The dynamical motion of a rolling cylinder and its stability analysis: analytical and numerical investigation

Abstract

The present paper addresses the dynamical motion of two degrees-of-freedom (DOF) auto-parametric system consisting of a connected rolling cylinder with a damped spring. This motion has been considered under the action of an excitation force. Lagrange's equations from second kind are utilized to obtain the governing system of motion. The uniform approximate solutions of this system are acquired up to higher order of approximation using the technique of multiple scales in view of the abolition of emerging secular terms. All resonance cases are characterized, and the primary and internal resonances are examined simultaneously to set up the corresponding modulation equations and the solvability conditions. The time histories of the amplitudes, modified phases, and the obtained solutions are graphed to illustrate the system's motion at any given time. The nonlinear stability approach of Routh–Hurwitz is used to examine the stability of the system, and the different zones of stability and instability are drawn and discussed. The characteristics of the nonlinear amplitude for the modulation equations are investigated and described, as well as their stabilities. The gained results can be considered novel and original, where the methodology was applied to a specific dynamical system.

Dynamical Stability of a 3-DOF Auto-Parametric Vibrating System

“…Over the past few periods, a large number of publications addressing different types of dynamic dampers and associated subjects drew the attention of numerous researchers e.g. [4][5][6][7][8][9]. In [4], the author investigated the vibrational motion of an absorber with a rotating base to reduce the vertical excitations, from which the fundamental frequency has been derived.…”

“…In [7], the authors replaced the pendulum rigid arm with a spring pendulum with linear stiffness to constitute a 3DOF model and generalize the work in [6]. Recently, the stability of 2DOF auto-parametric systems has been analyzed in [8] and [9] for two dynamical models. The first one is the vertical movement of a spinning cylinder on a connected circular surface with a nonlinear damped spring, while the horizontal frictional motion of this surface was considered in [9].…”

“…Recently, the stability of 2DOF auto-parametric systems has been analyzed in [8] and [9] for two dynamical models. The first one is the vertical movement of a spinning cylinder on a connected circular surface with a nonlinear damped spring, while the horizontal frictional motion of this surface was considered in [9]. The analytic estimated solutions of these models up to desired orders of calculation were obtained using the approach of multiple scales [10], while the numerical solutions of the system in [9] were attained by applying the Runge-Kutta approach of fourth order [11].…”