Moment problem

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Graphs showing the lean and steer angle response of an otherwise uncontrolled simplified model of a typical bike, traveling at a forward speed in its stable range (in this case 6 m/s), to a positive steer torque (to the right) that begins as an impulse and then remains constant. It causes an initial steer angle to the right, a lean to the left, and eventually a steady-state lean to the left, steer angle to the left, and thus a turn to the left.

File:Inoutrak.gif

Countersteering is required to turn any tandem 2-wheeled vehicle

A hypothetical curve on dry asphalt

Countersteering is the technique used by single-track vehicle operators, such as cyclists and motorcyclists, to initiate a turn toward a given direction by momentarily steering counter to the desired direction ("steer left to turn right"). A more accurate term is deliberate countersteering when employed by a rider, though it is often shortened to just countersteering. To negotiate a turn successfully, the combined center of mass of the rider and the single-track vehicle must first be leaned in the direction of the turn, and steering briefly in the opposite direction causes that lean. This technique does not apply to conventional multiple-tracked vehicles such as trikes or sidecar-equipped bicycles and motorcycles.

Definition

The scientific literature does not provide a clear and comprehensive definition of countersteering. In fact, "a proper distinction between steer torque and steer angle ... is not always made."

It is important to distinguish the steering torque and steering angle necessary to establish the lean required for a given turn from the sustained steer torque and steer angle necessary to maintain a constant turn radius and lean angle until it is time to exit the turn. The initial steer torque and steer angle are both opposite the desired turn direction. The sustained steer angle is usually in the same direction as the turn, but may remain opposite to the direction of the turn, especially at high speeds. The sustained steer torque required to maintain that steer angle is usually opposite the turn direction. (See the graphs to the right.) The actual magnitude and orientation of both the sustained steer angle and sustained steer torque of a particular bike in a particular turn depend on forward speed, bike geometry, tire properties, and combined bike and rider mass distribution.

It is also important to distinguish between countersteering as a physical phenomenon and deliberate countersteering as a conscious rider technique for initiating a lean (the usual interpretation of the term).

The physical phenomenon always occurs in any cornering manoeuvre at more than walking pace and is also evident in situations where the bike and rider compensate for some outside influence such as an opportune side wind, although at low speeds it can be lost or hidden in the minute corrections made to maintain balance.

The technique of exploiting the physical phenomenon by deliberately countering the steering is achieved by applying a steering torque in the opposite direction of the road curve. For example, if a turn to the left is desired, it is started by an applying a torque on the handlebars to the right.

It is important to note that the rider is not causing countersteering, merely initiating it at a time of their choosing. Another way to cause the bike and rider to lean is by applying appropriate torques between the bike and rider similar to the way a gymnast can swing up from hanging straight down on uneven parallel bars, a person can start swinging on a swing from rest by pumping their legs, or a double inverted pendulum can be controlled with an actuator only at the elbow.

At the same time, the rider technique of applying pressure to the handlebars to initiate a lean is not always necessary, since, on a sufficiently light bike (especially a bicycle), the rider can initiate a lean and turn by shifting body weight, called counter-lean by some authors. Documented physical experimentation shows that on heavy bikes (many motorcycles) shifting body weight is less effective at initiating leans. This technique works by employing the natural tendency of most bikes to steer towards the direction they are leaned and is different from the acrobatic maneuver described above. Note that when applying a counter-lean technique, the countersteering as a physical phenomenon still occurs, the manner in which it is initiated by the rider is different.

Bikes vs automobiles

Automobiles steer by imposing a steer angle between the front and rear wheels. Broadly speaking, the response of the vehicle is related to that angle; the automobile is a "position-controlled" system in normal use.

Bikes steer by controlling roll angle. However, roll angle is not directly manipulated by the handlebars in the same way as steer angle in an automobile. Rather it is influenced indirectly by applying roll moments to the machine. These roll moments in turn come from side forces at the wheels, which are closely related to steering torques. Thus the bike is a "force-controlled" system in which the actual position of the handlebars is free.

How it works

A single-track vehicle such as a bicycle or a motorcycle is an inverted pendulum—it will fall over unless balanced.

The technique used by cyclists and motorcyclists to initiate turning in a given direction is to first apply a steering torque in the opposite direction. For example, if a turn to the left is desired, it is started by applying a torque on the handlebars to the right. This causes the front wheel to rotate about the steering axis to the right and the front tire will generate forces in the contact patch to the right. The machine as a whole steers to the right briefly, and because the forces in the contact patch are at ground level, this pulls the wheels "out from under" the bike to the right and causes it to lean to the left. Then the rider, or in most cases, the inherent stability of the bike provides the steering torque necessary to rotate the front wheel back to the left and in the direction of the desired turn. Finally, the bike begins a turn to the left. It is often boiled down to "push left to go left".

While this appears to be a complex sequence of motions, it is performed by every child who rides a bicycle. The entire sequence goes largely unnoticed by most riders, which is why some assert that they do not do it.

