Gearing reduction ratio


Harmonic Drive is the brand name of strain wave gear trademarked by the Harmonic Drive company, and invented in 1957 by C.W. Musser.

It is very commonly implemented in robotics today and used in aerospace as well, for gear reduction but may also be used to increase rotational speed, or for differential gearing.

The strain wave gearing theory is based on elastic dynamics and utilizes the flexibility of metal. The mechanism has three basic components: a wave generator, a flex spline, and a circular spline. More complex versions have a fourth component normally used to shorten the overall length or to increase the gear reduction within a smaller diameter, but still follow the same basic principles.

The wave generator is made up of two separate parts: an elliptical disk called a wave generator plug and an outer ball bearing. The gear plug is inserted into the bearing, giving the bearing an elliptical shape as well.

The flex spline is shaped like a shallow cup. The sides of the spline are very thin, but the bottom is relatively rigid. This results in significant flexibility of the walls at the open end due to the thin wall, and in the closed side being quite rigid and able to be tightly secured (to a shaft, for example). Teeth are positioned radially around the outside of the flex spline. The flex spline fits tightly over the wave generator, so that when the wave generator plug is rotated, the flex spline deforms to the shape of a rotating ellipse and does not slip over the outer elliptical ring of the ball bearing. The ball bearing lets the flex spine rotate independently to the wave generator’s shaft.

The circular spline is a rigid circular ring with teeth on the inside. The flex spline and wave generator are placed inside the circular spline, meshing the teeth of the flex spline and the circular spline. Because the flex spline is deformed into an elliptical shape, its teeth only actually mesh with the teeth of the circular spline in two regions on opposite sides of the flex spline (located on the major axis of the ellipse).

Assume that the wave generator is the input rotation. As the wave generator plug rotates, the flex spline teeth which are meshed with those of the circular spline slowly change position. The major axis of the flex spline’s ellipse rotates with wave generator, so the points where the teeth mesh revolve around the center point at the same rate as the wave generator’s shaft. The key to the design of the strain wave gear is that there are fewer teeth (often for example two fewer) on the flex spline than there are on the circular spline. This means that for every full rotation of the wave generator, the flex spline would be required to rotate a slight amount (two teeth in this example) backward relative to the circular spline. Thus the rotation action of the wave generator results in a much slower rotation of the flex spline in the opposite direction.

For a strain wave gearing mechanism, the gearing reduction ratio can be calculated from the number of teeth on each gear.

Related formulas


reductionratiogearing reduction ratio (dimensionless)
flexsplineteethnumber of flex spline teeth (dimensionless)
circularsplineteethnumber of circular spline teeth (dimensionless)