A deeper look at bicycle stability and maneuverability

Now that I've got some free time, I think that I should explain better the "A minimal bike geometry" post. In that post I said that dynamical behaviour of a bike can be defined using just 4 parameters ("modified" trail, chainstay length, wheelbase and BB offset). One might argue that such a definition of bike geometry is arbitrary but it steams from a deep analysis of steering and bike balance mechanics.

I haven't seen yet a complete enough analysis/explanation of the relation between bike geometry parameters, balance mechanics and rider input so this will be my aim in the following lines. As a first step, let's consider the typical 4 body (two wheels, frame and fork) singletrack vehicle model. Assuming RWS, this model has 3 DOF tipically defined as the rotation of the rear wheel, the roll angle of the frame and the steering angle.

At this moment, it's interesting to present the rider as both a controller (due to his capability to control the steering motion and displace his weight laterally to control the rolling motion) and a disturber of the balance (due to the pedalling loads). The controlling actions of the rider induce loads and moments in the wheels that tend to stabilize the whole system that has certain inertial characteristics.

Once this is clear, let's pass to analyze the two basic states of motion of the bike: rectilinear motion while pedaling and turning without pedaling. Motion while pedaling standing is more obvious but it's also present when pedaling seated.

Pedaling motion

1- Starting with the crankset horizontal and the left leg doing the downstroke movement, the pedaling loads generate a moment that makes the bike roll to the left. To compensate this, the rider applies higher vertical forces to the left side of the handlebar and steers to the right. Due to this steering moment and as the system tends to decrease its potential energy, a clockwise pitch motion that is only possible if the bike rolls to the right starts. As a result, the COG of the system displaces to the right and the gravity induces a moment contrary to the one generated by pedaling loads.

Loads:
- Roll motion. Encouraging desired motion: moment due to gravity and handlebar loads. Against desired motion: moment due to pedaling loads, gyroscopic moments, overturning moment in both wheels and centrifugal forces due to the spiral motion of the COG
- Steering motion. Encouraging desired motion: steering moment. Against desired motion. steering moment due to the product of lateral forces in the tire and trail (variable).

2- As the left leg approaches its lowest position, the roll angle reaches its highest value (to the right) and the steering angle is null. The pedaling loads become smaller and the rider starts to steer the bike to the left to reduce the rolling moment generated by gravity, causing a counterclockwise pitch motion that increases the height of the COG of the system. In this moment, steering and roll angles have different sign, something that reduces the leverage of the rolling moment due to gravity so the roll angle starts to decrease helped by handlebar loads. When the roll angle is 0º, the steering angle is maximum and the cycle starts again.

Loads:
- Roll motion. Encouraging desired motion: moment due to pedaling loads, overturning moment in both wheels and moment due to handlebar loads. Against desired motion: moment due to gravity, gyroscopic moments and centrifugal forces.
- Steering motion. Encouraging desired motion: steering moment. Against desired motion: steering moment due to the product of lateral forces in the tire and trail (variable).

The followings plot will help you to understand how it works. In the first one, cadence is constant and equal to 90rpm and roll and steering angles are assumed to be sinusoidal. As you can see, roll and steering angle are π/2 out of phase.

This one shows the relation between pitch and roll and steering angles for a typical bike. As you can see, roll and steering angle naturally tend to have the same value to minimize potential energy.

Turning without pedaling

In the case of turning without pedaling, the rider starts by countersteering to induce gyroscopic, steering (due to the lateral forces in the contact patch) and overturning moments that tries to turn the bike in the desired direction. When the rider has defined his desired trajectory and calculated the path radius needed, he modifies steering and roll angles in order to 1) ensure that the path of the wheels is the desired 2) turn in steady state conditions (moments due to gravity equal to moments due to centrifugal forces). Once the bike approaches the exit of the curve, the rider countersteers again inducing a counterclockwise pitching motion that increase the height of the COG. Thanks to the same mechanism previously explained in the second part of the motion while pedaling, the roll angle tends to zero and the bike return to the vertical position

Once we have analyzed the way the bike is controlled, we can define the parameters that define the manouverability and stability of a bike. Leaving the influence of the components aside, these are the following ones:

1) The vertical position of the COG of the system. This parameter modifies the rolling moment generated by the gravity for a given roll angle. It's just a function of the BB offset for a given rider.

