a beautiful question with no good answer. bicycle design has remained largely uncharged for at least a hundred years, i fell down a youtube rabbit hole on this subject a bit ago, and well everything plays a part sure. but i bike without a stearing mechanism is also self balancing, or a bike with out gyroscopic force will also balance itself, a bike without a rider can balance itself while in motion.
Really there is no complete answer!
bicycle is stable thanks to a combination of the front wheel touching the ground behind a backwards tilt steering axis, the center of mass of the front wheel and handlebars being located in front of the steering axis, and the gyroscopic precession of the front wheel.
In High school my science teacher brought out a bicycle wheel that had a handle through the middle. He had you turn the wheel while not in motion (moves easily) and then he would spin the wheel and have you try again (much more difficult) to demonstrate the different force while the wheel is in motion. It was interesting.
There are several functions that contribute to bike stabilty. First, the line thru the steering tube extends down to the ground a couple of inches ahead of the tire contact point. So if you lean to the right, that distance makes the front wheel turn to the right and if you’re rolling forward, the wheel rolls in under you and sets you back up straight.
Also, the tire has width, so when you lean to the right, you’re rolling on the right side of the tire. It has rolling friction so it tends to steer to the right same as the above.
Third, if you’re moving at a good clip, the front wheel becomes a gyroscope. Lean to the right and the wheel transfers the torque into steering to the right.
Finally there’s you, correcting any leftover imbalance. You can launch the bike down a hill by itself and it will stay up, but it might hit something
When you’re in motion on a bike, there are two forces acting upon you.
Gravity
&
Velocity.
In order to balance on a bike whilst static, you need to centre your body weight directly through the centre of the bike to the floor. This is because gravity pulls you down.
In order to balance on a bike in motion, you need to centre your bodyweight broadly on the bike relative to the speed. This is because velocity pushes you forward.
In order to not fall off a bike, you need to apply more velocity than gravity to the bike and yourself.
Think of it like this… When you jump, you’re pushing yourself off the ground. How can you do this if gravity is pulling you down? Well, you’ve applied more force upwards than the force pulling you down. However, once you’ve jumped the ability to add more force going up stops, and so you must come back down.
So, back to the bike… Once you’re in motion you need to apply more force to keep yourself going forward (velocity). If you apply more force to go forward than the force pulling you down, you’ll suddenly find it much easier to balance.
This is because you’re no longer trying to balance yourself and the bike, just yourself on a bike that’s being driven forward rather than down.
The fundamental geometry of a bike causes it to self correct when it’s in motion. It may not be immediately apparent but when you turn the handlebars the ground resistance against the front wheel turns it in such a way that it’s restored to upright and straight travel. This is mainly achieved through the angle of the front forks relative to a perpendicular to the ground axis.
This can be easily seen if you send a bike careening down a road at speed without a rider. It might wobble yes but it generally comes back to being upright, and you can see it happening as the bike leans and weaves but ultimately comes back to going straight.
When you’re not moving you lose this force acting on the bike so it cannot stay upright.
The mech ice of having your weight on top of the bike as well as the angle of the front wheel fork is self-correcting.
The first thing to notice about a bike is that leaning your weight turns the wheel in that direction, causing you to turn that way.
The next thing to notice is how you actually use the handlebars to steer. Next time you’re on a bike coasting at a reasonable speed take your hands off the handlebars if you can, or just hold them very lightly. Then pull one handlebar slightly: you’ll notice that when you pull the handlebar it will send the wheel in that direction, but not your weight. This will initiate a lean in the opposite direction. Thus to turn left, you actually steer right. Turning and controlling the bike is all about moving your weight, and we just use the handlebars to control our weight.
This is constantly happening as you ride: you lose balance to your left and start leaning left, the bike naturally turns left putting the bike back under your weight, stabilising it.
While riding slow or being stopped, the bike is itself unstable, but when moving at a reasonable speed, even a riderless bike can keep itself balanced.
As to why a riderless bike can keep itself balanced when moving, that’s sort of still an open problem. Part of it is because the wheels of the bike act as gyroscopes. Part of it is that the point of contact of the front wheel is behind where its axis of rotation intersects the ground. Part of it is a complicated thing with how the shape of the wheel acts. But even those explanations together don’t give the entire answer
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