Dynamics of a coupled mechanical system containing a spherical pendulum and a fractional damper

Abstract

The presented work deals with nonlinear dynamics of a three degree of freedom system with a spherical pendulum and a damper of the fractional type. Vibrations in the vicinity of the internal and external resonance are considered. The system consists of a block suspended from a linear spring and a fractional damper, and a spherical pendulum suspended from the block. The viscoelastic properties of the damper are described using the Caputo fractional derivative. The fractional derivative of an order of $$0 < \alpha \le 1$$ 0 < α ≤ 1 is assumed. The impact of a fractional order derivative on the system with a spherical pendulum is studied. Time histories, the internal and external resonance, bifurcation diagrams, Poincaré maps and the Lyapunov exponents have been calculated for various orders of a fractional derivative. Chaotic motion has been found for some system parameters.

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

“…The presented work is a continuation of previous works by authors dealing with the system of three degrees of freedom with a spherical pendulum [1][2][3]. In this work, a system consisting of a mass element and a spherical pendulum swung from the mass element is examined.…”

“…To the our knowledge, thus far only by the authors have performed the study of vibrations of a spherical pendulum with fractional damping [3]. In the aforementioned work, the authors assumed fractional damping only in the damper attached to the mass element [3]. The effect of a fractional-order derivative on the system vibrations was analyzed using numerical calculations.…”

Spatial transportation of the beam on a bifilar fastening

“…In a number of works, a model of a spherical pendulum with a movable suspension point is considered a 3D model for cargo movement by a crane, taking into account the elastic properties of the rope and the action of an external wind load [21][22][23][24][25][26][27][28][29][30][31][32][33].…”

Solution of mass-spring-damper fractional systems using Caputo derivative and orthogonal collocation

“…Parovik (2020) realized a numerical analysis considering fractional linear oscillators using the Finite Differences Method and Gerasimov and Caputo derivatives. The nonlinear response of an auto parametric system of three degrees of freedom represented by a spherical pendulum and a damper of fractional type is investigated by Freundlich and Sado (2020).…”