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<font class="text1" size="2" color="#FFFFFF"><b> Topic: Mechanical Pendulum Gear Clocks, Escapement</b> (Read 41965 times)</font></td>
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<font size="2"><b>Mechanical Pendulum Gear Clocks, Escapement</b></font> <font size="1"><a href="http://www.ocf.berkeley.edu/~wwu/YaBBAttachments/dbe.scad.pdf" target="_blank"><img src="escapement_files/paperclip.gif" border="0" align="absmiddle"> dbe.scad.pdf</a></font><br>
<font size="1">« <b> on:</b> Jul 18<sup>th</sup>, 2015, 1:57pm »</font></td>
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<a href="https://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_general;action=display;num=1437175174" target="_blank"><b>Mechanical Pendulum Gear Clocks</b></a> <br> <br> <font color="Cyan"><b>Escapement</b></font> <br> <br> 1) Graham <br> 2) Major Parts <br> 3) Cycle <br> 4) Purpose <br> 5) Escape Wheel <br> 6) Draw the Wheel <br> 7) Draw the Fork <br> <br> <br> <font color="Cyan"><b>Graham</b></font> <br> <br>
In the introductory chapter we have modularized a mechanical gear clock
into three major parts - a time intervals producer (pendulum), an
intervals manager (escapement) and a time displayer (hands and face). We
have covered the pendulum in a dedicated chapter and now we know how to
calculate the pendulum's length for a chosen period <b>T</b>. In this
chapter we will investigate the inner workings of an escapement
mechanism which is used to convert the back and forth swinging motion of
a pendulum into a predictable circular motion of the gears. <br> <br>
Over the years many different types of escapement mechanisms were
invented. The purpose or area of application usually defined the
concrete implementation. For example, a bell tower clock positioned at a
considerable height above ground is subject to sudden wind gusts or
snow accumulation which could be damaging to the mechanism if the
appropriate steps are not taken. The need for a sturdy assembly for such
situations was answered with <i>Gravity</i> and <i>Pin Wheel</i> escapements. <br> <br>
In general an escapement mechanism carries two names - of its inventor,
if known, and of its characteristic feature. For example, verge or
crown-wheel, Hooke anchor or recoil, Graham deadbeat, Dennison gravity,
Lepaute pin wheel, Brocot pin pallet, Mudge gravity, Grimthorpe
three-legged gravity, Riefler, Harrison grasshopper, Bloxham and so on. <br> <br> There are two reasons why we will examine the inner workings of the <i>Deadbeat</i> or <i>Graham Escapement</i>
invented as the name implies by the British watchmaker George Graham
(1673-1751) in 1715. Time being on our side the deadbeat escapement has
been historically proven to produce accurate results. And it does not
require expensive industrial machinery to manufacture - it is well
within our reach, after some practice, to have it ticking within couple
of hours. In addition George Graham was one of those simply decent human
beings whose technical talents were matched by an equal amount of
generosity - he refused to patent his invention reasoning that other
clock makers should also be able to use it free of charge. <br> <br> <br> <font color="Cyan"><b>Major Parts</b></font> <br> <br> As a first much simplified cut the two major parts of most of popular escapements are an <i>Anchor</i> or a <i>Fork</i> and an <i>Escape Wheel</i>. Let us examine these parts as they are implemented in the deadbeat escapement. <br> <br>
A fork and a pendulum are coaxal. The net affect of such an arrangement
is that they both swing in unison. It further means that if a pendulum
reaches its extreme position - so does the fork. Though most escapement
forks are called forks they are actually two-tooth spur gears in
disguise. As such they must mesh with the escape wheel to operate
properly. I have used the OpenSCAD software to create the following
drawing. It is based on the 2-D geometric constructions of the escape
wheel and the fork at the end of this chapter. If you want to
understand, modify and improve the attached OpenSCAD program it is best
to work through the 2-D steps first: <br> <br> <img src="escapement_files/rlu_pgc04.png" alt="" border="0"> <br> <br> The two major parts of a fork are a <i>Crutch</i> or an <i>Arm</i> and a <i>Pallet</i>.
A fork has two arms and two pallets - one per arm. The angle under
which the pallets are seen from the center of the escape wheel depends
on a variable set by the author - the number of teeth of the escape
wheel embraced or hugged by the fork. The more teeth are embraced - the
larger the angle and conversely. <br> <br> The escape wheel can
rotate in either of two directions - clockwise or counterclockwise, the
former by far being the most popular one. If the escape wheel rotates
clockwise, as depicted in the drawing below, then the pallet on the left
is called an <i>Entry Pallet</i>, <b>P<sub>n</sub></b>, while the pallet on the right is called an <i>Exit Pallet</i>, <b>P<sub>x</sub></b>.
If the escape wheel rotates counterclockwise, a less popular but
nonetheless an implementable choice, then the pallet on the right would
be an Entry Pallet while the pallet on the left would be an Exit Pallet
with the remaining definitions that follow reversed accordingly. The
direction of rotation of the escape wheel must be decided on ahead of
time because it affects the fork's final configuration which is not
symmetrical (to avoid the clutter the pendulum is not shown): <br> <br> <img src="escapement_files/rlu_pgc05.png" alt="" border="0"> <br> <br> Each pallet has two special surfaces or, more commonly, simply <i>Faces</i>. One face is called <i>Dead</i>, or <i>Stopping</i>, or <i>Locking</i> or <i>Resting Face</i>, the other is called an <i>Impulse Face</i>.
