J
Joe Riel
Guest
While replacing a spoke on my Moulton---it has small diameter wheels, the spokes are approximatly
half the length of a 700C wheel---I wondered whether the force required to properly stress relieve
it, see Jobst Brandt's book "The Bicycle Wheel," was dependent on the spoke length. Intuitively it
seemed as though I should squeeze substantially harder to achieve the same tension increase.
Let
L = nominal length of spoke F = sideways force applied at center of spoke T = increase in spoke
tension D = sideways displacement of the spoke K = series compliance of the structure dL = strain
of structure
From statics, and using a small angle approximation
(1) F = 4*T*D/L Brandt's book [revised ed.] has an error
From geometry
(2) dL = L - 2*sqrt(L^2/4 - D^2)
Assuming D << L, this is well approximated by
(3) dL/L = 2*D^2/L^2
From elasticity, with K being the sum of compliances of the wheel structure and the spoke
(4) dL/L = K*T
Combining (1), (3) and (4) gives
(5) T = 1/2*K^(1/3)*F^(2/3)
L does not appear in (5), that is, the increase in tension of a spoke due to deflecting it
horizontally by a fixed force is independent of its length. There is a relatively small (1/3 power)
dependency on K, the compliance of the structure, which depends on a number of factors (number of
spokes, spoke thickness, rim shape).
Joe Riel
half the length of a 700C wheel---I wondered whether the force required to properly stress relieve
it, see Jobst Brandt's book "The Bicycle Wheel," was dependent on the spoke length. Intuitively it
seemed as though I should squeeze substantially harder to achieve the same tension increase.
Let
L = nominal length of spoke F = sideways force applied at center of spoke T = increase in spoke
tension D = sideways displacement of the spoke K = series compliance of the structure dL = strain
of structure
From statics, and using a small angle approximation
(1) F = 4*T*D/L Brandt's book [revised ed.] has an error
From geometry
(2) dL = L - 2*sqrt(L^2/4 - D^2)
Assuming D << L, this is well approximated by
(3) dL/L = 2*D^2/L^2
From elasticity, with K being the sum of compliances of the wheel structure and the spoke
(4) dL/L = K*T
Combining (1), (3) and (4) gives
(5) T = 1/2*K^(1/3)*F^(2/3)
L does not appear in (5), that is, the increase in tension of a spoke due to deflecting it
horizontally by a fixed force is independent of its length. There is a relatively small (1/3 power)
dependency on K, the compliance of the structure, which depends on a number of factors (number of
spokes, spoke thickness, rim shape).
Joe Riel