Equivalent Statistical Quadratization for Multi-Degree-of Freedom Nonlinear Systems

Summary

The statistical linearization method is often inadequate for estimating spectral properties of random responses of nonlinear systems. This is sometimes due to the fact that the power spectra of responses of linear systems span only the frequency range of the excitation spectrum, whereas significant responses outside this range are possible for nonlinear systems. Recently, the concept of a statistical “quadratization” method was introduced to address this shortcoming of the linearization methods. The effectiveness of statistical quadratization was demonstrated on several single-degree-of-freedom systems. In this paper the method is generalized to multi-degree-of-freedom systems. The statistical quadratization solution procedure involves replacing the nonlinear system with an “equivalent” system with polynomial nonlinearities up to quadratic order. The nonlinear equivalent system has a form whose solutions can be approximated by using the Volterra series method. The non-Gaussian joint response probability distribution is approximated by a third order Gram-Charlier expansion. The method is formulated for systems with general nonlinearities and with nonlinearities of a special form. To demonstrate the method, solutions are obtained for a specific system. The corresponding results compare well with Monte-Carlo simulation data. Further, it is shown that the quadratization method is notably superior to the linearization method for the considered system.