General Cycling Discussion - Effect of a smaller wheel on bumps

So I would like to know how much of an effect wheel size has on ride quality. Ride quality being the time and amplitude of the accelerative forces generated by bumps in the pavement.

Assume a rigid wheels of iso 622 and iso 540, 25mm tires at 100 psi, forward speed of 25kph, 75kg bike and rider, standard 1g of gravity. Rider can be rigidly attached or not, and energy loss can be figured in or not, please specify though.

The obstacles will be a 3mm 6mm 15mm and 30mm step up, 3mm 6mm 15mm and 45mm step down, and a 100mm wide bottomless gap.

If anyone would like to tackle this physics problem then go for it.

:trainwreck:

:popcorn

I worked a most marvelous solution on the margin over there ====>>>
But there wasn't enough room to post in the forum.
But I proved that bigger bumps bump you more, and
that bigger wheels bump you less.

when you hit a bump, the bump applies a force directed towards the hub of the wheel. I'm assuming that the more annoying portion of this force is the part thats directed horizontally and in the opposite direction of travel. See the figure of trig terms here (2nd one down on the right):
http://en.wikipedia.org/wiki/Trigonometry

Imagine that the "wheel" is rolling up the page and encounters the bump. So we have
bump size = Radius - cos(theta) = the green section
and we solve for theta so that
theta = inverse cosine (1 - (bump / Radius))

once you have the angle, the horizontal portion of the force is:
Force * sin(theta)
Lets assume the force is primarily based on the speed, wheel inflation, and weight, and is accordingly a fixed value, its just being applied at a different angle due to the wheel size. I came up with the smaller wheel applying roughly 7% more force than the bigger wheel for each of your "step ups". (between 6.77 and 7.09 percent, smaller value for larger bump)

So would it be resonable to assume that 7% more horizantal force would mean the wheel takes about 7% less time to rise from first contact to full bump elevation. Or is it just a different acceleration curve?

no, because thats an instantaneous force when it first hits the bump. It will be reduced as soon as the wheel starts moving up and over the bump (i think).

there are more difficulties to consider, its not really an easy problem from what i can tell. The bigger wheel receives less force in the reverse direction. It also spreads out the impact over a slightly longer travel path, which means less force required to do the same amount of work. I think i'll get back to my day job :)

Yes. Now who can say by how much? Anyone have the acceleration sensors and high speed cameras to do this the empirical way?

Somebody else can do the math, but as a practical matter, you're not going to notice much difference over bumps. Grant Petersen at Rivendell is the most recent champion of 650B wheels, and I got wheels from him to convert an old Bridgestone my wife didn't use much because the widest 700s it could fit were 25mm. The smaller wheels gave more clearance for fatter tires, and it transformed a bike she didn't like at all to one she loves. It's 12cm too small for me, so I can't really get a good feel for specifics, but she's an experienced rider, and she says it's more comfortable, more stable and no slower. whetever it loses, theoretically, in hitting obstacles seems to be more than countered by the increased volume of the tires.
You can read Grant's pitch at www.rivbike.com

I am more interested in comfort than efficiency but either way it must be apples to apples (same size tires and pressures, say 25mm 100psi) ISO 584 [650b] with a larger tire is close to the same dia. and has more cushion. I selected 3mm bumps to simulate a rough chip seal, 30-45mm big bumps for pavement edges, stuff in between for graphing. 100mm wide gap for a small potholes, cattle grate, or storm drain.

Small or big, wheels have no effect on bumps ;)

Small or big, wheels have no effect on bumps ;)
These wheels do... 175 short tons :D
http://www.mackenziechamber.bc.ca/images/tree_c2.jpg