Statistical Linearization of Nonlinear Structural Systems with Singular Matrices

Random vibration of linear and nonlinear structural systems with singular matrices: A frequency domain approach

“…(15) can be found in Refs. [17,18], and is also outlined, for convenience, in Appendix A. Next, focusing on the frequency domain, the problem of determining the FRF matrix of a system with singular mass, damping and stiffness matrices is considered.…”

“…(53) was utilized in Ref. [18], and can be construed as a direct generalization of the standard relationship for non-singular matrices [6,27]. Nevertheless, the step of arbitrarily choosing the solution of Eq.…”

“…Following Ref. [18], the basic steps for determining the equivalent linear matrices are concisely reviewed next for completeness. Further, to account for singular matrices, a generalization of a formula [6,27] based on a Gaussian response assumption and related to the expectation of the derivatives of the nonlinear function Φ x is proved for the first time in the literature.…”

“…Next, relying on the M-P FRF the celebrated standard input-output (excitation-response) relationship in the frequency domain is generalized for determining the system response power spectrum. Finally, the above derived frequency domain relationship is utilized in conjunction with a recently developed statistical linearization technique [18] for determining the response statistics of nonlinear systems with singular matrices. The validity of the herein developed frequency domain random vibration techniques is demonstrated by pertinent numerical examples including several linear and nonlinear systems with singular matrices.…”

“…Subsequently, the authors developed in Refs. [17,18] generalized random vibration time-domain techniques for determining the response of linear and nonlinear structural systems subject to stochastic excitations; see also Ref. [19] for an alternative treatment based on polynomial matrix theory.…”