Nondeterministic dynamics of a mechanical system

A mechanical system is presented exhibiting a nondeterministic singularity, that is, a point in an otherwise deterministic system where forward time trajectories become nonunique. A Coulomb friction force applies linear and angular forces to a wheel mounted on a turntable. In certain configurations, the friction force is not uniquely determined. When the dynamics evolves past the singularity and the mechanism slips, the future state becomes uncertain up to a set of possible values. For certain parameters, the system repeatedly returns to the singularity, giving recurrent yet unpredictable behavior that constitutes nondeterministic chaotic dynamics. The robustness of the phenomenon is such that we expect it to persist with more sophisticated friction models, manifesting as extreme sensitivity to initial conditions, and complex global dynamics attributable to a local loss of determinism in the limit of discontinuous friction.

Figure 1

Mechanical model of the system shown from the top (a) and from the side (b), with a conceptual sketch (c). The bottom disk is rotating with a constant angular speed ${\omega }_0$. The top disk rotates around the same center pin, but it is connected to the bottom disk through a viscous friction coefficient $c_1$. A slider is fitted in the top disk, forming a single degree of freedom oscillator. The slider contains a wheel mounted at an angle $\gamma$ to the slider direction. The wheel is in contact with the bottom disk and Coulomb friction acts between them.

Figure 2

Typical dynamics as the switching surface: crossing, sliding, and escaping. The limit of Eqs. (1) and (2) as $(r,v,\omega )$ approaches $\Sigma$ are labeled as ${\mathbf{f}}_{\pm }={lim}_{h\to 0\pm }(\dot{r},\dot{v},\dot{\omega })$. The vector field ${\mathbf{f}}_s$ that governs sticking will be defined in Sec. 4.

Figure 3

Nondeterminism at a Teixeira singularity. (a) Slipping (single-headed arrows) follows $\mathbf{f}(\mathbf{x},+\mu )$ or $\mathbf{f}(\mathbf{x},-\mu )$ either side of the switching surface. Sticking (double arrows) follows $(\dot{\overline{\xi }},\dot{\overline{\eta }})$ in the sliding and escaping regions (shaded). In the case shown, the sticking flow passes through the singularity nondeterministically. (b) Sketch of nondeterministic chaos in the mechanical system. The small shaded “bowtie” shows the local region of the singularity (shown left). A set of trajectories loop around via a sequence of slipping (white bands) with switches, connecting to the singularity via sticking (dark band) in both forward and backward time.

Figure 4

The shaded region in panel (a) represents parameter values where the system has a Teixeira singularity (case 1 in Sec. 4). This is the overlap of several conditions: (b) ${\mathcal{K}}^{--}>0$, (c) ${\mathcal{K}}^{++}<0$, (d) ${\mathcal{J}}_1<0$, (e) ${\mathcal{J}}_2<0$, (f) ${\mathcal{J}}_1{\mathcal{J}}_2-1>0$, and in (g) showing both $k_2>0$ (shaded) and $\kappa >0$ (hatched).

Figure 5

Nondeterministic chaos in the wheel-and-disk assembly for parameter values given in the text. (a) A sketch of the sticking flow, showing sliding regions (S), escaping region (E), and crossing regions (C). A Teixeira singularity is seen at $(r,\omega )=(0.1859,-1.037)$, with the sticking flow passing through it along a weak eigenvector. The three dimensional blocks (b) and (c) show a simulation of different trajectories with initial conditions near the escaping region [regions on $v=0$ correspond to those in panel (a)]. These represent the different trajectories followed after passing through the singularity. The trajectories wind around the escaping region causing the dense shell seen in $r<0.1859$.

Figure 6

Free-body diagrams of the mechanical system.

Figure 7

Contact forces on a rolling wheel. The ground surface is moving with relative velocity $v$ under the wheel, whose lateral deformation is described by $q(x,t)$ within the contact patch spreading from $x=-a$ to $a$.

Figure 8

Plot of the friction force (dashed red lines) and moment (solid blue lines) as a function of slip angle $\phi$. Parameters for panel (a) are $k=1$, $\delta =1$, $\sigma =1$, and $a=1$, and for panel (b) are $k={10}^3$, $\delta ={10}^{-3}$, $\sigma ={10}^{-3}$, and $a=1$.