The stopping face is a portion of a right circular cylinder whose axis
of rotation coincides with the axis of the fork. The impulse face is a
plane formed by the intersection of the above (stopping) cylinder with
another plane running parallel to the fork's or cylinder's axis of
rotation. <br> <br> In this arrangement the stopping face <b>S<sub>n</sub></b> of the <i>entry</i> pallet is located on its <i>outer</i> side while the stopping face <b>S<sub>x</sub></b> of the <i>exit</i> pallet is located on its <i>inner</i>
side. As such the pallets are not entirely symmetrical. The absence of
symmetry is explained by the fact that two different types of motion are
forced to interact with each other - circular, in one direction only,
of the escape wheel and oscillating, back and forth, of the fork. <br> <br> The intersection of the stopping and impulse faces forms a <i>Stop</i>, or <i>Lock</i>, or <i>Rest</i> or <i>Dead Corner</i>. The intersection of the impulse face with the face opposite to the stopping face forms the <i>Discharge Corner</i>. <br> <br>
From the above definitions of the stopping and impulse faces it follows
that it is possible to make either of them equidistant from the center
of the fork. As such there are two major design variations of the fork -
when the <i>impulse</i> faces are equidistant from the axis of the fork or when the <i>stopping</i> faces are equidistant from the axis of the fork.
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<font size="2"><b>Re: Mechanical Pendulum Gear Clocks, Escapement</b></font> <br>
<font size="1">« <b>Reply #1 on:</b> Jul 18<sup>th</sup>, 2015, 2:01pm »</font></td>
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If the <i>impulse</i> faces are equidistant from the axis of the fork then its arms have <i>equal</i> lengths while if the <i>stopping</i> faces are equidistant from the center of the fork then its arms have <i>different</i>
lengths. The drawing above depicts a fork with the equidistant impulse
faces. Historically there was much debate about which arrangement is
better but by now it has been established that there is no major
difference between the two performance wise so pick the most
technologically convenient one. I will cover the construction of both
types of forks. <br> <br> Unlike the fork the escape wheel is not
attached to the pendulum's axis but rather sits on a separate axis of
its own. Normally the escape wheel is set in motion with a weight
attached to it with a string. As gravity pulls the weight down the
weight pulls on the string which in turn pulls on and spins the wheel.
After the weight reaches the floor it stops pulling on the escape wheel
which eventually comes to a complete stop. Rewind the string to set the
escape wheel in motion again: <br> <br> <img src="escapement_files/rlu_pgc06.png" alt="" border="0"> <br> <br> <br> <font color="Cyan"><b>Cycle of Operation</b></font> <br> <br>
To better understand the cycle of operation of an escapement mechanism
let us conduct the following thought experiment. Imagine a fork swinging
back and forth by itself in one location and an escape wheel spinning
by itself in a different location so that the two are not in contact.
Observe that we have two independent motions occurring simultaneously.
Nothing particularly interesting is happening. Now let us bring the
swinging fork and the spinning wheel closer and closer together until
they come into contact in the same plane as spur gears should, engage
and start a two-way interaction with each other. The way to determine
the exact distance between the centers of the escape wheel and the fork
will be explained below. <br> <br> Now on the one hand, during the
moments of their physical contact, the fork puts an end to the escape
gear's freewheeling by forcing it to turn at a specific rate which we
can calculate ahead of time. On the other hand, the escape wheel passes
the potential energy of the attached weight on to the fork in the form
of a gentle nudge. The fork passes that energy on to the axis which in
turn passes it on to the pendulum thus keeping it swinging at a constant
rate. <br> <br> The upcoming terminology may sound very confusing
for a novice because its use is overloaded - the same words are used to
describe <b>a process</b> or <b>an event</b> - as verbs - and define <b>magnitudes</b> - as nouns. In particular this applies to <i>Lock</i>, <i>Lift</i> and <i>Drop</i>. Not only that - the word <i>drop</i> is used in two different contexts. <br> <br>
Before we deep dive into more details it should be duly noted that from
this point onward, in various procedures, steps and definitions I will
explicitly avoid specifying any measurements in <i>linear</i> magnitudes. For these procedures, steps and definitions to be truly generic they must be framed in the independent terms - <i>angular</i> in this case. The gears of <i>any physical size</i> can then be drawn - small, medium or large and anything in between. <br> <br> <img src="escapement_files/rlu_pgc07.png" alt="" border="0"> <br> <br>
