In Theory

Wheel diameter and performance: part 3

A lady in old-fashioned clothing next to a penny-farthing

In this installment of the series we take a look at gyroscopic effects caused by wheel rotation.

The influence of a rotating wheel on a two-wheeled vehicle and its dynamics is not a straightforward one and is a subject of many myths and urban legends in the cycling community. If you’ve been into cycling long enough, you’ve probably heard the phrase ‘rotating mass’ more than once, and with any luck you’ve also seen a video or two about it.

But the effect of wheel mass on acceleration is just barely scratching the surface of the problem.

The wheel gyroscope

Our observation of the universe has led us to believe that the space we live in is isotropic. In other words, its properties are the same regardless of which direction you look and no direction is singled out as a preferred one. As a result, the angular momentum of a rotating body must be conserved if no external moments are applied to it, as proved by one Emmy Noether in 1915. Furthermore, the greater the angular momentum, the greater the moment required to achieve the same effect.

The tendency of a rotating body to remain in the same orientation in space is called the gyroscopic effect and can be quite counterintuitive, hence it’s often presented during lectures on classical mechanics. And if you imagine a bike in motion, there are actually two such bodies – the front and the back wheel.

Every time a rider tries to twist the handlebar or lean the bike, they will have to overcome the resulting gyroscopic effect. It’s probably everyone’s intuition that smaller and lighter wheels will be easier to steer, but by how much exactly?

Euler equations

Body-fixed coordinate system (unprimed) and inertial coordinate system (primed). The x axis of the body-fixed system is chosen such that it is parallel to the ground and pointing forward when the bike is upright. The y axis point vertically up, and the z axis coincides with the axis of rotation of the wheel.

To estimate the torques necessary to steer the bike we will utilize the so-called Euler equations. They relate moments of forces acting on a rigid body with its angular velocity in a frame of reference fixed to the body – see the picture above. If you got lost reading that, don’t worry – it’s a topic for a post on its own. What’s more important is that if we choose the reference frame to coincide with the principal axes of the wheel as shown in the figure below, the Euler equations have the following form:

(1)   \begin{align*}I_x \dot{\omega}_x+(I_z - I_y)\omega_y \omega_z &= M_x \\I_y \dot{\omega}_y+(I_x - I_z)\omega_z \omega_x &= M_y \\I_z \dot{\omega}_z+(I_y - I_x)\omega_x \omega_y &= M_z\end{align*}

where I_k denote moments of inertia, \omega_k are components of the angular velocity, and M_k are torques along respective axes. The index k can naturally be either x, y, or z.

The model

Theory is fun (at least for deviants like you and me), but at the end of the day, we need some numerical values to plug into the equations. Unfortunately, finding real-world data on bike wheel inertia proved to be an almost impossible task, and I really don’t want to collect it myself. We could, however, estimate it by reducing the wheel to an assembly of simple geometric shapes plus a hub. We can easily calculate inertia for the former, while a CAD package will provide the inertia of the hub1.

The tables below list all the major components of a bike wheel with their masses and moments of inertia along all three axes. Both sizes have the same hub and the same disc rotor (Shimano XTR RT-MT900, 180 mm diameter), as those are wheel-size independent. The rim is in both cases a DT Swiis XRC 320, the spokes are DT Swiss Competition 2.0/1.8, and the tires are Schwalbe Rocket Ron in Super Race trim with Addix compound. The sealant masses were taken as recommended by Muc Off.

disc rotor1080.0002190.000437
Mass and moment of inertia contributions for a 26-inch wheel. Mass in grams, moments of inertia in kg·m².
disc rotor1080.0002190.000437
Mass and moment of inertia contributions for a 29-inch wheel. Mass in grams, moments of inertia in kg·m².

As you can see, the biggest contribution to the total moment of inertia is due to the tire, the rim, and the sealant: around 96% of the total in both cases. On the other hand, the hub’s contribution is minimal as its mass is low and concentrated near the rotation axis2.

