CLOCK AND WATCH ESCAPEMENT MECHANICS (1997)

Section 1: Clock Escapements.

1: Efficiency and Power Losses.

Page 4

2: Drawing an Escape Wheel.

3: Drawing the Graham Pallets.

4: The Graham Explained.

5: The Importance of the Simulation.

6: The Recoil Escapements.

7: The Brocot Escapement.

8: The Pin Wheel Escapement.

9: Other Design Considerations.

10: The Graham Reconsidered.

11: Efficiency in Numbers.

The first nine chapters are explained without math. These chapters are less

complicated, and they introduce the reader to the logic behind the drawing techniques, so

as to serve the needs of those who want to learn about escapements but would not be

drawing them on their own computers. The tenth chapter explains the math behind the

previous chapters. This way the reader has the option whether or not to become involved

with the math.

References:

Britten's Watch & Clock Maker's Handbook, Dictionary and Guide (1978 Edition).

Donald de Carle:

Watch and Clock Encyclopedia (1977 Edition).

Henry B. Fried:

The Watch Escapement (1974 Edition).

Section 2: Watch Escapements.

12: Drawing the Club-Tooth Escape Wheel.

Page 41

13: Drawing the Pallets.

14: Changing the Design.

15: Improving the Design.

16: The English Lever.

17: The Pin Pallet Escapement.

18: The Cylinder Escapement.

19: The Duplex Escapement.

20: The Chronometer Escapement.

21: Daniel's Independent Double-Wheel Escapement.

22: The Double-Roller.

Chapters 12 to 18 are similarly presented in order to introduce the reader to the

logic behind the drawings, though these drawings are more involved and require some

understanding of watch theory, such as lock, drop, draw, and impulse. Chapter 19 and

beyond become more involved with the math that is required to create the drawings. The

reader who does not want to become involved with the math could benefit from the logic

of the latter chapters by passing over the math.

I would like to thank the following for their invaluable assistance and encour-

agement:

Daniel Henderson

Cecil Mulholland

Roy Hovey

Steve Conover

Nino Gonzales

Harry and Sue Wysong

Introduction.

Graham escapement

Discover how much more there is

to know about escapements by creat-

ing your own drawings. The purpose

of this project is to introduce the

reader, whether professional or hob-

byist, to a hands-on method for draw-

ing a mechanical clock or watch es-

capement. While it is more obvious

that a manufacturer needs to know

how to design a pallet, the repairman

could do a better repair with an im-

proved understanding of design the-

ory. The ability to draw an escape-

ment enables one to experiment more

easily with the effects of changing the

variables and to compare different

Strip-pallet recoil

types of escapements, their similari-

ties and differences.

The most important reason for

careful attention to design is effi-

ciency. The escapements most fre-

quently encountered at the bench are

the Recoil, the Graham

(or

dead-beat), and the Swiss Lever.

These all have efficiencies below

50%. This means that more than half

the power is lost in the escapement

alone, after all the power losses in the

gear train.

modern

Swiss

Lever

These drawings are not dissimilar to what I

have seen in escapement literature. They do not

reveal the methods by which they were de-

signed, why the lines and curves are positioned

so. If these drawings ignite your curiosity, this

project is for you.

1: Efficiency and Power Losses.

A clock is usually taken to the shop for repair because the clock fails to keep

running. It runs for a while and stops because not enough power reaches the pendulum to

keep it running. One way to get it running is to double the weight, but this causes enor-

mous wear and consequent damage in the long run. The other way is to overhaul the

clock. The job of the repairman is to do whatever is necessary to minimize the power

losses between the weight and the pendulum. The clock is cleaned. Pivots are polished.

Worn bushings are replaced. Adjustments are made to the escapement to improve its

action. The clock is carefully lubricated. Other repairs are made as needed. These all

have the ultimate goal of reducing power losses.

There are two kinds of power losses in a clock: frictional losses and losses caused by

the action of the escapement (which result in additional frictional losses). Frictional

losses are easily understood. The lubricant has failed, causing drag. The bushings are

worn, so the gear and pinion teeth grind together. The pivots are scored, causing

frictional losses from rough surfaces. Dirt particles become imbedded in the worn

bushing, causing binding of the pivot, and so on.

Power losses caused by the action of the escapement are less obvious. The following

reasons are explained in detail in subsequent chapters. If the escape wheel rotates clock-

wise, the force it exerts on the pallet would be in a different direction to that of the

pallet's movement. The greater the angle between the two directions, the greater the loss

of power as it is transferred from the escape wheel to the pallet. Consider that the pallets

rotate clockwise as the escape tooth pushes on the entry pallet, and counterclockwise as

the tooth pushes on the exit pallet, yet the escape wheel continues to rotate in the same

clockwise direction.

As the escape tooth pushes on the pallet, the tooth exerts a force in the

same direction as its direction of travel at that point. If the point of contact

on the pallet were at 90º to this direction, such as on the locking face of the

Graham pallet, the escape tooth would not move forwards and no power

would be transferred to the pallet.