In the series of sketches above the escape wheel always runs clockwise
and so are the progressive positions of the single depicted tooth. Let
us start at the leftmost sketch when the tip <b>T</b> of the wheel's tooth just kissed the entry pallet <b>P<sub>n</sub></b> on the (outer) stopping face <b>S<sub>n</sub></b>. The reason this surface is called <i>stopping</i> is because for a brief moment the escape wheel stops rotating since <b>S<sub>n</sub></b> is in its way. <br> <br> The exact <i>manner</i> in which the wheel stops is very important - it does so without any recoil which the earlier <i>anchor</i>
design suffered from. The absence of recoil during the fork-wheel
interaction is one of the key features of Graham's invention. Do keep in
mind that even though the <i>wheel</i> stops - the <i>pendulum</i> does not as it <i>and the pallet of the fork</i> keep tracing out circular paths. <br> <br> As <i>a process</i> or as <i>an event</i> the above kissing moment, which occurs very close to the pendulum's right extreme point, is called a <i>Lock</i>. By analogy with the name of the responsible surface we might as well call it <i>Stop</i>, <i>Rest</i> or <i>Death</i>. <br> <br>
As a purely geometric idea a point or an edge are very convenient
notions but in practice the tip of the tooth of the escape wheel should
not come into contact with the stopping face <b>S<sub>n</sub></b> dead
on a point or an edge. It should also not land on the impulse face
either - that as is easy to imagine would be really bad. So in practice,
for a safe mechanism operation, we want the tip of the tooth to land
somewhere above the lock <i>corner</i> <b>C<sub>n</sub></b> at a point we can call a <i>Lock Point</i> <b>L<sub>n</sub></b>. As <i>a magnitude</i> then a Lock is the angle between <b>C<sub>n</sub></b> and <b>L<sub>n</sub></b> measured along the circumference of the stopping surface the center of which is located at <b>F</b>. In the drawing below I exaggerated that angle greatly for demonstration purposes. In practice it is usually about <b>1</b> degree: <br> <br> <img src="escapement_files/rlu_pgc08.png" alt="" border="0"> <br> <br>
After the pendulum reaches its right extreme point and flips its swing
from counterclockwise to clockwise the entry pallet's (outer) stopping
face <b>S<sub>n</sub></b> glides upwards for a while, along the stationary tip <b>T</b>, tracing a circular arc, until the entry pallet's lock corner <b>C<sub>n</sub></b> comes into contact with <b>T</b>. Note that in reality <b>C<sub>n</sub></b> is an edge - an intersection of a right circular cylinder with a plane running parallel to the cylinder's axis of rotation. <br> <br> At that moment the wheel resumes its clockwise rotation as nothing stands in its way. The tip <b>T</b> now kisses the impulse face of the entry pallet <b>I<sub>n</sub></b>, see the second sketch above. As <i>a process</i> it is called a <i>Lift</i>. During that process as <b>T</b> runs from a lock to a discharge corner, from <b>C<sub>n</sub></b> to <b>D<sub>n</sub></b>, the fork sweeps a certain angle. A Lift as <i>a magnitude</i> is that angle: <br> <br> <img src="escapement_files/rlu_pgc09.png" alt="" border="0">
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<font size="2"><b>Re: Mechanical Pendulum Gear Clocks, Escapement</b></font> <br>
<font size="1">« <b>Reply #2 on:</b> Jul 18<sup>th</sup>, 2015, 2:03pm »</font></td>
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It is during that lift, as both the wheel, <b>T</b>, and the fork, <b>P<sub>n</sub></b>, run clockwise, the potential energy is transferred <i>from</i> the wheel <i>to</i> the pallet and hence to the pendulum. That is why <b>I<sub>n</sub></b> is called an <i>impulse</i> face. To reiterate, the energy is transferred from the wheel to the pallet on the pallet's upswing, against gravity. When <b>T</b> reaches <b>D<sub>n</sub></b>,
the discharge corner, it breaks its contact with the fork. The
arithmetic sum of the above two angles, the lock and the lift (or actual
lift) is called a <i>Total Lift</i> or an <i>Escape Angle</i>. <br> <br>
At this point the escape wheel runs free but only for a short while as
it is soon stopped by the opposite, exit, pallet because the entry
pallet swung almost all the way up corresponding to the event of the
pendulum almost reaching its left extreme point. This is the exact
moment captured in the last sketch above - the angle swept by the escape
wheel while running free of contact with either pallet. As <i>a process</i> this freewheeling is called a <i>Drop</i>. It is bracketed by two events - <b>T</b>
breaking contact with the entry pallet and the escape wheel as a whole
being stopped completely as its different tooth comes into contact with
the exit pallet. <br> <br> During this process the escape wheel or simply <b>T</b> sweeps a certain angle. A Drop as <i>a magnitude</i> is that angle - it is highlighted with a different color in the sketch. <br> <br> The drop as <i>a process</i>
occurs once per beat or twice per pendulum's period of oscillation.
