# [time-nuts] Pendulum clocks based on John Harrison's writing

Tom Van Baak tvb at LeapSecond.com
Sun Jan 24 08:44:09 EST 2016

```Hi Chris,

> I like that concept of "P" in the "Pendulum Accuracy, part 1" paper.  I
> had not seen it described like that before.

Right. Over the past couple of decades the role of Q in pendulum clocks sometimes became more confusing or divisive than helpful. So I introduced the P (purity) variable to compliment Q (quality). That enabled Q to be static and pushed the measure of subtle variations into another variable, P. Part 1 of the paper is here:

http://leapsecond.com/hsn2006/pendulum-accuracy-1.pdf

> Is it accurate to think of P as related to the instantaneous variation in
> Q?  In the paper it was notated as deltaE/sigmaE, but isn't that also
> deltaE/delta(deltaE) ?  Kind of like the second derivative of energy flow
> in the system.
>
> --
> Chris Caudle

I would not want to go with "instantaneous variation in Q", as if Q itself was the root source of clock instability. I think that overloads the notion of Q too much and contributes to the confusion about the definition or role of Q (quality).

P (purity) fills a similar role as S/N in atomic clock stability prediction equations. That is, the short-term stability is a combination of *both* Q and S/N. Long-term stability goes down at sqrt(tau), subject to environmental effects.

My delta and sigma notation is a bit muddy so I'm not clear if delta(delta(x)) is helpful. It's clearer if you use RMS or ADEV(tau) statistics but for that audience I wanted to keep it simple.

The basic idea is there is a store of energy, and there is energy flow during each swing in a pendulum. The key to stability is not so much the total energy, nor the amount of energy gain and loss, but in the consistency of gain and loss. You'll recognize that we do the same thing with ADEV; it's not the phase, nor the rate, but variations in rate that are a measure of stability.

You have to be careful with your "second derivative of energy flow" wording because the word "flow" already implies a derivative. I think you could say 2nd derivative of energy or 1st derivative of energy flow, though.

This is getting OT, so follow-up off-list.

Thanks,
/tvb

```