Analysis of nonlinear vibrations of a two-degree-of-freedom mechanical system with damping modelled by a fractional derivative

Abstract

Free damped vibrations of a mechanical two-degree-of-freedom system are considered under the conditions of one-to-one or two-to-one internal resonance, i.e., when natural frequencies of two modes – a mode of vertical vibrations and a mode of pendulum vibrations – are approximately equal to each other or when one natural frequency is nearly twice as large as another natural frequency. Damping features of the system are defined by the fractional derivatives with fractional parameters (the orders of the fractional derivatives) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. The model put forward allows one to obtain the damping coefficient dependent on the natural frequency of vibrations, so it has been shown that the amplitudes of vertical and pendulum vibrations attenuate by an exponential law with damping ratios which are exponential functions of the natural frequencies. Damped soliton-like solutions have been found analytically.