## Clock Gear Math

Here is what we are trying to do:

Make a device where the minute shaft makes one revolution every 3600 seconds.

That's it.

Early clocks behaved this way with just a minute hand.

It wasn't until much later that clocks were made with hour and second hands.

Let's ignore the second and hour hands for a moment to make the calculations easier.

Other hands are merely gear ratios to the minute hand.

In our example the minute hand is the shaft with the 90 tooth gear.

This shaft makes one rotation every 3600 seconds or once an hour.

Another simplification to make understanding the calculations clearer is to keep the units consistent.

In our case we will use seconds as our unit of time.

It is very easy to get caught up in minutes and hours getting all confused with dividing and multiplying by 60 that your head will spin.

For example the minute hand rotates once an hour. Why isn't it called the hour hand. Because the hour hand rotates once every 12 hours. That's why. Yikes.

In our example the Escapement shaft makes one revolution every 60 seconds.

(It doesn't have to but many clocks use this shaft as the second hand.)

Then the gear ratio between the escapement wheel and the minute hand is then 3600 divided by 60.

Or 60:1.

Multiply the driving Wheels and divide by the pinions to get a 60 :1 ratio.

In our example (90X64)/(8X12)=60

In our Clock we also want one tick or tock sound per second.

Pendulums:

A 1 meter pendulum makes 1 tick and 1 tock sound every 2 seconds

Another way to look at this is a 1 meter pendulum makes a sound every 1 second

length of pendulum = g(T/(2 *pi))squared

where g is gravitational constant 9.8m/s/s

T is the period of the pendulum (in our case 2 seconds)

an pi is 3.14159

Therefore:

30 tooth escapement wheel gives (30 ticks +30 tocks)per rev

Or looking at this mathematicaly,

divide the time it takes for the escapement shaft to make one revolution

by the number of teeth on the escapement wheel to

determine the period of the pendulum.

60 sec/30 teeth=2 seconds

### The Hour Hand

The hour hand is simply a second set of gears that run 12:1 to the minute hand.

Once again multiplying the driving gears and dividing by the pinions

(48*40_/(10*16)=12

The Hour hand slides on a tube over the minute hand.

OK, this is fine for Gary's Clock but what if I want to make a wall clock where the pendulum isn't so long?

Maybe you've seen other clocks and wondered about the math behind the gear ratios.

Here is an explanation.....

Some common gear practices.

The smallest number of teeth on the pinion should not be less than 8.

An even better scenario is where the pinions are not smaller than 12 teeth.

Small tooth pinions less than 8 teeth do not engage very well.

See the gear animation page for more evidence.

It is also uncommon to see gear ratios larger than 10 to 1 for regular spur gears.

We could make a clock with a 600 tooth gear and 10 tooth pinion as an example and still get a 60 to 1 ratio.

A 600 tooth gear is pretty large. The 600 tooth gear with teeth similar in shape to our example would be 60 inches in diameter.

A bit large.

Let's look at two common gear trains.

One with 3 shafts between the minute hand and escapement and another with 4 shafts.

### Case 1- 3 shafts.

This arrangement is the same as the one used in Gary's Clock.

Remember the minute hand makes one rotation every 3600 seconds.

The escapement shaft rotates faster than the minute hand by a factor determined by the gear ratio.

Multiply the driving gears and divide by the pinions to get gear ratio.

Time for the escapement wheel to rotate is then 3600 divided by this ratio.

The period of the pendulum is then the time for one rotation of the escapement wheel divided by the number of teeth on the escapement wheel.

### Case 2- 4 shafts

Wall clocks do not usually have 1 meter long pendulums.

Cuckoo clocks for example have 300mm or shorter pendulums.

You may also notice that shorter pendulums move faster.

So if we add an extra gear reduction shaft we'd expect the escapement wheel to rotate much quicker and then require a shorter pendulum.

The other way to shorten the pendulum is to increase the number of teeth on the escapement wheel.

I have prepared an excel spreadsheet that I hope makes understanding how the gear ratios and the number of teeth on the escapement wheel affect the pendulum length.

clock-gear-calculations-excel.zip

Change the values in the green cells. The pendulum length is then calculated in the red cells.

An example for Gary's clock is given as well as for a paper clock many of you may have seen.

Enter your own values in the spare rows.

This excel file updated 27 sep03. Should be easier to use with embedded pictures to explain how the calculations apply to the gear train.

How Stuff Works-Gears

for a more complete explanation on gear ratio theory.