Stochastic Averaging of Energy Envelope

Abstract

The method of stochastic averaging, as applied previously to analyze the energy envelope of a nonlinear system under random excitations of independent ideal Gaussian white noises, is generalized to include the case of correlated physical white noises. The excitations can be additive or multiplicative, or both. The key to the generalization is to separate the well‐known Wong‐Zakai correction terms into two parts and identify them as additional contributions to damping and restoring force, respectively. It is shown that this generalized procedure yields the same stationary joint probability density for the displacement and velocity as that obtained from another procedure, called dissipation energy balancing. Furthermore, the second‐moment stability conditions obtained for a reduced linear system are exact. Two examples are given for illustration.