Response Determination of Nonlinear Systems with Singular Matrices Subject to Combined Stochastic and Deterministic Excitations

A new technique is proposed for determining the response of multi-degree-of-freedom nonlinear systems with singular parameter matrices subject to combined stochastic and deterministic excitations. Singular matrices in the governing equations of motion potentially account for the presence of constraint equations in the system. They also appear when a redundant coordinates modeling is adopted to derive the equations of motion of complex multibody systems. Since the system is subject to both stochastic and deterministic excitations, its response also has two components, namely a deterministic and a stochastic component. Therefore, using the harmonic balance method to treat the deterministic component leads to an overdetermined system of equations to be solved for computing the associated coefficients. Then the generalized statistical linearization method for deriving the stochastic response of nonlinear systems with singular matrices, in conjunction with an averaging treatment, are utilized to determine the stochastic component of the response. The validity of the proposed technique is demonstrated by pertinent numerical examples.