Once when one tooth is about to kiss the exit pallet while another tooth
turns away from the discharge corner of the entry pallet and another
time when one tooth is about to kiss the entry pallet while another
tooth turns away from the discharge corner of the exit pallet. <br> <br> After the drop as a process is over the above tooth-pallet interaction scenario plays out again. The wheel's tooth <b>T</b> kisses the exit pallet's (inner) stopping face <b>S<sub>x</sub></b>, the lock. The pendulum's swing flips from clockwise to counterclockwise. The lock corner <b>C<sub>x</sub></b> glides into contact with <b>T</b> which resumes its clockwise run giving <b>I<sub>x</sub></b> a gentle nudge upwards against gravity, the lift. Note that during this love affair the fork runs counterclockwise while <b>T</b>
runs clockwise until they part their ways. The wheel revolves free for a
short while, the drop, but is soon trapped by the entry pallet and the
whole process repeats. <br> <br> To sum up the above cycle at the
intuitive, gut level this dynamic duo goes through lock, lift and drop
in a potentially never ending succession. The lock and the drop as the
magnitudes should be as small as technologically possible. <br> <br>
Keep in mind that not only the stopping faces of the entry and exit
pallets are situated differently but the orientation of the impulse
faces is different as well. When the pendulum (and the fork) is at rest
the impulse face of the entry pallet is closer to a horizontal while the
impulse face of the exit pallet is closer to a vertical. <br> <br> <br> <font color="Cyan"><b>Purpose</b></font> <br> <br> Now that we know <i>how</i> the escapement mechanism works let us investigate <i>why</i>
is it needed, what is its purpose. Since we already have a pendulum
that swings at a known rate we wish to somehow exploit that regularity
by transforming its choppiness into continuousness. That is where the
escapement mechanism comes in. <br> <br> It actually serves two purposes - to <i>convert</i> and to <i>maintain</i>.
As a pure time intervals manager the escapement is needed to convert
the back and forth motion of a pendulum into a controlled circular
motion of the gears. To put it even simpler - escapement translates so
many swings into that many rotations. The swings are of the pendulum,
the rotations are of the gears. <br> <br> Let us break down the motion of a pendulum in a more atomic way and introduce a formal concept of a <i>Swing</i> or a <i>Beat</i>. Let us further agree that such a beat is <i>an event</i>
of a pendulum reaching one of its extreme points after departing from
the opposite extreme point. Using the drawing from the Pendulum chapter a
beat corresponds to our pendulum reaching the point <b>B</b> after departing from the point <b>A</b>. To make it perfectly clear a beat is not a <i>time</i> but an <i>event</i> which occurs after some amount of time goes by.
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<font size="2"><b>Re: Mechanical Pendulum Gear Clocks, Escapement</b></font> <br>
<font size="1">« <b>Reply #3 on:</b> Jul 18<sup>th</sup>, 2015, 2:07pm »</font></td>
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With the above definition in place we can capture a unique
characteristic of any pendulum numerically via its rate or frequency -
in the number of beats per unit of time. One convenient unit of time is
one minute or <b>60</b> seconds. Since in time one beat corresponds to half the period the number of beats per minute <b>B<sub>m</sub></b> then is: <br> <br> <center><font color="Cyan">B<sub>m</sub> = 60/(T/2) = 120/T </font><font color="Yellow">(e.1)</font></center> <br> <br> where <b>T</b> is given in seconds. Our <b>2</b>-second pendulum then makes <b>60</b> beats per minute, for example. <br> <br> Using the above terms we can say that the escapement mechanism converts a known number of pendulum's beats per minute <b>B<sub>m</sub></b> into a known and adjustable on demand rate of revolution of a single gear <s>number of full revolutions <b>R<sub>m</sub></b> of a single gear per minute</s>. If we call this special (conversion) gear an <i>Escape Wheel</i>
and to that wheel we connect some number of gears the last one of which
rotates at the exact rate of one full revolution per minute then we can
attach a hand to that wheel which gives us a way to tell seconds. <br> <br>
If to the above second gear we connect some number of gears the last
one of which rotates at the exact rate of one full revolution per hour
(or <b>60</b> times slower) and to that wheel we attach a hand then we have a way to tell minutes. <br> <br>
Finally, if to the above minute gear we connect some number gears the
last one of which rotates at the exact rate of one full revolution in <b>12</b> hours (or <b>12</b>
times slower) and to that wheel we attach a hand then we have a way to
tell hours, the clock is complete and we can ... sell it. Just kidding. <br> <br> Our <i>idealized</i> pendulum may swing forever but a <i>real</i>
one of course will not. The friction and the air resistance will eat
away the initial energy given to the pendulum by our muscle power. If
left unattended such a pendulum will not swing at a constant rate - it
will slow down over time until it stops completely. <br> <br> In
addition to a conversion role the escapement mechanism is used to
address this gradual slow down phenomena by taking on an extra role of a
constant rate of swings maintainer. We have covered the exact way this
is done in the cycle of operation section above. The energy transfer
from the wheel to the fork in the form of gentle pushes or nudges must
be carefully orchestrated. These nudges can not be haphazard and
frivolous but rather must be doled out in precise and delicate dosages -
not too large as they will disrupt the workings of the entire mechanism
