptotic method is proposed and applied to evaluate the frequency of self-excited oscillation in a Van-Der-Pol oscillator with large cubic restoring nonlinearity. The result obtained can be used for large enough nonlinearity. 1 Introduction In [3], to evaluate the frequency of steady self-excited oscillation in a Van-Der-Pol oscillator with large cubic restoring nonlinearity, a modified Poincare method has been presented . There, assuming that the strongly nonlinear oscillator of interest is near certain linear one with unknown (to be evaluated) frequency, a formal small parameter is introduced and the governing different ial equation is written in the form corresponding to weakly nonlinear system. In this article, based on the same assumption and also on the variant of the asymptotic procedure presented in [1, 2], a modified asymptotic method is proposed to examine t he problem considered . The results obtained are identical with those given in [3] . Compared wit h the modified Poincare method, the modified asymptotic method is a little more complicated. T his results from the fact that in t he asymptotic method, we have to establish not only the algebraic equations determining steady state but the differential equations governing the amplitude-phase variation of the "general" solution. However, as compensation, t he stability study is rather simple. 2 Systems under consideration. Frequency from the usual asymptotic method Consider a Van-Der-Pol oscillator described by the differential equation x+ x = - 1x 3 + h(l - x 2 )x, () where the significations of all the notations have b een explained in [3] ; h > 0 is assumed t o be sm all but r > 0 may be large enough. The problem p osed is to evaluate the frequency of steady .