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On a variant of the asymptotic procedure

A variant of the asymptotic method is proposed to construct steady solution of weakly nonlinear non-autonomous oscillating systems. The amplitude and the dephase angle of order c0 are used as variables, the uniqueness of the asymptotic expansions is assured by stationarity conditions. | Vietnam Journal' of Mechanics, VAST Vol. 26, 2004, No 2 (76 - 82) ON A VARIANT OF THE ASYMPTOTIC PROCEDURE (II: WEAKLY NONLINEAR NON-AUTONOMOUS SYSTEMS) NGUYEN VAN DINH Institute of Mechanics ABSTRACT. A variant of the asymptotic method is proposed to construct steady solution of weakly nonlinear non-autonomous oscillating systems. The amplitude and the dephase angle of order c 0 are used as variables, the uniqueness of the asymptotic expansions is assured by stationarity conditions . 1 Introduction In [3] , to determine steady state in weakly nonlinear autonomous oscillating systems, a variant of the asymptotic procedure has been proposed, consisting of two modifications: - the approximate amplitude a of order c 0 of the first harmonic is chosen as variable in asymptotic expansions and - the arbitrariness of the latter is removed by initial conditions and by stationarity conditions. In the present paper, the case of non-autonomous system is considered. Besides the mentioned amplitude of order co, the dephase angle of same order is used as the second variable and the additional stationarity conditions are used to assure the uniqueness of the asymptotic expansions. It is shows that steady state (stationary oscillation) can be successively determined in each step of approximation and the solution obtained is identical with that given by the Poincare method. e 2 Systems under consideration. The usual asymptotic procedure Consider a weakly nonlinear non-autonomous oscillating system described by the differential equation: x + w2x = cf(x,±, wt), (1.1) where w is the exciting frequency; f(x , x,wt) is a function of (x,x,wt) , 27r-periodic with respect to wt; the significations of other notations have been explained in [3]. Fc;>r simplicity, f (x, x, wt) is assumed to be a finite Fourier series in t with polynomial in ( x, ±) coefficients. To be able to make a comparison, the usual asymptotic procedure is briefly recalled [l]. First, following asymptotic .