Abstract
A semianalytical algorithm is proposed for the solutions and their stability of a piecewise nonlinear system. The conventional harmonic balance method is modified by the introduction of Toeplitz Jacobian matrices (TJM) and by the alternative applications of fast Fourier transformation (FFT) and its inverse. The TJM/FFT method substantially reduces the amount of computation and circumvents the necessary numerical differentiation for the Jacobian. An arc-length algorithm and a branch switching procedure are incorporated so that the secondary branches can be independently traced. Oscillators with piecewise nonlinear characteristics are taken as illustrative examples. Flip, fold, and Hopf bifurcations are of interest.
Highlights
Many physical systems exhibit complex nonlinearities that cannot be described by smooth analytical functions
The use of the Toeplitz Jacobian matrices (TJM) and fast Fourier transform (FFT) (TJM/FFT) greatly reduces the amount of the mUltiplication operations included in the MIHB
The FFT/IFFT algorithm with a Toeplitz matrix is applied to determine the explicit form of the Jacobian matrix required by Newton-Raphson iterations
Summary
Many physical systems exhibit complex nonlinearities that cannot be described by smooth analytical functions. An analytical method to find the steady-state response of forced piecewise linear oscillators was proposed by Masri (1978) in which boundary values at the contact points are matched and the resulting nonlinear algebraic equations are solved using Newtonian iteration. Further development of this method was reported by Bapat and Sankar (1986), Choi and Noah (1988, 1989), and Natsiavas (1989). The use of the TJM and FFT (TJM/FFT) greatly reduces the amount of the mUltiplication operations included in the MIHB
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