After the recent comments on the difficulty of adding dissipation in
the L. formulation**, I hesitantly added ViscousDampingCoefficient*
(V1) to
X1 (double dot) = F(X1, V1, K, X2, V2, L, m1, m2 ) where X1 is the
position of the mass (m1) of the spring oscillator, V1 it's speed,
and X2 is the angle of the coupled pendulum, its bob mass (m2) and V2
its angular speed. the m1 is the support for the suspension (a
rotary motion detector) of the pendulum. K is the spring constant
and L the rod length. There is a corresponding X2 (double dot), but
I ignore its dissipation, because the pendulum's Q is > 1k. The Q of
the spring oscillator is very low. (~ <100).
The two X's (double dotted) were found by plugging and chugging into
the formula the L=T-U. Separating the accelerations was VERY time
consuming! I checked the result by linearizing the sines and
cosines and comparing to AP French's small angle derivation***. Also
comparing the exptl. curves (data) to that predicted by my analytical
result is "reasonable" To begin w/, I'm using the linear
dissipation, but the exptl. trials show the dissipation is mainly
Coulomb. If what I've done (added the VDC*v1) is valid, I'll try the
Coulomb (signum[V1]*Cdc)?.
comments please.
** Is this only in the before plugging and chugging?
*** Initial trial angles ~ > one radian.
bc over his head in theory.
p.s. this work is of horological interest, as a principle problem
with very high Q pendulum clocks is support movement; both due to
very heavy bobs and microseisms.