Driving physics (bicycle)

The driving on a bicycle is a dynamic process with the result that a single-track bike from falling over with his sitting on his driver.

Conditions for this are:

a minimum speed
the cyclist's ability to balance (he intuitively steers to the side where the bike is about to fall).

In addition, the gyroscopic effect facilitates riding, especially hands-free riding, thanks to the automatic steering of the front wheel when the bike is tilted.

The bike in unstable equilibrium

A bicycle touches the ground in two places - the contact surfaces of the tires. Even a slight deviation of the frame plane from the vertical direction leads to tipping over when the bicycle is stationary. As soon as the center of gravity is no longer above the support surface that encompasses and connects the contact surfaces, the wheel tips over.

By alternately turning the handlebars vigorously, the wheel can be rotated a little about the contact point of the rear wheel, whereby the support surface moves relative to the driver. This is equivalent to a shift in the driver's center of gravity. Only experienced people can balance on a stationary bike for a long time without getting off. Since these problems do not exist when driving straight ahead, the driving dynamics must be decisive.

Balance while driving

A tipping over in one direction while driving is counteracted by the fact that the handlebars deflect in the same direction, initiate a short curve and the bike is now raised to the other side by the centrifugal force . Tipping over can hardly be avoided, the handlebars have to be turned in the other direction and so on.

Driving straight ahead is equivalent to a barely noticeable pendulum around the equilibrium position between tilting and getting back upright. When driving slowly, the oscillation manifests itself in strong, alternating steering deflections.

That the generation of centrifugal forces to control the tilting angle is sufficient to steer a single-track vehicle, tests with specially equipped wheels, but also in practice with other single-track vehicles, in which the gyroscopic moments play no role or a minor role, such as velogemel , monoskibob or scooter . The gyroscopic effects become more important when riding hands-free, but they also help to stabilize the bike when riding normally.

Cornering

Entering a curve, exiting a curve

A curve is not initiated directly by turning the handlebar in the desired direction. Wheel tracks on sand or snow show that there is initially a slight steering movement in the opposite direction. If you simply turn left for a left turn, then the contact surface of the tire moves to the left, away from the center of gravity. This causes an inclination to the right, which is subsequently increased by gravity. In order to make a left turn, however, an incline to the left is always necessary so that the wheel does not tip outwards.

Another way to initiate cornering is to shift your weight slightly towards the inside of the curve. The inclined position is then stabilized by steering. This achieves a new equilibrium in which the overturning moment is balanced out by centrifugal force.

To end cornering, you first have to turn in a little more than what corresponds to the transverse slope. The moment from the centrifugal force becomes greater than the moment from the weight force, which causes the wheel to straighten up.

Determination of the angle of inclination

Two-wheelers in stationary cornering

A curve can be viewed as part of a circular path. If the driver lies down in the curve, the angle of inclination depends on the driving speed and the curve radius. The faster the journey and the tighter the curve, the greater the angle of inclination to be taken. This can be clearly determined: The line connecting the center of gravity and the support surface must run in the direction of the resultant of centrifugal force and weight . For the angle of inclination between the resultant and the vertical, the following applies: ${\ displaystyle F _ {\ text {R}}}$ ${\ displaystyle F _ {\ text {Zf}}}$ ${\ displaystyle F _ {\ text {G}}}$ ${\ displaystyle \ alpha}$

{\ displaystyle \ tan {\ alpha} = {\ frac {F _ {\ text {Zf}}} {F_ {G}}} = {\ frac {v ^ {2}} {r \; g}}}

This includes the speed, the curve radius, and the acceleration due to gravity . ${\ displaystyle v}$ ${\ displaystyle r}$ ${\ displaystyle g}$

Since the ratio of centripetal force and normal force is at the same time , the result for the necessary coefficient of adhesion on a level road is: ${\ displaystyle \ tan {\ alpha}}$

