On Duffing oscillators with slowly varying parameters

Abstract

A method of Krylov-Bogoliubov type, which gives the approximate solution in terms of Jacobi elliptic functions, is used for the study of perturbed Duffing oscillators with slowly varying parameters: d dt [μ(τ) x ̈] + c1(τ)x + c3(τ)x3 + εf(x, x ̈, τ) = 0. This method is a natural generalization of the usual Krylov-Bogoliubov method that is only valid when c3(τ)x3 is of ε order. Two examples are given. One is a pure cubic oscillator (c1 = 0) with variable mass and linear damping, f(x, x ̈) = x ̈, for which a simple accurate approximate solution is found. The other is a pendulum with variable length and damping proportional to the velocity for which an approximate analytical expression for the rate of variation of the oscillation amplitude is obtained, and successfully compared with the numerically calculated result and with that obtained using the normal Krylov-Bogoliubov method. © 1991.