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Exact analytical solutions for forced cubic restoring force oscillator

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Abstract

A strongly nonlinear oscillator is considered in which the restoring force is a purely cubic function of the displacement variable. Its forced undamped oscillation response to non-harmonic periodic loading is studied. The loading function is derived from the free oscillation response whose time course follows a Jacobi elliptic function. It is chosen such that exact analytical solutions are obtained for the steady-state response and the amplitude–frequency relation. The equation describing the amplitude–frequency relation is a cubic polynomial equation. Its solutions are presented and further discussed by means of diagrams that illustrate the equilibrium of dynamic forces. Furthermore, results of a numerical study are presented concerning the stability of the identified analytical steady-state solutions. The numerical study also reveals the existence of a subharmonic steady-state response with a period three times the period of the loading function. The general approach of using non-harmonic loading functions is transferable to other types of nonlinear oscillators.

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Acknowledgments

The numerical study was performed by the author’s students Richard Bäumer M.Sc., and Hannah Ziems B.Sc., which is gratefully acknowledged.

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  1. Hamburg University of Technology, Denickestr. 17, 21073, Hamburg, Germany

    Uwe Starossek

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  1. Uwe Starossek
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Correspondence to Uwe Starossek.

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Starossek, U. Exact analytical solutions for forced cubic restoring force oscillator. Nonlinear Dyn 83, 2349–2359 (2016). https://doi.org/10.1007/s11071-015-2486-2

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  • DOI: https://doi.org/10.1007/s11071-015-2486-2

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