Accession Number : AD0721704

Title :   On Infinitesimal Stability and Instability of Pendulum Type Oscillations.

Descriptive Note : Research rept.,

Corporate Author : POLYTECHNIC INST OF BROOKLYN N Y DEPT OF AEROSPACE ENGINEERING AND APPLIED MECHANICS

Personal Author(s) : Morduchow,Morris

Report Date : FEB 1971

Pagination or Media Count : 23

Abstract : 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)

Descriptors :   (*OSCILLATION, EQUATIONS OF MOTION), VIBRATION, NONLINEAR SYSTEMS, PERTURBATION THEORY, DIFFERENTIAL EQUATIONS, STABILITY

Subject Categories : Numerical Mathematics
      Mechanics

Distribution Statement : APPROVED FOR PUBLIC RELEASE