Analysis of nonlinear damped vibrations of a two-degree-of-freedom mechanical system by use of fractional calculus

Free damped vibrations of a mechanical two-degree-of-freedom system are considered under the conditions of the internal resonance one-to-one, 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. Damping features of the system are defined by a fractional derivative with a fractional parameter ( the order of the fractional derivative) 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 the common damping ratio which is an exponential function of the natural frequency.