and not too small as they will have no meaningful effect. Put it
another way, the escapement mechanism releases the potential energy of
the weight or the elastic potential energy of a spring in small chunks
(quanta) over a prolonged period of time. The devil here is in the
construction details and that is what we will look at next. <br> <br> <br> <font color="Cyan"><b>The Deadbeat Escape Wheel</b></font> <br> <br> The deadbeat escape wheel is a special <b>n</b>-tooth
spur gear that must mesh with another special two-tooth spur gear - a
fork. If at this point you are not familiar with gears you should
acquire some basic knowledge about them out of band since this knowledge
will be much needed. I list a few gear-related books at the end of the
Hands chapter but the crash course on gears as they relate to us is
this. <br> <br> Conceptually gears are levers. Practically they
can be classified roughly by the relative position of their axes in
space - parallel, intersecting, neither. Physically gears are
implemented based on surfaces capable of rolling without slipping. The
most popular surfaces are right cylinders, truncated cones and
hyperboloids, circular or elliptic. If two engaged gears rotate smoothly
without blocking and vibration it is said that they <i>mesh</i>. Two or more gears connected for a purpose are called a <i>Gear Train</i>. The purpose usually is to convert a known input into a desired output. <br> <br> A gear to which an input or power is applied directly is called a <i>Driving</i> or an <i>Input Gear</i> and a gear affected by it is called a <i>Driven</i> or a <i>Follower</i> or an <i>Output Gear</i>. A <i>Gear Ratio</i> <b>r<sub>io</sub></b> of two gears is: <br> <br> <center><font color="Cyan">r<sub>io</sub> = <img src="escapement_files/omega.gif" alt="" border="0"><sub>i</sub>/<img src="escapement_files/omega.gif" alt="" border="0"><sub>o</sub> = N<sub>o</sub>/N<sub>i</sub> </font><font color="Yellow">(e.2)</font></center> <br> <br> where <img src="escapement_files/omega.gif" alt="" border="0"> designates an angular velocity and <b>N</b>
designates a tooth count of the gears. If in a gear train we carve out a
path from the input gear to the gear of interest, possibly the last
one, and calculate the gear ratio of all the pairs in between then by
taking the product of all these ratios we will arrive at the gear ratio
of an aggregate - the whole or part of a train. To calculate the gear
ratio of a current pair in the above set make the output gear of the
previous pair an input gear for the current one. We will look at the
gear trains in a bit more detail in the Hands chapter. <br> <br> <br> [e]See crossed out text above[/e]
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<td align="left"><font size="1">« <i>Last Edit: Jul 25<sup>th</sup>, 2015, 3:54pm by rloginunix</i> »</font></td>
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<font size="2"><b>Re: Mechanical Pendulum Gear Clocks, Escapement</b></font> <br>
<font size="1">« <b>Reply #4 on:</b> Jul 18<sup>th</sup>, 2015, 2:12pm »</font></td>
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In clock making we are interested in a special type of gears
that mesh in the same plane, have parallel axes and straight teeth.
Such gears are called <i>Spur Gears</i>. They are distinguished further by the shape of their teeth or by a particular type of a plane curve used to construct them. A <i>Cycloid</i> was an early choice but later on an <i>Involute of a Circle</i>
became very popular as its use lead to gears that performed better
under high speeds and heavy loads. An involute of a circle is a plane
curve traced by a point at the end of a taut string as it unwinds off of
or winds around the circumference of a stationary circle. Clock gear
trains, however, are unlike most of their industrial cousins. They turn
very slowly under very light loads. As such it is perfectly fine to use
cycloidal or involute (or their combination) tooth profiles in your
projects. <br> <br> The deadbeat escape wheel, as I mentioned
before, is a special animal as its teeth can be made triangular in shape
and in theory arbitrary in number. In practice, however, certain tooth
counts have an advantage over the others. To see why let us task
ourselves with a requirement of designing a clock with the smallest
number of gears practically possible. The question then is - how many
teeth must the escape wheel have for it to make one full revolution in
exactly one minute? If we find the answer to that question then we can
attach a second hand to the escape wheel directly and use it to drive
the minute and hour gears. <br> <br> Since the escapement
mechanism is designed in such a way that each extremity of the fork
engages one tooth of the escape wheel once per period of oscillation
then we can equate the number of such engagements to the number of teeth
of the escape wheel <b>N<sub>m</sub></b> where sub index <b>m</b> designates the chosen time interval of one minute or <b>60</b> seconds. It follows then that if one engagement occurs in <b>T</b> seconds then <b>N<sub>m</sub></b> engagements occur in a given time interval or <b>60</b> seconds: <br> <br> <center><font color="Cyan">N<sub>m</sub> * T = 60</font></center> <br> <center><font color="Cyan">N<sub>m</sub> = 60/T </font><font color="Yellow">(e.3)</font></center> <br> <br> Our <b>2</b>-second pendulum then should have a <b>30</b>-tooth
escape wheel which will make one full revolution in one minute. Keeping
