Nonlinear Dynamics

, Volume 83, Issue 4, pp 2349–2359 | Cite as

Exact analytical solutions for forced cubic restoring force oscillator

  • Uwe Starossek
Original Paper


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.


Nonlinear oscillator Non-harmonic periodic loading Jacobi elliptic functions Amplitude–frequency relation 



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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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Hamburg University of TechnologyHamburgGermany

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