Perturbation analysis of the effective equation for two coupled periodically driven oscillators

Abstract

Dynamics of two coupled periodically driven oscillators is analyzed via approximate effective equation of motion. The internal motion is separated off exactly and then approximate equation of motion is derived. Perturbation analysis of the effective equation is used to study the dynamics of the initial dynamical system.

Exact Nonlinear Fourth-order Equation for Two Coupled Oscillators: Metamorphoses of Resonance Curves

“…where F = m e ω 2 f , µ = mm e /M and ǫ = m e /M is a nondimensional parameter [9,10], see also Ref. [14] where separation of variables for a more general system of coupled equations was described.…”

“…Starting from equations (1), (2) we derived the exact fourth-order nonlinear equation for internal motion as well as approximate second-order effective equation [9,10] (this approximation performs well for m2 m1 ≪ 1). Applying the Krylov-Bogoliubov-Mitropolsky (KBM) method to the effective equation we have computed and studied the corresponding nonlinear resonances.…”

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“…It is possible to simplify the problem eliminating the variable x in (2.1) to obtain the exact fourth-order equation for the variable y only, describing relative motion of the mass m e [8], see also Ref. [11], where separation of variables for a more general system of coupled equations was described.…”

“…We simplified the problem described by equations (1.1), (1.2) deriving the exact fourth-order nonlinear equation for internal motion as well as approximate second-order effective equation in [8].…”

“…[8] and [9] for the cases of 1 : 1 and 1 : 3 resonances, respectively). Dependence of the amplitude A of nonlinear resonances on the frequency ω is significantly more complicated than in the case of Duffing oscillator and this leads to new nonlinear phenomena.…”