Stability Boundary for Pseudo-Random Parametric Excitation of a Linear Oscillator

Abstract

Stability bounds are outlined for the null solution of the equation describing the response of a linear damped oscillator excited through periodic coefficients, the excitation being a form of Rice noise comprising equal amplitude sinusoids with frequencies at equal intervals in the vicinity of twice the natural frequency of the system, but with pseudo-random initial phases. Stability was investigated by the monodromy matrix method, which is exact apart from errors due to numerical integration, and by the approximate method due to R. A. Struble, which replaces the dependent variable by its amplitude and a phase variable. Struble’s method gives the main features of the stability diagram and leads to faster and more robust numerical integration with potential advantages for nonlinear and several degree-of-freedom systems, but loses much of the detail. When the frequency spacing is relatively large, the stability diagram is closely related to that for Mathieu’s equation, but the detailed shape becomes very complicated as the frequency spacing decreases. Quantitative comparison with the corresponding boundary for Gaussian white noise excitation shows very approximate equivalence.