Bifurcations and synchronization of coupled translational-rotational stick-slip oscillators

TL;DR

This work analyzes two-dimensional stick-slip oscillations of translational-rotational disks on moving belts with finite-area friction under non-ideal energy input. Using Contensou–Zhuravlev Padé-based smooth friction models, the authors derive a dimensionless, coupled set of equations for a single oscillator and for chains driven by a common DC motor, enabling efficient bifurcation and synchronization analysis. The study reveals that stiffness asymmetry activates rotational-translational coupling, producing amplitude jumps and period-adding cascades up to period-28, while a DC-motor drive lowers bifurcation thresholds and smooths stick-slip transitions. For multi-oscillator chains, the work demonstrates robust synchronization in symmetric configurations and rich desynchronization and period-doubling phenomena in asymmetric ones, highlighting practical implications for clutches, brakes, and precision actuators.

Abstract

This paper presents mathematical modeling and numerical analysis of bifurcation and synchronization phenomena in a system of coupled oscillators driven by a finite-power energy source and generating two-dimensional stick-slip translational-rotational vibrations. The mechanical system consists of rigid disks placed on moving belts and asymmetrically connected to a support via springs, with each disk simultaneously performing translational and rotational motion. The belts are driven by a common DC motor. The disk contacts the belt over a finite contact area, resulting in mutually coupled frictional force and torque through their dependence on the linear and angular slip velocity. The paper utilizes special approximations of the resultant frictional forces and torques based on generalizations of Padé's developements, in which special smoothing elements are introduced to avoid singularities when the relative motion between the disk and belt disappears. Furthermore, the model also provides a smooth approximation of the friction model, in which static friction is greater than kinetic friction. Numerical analysis was conducted based on direct numerical simulations, Poincare maps, and bifurcation diagrams. It was shown that the system can exhibit a two-dimensional stick-slip phenomenon, and the system exhibits predominantly periodic dynamics, sometimes with a very long period. In this work, a system of two oscillators driven by a common, limited-power motor was studied, which can synchronize and oscillate in phase or in counterphase.

Figures (18)

• Figure 1: 2DOF system consisting of a single plane disk situated elastically on a belt moving with constant velocity op2.
• Figure 2: Bifurcation diagrams of the Poincaré map defined by the local minima of $x$, as the friction coefficient ratio ($\eta$) is swept as the control parameter: (a) displacement $x$; (b) rotational coordinate $\varphi$; For each $\eta$, integration is restarted from $(x_0,\,\dot{x}_0,\,\varphi_0,\,\dot{\varphi}_0) = (0,\,0,\,0,\,0)$; Parameters follow Eq. \ref{['eq:Literature parameter values']}.
• Figure 3: Time history (a), phase plots (b--d), and corresponding Poincaré points at $\eta = 2.7$; Integration is performed from the initial condition $(x_{0}, \dot{x}_{0}, \varphi_{0}, \dot{\varphi}_{0}) = (0,0,0,0)$; Other parameters follow Eq. \ref{['eq:Literature parameter values']}.
• Figure 4: Bifurcation diagram of the Poincaré map defined by the local minima of $x$, as the friction coefficient ratio ($\eta$) is swept as the control parameter for the symmetric spring configuration ($\hat{k}_1 = \hat{k}_2 = 5000$ and thus $k_{12} = 0$); For each $\eta$, integration is restarted from $(x_0,\,\dot{x}_0,\,\varphi_0,\,\dot{\varphi}_0) = (0,\,0,\,0,\,0)$; Other parameters follow Eq. \ref{['eq:nondimensional parameter values']}.
• Figure 5: Time history (a) and phase plot (b) with the corresponding Poincaré point at $\eta = 9.95$, for the symmetric spring configuration ($\hat{k}_1 = \hat{k}_2 = 5000$ N/m, so $k_{12} = 0$); Integration is started from $(x_{0},\,\dot{x}_{0},\,\varphi_{0},\,\dot{\varphi}_{0}) = (0,\,0,\,0,\,0)$; Other parameters follow Eq. \ref{['eq:nondimensional parameter values']}.