### Wonky Wheels on Wacky Roads

Tricycles with three square wheels, each a different size. Image from the Museum of Mathematics, New York.

Imagine trying to cycle along a road with a wavy surface. Could anything be done to minimise the ups-and-downs? In general, this would be very difficult, but in ideal cases a simple solution might be possible.

Elliptic Wheels

We suppose that the road runs along the ${x}$-axis, with its height varying like a sine wave

$\displaystyle z = h \cos kx \,,$

where ${h}$ is the amplitude of the wave and ${k=2\pi/L}$ is the wavenumber with ${L}$ the wavelength.

Elliptical wheel rolling along a sine curve. The centre height is equal at the crest ${x=0}$ and the trough ${x=L/2}$.

If the bicycle wheels are circular, their centres will rise and fall with the road, giving the rider an uncomfortable journey. So, we make the wheels elliptical. In standard position, the equation for an ellipse is

$\displaystyle \frac{x^2}{a^2} + \frac{z^2}{b^2} = 1$

We want to choose the shape parameters ${a}$ and ${b}$ to satisfy the following condition:

• {The centre of the ellipse remains at a constant height as the wheel rolls along.}

Let us calculate the distance along the road surface over a full wavelength. The arclength is ${\mathrm{d}s^2 = \mathrm{d}x^2 + \mathrm{d}z^2 = (1+z^{\prime 2}) \mathrm{d}x^2}$ where ${z^\prime = \mathrm{d}z/\mathrm{d}x}$. Since ${z^\prime = - kh\sin kx}$, this means that the road distance over a span of length ${L}$ is

$\displaystyle S = \int_0^L \sqrt{ 1+(kh)^2 \sin^2 kx }\,\mathrm{d}x$

Assuming that ${h\ll L}$ so that ${\varepsilon = kh \ll 1}$ we can approximate this by the first two terms in an ${\varepsilon}$-expansion:

$\displaystyle S \approx [ 1 + \textstyle{\frac{1}{4}}\varepsilon^2 ] L$

We require the wheel to make one revolution for every wavelength. Thus, the circumference must be equal to ${S}$. We use the approximate formula ${C = \pi(a+b)}$, which has good accuracy for elliptical eccentricity ${e \lesssim \frac{2}{3}}$ [TM1]. Then we have

$\displaystyle \pi (a + b) \approx S \ \ \ \ \ (1)$

We assume that, when the centre of the ellipse is above a crest in the road, ${kx\equiv 0\, (\mbox{mod}\,2\pi)}$, the long axis is horizontal and the centre is at height ${z=b + h}$. When the centre is above a trough, ${kx\equiv \pi\, (\mbox{mod}\,2\pi)}$, the long axis is vertical and the centre is at height ${z=a - h}$. For these heights to be equal, we require

$\displaystyle a - b = 2h \ \ \ \ \ (2)$

Eqns. (1) and (2) allow us to compute ${a}$ and ${b}$ in terms of ${h}$ and ${\varepsilon}$.

The parameter values for the example shown in the above Figure are ${L=2\pi}$, ${k=1}$, ${h=0.2}$ and ${\varepsilon=0.2}$. The road distance over length ${L}$ is ${S=6.34555}$ and the second-order approximation is ${S=6.34602}$. The values for the semi-axes of the ellipse are ${a=1.21}$, ${b=0.81}$ and the eccentricity is ${e=0.743}$.

There is no guarantee that the axle-height of an ellipse rolling on a sine wave remains constant; indeed, we should expect some variation but, for the chosen parameters, it should be small. An exact analysis would reveal the precise variation of height over a wavelength. No doubt, it would involve the Jacobian elliptic functions, “sun”, “cun” and “dun” [TM2].

Square Wheels

If you think elliptical wheels are strange, take a look at the figure at the head of this article. it is from from MoMath, The Museum of Mathematics in New York, where visitors can take a smooth ride on a tricycle with square wheels [SWT].

The bumps in the cleverly-designed track, made using inverted catenaries, ensure that the rider follows a more-or-less horizontal path. The three wheels of the tricycle have different sizes, as they vary with the distance from the centre of the track. The rear wheels are connected through a gearbox, that compensates for the different sizes of the wheels and keeps them in sync. Ingenious!

Square wheel on a road of circular arcs. Left: starting position at trough. Right: wheel at trough, crest and following trough.

We did a quick analysis of square wheels on a surface comprising a sequence of circular quadrants; see Figure above. The side length of the square wheels is equal to the arc-length of the quadrants, so that the wheel turns through a right angle as the contact point moves from one groove to the next. The values of the radius and arc-length were ${r=1}$ and ${a= \pi r/2}$. The minimum height of the axle, when the square is at a crest, is ${(1+\pi/4)r}$. The maximum height, with the corner in a groove, is ${(1+\pi/2)r/\sqrt{2}}$. For the chosen parameters, the minimum axle-height was ${1.785}$ and the maximum height was ${1.818}$, a ${2\%}$ difference. The slight difference, visible in the  Figure, would probably not cause too much discomfort at low speed.

Stan Wagon, in his book Mathematica in Action [WAG], discusses how the centre of a square wheel will remain at a constant height on a road where the circular quadrants are replaced by segments of a catenary (${z = z_0 -h\cosh kx}$). The Figure below [from WAG] shows the trajectory of a corner point of the wheel and the horizontal line of the centre. The photo shows the author riding the square-wheeled trike. For further details, see Wagon (2010).

Trajectory of a point on rim of square wheeled trike on a road of catenary arcs.

Trajectory of a point on rim of square wheeled trike on a road of catenary arcs.

Sources

${\bullet}$ [TM1] ThatsMaths post, July 8, 2021: Approximating the Circumference of an Ellipse. URL

${\bullet}$ [TM2] ThatsMaths post, November 14, 2019: Elliptic Trigonometry: Fun with “sun” “cun” and “dun”.  URL

${\bullet}$ [SWT] Mathcom Wiki: Square Wheeled Tricycle: URL

${\bullet}$ [WAG] Wagon, Stan, 2010: Mathematica in Action. Springer, 578pp. ISBN: 978-0-3877-5366-9.