On Infinitesimal Stability and Instability of Pendulum Type Oscillations

摘要

For the pendulum type of oscillations governed by the equation (the second derivative of x with respect to t) + phi (x) = 0, with phi(x) an odd function, it is shown that according to the linearized disturbance equation, stability is predicted if and only if dT/d alpha = 0, where T is the period and alpha is the amplitude of the nonlinear steady-state oscillations. From this it follows that for a given nonlinear function phi(x), infinitesimal stability can at most be predicted only for certain discrete values of alpha. It is shown analytically that for a simple pendulum, a power-law spring and a cubic hard or soft spring, the oscillations are infinitesimally unstable for all alpha. It is further shown, however, that particular cases of nonlinear restoring forces do exist for which infinitesimal stability is predicted for certain alpha's, in contrast to the actual Liapunov instability in these cases. (Author)