If the point of contact were in the same direction as the escape tooth,

so that the pallet's impulse face were parallel to the path of the tooth, the

tooth would pass by freely and provide no impulse to the pallet.

No power is transferred to the pallet when the angle is 90º or 0º. An angle in

between is needed: the angle that maximizes the power transfer. Find what direction the

escape tooth is moving in as it passes over the pallet, and the direction of the pallet. Then

determine the direction in which the pallet should receive power from the tooth. The di-

rection should be half way between those of the tooth and of the pallet. For maximum

efficiency, the pallet's impulse face needs to be at right angles (90º) to this direction.

If the angle between the directions of the tooth and of the pallet were 90º, the maxi-

mum achievable efficiency would be only 50%. This is achieved when the impulse face's

angle is at 45º to the direction of the tooth's travel. If the impulse face's angle were 25º,

the efficiency would be merely 38%. That is a 24% power loss caused by improper de-

sign. If the repairman could see this, he could adjust the impulse face's angle more

closely to what it should be, and maximize the pallet's efficiency, given the original

design he has to work with. Ideally, the impulse face's angle should be at 90º to the angle

half way between the directions of the tooth and of the pallet.

You may not find this easy to understand: power losses by escapement design are

less obvious. This will become clearer in the next chapters, as we draw the Graham

escapement. It should, however, have become clear to you how important the design of

the escapement is. I have seen clocks in which one pallet received a negligible impulse

from the escape wheel, so the maximum achievable efficiency was only 25%. Why

would they would not keep running?

2: Drawing an Escape Wheel.

I am currently using an IBM with Windows and a drawing program called Key-

Draw. If you have AutoCAD, you are well equipped. The most important features you

need are the ability to draw lines and circles on a grid, and the ability to rotate the lines

by angles that you determine. You may have a different method. This is just one way to

do it.

First, draw a large circle. A diameter of 6

inches worked well for me. Draw another

circle of 4.5 inch diameter and center it inside

the first. Draw a horizontal line across this

circle, bisecting it. Draw a tooth on one side

until you are satisfied: I chose a line at 15º and

another at 25º from horizontal, positioned to

fit, and with a small gap to allow for tooth

thickness. Duplicate, rotate, and place a tooth

on the other side. Your drawing should look

like this:

Move the inner circle outside. Group the

outer circle with the line and the two teeth,

duplicate and rotate by 12º. Duplicate the new

group and rotate it by 12º again. Repeat this

until you have thirty teeth. Ungroup all the

elements and remove the outer circles, one at a

time, until only one is left. Place the inner circle

in its original position, and group all the teeth

and circles. The reason for placing a tooth in a

circle before rotating is because the circle makes

the tooth rotate about the center of the circle, in

order to achieve the desired result.

Rotate the groups by 12º at a time because there are 360º in a full circle, so if you

want a 30 tooth circle: 360 / 30 = 12º. If you want a 48 tooth circle, divide 360 by 48 to

get 7.5º. Thus escape wheels with different numbers of teeth could easily be made.

Escape wheels with the teeth pointing in the other direction could be made by flipping

the image, or with teeth of different shapes, such as the club-tooth escape wheel in Swiss

watches.

3: Drawing the Graham Pallets.

Over the escape wheel, draw

two radius lines at

90º to one

another. Draw two more lines at

90º to the radius lines and place

them at the edge of the circle. The

point where these new lines inter-

sect is the center of the pallet cir-

cle, the radius of which should

measure three inches. Draw the

pallet circle about this point. You

now have four radius lines. Draw

two more radius lines next to each

one, positioned at 3º on either side

of each. I chose to extend all the

radius lines:

Since the escape wheel rotates clockwise, the entry

pallet is on the left side of this drawing. Draw a horizontal

line between the three lines.

The exit pallet is on the right side: draw a vertical line

between the three lines there.

It is important to draw the locking

faces, which will consist of curves

drawn over circles, such as to preserve

the dead-beat nature of the design. Draw

two more pallet circles with diameters of

5.69 and 6.31 inches. Once these lines

have been completed, the rest of the

pallet could be drawn any way you wish.

I drew the this one by drawing a hori-

zontal line across the pallet circle, then

rotating it by

24º, duplicating it and

placing one line 0.25 inches above the

center line and the other 0.25 below. I

grouped these two lines, duplicated

them, flipped them horizontally, and

placed one pair on the other side.

Remove the excess lines to get the

result. This is the ideal Graham design as its

angles maximize the power transferred from

the escape wheel to the pallets.

4: The Graham Explained.

As the escape wheel rotates, it exerts a

force upon anything it encounters in its path.

The direction of this force depends on what

point on the escape wheel the pallet is

located, as shown here: the arrows show the

different directions of force for three points

on the escape wheel.