{\ displaystyle \ mu = \ tan {\ alpha}}

The static friction of the wheels now determines the maximum angle of inclination which, if exceeded, leads to a wheel slipping and falling. Braking is therefore necessary before tight bends and on greasy , gravelly or smooth surfaces, because otherwise the friction will not be sufficient to generate the required centripetal force. The maximum angle of inclination is limited by the coefficient of static friction : ${\ displaystyle \ mu _ {H}}$

{\ displaystyle \ tan \ alpha \ leq \ mu _ {H}}

With the linear single-track model , an equation for the relationship between steering angle and path curvature can be derived:

{\ displaystyle {\ frac {1} {r}} \ approx {\ frac {\ delta} {L}}}

This includes the wheelbase and the steering angle. The following results for the angle of inclination: ${\ displaystyle L}$ ${\ displaystyle \ delta}$

{\ displaystyle \ tan \ alpha \ approx {\ frac {v ^ {2}} {L \, g}} \ cdot \ delta}

The equation also explains why a minimum speed is required when cycling. At very low speeds, even small angles of inclination must be compensated for by large steering angles or overcompensated when driving straight ahead. Above the minimum speed, the centripetal force directed towards the center of the circle can be applied through smaller, metered steering turns.

Cant

The curve radius can be reduced considerably if the roadway is not level, but rather inclines downwards towards the center of the curve ( superelevation ). If the required angle of inclination is the same as the superelevation angle, the runway can be driven on without lateral forces . Cyclo-cross riders and mountain bikers as well as track cyclists use this help :

In cyclocross and mountain bike sports you use z. B. furrowed curves, which have a cant in order to drive through curves faster.
In track cycling , the cycling tracks generally have superelevations in the curves between 30 degrees (long open-air cement tracks with greater static friction) and usually 45 degrees superelevation angle (in exceptional cases even above: the no longer existing tracks in Münster and Frankfurt am Main had superelevations of over 55 Degree).

Driver influence when cornering

Fine-tuning is left to the driver when cornering, without which a controlled drive would not be possible. In sporty cycling ( cycling ), additional techniques are essential to successfully negotiate bends. For example, the driver has to build up body tension , which is achieved by pressing down the almost stretched outside leg (pedals at the lowest point). In mountain biking, on the other hand, where it is more about a quick shift of the body's center of gravity due to the nature of the ground, a horizontal position of the pedals has proven to be more useful.

Evasive maneuvers

In the case of curves that are driven in the course of short evasive maneuvers, the counter-steering technique to initiate tipping is not necessary if the driver then wishes to continue driving on the original route. Instead of the technology described, the rider steers the bike past the obstacle while his body's center of gravity moves almost straight ahead. As a result, this technique is only suitable for avoiding obstacles close to the ground, potholes, etc. If used in the wrong situation, it will lead to serious falls. The driver does not make the decision about the technology consciously, but intuitively in a matter of tenths of a second.

Hands-free driving

Bicycle with coordinate axes

When driving hands-free, tilting the body to the side causes a steering movement. The more you tilt the bike, the more steering angle there will be, to a certain extent. Hands-free driving is almost impossible when driving slowly. Only the caster described below and the gyroscopic forces make this possible by triggering a controllable steering angle when the wheel is tilted and thus bringing the wheel back into the straight line of travel or enabling stable cornering. The self-adjusting steering angle also supports non-hands-free driving.

Gyroscopic effects - stabilization through precession

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Tilting the bike causes the handlebars to turn

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Steering the bicycle causes a tilting movement

The front wheel is a symmetrical, nutation-free top; the angular momentum , rotation and figure axes are identical. If you try to tilt such a rotating front wheel around the horizontal on a swivel chair, you notice that the chair begins to rotate. In order to formulate the effect in general, it is first assumed that such a top rotates around the axis of rotation. From now on it will keep moving. If it is then rotated a bit about a vertical line without any other influence, it also rotates a bit about the longitudinal axis. The effect is called precession .

The model can be applied to the front wheel in 2 ways.

If the wheel tilts a little around the red longitudinal axis, the wheel turns a little around the blue vertical axis. A tilting of the front wheel and with it the whole bike causes a steering, the handlebar rotates in the direction of the inclination.

