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Bifurcation analysis of the problem of a "rubber" ellipsoid of revolution rolling on a plane

Alexander Kilin, Elena Pivovarova

TL;DR

This work analyzes the rubber rolling (no-slip, no-spin) of a body of revolution, specifically an ellipsoid, on a horizontal plane. By exploiting four integrals of motion ($\mathcal{E}$, $\mathcal{F}_0$, $\mathcal{F}_1$, $\mathcal{F}_2$) and reducing to a fixed level set, the authors reduce the dynamics to one degree of freedom and derive explicit reduced equations for the ellipsoid. They construct complete bifurcation diagrams in the first-integral plane, classify permanent rotations, and map the resulting phase portraits, revealing conditions for circular, pendular, bounded, and unbounded motions. The results yield a detailed qualitative picture of trajectories, including resonance phenomena and stability boundaries, with potential applications to robotic systems using rubber-rolling models. Overall, the paper provides a thorough, integrable-framework-based taxonomy of the ellipsoid’s rolling dynamics on a plane under rubber constraints.

Abstract

This paper is concerned with the problem of an ellipsoid of revolution rolling on a horizontal plane under the assumption that there is no slipping at the point of contact and no spinning about the vertical. A reduction of the equations of motion to a fixed level set of first integrals is performed. Permanent rotations corresponding to the rolling of an ellipsoid in a circle or in a straight line are found. A linear stability analysis of permanent rotations is carried out. A complete classification of possible trajectories of the reduced system is performed using a bifurcation analysis. A classification of the trajectories of the center of mass of the ellipsoid depending on parameter values and initial conditions is performed.

Bifurcation analysis of the problem of a "rubber" ellipsoid of revolution rolling on a plane

TL;DR

This work analyzes the rubber rolling (no-slip, no-spin) of a body of revolution, specifically an ellipsoid, on a horizontal plane. By exploiting four integrals of motion (, , , ) and reducing to a fixed level set, the authors reduce the dynamics to one degree of freedom and derive explicit reduced equations for the ellipsoid. They construct complete bifurcation diagrams in the first-integral plane, classify permanent rotations, and map the resulting phase portraits, revealing conditions for circular, pendular, bounded, and unbounded motions. The results yield a detailed qualitative picture of trajectories, including resonance phenomena and stability boundaries, with potential applications to robotic systems using rubber-rolling models. Overall, the paper provides a thorough, integrable-framework-based taxonomy of the ellipsoid’s rolling dynamics on a plane under rubber constraints.

Abstract

This paper is concerned with the problem of an ellipsoid of revolution rolling on a horizontal plane under the assumption that there is no slipping at the point of contact and no spinning about the vertical. A reduction of the equations of motion to a fixed level set of first integrals is performed. Permanent rotations corresponding to the rolling of an ellipsoid in a circle or in a straight line are found. A linear stability analysis of permanent rotations is carried out. A complete classification of possible trajectories of the reduced system is performed using a bifurcation analysis. A classification of the trajectories of the center of mass of the ellipsoid depending on parameter values and initial conditions is performed.
Paper Structure (23 sections, 5 theorems, 69 equations, 14 figures)

This paper contains 23 sections, 5 theorems, 69 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Equations of motion for a body of revolution on a plane
  3. Configuration space and kinematic relations
  4. Constraint equations and dynamical equations
  5. Invariants and symmetries
  6. Reduction to a system with one degree of freedom
  7. Explicit form of the equations of motion for an ellipsoid of revolution
  8. Bifurcation analysis
  9. General definitions
  10. Critical sets
  11. The critical subset $S_1$
  12. The critical subset $S_0$
  13. Example of a bifurcation diagram
  14. Bifurcations of critical sets and types of bifurcation diagrams
  15. Bifurcations of the critical subset $S_0$
  16. ...and 8 more sections

Key Result

Proposition 1

In the case $\alpha\neq0$ the critical subset $S_1$ of the integral map $\Phi$ has the form where the dependence $\boldsymbol\gamma(\vartheta,\varphi)$ is defined by the expressions newvars, $\omega_0(\vartheta)$ has the form $\varphi\in[0,2\pi)$, and the range of angle $\vartheta$ is determined from the condition that the expression usl2 be nonnegative for $\omega_0^2$.

Figures (14)

  • Figure 1: A body of revolution on a plane.
  • Figure 2: (a) Diagrammatic representation of an ellipsoid on a plane. The plane $\rm\Sigma$ is a vertical plane passing through the symmetry axis of the ellipsoid. The red elliptic curve is the intersection of the ellipsoid and the plane $\rm\Sigma$. (b) Representation of the ellipsoid on the plane in a meridional section.
  • Figure 3: Example of permanent rotations of an ellipsoid
  • Figure 4: Regions of positive definiteness of the expression \ref{['usl2']} on the parameter plane $(\beta^2,\vartheta)$ for a fixed value of $\alpha$.
  • Figure 5: Example of a bifurcation diagram. This type of diagram is characteristic of an "oblate" ellipsoid with a fairly small displacement of the center of mass $\alpha$.
  • ...and 9 more figures

Theorems & Definitions (16)

  • Remark
  • Remark
  • Remark
  • Remark
  • Remark
  • Remark
  • Proposition
  • Proposition
  • proof
  • Remark
  • ...and 6 more