How is the suspension frequency response effected by the wheel diameter?

Question

Imagine riding off a curb, crossing a pothole, or hitting a pebble. Would you prefer to ride that with a toy bike or a penny-farthing? Regardless of preference, the penny-farthing would have a smoother ride, but how much smoother?

I want to know how much smoother a ride (less vertical acceleration) is when using a larger wheel diameter assuming the wheels are perfectly balanced and rigid. If I give a Power Spectral Density function of a road, how would the wheel diameter change its effect on the suspension?

I would expect that a wheel of diameter d bound the Amplitude A at distance frequency f to be

A < π^2/(2d*f^2) if A << d.

I would like any insights, and if you have questions, I answer the best I can.

Answer 1

You say some the wheels are rigid, does that include the tires? If so, then the vertical displacement of the bike and cyclists will be the same on either.

On a real bicycle, we use pneumatic tires which of course help mitigate the bumps. However, there exists a combination of tire width and pressures for each wheel that would again result in the same vertical displacement of the cyclist.

How are you defining smoothness of the ride? I assume you mean less vertical displacement for the cyclist, but your question is unclear.

If the moment of inertia is the same the bigger wheel will have a smaller horizontal deceleration from the pebble than the small wheel, due to the angle of stack of the pebble force being smaller. That is often referred to as rolling better or carrying speed.

If the moment of inertia is higher for the smaller wheel, it could actually have a smaller horizontal deceleration despite experiencing a higher horizontal force.