Bicycle and motorcycle dynamics

Bicycle and motorcycle dynamics is the science of how bicycles and motorcycles move: balance, steer, brake, etc. It is concerned with not just the motions of bikes and their parts but also the forces on them. It is studied by manufacturers and at universities around the world: for example Padua University, Italy; Cornell University, USA; and Delft University of Technology, The Netherlands.

Experimentation and mathematical analysis have shown that a bike stays upright when it is steered to keep its center of gravity over its wheels. This steering is usually supplied by a rider, or in certain circumstances, by the bike itself.

While remaining upright may be the primary goal of beginning riders, a bike must lean in order to turn: the higher the speed or smaller the turn radius, the more lean required. This is necessary in order to balance centrifugal forces due to the turn with gravitational forces due to leaning.

Finally, as far as braking is concerned, depending on the the location of the combined center of mass of the bike and rider with respect to where the front wheel contacts the ground, bikes can either skid the front wheel or flip the bike and rider over the front wheel.

Balance

Bikes remain upright when they are steered so that the ground reaction forces (vertical and horizontal) exactly balance all the other forces (or accelerations, depending on the reference frame used) it experiences such as gravitational, inertial if in a turn, and aerodynamic if in a crosswind. This steering may be easily supplied by a rider, or in certain circumstances, by the bike itself.

This self-stability is generated by a combination of several effects that depend on the geometry, mass distribution, and forward speed of the bike. Tires, suspension, steering damping, and frame flex can also influence this self-stability, especially in motorcycles.

Lock the steering of a bike and it is virtually impossible to ride. Cancel the gyroscopic effect of rotating bike wheels by adding counter-rotating wheels, and it can still be easily ridden. For examples of the latter, read "Bicycle Science" by Dr. Klein or "The Stability of the Bicycle" by David Jones.

Trail

A factor that influences how easy or hard a bike will be to ride is called ‘trail’, the distance by which the front wheel ground contact point trails behind where a line through the steering axis intersects the ground. In traditional bike designs, with a steering axis tilted back from the vertical, trail causes the front wheel to steer into the direction of a lean, independent of forward speed. This can be seen by pushing a stationary bike to one side. The front wheel will usually also steer to that side. In a lean, gravity provides this force.

The more trail a bike has, the more stable it feels. Bikes with negative trail, while still ridable, feel very unstable. Bikes with too much trail feel difficult to steer. In bicycles, fork 'rake' or 'offset', often a curve in the fork blades forward of the steering axis, is used to diminish trail.Zinn. In motorcycles, 'rake' instead refers to the head angle, and 'offset' created by the triple tree, is used to diminish trail.Foale.

Mass distribution

Another factor that can also contribute to the self-stability of traditional bike designs is the distribution of mass in the steering mechanism (front wheel, fork, handle bars, etc.) If the center of gravity for the steering mechanism is forward of the steering axis, then the pull of gravity will also cause the front wheel to steer in the direction of a lean. This can be seen by leaning a stationary bike to one side. The front wheel will usually also steer to that side independent of any interaction with the ground.Hand, 108.

Non standard bike configurations, such as long-wheel-base recumbents that have a long steering boom, may have the steering mechanism center of gravity so far behind the steering axis, that in a lean, this factor over-powers all others, the front wheel actually steers away from the lean, and they do not exhibit self-stability at any forward speed.

Other, even more subtle effects, such as the fore-to-aft position of the center of gravity, and a slight lowering of the center of gravity, and thus potential energy, as the front wheel is steered away from straight ahead, also contribute to the dynamic behaviour of a bike.Hand, 157.

Gyroscopic effects

The role that the gyroscopic effect plays, in most bike designs, is simply to help steer the front wheel into the direction of a lean. This phenomenon is called precession and the rate at which an object precesses is inversely proportional to its rate of spin. The slower a front wheel spins, the faster it will precess when the bike leans, and visa-versa.Fajans, 656.

The rear wheel is prevented from precessing as the front wheel does by friction of the tires on the ground, and so continues to lean as though it were not spinning at all. This can easily be demonstrated by setting a spinning toy gyroscope on its side and then preventing it from precessing. It immediately falls over. A quick analysis with Euler's equations for 3D rigid-body motion confirms that it falls at the same rate it would if it were not spinning at all. To emphasize — gyroscopic forces do not provide any resistance to tipping at all, at any speed.

