Resonances in Duffing’s Problem

Abstract

The forced oscillation of a pendulum is given by the solution of the differential equation,

$$ \ddot{x} + \alpha \sin x = \beta \cos \omega t\quad (\alpha ,\beta ,\omega > 0) $$

and is called the Duffing’s problem. The general solution of (1) is not periodic in general, but (1) admits the periodic solutions of the type

$$ x\sum\limits_{{j = 0}} {{{A}_{{2j + 1}}}\cos \left[ {(2j + 1)\frac{\omega }{n}t} \right],n = 1,3,5,...} $$

When n=1, the solution is called the harmonic solution, while the other cases n = 3, 5, …, are called the subharmonic solutions of the orders 1/3, 1/5, … respectively. The conditions for the harmonic or the subharmonic solutions are referred as the harmonic or the subharmonic responses. For example, the harmonic response is expressed as

$$ {{\omega }^{2}}{{A}_{1}} - 2\alpha {{J}_{1}}\left( {{{A}_{1}}} \right) + \beta = 0 $$

.