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Let me suppose I have arranged a pendulum consisting of a
weightless rod with a massive bob on its end in a vacuum chamber.
The pendulum is suspended by a flexure made of a non-ideal material,
like a spring steel blade, which is fixed to a rigid immobile structure.
Let me suppose I provide a fixed kinetic energy increment each time
the pendulum crosses the vertical.
Absent electrical or magnetic fields, I can confidently assert that
the sole loss mechanism is at the flexure.
The question with which you seem to be wrestling presently, is the
functional dependency of losses at the flexure.
It is certainly reasonable to appeal to experiment for these data.
It will be entirely unsurprising if the flexure loss is proportional to
maximum deflection i.e. amplitude.
[Lacking this physical feature, a pendulum's amplitude would
increase without limit.]
It will be reasonable to suppose (though needing experimental support)
that the flexure's loss is also proportional to the pendulum's weight.
This is certainly what an engineer would assume, prima facie.
Very well then....for this preceding assumption, one can say
that if one doubles the bob weight mass, one doubles the flexure loss
component due to weight. This has the effect of reducing the amplitude
arising from the constant energy uptake per cycle.
However, the amplitude dependant loss component at the flexure
is reduced in this case.
But you were not concerned with amplitude as maintained, but
amplitude rate of decay.
Given that flexure energy loss is in fact proportional to deflection
i.e amplitude and also to pendulum weight, then it is evident that
amplitude decay rate is increased by increased pendulum mass.
All of which is to confirm that your original assertion given below
"For linear damping the decay constant is independent of the mass."
is inapplicable, for my reasonable assumptions.
Brian W
p.s. the math can be correct but inapplicable, if the model is
wrongly conceived?
At 01:52 PM 6/9/2007, you wrote:
Brian!
For linear damping the decay constant is independent of the mass. The
confusion is I'm probably using the wrong term. Should be using the
term "resistance"?, i.e. the term in front of the speed in the eq. of
motion. Equating the torques w/ the moment of inertia times the angular
acceleration: I alpha = - k*l*l theta dot - mglsin theta See. No m
in front of the damping term?
Simplifying (and linearizing) results in theta double dot + theta * g/l
+ k/m theta dot The loss part of the soln:
A = A(0) * exp(- kt/2m) This is what I want i.e. data on amplitude
change w/ time as a function of the mass. No hay nada, for bc's
search. Simanek has this in his lab manual, but retired w/o saving any
of his students' data.
bc, prays he has it "right".
p.s. In a synchronome the steady state amplitude, even when mounted
rigidly on a "brick" wall changes markedly w/ bob mass. This
theoretically won't happen if the support is absolutely rigid, as then
the amplitude, which is the measure of the total energy, equals the
input from the escapement. In the synchronome (and any other pendulum
clock?) the loss increases w/ bob mass from the support, or at least
that's the result found recently both by free decay and the amplitude.
A few other physicists have examined, of at least noticed, this also --
starting w/ Huygens.
* For a simple pendulum. I have to cheque my maths, but my impression
is, for a physical p,. it will change, because the equivalent length
changes. It will lengthen as the mass increase increasingly swamps the
moment of inertia of the pendulum's rod. This effect is rather minor
compared to the support loss.
Brian Whatcott wrote:
At 11:09 AM 6/9/2007, you wrote:
PHYS-L, PHYSHARE, and TAP-L, people!I expect I am entirely missing bc's point.
There is a controversy among the horological community on the effect of
pendulum mass on the steady state amplitude of an escapement driven
clock, in particular the impulsing by the gravity arm of a Synchronome
clock. Theoretically for a simple pendulum clock with a decay constant
independent of the mass, there is, obviously, no effect. However, the
free decay (they call it "run down") is, theoretically much effected;
the time is proportional to the mass. I have searched somewhat
diligently and find no data (experimental) relating to this effect. Do
any of you have such?
bc, frustrated.
This appears to be a question centering on mechanical Q: the ratio of
energy stored to the energy lost per cycle.
It seems like increasing the bob weight should increase Q,
But this may well increase the bob weight drag,
which should decrease Q. Hence the decay constant of a pendulum
should not be independent of its mass.
As to experimental data - this is the grist of engineering labs
such as this one:
<http://www.physics.odu.edu/hyde/Teaching/Fall04/Lectures/ResonanceLab.html>
Brian W
Brian Whatcott Altus OK Eureka!
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