After 1,935,960 hours, we solved the mystery of bicycle balance

That day, a colleague sent me a video and said he wanted to test me.

Source: Bilibili @TheQ Handicraft

"Why can't the wheel be balanced if it's taken apart?"

Can bicycle problems stump me, a little veteran rider with twelve years of experience?

To understand why this type of bicycle cannot balance, we just need to analyze why ordinary bicycles can balance.

The reason why ordinary bicycles can maintain balance is because...

Because of why?

If you ask me how to control the balance of a bicycle, I know this question very well and can even write a paper about it.

But why can a bicycle balance?

Could it be that I have been riding a fake bicycle for all these years?

With this question, I tried to look up information to find the answer. I was surprised to find that the problem of bicycle balance has troubled the scientific community for hundreds of years. It is really not simple.

Part 1

Bicycle: I bet you don't understand my balance

It is said that the earliest bicycle was invented in 1790 by a Frenchman who thought that a four-wheeled carriage took up too much space, so he simply removed half of it and left only two wheels. This was the prototype of the bicycle.

Wooden draisine, an early bicycle that appeared around 1820, was powered entirely by pedaling, but the front wheel could be steered | Source: Reference 4

After hundreds of years of development, bicycles have gone through various strange designs and finally became the shape we see today.

It is such a familiar means of transportation that has puzzled scientists.

If you ask how air conditioning controls temperature, scientists can tell you about the Carnot cycle.

Schematic diagram of reverse Carnot cycle refrigeration | Source: benchmark-id.com

If you ask about the principle of remote control, scientists can explain it from infrared rays and coding sequences.

Infrared detector | Source: Baidu Images

However, as a product of life experience, when the bicycle was invented, no one considered why it could be balanced. When later generations tried to analyze it, they found one problem after another.

Why can a bicycle only stay upright after it is started? What role does speed play in this? Why can a bicycle remain balanced even without anyone controlling it when it is pushed hard?

A mature bicycle should learn to drive itself | Source: Reference 3

A hundred years after the invention of the bicycle, British mathematician Francis Whipple derived the Whipple model , which uses four rigid bodies to represent the bicycle and introduces 25 parameters such as mass and wheel radius to describe the motion of the bicycle. It is one of the earliest and most enduring bicycle models in the world.

Whipple model, describing the motion of a bicycle using the motion of four rigid bodies: R, B, H, and F | Source: Reference 1

Unfortunately, this model only provides us with a simulation method but does not point out the principles behind it.

Part 2

Sommerfeld et al.: Gyroscopic effect can be solved

By the 20th century, the problem of bicycle balance remained unresolved and attracted the attention of a large number of scientists, including an old acquaintance in our physics world, Arnold Sommerfeld.

Arnold Sommerfeld, a famous physicist, one of the founders of quantum mechanics and atomic physics, discovered the fine structure constant. | Source: Baidu Image

He and two other scientists, Felix Klein and Fritz Noether, came up with an explanation: the gyroscopic effect .

What is the gyroscopic effect?

The gyroscopic effect is actually a manifestation of the angular momentum theorem. Under the action of external torque, the angular momentum of a rotating object changes, generating precession angular momentum. In other words, under the action of the gyroscopic effect, a rotating object will not fall directly, but will change its direction of rotation.

You must have played with a spinning top when you were a child. When a spinning top spins, it rotates around an axis, and at the same time, this axis also rotates around a vertical axis. This phenomenon in which the axes of rotation of a rotating object rotate at the same time is called precession .

Source: Bilibili "Inception"

The precession state of a gyroscope is very stable, and it can maintain balance even if a certain external force is applied. This is the gyroscopic effect. For example, a gyroscope uses the gyroscopic effect to keep a very large mass from falling.

The wheel maintains its stability under the action of gyroscopic effect | Source: Bilibili @NASA Chinese synchronization

In fact, the animated picture above has explained it very clearly. Under the influence of the gyroscopic effect, the wheel overcomes gravity, forms precession, and maintains its own balance by changing direction.

This state of balance is related to the speed of the wheels. The faster the speed, the smaller the inclination of the wheels and the more stable the body. So when we ride a bicycle, we will find that the faster we ride, the easier it is to keep balance.

Now let’s go back to the original question, why can’t this car be balanced?

After the front wheel is disassembled, only one-third of it can be in contact with the ground at the same time. The other parts have no direct external torque, cannot form precession, and have no gyroscopic effect to keep it stable.

Part 3

Jones: It also depends on the "steering wheel tilt"

In fact, so far, the gyroscopic effect explains the question at the beginning and has convinced the editor well.

Do you think this is the end? NO!

Sixty years after Sommerfeld and others proposed the explanation of the gyroscopic effect, a chemist named David Jones published an article that overturned the explanation of the gyroscopic effect.

Didn't it say that the gyroscopic effect is applicable to common car models? Jones said, then I will design a bicycle that is not applicable. So there is the bicycle in the picture below.

Source: Reference 3

This bicycle has two front wheels, one large and one small. Through the transmission between the large and small front wheels, the two front wheels rotate in opposite directions . By controlling the size ratio, the angular momentum of the two front wheels can be equal in magnitude and opposite in direction, that is, completely canceled out!