At slow forward speeds, the precession of the front wheel is usually too quick, contributing to an uncontrolled bike's tendency to over-steer, start to lean the other way, and eventually fall over after one or more oscillations. At high forward speeds, the precession is usually too slow, contributing to an uncontrolled bike's tendency to under-steer and eventually fall over without ever having reached the upright position. This instability is very slow, on the order of seconds, and is trivial to counteract for most riders. Thus bikes may feel more stable to a rider even if they are actually not self-stable and would eventually fall over if uncontrolled.

Self stability

Between these two extremes, there may be a range of forward speeds for a particular bike design at which all the effects described above are just right to steer an uncontrolled bike back to vertical.Schwab, Meijaard, and Papadopoulos, 4. However, as Jones and Klein have shown, even without self-stability a bike may be easily ridden simply by steering it to keep it over its wheels.

See a video of a riderless bicycle exhibiting this self-stability.

Wobble and shimmy

At higher speeds bikes can also experience speed wobbles or shimmies, where the front wheel spontaneously oscillates to the left and right. While the wobbles can be easily remedied by slowing down, adjusting position, or relaxing one's grip on the handlebars, speed wobbles can be fatal.Kettler.

This shimmy is often seen in shopping cart front wheels. Some otherwise minor irregularity accelerates the wheel to one side. The restoring force is applied in phase with the progress of the irregularity, and the wheel turns to the other side where the process is repeated. If there is insufficient damping in the steering the oscillation will increase until system failure. Speed changes, making the bike stiffer or lighter, or increasing the stiffness of the steering (of which the rider is the main component) can change the oscillation frequency, though only speed change is applicable in the situation.

Stability with full suspension

Full suspension bikes, especially motorcycles, face three common stability problems:

'Dropping the bike' (ie. having the bike fall over) can occur at low speeds; this usually happens when parking the bike, using the front brake at low speeds with the front wheel out of line with the rear wheel, or allowing a stall of the bike at low speed.

Wobbling (AVI movie) is a high frequency (7-9 Hz) oscillation of the front wheel. It is often relatively harmless but annoying (and quite frightening if not previously experienced). It can appear at moderate speeds.

Weaving (AVI movie) is a low frequency (2-3 Hz) oscillation of the whole vehicle. It can become unstable at higher speeds with fatal results.

The design characteristics of a bike can affect the stability in the following ways:Evangelou, 159.

The weave oscillations dampen out once the rider reduces the roll angle.

Tire characteristics and inflation pressures are important variables in the behaviour of a motorcycle at high speeds.

From a stability point of view it is desirable to make the lateral stiffness as large as possible, with the possibility of an optimum value for the torsional stiffness of the rear frame.

Common levels of lateral stiffness at the wheel spindle deteriorates the wobble-mode damping substantially with significant changes in the wobble frequency as well, and slight reduction in the weave-mode damping at high speeds.

Lateral distortion should be opposed as much as possible by locating the front fork torsional axis as low as possible.

The largest contribution to the weave damping comes from the cornering and camber stiffnesses and relaxation length of the rear tire and not so much from the same parameters of the front tire.

Amongst others, stiff frames, a long wheelbase, a long trail and a flat steering-head angle were found to increase weave-mode damping.

Degraded damping of the rear suspension, rear loading and increased speed amplifies cornering weave tendencies.

Rear load assemblies with appropriate stiffness and damping were successful in damping out weave and wobble oscillations.

Turning

Bikes must lean in order to turn. This is necessary to keep all the relevant forces in balance: gravitational, inertial, frictional, and ground support. The angle of lean, $\theta \,$ , can easily be calculated using the laws of circular motion:

\theta = \arctan \left (\frac{v^2}{gr} \right )

where

v \,

is the forward speed,

r \,

is the radius of the turn, and

g \,

is the acceleration of gravity.Fajans, 654.

