A statistical nonlinearization technique for multi-degrees of freedom nonlinear systems under white noise excitations

Abstract

A statistical nonlinearization (SNL) technique is proposed for the solution of the joint probability density function of general multi-degrees of freedom (mdof) nonlinear systems under stationary white noise excitations. The nonlinearities associated with damping and restoring forces as well as the intensities of excitations are not necessarily small. It is based on the application of the exact solution of the joint probability density function of a mdof nonlinear systems under stationary white noise excitations. This exact solution is different from that of Caughey in that in the present exact solution the ratios of damping coefficients to applied white noise excitations are not identical. The exact solution is also different from those of Cai and Lin, and Zhu and Huang in that the present exact solution is obtained directly from the theory of differential equations while that of Cai and Lin requires satisfaction of a relatively restrictive criterion. Furthermore, the Hamiltonian formulation is applied by Zhu and Huang. Their solution depends on the number of independent integrals of motion, for example. Results by applying the proposed technique for a general two-degrees of freedom nonlinear system are compared with those obtained by Monte Carlo simulation (MCS). It is concluded that the technique is accurate, simple to implement and is applicable to mdof systems with both nonlinear damping and nonlinear restoring forces.