The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS7.001

Harmonic Drive. Evolution of Wave Gear Design

G. Timofeev1 O. Egorova2 M. Samoilova3

All: Department of Theory of Mechanisms and Machines (TMM) National Research University

Bauman Moscow State Technical University (BMSTU), Moscow, RUSSIA

Abstract: This paper presents a brief history of wave transmission evolution and Harmonic Drive in particular. The invention became a breakthrough in mechanical design and has been attracting the close interest of scientists and engineers for more than fifty years. Contrary to the traditional gearing systems and the laws of “rigid-body mechanics”, the new «Strain Wave Gearing» (SWG) patented by Walton Musser (1959) was based on the use of flexible gear that became the main idea and the driving force of his invention. Nowadays Harmonic Drive is a brand of strain wave gearing. It took just a few years for the SWG to receive recognition - in 1962 one of wave gears was already used in the sputnik. Over the next years’ studies in the world, including Russia, led to the design of many new wave gear systems which have high performance characteristics: high positioning accuracy, high repeatability, high reduction ratio, high torque capacity and efficiency, compactness and light weight. Due to their advantages the wave gears are widely used in aerospace and military applications worldwide. Keywords: History of MMS, wave transmission, Harmonic Drive, Musser, Responsin, flexible gear (flex spline), low-speed motor, Moskvitin, high performance characteristics

I. Introduction The oldest and most versatile mechanism created by human genius at the dawn of civilization was a mechanical transmission: chain, belt, friction, gear, and etc. It is used everywhere and is an integral part of many machines too. Interestingly, the most daring visionaries who look ahead many centuries and picture life in the distant worlds, do not ever try to portray the future technology without mechanical transmissions. They assign them a function, perhaps, as important as they play today. Is the science fiction right? It’s hard to say. Anyway, from today’s perspective the further development of the scientific and technological progress without mechanical transmission is unthinkable.

Among all mechanical transmissions, gears are most widely used as they are compact, reliable and have a long life. Their history goes back millennia: mention of iron and bronze gears occurs in "Mechanical problems" by Aristotle; a detailed description of gears can be also found in the writings of Heron of Alexandria "Mechanics", which is considered an encyclopedia of ancient art [1]. Archaeological excavation proved that the first simple gears were actually used for rotary motion conversion over 3000 years ago. Ancient Egyptians, for example, for irrigation of agricultural land used wooden constructions that had a gear and a wheel with buckets attached to it.

1timga@bmstu.ru 2tmm-olgaegorova@yandex.ru 3cm2003@list.ru

In ancient Greece and the Roman Empire they used wooden wedge gear teeth. In the Middle Ages gears were commonly utilized in water and wind mills. During the Industrial Revolution, gears were used to actuate a variety of machines, such as mine fans, lifts, pumps, and etc. In the middle of the 18th century the involute gear profile appeared, originally designed by Leonhard Euler (1707-1783), a great mathematician and scientist of Swiss origin, who in 1726 was offered a position at the St-Petersburg Academy of Science (Fig.1.a,b), and then spent two long periods of 14 and 17 years in Russia [2-8]. In connection with the design of water turbines Euler developed optimal profiles for teeth in cogwheels that transmit motion with a minimal resistance and noise. These profiles are involutes of a circle. When this is true, the gears obey the Fundamental Law of Gearing [3]:

“The angular velocity ratio between two gears of a gearset must remain constant throughout the mesh”.

Thus, involute tooth profiles provide a uniform rotational speed ratio. 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. Euler not only is the inventor of this kind of gear, but he also anticipated the underlying geometric equations now usually called the Euler–Savary equations [4]. The technical realization of this design took shape only later in what is called the involute gear that became the most commonly used, even today [5].

a) b) Fig.1. a) St-Petersburg Academy of Science; b) Leonhard Euler (1753)

Next centuries saw new technologies in the sphere of gears to be developed to meet higher requirements of the customers. High reduction ratio was connected to the advent of high-speed rotation machines (steam and gas turbines, electric motors) and mechanisms designed for a very low rate of displacement.

