a mechanical gear, friction, or worm transmission is which rotation is transmitted and trans-formed by the cyclical excitation of strain waves in a so-called flexible element (hence the name“wave transmission”). The American engineer W. Musser invented the wave transmission in 1959.
The most common wave transmission is the gear type (see Figure 1), which usually consists of a rigid element—a gear wheel with internal teeth, which is immovably fastened to the transmission’s housing; a flexible element—a cylindrical, thin-walled pinion in the shape of a cup with external teeth
that are somewhat fewer in number than the teeth on the rigid wheel (the cup is attached to the output shaft and is positioned inside the rigid wheel); and a generator of strain waves (wave former)—an oval cam with a ball bearing mounted on it. The generator is fitted coaxially within the flexible wheel and stretches it during rotation. The number of strain waves is equal to the number of lobes on the cam. At the crests of the waves the teeth on the flexible wheel are fully engaged with the teeth on the rigid wheel, and in the troughs they are completely disengaged. Upon rotation of the generator, the strain waves move with the same angular velocity—that is, traveling waves are excited in the flexible wheel, and meshing occurs at their peaks. The difference between the number of teeth on the rigid and flexible wheels is usually equal to—less frequently, a multiple of—the number of strain waves. Depending on the number of waves, a wave transmission is called a one-wave, two-wave, or three-wave type. If, for example, the number of teeth on the flexible wheel Zf- 200, and on the rigid wheel Zf = 202, it is a two-wave transmission (Figure 2) with a wave generator in the form of a carrier with two rollers; as the generator rotates clockwise, the first tooth on the flexible wheel will enter the first space on the rigid wheel, the second tooth will enter the second space, and so on up to the 200th tooth and space. As the generator rotates further, the first tooth of the flexible wheel will enter the 201 st space, the second will enter the 202nd, and the third will enter the first space of the rigid wheel (Figure 2,d). Thus, after one complete revolution of the generator, the flexible wheel is displaced relative to the rigid wheel by two teeth or an angle 0 = 200 x 360 = 3.6° (Figure 2,c) in the opposite direction—that is, the gear ratio i = Zf/2.
In the general case the gear ratio of a wave transmission with a rotating flexible wheel is 
Gear wave transmissions are also used with a fixed flexible wheel and a
rotating rigid wheel. In this 
case and the generator and the output shaft rotate in the same direction. One of the principal features of the wave transmission is the possibility of obtaining a high gear ratio in one stage. The wave reducers being produced serially in the United States (1970) have gear ratios of 60-320 : 1. Because of the small difference between the diameters of the flexible and rigid wheels and the flexibility of one of the elements, 10-50 per-cent of the teeth mesh at any one time—that is, multiple-pair engagement takes place—which makes possible the use of a wheel with a low engagement factor. Wave transmissions are able to transmit a torque several times greater than other gear transmissions of the same size and weight, and they are considerably more compact than gear transmissions of other types with the same load capacity. The efficiency of gear wave transmissions is usually 80-92 percent. They are smooth and shockless, have greater kinematic precision, and permit engagement without backlash. They can operate both as underdrive (reducer) and overdrive (multiplier) transmissions. The flexible wheels of wave transmissions are usually made of metal with a high endurance limit or of various plastics obtained by injection molding. There are designs for gear wave transmissions with an externally mounted wave generator; in this case the rigid wheel is positioned inside of the flexible wheel. The flexible wheels of wave transmissions are fabricated in the form of diaphragms, cones, spheres, bells, narrow rings, or tubes, which are joined to the output shaft by splines. Wave transmissions may also have pneumatic and hydraulic wave generation, in which radially positioned plungers supplied through a pressure-distributing device perform the function of the cam. This type of trans-mission has low inertia because there is no rapidly rotating generator. It is possible to transmit rotation with a wave transmission through a blank metal wall into or from a closed, hermetically sealed space. The flexible wheel of a sealed wave transmission is usually in the form of a bell with two headers, one of which is secured to the transmission’s housing. The wave generator is inside the bell, and the rigid wheel, which is attached to the output shaft, is outside. Another design possibility is a sealed wave transmission with the rigid wheel inside and the generator outside. The so-called responsin occupies a special place among gear wave transmissions. A prototype of this device is the low-speed electric motor with a flexible rotor invented by the Soviet engineer A. I. Moskvitin in 1944 for a reducerless drive. There are no rapidly rotating parts in a responsin, and consequently no known actuator used in servo systems and similar mechanisms can equal it in speed of operation.
The friction wave transmission has smooth contact surfaces on the flexible and rigid elements. The gear ratio of friction wave transmissions is given by 
where Pfand Pr are the perimeters of the contacting surfaces of the flexible and rigid elements respectively. They are used as speed regulators (see Figure 3).
A hollow screw (Figure 4) or thin-walled nut may be the flexible element in a worm wave transmission. Correspondingly, the wave generator is located inside or outside the flexible element. Depending on the relationship of the thread parameters for the screw and the nut, the rotation of the generator in worm wave transmissions is converted into translational or helical motion of the output element. Worm wave transmissions are used mainly to transmit motion into a hermetically sealed space and for very slow displacements.
Sometimes wave couplings, which transmit rotation through a cylindrical shell to a hermetically sealed space and have a gear ratio of 1 : 1, are also included among wave transmissions.
Wave transmissions are used in various branches of technology: drives for hoisting machinery, conveyors, machine tools, aviation and space technology, precision instruments, actuators for remote and automatic control systems, and drives for highly directional radar antennas of systems used to observe cosmic objects. Sealed wave transmissions convey rotation into hermetically sealed enclosures containing chemically aggressive and radioactive mediums and into high-pressure and high-vacuum chambers, and also provide drives for sealed valves. For example, in the American Centaur rocket (1960’s) a hermetically sealed wave transmission is used in the valve mechanism of the liquid oxygen system thereby eliminating oxygen leakage and decreasing the danger of explosion and fire.
REFERENCES
Tseitlin, N. I., and E. M. Tsukerman.“Volnovye peredachi.” Voprosy raketnoi tekhniki, 1965, no. 8.Ekspress-informatsiia: Seriia detail mashin, 1968, no. 11.
Ginzburg, E. G. Volnovye zubchatye peredachi. Moscow, 1969.
IU. B. SINKEVICH
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