NORTH

If the top of the page were North,

PALLETS

the escape wheel would push on the en-

try pallet towards the North-East. This

force has size and direction. I have la-

beled it 'Fe1,' or the force exerted by the

escape wheel. At this point, the pallet

Fe1

rotates clockwise: a force should push it

Fp1

Fp2

in a North-West direction, marked 'Fp1'

WEST

entry

side

exit

side

EAST

(force to pallet). The angle between Fe1

Fe2

and Fp1 is

90º. If a third line were

introduced, labeled 'Fi'

(force of

impulse), with an angle half way in

between, you could think of an impulse

force in that direction:

ESCAPE

WHEEL

Fp1

Fe1

SOUTH

We have Fi because a force cannot be rotated by 90º in the Graham escapement,

which is why no power is transferred to the pallet from the escape wheel when the

pallet's impulse face is at 90º or 0º to Fe. It could be rotated twice by 45º, though. Fi is at

90º to the impulse face's angle, so if the impulse face's angle were changed, Fi would

change. As Fe1 goes North-East, imagine that a portion of that force is received by the

impulse face at right angles to its angle: Fi due North. Then imagine that a portion of Fi

acts in a North-West direction, Fp1, to rotate the pallet. The portion of Fe1 not received

in Fi is lost, and so is the portion of Fi not received in Fp1, which means that power is

lost in each step.

If the lines were drawn to scale, you could see

91%

how much power is lost. Scale drawings could be

100%

used to show the effect of angle on Fi and Fp:

Fe1

changing the angle of the impulse face has a

38%

dramatic effect on the force Fp1 that rotates the

Fp1

pallet, even though the angle between Fe1 and Fp1

remains unchanged at 90º. If the angle FeFi were

25º, the efficiency would be 38%.

impulse

face

71%

100%

Fe1

If the angle were 45º, the efficiency would be

50%

50%.

Fp1

impulse

face

100%

Fe1

If the angle were 65º, the efficiency would be

42%

38%.

38%

Fp1

impulse

face

The greatest efficiency is achieved

when the angle of Fi is half way between

those of Fe1 and Fp1.

10 20 30 40 50 60 70 80 90

angle between Fe and Fi

As the angle FeFp increases, the maximum

achievable efficiency decreases, so the smallest angle

possible is preferred. The loss of efficiency on one

side is equal to the gain in efficiency on the other

side, but there is no advantage in an unequal ar-

rangement. When the efficiencies are unequal, the

push the pendulum receives on one side is different

versus the other side, instead of the same. In the Gra-

ham escapement, therefore, the angle FeFp should be

90º because of symmetry. This is illustrated by the

following drawing, which should be familiar to many

clockmakers.

This approach could be used to determine the

distance between the escape circle center and the

pallets' circle center for the tooth span (the number

of teeth between the pallets) of your choice: choose a 4.5 tooth span for a wide pen-

dulum swing, or an 11.5 tooth span for a narrow swing. It is generally accepted that a

7.5 tooth span gives the most desirable results in practice for a 30 tooth escape wheel.

5: The Importance of the Simulation.

You need to simulate the action of the escape-

ment on a computer to determine if the drawing works

in practice, without having to make the parts first to

find out. I was able to rotate the escape wheel by one

degree at a time, and the pallet in the same manner, to

simulate the action in practice. There was binding

because there was no inside drop.

There was also no outside drop.

The creation of drop results in the escape tooth

landing on the pallet's impulse face, which causes

recoil action. In order to avoid this, the pallets must

be designed with impulse face angles that result in

lowered efficiency. See the drawing on the next

page. This is the modified Graham design as the

ideal design had to be modified before it could be

used in a simulation.

In theory, there should be 1º of lock and 1º of

drop. However, the simulation works better with 2º

of lock and drop.

6: The Recoil Escapements.

Once you have the Graham escapement drawings, drawing the recoil escapements is

easier because the images could be superimposed for reference.

With the recoil escapements, there is no need to adjust for lock, only drop. This

really simplifies the issue. Just create some inside and outside drop, and make sure that

the angle FeFi is 45º. Since no adjustment is made for lock, use the ideal Graham

drawing for reference. Shorten the entry pallet slightly and move the exit pallet slightly

outwards, parallel to the impulse face of the Graham exit pallet:

Notice that the escape wheel has been changed.

Once the Graham is removed:

Now for another recoil pallet, this time superimposing one recoil pallet over another:

This design is similar to one I saw recently in an antique British grandfather clock:

The following is the most important fact about recoil escapements: the angle FeFi

must be 45º, regardless of the angle FeFp. This is because of recoil action. If the effi-

ciency were 50%, or 1/2, moving forwards, then the deceleration of recoil would have

twice the magnitude of Fe (in the opposite direction). Some power is "stored" in the re-

coil action because the escape wheel moves back a little, as if it were winding the clock.

However, the power stored is only 50% of Fe, so we have a very significant power loss,

even under the best conditions.