Another requirement - high torque capacity, was caused by the growing power of machines, increasing vehicle capacity or tremendous forces involved at forging, rolling, extrusion and other manufacturing operations. Additionally to the above mentioned requirements, the gears had to have high precision, high mechanical efficiency and high reliability, compactness and light weight. Logically, the engineers started using new high-strength materials to provide strength and surface hardness of the tooth; and new high technologies to ensure the exact profile and high accuracy of the tooth surface during the cutting process. A lot of different gear types were developed: involute, cylindrical and conical, parallel helical, herring-bone, cycloidal, spiral cylindrical, cycloidal involute, spur, spherical involute, crossed helical, worm gear and many others. Still, each of them could only meet one or two of the customer’s requirements, but not all. Thus, cylindrical and conical gears have high mechanical efficiency, but do not provide high reduction ratio when using a single-stage transmission (not more than 6-8). For higher reduction ratios more than a dozen pairs of gears sometimes have to be connected in a train. It creates a cumbersome multistage transmission and the unjustified increase in mass, size and cost that in turn decreases efficiency and reliability of a gearing. Single-stage cylindrical worm gearing provides a reduction ratio of about 80, but it has very low efficiency. Increasing the accuracy of transmission by reducing the meshing clearance also leads to higher cost, etc. It became clear that engineers needed to look for some innovative solutions. And suddenly ... In January 1955 an American inventor Clarence Walton Musser announced, and in 1957 received a patent for a mechanism called "Strain Wave Gearing”, the main idea of which was based on the use of a flexible gear wheel (Fig. 2). Official information about the patent was issued in 1959 [9,30]. At that time even the combination of words "Strain Wave Gear" (SWG) was a puzzle and caused a lot of questions as the mechanical engineers were accustomed to the laws of “rigid-body mechanics” [10] and traditional gear systems. That means that the flexibility was driven out of the gear design. They say that Albert Einstein (1879-1955), one of the most famous theoretical physicists, said once:

Fig.2. W. Musser, US patent 2,906,143 [9]

“Everyone knows it is impossible. But here comes an ignoramus who does not know that and… – makes the discovery!” [11].

Up to 1955 all mechanical engineers knew for sure: the deformation of gears - the enemy, and it must be fought! But Walton Musser offered the flexibility of gears to solve the "intractable" problems! Contrary to the traditional gearing systems, Musser’s invention (SWG) was based on dynamics of an elastic body.

II. C.W. Musser. Harmonic DriveA. Brief biography

"Nothing is impossible if you are willing to put in the time and effort." (Musser’s motto)

Clarence Walton Musser (1909-1998) (Fig.3) was a professional inventor [12-14] akin to the world famous Thomas Edison (1847-1931) awarded in 1928 the highest civilian award in the United States – Congressional Gold Medal. In 1930 Edison became a foreign honorary member of the USSR Academy of Sciences.

Fig.3. Clarence Walton Musser (1909-1998)

According to the official [17, 30] biography, Musser is credited with over 250 major inventions and discoveries. Some of his inventions were the Army recoilless rifle, aircraft personnel catapults, instrumentation for underwater detonation testing, and major phase of the Marines’ Ontos vehicle. His varied career includes experience in many diverse fields. He was research advisor to the Department of Defense for 15 years. With other industrial and government organizations, he served as chief engineer, director of research, and consultant. He held professional engineering licenses in Pennsylvania and Massachusetts (Fig.4). As a research adviser to United Shoe Machinery Corp., he explored non-rigid-body mechanics, using controlled deflection as an operating medium. His biggest invention [9] is “Strain Wave Gearing” (US patent 2,906,143), later called as «Harmonic Drive». The invention is included in the list of 5000 events that fully characterize the world scientific and technological progress of the XX century. During his life Walton Musser totally received more than 70 patents for wave gearings in 15 countries.

B. Harmonic Drive Harmonic Drive now is a brand of strain wave gearing. It is typically used for gear reduction but may also be used to increase rotational speed, or for differential gearing. In 1958 Harmonic Drive Division [15] was established as a division of United Shoe Machinery Corp. (USM) in Beverly, Massachusetts and manufactured a variety of gear reducers based on the strain wave gear principle, invented by Walton Musser.

Fig. 4. The first ever set of Harmonic Drive was built here [30]

In 1967 a license agreement was signed between the American patent-holder and a Japanese company, which later, in 1970, traded under the name of Harmonic Drive Systems Inc. Thus, Harmonic Drive Systems Inc. was formed in Japan as a joint venture between Hasegawa Gear Works of Japan and USM in the USA. The same year, 1970, Harmonic Drive AG was established in Germany and for more than 40 years its precision gears have been able to satisfy the highest requirements of aerospace and defense industries. In 2006, Harmonic Drive LLC was established as a joint venture between Harmonic Drive Systems, Inc., and Harmonic Drive Technologies Nabtesco Inc. (who was the direct descendent of USM Corporation). Today the three group companies composed of Harmonic Drive Systems in Japan, Harmonic Drive LLC in the USA and Harmonic Drive AG in Germany [16] engineer and manufacture precision Harmonic Drive gearings that are used in many spokes: industrial robots [15], machine tools, medical equipment, auto industry, communications equipment, printing presses, high-precision electronic component production, aerospace, industrial motion control and more. Harmonic Drives have worldwide applications, and in Russia as well. For many years the Russian scientific schools have been engaged in research and design of new wave gear systems.

III. Strain Wave Gearing MechanismA. Basic Principle In 1960, almost 5 years after the basic patent was filed, Walton Musser published a paper “Breakthrough in Mechanical Design: The Harmonic Drive” where he for the first time described how it works: «A continuous deflection wave generated in a flexible spline element achieves high mechanical leverage between concentric parts» [17]. The SWG theory is based on elastic dynamics and utilizes the flexibility of metal. That is the main feature that defines all of unique properties of the transmission: high reduction ratios in a single stage, very low backlash, high efficiency, compact size and light weight.

