Non-stationary response of nonlinear systems with singular parameter matrices subject to combined deterministic and stochastic excitation

A new technique is proposed for determining the response of multi-degree-of-freedom nonlinear systems with singular parameter matrices subject to combined deterministic and non-stationary stochastic excitation. Singular matrices in the governing equations of motion potentially account for the presence of constraints equations in the system. Further, they also appear when a redundant coordinates modeling is adopted to derive the equations of motion of complex multi-body systems. In this regard, the system response is decomposed into a deterministic and a stochastic component corresponding to the two components of the excitation. Then, two sets of differential equations are formulated and solved simultaneously to compute the system response. The first set pertains to the deterministic response component, whereas the second one pertains to the stochastic component of the response. The latter is derived by utilizing the generalized statistical linearization method for systems with singular matrices, while a formula for determining the time-dependent equivalent elements of the generalized statistical linearization methodology is also derived. The efficiency of the proposed technique is demonstrated by pertinent numerical examples. Specifically, a vibration energy harvesting device subject to combined deterministic and modulated white noise excitation and a structural nonlinear system with singular parameter matrices subject to combined deterministic and modulated white and colored noise excitations are considered.