Modern Gear Tooth Profile Design
By TK NG
Gear is a piece of mechanical part for transmission of power and motion. It has been in existence for over thousands of years in human history. You can find it in all sorts of things nowadays, such as cars, watches, toys, etc. It could also be found in devices of the ancient time, including the famous South-Pointing-Chariot (指南車) of China.
Modern and ancient gears differ in both design and construction. Ancient gears were mostly fabricated from wooden materials and with simple tooth forms. Modern gears employ various materials and specific tooth profiles to enhance their transmission efficiency, motion accuracy, noise performance, durability, etc.
Means of Power and Motion Transmission
Power and motion can be transmitted through various ways and means. A simple way is shown in Fig 1a. Transmission is effected through a single point of contact between the drive and driven circular discs if the friction there is large enough. Fig 1b shows another arrangement. It represents two pulleys (in blue) with a cross-coupled belt (in red) wrapped around them. The much larger extent of contact between the belt and pulleys results in more friction for power and motion transmission.
For a belt and pulley system, slippage can still occur when frictional force between the belt and any of the pulleys is less than that required for the power to be transmitted. The meshed gear set shown in Fig 1c is another alternative, which can avoid slippage to ensure smooth transmission.
But how can the latter means work better than the former ones? The secret lies in the form of the gear tooth, i.e., the gear tooth profile.
Gear Tooth Profile
The tooth profile of a gear is part of an involute or cycloidal curve. A gear is referred to as involute or cycloidal type according to its tooth form. Gears in most applications are involute type while cycloid gears are mainly used for clocks and watches.
Let us focus on the more widely used involute gears. You can imagine an involute curve is produced by tracing the end of a taut string when being unwound from a cylinder. The circle of the cylinder is referred to as the base circle. As the string is kept in tension throughout the tracing process, it is always tangential to the base circle. Fig 2 shows an involute curve formed from a base circle (with the unwound imaginary taut string marked in red).
The involute tooth profile is made up of two involute curves formed from the same base circle but in opposite directions (see Fig 3). The tip of the tooth (inside dotted circle) is trimmed flat in practice.
Referring to Fig 4, a property of the involute curve is that a tangent (in green) to the curve at any point is perpendicular to the taut line unwound from the base circle (known as the radius of curvature of the involute curve), which is tangential to the base circle.
In Fig 5 where the larger gear is driven by the smaller one, the common tangent AA’ to the involute curve profiles of the meshing teeth is perpendicular to the lines BB’ and CC’, which are tangents to their respective base circles (in blue). As the two lines meet at the meshing point, they must form a straight line which is the common tangent of the two base circles.
Fig 6 shows two gears with a pair of teeth meshing at three different locations when the smaller driving gear turns counterclockwise. Since the tooth profiles of the gears are in the form of involute curve, there must always be a common tangent to the meshing teeth (shown in green lines) no matter where the meshing location is. Each green line is perpendicular to the common tangent of the base circles (red line). Obviously, the three red lines are the same line since there is only one common tangent of that orientation to the base circles Note. All the meshing locations (green dots) therefore lie on the red common tangent, which is known as the line of action.
Looking back at Fig 1b and comparing it with Fig 5, you may immediately realize the resemblance between a cross-coupled belt & pulley system and a meshed gear set. The line of action of two meshed gears in effect replaces the belt coupling two pulleys of the same sizes as the respective gear base circles. A set of gears with involute tooth profiles hence provides a better alternative to the cross-coupled belt & pulley configuration for power and motion transmission, avoiding possibility of slippage and the need of belt replacement due to wear and tear.
Gear has a long history of application, and various gear types are in use. The design of tooth profiles and manufacturing of modern gears involve profound knowledge and technologies, although people may just regard them as simple mechanical parts only. If you are interested in the technicalities of different gear types, you can visit https://khkgears.net/new/gear_knowledge/ and https://drivetrainhub.com/notebooks/#toc for further reading.
Note
For two circles located side by side with similar configuration as the gear base circles, there are four common tangents at different locations as shown in solid and dotted red lines below. 
