Re: [Phys-L] Help w/ Euler Cromer algo. (train of consciousness)
On 9/21/2014 12:58 PM, Bernard Cleyet wrote:
As usual, by being a telegrapher I failed to obtain a complete answer.
The “motivation” was to determine if a typical clock’s pendulum
exhibited SHM. [Of course it doesn’t, but to what degree? —] IIUC,
many horologists think it doesn’t (significantly — whatever that
means) One, using a Renishaw found it is SHM some time ago, but didn’t
write it up, so we don’t know w/ in what accuracy. Hence the cos fit.
All this may be “moot”, as the deviation from “harmonicity” as
measured by the fit using the non-lin. diff. eq. [done after my post]
“swamps” the inaccuracy of the LPA by at least two orders! I expect
the fit of the diff. eq. including a drive numerically modeled will
result in further deviation. One poss. prob. A not very gud mantel
clock’s p.’s period on average over a week deviates only ~ one PPM.
This means that the horological definition of harmonicity is in
position only, not period. So “my” fit comparison must not include
variation of period.
Always interesting to understand the outcome that Bernard desires.
Suppose I compare the vertical position of a mass in an ideal suspended
mass spring arrangement with the
circumferential position of an ideal pendulum bob. The one has simple
harmonic motion; the other has motion that differs from SHM by an amount
that increases with increased maximal amplitude of the bob.
I can arrange a model to represent both systems having the same
frequency, perhaps 0.5 Hertz for a 1 second pendulum. Such a pendulum
can be arranged to hold its frequency for a given amplitude. The
concept of Q loses value where no losses exist. Still, I think that
Bernard would like to know what exactly are the departures from
sinusoidal displacement over a range of let us say one thousand points
in the swing. If this is truly the case, it is soon tabulated or
expressed in some suitable form.
But Bernard explicitly mentions Q and so he is telling us that he is
interested in the case where some energy is lost in each swing, and is
made up at each swing, by some impulsive means. However, he may rather
be interested in the discrepancy between the table of displacements of a
High-Q pendulum as its amplitude decays, and the data for a sine curve
at equal times. It is evident that the frequency of the pendulum varies
with varying max amplitude, and so he would expect increasing variation
between a cosine of fixed frequency and the pendulum's decaying
amplitude (and hence changing frequency).
This does not seem like his motivation (but I could be wrong): I prefer
to think that Bernard would like to compare a table of displacements of
a pendulum over one cycle, where the losses of whatever kind, are made
good by some escapement action, so as to maintain the peak amplitude
constant and comparing its displacements with an ideal spring and mass
cycling at the same frequency.
The resulting table of bob displacements would show an increased
discrepancy during the impulsive drive, so that would be the other
departure from SHM that he desires to evaluate, as it seems to me.
This sort of question is clearly open to resolution for given pendulum
amplitude, frequency, Q and one or two other parameters. Or is the
interest in making a home-brew code work? Not sure about that.
Brian Whatcott