tailieunhanh - On a variant of the asymptotic procedure (For weakly nonlinear autonomous systems)
In this article, a variant of the asymptotic procedure is presented and applied to determine stationary oscillations in weakly nonlinear autonomous system with given initial conditions. Instead of the full amplitude, the approximate amplitude of order c0 is used and by this, stationary oscillations can easily and successively be determined in each step of approximation, although various types of initial conditions may be imposed. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 25, 2003, No 2 (77 - 84) ON A VARIANT OF THE ASYMPTOTIC PROCEDURE (FOR WEAKLY NONLINEAR AUTONOMOUS SYSTEMS) NGUYEN VAN DINH Institute of Mechanics As well-known, usually, in the asymptotic (Krylov-Bogoliubov-Mitropolski) method, the full amplitude (a) of the first harmonic is used as variable in asymptotic expansions [1]. In the first approximation, the equations of stationary oscillations are rather simple, however, in higher approximation, these equations often become very complicated, especially when initial conditions are imposed. In this article, a variant of the asymptotic procedure is presented and applied to determine stationary oscillations in weakly nonlinear autonomous system with given initial conditions. Instead of the full amplitude, the approximate amplitude of order c0 is used and by this, stationary oscillations can easily and successively be determined in each step of approximation, although various types of initial conditions may be imposed. It is interesting to note that the results obtained are identical with those given by the Poincare method [2] . 1. Systems under consideration - The usual asymptotic procedure Consider weakly nonlinear autonomous oscillating systems described by following differential equations: x+ x = x+ x = cf(x), cf(x, x), () () where x is oscillatory variable; overdots denote differentiation with respect to time t; 1 is own frequency; f (x) and f (x, x)-for simplicity-are polynomials of their variables; c is a small formal parameter. The equation () represents weakly nonlinear conservative systems, the equation () represents weakly nonlinear self-excited systems. The problem of interest is to determine stationary oscillations (free or self-excited oscillations) satisfying initial conditions: for () : x(O) = Xo, for () : x(O) = 0. x(O) = 0, () . () For the sake of comparison, the usual procedure of the asymptotic method is .
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