Exact analytical solutions for forced cubic restoring force oscillator
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.
KeywordsNonlinear 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.
- 5.Belhaq, M., Fiedler, B., Lakrad, F.: Homoclinic connections in strongly self-excited nonlinear oscillators: the Melnikov function and the elliptic Lindstedt–Poincaré method. Nonlinear Dyn. 23, 67–86 (2000)Google Scholar
- 8.Coppola, V.T.: Averaging of Strongly Nonlinear Oscillators Using Elliptic Functions. Ph.D. dissertation, Cornell University, Ithaca, August (1989)Google Scholar
- 11.Hsu, C.S.: On the application of elliptic functions in non-linear forced oscillations. Q. Appl. Math. 17, 393–407 (1960)Google Scholar
- 13.Milne-Thomson, L.M.: Jacobian elliptic functions and theta functions. In: Abramowitz, M., Stegun, I.A. (eds.) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, pp. 567–585. Dover, New York (1972)Google Scholar
- 15.Bronstein, I.N., Semendjajew, K.A., Musiol, G., Mühlig, H.: Taschenbuch der Mathematik, 8th edn. Harri Deutsch, Frankfurt am Main (2012)Google Scholar