If the impulse face's angle were 15º, the efficiency would be 25%, or 1/4. The decel-

eration of recoil would be four times as large as Fe in the opposite direction. Having the

impulse face's angle at 15º causes two problems: (1) more efficiency is lost moving

forwards because of the incorrect angle, and (2) much more power is lost in recoil (going

backwards).

This example demonstrates how important the impulse face's angle is in a recoil es-

capement. The angle FeFi must be 45º to minimize power losses in recoil. Therefore, the

angle FeFp should be 90º in order to maximize efficiency moving forwards. How many

recoil escapements have you seen that had the correct angles?

7: The Brocot Escapement.

Drawing the Brocot pallets is made easier by

using a Graham drawing as background. Choose

the modified Graham, which works in the simula-

tion, because both lock and drop must be consid-

ered. Quite simply, the radius of the pallet is equal

to the thickness of the modified Graham pallet.

Then just position it over the Graham drawing so

that the lines meet.

This escape wheel drawing is borrowed from

the recoil escapement and flipped over horizon-

tally. The rest of the Brocot pallet could be drawn

as you wish:

The Brocot pallets require both lock and drop. Correctly adjusted, they could behave

much like a Graham escapement. Since the pallet is essentially half a circle, the impulse

surface is a quarter of a circle. The impulse angle changes as the tooth slides across the

impulse face. The impulse angle is therefore very inefficient at first, becoming more effi-

cient until reaching a peak, and then losing efficiency towards the end of the stroke.

Compare the Graham, which has a straight and horizontal line for Fp, with this.

0.500

0.250

0.000

45.000

90.000

angle (in degrees)

The impulse face's angle changes un-

0.008

evenly over the time it takes the escape tooth

to pass over the impulse face. If you consider

the displacement (or distance moved) of the

0.004

pallet in the direction of Fp, and multiply this

by the force Fp that rotates the pallet at each

instant in time, you get a 'work done' curve.

0.000

This curve looks very different and reveals

0.000

1.000

how inefficient the Brocot design really is.

time (index)

8: The Pin Wheel Escapement.

Drawing the pin wheel escapement proved to be a considerable challenge. A simu-

lation of the action of the pin wheel escapement is interesting when compared with that

of the Graham escapement: the action is different because of the directions of the forces

Fe and Fp, but the results are the same.

You will have noticed that, in

the Graham escapement, the angle

Fe1Fe2 was 90º, and so was the angle

Fp1Fp2. In the pin wheel escapement,

the angle Fe1Fe2 is zero (both go in

the same direction), and the angle

Fp1Fp2 is 180º. However, the angle

Fe1Fp1 is 90º, and so is the angle

Fe2Fp2. Another consideration in this

drawing is that the thickness of each

pallet is equal to half the arc between

two escape wheel teeth, as in the Gra-

ham escapement, minus enough pallet

thickness to allow for drop. The angle

of the pallet's impulse face needs to be

45º relative to Fe. The pallet locking

Fp1

Fp2

face should be designed about circles,

as in the Graham, in order to preserve

the dead-beat nature of the action.

Fe1 Fe2

Changing the pallet circle's diameter does

not affect the number of teeth between the

pallets, as it does in the Graham. However, it

does affect the angle of swing of the pallets

(from side to side). If the pallet circle radius

were increased, the angle of swing could be

decreased, which is desirable for a fine

regulator. This would make it possible to

minimize the circular error in the movement

of the pendulum.

The lack of popularity of the pin

wheel escapement could be explained in the

difficulty in manufacturing the pallets because

of their intolerance for error: if the design

were not perfect, it probably would not work

at all, which becomes very obvious when preparing drawings for a simulation. The recoil

escapement, on the other hand, could be imprecisely designed, and it would be much

more likely to work.

9: Other Design Considerations.

In previous chapters, all the pallet designs were created with an emphasis on sym-

metry so that the impulse received by the pendulum would be equal in each direction. All

designs, whether for clocks or watches, should be based on the same method of vector

analysis. This should be clear because of the simple method of drawing one type of

escapement over another, as shown in chapters 6 and 7. Draw the impulse face first, and

then the rest of the pallet.

In watch theory, the symmetrical design is referred

to as either a "circular" design or an "equidistant im-

pulse" design. In the circular design, the impulse faces

Graham escapement

are bisected by the same circle. However, the entry

circular design

pallet's locking face is outside the circle, and the exit

pallet's locking face is inside the circle. The locking

faces clearly are not symmetrical. This problem is

unimportant in pendulum clocks, but it is an issue in

watches.

If the pallets were modified to make them with

Graham escapement

equidistant lock, the locking faces would be drawn on

equidistant lock design

the same circle. However, the pendulum would receive

unequal impulses in each direction.

A pallet with equidistant drop could similarly be designed, but it has no practical

application in horology.

The Graham pallet has curved locking faces in order to achieve what we call a

"dead-beat," where the escape wheel does not move either forwards or backwards during

lock. When the escape wheel is pushed backwards, there is recoil. In clocks, a dead-beat

action is preferred because the movement of the pallets is controlled by the pendulum at

the point where, for example, the crutch pin goes into the suspension leader. At no time

are the pallets independent of the pendulum.

