1. Wheels and pinions

2. Definitions
Wheel - A circle with a toothed profile turning on a central axis. Pinion - A small toothed wheel that works in unison with a larger toothed surface such as a wheel or rack. A toothed wheel with less than 20 leaves. Arbor - The central axis of a wheel. Pivot - The cylindrical bearing surface of a wheel or pinion that rides in a hole. Addendum - The part of each tooth between its tip and the pitch circle. Dedendum - The part of each tooth that lies inside the pitch circle. Cycloid - A point on a circumference of a circle that rolls without slipping along a straight line. Used to define the face of the tooth. Epicycloid - A point on a circle which rolls without slipping on the outside of a fixed circle. Also used to define the face of the tooth. Hypocycloid - A point on a circle which rolls without slipping on the inside of a fixed circle. The straight flanks of the teeth of wheels and pinions are hypocycloids. Used to define the flank of the tooth. Involute - A curve traced by a point on a straight line that rolls on a curve without sliding. Involute gearing is used for ratchet and transmission wheels (winding wheels under high pressure) Ogival - Pointed arch, term for the tip or point of a pinion or wheel tooth. Root - Term for the circumference of a circle that makes up the base or root of a pair of teeth Module - A standard used to determine the dimensions of a gear. It is the measure of the ratio between the pitch circle of a wheel and its number of teeth. In order for two toothed objects to mesh with one another the module must be the same.

3. Wheels and pinions
Watch wheels are generally made of brass or plastic while pinions are made from hardened steel. This arrangement is used because the difference in hardness of the two wheels creates a smooth transfer of energy and helps prevent wear. Additionally, the way wheels and pinions are joined is either through riveting or friction fitting and the softer brass accepts steel in both these methods very efficiently. Plastic wheels are injection molded. These are most commonly seen in quartz watches, but occasionally they are used in mechanical watches. Plastic has the benefit of being soft enough to give some cushioning and often times negates the need for oil. Another advantage is that a wheel and pinion assemble that is injection molded has no need for a joining method like riveting or friction fitting because the part is made in one piece.

4. Wheels and pinions
Wheels turn on a central axis called a pivot. Pivots are either a cylinder shape called a shouldered pivot or a tapered cylinder called a conical pivot. Conical pivots are used in conjunction with a rounded olive shaped hole jewel with a cap jewel. this design is utilized on very small pivots like balance wheels and escape wheels because their pivots must be smaller to reduce friction and therefore need to be a stronger shape. Shouldered pivots are used in most other applications.
Conical pivot
Shouldered pivot

5. Pivots Bad Pivot Good Pivot
Pivots are the small cylinders on the end of arbors that serve as the axis of rotation for wheels and pinions. Pivots are burnished or pressure polished in order to provide a smooth flow of energy. The pivots must be very smooth and highly polished to reduce the friction coefficient between the pivot and its bearing. In addition, burnishing compresses the structure of the steel microscopically, causing the steel to greatly increase in hardness. This process creates a pivot that is much more wear resistant.
Bad Pivot
Good Pivot

6. Wheels and pinions
Tooth profiles The shape that teeth on wheels and pinions are cut to is the product of years of careful research and experimentation. Many companies have their own proprietary profiles that only they use. The roots of teeth used to be square while the tips were almost a perfect circle. Modern technology has allowed for shapes that are much more efficient and reduce errors.
By using dimensions derived from the cycloidal and involute curves, Tooth profiles allow for constant contact from one tooth to the next so there is no loss of energy caused by the teeth dropping onto one another, and the risk of butting or binding is nearly non existent.

7. Wheels and pinions
Module: - A standard used to determine the dimensions of a gear. It is the measure of the ratio between the pitch circle of a wheel and its number of teeth. In order for two toothed objects to mesh with one another the module must be the same.  A table like the one pictured below is used to find a corresponding cutter that is most ideal for the size and number of teeth needed for a given set.

8. Gearing systems
There are three types of gear trains, each with their own intended use. Intermediate wheel trains: A series of toothed wheels that all share the same circular pitch. Teeth drive each other at a 1:1 ratio with each wheel in the train reversing its direction of rotation. The number of rotations from one wheel to the next can increase or decrease depending on the number of teeth on each wheel. Commonly used in Setting mechanisms, winding trains and calendar mechanisms. Calculating the rate of rotation can be derived using the following formula:
𝑁4 𝑁1 = 𝑍1 𝑍4

9. Gearing systems 𝑁6 𝑁1 = 𝑍1 ∙ 𝑍3 ∙ 𝑍5 𝑍2 ∙ 𝑍4 ∙ 𝑍6
Multiplication train: A train in which the wheels are the drivers and the pinions are driven. This results in the rotational speed from one wheel to the next being increased. As a multiplication train progresses, the rotational speed increases and the torque will inversely decrease. This kind of gearing is most commonly seen in going trains. The rate of rotation can be derived using the following formula:
𝑁6 𝑁1 = 𝑍1 ∙ 𝑍3 ∙ 𝑍5 𝑍2 ∙ 𝑍4 ∙ 𝑍6

10. Gearing systems
Reduction gear train: A train in which the are pinions the drivers and the wheels are driven. This results in the rotational speed from one wheel to the next being decreased. As a reduction train progresses, the rotational speed decreases and the torque will inversely increase. This kind of gearing is most commonly seen in going trains of quartz watches, motion works and automatic winding systems. The rate of rotation can be derived using the same formulas a multiplication train.

11. Wheel and pinion cutting
Wheels and pinions are cut using profile cutters on a milling attachment. The diameter of the stock is cut to accommodate the number of teeth being cut. In order for all of the teeth to be complete with no gaps or difference in thickness, the ration of diameter to tooth size and number must be perfectly calculated.