# HOW TO SIZE A GRAHAM ESCAPEMENT

#### pepi

##### Registered User
NAWCC Member
I am looking for a criterion to size the geometric characteristics of the Graham escapement. This is what seems to make sense to me if I disregard all escapement friction in a small angle approximation.

In order of appearance:

1) The pendulum.
The ingredients that matter for our purpose are the energy lost per cycle, ΔE, equal to the one provided by the escapement, and the βact angle, that portion of the pendulum swing angle during which the escapement is active.
The complete amplitude of oscillation β = βact + βpas, has a passive part too, βpass, the dead beat.
If I understand well βpass is a safety measure, to guarantee the functioning of the escapement, in principle it could be as small as acceptable.
From the escapement/spring viewpoint, the smallest possible β would be advantageous but regularity considerations require a minimum β of 1°to 2° and in turn a similar minimum βact.
The energy E of the pendulum does not enter in the problem of the dimensioning of the escapement.
The relevant energy requirement is ΔE = Pdiss * Period and both Period and Pdiss, the pendulum-only dissipated power, dont depend on the bob mass but on the aerodynamics of the bob and the friction of the suspension.
To determine ΔE in practice one would use ΔE = 2π E / Q rather than Pdiss * Period since Q = 2π E / ΔE is easy to measure from the oscillation damping time of the pendulum in free fall and E, from the maximum speed of the pendulum, g P β / 2π, at steady state amplitude.
2) I neglect the crutch and consider the anchor of radius Ra to be integral and coaxial with the pendulum rod. The active work displacement of the anchor is
h = Ra * βact (1)
and the energy transmitted from the anchor to the pendulum is
ΔE = Fh * h (2)

3) The impulse plane transforms Fl, the tangent force of the RdS, into Fh according to the relation
Fh * h = Fl * l, the frictionless wedge advantage, so eq. (2) becomes:
ΔE = Fh * h = Fl * l (3)
4) the train applies to the Escapement Wheel a torque Cew = Rew * Fl, Rew the EW radius, therefore Fl = Cew / Rew and eq. (3) becomes:
ΔE = Fl * l = Cew * l / Rew (4)
5) N teeth encircle the EW, there are about two impulse planes of width l per tooth, hence
l = 2π Rew / (2 * N + 1) = π Rew / (N + 1/2) (5)
then (4) becomes:
ΔE = Cew * l / Rew = π Cew / (N + 1/2) (6)
If I am not mistaken
a) since ΔE is prescribed and Cew only depends on the spring / train characteristics, eq. (6) prescribes the number of teeth N.
b) Once the radius of the escapement wheel Rew and the number of teeth prescribed by (6) have been chosen, eq. (5) defines the side dimension of the impulse surface l.
c) Tan (alpha) = h / l is defined by (1) and by (5) Tan (alpha) = Ra * βact * (Nd + 1/2) / π Rew.
This eq. establishes a relationship between alpha and the anchor radius:
Tan (alpha) / Ra = βact * (N + 1/2) / π Rew = 1/π * Ra/Rew * βact * (N+1/2) .

At this point it seems to me that all the parameters of the escapement are determined, or linked to the two wheels radii, Rew and Ra. The size of alpha, the impulse plane angle, only depends on the Ra/ Rew ratio. Does all of this fit your knowledge of pendulums in practice? Does anybody know of a similar analysis?