the one full revolution per minute requirement the <b>1</b>-second pendulum should have a <b>60</b>-tooth escape wheel, the <b>4</b>-second pendulum should have a <b>15</b>-tooth wheel and so on. <br> <br> Now that we have a connection between <b>T</b> and <b>N<sub>m</sub></b> instead of deciding on the value of <b>T</b> we may decide on the value of <b>N<sub>m</sub></b>. If we solve (e.3) for <b>T</b> and substitute it into the length of pendulum's rod formula we can dispense with the period altogether and calculate <b>L</b> as a function of <b>N<sub>m</sub></b>: <br> <br> <center><font color="Cyan">T(N<sub>m</sub>) = 60/N<sub>m</sub></font></center> <br> <center><font color="Cyan">L(N<sub>m</sub>) = gT<sup>2</sup>/(4<img src="escapement_files/pi.gif" alt="" border="0"><sup>2</sup>S<sup>2</sup>) = 3600g/(4<img src="escapement_files/pi.gif" alt="" border="0"><sup>2</sup>N<sub>m</sub><sup>2</sup>S<sup>2</sup>)</font></center> <br> <center><font color="Cyan">L(N<sub>m</sub>) = 900g/(<img src="escapement_files/pi.gif" alt="" border="0"> N<sub>m</sub> S)<sup>2</sup> </font><font color="Yellow">(e.4)</font></center> <br> <br> For example, if we take <b>S = 1</b> and wish to have a <b>40</b>-tooth escape wheel then a rough estimate for <b>L</b> for NYC, NY, US is <b>55.87</b> centimeters. For a <b>50</b>-tooth wheel we get roughly <b>35.76</b> centimeters and so on. <br> <br> As we can see we can drive the design of our clock by a number of parameters. We can fix <b>T</b> to obtain <b>N<sub>m</sub></b> and <b>L</b>, we can fix <b>N<sub>m</sub></b> to obtain <b>L</b> and <b>T</b> or we can fix <b>T</b> to obtain <b>L</b> and pick an arbitrary number of teeth for the escape wheel, <b>N</b>, without imposing the one full revolution per minute requirement on it. <br> <br> In the latter case let us say we want a <b>50</b>-tooth escape wheel for a <b>2</b>-second pendulum. In this case the wheel will make one full revolution every <b>50*2 = 100</b>
seconds. As an option, since we the designers are in control, we may
choose to not have the second hand at all. Since there are <b>3600</b> seconds in one hour our <b>50</b>-tooth escape wheel will make <b>3600:100 = 36</b> full revolutions. Consequently, to the above <b>50</b>-tooth escape wheel we attach a <b>36:1</b>
gear train the last gear of which will make one full revolution in
exactly one hour, we attach the minute hand to that gear and proceed on
to the hour hand as usual. <br> <br> Keep in mind that whether we
impose the one full revolution per minute requirement on the escape
wheel or not it still has to interact with the fork. As such between two
consecutive engagements <i>with the same pallet</i>, entry or exit, or per period of oscillation the escape wheel will turn: <br> <br> <center><font color="Cyan"><img src="escapement_files/theta.gif" alt="" border="0"><sub>T</sub> = 360/N </font><font color="Yellow">(e.5)</font></center> <br> <br> degrees where <b>N</b> is the escape wheel's tooth count. However, between two consecutive engagements <i>with the opposite pallets</i>, first entry and then exit, or per beat the escape wheel will turn half of the above value: <br> <br> <center><font color="Cyan"><img src="escapement_files/theta.gif" alt="" border="0"><sub>b</sub> = <img src="escapement_files/theta.gif" alt="" border="0"><sub>T</sub>/2 = 180/N </font><font color="Yellow">(e.6)</font></center> <br> <br> It follows then that if we attach a second hand directly to the escape wheel then per beat that hand will sweep <img src="escapement_files/theta.gif" alt="" border="0"><sub>b</sub>
degrees. To translate that angular measure into the corresponding
number of seconds swept by the second hand per beat we need to divide <img src="escapement_files/theta.gif" alt="" border="0"><sub>b</sub> by <b>6</b> degrees - the angular distance between the one-second/one-minute gradation marks on a traditional clock face: <br> <br> <center><font color="Cyan">sec<sub>b</sub> = <img src="escapement_files/theta.gif" alt="" border="0"><sub>b</sub>/6 = 30/N </font><font color="Yellow">(e.7)</font></center>
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<font size="2"><b>Re: Mechanical Pendulum Gear Clocks, Escapement</b></font> <br>
<font size="1">« <b>Reply #5 on:</b> Jul 18<sup>th</sup>, 2015, 2:15pm »</font></td>
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It should be clear now why some tooth counts for an escape wheel are more advantageous than the others. For our <b>2</b>-second pendulum the <b>30</b>-tooth escape wheel will turn <b>6</b>
degrees per beat making the second hand attached to it sweep exactly
one-second intervals - a perfect fit as long as we align the clock face
with gradations on it properly. For the upcoming drawing procedure,
however, it does not really matter exactly <i>how</i> you came up with the escape wheel's tooth count. You just have to settle on <i>a number</i>. <br> <br> Before we move on to the drawing steps we will need to get acquainted with a small amount of basic terminology. <br> <br> <br> <i>Wheel Anatomy</i> <br> <br> Since all our gears are spur gears when I say <i>"a gear"</i> it should be taken to mean <i>"a spur gear"</i>
unless otherwise noted. A spur gear is based on a right circular
cylinder or simply cylinder from now on. If we cut such a cylinder with a
plane perpendicular to the cylinder's axis of rotation a circular
surface will result. This surface is called a <i>Face</i> of a gear. The
edges of the gear's teeth are straight lines that run parallel to the
cylinder's axis of rotation and as such they are perpendicular to the
gear's face. That is what is meant when we say that the teeth of a spur
gear are <i>straight</i>. <br> <br> The largest circle of a gear's face is called an <i>Outer</i> or an <i>Addendum</i> or a <i>Tip Circle</i>.
Its radius is a linear distance from the center of a gear to the outer
tips of its teeth. Since we are dealing with a circle, by definition all
the outer tips of all the teeth are equidistant from the center of the
gear: <br> <br> <img src="escapement_files/rlu_pgc10.png" alt="" border="0"> <br> <br>
On the same face of a gear there is a circle on the circumference of
which the roots of the gear's teeth sit. That circle is called a <i>Root</i> or a <i>Dedendum Circle</i>.
