Re: [Phys-l] Q, mass (or L), and the driven response of an oscillator

[Bernard Cleyet <bernardcleyet@redshift.com>]

On 2012, Mar 23, , at 14:49, John Denker wrote:

> 7) Convince yourself that changing L and/or C (at constant R) 
> changes ω_0, but leaves the maximum response the same, namely 1/R.

This was the prob.  I find this not the case according to several texts for example a spring oscillator.

I have "put" here:

http://www.cleyet.org/Someone_is_Wrong/Driven%20Harmonic%20Oscillator/

graphs showing this.

My prob. was it wasn't intuitive.  After mulling over JD's post, I guess the amplitude variation is due to the inertia of the increased mass (also L).   It's is interesting to note the mathematical model shows the maximum or resonance for the kinetic energy is the natural undamped frequency and the potential energy resonance is the same as the amplitude resonance frequency for an obvious reason!

The read me in the driven harmonic oscillator file:
--------------------------------
Graphs of the Harmonically Driven Linearly Damped Spring Oscillator Responses as a Function of their Masses

The equation used is (3.59) of Thornton | Marion1and the Qs are calculated using equation (3.64)2.  

I have “put” them in response to both questions discussed on the physics list (phys-l), during personal communications, and published in the Horological Science Newsletter.  

The graphs show, contrary to some person’s belief, that 

One:  The maximum amplitude does not occur at the undamped natural 	frequency, and  
Two:  Increasing mass results in decreased maximum amplitude.

Note, however, with increasing mass (and resulting Q) the maximum frequency is asymptotic to the natural frequency, and the maximum amplitude is also asymptotic.  

Since periodic forcing modeling, using either a Laplace transformation or delta function, results similarly to harmonic forcing3, I suspect a similar result, which is horologically interesting. 

bc

1 p. 118, 5th ed. 2 p. 121, op. cit. 3 Baker and Blackburn, The Pendulum, pp. 37 ff.4 p. 121 op. cit.

p.s.  T | M’s graph4 of the amplitude (D) as a function of the driving frequency for various Q’s is quite misleading.  It shows D increasing with increasing Q.  As I’ve shown this is not generally true.  This error is very common.  The amplitude (A) in equation (3.59) is , not just the driving amplitude, which A implies.  I’ve made this explicit in the foot note 1 above.  Of course, if the increased Q is due to a decreased damping coefficient then the D will increase.  

1 p. 118, 5th ed. 2 p. 121, op. cit. 3 Baker and Blackburn, The Pendulum, pp. 37 ff.4 p. 121 op. cit.