Summary and Conclusions

Abstract

A method called equivalent stochastic quadratization has been proposed for stationary response analyses of nonlinear systems. The procedure can be viewed as an extension of the equivalent stochastic linearization method. Equations have been developed for general single and multi-degree-of-freedom systems subject to filtered gaussian white-noise excitation. An “equivalent” quadratic system is constructed by replacing the. nonlinearity with polynomials up to quadratic order. The equivalent system has a form whose solution can be approximated by using the Volterra series method. The coefficients of the polynomials are determined from a mean square minimization. An iterative method is required. A third order Gram-Charlier expansion is used to describe the non-gaussian response probability distribution. Numerical results are presented for one and two-degree-of-freedom models with quadratic type nonlinearities. The results are compared to simulation to verify their reliability. The proposed method offers a notable improvement over linearization when the nonlinearity is non-symmetric and the excitation frequencies are banded in a range outside the natural frequency of the system. In this case, significant resonance responses can occur which are not accounted for by linearization. When the nonlinearity is symmetric the quadratization solution reduces to the linearization solution.