Closed Form Solutions for Nonlinear Oscillators Under Discontinuous and Impulsive Periodic Excitations
Mechanical Engineering, Wayne State University, Detroit, MI 48202, USA
Symmetry 2019, 11(11), 1420; https://doi.org/10.3390/sym11111420
Received: 9 October 2019 / Revised: 10 November 2019 / Accepted: 12 November 2019 / Published: 16 November 2019
(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)
Periodic responses of linear and nonlinear systems under discontinuous and impulsive excitations are analyzed with non-smooth temporal transformations incorporating temporal symmetries of periodic processes. The related analytical manipulations are illustrated on a strongly nonlinear oscillator whose free vibrations admit an exact description in terms of elementary functions. As a result, closed form analytical solutions for the non-autonomous strongly nonlinear case are obtained. Conditions of existence for such solutions are represented as a family of period-amplitude curves. The family is represented by different couples of solutions associated with different numbers of vibration half cycles between any two consecutive pulses. Poincaré sections showed that the oscillator can respond quite chaotically when shifting from the period-amplitude curves.
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MDPI and ACS Style
Pilipchuk, V. Closed Form Solutions for Nonlinear Oscillators Under Discontinuous and Impulsive Periodic Excitations. Symmetry 2019, 11, 1420.
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