View Single Post
Old 10-11-11 | 03:30 PM
  #2  
gerundium
Senior Member
 
Joined: Sep 2011
Posts: 329
Likes: 11
From: Netherlands

Bikes: BMC

all of you have seen newtons law, F = M*A

M = mass
F = the net force on the mass
A = acceleration.

for rotational systems a similar relationship holds.

M = I*a

M = the net torque on the system
a = the angular acceleration.
I = moment of inertia

For a wheel spinning up the mass is not as important as it's moment of inertia. more about that can be found here: Wiki but this contains mostly definitions and is hard to grasp without a solid mathematical base. for a simple approximation we can estimate the moment of inertia by considering it a combination of a thin circle (the rim) that rotates around it's center axis and a set of rods rotating around it's outer point. this ignores the hub, but that is not a problem because in the definition of the moment of inertia you multiply by the distance of a mass from the point of rotation squared. this means the hub has a very small impact on the total compared to the rim (due to large distance it is away from the center axis. This calculation is something i think is reasonable for a ballpark figure but i am not completely sure, a highly skilled mechanical engineer will probably weigh in to correct me if i am wrong. Civil engineering student myself so moving systems are a little rusty.

my approximation yields the following

I(rim) ~= m*r^2
m = mass
r = radius of the rim

I(spokes) ~= n*(1/3)*m*l^2
n = the number of spokes
m = mass of a single spoke
l = length of a single spoke
(this approximation assumes the end of the spoke to be in the center axis, which i will leave as is for simplicity sake)

I(hub) ~ 0
ignoring this for simplicity

I(total) = I(rim)+ I(spokes)+ I(hub) = n*(1/3)*m(spoke)*l^2 + m(rim)*r^2

you can try filling in this above relationship yourself if you know the mass of your rim and spoke seperately, and then you can compare between different wheels by comparing the approximation. Do note though that we have approximated the rim as a thin circle so for high rims like Zipp 404's for example this is incorrect and another more complicated approximation is in order and one that i can't come up with so easily (for those interested you would need to take the moment of inertia of a solid disk with the radius equal to the radius of the outer side of the ring and subtract a solid disk with the radius of the inner radius of the rim.)

Last edited by gerundium; 10-11-11 at 03:36 PM.
gerundium is offline  
Reply