Why is a bicycle stable?

Kvant 1970, No. 12 p. 24-27 Tag GK5C

Rough automatic draft, low quality: expect mistakes and awkward wording. It will be replaced by a better translation. Original →

WHY
SUSTAINABLE
BIKE

Almost everyone knows how to ride a bicycle, but, apparently, they still don’t know how they do it. Every simplicity and ease hides unexplored subtleties, and I spent a lot of time and effort trying to uncover the reason for the bike’s stability. Publications on this topic, as a rule, are fragmentary and do not contain experimental testing of the theory. In my research, I hoped to find out the reason for the stability of shared bicycles using modified bicycle designs. The first attempts to create a bicycle on which one cannot ride ended in failure. This led me to closely study the shape of the front wheel fork. After this, with the help of calculations on an electronic computer, I designed and built a perfect unstable bicycle.

This article is based on the article “The Stability of the Bicycle” by D. E. H. Jones, published in Physics Today No. 4, 1970.

Translation and revision of the article by D. E. H. Jones was carried out by V. N. Berezny and G. N. Sukhova.

Existing theories

Most mechanics textbooks and manuals on cycling either do not consider the issue of their stability at all, or consider it too simple. It is believed that the bicycle is balanced by the success of the bicycle: when the bicycle perceives that the car is falling, it turns it in the direction of the fall. The bicycle spins and rides along a curve. This helps him return to an upright position and avoid falling.

It’s not hard to understand why this happens. Let’s consider the forces acting on a bicycle with a VELOSPEED (Fig. 1). F is the reaction of the earth (it is equal to the sum of the friction force Ftr and the normal reaction force N), P is the gravitational force. The resultant of these forces is equal to P·ctgα (where α is the angle of inclination of the car) and is directed towards the center of the circle along which the VELOSPEED travels, communicating

Fig. 1.

changing the center of gravity of the bicycle with the cyclist’s acceleration

$a = \frac{P}{M} \text{ctg} \alpha = g \text{ctg} \alpha.$

Since friction is usually high and the wheels of a bicycle do not slide on the ground, the acceleration of the point of contact of the wheel with the ground is determined by the radius of the trajectory and is equal to $\frac{v^2}{R}$ . If the angle of inclination of the cyclist is such that the acceleration of the center of gravity of the car with the center is equal to the centripetal acceleration of the point of contact of the wheel with the ground, then the cyclist does not fall or straighten out: the force $F$ passes through the center of gravity of the system; if $a > \frac{v^2}{R}$ , then the cyclist will fall (the force $F$ passes above the center of gravity of the system); if $a < \frac{v^2}{R}$ , that is, $R < \frac{v^2}{g \text{ctg} \alpha}$ , then the bicycle will flatten (the force $F$ passes below the center of gravity of the system), creating a moment that rotates the system around an axis passing through the center of mass).

If the cyclist is leaning at an angle $\alpha$ , then in order for the car to straighten out, it must be directed in a circle with a radius $R < \frac{v^2}{g \text{ctg} \alpha}$ .

The higher the speed, the larger the radius of the circle along which you need to ride in order not to fall. This means that at higher speeds the bikes will level out if the handlebars are turned at a relatively small angle. Therefore, the faster the bicycle moves, the easier it is to control.

However, observations reveal convincing possibilities against the theory, in which it is believed that the bicycle is stable only thanks to the driver. After all, if you push a bicycle without a cyclist, it will travel a fairly long distance, although it is not controlled, and will control it in about 20 seconds, and not in 2 seconds, like a stationary bicycle. It is clear that the car is stable not only due to the efforts of the bicycle;

Fig. 2.

dista. In addition, when riding fast, the bike is only stable, making it difficult to tip over even if you want to.

A more complex theory explaining the stability of a bicycle takes into account the hygroscopic effect of its front wheel.

A gyroscope is usually called a body that rotates quite quickly around its axis. A classic example of hyprosopp is ordinary fiber. When you try to tilt the spinning wolf’s axis, you will feel resistance. The axis of the wolf begins to move not in the direction in which you are pushing it, but in a perpendicular direction. This phenomenon is called the “gyrioscopic furnace effect.” Thanks to it, the axis of the spinning wolf usually moves along a cone, as shown in figure 2: under the influence of gravity directed vertically, the axis turns in a horizontal plane. The direction of rotation of the axis depends on the direction of rotation of the wolf around the axis.

