SH driven oscillator amplitude at resonance equation

[bgc]

I found via this forum the hint to use the inverse squared equation to differentiate to find the resonance frequency from the amplitude equation (equilibrium not transient solution). Thank you! (AlephZero?)

When substituting the resulting frequency for the resonance into the amplitude equation, I find the amplitude is a function of the mass of the mechanical oscillator. This violates centuries of horological experience!

bgc

p.s. Amp. (at resonance) = F/m / [dissipation constant/m * {W(0)^2 - (d.c./(m*2)^2)}^0.5]

The first two m's cancel (good), but there is another, bad! [The dissipation constant /m is the coefficient of the speed term in the driven damped (linear) harmonic oscillator differential equation; F is the amplitude of the sinusoidal forcing function]

 

[Philip Wood]

I've just checked your amplitude at resonance and agree with you except for a factor of 2 in the second term under the square root in the denominator. This may well be a slip on my part. In any case, I don't think it affects the point you're making.

First thing: this second term is usually very small compared with $\omega$₀².

But you think there should be no dependence at all on m?

I can't see anything wrong with such a dependence. For example, m certainly affects the half life of ordinary damped shm. A wooden pendulum of the same size and shape as a lead pendulum will take a shorter time to lose (say) half its amplitude. And I don't understand your reference to centuries of horological experience. Can you be more specific?

 

[bgc]

P.W.!


Horologist have tried to increase the amplitude of a 'failing" clock by adding mass to the bob. I didn't believe a friend's claim the amplitude was independent of bob mass. So I reviewed the differential equation and solution. The m's cancelled. Then a friend gave me an electromagnetic clock drive and I verified the analytic conclusion.

Description here:

http://www.cleyet.org/Pendula, Horo...; Energy Stored vs. Dissipation corrected.pdf

bc

p.s. it's difficult to compare various authors solutions (equations), because there is no standard differential equation.

 

[Philip Wood]

The factor of 2 was indeed a slip on my part. I agree completely with your formula. One interesting point is that increasing m actually decreases the amplitude at resonance. Have to admit I find this counterintuitive. But then, of course, it is a very small effect. So much so, that I wouldn't have thought that changing the mass of the pendulum in a clock would make any significant to its amplitude - at least not due to the effect given by the formula.

Of course, in a traditional escapement-driven pendulum clock, the driving force is far from sinusoidal, and it depends to some extent on the pendulum's amplitude, so we can't apply the formula with any confidence, especially if we're thinking of small effects such as the one we've been discussing.

Thanks for the link: some interesting experimentation.

Sorry not to have helped. I'm sure you know much more about theoretical and practical horology than I do.

 

[bgc]

Sorry not to have helped. I'm sure you know much more about theoretical and practical horology than I do.

Au contraire! You confirmed and did help.

bc more later?