On the other hand, if you turn the rotating front wheel with the handlebar to the left around the blue vertical axis, it tilts to the right around the red longitudinal axis. Steering causes the bike to tip in the opposite direction.

Design features of a bicycle that affect riding

Bicycle: positive caster, wheelbase

trailing

The caster is the distance between the front wheel contact point and the point where the imaginary extension of the steering axis hits the ground, the so-called toe point. The caster is determined geometrically by the wheel radius, the steering head angle (between the steering axis and the floor) and the fork bend (vertical distance from the hub to the steering axis). The name comes from the fact that the wheel “follows” the track point when it is turned. If the caster is positive, the track point is located in front of the touchdown point in the direction of travel as shown in the figure. The size of the trail is usually between five and seven and a half centimeters.

The caster is probably the most important structural support in the effort to prevent falling while driving straight ahead. It works in the following ways:

The gravity of the front wheel and the handlebars acting on the front axle when the wheel is inclined causes the handlebar to be turned in the direction of the incline. This effect is clearly visible if you hold the saddle firmly and tilt the wheel, the front wheel turns. The moment of the wheel contact force around the steering axis is also turning. Both effects increase with the tilt angle and support the driver. When the handlebars are turned, cornering is initiated in which centrifugal force straightens the wheel.
The cornering force creates a restoring moment. When cornering in a stationary position, the resultant of the side force and wheel load is roughly in the plane of the front wheel. The moments of the two forces around the steering axis cancel each other out. Overall, the handlebar is almost free of moments.

The experiment by chemist David EH Jones shows that driving with a negative trail is difficult. In 1970 he tried to design a bicycle that could not be driven. Most of the bike types developed were more or less still usable. Only a bike with negative caster was "very tricky" to steer and had negligible self-stabilization.

wheelbase

A bicycle with two wheels hits the ground at two points. The distance between these points is called the wheelbase . The reaction of the bicycle to a steering angle is approximately inversely proportional to the wheelbase. This explains the somewhat sluggish behavior of bikes with a long wheelbase, such as tandems .

A bicycle with the wheels widely spaced is less agile, but stays true to direction. If the wheels are close together, it reacts more strongly to steering movements, but this results in a rather nervous straight-line stability. Its maneuverability is used on racing bikes .

Above all, bicycles that are not designed to be particularly sporty have a wheelbase of well over one meter, a tandem even two. Racing bikes used in competitions have i. d. Usually a wheelbase of 97 to 100 centimeters.

To measure the wheelbase, measure the distance between the wheel center points (hub axle center point) with the handlebars pointing straight ahead, which have the same distance as the contact points on the ground, provided that the front and rear wheels have the same radius.

Wheel size and weight

The larger the diameter and heavier the wheels, the greater the gyroscopic moments. With a normal utility bike ( wheel diameter 60 cm, weight 1 kg) the gyroscopic effects are about five times as great as with a children's bike (30 cm; 0.4 kg). Correctly, of course, the gyroscopic torque does not depend on the mass of the impeller, but on the distribution of the mass in the impeller (moment of inertia with respect to the axle); For example, a wheel with a "heavy" rim has a higher gyroscopic torque than an equally heavy wheel with a "light" rim with a "heavy" gear hub. Bicycles, however, are designed to save energy and are therefore as light as possible.

Sitting position

If the driver shifts his weight to the rear wheel, less steering forces are required. However, this leads to oversteer and fluttering driving behavior due to directional corrections that are too far or too fast. If you lean forward and put pressure on the front wheel, greater steering forces are required. You understeer and your driving behavior fluctuates due to late and minor corrections.

Experience has shown that balanced riding behavior is guaranteed when 55 to 60% of the total weight of the bike and rider are on the rear wheel.