For example, a bike in a 10 meter (~32.8 ft) radius steady-state turn at 10 meters per second (~22.4 mph) must be at about a 45° angle. There is some ability for a rider to lean with respect to the bike in order to keep either their torso or the bike more or less upright if desired. The only angle that really matters is the one between the horizontal and the plane between the tire contacts and location of the center of gravity of the combined bike and rider.

Countersteering

In order to initiate a turn, a bike must momentarily steer in the opposite direction. This is often referred to as countersteering. This brief turn moves the wheels out from directly underneath the center of gravity, and thus causes a lean in the desired direction. As there is no other way, short of an opportune side wind or some other external influence, to create the force necessary to lean the bike, countersteering, whether the rider realizes it or not, happens in every turn.Code.

As the lean approaches the desired angle, the front wheel must be steered more or less in the direction of the turn, depending on the forward speed, the turn radius, and the need to maintain the necessary lean angle. Once in a turn, the radius can only be changed with an appropriate change in lean angle, and this can only be accomplished by additional countersteering: out of the turn to increase lean and decrease radius, and into the turn to decrease lean and increase radius. To exit the turn, the bike must again countersteer and momentarily steer more into the turn to decrease the radius to increase inertial forces in order decrease the angle of lean.

No Hands

While countersteering is usually initiated by applying torque directly to the handlebars, on lighter vehicles such as bicycles, it can also be accomplished "no-hands" by shifting the rider's weight. If the rider leans to the right, relative to the bike, the bike will necessarily lean to the left in order to conserve angular momentum: the combined center of mass may lower but will remain in the same vertical plane. This leftward lean of the bike will cause it to steer to the left, as described above in the balance section, and initiate a right hand turn just as if the rider had countersteered to the left by applying a torque directly to the handle bars.Fajans, 657.

Braking

Most of the braking power of standard upright bikes comes from the front wheel. If the brakes themselves are strong enough, the rear wheel is easy to skid, while the front wheel often has enough stopping power to flip the rider and bike over the front wheel. This is called a stoppie or an 'endo'. However, long and/or low bikes, such as cruiser motorcycles and recumbent bicycles, can also skid the front tire, causing loss of the ability to balance.

Mechanical analysis of the forces generated by a bike (assuming a wheelbase of $L \,$ , and a center of mass half way between the wheels and at height $h \,$ ) with both wheels locked reveals that normal (vertical) forces at the wheels are:Ruina and Pratap, 350.

N_r = mg(\frac{1}{2} - \mu \frac{h}{L})

for the rear wheel

N_f = mg(\frac{1}{2} + \mu \frac{h}{L})

for the front wheel

while the frictional (horizontal) forces are simply:

F_r = \mu N_r \,

for the rear wheel

F_f = \mu N_f \,

for the front wheel

where

\mu \,

is the coefficient of friction,

m \,

is the mass, and

g \,

is the acceleration of gravity.

So, if $\mu \frac{h}{L} >= \frac{1}{2}$ then the normal force of the rear wheel will go to zero and the bike will flip over.

Since the coefficient of friction of rubber on dry asphalt is between 0.5 and 0.8 Kurtus., if we use the lower value, and if the center of mass height is greater than or equal to the wheel base, then the front wheel can generate sufficient stopping force to flip the bike and rider over.

Theory

A bike is a nonholonomic system because its outcome is path-dependent. In order to know its exact configuration, especially location, it is necessary to know not only the configuration of its parts, but also their histories: how they moved over time. This makes mathematical analysis more complicated.Fajans, 654.

A bike is also an example of an inverted pendulum. Thus, just as a broomstick is easier to balance than a pencil, tall bikes can be easier to balance than short ones because their lean rate will be slower.Fajans.

Equations of motion

The equations of motion of an idealized bike, consisting of

a rigid frame
a rigid fork
two knife-edged, rigid wheels
all connected with frictionless bearings and rolling without friction or slip on a smooth horizontal surface
operating at or near the upright and straight ahead unstable equilibrium

can be represented by two linearized second-order ordinary differential equations:Hand, 37.

