Exact steady states of periodically forced and essentially nonlinear and damped oscillator.

• Exact fundamental resonances of the damped and forced truly nonlinear oscillator are computed. • Methodology that yields exact expression for the responses at resonance is developed. • An exact steady state response of the truly nonlinear oscillator needs a multi-harmonic excitation. • Time-periodic excitation is given as a sum of Ateb periodic functions. • For the fundamental resonance the frequency of the steady state response coincides with the applied excitation. In this paper steady states of a essentially nonlinear and damped oscillator excited with a time periodic function is investigated. The procedure for computing exact strongly nonlinear, damped resonances of a nonlinear oscillator, developed by Vakakis and Blanchard (2018), is used and extended to a more general class of strongly nonlinear oscillators. The method is applied for computing of the external excitation which produces the motion equal to that of the free strong nonlinear undamped oscillator. The obtained periodical excitation force is the sum of two Ateb periodic functions which correspond to the sum of various multi-harmonic forces. The amplitude-frequency diagrams are plotted and the resonant frequency is calculated. Motion around steady state is approximately determined by using the method of time variable amplitude and phase, An example of an excited and damped quadratic oscillator perturbed with a linear elastic force is considered. The approximate analytical solution is compared with exact numerical one. The solutions are in a good agreement. [ABSTRACT FROM AUTHOR]