# Blog: Vibrations

Date: 17 December 2015

Copyright © David Boettcher 2006 - 2017 all rights reserved.I make additions and corrections to this web site frequently, but because they are buried somewhere on one of the pages the changes are not very noticeable, so I decided to create this blog section to highlight new material. Here below you will find part of one of the pages that I have either changed or added to significantly.

The section below is from my second page about technical aspects of watch movements.

If you have any questions or comments, please don't hesitate to contact me via my Contact me page.

### Vibrations

Each swing of the balance from the centre to the point where it reverses and returns to the centre is called a vibration or beat. The Swiss use the term vibration, Americans use beat. Escapements are often characterised by their "vibrations per hour" (vph), sometimes shortened to simply "vibrations", or beats per hour (bph).

At the start of each vibration, as the balance swings through the centre, the impulse pin moves the lever and a tooth of the escape wheel is unlocked by a pallet. The shape of the pallet face and escape wheel tooth are designed so that after unlocking an impulse is given to the balance, replacing the energy lost by friction and keeping it turning. After the escape wheel has turned slightly, the other pallet locks a tooth which generates an audible tick.

Very old pocket watches ran at 14,400, 15,400 or 16,200 vibrations per hour. The 18,000 vph escapement eventually superseded all of these and for many decades this was standard, giving 5 vibrations or ticks per second. With 3,600 seconds in an hour, 5 ticks per second = 18,000 vph.

Higher frequency escapements were introduced to give mechanical watches greater resolution, which was useful for timing events accurately. These operated at 21,600 vph then 28,800 vph, giving 6 or 8 vibrations per second. When electric watches came along higher frequencies were introduced for mechanical watches such as the Girard Perregaux 36,000 vph movement of 1966 and the Zenith El Primero 36,000 vph movement of 1969.

In physics terms a cycle is a complete swing of the balance one way and then the other way, returning to where it started, so an 18,000 vph, 5 vibrations per second, balance cycles at 2.5 cycles per second, called Hertz (Hz).

The sketch shows the flow of power from the great wheel through the train to the escapement. The great wheel, which might be on a going barrel or a fusee, drives the pinion of the centre wheel. The minute hand is mounted on the arbor of the centre wheel so it must turn once an hour. The centre wheel turns the pinion of the third wheel, and the third wheel turns the pinion of the fourth wheel. The fourth wheel drives the pinion of the escape wheel, which is locked by the pallets on the lever until a swing of the balance unlocks a pallet and the escape wheel can advance by one tooth. From number of teeth on each wheel and pinion in the train you can work out how fast each wheel is turning and how many vibrations the escapement makes in an hour.

Typical trains might have the following tooth counts.

Vph | 18,000 | 14,400 | ||
---|---|---|---|---|

Wheel name | Wheel teeth | Pinion leaves | Wheel teeth | Pinion leaves |

Centre wheel | 80 | 64 | ||

Third wheel | 60 | 10 | 60 | 8 |

Fourth wheel | 70 | 8 | 56 | 8 |

Escape wheel | 15 | 7 | 15 | 7 |

The relative speed of two wheels is given by the ratio of the teeth on the wheel and the teeth on the pinion it is driven by or driving. For example, in this train, the third wheel will turn 8 times as fast as the centre wheel, because 80 teeth on the centre wheel engage with 10 teeth on the third wheel pinion. The third wheel pinion will make one revolution when 10 teeth of the centre wheel have turned with its 10 teeth, and as the centre wheel has 80 teeth it will rotate once for 8 turns of the third wheel.

The escape wheel has 15 teeth, and for the wheel to make one complete revolution each tooth has to pass both the entry and the exit pallet in getting back to where it started from, so each tooth on the escape wheel needs two ticks to get back to where it started. For an 18,000 vph train, which ticks 5 times per second, a 15 tooth escape wheel will take 30 ticks to make one revolution. Each tick takes 1/5 seconds, so a complete revolution of the escape wheel takes 30 x 1/5 = 6 seconds.

The fourth wheel engages with the escape wheel pinion, which has 7 teeth. The fourth wheel has 70 teeth, so it will turn once for every 10 revolutions of the escape wheel, that is once every 60 seconds. That's good, because the fourth wheel arbor also carries the seconds hand, so it is useful that it turns once a minute.

Knowing that the centre wheel rotates one an hour allows us to work out the vibrations and frequency of the escapement using the following formula, where w is the number of teeth on a wheel and p is the number of teeth on a pinion, so tw is the tooth count of the third wheel and tp is the number of leaves on its pinion. Wheels have teeth but pinions have leaves. Why? I have no idea. The 2 on the top is there because each tooth of the escape wheel has to escape twice during one revolution, the 3600 on the bottom is there because the centre wheel revolves once in one hour, which is 3,600 seconds.

The calculator below lets you enter all of these wheel and pinion counts and calculates the escapement's vibrations for you.

#### Watch with seconds

It isn't always necessary to count all the wheel and pinion teeth if you just need to work out how many vibrations a movement does. This is because the ratio of the fourth and centre arbors is usually fixed - if the watch has a seconds hand the fourth arbor turns once a minute to drive the seconds hand, the centre arbor turns once an hour and drives the minute hand. So the ratio between the centre and fourth arbors must be 60, no matter what the counts are of the centre wheel, third wheel, third pinion and fourth pinion are. We can therefore simplify our calculator and just enter the counts of the fourth wheel, escape pinion and escape wheel, like this. The 60 on the bottom is because the seconds hand takes 60 seconds to revolve.

If you have any questions or comments, please don't hesitate to contact me via my Contact me page. Back to the top of the page.

Copyright © David Boettcher 2006 - 2017 all rights reserved. This page updated January 2016. W3CMVS.