Machine Design

Sometimes it pays to be eccentric

There's more than one way to get high positional accuracy, minimal backlash, and high reduction rations in small packages.

An unassembled harmonic-drive gear from HD Systems Inc., Hauppauge, N.Y., shows the three main components. The reducer includes a wave generator, far left, which engages the flexspline, center, with the circular spline in two regions on opposite sides of the ellipse's major axis.

Harmonic-drive pancake gears from Harmonic Drive Technologies, a Peabody, Mass., harmonic-gear manufacturer, are nearly flat for applications that require high reduction ratios in small packages. HDF models, for example, are as thin as 11 mm and have torque ratings from 56 to 3,180 lb-in. with higher peak ratings.

Dojen cycloidal speed reducers, from Mectrol Corp., Salem, N.H., maintain constant contact between all cam lobes and followers. This creates high shock-load capacity and consistent torsional stiffness.

Designing a system that demands minimal backlash, tight space constraints, or high reduction ratios, often calls for looking beyond traditional spur or helical gearboxes. In fact, less common power-transmission devices can often solve all of these design constraints. Several technologies produce high reduction ratios in small lightweight packages. They do it using clever, reliable kinematic principles instead of conventional methods.

All three methods highlighted in this article use some type of noncircular rotation to generate reductions. Harmonic gears use an elliptical disc to flex a geared ring that meshes with a stiff outer ring. Cycloidal speed reducers use cams and needle rollers to generate reductions. Bearing reducers have several sets of bearings that serve different functions, including letting the reducers carry radial and axial loads.

For each device in this article, either of the three main components can act as either the input, output, or fixed member. This increases the reducers' versatility. Because the components are reversible, they can be used as speed reducers, increasers, or differentials. If an application requires the component to hold a load, however, a brake must be specified.

Although all three types of reducers offer high reductions, small packages, and high accuracy compared to traditional gear reducers, each has something special to offer. Harmonic-drive gears, for example, provide high reduction ratios in a single stage and good torque-to-weight ratios. Cycloidal speed reducers offer extremely high stiffness and high shock-load capacity. And bearing reducers operate with zero sliding friction and their crossed-roller bearings let them carry radial and axial loads and bending moments.

Harmonic-drive gears are probably the most widely known of the reducers in this article. The devices provide high torque capacity, torsional stiffness, and positioning accuracy. These characteristics make harmonic-drive gears ideal for applications where load and driving speed are seldom constant, such as servodrives. Harmonic-drive gears have single-stage reduction ratios to a lofty 320:1, efficiencies to 90%, and high torque-to-weight ratios.

The three main components of harmonic-drive gears are a circular spline, a flexspline, and a wave generator. The wave generator is basically a ball bearing with thin races fitted onto an elliptical plug. Bearings in the wave generator let it serve as an efficient torque converter.

The flexspline is a flexible, thin-walled cylindrical cup with external teeth. The flexspline fits over the wave generator, which holds the flexspline in an elliptical shape. Although it flexes continuously during operation, the deflection is well within the mater-ial's fatigue limits.

The circular spline is a rigid thick-walled ring with internal teeth and a slightly larger pitch diameter than the circular spline. The circular spline engages with the teeth of the flexspline across the major axis of the wave generator.

The flexspline usually has two fewer teeth than the circular spline. The elliptical shape of the wave generator causes the teeth on the flexspline to engage the circular spline in the two regions across the major axis of the ellipse.

As the wave generator rotates, the zone where the flexspline engages with the circular spline travels with the major elliptical axis. For each 180° clockwise movement of the wave generator, the flexspline moves counterclockwise by one tooth relative to the circular spline.

Each complete clockwise rotation of the wave generator results in the flexspline moving counterclockwise by two teeth from its original position relative to the circular spline. This results in a reduction ratio equal to one half the number of teeth on the output element, in this case the flexspline. For instance, if a flexspline has 160 teeth, the output ratio is 80:1.

Harmonic-drive gears from HD Systems Inc., Hauppauge, N.Y., have what's called an "S-tooth profile" that is said to optimize the way the mating gears mesh. Original harmonic gear teeth have a much lower contact ratio than S-tooth gears, according to HD Systems.

The S-tooth profile increases the size of the tooth-engagement region compared with conventional harmonic-gear teeth, which have about 15% of the total number of teeth in contact. S-tooth gears, on the other hand, have up to 30% of the teeth in contact. This spreads the torque load over more teeth and is said to increase torsional stiffness by 100% in the low and midtorque ranges.

