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Figure 2-22 Exploded view of a typical harmonic drive showing its principal parts. The flexspline has a smaller outside diameter than the inside diameter of the circular spline, so the elliptical wave generator distorts the flexspline so that its teeth, 180 apart, mesh.
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tooth gears that coupled the power of water wheel to millstones for grinding corn. Those gears offered about a 5 to 1 reduction. In about 300 B.C., Aristotle, the Greek philosopher and mathematician, wrote about toothed gears made from bronze. In 1556, the Saxon physician, Agricola, described geared, horsedrawn windlasses for hauling heavy loads out of mines in Bohemia. Heavy-duty cast-iron gear wheels were first introduced in the mideighteenth century, but before that time gears made from brass and other metals were included in small machines, clocks, and military equipment. The harmonic drive is based on a principle called strain-wave gearing, a name derived from the operation of its primary torque-transmitting element, the flexspline. Figure 2-22 shows the three basic elements of the harmonic drive: the rigid circular spline, the fliexible flexspline, and the ellipse-shaped wave generator. The circular spline is a nonrotating, thick-walled, solid ring with internal teeth. By contrast, a flexspline is a thin-walled, flexible metal cup with external teeth. Smaller in external diameter than the inside diameter of the circular spline, the flexspline must be deformed by the wave generator if its external teeth are to engage the internal teeth of the circular spline. When the elliptical cam wave generator is inserted into the bore of the flexspline, it is formed into an elliptical shape. Because the major axis of the wave generator is nearly equal to the inside diameter of the circular
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Figure 2-23 Schematic of a typical harmonic drive showing the mechanical relationship between the two splines and the wave generator.
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spline, external teeth of the flexspline that are 180 apart will engage the internal circular-spline teeth. Modern wave generators are enclosed in a ball-bearing assembly that functions as the rotating input element. When the wave generator transfers its elliptical shape to the flexspline and the external circular spline teeth have engaged the internal circular spline teeth at two opposing locations, a positive gear mesh occurs at those engagement points. The shaft attached to the flexspline is the rotating output element. Figure 2-23 is a schematic presentation of harmonic gearing in a section view. The flexspline typically has two fewer external teeth than the number of internal teeth on the circular spline. The keyway of the input shaft is at its zero-degree or 12 o clock position. The small circles around the shaft are the ball bearings of the wave generator. Figure 2-24 is a schematic view of a harmonic drive in three operating positions. In Figure 2-24A, the inside and outside arrows are aligned. The inside arrow indicates that the wave generator is in its 12 o clock position with respect to the circular spline, prior to its clockwise rotation.
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Figure 2-24 Three positions of the wave generator: (A) the 12 o clock or zero degree position; (B) the 3 o clock or 90 position; and (C) the 360 position showing a two-tooth displacement.
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Because of the elliptical shape of the wave generator, full tooth engagement occurs only at the two areas directly in line with the major axis of the ellipse (the vertical axis of the diagram). The teeth in line with the minor axis are completely disengaged. As the wave generator rotates 90 clockwise, as shown in Figure 2-24B, the inside arrow is still pointing at the same flexspline tooth, which has begun its counterclockwise rotation. Without full tooth disengagement at the areas of the minor axis, this rotation would not be possible. At the position shown in Figure 2-24C, the wave generator has made one complete revolution and is back at its 12 o clock position. The inside arrow of the flexspline indicates a two-tooth per revolution displacement counterclockwise. From this one revolution motion the reduction ratio equation can be written as:
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