The response of linear oscillators coupled to MDOF nonlinear attachment with combined seismic excitation

The system of linear oscillators coupled to multi-degree-of-freedom (MDOF) nonlinear attachment with combined periodic and stochastic seismic excitation is studied in this paper. The MDOF nonlinear attachment passively absorbs broadband energy from the linear system in a unidirectional and irreversible way numerically, which acts as a nonlinear energy sink (NES) in essence. The typical cubic nonlinear terms and sinusoidal nonlinear terms are considered, respectively. The periodic generalized harmonic wavelet (PGHW) and wavelet-Galerkin methods are used for motion equations, then a set of nonlinear algebraic equations can be obtained to get the wavelet coefficient of response. The quasi-Newton method is selected to solve these equations, which is more effective than the Newton method. In the numerical example, the algorithms of Single rank inverse Broyden (Rank-1 iB), Rank-2 Broyden-Fletcher-Shanno (Rank-2 BFS), Davidon-Fletcher-Powell (DFP), and Broyden-Fletcher-Goldfarb-Shanno (BFGS) are used to estimate the Jacobian matrix. The number of iterations, calculation time, and relative error are listed and compared, and it is concluded that the algorithm of Rank-1 iB is a better one for solving this problem in both cases. Finally, the displacement and estimated power spectral density (EPSD) of the response are obtained and shown.