SUBHARMONICS IN A NON-LINEAR SYSTEM WITH UNSYMMETRICAL RESTORING FORCE

Abstract

The behaviour of an oscillatory system which is positively damped but in which the restoring force is unsymmetrical about the equilibrium position is investigated. It is shown that if a periodic external force whose period is approximately half the natural period is applied, the system may oscillate with twice the period of the external force (such oscillations are called subharmonics of order 2). The sub-harmonics only occur when the amplitude of the external force reaches a certain critical value; if the forcing period is slightly greater than half the natural period, there is a second critical amplitude beyond which the ordinary forced oscillation (with the same period as the external force) becomes unstable. In the range between the two critical amplitudes, therefore, the forced oscillation or the subharmonics may occur (which of them occurs will depend on the initial displacement and velocity of the system), whilst beyond the second critical amplitude only the subharmonics can occur. This gives rise to a ‘hysteresis’ effect when the amplitude of the external force is increased and then decreased again.