Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gears and pulleys in non-Euclidean space forms

Heleno S. Cunha, Lucas H. R. de Souza, Sergio A. P. Prado

Abstract

In this article we study, in their non-Euclidean versions, two important mechanical systems that are very common in numerous devices. More precisely, we study the laws governing the movement of pulley and gear systems in spherical and hyperbolic geometries. And curiously, we were able to see an interesting similarity between the determined laws.

Gears and pulleys in non-Euclidean space forms

Abstract

In this article we study, in their non-Euclidean versions, two important mechanical systems that are very common in numerous devices. More precisely, we study the laws governing the movement of pulley and gear systems in spherical and hyperbolic geometries. And curiously, we were able to see an interesting similarity between the determined laws.
Paper Structure (12 sections, 12 theorems, 32 equations, 5 figures)

This paper contains 12 sections, 12 theorems, 32 equations, 5 figures.

Table of Contents

  1. Transforming the physical problems into geometric problems
  2. Gears
  3. Pulleys
  4. The hyperbolic plane
  5. The Poincaré-disk model
  6. Gears at the Poincaré-disk
  7. Pulleys at the Poincaré-disk
  8. The hyperboloid model
  9. Pulleys at the hyperboloid model
  10. The sphere
  11. Gears at the sphere
  12. Pulleys at the sphere

Key Result

Proposition 1.1

Let $C$ be a gear with center $p \in C$ and $\sigma: \{0,...,s\} \rightarrow \mathbb{Z}$ a map. Then there exist a unique rotation movement such that its associated winding map with respect to $p$ is $\sigma$.

Figures (5)

  • Figure 1: A system of two gears in the Euclidean space attached to each other.
  • Figure 2: A system of two pulleys in the Euclidean space attached by a tensioned belt.
  • Figure 3: The first image is a representation of two gears in the Poincaré-disk model of the hyperbolic plane. The representation of the gears' teeth is merely illustrative, as considered in Section 1.1. The second image is a representation of two pulleys in the Poincaré-disk model of the hyperbolic plane attached by a tensioned belt that satisfies the conditions of the theorem above. The belt is represented here by two geodesics that are tangent to both circles and it is merely illustrative.
  • Figure 4: The first image is a representation of two gears in the sphere. The representation of the gears' teeth is merely illustrative, as considered in Section 1.1. The second image is a representation of two pulleys in the sphere attached by a tensioned belt that satisfies the conditions of the theorem above. The belt is represented here by two geodesics that are tangent to both circles and it is merely illustrative.
  • Figure 5: The second figure represents the plane that contains all triangles of the first one

Theorems & Definitions (26)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • proof
  • Theorem 1.5
  • proof
  • ...and 16 more