The S-tooth profile also has a larger tooth-root radius than conventional spur gears, which produces a higher allowable stress and a corresponding increase in torque capacity. The enlarged region of tooth engagement loads the wave generator more evenly, lengthening the life of harmonic-drive gears.

The advantages of the S-tooth profile make harmonic-drive gears a good choice for applications requiring high rigidity and peak torque capacities, such as servo applications. The design has also recently helped harmonic-drive manufacturers drop reduction ratios down to 30:1 from previous limits of 50:1. This opens the way for higher-speed applications than in the past.

The high torque-to-weight ratio of harmonic-drive gears makes them the only choice in some applications. "One particular aerospace customer," reports a company spokesperson for HD Systems, "doesn't really care so much about positional accuracy. They like the fact that they can take a small motor and run it through a single-stage 160:1 harmonic-drive gear reducer in a very lightweight frame."

Harmonic gears have several other performance advantages over conventional gearboxes. For example, the reducers sport negligible backlash and excellent positioning accuracy and repeatability. And the relative motion between the circular spline and flexspline happens on the minor axis where there is no contact. This feature, and the way the teeth engage, create minimal friction so backlash remains negligible throughout the life of harmonic gears.

Gear teeth aren't the only way to generate speed reductions. For instance, one type of cycloidal speed reducer uses rolling elements and cams to transmit torque and provide speed reduction. Every rolling element is always in contact with a cam, thereby producing high stiffness with zero backlash.

Dojen speed reducers from Mectrol Corp., Salem, N.H., use a single-piece, dual-track cam that acts as a speed reducer. The number of cam lobes ranges from 8 to 15, depending on the needed reduction ratio. Each set of cams has one fewer lobe than its mating set of cam followers.

An eccentric input shaft operates similar to an automobile crankshaft. When the input shaft rotates it drives both tracks of cam lobes in an orbit within the cam followers. A mechanical phase shift occurs between each set of cam followers and lobes because there is one less cam lobe than mating follower. If the input shaft is acting as the input, the set of cam followers in the housing does not rotate. In this case the set of cam followers on the output side rotates and causes the output shaft to rotate.

The two phase shifts that happen between each set of cam lobes and followers produce specific, fixed reductions. The reduction ratio is the difference in angular displacement between the two sets of cam followers. More cam lobes generally produce higher reduction ratios but there's no simple relationship. For instance, it takes 26 total lobes to make a 25:1 ratio while 31 lobes produce a 105:1 ratio.

The cam followers are precision-ground needle bearings preloaded against the cam, thus creating an interference fit. This eliminates backlash and provides a rigid transmission. The output shaft rotates within a high-capacity output bearing.

Cycloidal speed reducers have several performance advantages over conventional gearboxes. For instance, according to John D'Amico, product manager of speed reducers at Mectrol, the company's Dojen cycloidal reducers work with high torsional stiffness, zero backlash, and high shock-load tolerance.

Although gearbox designers look for ways to eliminate backlash using methods such as preloading, all gear-boxes need some space between mating teeth to let the gears move. And even when spur gears are designed with minimal backlash, it often increases as the gears wear. Cycloidal reducers with cantilevered cam followers, however, allow for a controlled preload between the cams and followers, which is said to eliminate backlash. Needle bearings on the followers let the mating parts move freely. The cantilevered design maintains preload throughout an L10 bearing life.

"One of the most unique features we offer is true zero backlash," says D'Amico. "A lot of companies promote zero backlash but ours is zero wherever you measure it. There's no backlash whether you measure it anywhere around the 360° of output or under any amount of load. It also remains zero throughout the life of the unit. It doesn't pick up backlash because the cantilever design stays springy against the cam."

Keeping all followers in contact with cams also boosts torsional stiffness. With the preloaded design all cam followers equally share the load, which guarantees consistently high torsional rigidity, regardless of load position. The load-sharing feature also lets cycloidal reducers withstand high shock and reversal loads.

The mechanisms are manufactured to exacting standards to ensure accurate performance. For instance, the cam-follower bearings are precision class 6 and cam-follower bolt circles are held concentric to 0.0002 in. This gives positional accuracy down to seconds of arc.

One practical advantage of cycloidal speed reducers is their low-profile design. Package sizes range from 4 to 20-in. diameters and 11/4 to 3 1/2-in. lengths. The reducers have hollow-bore input options for mounting directly to motor shafts. And where space is critical, hollow-bore reducers can connect to pancake motors to save even more space.