Case 1: turning the handlebar

Let’s begin with a very simple case of turning the handlebars with rotational speed \omega while the front wheel is spinning at a constant speed and the frame is not moving sideways. This is similar to the moment a turn is initiated. Under our assumptions, \omega_x=-\omega \sin{\alpha}, \omega_y=\omega \cos{\alpha}, and \omega_z=v/r where \alpha is the steerer tuba angle, v is the bike speed, and r the wheel radius. Furthermore, as for a wheel I_x=I_y, the Euler equations reduce to

(2)   \begin{align*}-I_x \dot{\omega} \sin{\alpha}+(I_z-I_y) \omega \omega_z \cos{\alpha}&= M_x \\I_y \dot{\omega} \cos{\alpha}-(I_x-I_z) \omega \omega_z \sin{\alpha} &= M_y \\0 &= M_z\end{align*}

It might come as a surprise that there is not one, but two non-vanishing moments – along the x and y axes. That means turning the bars causes a moment trying to flip the bike around the x axis. How big is that moment? Assuming that the bike is travelling at 8 m/s, the handlebar turning speed is such that the full turn takes two seconds and is achieved within 0.2 seconds, the peak moment of forces along the x axis is equal to 1.2 N·m for 26-inch and 1.52 N·m for 29-inch wheels. The absolute difference is not very big, but still noticeable.

But how about the moment on the handlebars? Here the calculation is slightly more ivolved, as we need to transform from the coordinate system fixed to the wheel to the one fixed to the handlebars, which includes a translation as well as a rotation3. However, the final result reads 4.97 N·m for smaller and 6.02 N·m for bigger wheels, assuming the bike has geometry of a Trek Procaliber 9.6 and the same values for handlebar turning and acceleration as in the previous paragraph. As you can see, the difference is quite substantial.

Case 2: steady turning

If you ride a motorcycle, especially on track, you might have noticed that torque must be applied to the handlebar during turning. There are several causes for this, one of which is due to gyroscopic effects. Moreover, another moment develops when that tries to flip the bike towards the inside of the curve. A full derivation is quite lengthy, and perhaps better saved for a separate post. The end result, however, depends on angular velocity in the corner and lean angle. In general, the higher the angular velocity and the smaller the lean angle, the bigger the resulting flipping torque.

So how big is this torque? If we were to make a right-angle turn in one second at 8 m/s and a modest lean angle of 18 degrees, we’d get 4.22 N·m for 26-inch and 5.57 N·m for 29-inch wheels. However, if we managed to negotiate this corner in only 0.5 seconds at 10 m/s and a higher lean angle of 30 degrees, it would increase to 9.3 and 12.2 N·m, respectively.

But what about handlebar moments? Well, in the slower case we’d get 0.65 N·m for smaller, and 0.86 for bigger wheels. The faster case increases those to 0.74 and 0.97 N·m, respectively.

Case 3: unsteady turning

We finally arrived at the most complicated case one of which baffled me when I started riding a motorcycle. You see, when you start leaning into the corner, the front wheel has a tendency to turn in the same direction. If you start leaning right, it will want to turn clockwise. If you start leaning left, the handlebar rotation will be anti-clockwise. And the reason? Euler equations.

The magnitude of the effect depends on the lean angle, roll and yaw rates, and their respective time derivatives. The mathematical formulas are rather long, but in general, the moments grow higher with increasing rotation rates and, up to a degree, lean angle. However, it’s difficult to give any absolute estimate without real-world data. What we could do, however, is compare relative torque values for the same cornering parameters. The 26-inch wheel performs significantly better than the 29-inch one: the torque along the x axis is about 30% smaller, while the difference around the y axis is even bigger, between 30 and 50%.

Are bigger wheels worth it?

We’ve seen that 29-inch wheels come with a natural penalty of higher gyroscopic effects compared to their smaller, 26-inch counterparts. The difference may not be huge, but is still perceptible. The higher torque necessary to initiate and sustain a turn contributes to faster fatigue. Higher torques also result in higher forces between the tire and the ground. All that is very unlikely to be felt by a weekend warrior rider, but in competition on courses requiring frequent cornering, it might make a difference between winning and losing.

  1. If at this point you scratch your head asking yourself why couldn’t we do the same for the entire wheel, the answer is simple: yes, we could. But I didn’t have a CAD model of an entire wheel and was too lazy to actually prepare one myself. ↩︎
  2. I must admit that the moment of inertia of the hub seems to be a bit too low. I’d expect a value of the order 10⁻⁵, but that still would be four order of magnitude smaller than the total moment of inertia of the wheel. ↩︎
  3. Let’s save the derivation of that for a different occasion. ↩︎

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