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Bicycle physics is a broad and complex subject, perhaps more so than one can imagine. Although the number of components of a bicycle is small, the interaction between them and the dynamic principles involved, is complicated. This is especially true with regards to bicycle stability, which is the result of a complex dynamic interaction within the bike-rider system.

On this page I will explain some of the main aspects of the physics of bicycles, which should give the reader a greater appreciation of how bicycles work, from a physics perspective.

Bicycles are inherently stable when riding. Even riderless bicycles are stable if given enough forward velocity. Much effort has gone into analyzing the factors which make a bicycle stable. It has been determined that "trail" (shown below) is often an important contributor to bicycle stability. For the traditional bicycle design, if trail is positive, meaning the projection of the steering axis with the ground is in front of the contact point of front wheel and ground, then the bicycle is more stable when riding (i.e. it's less likely to fall over when riding it). If this projection is behind the contact point (negative trail) then the bicycle is less stable and the bicycle is more likely to fall down when riding it.

Based on the geometric parameters shown, the mathematical formula for trail is:

where

When analyzing bicycle stability it is common to use two parameters; the lean angle and steering angle of the bike. The lean angle is the left and right angle the bike frame makes with a vertical plane, and the steering angle is the angle the front wheel makes with the plane of the bike (containing the bike frame). The figure below illustrates the lean and steering angle.

where

As mentioned, analyzing bicycle stability is a complex undertaking involving large and "messy" equations. There are too many physical interactions taking place between the various bicycle components (namely the front and back wheel, steering column, and bicycle frame) to allow for a completely intuitive explanation. To gain an appreciable understanding of bicycle stability it is best to do a full dynamics analysis and then base your understanding on the results of this analysis.

It is common to analyze bicycle stability using a "riderless" assumption. This means that the bicycle is modeled with just the bike by itself (with no rider). This greatly simplifies the analysis, and consequently it is often assumed that a stable riderless bicycle will also be stable with a rider present. This can be a reasonable assumption but unfortunately it ignores the "input" of the rider which also affects how stable a bike is during its use.

Nevertheless, in the absence of something better it is common to perform stability analyses on riderless bicycles. If you want to see the full analysis of a riderless bicycle see the following paper:

A common belief is that gyroscopic effects by themselves are what makes a bike stable. This is actually not the case. Although gyroscopic effects do play a role, they are merely part of a much larger dynamic interaction taking place between the various bicycle components, which altogether is what ultimately makes a bicycle stable during riding. The design of a bicycle, and the configuration of the different components, have been optimized through the ages (largely through trial and error), to make it as stable as possible.

As mentioned, gyroscopic effects are not the main contribution to bicycle stability, but it is still informative to see

Let's say a riderless bicycle is moving at a certain velocity. Let's further say that the bicycle leans right (positive

With the front wheel steering right, the bicycle then travels in a circular path (towards the right). This decreases

The entire physical interactions taking place are actually more complex than the scenario given above, especially due to oscillations in

Source: http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics. Author: http://en.wikipedia.org/wiki/User:Furmanj

When riding a bicycle it is necessary to lean into a turn in order to compensate for the effect of centripetal acceleration. The inward lean balances the centripetal acceleration which makes the turn possible without falling over.

To analyze the physics behind a lean consider the following schematic.

Where:

Since there is no acceleration in the vertical direction the sum of the vertical forces is zero. Thus,

Apply Newton's second law in the horizontal direction:

where

Sum the moments about point

(Note that we are ignoring three-dimensional effects in this equation. They are assumed to be negligible).

Combine the above three equations to find an expression for the angle of lean

In the next section we will look at forces and power.

The figure below shows a bicycle going uphill at an angle of inclination

To propel the bicycle uphill the rider must push down on the pedals. The pedals are offset 180° which means that only one pedal can be pushed at a time, from the top position to the bottom position, and then switching to the other pedal.

Given a force

We can perform a torque analysis with good accuracy based on the assumption that acceleration (linear and angular) is negligible. Hence, we can treat this as a static problem.

Consider the figure below, with forces and radial dimensions shown.

Where:

Using the static equilibrium assumption, we can write the following torque equations:

and

Since

The force

Where:

The first term on the right side of the above equation is the gravity contribution. The second term is the rolling resistance contribution. The third term is the air drag contribution.

To evaluate the power

For a flat surface (no incline) set

and

We can also solve for the terminal velocity of a bicycle coasting down a hill with given inclination angle of

Naturally, when riding a bike we wish to keep resistive forces opposing motion as low as possible. This is accomplished by keeping the tires well pressurized (which minimizes rolling resistance) and keeping the frontal area

Try out this fun experiment related to bicycle physics (shown below). Stand a bicycle upright and orient one of the pedals so that it's at the bottom position. Next, push left on the pedal. Which way does the bicycle move?

Answer: The bicycle moves to the left (which might be non-intuitive to you). Even though the force you are applying to the pedal turns the main crank clockwise, which is the direction required to move the bike to the right, the bicycle ends up moving left. This is because the external force

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