2) The weight distribution between both wheels. Steering moment is proportional to the product between front wheel loads and "modified" trail. Weight distribution is a function of chainstay length and wheelbase for a given rider.

3) Head tube angle (HTA). The axis of precession of the wheel as a gyroscope is the axis of the steering column so HTA has an effect on gyroscopic moments.
4) How camber of the front wheel changes when roll and steering angles are modified. Wheel loads, specially overturning moment, depend on camber.

5) The relation between path radius and roll and steering angles. In short, how much you need to steer and make the bike roll for a given radius of a curve.
6) The relation between "modified" trail and roll and steering angles. This the leverage of steering moments due to loads in the wheel-ground contact.

7) The relation between frame pitch and roll and steering angles. The most obvious effect of this can be seen during the turn before a final sprint. Riders usually lower their upper body to minimize the rolling moment due to centrifugal loads. If a bike pitches clockwise more than other for a given path radius (or, alternatively, for given steering and roll angles), COG height decreases and the rider can make the turn faster. I think that this is well related to what is tipically called "a bike that just wants to go" or "aggresivity in the corners".

From 1-2 we can say that BB offset, chainstay length and wheelbase are needed to define the behaviour of the bike.

About 3), it has been said many times that gyroscopic moments are very important for the stability of a bike but it has been shown both analitically and experimentally (with a counter-rotating wheel) that they have much lower influence than gravity-induced ones. We can then neglect the influence of head tube angle for now.

Regarding 4), camber generates an overturning moment that affects the rolling motion of the bike. Camber of the front wheel for a given combination of steering and roll angles is affected by head tube angle but is it significative compared to the rolling moment due to gravity? To check this, I have plotted the ratio between rolling moments due to gravity and total (both wheels) overturning moment assuming a small steering angle. As you can see, the contribution of overturning moments to total rolling moments is negligeable so the role of the head tube angle can be neglected for the moment.

800N rider+bike. 25c tire. Considering lateral force significatively smaller than vertical force.

For 5-7, I could have developed expressions for these relations using a 4-body model but I have consulted the very interesting Motorcycle Dynamics where they are already derived. Apart from the parameters that I have already defined as necessary (BB offset, chainstay length and wheelbase), there are only two that have an influence on these relations: "modified" trail and head tube angle (HTA). To check how these dimensions affects parameters 5-7, I will define a typical bike. The geometric parameters of the typical bike are the following ones:

Regarding 5), path radius is mainly a function of steering angle and it's independent of "modified" trail. The very strong dependence on steering angle makes that important changes in the HTA doesn't affect significatively the way the bike handles because the rider can compensate those modifications by changes of the steering angle of the order of thousandths of degree. As you can see in the following plots, path radius changes due HTA are very small compared to those caused by steering motion so we can say that path radius is nearly independent of HTA.

About 6), "modified" trail is the most important parameter that affects steering motion. Although trail is defined when both roll and steering angles are null, it isn't constant and changes due to those two rider inputs. To check the influence of HTA and "modified" trail at standstill in the evolution of this parameter, I have done a small sensitivity study by changing baseline values of each parameter by 6%. As you can see in the following plots, "modified" trail in standstill condition has much  higher influence on steering moments than HTA.

Focus on the range with same sign steering and roll angles 

Regarding 7), knowing that pitch angle is a linear function of "modified" trail and a trigonometric function of HTA, the changes due to HTA are significantly damped. For example, for a given combination of roll and steering angles, a 6% variation of "modified" trail with respect to the baseline configuration will change pitch angle by the same amount while the same variation of HTA will change pitch angle by only 1.7%.

Taking into account all this, we can say that "modified" trail plays a much more important role on bicycle's stability and maneuverability than HTA. Consequently, a definition of bicycle geometry using 4 parameters ("modified" trail, chainstay length, wheelbase and BB offset) is able to fully capture all the mechanisms that have a significative role in the behaviour of the bike.

A very long post. I hope you have found it interesting.