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Interaction between nonlinear parametric and forced oscillations

The interaction of nonlinear oscillations is an important and interesting problem, which has attracted the attention of many researchers. Minorsky N. [5] has stated "Perhaps the whole theory of nonlinear oscillations could be formed on the basis of interaction". | Vietnam Journal of Mechanics, NCNST of Vietnam T. XX, 1998, No 3 (16- 23) INTERACTION BETWEEN NONLINEAR PARAMETRIC AND FORCED OSCILLATIONS NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI The interaction of nonlinear oscillations is an important and interesting problem, which has attracted the attention of many researchers. Minorsky N. [5] has stated "Perhaps the whole theory of nonlinear oscillations could be formed on the basis of interaction". The interaction between the forced and "linear" parametric oscillations when the coefficient of the harmonic function of time is linear relative to the position has been studied in [1, 4]. In this paper this kind of interaction is considered for "nonlinear" parametric oscillation with cubic nonlinearity of the modulation depth. The asymptotic method of nonlinear mechanics [1] is used. Our attention is focused on the stationary oscillations and their stability. Different resonance curves are obtained. 1. Equation of motion and approximate solution Let us consider a nonlinear system governed by the differential equation x + w2 x = 0 is the small parameter; h 2': 0 is the damping coefficient; 1 > 0, p > 0, r > 0, w > 0 are the constant parameters;"~= w2 -1 is the detuning parameter, where the natural frequency is equal to unity; and 8 2': 0 is the phase shift between two excitations. The frequency of the forced excitation is nearly equal to the own frequency w, and the frequency of the nonlinear parametric excitation is nearly twice as large. So, both excitations are in fundamental resonance. They will interact one to another. Introducing new variables a and x =a cosO. t/J instead of x and ± = -awsin8, 8 = wt :i; as follows, + 7/J, (1.2) we have a system of two equations which is fully equivalent to (1.1) " . -da =. --Fsm8 dt w ' d,P " a-= --Fcos8 dt w ' 16 (1.3) where F = Ax- hx -1x 3 + 2px 3 cos 2wt + rcos(wt- 6). The equations (1.3) belong to the standard form, for which the asymptotic method is applied .