Why don’t bicycles fall over
Have you ever wondered what keeps bicycles from falling over? They only have two wheels and they are rather narrow wheels at that. We only have two feet but at least our feet are moderately broad and flat. In this article I’m going to explore this question, and use some basic physics and control theory to explain what really keeps bikes up. I will use the concepts from Newton’s laws of motion, but hopefully it reasonably readable to most people who are likely to find the topic interesting.
Paradox 1: Bikes are Unstable
Stability is an important concept in maths and science and has some rather abstract mathematical definitions, but it’s also a very intuitive concept, so you do not need to be familiar with the theory of stability to have an understanding of it. The scientific definition is not particularly different from the common understanding. Basically if you can give a system a little nudge and it doesn’t react very much, then it’s stable. If you give it a little nudge and it completely changes course or collapses then it’s unstable.
A pendulum is stable system. It may swing from side to side, but it will always settle down in time. The mass on the end of the string or rod will naturally return to being below the pivot point. A bike however, bares a similarity to an inverted pendulum. The rider sites on-top of the bike which is free to pivot from side to side. With just two narrow wheels, the bike can fall to the side just as a pendulum can swing to the side, but unlike a normal pendulum the weight is above the pivot point. We are over-simplifying in this analogy, a pendulum doesn’t involves wheels that go around and in our case the weight is a person who can move their arms and legs, but a simple model is a good place to start to build up our understanding of the physics. We will gradually add back in the more complex features of the bike as we consider why they don’t actually fall over.
A pendulum system is very simple to analyse. There is a mass on a string. The string is held at the top, but is free to pivot, allowing the weight to move in an arc. Gravity acts on the weight, and provides a force pulling the weight straight down towards the earth. The string is placed under strain by this weight, but resists being stretched. It bears the weight, and prevents it from falling by providing force upwards that counteracts the gravity on the weight. When the weight is given a nudge to the side, it starts to swing, and as it does so it changes the angle of the string. The string still has the tension in it, but now it’s not pulling straight up, it’s got an angle, so is pulling slightly to the side. It’s pulling the weight back towards it’s centre position. We call this a restoring force as it acts to restore the system to it’s equilibrium. A pendulum is stable because it has a restoring force. It’s slightly more complicated than that, I have not discussed how a pendulum continues to oscillate from side to side before settling down, but let’s ignore that for now.
Thinking about a bike as an upside down pendulum, we can go through the same thought process. The rider has gravity pulling them down. The bike, is under the rider, so rather than having a tension like the string and pulling up, the bike is in compression and hold the rider up by pushing against the gravity. The same concept but in reverse. Now when we give our hypothetical rider a bit of a nudge, they will lean to the side a bit. Gravity will still pull straight down, but now our bike is pushing up and pushing the rider slightly to the side. Unfortunately this pushing to the side force is not helping return our rider to the upright. It is not a restoring force at all, it’s a de-stabilising force.
So we have our first paradox, bikes are fundamentally unstable, yet they don’t tend to fall over, too often anyway. Before we solve the paradox, I’m going to introduce a second paradox.
Paradox 2: The rider cannot impart an external force
Let’s now consider what the rider could do provide that restoring force. Perhaps it’s the movement of the rider that stabilises everything. The rider can move their arms and legs, they can move their weight around and adjust the angle of the bike. Perhaps we need to move beyond seeing them as one mass that is simply, passively pulled down by gravity. But we need to be careful here. Newton’s first law of motion states: "an object at rest will stay at rest, and an object in motion will stay in motion unless acted on by a net external force". In our context we can interpret this as saying that if the bike was in motion, specifically if it was starting to topple, then it will continue to topple unless acted on by a net external force. The external part if crucial. No matter how much the rider is throwing the body to one side or the other, they are not providing an external force, they are just creating internal forces within the bike-rider system. If the bike is toppling over, the rider and bike will continue to topple according to the simple laws of motion. A bystander providing a shoulder to grab hold of, would be an external force, but that’s not really what we were looking for as our answer.
This notion of external forces, and the futility of internal forces, can be illustrated by thinking about an animal jumping through the air. When a cat leaps from a tree, it can move it’s legs and tail about in mid-air, and manipulate its shape, but it can’t provide an external force. If you plotted its center of gravity it would follow a simple, smooth trajectory. The overall motion of the cat is unaffected by the internal forces moving is legs. That cat can’t affect how fast is falls, just what way up it lands. Our bike rider, can control how he falls, he can chose to stick an arm out, but there must be a mechanism we have not yet considered that allows him to arrest his topple and stay upright.
The steering: moving the bike to the side
How can it be then, that we see inherently unstable systems riding around town with riders on-top unable to apply any external force to keep them upright? There is one important factor we have yet to consider. The rider can do more than move their weight around to try and stay upright, they can turn the handlebars.
So what effect does turning the handle bars have on bike and how might it help the rider stay upright? When the bike is moving the wheels are spinning and spinning objects do strange things when you turn them. Each wheel of a bike is in effect a large gyroscope. However the gyroscopic is not essential to understanding how a rider keeps a bike upright. The main effect of the steering is really rather simple: it moves the wheels to the side.
If you get a long stick, a broom or a hockey stick for example, and try to place it upright on the palm of you hand, you notice two things. Firstly, it’s precarious, it’s fundamentally unstable, it wants to fall over. But secondly you notice that, with some practice, it is possible to balance it. When it starts to fall to the left, you quickly shift you hand to the left, and it arrests it’s fall. The mechanism that you use to balance a hockey stick is in-fact the answer that we have been looking for. Bikes are kept upright by exactly the same mechanism. The rider can steer the bike, they can move the wheels to the side by turning the handlebars. If they are leaning to the left they can turn to the left and get the wheels to move over. It takes a bit of practice, but just like balancing a hockey stick you get the knack quick enough.
Resolving Paradox 1: Bikes really are unstable
Now that we have the kernel of an explanation, let’s go back and see how it sheds light on our previous considerations and paradoxes. First lets consider if we still think that bikes are unstable. Well, is a hockey stick unstable, yes, but it can be kept upright by a fast reacting hand. A bike is the same, it really is unstable, but it can be kept upright by a fast moving hand. The nuance here is that the fast moving hand is actually on the bike. So the bike itself is unstable and if you put a manakin on top of a bike it’s still unstable, but if you put a competent rider on top, and they apply their balancing skill, then you create a stable bike-rider system. There is an important principle here, it is possible to for a controller to stabilise an inherently unstable system.
Interestingly, a manakin on a bike can actually be stable, I should correct myself. But that is getting into the details of how a bike steers itself without a rider, and that should be the topic for another day. Certainly a manakin that held the handlebars firm and preventing any steering would very quickly end up on the ground.
Resolving Paradox 2: The rider cannot impart an external force
It is true that the rider cannot directly impart an external force on themselves. But what they can do is turn the handlebars so that fairly quickly the wheels have tracked over to the side. This has the effect that the bike now does impart a different external force on the rider. Similarly a cat flying through the air cannot affect how fast it falls, but a bird can. A bird does not fly simply by moving the weight of it’s wings. It fly by using it’s wind to disrupt the air flow, and it’s the air that imparts the external force that keeps them up.
Further reading: I have only touched on the simplest aspects of bike dynamics. To get a peek into the full breadth of what is going on when you consider all the moving parts, take a look at Andy Ruina’s website. He has studied bicycle dynamics in great detail and has some great resources there. Link