Stochastic cumulant analysis of MDOF systems with polynomial-type nonlinearities

A new methodology is presented for formulating the equations governing the evolution of the response cumulants of MDOF nonlinear dynamic systems subjected to external delta-correlated processes. The system nonlinearities are represented by polynomial terms involving the system variables. Kronecker algebra and matrix calculus are used as efficient mathematical tools to organize the information and present the cumulant equations in a compact form. Appropriate recursive relationships are developed to relate the joint cumulants involving Kronecker powers of the response vector variables to the joint cumulants involving the original response vector variables. It is found that the differential equations governing the system of cumulants has a form similar to the state space form of equations for the original dynamic system. The state and excitation matrices describing the system of cumulants are obtained, respectively, from the state and excitation matrices describing the original dynamic system. Cumulant-neglect closure techniques, in which the joint cumulants of the response variables with order higher than a specified order are neglected, can be directly incorporated and efficiently used in the analysis to truncate the infinite hierarchy of the resulting system of cumulant equations. Examples are presented to illustrate the use of the formulation, as well as to investigate convergence and accuracy issues related to the higher-order cumulant-neglect closure scheme.