A Discovery... thanks to Oxygon
Two equal weights, one movable in and out along a spoke, the second movable side to side straight line along a cord (a trig. term, NOT a string). The two weights are interconnected to each other so when one move the other moves, each the same distance. Gravity will shift the weights at about 45 degrees of wheel rotation past top and past bottom.
The discovery... the wheel will be very slightly out of balance! The below calculation comes very close to equaling the OB (Out of Balance torque) found using a computer program that sums the torques at each 1/10 degree full circle.
r = radius from center of wheel to center of small weight.
d = movement distance of a weight
a = angle that the small weight is on either side of the spoke.
OB = Out of Balance torque per one unit of mass
a = ATN( d / 2 / r )
OB = r * ( TAN(a) - SIN(a) ) * 2 * 2
A bigger angle produces a bigger OB so it is best to have the inner weight close to the center. The distance of the outer weight from the axle makes no difference.
Sample numbers...
r = 10
d = 1
a = 2.8624
OB = .0025
Another sample...
r = 4
d = 1
a = 7.125
OB = .0154
I have NOT looked at inertia, centifugal forces, etc. Maybe they cancel out the effect? And I hope I've not made some type of mistake!