Fig. 3.

The greater the mass of the top and the speed of its rotation, the greater the gyroscopic effect: the faster the axis of the top rotates when trying to tilt it. The rapidly rotating front wheel is the same as a top. When tilting forward, forces act on the wheel axle, tending to rotate it along with the forward motion. In this case, the axle, fork and steering wheel, due to the gyroscopic effect, must turn in a plane perpendicular to the plane of acceleration (Fig. 3). It turns around and thanks to this it returns to a vertical position.

The appeal of this theory is that it is imitated by a rolling hoop, which is truly stable for this very reason. Vvosped in this interpretation is simply a hoop with a sight.

Some have theoretically expressed doubts about this theory. They felt that the weight of the wheel was too small to create the gyroscopic effect sufficient to level a heavy wheel.

Anti-gyroscopic acceleration

I began my series of stability experiments with rather vague ideas about the theories discussed here.

It occurred to me that it would be interesting to build an unstable ride that would knock the most experienced rider out of the water. First, I modified the standard vsped by placing a second wheel on the front fork, off the ground, so that this wheel could be rotated in the opposite direction to the direction of rotation of the front wheel in the vsed. This was supposed to eliminate the influence of the gyroscopic effect. I called this device “uncontrolled vsped”—HB-I (Fig. 4, 6). He did not justify the hopes that arose in him; it could be easily controlled both when the additional wheel rotated in any direction and when it remained stationary. The NB-I behaved somewhat awkwardly (I attributed this to the increased weight of the fork and front wheel), but this did not make it too difficult to control even with my average driving abilities and low speeds. This means that the gyrosoptic effect had very little effect on forward control even at low speeds.

The unexpected result inspired me. If the vvosped, considered as a hoop with a sight, is not stabilized due to the gyrosoptic effect, then perhaps the hoop is not a gyrosopp at all?

I continued the experiment with an additional HB-I wheel, attaching to it an internal hoop that could rotate in the opposite direction. The device fell safely when I encouraged it to roll. This means that the hoop behaves like a girrosopp.

Another, more severe test was carried out: will the HB-I be controlled if the driver’s hands are removed from the steering wheel? According to what is probably the only theory of driving forward control “without hands,” it was assumed that the driver tilts the frame with angular movements of his body and thus steers using the rotation of the front

Fig. 4.

its wheels due to the gyroscopic effect.

Carefully and with great fears for the result, I conducted an experiment during which the bicycle was driven downhill; thereby eliminating the influence of the bicycle on teachers. HB-I is not very convenient for “hands-free” riding, even in the “static” version. Something was lost in balance, and in the case when the front and additional wheels turned in different directions, it was difficult to control, but still possible.

This makes me think that there is another force that appears when the bicycle is moving.

Then I tried to roll the HB-I without a driver. When the additional wheel rotated in the opposite direction to the direction of rotation of the front wheel of the bicycle, the bicycle fell awkwardly; when the wheel rotated in the same direction as the front wheel, the bicycle moved quickly, having previously run a circle.

It became clear to me that a lightweight, riderless bike could stabilize the gyroscopic ones, but a heavier model could not. Thus, it is still possible to solve the problem of stability of a bicycle weighted by a driver.

More theories

In the preliminary stages of the research, I put it on everyone I knew, explaining to them the theory of the bicycle. In addition to the two well-known theories that I have already mentioned, I also proposed new ones.

The bike stays upright thanks to the thickness of its tires, meaning it is like a thin inflatable roller.
If the bicycle leans, the point of contact of the front wheel with the ground moves to one side of its plane, creating a frictional moment that turns the leaning wheel. This keeps the bike level.
The contact point of the bicycle tire with the ground is located behind the handlebar control axis (Fig. 4). As a result, when the bicycle leans, a ground reaction force F will occur, which turns the front wheel around the steering axis.

One might assume that theory 1 is not that important. I was also suspicious of theory 2, since a force acting over less than half the width of the tire should produce a very small torque depending only on how tightly the tire was inflated.

Theory 3 raised questions about shape