Toeplitz Jacobian method for nonlinear double-periodic excitations

Abstract

The Toeplitz Jacobian matrix method is an efficient algorithm for computing the steady state solutions of nonlinear periodic vibration. In this paper, the method is generalized by using multiple time scales to double-periodic solutions in a multi-frequency excited system. The method is combined with a standard multi-dimensional FFT algorithm to accurately simulate the nonlinear oscillators with widely separated frequencies. The continuation technique can also be incorporated with the Newton–Raphson iteration to further increase its efficiency, and to achieve the complete frequency response characteristics.