Nonlinear analytical multi-dimensional convolution solution of the second order system

In this paper, a procedure to analytically develop an approximate solution for the prototypical nonlinear mass–spring–damper system based on multi-dimensional convolution expansion theory is offered. The nonlinearity herein is mathematically considered in quadratic and bilinear terms. A variational expansion methodology, one of the most efficient analytical Volterra techniques, is used to develop an analytical two-term Volterra series. The resultant model is given in the form of first and second kernels. This analytical solution is visualized in the time domain followed by a parametric study for understanding the influence of each nonlinear/linear term appearing in the kernel structure. An analytical nonlinear step response is also conducted to characterize the overall system response from the fundamental components. The developed analytical step response provides an illumination for the source of differences between nonlinear and linear responses such as initial departure time, settling time, and steady value. Feasibility of the proposed implementation is assessed by numerical examples. The developed kernel-based model shows the ability to predict, understand, and analyze the system behavior beyond that attainable by the linear-based model.