It is often claimed that two-wheeled vehicles can be steered using only weight shifts. While this is true for small "trim" inputs to direction, complex maneuvers are not possible using weightshifting alone because even for a light machine there is insufficient control authority. Although on a sufficiently light bike (especially a bicycle), the rider can initiate a lean and turn by shifting body weight, there is no evidence that complex maneuvers can be performed by bodyweight alone.

It is also important to distinguish the steering torque necessary to initiate the lean required for a given turn from the sustained steering torque and steering angle necessary to maintain a constant radius and lean angle until it is time to exit the turn. The initial steer torque and angle are both opposite the desired turn direction. The sustained steer angle is in the same direction as the turn. The sustained steer torque required to maintain that steer angle is either with or opposite the turn direction depending on forward speed, bike geometry, and combined bike and rider mass distribution.

Deliberate countersteering is necessary to adjust the angle of lean of a bike and works in the opposite sense to the handlebar input while conventional steering in the direction of the turn is used to negotiate the turn.

Need to lean to turn

A bike can negotiate a curve only when the combined center of mass of bike and rider leans toward the inside of the turn at an angle appropriate for the velocity and the radius of the turn:

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

where $v$ is the forward speed, $r$ is the radius of the turn and $g$ is the acceleration of gravity.

Higher speeds and tighter turns require greater lean angles. If the mass is not first leaned into the turn, the inertia of the rider and bike will cause them to continue in a straight line as the tires track out from under them along the curve. The transition of riding in a straight line to negotiating a turn is a process of leaning the bike into the turn, and the most practical way to cause that lean (of the combined center of mass of bike and rider) is to move the support points in the opposite direction first. The rider can shift his weight of course, but any force used to move one way laterally pushes the bike laterally the opposite direction with equal force. That makes the bike lean (and can affect the steering), but it does not change the combined center of mass of bike and rider.

Lean by countersteering

When riding a bicycle or a motorcycle, deliberate countersteering is a method of initiating a turn by a small, momentary turn of the front wheel, usually via the handlebars, in the opposite (counter) direction. This moves the pivot point (the wheels' contact patches) out from under the center of mass to establish the lean angle for a turn. While exploitable at all speeds, the need to deliberately countersteer becomes more noticeable as speed increases.

Hence, to turn to the right, the rider first throws the bike off balance by momentarily pointing the front wheel slightly to the left. The center of mass of the bike plus rider will continue in a straight line, but the contact patches of the tires move to the left with respect to this straight line.

Once lean is achieved

As the desired angle is approached, the front wheel must usually be steered into the turn to maintain that angle or the bike will continue to lean with gravity, increasing in rate, until the side contacts the ground. This process often requires little or no physical effort, because the geometry of the steering system of most bikes is designed in such a way that the front wheel has a strong tendency to steer in the direction of a lean.

The actual torque the rider must apply to the handlebars to maintain a steady-state turn is a complex function of bike geometry, mass distribution, rider position, tire properties, turn radius, and forward speed. At low speeds, the steering torque necessary from the rider is usually negative, that is opposite the direction of the turn, even when the steering angle is in the direction of the turn. At higher speeds, the direction of the necessary input torque often becomes positive, that is in the same direction as the turn.

Adjusting or exiting a turn

Once in a turn, deliberate countersteering is again required to make changes to its shape. The only way to decrease the radius at the same speed is to increase the lean angle, and the most effective way to increase the lean angle, is again to momentarily steer opposite to the direction of the curve. To the untrained, this can be extremely counter-intuitive.

To exit a turn, a deliberate countersteer by momentarily steering further in the direction of the turn will tilt the bike back upright.

At low speeds

At low speeds countersteering is equally necessary, but the countersteering is then so subtle that it is hidden by the continuous corrections that are made in balancing the bike, often falling below a just noticeable difference or threshold of perception of the rider. Countersteering at low speed may be further concealed by the ensuing much larger steering angle possible in the direction of the turn.

Unthinking behavior

Countersteering is indispensable for bike steering. Most people are not aware that they employ countersteering when riding their bike any more than they are aware of the physics of walking. They have learned to apply the required countersteering without thinking.

As is well known in bicycle racing, the countersteering phenomenon becomes evident when there is an obstacle preventing the wheel from countersteering (e.g., when closely overlapping wheels or riding very close to a curb). In these situations, the way to initiate a turn with the handlebars away from the obstacle is to countersteer towards obstacle to avoid crashing into it. Lack of understanding of this principle leads to accidents in novice bicycle races.

Gyroscopic effects

One effect of turning the front wheel is a roll moment caused by gyroscopic precession. The magnitude of this moment is proportional to the moment of inertia of the front wheel, its spin rate (forward motion), the rate that the rider turns the front wheel by applying a torque to the handlebars, and the cosine of the angle between the steering axis and the vertical.