The radius of a root circle is also a linear measure. The difference
between the radii of the outer and root circles which determines a
linear height of the gear's teeth is called a <i>Whole Depth</i>. <br> <br> <br> <i>Tooth Anatomy</i> <br> <br>
It is possible to use a number of different tooth profiles for a
deadbeat escape wheel. We will settle on a triangular profile - a shape
trapped between two straight lines - the <i>Leading Line</i> and the <i>Trailing Line</i>.
The tip of a tooth is located on the circumference of the tip circle.
The root of a tooth is located on the circumference of the root circle. A
small portion of the leading line close to the tip of a tooth comes
into a direct contact with the pallets. No points on the trailing line
come into contact with the pallets. <br> <br> Let us call the angle between the tip radius and the trailing line the <i>Trailing Angle</i> <img src="escapement_files/theta.gif" alt="" border="0"><sub>t</sub>: <br> <br> <img src="escapement_files/rlu_pgc11.png" alt="" border="0"> <br> <br>
Since in practice the very tip of a tooth is not exactly sharp let us
call the angle under which the two tooth tip edges are seen from the
center of the gear the <i>Central Angle</i> <img src="escapement_files/theta.gif" alt="" border="0"><sub>c</sub>.
In the drawing below the magnitude of the central angle is exaggerated
greatly for demonstration purposes. In practice it is about <b>0.5</b> degrees. Let us further agree that the two corresponding points (edges) on the circumference of the tip circle are the <i>Trailing Edge</i> and the <i>Leading Edge</i>: <br> <br> <img src="escapement_files/rlu_pgc12.png" alt="" border="0"> <br> <br>
Lastly, we will undercut our teeth by an arbitrary angle which the
leading line forms with the tip radius of a gear. Let us call that angle
the <i>Leading Angle</i> <img src="escapement_files/theta.gif" alt="" border="0"><sub>l</sub>.
The purpose of the undercutting is to make the surface area of contact
between the leading edge of a tooth and a pallet smaller which in turn
will make the friction between the two smaller. The undercutting also
makes the teeth of the escape wheel <i>lean in the direction of rotation</i>: <br> <br> <img src="escapement_files/rlu_pgc13.png" alt="" border="0"> <br> <br> Using the triangles involved we can work out the magnitude of the <i>Included Angle</i> of a tooth <img src="escapement_files/theta.gif" alt="" border="0"><sub>i</sub> - the angle between the trailing and leading lines: <br> <br> <img src="escapement_files/rlu_pgc14.png" alt="" border="0"> <br> <br> <center><font color="Cyan"><img src="escapement_files/theta.gif" alt="" border="0"><sub>i</sub> + (180 - <img src="escapement_files/theta.gif" alt="" border="0"><sub>t</sub>) + (<img src="escapement_files/theta.gif" alt="" border="0"><sub>c</sub> + <img src="escapement_files/theta.gif" alt="" border="0"><sub>l</sub>) = 180</font></center> <br> <center><font color="Cyan"><img src="escapement_files/theta.gif" alt="" border="0"><sub>i</sub> = <img src="escapement_files/theta.gif" alt="" border="0"><sub>t</sub> - <img src="escapement_files/theta.gif" alt="" border="0"><sub>c</sub> - <img src="escapement_files/theta.gif" alt="" border="0"><sub>l</sub></font></center> <br> <br>
Let us also agree to call the central angle under which two like edges
of two adjacent teeth are seen from the center of the escape wheel the <i>Angular Pitch</i> <b>ap</b>: <br> <br> <img src="escapement_files/rlu_pgc15.png" alt="" border="0"> <br> <br> The Angular Pitch is derived by dividing <b>360</b> degrees by the number of teeth of the escape wheel. <br> <br> With the above definitions in place the tooth, the wheel and the fork construction procedures become transparent.
