I’m looking for something more than just simply the ‘gyroscopic effect’ as this is too vague. Why is it that whilst doing slow speed manoeuvres, it is infinitely easier to do with larger wheels than with smaller wheels?
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This is a question with more behind it than you might expect. It’s true the wheels on a bike act as gyroscopes (i.e. they spin, and something that spins is harder to pull to one side), but there are more effects that increase stability.
If you’ve ever held a spinning gyroscope, you’ll notice that the faster it spins, the more it wants to keep spinning in that same plane. When you do put a force on their axis and pull the spinning disk out of its plane, it doesn’t just go to the side, it also wants to go in another direction. It wants to wobble. In the case of a bicycle wheel, if you were to tip start tipping over, that effect causes the wheel to turn toward the side you’re falling to. Meaning you’re steering into the fall, which naturally causes the bike to go upright again. It’s a small effect but it plays constantly, the moment you go even a degree out of straight up, your wheels will slightly steer into that, like balancing a broom upright on your hand. The larger the wheel, the more effective that is, so the more stable.
But a bike’s stability is not just due to the gyroscope effect of the wheels. There have been test setups built that use multiple counter-spinning wheels to counteract the gyroscope, and they still stay upright. The reason is that the front wheel steers, with the back wheel trailing behind it. When you have a trailing wheel, it follows the path the bike goes, but it will add a resistance to going left or right. As you can imagine, a larger wheel makes for a better trailing wheel.
And then, even if you eliminate that trail, too, a bike can STILL stay upright. Bikes are built so that the main weight of it is on the back wheel, and the steering is behind the front wheel. Essentially, this means the back wheel will fall slower, as there’s more weight on it to move, and that again means the front wheel is quicker to turn into the fall, setting you upright again. There too, the larger the wheels, the further apart they are, and since we’re still sitting closer to the back wheel, it makes the weight distribution difference even bigger.
In conclusion: bicycles are really really really complicated and even this explanation isn’t complete on how they work, it’s an area of active research, believe it or not. Bigger wheels help in a lot of ways.
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