Modern Swiss watches have pallets that are independent of the balance wheel's

movement most of the time: this system is referred to as the "detached lever." The single

and double-rollers are designed so that the pallet fork would not accidentally jump across

to the wrong side of the roller jewel if the watch were jolted by a fall, for example. In

addition to the roller table, it is necessary to keep the pallet fork over to the side, in its

place, until the roller jewel returns. If the escape wheel were allowed to move forwards

slightly, beyond the entrance corner, as the pallet moves over, then it would be necessary

to move the escape wheel backwards by the same amount when the roller jewel returns to

unlock the pallet. The need to push the escape wheel backwards slightly results in a small

binding action, which encourages the pallet to stay in its place. Watch pallets, therefore,

instead of having curved locking faces, have flat locking faces at an angle of about 15º

from the escape circle radius at the pallet entrance corner, such as to make the escape

wheel rotate by about 1º extra. Watchmakers call this "Draw."

An equidistant lock design is important in watches because of the need for

symmetrical lock. In a circular design, the locking faces are at unequal distances from the

pallet center, causing a need for unequal torque to unlock, torque that adversely affects

the oscillation of the balance wheel. Any asymmetry in the oscillation of the balance

wheel could add to a factor that causes positional error, such as the poise of the balance

wheel.

Since clocks do not have pallets that are

independent from the pendulum, there is no

need for draw. Therefore, the equidistant

lock design is of no advantage in pendulum

clocks. The equidistant impulse design is the

obvious choice for the Graham, Recoil, and

Brocot escapements. (It cannot be applied to

the Pin Wheel escapement, as its drawing

demonstrates.)

The same principles of lock and draw

apply to pin pallet escapements in watches

and clocks with balance wheels, but the

impulse and locking faces and the angle of

draw are designed into the escape wheel's

tooth rather than the pallet. (A pin pallet es-

capement for a pendulum clock, such as the

Brocot, requires no draw and should have a

dead-beat action.)

If you look at a clock with a floating

balance, such as a Hermle, you would see that the impulse face is part of the escape

tooth. In a drawing, the pallet radius lines bisect the pallet pins. The escape radius lines

bisect the impulse faces of the teeth. The locking faces of the escape teeth are not parallel

to the escape radius lines, but appear to lean forwards to create draw. The length of the

escape impulse face lines is shortened to allow for at least 1º of drop, and these lines are

at 45º to the escape circle's radius lines that bisect each of them.

By drawing the different escapements, you could see how the principles, by which

they are formed, apply to all of them. They look different, but they actually behave in

similar ways.

10: The Graham Reconsidered.

The Graham design in chapter

5 did not maximize efficiency,

whereas the drawing in chapter 3,

before modification, had the maxi-

mum achievable efficiency. The

modified Graham was less efficient

because of lock: if modified with

maximum efficiency, the escape

tooth would always land on the

pallet's entrance corner. The reason

the Graham pallets could not be

modified as efficiently as possible,

whereas an efficient watch escape-

ment could be created

(and it

would work in a simulation), is be-

cause of the design of the escape

wheel. The watch escape wheel has

an impulse face of its own and its

let-off corner is above its entrance

corner, which creates lock. The

back of the tooth makes it possible

to reduce drop without binding.

The watch escape wheel offers

these two design advantages. If the

Graham escapement were designed

with a club-tooth type escape

wheel, the pallets could easily be

modified in a more efficient design that would work in a simulation. These pallets

would be much thinner and of equidistant impulse.

There are other ways to change the design of the Graham. The 30 tooth escape

wheel is the most widely accepted choice because grandfather clocks with the

one-second pendulum could display a second hand moving by one second at a time.

If the issue of efficiency were considered, the 15 tooth escape wheel would be more

efficient by 8.33%. Since the drop requirement is the same for both the 15 and 30

tooth designs, the 15 tooth design loses 1º for drop out of every 12º, but the 30 tooth

design loses 1º out of every 6º. If the Graham were designed with a 15 tooth escape

wheel, its efficiency could be improved significantly.

However, the 15 tooth escape wheel

rotates by 12º per beat, so if the pallet

circle's radius were equal to the escape

circle's radius, the pendulum would have a

greater arc of swing, which would be

undesirable. By increasing the pallet

circle's radius to approximately double the

r = 3 tan 66 = 6.74

escape circle's radius, (because the angle

of rotation per beat is doubled), the arc of

swing of the pendulum could remain

virtually unchanged. With a

5.5 tooth

span, the pallet circle's radius increases

from 3" to 6.74". Thicker, stronger pallets

and escape teeth could then be designed.

This drawing shows how the impulse

face of the escape tooth extends beyond

r=3

the circumference of the escape circle,

which passes through the tooth's entrance

corner. This is how lock could be created

and yet maximize the efficiency of the

pallet impulse face's angle.

