The Effect of Damping on the Energy Transfer in the Spherical Pendulum with Fractional Damping in a Pivot Point

Nonlinear vibrations of a system with three degrees of freedom with a spherical pendulum are investigated. The system contains an oscillator and a spherical pendulum suspended from the oscillator. The damping at the pendulum pivot point is assumed to be modelled by a fractional derivative. The viscoelastic damping properties are described using the fractional Caputo derivative of order \(0 <\alpha \le 1\) . Vibrations in the vicinity of the internal and external resonance are considered. The effect of the order of the fractional derivative on the vibrations of the autoparametric system is studied. Responses of the system, the internal and external resonance, bifurcation diagrams, Poincaré maps and the Lyapunov exponents have been calculated for various orders of fractional derivatives. Chaotic motion has been found for some system parameters.