Key words: triangle wheel,
If a wheel can be square and run along a specially built track can a triangular wheel do the same?
Is it even possible?
An equilateral triangle wheel
Placing the wheel on the track
and using the SINE rule as for the square wheel
$$ \small \frac{l}{\sin(\frac{\pi}{6})}=\frac{ ^{2 \sqrt{3}} \ / _3 }{\sin(\frac{\pi}{2}+\theta)}=\frac{ ^{2 \sqrt{3}} \ / _3}{\cos(\theta)}$$
gives
$$ \small l =\frac{2 \sqrt{3}}{3} \frac{ \sin\frac{\pi}{6}} {\cos \theta}= \frac{\sqrt{3}}{3}\sec \theta \Rightarrow \small y = \frac{ 2\sqrt{3}}{3} – \frac{\sqrt{3}}{3}\sec \theta $$
A differential equation for \( \small x \) in terms of \( \small \theta \)
$$ \small \frac {dy}{dx} = \frac{dy / d\theta}{dx / d\theta} \ \Rightarrow \ \tan \theta = \ – \frac{\sqrt{3}}{3} \frac{\tan \theta \sec \theta}{dx / d\theta} \ \Rightarrow \ \frac{dx}{d \theta} = \ – \frac{\sqrt{3}}{3}\sec \theta $$
and solving for \( \small x \) gives (the constant of integration is 0 again)
$$ \small x = \ – \frac{\sqrt{3}}{3}\ln ( \sec {\theta} + \tan {\theta} ) $$
with a bit or rearranging before taking the exponential function of both sides gives
$$ \small e^{- \sqrt{3}x} = \sec {\theta} + \tan {\theta} \ , \ e^{\sqrt{3}x} = \sec {\theta} \ – \tan {\theta} $$
so for the triangular wheel \( \small \ \sec \theta = \cosh (\sqrt{3}x) \)
and finally
$$ \small y = \frac{2 \sqrt{3}}{3} – \frac{\sqrt{3}}{3} \cosh (\sqrt{3}x)$$
The Track
Repeating the analysis for the square wheel but omitting the working gives
- Width of base: \( \small x = \frac{\ln (2 \pm \sqrt{3})}{\sqrt{3}} = \pm 0.76035 \)
- Length: L=2
- Height: \( \small y = \frac{\sqrt{3}}{3} \approx 0.57735\)
Putting a few segment together gives
But pause a moment. Can the wheel move along the track smoothly. While it can be shown it (the track segments) make 60° angles with the ground leaving a snug 60° for the corner of the wheel I used Excel to program the position of a corner of the wheel (blue) as it moves along the track (red)
I contend that the wheel works it way into the join between two segments and there it gets stuck and can only get out again with some force and hardly smoothly.
Anyone want to take the challenge and build a triangle wheel and appropriate track? Probably a waste of time and effort!
Ah, but what about other shapes? A pentagonal wheel or a hexagonal one or any kind of polygonal wheel? I’ll look into that NEXT
The ABC series …
1. Arches and Bridges
- Arches, Bridges & Chains
- All Aqueducts lead to Rome
- Arches & Bridges
- Suspension Bridges
2. Hanging Cables or Chains
- Hanging Cables
- Catenary Equations
- Power lines
3. Strange Tracks
- Square Wheels
- Triangular Wheels
- Polygonal Wheels