This design has practical limitations. It

would require either a two-second pendulum or

that a different set of gear train ratios be used

to keep the one-second pendulum. An ap-

propriate gear train combination could be

found in the De Carle "Watch and Clock Ency-

clopedia." If a two-second pendulum were

used, a 60 second dial could be used, but the

second hand would move forwards every two

seconds.

Nevertheless, this example should be

useful to clockmakers interested in escapement

design because an efficiency improvement of

over 8% is well worth considering.

11: Efficiency in Numbers.

This chapter addresses the math behind the drawings: first, how changing the

directions of the forces results in reductions in magnitude; how the efficiency of an

escapement design could be calculated; and how to design a pallet blank, by using these

calculations, for a clock that is missing the pallets.

1.0

Consider the coordinates of a quarter circle. If

you take a horizontal line one inch long and rotate it

by one degree at a time until you trace a quarter of a

circle, you have points on a drawing which shows the

0.5

position of each point relative to a horizontal line and

a vertical line.

0.0

0.5

1.0

In the above graph, the value of each

(X,Y)

point along the X line is given by:

X= cos (A)

where A is the angle of the line from the

horizontal line. The value of each point along

H = 1

the Y line is given by:

Y = H x SIN(A)

Y= sin (A)

So each point has a coordinate (X, Y), which

could be seen as (cos (A), sin (A)).

angle A

90º

(0,0)

X = H x COS(A)

PALLETS

This could be applied to clock escape-

ments. In chapter 4, we considered the action

Fe1

of the escape wheel's tooth as it pushed upon

Fp1

the entry pallet. The tooth exerted a force Fe

in a North-East direction. Since the impulse

impulse

face

face was horizontal, it received the impulse

due North.

ESCAPE

WHEEL

If the lines were positioned to create a triangle,

they could be used for calculation. Fe (100%) goes

North-East. Fi is the portion acting due North:

Fi = Fe x cos (45) = 100% x 0.707

= 70.7%

29.3% efficiency has just been lost!

As the pallet is pushed by the tooth, it rotates

clockwise, so a portion of Fi is received by the

pallet to push it in a North-West direction.

As before, calculate the size of Fp:

Fp = Fi x cos (45) = 70.7% x 0.707

= 50%

Fp is therefore half as large as Fe. The efficiency is

50%.

This...

could be used to create this:

Fe1

Fp1

impulse

face

Fe1

If the angle of the pallet's impulse

face were changed, the direction of Fi

would change, and therefore the pro-

Fp1

portion of Fe that could be used to

rotate the pallet:

impulse

face

Fi = Fe x cos (65) = 100% x 0.423

= 42.3%

The angle FeFp is 90º, and the angle FeFi is 65º, so the angle FiFp is 25º.

Fp = Fi x cos (90-65) = 42.3% x cos (25)

= 42.3% x 0.906

= 38.3%

This is less efficient by almost 24% (divide 38.3 by 50) because Fi is not

half way between Fe and Fp.

When the angle between two vectors is small,

the loss of efficiency is small: when the angle

FiFp is only 25º, the efficiency drops from 42.3%

38.3%, a decline much smaller than occurs

between Fe (100%) and Fi (42.3%) when the

angle is larger (65º). Therefore, the angle FeFp

should be as small as possible. The effects of

angles could be computed on a spreadsheet. Look

at the chart after this page. As the angle FeFi

increases, the efficiency increases until Fi is half

way between Fe and Fp, beyond which the effi-

ciency decreases. The graph was created using the

10 20 30 40 50 60 70 80 90

data in this chart. In order for a computer to do

angle between Fe and Fi

the calculations, the angles need to be expressed

in radians: multiply the angles by pi

(3.1415) and divide by 180.

The same chart could be used to see what happens when the angle FeFp is changed.

The second chart on the following page shows that as the angle FeFp increases, the maxi-

mum achievable efficiency decreases. The maximum efficiency consistently occurs when

Fi is half way between Fe and Fp. This chart was used to create the graph on the page

after it. The graph illustrates how the maximum efficiency changes.

Use this chart to determine the effect of having an angle greater than 90º between Fe

and Fp on the entry side. Then determine the effect of having an angle smaller than 90º

to the same extent on the exit side. (See page 10.) You would see that the gain in

efficiency on the exit side is offset by the loss of efficiency on the entry side, so the

overall efficiency remains the same. Since there is no advantage in this asymmetry, an

angle of 90º on both the entry and the exit sides is preferred.

Anyone with a computer spreadsheet could create this chart. I am using MS Works.

The chart on the page following the graph shows the formulas used. Enter one of the Fp

formulas in the appropriate cell, and then use the "fill-right" and "fill-down" functions to

complete the chart. You could create a chart showing the angles increasing by 1º at a

time. (I chose 2º so as to fit the chart on one page.)