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<font size="2"><b>Re: Mechanical Pendulum Gear Clocks, Escapement</b></font> <br>
<font size="1">« <b>Reply #6 on:</b> Jul 18<sup>th</sup>, 2015, 2:19pm »</font></td>
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<font color="Cyan"><b>To Draw the Deadbeat Escape Wheel</b></font> <br> <br>
In the long run, in a steady state of making clocks it is very likely
that you will be using software to draw the gears. However, it will not
hurt to know how to do it manually yourself. <br> <br> Before drawing a deadbeat escape wheel choose: <br> <br> 1) Direction of rotation. <br> 2) Tooth count <b>N</b>. <br> 3) Tip radius <b>R<sub>t</sub></b>. <br> 4) Root radius <b>R<sub>r</sub></b>. <br> 5) Tooth Trailing Angle <img src="escapement_files/theta.gif" alt="" border="0"><sub>t</sub>. <br> 6) Tooth Central Angle <img src="escapement_files/theta.gif" alt="" border="0"><sub>c</sub>. <br> 7) Tooth Leading Angle <img src="escapement_files/theta.gif" alt="" border="0"><sub>l</sub>. <br> <br> For the following demonstration the choices are: <br> <br> 1) Direction of rotation: clockwise. <br> 2) Tooth count <b>N = 30</b>. <br> 3) Tip radius <b>R<sub>t</sub></b>. <br> 4) Root radius <b>R<sub>r</sub> = 0.75*R<sub>t</sub></b>. <br> 5) Tooth Trailing Angle <b><img src="escapement_files/theta.gif" alt="" border="0"><sub>t</sub> = 18.5</b> degrees. <br> 6) Tooth Central Angle <b><img src="escapement_files/theta.gif" alt="" border="0"><sub>c</sub> = 0.5</b> degrees. <br> 7) Tooth Leading Angle <b><img src="escapement_files/theta.gif" alt="" border="0"><sub>l</sub> = 6</b> degrees. <br> It follows then that <b><img src="escapement_files/theta.gif" alt="" border="0"><sub>i</sub> = 18.5 - 0.5 - 6 = 12</b> degrees and <b>ap = 360:N = 12</b> degrees. <br> <br> Procedure: <br> <br> 1) Draw a straight line of centers <b>c</b>. The centers of the escape wheel and the fork will be on it, pick a convenient orientation. <br> <br> 2) With the center <b>W</b> on <b>c</b> and the radius <b>R<sub>t</sub></b> draw the tip circle <b>t</b>. <br> <br> 3) With the center at <b>W</b> and the radius <b>R<sub>r</sub></b> draw the root circle <b>r</b>. <br> <br> 4) Let <b>A</b> be one of the intersections points of <b>c</b> and <b>t</b>: <br> <br> <img src="escapement_files/rlu_pgc16.png" alt="" border="0"> <br> <br> 5) Construct one tooth starting with the Trailing Angle. With <b>W</b> as a leg point and <b>A</b> as a vertex draw the Trailing Angle <b><img src="escapement_files/theta.gif" alt="" border="0"><sub>t</sub></b> in the chosen direction, clockwise in this case. Let <b>D</b> be the intersection of the remaining side <img src="escapement_files/tau.gif" alt="" border="0"> of <img src="escapement_files/theta.gif" alt="" border="0"><sub>t</sub> with the root circle <b>r</b> so that <b><img src="escapement_files/theta.gif" alt="" border="0"><sub>t</sub> = <img src="escapement_files/angle.gif" alt="" border="0"> WAD</b>. <br> <br> 6) With <b>A</b> as a leg point and <b>W</b> as a vertex draw the Central Angle <img src="escapement_files/theta.gif" alt="" border="0"><sub>c</sub> in the chosen direction, clockwise in this case. Let <b>B</b> be the intersection of the remaining side <img src="escapement_files/sigma.gif" alt="" border="0"> of <img src="escapement_files/theta.gif" alt="" border="0"><sub>c</sub> with the tip circle <b>t</b> so that <b><img src="escapement_files/theta.gif" alt="" border="0"><sub>c</sub> = <img src="escapement_files/angle.gif" alt="" border="0"> AWB</b>. <br> <br> 7) With <b>W</b> as a leg point and <b>B</b> as a vertex draw the Leading Angle <img src="escapement_files/theta.gif" alt="" border="0"><sub>l</sub> in the chosen direction, clockwise in this case. Let <b>C</b> be the intersection of the remaining side <img src="escapement_files/lambda.gif" alt="" border="0"> of <img src="escapement_files/theta.gif" alt="" border="0"><sub>l</sub> with the root circle <b>r</b> so that <b><img src="escapement_files/theta.gif" alt="" border="0"><sub>l</sub> = <img src="escapement_files/angle.gif" alt="" border="0"> WBC</b>: <br> <br> <img src="escapement_files/rlu_pgc17.png" alt="" border="0"> <br> <br> At this point the shape <b>ABCD</b> delineates one tooth. <b>DA</b> is the trailing line (plane), <b>A</b> is the trailing edge, <b>B</b> is the leading edge, <b>BC</b> is the leading line (plane) and <b>DC</b> is the root: <br> <br> <img src="escapement_files/rlu_pgc18.png" alt="" border="0"> <br> <br> Using your software's features repeat the steps 4-7 the remaining <b>N - 1</b> times rotating the reference straight line <b>c</b> by the magnitude of the Angular Pitch <b>ap</b> about the center of the escape wheel <b>W</b> each time if needed. Most types of graphing software have a feature called <i>Rotate</i>. You can either rotate the whole tooth or if your software can not do that then rotate each point separately. <br> <br>
Once all the teeth are in place make sure that you have marked the
center of the escape wheel clearly. Draw a small circle around the
center of the wheel to represent the opening for its shaft: <br> <br> <img src="escapement_files/rlu_pgc19.png" alt="" border="0"> <br> <br>
This is where the science mostly ends and the art mostly begins. Draw
the spokes of the wheel making them regular in shape or ornamental -
your imagination is the only limit at this point. Keep in mind that the
escape wheel should possess two conflicting qualities. On the one hand,
it should be made as light as possible - hollow it out a lot and on the
other hand, the wheel and its teeth should be sturdy enough to withstand
the many many happy hours of operation. In other words balance the
light and hollow against the sturdy and true.
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<font size="2"><b>Re: Mechanical Pendulum Gear Clocks, Escapement</b></font> <br>
<font size="1">« <b>Reply #7 on:</b> Jul 18<sup>th</sup>, 2015, 2:25pm »</font></td>
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