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?

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?