#### howtorepairpendulumclocks

##### Registered User
I think you may need to talk to the horological science chapter...#161 ?

#### John MacArthur

##### Registered User
NAWCC Member
You might look into "Science of Clocks and Watches, Rawlings. His analysis is somewhat disputed, but he gets closer to what you are looking at than others. The best books on layout of Grahams are Gazely "Practical clock escapements" and Goodrich "The Modern Clock". From the looks of things you have it about right. Most makers don't try to calculate the energy required to run a given clock; they try different weights until the clock runs reliably. There are too many other variables to try to figure it mathematically. Is this clock under construction? If so, show us some pics please!
Johnny

#### pepi

##### Registered User
NAWCC Member
Most makers don't try to calculate the energy required to run a given clock; they try different weights until the clock runs reliably. There are too many other variables to try to figure it mathematically.
Johnny
thanks Johnny, your comments they have been very interesting and useful, I am now trying to read the classics. I have found Guy D. Aydlett here on NAWCC, very thorough but old.
About the larger logic I find mine simpler, more logical and more economical than constructing the escapement and then changing weights. What I am proposing is to build just the pendulum first, measure its Q, which is very easy and precise from free fall amplitude decay, decide the steady state angle amplitude on theoretical regularity ground, or previous experiments, and then all the rest of the design of the escapement follows as explained above. Apart for the size of the escapement, or the anchor, wheel which will come from the size of the clock case for max period reproducibility and least wear. The most difficult question to me is still how to decide on amplitude. What limits the amplitude range of choice from below? I have built a precision 1 s clock which is magnetically impulsed, it makes it very easy to control amplitude to 0 and it's quite clear that the standard deviation of the period grows fast below say one deg. Why? Q doesn't change much with amplitude but somewhat improves towards lower amplitudes. How should one find the right compromise between loss of isochronicity and this high frequency period fluctuations I see? And high frequency fluctuations don't necessarily imply bad regularity where it counts the most.
p.-

PS I am helping a friend who does excellent mechanical work to design his next Graham clock. His first is here and is not showing a good performance,. The cause is large escapement error induced by spring force variations because of low Q related to the roller bearing suspension.

#### John MacArthur

##### Registered User
NAWCC Member
Pepi - what limits the amplitude is the combined angles of impulse added to the combined angles of lock. My early clocks had impulse angles of two degrees, and a little more than half a degree of lock. They take a fairly large swing. My more recent ones have 1.5 degrees of impulse and at most half a degree of lock. They have a smaller arc of swing. Some of the fine French and German regulators have only one degree or less of impulse. Aydlett is excellent - I'm glad you found his treatise. "Accurate Clock Pendulums" by Matthys is excellent on the problems presented in pendulum fluctuations. Timekeeping drifts in fine clocks have been driving clockmakers nuts for centuries!
Johnny

#### pepi

##### Registered User
NAWCC Member
what limits the amplitude is the combined angles of impulse
My question refers to the inferior limit not the superior. Your comments illustrate my point perfectly, you are quoting pendulum amplitude differences by a factor of 3, from less than 1 to 2.5 deg in the same category of clocks, the 1 s high precision regulators. Energy goes with the square of the amplitude, Q is roughly the same in all clocks of the quality you mention, it follows that the power of the escapement, the engine which powers all clocks, can vary by an order of magnitude for no apparent reason. I think we are missing a criterion to determine the amplitude of the swing, we don't even have a reason to believe that, the rest being constant, a bigger amplitude is better than a smaller one.
There is some discussion about spontaneous microscopic oscillations on HSN , probably P. Boucheron I don't remember, which might be pertinent, I read it and to my understanding it wasn't helpful. To me this subject seems an open and relevant question in the design of any escapement.
About amplitude there is also the approach John Harrison took - is there a way to ‘play’ error sources against each other to reduce their net effect but that's not what we are discussing here now which is the idealized case where there are no temperature/pressure variations to compensate, but maybe somebody in the past was thinking about this and we forgot.

#### John MacArthur

##### Registered User
NAWCC Member
As to the power of the escapement varying by an order of magnitude: Since the escape wheel is "geared up" by a factor (usually) of some 43k, the minute changes in lubricant viscosity make a large difference in rotating resistance, as many, many clockmakers have noted over the centuries. I am currently doing some experiments making bushings out of PEEK which is supposed to have self-lubricating properties and is not nearly so prone to displacement as Teflon. We'll see....
Johnny

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