Forced harmonic response analysis of nonlinear structures using describing functions

The dynamic response of multiple-DOF nonlinear structures is usually determined by numerical integration of the equations of motion, an approach which is computationally very expensive for steady-state response analysis of large structures. In this paper, an alternative semianalytical quasi-linear method based on the describing function formulation is proposed for the harmonic response analysis of structures with symmetrical nonlinearities. The equations of motion are converted to a set of nonlinear algebraic equations and the solution is obtained iteratively. The linear and nonlinear parts of the structure are dealt with separately, the former being represented by a constant linear receptance matrix and the latter by the generalized quasi-linear matrix which is updated at each iteration. A special technique that reduces the computation time significantly when the nonlinearities are localized is used with success to analyze large structures. The proposed method is fully compatible with standard modal analysis procedures. Several examples dealing with cubic stiffness, piecewise linear stiffness, and Coulomb friction type of nonlinearities are presented in the case of a 10-DOF structure. (Author (revised))