Reduction ratios don't affect the size of reducer packages either. Ratios typically range from 9:1 to 256:1 with no change in package size. Units can handle input speeds to 8,000 rpm. According to Mectrol designers, the mechanisms also operate more quietly than gearboxes because of the rolling contact between mating cams and followers as opposed to sliding contact between gear teeth.

Bearing reducers provide another alternative to conventional gear-boxes. The compact mechanisms use high-precision roller bearings to provide reduction. Crossed-roller bearings that support the outer case carry radial and axial loads and bending moments. The reducers have fixed gear ratios ranging from 1:31 to 1:191 with outside diameters down to 63 mm.

The reducers, called TwinSpin bearing reducers, are developed and manufactured by Spinea Ltd., Kosice, Slovak Republic. The mechanisms offer up to 90% efficiency and positioning accuracy below 1 arcmin. They also operate quietly and have a low size-to-torque ratio.

Bearing reducers have several technical advantages over conventional gearboxes, according to Spinea engineers. For instance, they run at low temperatures with low noise and vibration levels. They also offer high torque ratings, no sliding friction, and high transmission efficiency for their size. Because of pure rolling friction between moving parts, there is little energy loss and component wear. This not only improves efficiency but also extends the reducers' life.

Because bearing reducers absorb radial and axial loads, designers can mount driven components directly to the output side of the reducers, eliminating the need for shafts supported with bearings, as in conventional gearboxes.

Bearing reducers are used in a variety of industries, including factory automation, medical, robotics, machine tool, and semiconductor manufacturing. The mechanisms can be used in many unique applications, such as directly as joints on robots, in indexing tables, or even as wheel gears in transporting systems.


This shows how harmonic gear teeth mesh. The gear ratio for harmonic-drive gears is calculated with:

Rg = Tf/(Tc - Tf) where Tf is the number of teeth in the flexspline and Tc is the number of teeth in the circular spline. For instance, a harmonic gear with 160 teeth on the flexspline and 162 teeth on the circular spline has the ratio:

Rg = 160/(162 160)

= 160/2 = 80:1


Flexspline teeth must be concentric with teeth on the circular spline, otherwise the flexspline is in what's called a dedoidal condition. This results from off-center installation or exceeding the torque limit of mounting screws. A dedoidal mesh produces excessive noise and vibration, and leads to early gear failure.

Experts recommend measuring the gap on each side of the flexspline to ensure an equal gap, as shown above. If you cannot access the gap in your application, insert a probe in an access hole in the housing (a recommended option) to measure deflection of the flexspline. There should be two equal deflections during one revolution, as the top graph shows. If there is only one deflection, as in the bottom graph, a dedoidal condition exists.


TwinSpin bearing reducers from Spinea Ltd., Kosice, Slovak Republic, use several sets of roller bearings to generate reductions. Their nearly 50% contact ratio lets the reducers transmit high torque with minimal backlash.

The main components of bearing reducers include an outer case, two output flanges, an input shaft, two geared members, two transmission members, and several sets of roller bearings that perform different functions. Although bearing reducers can be used in a variety of input/output combinations, this description focuses on the input shaft acting as the input and the flanges acting as the output, while the case is held stationary.

The input shaft has an eccentric shape similar to an automobile crankshaft. Roller bearings rest between the eccentrics of the input shaft and the inner diameter of the geared members. The eccentrics of the input shaft cause the geared members to oscillate. As the geared members oscillate, cycloidal teeth on their outer diameter periodically engage with needle rollers that rest in semicircular grooves on the inner diameter of the case.

The oscillating gears rotate while also moving radially. Transmission members also oscillate, but the output flanges can only rotate. Roller bearings between transmission-member arms and the gears and flanges let the transmission members and gears oscillate while letting the output flanges only rotate.

The transmission members rest between bosses (with roller bearings at the interface) on the external gears and flanges. The reducers are designed so two transmission-member arms opposite each other rest between two bosses on an external gear, while the other two transmission-member arms rest between two bosses on an output flange.

The flanges are bolted together and supported by crossed roller bearings between the inner diameter of the case and the outer diameter of the flanges. The crossed bearings support radial and axial loads, letting the reducers carry thrust loads and bending moments instead of only radial loads. The reducers are mounted on driven equipment through holes in the output flange and mounted to fixed equipment through holes in the case.

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