Fe = vector force by escape wheel = 100%

Fi = vector force received by pallet impulse face

Fp = vector force acting to rotate pallet

angle between Fe and Fp in degrees

(radians)

1.571

angle between Fe and Fi

degrees

(radians)

0.000

100.0

0.0

0.017

100.0

1.7

0.052

99.9

5.2

0.087

99.6

8.7

0.122

99.3

12.1

0.157

98.8

15.5

0.192

98.2

18.7

0.227

97.4

21.9

0.262

96.6

25.0

0.297

95.6

28.0

0.332

94.6

30.8

0.367

93.4

33.5

0.401

92.1

36.0

0.436

90.6

38.3

0.471

89.1

40.5

0.506

87.5

42.4

0.541

85.7

44.1

0.576

83.9

45.7

0.611

81.9

47.0

0.646

79.9

48.1

0.681

77.7

48.9

0.716

75.5

49.5

0.750

73.1

49.9

0.785

70.7

50.0

0.820

68.2

49.9

0.855

65.6

49.5

0.890

62.9

48.9

0.925

60.2

48.1

0.960

57.4

47.0

0.995

54.5

45.7

1.030

51.5

44.1

1.065

48.5

42.4

1.100

45.4

40.5

1.134

42.3

38.3

1.169

39.1

36.0

1.204

35.8

33.5

1.239

32.6

30.8

1.274

29.2

28.0

1.309

25.9

25.0

1.344

22.5

21.9

1.379

19.1

18.7

1.414

15.6

15.5

1.449

12.2

12.1

1.484

8.7

1.518

5.2

1.553

1.7

1.571

0.0

Fe = vector force by escape wheel = 100%

Fi = vector force received by pallet impulse face

Fp = vector force acting to rotate pallet

angle between Fe and Fp in degrees

106

(radians)

1.571

1.710

1.850

angle between Fe and Fi

degrees

(radians)

0.000

100.0

0.0

-13.9

-27.6

0.017

100.0

1.7

-12.2

-25.9

0.052

99.9

5.2

-8.7

-22.5

0.087

99.6

8.7

-5.2

-19.0

0.122

99.3

12.1

-1.7

-15.5

0.157

98.8

15.5

1.7

-12.0

0.192

98.2

18.7

5.1

-8.6

0.227

97.4

21.9

8.5

-5.1

0.262

96.6

25.0

11.8

-1.7

0.297

95.6

28.0

15.0

1.7

0.332

94.6

30.8

18.0

4.9

0.367

93.4

33.5

21.0

8.1

0.401

92.1

36.0

23.8

11.2

0.436

90.6

38.3

26.5

14.2

0.471

89.1

40.5

29.0

17.0

0.506

87.5

42.4

31.3

19.7

0.541

85.7

44.1

33.5

22.2

0.576

83.9

45.7

35.4

24.5

0.611

81.9

47.0

37.2

26.7

0.646

79.9

48.1

38.7

28.6

0.681

77.7

48.9

40.0

30.4

0.716

75.5

49.5

41.1

31.9

0.750

73.1

49.9

41.9

33.2

0.785

70.7

50.0

42.6

34.3

0.820

68.2

49.9

42.9

35.1

0.855

65.6

49.5

43.0

35.7

0.890

62.9

48.9

42.9

36.1

0.925

60.2

48.1

42.6

36.2

0.960

57.4

47.0

41.9

36.1

0.995

54.5

45.7

41.1

35.7

1.030

51.5

44.1

40.0

35.1

1.065

48.5

42.4

38.7

34.3

1.100

45.4

40.5

37.2

33.2

1.134

42.3

38.3

35.4

31.9

1.169

39.1

36.0

33.5

30.4

1.204

35.8

33.5

31.3

28.6

1.239

32.6

30.8

29.0

26.7

1.274

29.2

28.0

26.5

24.5

1.309

25.9

25.0

23.8

22.2

1.344

22.5

21.9

21.0

19.7

1.379

19.1

18.7

18.0

17.0

1.414

15.6

15.5

15.0

14.2

1.449

12.2

12.1

11.8

11.2

1.484

8.7

8.5

8.1

1.518

5.2

5.1

4.9

1.553

1.7

1.571

0.0

These figures consider the action of the ideal Graham

escapement. In this case, the 30 tooth escape wheel rotates

by 6º and the pallet rotates by 6º as the escape tooth pushes

on the pallet's impulse face.

In practice, if

1º were lost for drop, the triangle

would become smaller.

To calculate the work done, multiply the force, which was calculated to be 50% (or

0.5), by the distance (in the direction of Fp) that the pallet is pushed during impulse.

Take the distance as an index of 1 (or 100%), so the work done is: 1 x 0.5 = 0.5. Divide

the distance into six parts because there are 6º of rotation and you need to compare it

with the distance in the practical pallet, after losing 1º to drop. In the practical example,

the distance is: 1 x 5/6 = 0.833. The work done is: 0.5 x 0.833 = 0.417. Therefore, over

16% efficiency is lost to drop.

