Imperfections, impacts, and the singularity of Euler's disk

The motion of a rigid, spinning disk on a flat surface ends with a dissipation-induced finite-time singularity. The problem of finding the dominant energy absorption mechanism during the last phase of the motion generated a lively debate during the past two decades. Various candidates including air drag and different types of friction have been considered, nevertheless impacts have not been examined until now. We investigate the effect of impacts caused by geometric imperfections of the disk and of the underlying flat surface, through analyzing the dynamics of polygonal disks with unilateral point contacts. Similarly to earlier works, we determine the rate of energy absorption under the assumption of a regular pattern of motion analogous to precession-free motion of a rolling disk. In addition, we demonstrate that the asymptotic stability of this motion depends on parameters of the impact model. In the case of instability, the emerging irregular motion is investigated numerically. We conclude that there exists a range of model parameters (small radii of gyration or small restitution coefficients) in which absorption by impacts dominates all previously investigated mechanisms during the last phase of motion. Nevertheless the parameter values associated with a homogeneous disk on a hard surface are typically not in this range, hence the effect of impacts is in that case not dominant.

Figure 1

Values of $\beta$ depending on the physical properties of the polygonal model. Left: $n=3$, right: $n=25$.

Figure 2

Border of feasible and infeasible regions for self similar motion with $\gamma =0$. Notice that homogeneous disk corresponds to $\rho \approx 1/2$, in which case the non-trivial self-similar solution is feasible for all $n>3$.

Figure 3

Trival (stars), and feasible (full circle) or infeasible (empty circle) non-trivial solutions to the equations of self-similar motion. Parameters from top left to bottom right: $n=3$ and $\rho =0.25$, $n=3$ and $\rho =0.5$, $n=25$ and $\rho =0.25$, $n=25$ and $\rho =0.5$.

Figure 4

Feasibility and stability of self-similar motion. At least one non-trivial solution exists for all parameter values. One feasible nontrivial solution exists above the dotted curves (each curve corresponds to a specific value of $n$), and the feasible solution is stable for parameter values on the left of the dashed lines. Trivial solutions exist for parameter values below the solid lines. Notice that there is a dotted curve right next to each solid curve.

Figure 5

Log-log plot of energy versus time remaining to singularity obtained by numerical simulation of motion for various values of $n$ and $\gamma$. The parameters were, from top left to bottom right: $n=20$ $\gamma =0.6$, $n=20$ $\gamma =0.7$, $n=25$ $\gamma =0.5$, $n=25$ $\gamma =0.6$, $n=30$ $\gamma =0.5$, and $n=30$ $\gamma =0.6$.

Figure 6

Sketch of the theoretical model accounting for irregularity of the support surface.

Figure 7

Exponents of the falling rod when gravity is negligible.

Figure 8

Numerically obtained exponents in the case of $n=3$. Lack of values means simultaneous impacts. The curves labeled by CC and PCC are the boundaries of parameter regimes with the CC and PCC properties.

Figure 9

Sections of Fig. 8 (dots) with predictions of Eq. (75) (dashed curves).