This paper deals with the harmonic oscillations of periodically excited nonlinear systems where hysteresis is simulated via fractional operator representations. Employing a diophantine version of the fractional operational powers, the energy constrained Lindstedt–Poincaré perturbation procedure is utilized to establish the harmonic solution. The constrained perturbation procedure was employed since it allows for the handling of strong damping and exciting forces over the full span of the driving frequency range. Based on the approach taken, the long time behavior of the fractionally damped Duffing's equation is studied in detail. Of special interest is the determination of the influence of fractional order on the frequency amplitude response behavior.
Padovan, J. and Sawicki, J.T., Non-Linear Vibrations of Fractionally Damped Systems, Nonlinear Dynamics, Kluwer Academic Publishers, 16, pp.321-336, 1998
The final publication is available at Springer via http://dx.doi.org/10.1023/A:1008289024058