A new semi-analytical approach for quasi-periodic vibrations of nonlinear systems

This paper presents a new semi-analytical approach, namely the enhanced time-domain minimum residual method to solve the semi-analytical quasi-periodic solution for nonlinear system. This approach does not require multiple numerical integration and can be applied to strongly nonlinear systems. The approach is mainly three-fold. Firstly, the semi-analytical solution of the nonlinear quasi-periodic system is expanded into a set of N trigonometric series with unknown coefficients, i.e., x(t) approximate to Sigma(N)(k-1) [b(k) cos (omega(k)t) + c(k) sin (omega(k)t)]. Then, the problem of solving quasi-periodic solution can be expressed as: determining the coefficients of the trigonometric series so that the residual objective function R = M<(x)over dot> + C<(x)over dot> + Kx + N(<(x)over dot>, <(x)over dot>, x, t) - F (t) is minimum over a period, i.e., min(a)(is an element of A) integral(T)(0) R(a, t)(T) R(a, t)dt. Finally, the nonlinear minimum optimization problem is solved iteratively through the enhanced response sensitivity approach. Moreover, the adaptive zero-setting curve is introduced to accelerate the convergence. Two numerical examples, a van der Pol-Duffing system and a nonlinear energy sink system are adopted to verify the feasibility of the proposed approach. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This paper introduces an enhanced time-domain minimum residual method for solving the semi-analytical quasi-periodic solution for nonlinear systems. By iteratively solving the nonlinear minimum optimization problem and introducing an adaptive zero-setting curve, the approach accelerates convergence and has been verified through numerical examples.