Nonstationary Response of Linear Systems under Uncorrelated Parametric and External Excitations

Abstract

An explicit solution for the nonstationary second moment response and the evolutionary spectral density of the response of linear systems subjected to uncorrelated parametric and external excitations is obtained by a simplified state-variable approach. The excitations are modeled as suddenly applied white noise processes. The solution is expressed as a sum of the solution for the external excitation alone and a correction term which accounts for the effect of the parametric excitation. In the limiting stationary case, the total solution is proportional to the solution for the external solution alone. However, the proportionality does not hold during the nonstationary stage. The solution presented is conditionally stable. The effect of the parametric excitation on the second moment response is significant in certain cases, especially near the region of instablity where the intensity of the parametric excitation could be very small relative to that of the external excitation. It is also shown that the solution under the uncorrelated parametric and external excitations is exponentially stable for a general form of envelopes employed in engineering practice. Some numerical results are presented.