Bouncing at speed can happen when you are pushing the limits of your current level of ability. What happens is you think you’re pedaling faster, but you’re just pedaling harder!
When you are pedaling as fast/hard as you can, the pedal stroke is often reduced to a hard downward push rather than a smooth circular movement.
What then happens is that the entire mass of your pedal, crank, foot, ankle, shin, knee and thigh moves up and down rather faster than you can control it. What stops your foot carrying on down towards the ground is no longer fine motor control in your leg, but the simple fact that the pedal is in the way!
Add to this effect Newton’s famous “equal and opposite reaction” and you will see that as your pedal, crank, foot, ankle, shin, knee and thigh move downwards, your body will tend to move upwards. (Yes, I know that there is a {pedal…thigh} set of approximately equal weight moving upwards at the same time, but there is still an effect on the rest of your body.
So, the pedal strokes become jerky, and you start to bounce a little on your seat, and the combined effect compresses and releases the tyre, magnifying the effect - if it’s a big fat slightly squishy tyre, even more so. (This is related to why units of soldiers have to break step as they march across a bridge, to stop the oscillation making the bridge collapse.)
So, how do we deal with this?
In no particular order:
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Shorter cranks will mean that the mass of your {pedal…thigh} is moving less distance up and down. At the same wheel speed, therefore, there will be less up and down oscillation - the bouncing you were complaining about.
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Shorter cranks also change the leverage, so that the wheel has more control over your legs! Yes, really! It is harder to accelerate or decelerate the wheel if you have short cranks. This means that the flywheel effect of the wheel becomes more noticeable, and will help to smooth out the jerkiness of your pedal strokes.
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Possibly: adjust the height of your saddle. If you can get into a position where you are sure that your weight is resting on your saddle throughout the pedal stroke, that will stop your body mass bouncing up and down.
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Possibly, hold the handle or front of the saddle. This will help you to concentrate on staying still in relation to the seat.
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Build up to speed. Just going flat out will teach you bad habits.
Do it in stages: ride steadily, then gradually accelerate for 50 metres or so then back off a bit. Regain your composure, then bring the speed back up for 50 metres or so. After a few goes at this, bring the speed up, and keep it up for 100 metres or so, and near to the end of that 100 metre section, accelerate just that little bit faster.
What will happen over time is that your cruising speed will increase, meaning that your 50 metre burst speed will increase, and your 100 metre burst speed will increase and so on. A high top speed has to be built on a platform of sustainable fast cruising speed. You have to build that platform up gradually.
Have fun.