If the angular momentum is gone, how can there be any gyroscopic effect? ​​Doesn’t this prove that it is wrong to use the gyroscopic effect to explain the problem of bicycle balance?

Of course, Jones is not the kind of person who only cares about overthrowing without thinking about rebuilding. He proposed a new theory - the Caster Effects.

In this theory, there is an important concept: the front wheel track (the trail) . The front wheel track refers to the distance between the intersection of the extension line of the front wheel steering axis and the ground and the contact point between the front wheel and the ground. If the intersection of the steering axis is before the contact point, the front wheel track is positive; otherwise it is negative.

The red line c is the front wheel track

Jones pointed out that when a bicycle starts to tip over, the positive front wheel trajectory will cause the front wheel of the bicycle to tilt backward under the action of gravity, driving the front of the bicycle to turn. Through this turn, the bicycle returns the center of gravity to the middle of the body and returns to a balanced state.

The longer the front wheel trail, the more stable the bike will be (although also more difficult to ride); negative front wheel trail will make the bike unbalanced.

The blue line represents the extension line of the steering axis, and the black line represents the ground

Now let's go back to the problem we had at the beginning. After the wheels were split into three parts, the excessively long front wheel caused the front wheel trajectory of the car to become negative most of the time , so it was impossible to maintain balance.

Part 4

Schwab et al.: You are all right, but...

It seems that the "front wheel track" can indeed be used very conveniently to judge and explain the balance problem of a bicycle.

Jones was also very proud of his theory. In his memoirs published 40 years later, he regarded this theory as one of his great achievements and declared: "I am now hailed as the father of modern bicycle theory."

I am now hailed as the father of modern bicycle theory.

——David Jones

However, what the scientific community lacks the least is reversals.

In 2011, the year after Jones’s memoir was published, an article titled “A Bicycle Can Be Self-Stable Without Gyroscopic or Caster Effects” was published in Science magazine.

The research team in this article did what Jones did: design a bicycle that could remain stable after eliminating the above effects.

In Jones's theory of steering wheel caster, the front wheel casts backward under the action of gravity, so the weight of the entire vehicle body is redistributed to offset the negative front wheel trajectory. Then the gyroscopic effect is eliminated, and the following bicycle is obtained.

Source: Reference 1

The front and rear wheels are designed as two wheels, one above the other. The two wheels turn in opposite directions, and their angular momentum cancels each other out. The intersection of the extended line of the steering axis and the ground is after the contact point between the wheel and the ground, and the trajectory of the front wheel is negative.

According to the two theories mentioned above, this car cannot maintain balance, but...

The car is not only balanced, but also very stable.

The author of the article explains that by moving the car's center of gravity forward, the impact of the negative front wheel trajectory can be offset, so that when the car tip over, the front wheels will still tilt backward to drive the car to turn, bringing the car's center of gravity back to the middle and restoring balance.

This is the third way a bicycle maintains balance - by changing the mass distribution .

But this article does not deny the first two explanations .

This is also the breakthrough point of this article. It believes that the three theories of gyroscopic effect, steering wheel tilt effect and change of mass distribution mentioned in the article can explain the balance problem of bicycles and are sufficient but not necessary conditions for bicycle balance.

Careful readers may have discovered that the three theories have one thing in common: turning the tilt of the vehicle body into steering , which is the fundamental reason why bicycles can maintain balance.

When any one of the above three theories is met, it can be inferred that the car can maintain balance; and when none of these three are met, the car can also maintain balance as long as the tilting of the car body can be converted into turning.

Part.5

Rear Wheel: Does anyone remember me?

At this point, the bicycle balance problem has been solved, but not completely.

We know that the fundamental reason why a bicycle maintains balance is to transform leaning into turning , and we also know three theories that can achieve this transformation and the corresponding bicycle designs.

But we still don’t know if there are more theories that can achieve this transformation, or if there is a theory that can provide a necessary and sufficient solution to bicycle balance. We still lack a “grand unified theory” of bicycle balance.

Oh, and I'd like to remind you not to forget the rear wheel. Although it doesn't seem to exist in any of the three theories (and it really has nothing to do with it), it is still an indispensable part of the bicycle. Without it, the bicycle cannot... can a unicycle also be ridden? That's fine.

Back to the question at the beginning, has the problem of the bicycle not being able to balance been solved?

Solved, UP restored the front wheel to its original state.

References:

1. Borrell, B. The bicycle problem that nearly broke mathematics. Nature 535, 338–341 (2016).

2. Kooijman JD, Meijaard JP, Papadopoulos JM, Ruina A, Schwab AL. A bicycle can be self-stable without gyroscopic or caster effects. Science. 2011 Apr 15;332(6027):339-42.

3. https://mp.weixin.qq.com/s/Z7c5lJJhnG_Hvq-aSnHxDw?scene=25#wechat_redirect

4. https://encyclopedia.thefreedictionary.com/bicycle Source: Institute of Physics, Chinese Academy of Sciences

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