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First created: Sun Jun 10 12:44:42 EDT 2001
Last update: Sun Jun 10 12:44:42 EDT 2001

Breakdown

Overlooked in the simple model is the tension in the pendulum structure. If that tension exceeds what the coupling magnets can stand, the top and bottom pendulums decouple and there is a breakdown.

The top pendulum experiences a torque to return to the vertical, while the botom pendulum experiences a torque to fall away from the vertical. These torques are due to equal but opposite forces.
    Ftop(a) = m*g*sin(a)
    Fbot(a) = m*g*sin(b)
    a = b
(1)
where a and b are the top and bottom pendulum angles, respectively. The restoring magnetic force between the magnets is
    Fmag(a) = k/(r0 + h*a2)2
(1)
where r0 is the minimum distance between the magnets. There will be breakdown when
    Fmag(a) = Ftop(a)
    m*g*a =  k/r02*(1 - 2*h/r0*a2)
    m*g*a =  k/r02 + ...
    abreak =  k/(r02*m*g)
with the help of the oscillation period T
    T = 2*pi*r0*sqrt(h*m/k)
this can be expressed as
    abreak = (2*pi/T)2h/g
Increasing sensitivity (~ T), also brings about an earlier breakdown (abreak ~ 1/T2).

For T = 3 s, h = 25 cm, g = 980 cm/s2, this gives abreak*h = 4*25/980*h = 2.5 cm.

Which looks huge...