Average Power and Power Exchange in Oscillators

Abstract

Integral and algebraic equations for the computation of average kinetic, potential, and dissipative energies in systems of randomly excited coupled oscillators are developed. The latter equations are equivalent to those that appear in the study of the Lyapounov stability of linear dynamical systems. A numerical procedure comparable to the known engineering-literature computations of Lyapounov stability is developed and illustrated here. It is also compared with the solution of the stochastic dual of the Lyapounov-stability problem, the computation of “mean-square-response” matrices (Gersch). The known Lyapounov-stability solutions require inversion of a matrix whose order increases as the square of the number of degrees of freedom; the latter require the inversion of a matrix whose order increases linearly with the number of degrees of freedom. Exact results for average power exchange between two arbitrary oscillators and average power coupled into an arbitrary oscillator (in an n oscillator system) are obtained in terms of second moment (mean-square response) computations. In an illustrative example of a 3-degrees-of-freedom system, the exact expression for average power transfer is evaluated to permit an interpretation of earlier approximate analysis by Lyon and Newland on average energy transfer in complex structures.