The involute gear profile, originally designed by Leonhard Euler, is the most commonly used system for gearing today, with cycloidal gearing still used for some specialties such as clocks. In an involute gear, the profiles of the teeth are involutes of a circle. (The involute of a circle is the spiraling curve traced by the end of an imaginary taut string unwinding itself from that stationary circle called the base circle.)
Irrespective of whether a gear is spur or helical, in every plane of the involute gears the contact between a pair of gear teeth occurs at a single instantaneous point (see figure at right) where two involutes of the same spiral hand meet. Contact on the other side of the teeth is where both involutes are of the other spiral hand. Rotation of the gears causes the location of this contact point to move across the respective tooth surfaces. The tangent at any point of the curve is perpendicular to the generating line irrespective of the mounting distance of the gears. Thus the line of the force follows the generating line, and is thus tangent to the two base circles, and is known as the Line of Action (also called Pressure Line or Line of Contact). When this is true, the gears obey the Fundamental Law of Gearing:
The angular velocity ratio between two gears of a gearset must remain constant throughout the mesh.
This property is required for smooth transmission of power with minimal speed or torque variations as pairs of teeth go into or come out of mesh, but is not required for low-speed gearing.
Where the line of action crosses the line between the two centres it is called the Pitch Point of the gears, where there is no sliding contact.
The Pressure Angle is the acute angle between the line of action and a normal to the line connecting the gear centers. The pressure angle of the gear varies according to the position on the involute shape, but pairs of gears must have the same pressure angle for the teeth to mesh properly, so specific portions of the involute must be matched.
While manufacturers can produce any pressure angle, the most common stock gears have a 20° pressure angle, with 14½° and 25° pressure angle gears being much less common. Increasing the pressure angle increases the width of the base of the gear tooth, leading to greater strength and load carrying capacity. Decreasing the pressure angle provides lower backlash, smoother operation and less sensitivity to manufacturing errors.
Only used in limited situations are helical involute gears, where the spirals of the two involutes are of different 'hand' and the Line of Action is the external tangents to the base circles (like a normal belt drive whereas normal gears are like a crossed-belt drive), and the gears rotate in the same direction, and there is sliding at the contact point which gives inefficiency and thus can be used in limited slip differentials. These cannot be spur gears unless they comprise multiple sectors of gears, and are otherwise helical, but the meshing gears are of the same helix angle rather than of opposite hand.
- Goss, Geoff Application of analytical geometry to the form of gear teeth RESONANCE, September 2013, Volume 18, Issue 9, pp 817–831. *
- Norton, R.L., 2006, Machine Design: An Integrated Approach, 3rd Ed, Pearson/Prentice-Hall, ISBN 0-13-148190-8
- Juvinall, R.C. and K.M. Marshek, 2006, Fundamentals of Machine Component Design, 4th Ed, Wiley, ISBN 978-0-471-66177-1, p. 598
- Boston Gear Company, Open Gearing Catalog, http://bostongear.com/products/open/spur.html
- Professor Jacques Maurel, "Paradoxical Gears", http://www.jacquesmaurel.com/gears
- "Meshing gear members" Arthur J. Fahy, Neil Gillies US Patent 5129276
- "Differential device" Jacques Mercier, Daniel Valentin US Patent 4831890
- "Differential gear mechanism" Arthur J. Fahy, Neil Gillies US Patent 5071395
- Kinematic Models for Design Digital Library (KMODDL)
Movies and photos of hundreds of working mechanical-systems models at Cornell University. Also includes an e-book library of classic texts on mechanical design and engineering.
- Application of analytical geometry to the form of gear teeth