For a sample motorcycle moving at 22 m/s (50 mph) that has a front wheel with a moment of inertia of 0.6 kgm², turning the front wheel one degree in half a second generates a roll moment of 3.5 Nm. In comparison, the lateral force on the front tire as it tracks out from under the motorcycle reaches a maximum of 50 N. This, acting on the 0.6 m (2 ft) height of the center of mass, generates a roll moment of 30 Nm.

While the moment from gyroscopic forces is only 12% of this, it can play a significant part because it begins to act as soon as the rider applies the torque, instead of building up more slowly as the wheel out-tracks. This can be especially helpful in motorcycle racing.

No hands

This is how countersteering works when riding no-hands. To turn left, a rider applies a momentary torque, either at the seat via the legs or in the torso that causes the bike itself to lean to the right, called counter lean by some authors. The combined center of mass of the bike and rider is only lowered, of course. However, if the front of the bike is free to swivel about its steering axis, the lean to the right will cause it to steer to the right by some combination of gyroscopic precession (as mentioned above), ground reaction forces, gravitational force on an off-axis center of mass, or simply the inertia of an off-axis center of mass, depending on the exact geometry and mass distribution of the particular bike, and the amount of torque and the speed at which it is applied.

This countersteering to the right causes the ground contact to move to the right of the center of mass, as the bike moves forward, thus generating a leftward lean. Finally the front end steers to the left and the bike enters the left turn. The amount of leftward steering necessary to balance the leftward lean appropriate for the forward speed and radius of the turn is controlled by the torque generated by the rider, again either at the seat or in the torso.

To straighten back out of the turn, the rider simply reverses the procedure for entering it: cause the bike to lean farther to the left; this causes it to steer farther to the left, which moves the wheel contact patches farther to the left, eventually reducing the leftward lean and exiting the turn.

The reason this no-hands steering is less effective on heavy bikes, such as motorcycles, is that the rider weighs so much less than the bike that leaning the torso with respect to the bike does not cause the bike to lean far enough to generate anything but the shallowest turns. No-hands riders may be able to keep a heavy bike centered in a lane and negotiate shallow highway turns, but not much else.

Bicycles

Deliberate countersteering is generally not needed in normal bicycling, but it can be very useful—even potentially lifesaving—in emergency maneuvers, and is taught as the instant turn (a term coined by John Forester) in traffic cycling courses offered through the League of American Bicyclists. League certified instructors teach students to be prepared to make a sudden sharp turn to avoid, for example, being hit by a motorist who just overtaken them and then turned across their path, cutting the cyclist off. The instant turn is initiated by quickly jerking the bars to the left (the countersteer), which initiates the necessary lean to the right, and then turning sharply into the required turn to the right. Racing cyclists and cyclists performing fast hill descents may also use deliberate countersteering in order to initiate and manage the fast, precise turns necessary.

Motorcycles

Even more so than on a bicycle, mastering the technique of deliberately countersteering is essential for safe motorcycle riding, and as a result is a part of the safe riding courses run by the Motorcycle Safety Foundation and the Canada Safety Council. At the higher speeds that motorcycles commonly attain, it becomes increasingly impractical to steer by taking advantage of the minute and random corrections needed to maintain balance.

Much of the art of motorcycle cornering is learning how to effectively "push" the grips into corners and how to maintain proper lean angles through the turn. When the need for a quick swerve to one side suddenly arises in an emergency, it is essential to know, through prior practice, that the handlebars must be deliberately pressed away on that side instead of being pulled. Many accidents result when otherwise experienced riders who have never carefully developed this skill encounter an unexpected obstacle.

To encourage an understanding of the phenomenon of countersteering, the phrase positive steering is sometimes used, and is summed up in a simplified way as "Push the right-hand bar to steer right; push the left-hand bar to steer left".

Three wheelers

Typically, three wheelers like trikes and sidecar rigs do not countersteer. There are unconventional multi-tracked motorcycles like the three wheel Piaggio MP3 which use complex mechanical linkages to lean the two front wheels in parallel, so that it is countersteered in the same manner as a two wheeled motorcycle. In contrast, the three wheeled BRP Can-Am Spyder Roadster uses two front wheels which do not lean, and so it steers like a car.

Motorcycle speedway racing

Other uses

The term counter-steering is also used by some authors to refer to the need on bikes to steer in the opposite direction of the turn (negative steering angle) to maintain control in response to significant rear wheel slippage. Motorcycle speedway racing takes place on an oval track with a loose surface of dirt, cinders or shale. Riders slide their machines sideways, powersliding or broadsiding into the turns, using an extreme form of this type of countersteering that is maintained throughout the turn.

The term is also used in the discussion of the automobile driving technique called drifting.

The Wright Brothers

Wilbur Wright explained countersteering this way:

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See also

• Bicycle and motorcycle dynamics
• Motorcycle safety

References

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External links

• Balance and Steering, by Tony Foale
• "No B.S. Machine" physical experimentation
• "The Physics of Unicycling" 5 minutes, explains countersteering from fundamental physics principles, requires Adobe Flash

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