Compare the Graham's efficiency with that of the Brocot. This is more complicated

because the angle of the impulse face changes as the tooth slides across the pallet surface.

Since the Brocot impulse surface is a quarter of a circle, the cosine function could be

used to find the angles of the impulse face over the time it takes the tooth to slide from

one end of the pallet to the other. Then Fi and Fp could be calculated for each angle,

followed by the distance the pallet travels in the direction of Fp (in each instant of time).

The force Fp and the distance (X) could be multiplied to get the work done in each

instant, which, all added together, would give the total work done to rotate the pallet in

the direction of Fp. See the Brocot chart on the next page. The total at the bottom gives

the total work done as an index, which could be compared with the ideal Graham es-

capement. Since the Brocot has a work done index of 0.384 and the Graham of 0.5, the

Brocot is less efficient by over 23%!

To get a work done index for the Brocot, allow 1º of drop, as for the Graham. Just

multiply the index by the same factor:

0.384 x 5/6 = 0.32. Therefore, you could say that

the Brocot is only 32% efficient in practice (at best).

The Brocot chart could be used to create a graph. The Brocot graph, on the

following page, shows the work done to rotate the pallet in each instant (over the time it

takes the tooth to slide across the pallet).

BROCOT

Fe=1

time

angle (rad)

fv*dD

0.000

1.571

0.000

0.020

1.551

0.020

0.000

0.040

1.531

0.040

0.002

0.001

0.000

0.060

1.511

0.060

0.004

0.002

0.000

0.080

1.491

0.080

0.006

0.003

0.000

0.100

1.471

0.100

0.099

0.010

0.004

0.000

0.120

1.451

0.120

0.119

0.014

0.004

0.001

0.140

1.430

0.140

0.139

0.020

0.005

0.001

0.160

1.410

0.160

0.158

0.026

0.006

0.001

0.180

1.390

0.180

0.177

0.032

0.007

0.001

0.200

1.369

0.200

0.196

0.040

0.008

0.001

0.220

1.349

0.220

0.215

0.048

0.008

0.002

0.240

1.328

0.240

0.233

0.058

0.009

0.002

0.260

1.308

0.260

0.251

0.068

0.010

0.003

0.280

1.287

0.280

0.269

0.078

0.011

0.003

0.300

1.266

0.300

0.286

0.090

0.012

0.003

0.320

1.245

0.320

0.303

0.102

0.012

0.004

0.340

1.224

0.340

0.320

0.116

0.013

0.004

0.360

1.203

0.360

0.336

0.130

0.014

0.005

0.380

1.181

0.380

0.351

0.144

0.015

0.005

0.400

1.159

0.400

0.367

0.160

0.016

0.006

0.420

1.137

0.420

0.381

0.176

0.016

0.006

0.440

1.115

0.440

0.395

0.194

0.017

0.007

0.460

1.093

0.460

0.408

0.212

0.018

0.007

0.480

1.070

0.480

0.421

0.230

0.019

0.008

0.500

1.047

0.500

0.433

0.250

0.020

0.008

0.520

1.024

0.520

0.444

0.270

0.020

0.009

0.540

1.000

0.540

0.454

0.292

0.021

0.010

0.560

0.976

0.560

0.464

0.314

0.022

0.010

0.580

0.952

0.580

0.472

0.336

0.023

0.011

0.600

0.927

0.600

0.480

0.360

0.024

0.011

0.620

0.902

0.620

0.486

0.384

0.024

0.012

0.640

0.876

0.640

0.492

0.410

0.025

0.012

0.660

0.850

0.660

0.496

0.436

0.026

0.013

0.680

0.823

0.680

0.499

0.462

0.027

0.013

0.700

0.795

0.700

0.500

0.490

0.028

0.014

0.720

0.767

0.720

0.500

0.518

0.028

0.014

0.740

0.738

0.740

0.498

0.548

0.029

0.015

0.760

0.707

0.760

0.494

0.578

0.030

0.015

0.780

0.676

0.780

0.488

0.608

0.031

0.015

0.800

0.644

0.800

0.480

0.640

0.032

0.015

0.820

0.609

0.820

0.469

0.672

0.032

0.015

0.840

0.574

0.840

0.456

0.706

0.033

0.015

0.860

0.536

0.860

0.439

0.740

0.034

0.015

0.880

0.495

0.880

0.418

0.774

0.035

0.015

0.900

0.451

0.900

0.392

0.810

0.036

0.014

0.920

0.403

0.920

0.361

0.846

0.036

0.013

0.940

0.348

0.940

0.321

0.884

0.037

0.012

0.960

0.284

0.960

0.269

0.922

0.038

0.010

0.980

0.200

0.980

0.195

0.960

0.039

0.008

1.000

0.000

1.000

0.000

1.000

0.040

0.000

0.384

CLOCK AND WATCH ESCAPEMENT MECHANICS (1997) - page 1