Interaction between nonlinear parametric and forced oscillations

Abstract

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. Equation of motion and approximate solutionLet us consider a nonlinear system governed by the differential equationwhere<: > 0 is the small parameter; h 2' : 0 is the damping coefficient; 1 > 0, p > 0, r > 0, w > 0 are the constant parameters;"~= w 2 -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 t/J instead of x and :i; as follows,we have a system of two equations which is fully equivalent to (1.1)

The contributions of Professor Nguyen Van Dao in the Field of Nonlinear Oscillations

“…Nguyen Van Dao used the other way, arguing that the interactions in nonlinear systems have deep roots in parametric excitation terms. Using the asymptotic method, Nguyen Van Dao focused his research on parametric excitation and the interaction between parametric excitation and forced oscillation, interaction between parameter oscillation and self-excited oscillation, interaction between forced oscillation and self-attack oscillation [7,30,31]. Nguyen Van Dao collected his research papers on parametric excitation, nonlinear vibrations and the interaction between parametric excitation and other types of excitations in nonlinear systems and defended his advanced dissertation in Poland in 1976.…”

“…The investigation results on the interaction of the three basic equation systems mentioned above have been shown a relatively comprehensive picture of the vibration phenomena occurring in nonlinear systems. Nguyen Van Dao published over 100 scientific articles and many monographs [3][4][5][6][7][8][9]. Next, we review some outstanding results from his research works.…”