Dynamics of a mechanical system with a spherical pendulum subjected to fractional damping: analytical analysis

Abstract

In the paper, nonlinear vibrations of a system with three degrees of freedom having a spherical pendulum are considered. The system comprises a mass element suspended from a linear spring and a viscous damper, and a spherical pendulum swung from the mass element. It is assumed that the fractional viscous damping occurs in the viscous damper and at the pendulum pivot point. The viscoelastic properties of damping are assumed to be described using the Riemann–Liouville fractional derivative. The fractional derivative of an order of 0 < α≤ 1 is assumed. The nonlinear vibrations of the system near internal and external resonances are analyzed. The equations of motion of the analyzed system are solved using the multiple-scale method. The steady-state approximate solution is studied. The effect of a fractional-order derivative on the system vibrations is examined.