Frame size

Apart from the wheelbase, the frame size has no direct influence on the driving behavior. Nevertheless, the optimal adaptation to the body size makes it easier to control the bike. There are separate guidelines for each type of bicycle, for which your own stride length must be observed. We recommend choosing a smaller frame for a sporty driving style, and a larger frame for touring-oriented driving.

literature

Michael Gressmann: Bicycle Physics and Biomechanics . Moby Dick Verlag, Kiel 2002, ISBN 3-89595-023-8 .
Frank Bollerey: The problem of balance when cycling from a physical point of view . Thesis. University of Kassel, 1999.
Hans-Joachim Schlichting : On the balance problem when cycling. In: technic-didact. 9/4, 1984, p. 257. (Download 57 kB pdf)
H. Joachim Schlichting, Wilfried Suhr: Physics of Cycling: Modern Centaurs , Physics in Our Time, Volume 38, 2007, No. 4, pp. 184–188
H. Joachim Schlichting, Wilfried Suhr: The Physics of Cycling: Balance on Two Wheels , Physics in Our Time, Volume 38, 2007, No. 5, pp. 238–241
H. Joachim Schlichting, Wilfried Suhr: Physics of cycling: With pedal force against mountain and wind , Physics in our time, Volume 38, 2007, No. 6, pp. 294-298
H. Joachim Schlichting, Wilfried Suhr: The bicycle as an everyday means of transport: two-wheeled energy-saving box , physics in our time, Volume 39, 2008, No. 2, pp. 86-89
Gert Franke, Wilfried Suhr, Falk Rieß: Physical models of bicycle dynamics. Why is cycling so easy? , Pro Velo, Volume 21, 1990, Issue 5
David EH Jones : The Stability of the Bicycle , Physics Today, Vol. 23, April 1970, pp. 34-40
Roger Erb: On the problem of stability when cycling. In: MNU. 5/54, 2001, pp. 279-284. (on-line)
Felix Klein , Arnold Sommerfeld : About the theory of the top . Reprint of the first edition from 1897, 1898, 1903, 1910. Johnson Repr. U. a., New York et al. a. 1965, ISBN 0-384-29720-X .
FJW Whipple: The stability of motion of a bicycle , Quarterly Journal of Pure and Applied Math., Volume 30, 1899, p. 312

Web links

Wiktionary: Cycling - explanations of meanings, word origins, synonyms, translations

Commons : Cycling - Collection of Images

Wikibooks: Learning to ride a bike - learning and teaching materials

Individual evidence

↑ A single line of code was sufficient for the balance control. on: heise.de , November 2, 2012.

↑ New free-hand bike: It's the mass that matters. In: Spiegel online. April 15, 2011.

^ JDG Kooijman, AL Schwab, JP Meijaard, JM Papadopoulos, A. Ruina: A Bicycle Can Be Self-Stable Without Gyroscopic or Caster Effects . In: Science . tape 332 , no. 6027 , 2011, pp. 339–342 , doi : 10.1126 / science.1201959 . ,

↑ The suspension , chassis and suspension (“Why don't we just tip over with a single-track vehicle?”), Oelsumpfonline.de, 2004.

↑ Roger Erb: On the problem of stability when cycling . In: MNU - Association for the Promotion of Mint Teaching . No. 5/54 , 2001, pp. 279-284 . (online) (PDF 489 KB pp. 6-9)

^ David EH Jones: The Stability of the Bicycle. In: Physics Today. 23 (April 1970) pp. 34-40. ( PDF ( Memento of October 30, 2008 in the Internet Archive ), 9 MB, English).

[1] A single line of code was sufficient for the balance control. on: heise.de , November 2, 2012.

[2] New free-hand bike: It's the mass that matters. In: Spiegel online. April 15, 2011.

[3] JDG Kooijman, AL Schwab, JP Meijaard, JM Papadopoulos, A. Ruina: A Bicycle Can Be Self-Stable Without Gyroscopic or Caster Effects . In: Science . tape 332 , no. 6027 , 2011, pp. 339–342 , doi : 10.1126 / science.1201959 . ,

[4] The suspension , chassis and suspension (“Why don't we just tip over with a single-track vehicle?”), Oelsumpfonline.de, 2004.