This paper develops generalized analytical first and second Volterra kernels for the prototypic nonlinear mass–spring–damper system. 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 analytical first and second kernels are visualized in both the time and the frequency domains followed by a parametric study to understanding the influence of each nonlinear/linear term appearing in the kernel structure. An analytical nonlinear step and periodic responses are also conducted to characterize the overall system response from the fundamental components. The developed analytical responses provide an illumination for the source of differences between nonlinear and linear responses. 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.

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