Time-domain minimum residual method combined with energy balance for nonlinear conservative systems

Abstract

This paper presents a new semi-analytical method that combines the time-domain minimum residual method with energy balance for nonlinear conservative systems. Due to work done only by the conservative force, the total energy of the nonlinear conservative system will remain constant. The response frequency is also the function of the initial conditions and it is difficult to achieve the semi-analytical solutions accurately. The proposed method aims to quickly obtain the higher-order approximate semi-analytical periodic or quasi-periodic solutions of both weakly and strongly nonlinear conservative systems. To this end, the approximate analytical solution is expressed as a group of trigonometric series with undetermined coefficients firstly. Next, by deriving the assumed trigonometric series and substituting the state variables back into the original nonlinear conservative equation, the goal of solving the semi-analytical solution is transformed into an optimization problem that minimizes the residual objective function in a period. Finally, the energy constraints are introduced into the optimization process, and the enhanced response sensitivity approach is called to resolve the nonlinear objective function iteratively. A conservative Duffing system and a nonlinear symmetrically conservative two-mass system are taken as two numerical examples to illustrate the performance of the proposed method.