M_{\theta\theta}\theta_r'' +

K_{\theta\theta}\theta_r + M_{\theta\psi}\psi'' + C_{\theta\psi}\psi' + K_{\theta\psi}\psi = M_{\theta} \, (the 'lean' equation)

M_{\psi\psi}\psi'' +

C_{\psi\psi}\psi' + K_{\psi\psi}\psi + M_{\psi\theta}\theta_r'' + C_{\psi\theta}\theta_r' + K_{\psi\theta}\theta_r = M_{\psi} \, (the 'steer' equation)

Where

$M_{\theta\theta} \,$ , $M_{\theta\psi} \,$ , $M_{\psi\theta} \,$ , and $M_{\psi\psi} \,$ represent mass matrices based on the physical parameters of the bike.
$C_{\theta\theta} \,$ , $C_{\theta\psi} \,$ , $C_{\psi\theta} \,$ , and $C_{\psi\psi} \,$ represent 'damping' matrices (Note that for the upright bike $C_{\theta\theta} \,$ = 0, and so it does not appear)
$K_{\theta\theta} \,$ , $K_{\theta\psi} \,$ , $K_{\psi\theta} \,$ , and $K_{\psi\psi} \,$ represent 'stiffness' matrices
$\theta_r \,$ is the lean angle of the rear assembly
$\psi \,$ is the steer angle of the front assembly relative to the rear assembly
$M_{\theta} \,$ and $M_{\psi} \,$ are the moments (torques) applied at the rear assembly and the steering axis, respectively. For this analysis of an uncontrolled bike, these are both taken to be zero.
$'' \,$ and $' \,$ indicate the second and first derivatives with respect to time (often indicated by dots over the symbol, but that appears to be unavailable in the subset of TeX markup available in Wikipedia)

In this idealized and linearized model, there are many parameters (wheelbase, head angle, mass of each body, wheel radius, etc.), but only four significant variables: lean angle, lean rate, steer angle, and steer rate.

These equations have been verified by comparison with multiple numeric models derived completely independently.Schwab, Meijaard, and Papadopoulos, 7.

Eigenvalues

It is possible to calculate eigenvalues, one for each of the four significant variables, from these linearized equations to analyze the self-stability of a particular bike design.Schwab.

In the plot to the right, eigenvalues are calculated for forward speeds between 0 and 10 meters per second (~22.4 mph). When the real parts of all the eigenvalues (shown in dark blue) are all less than zero, the bike is self-stable (as described above).

When the imaginary parts of any of the eigenvalues (shown in cyan) are non-zero, the bike exhibits oscillation.

The forward speed at which oscillations do not increase, causing the uncontrolled bike to eventually fall over, is called the 'weave' speed.

The forward speed at which non-oscillatory leaning does not increase, causing the uncontrolled bike to eventually fall over, is called the 'capsize' speed.

Between these two speeds, if they both exist, is a range of forward speeds at which the particular bike design is self-stable. In the case of the bike whose eigenvalues are shown here, the self-stable range is from 5.2770 m/s to 7.9807 m/s.

Common misconception

Although bicycles and motorcycles can appear to be simple mechanisms with only four major moving parts (frame, fork, and two wheels), these parts are arranged in a way that makes them quite complicated to analyze correctly. For example, there are still plenty of instances online and in print of the common misconception that the gyroscopic effects of the wheels are what keep a bike upright.Fajans, 658.

Here are some examples of the common misconception still online, although some may have since been corrected.

And in print:

"Angular momentum and motorcycle counter-steering: A discussion and demonstration", A. J. Cox, Am. J. Phys. 66, 1018–1021 ~1998
"The motorcycle as a gyroscope", J. Higbie, Am. J. Phys. 42, 701–702
The Physics of Everyday Phenomena, W. T. Griffith, McGraw–Hill, New York, 1998, pp. 149–150.

Balance

Trail

Mass distribution

Gyroscopic effects

Self stability

Wobble and shimmy

Stability with full suspension

Turning

Countersteering

No Hands

Braking

Theory

Equations of motion

Eigenvalues

Common misconception

See also

Notes

References

External links

Further reading

Gourt Home::Gourt :: Bicycle and motorcycle dynamics

Balance

Trail

Mass distribution

Gyroscopic effects

Self stability

Wobble and shimmy

Stability with full suspension

Turning

Countersteering

No Hands

Braking

Theory

Equations of motion

Eigenvalues

Common misconception

See also

Notes

References

External links

Further reading