Response of a parametrically-excited system to a nonstationary excitation

Abstract

The response of a parametrically-excited system to a deterministic nonstationary excitation is studied. The system, which has a cubic nonlinearity, has one focus and two saddle points and can be used as a simple model of a ship in the head or follower seas. The method of multiple scales is applied to the governing equation to derive equations for the amplitude and phase of the response. These equations are used to find the stationary response of the system to stationary excitation. The stability of the stationary response is examined. The stability of stationary periodic solutions to the original governing equation is examined through a Floquet analysis. The response to a nonstationary excitation having (a) a frequency that varies linearly with time, or (b) an amplitude that varies linearly with time, is studied. The response is computed from digital computer integration of the equations found from the method of multiple scales and of the original governing equation. The response to nonstationary excitation has several unique characteristics, including penetration, jump-up, oscillation, and convergence to the stationary solution. The agreement between solutions found from the original governing equation and the method-of-multiple-scales equations is good. For some sweeps of the excitation frequency or amplitude, the response to nonstationary excitation found from the original governing equation exhibits behavior which is analogous to symmetry-breaking bifurcations, period-doubling bifurcations, chaos, and unboundedness in the stationary solution. The maximum response amplitude and the excitation frequency or amplitude at which the response goes unbounded is found as a function of sweep rate. The effect of initial conditions and noise on the response to nonstationary excitation is considered. The results of the digital-computer simulations are verified with an analog computer.