Stability Investigation of the Non-linear Periodic Oscillations

Abstract

The stability of the non-linear periodic oscillations is discussed by solving a variational equation which characterizes. small variations from the periodic states of equilibrium. This variational equation leads to a linear equation in which the coefficient is periodic in the time. If all solutions of this equation are bounded, then the oscillation is said to be stable, otherwise unstable. In order to establish the stability criterion, the characteristic exponents for the unbounded solutions are calculated by Whittaker's method. Then, the generalized stability condition is derived by comparing the said characteristic exponents with the damping of the system considered. Since the solutions of the variational equation have the form: e[μT] [sin(nζ-a)+···], our stability condition is secured not only for the unbounded solutions having the fundamental frequency (n=l), but also for the unbounded solutions with higher harmonic frequencies (n=2, 3, 4, •·····). Hence the generalized stability condition obtained in this way is particularly effective in studying the oscillations in which the higher harmonics are excited. Finally our investigation is compared with one of the stability conditions derived by Mandelstam and Papalexi for the subharmonic oscillations. In the appendix, the characteristic exponents are calculated at some length for the unbounded solutions of a variational equation in which the periodic coefficient involves sine series as well as cosine series.