Before Musser’s patent was issued the engineers tried

to improve certain performance characteristics by increasing the transverse angle of gearing through the use of wider wheel rims or increasing the number of teeth of the wheels. Still all rotating elements were assumed to remain rigid and to rotate about fixed axes. Departing radically from the traditional concepts, new Musser’s drive systems used controlled elastic deflection of one or more parts for transmission, conversion, or change of mechanical motion [17]. These systems achieve high mechanical leverage by generating a traveling deflection wave in a flexible element.

Fig. 5. Three basic components of SWG

The SWG mechanism has three basic components (Fig.5): a Wave Generator, a Flexspline, and a Circular Spline. A Wave Generator is a thin raced ball bearing fitted on elliptical plug serving as a high efficiency torque converter. A Circular Spline is a solid steel ring with internal teeth. A Flexspline is a flexible cylinder with external teeth and flanged mounting ring [16]. 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. Standard 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 Flexspline is like a shallow cup. The sides of the spline are very thin, but the bottom is thick and rigid. This results in significant flexibility of the walls at the open end due to the thin wall, but 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 Flexspline. The Flexspline fits tightly over the wave generator, so that when the wave generator plug is rotated, the Flexspline deforms to the shape of a rotating ellipse but does not rotate with the wave generator. The circular spline is a rigid circular ring with teeth on the inside. The Flexspline and wave generator are placed inside the circular spline, meshing the teeth of the Flexspline and the circular spline. Because the Flexspline has an elliptical shape, its teeth only actually mesh with the teeth of the circular spline in two regions on opposite sides of the Flexspline, along the major axis of the ellipse. Assume that the wave generator is the input rotation. As the wave generator plug rotates, the Flexspline teeth which are meshed with those of the circular spline change. The major axis of the Flexspline actually rotates with wave

generator, so the points where the teeth mesh revolve around the center point at the same rate as the wave generator (Fig.6).

Fig. 6. Principle of operation [18]

B. SWG Unique Properties Strain Wave Gearing is a special type of mechanical gear system that can improve certain characteristics compared to traditional gearing systems (such as helical or planetary gears). Musser claimed unique and inherent properties of SWG (Harmonic Drive) that become apparent [17]: - Tooth motion relationship between flexible and circular splines makes high-ratio speed reduction or speed increase in a single stage; - In operation, many spline teeth are in simultaneous engagement to carry torque loads. Teeth adjacent to load-bearing teeth are in near-engagement and provide a “reserve” capacity to accommodate shock overloads; - Spline teeth come into contact with an almost pure radial motion, and have essentially zero sliding velocity, even at high input speeds. Tooth friction losses and wear are thus negligible; - Spline teeth in contact are practically stationary. Dynamic loading, under normal operating conditions, is negligible, and splines are thus capable of transmitting torques nearly in proportion to their static strength; - Regions of tooth engagement and application of load torque are usually diametrically opposed, and result in a force couple that is symmetrical and balanced; - Diametrically opposed spline mesh and large number of teeth in simultaneous engagement result in a statistical averaging of errors in individual tooth shape and placement; - With the flexible spline formed as an integral section of a flexible cylindrical wall, positive transmission of mechanical motion through the wall can be achieved.

C. Harmonic Drive Reduction Ratio The key to the design of the Harmonic Drive is that a Flexspline has fewer teeth (two teeth, for example) than the circular spline. This means that for every full rotation of the wave generator, the Flexspline would be required to rotate a slight amount (two teeth, for example) backward relative to the circular spline. Thus the rotation action of the wave generator results in a much slower rotation of the Flexspline in the opposite direction. The gearing reduction ratio can be calculated from the number of teeth on each gear: for example, if there are 202 teeth on the circular spline and 200 on the Flexspline, the reduction ratio is (200 − 202)/200 = −0.01. Thus the Flexspline spins at 1/100 the speed of the wave generator plug and in the opposite direction. This allows different reduction ratios to be set without changing the mechanism's shape, increasing its weight, or adding stages. The range of possible gear ratios is limited by the tooth size limits for a given configuration. In his paper “Breakthrough in mechanical design: The Harmonic Drive” [17] Musser gave some examples of ratio calculations (Fig. 7).

Fig. 7. Musser drives’ ratio calculations [17]

IV. Low-Speed Electric MotorIt is interesting that on December 15, 1944 a Soviet

(Russian) engineer Anatoly Ivanovich Moskvitin appealed to the People's Commissariat of the USSR Electrical Industry in order to register his invention and later received the patent SU № 68211 (Fig.8) under the name of “Low-speed electric motor” [20]. The main operating principle of the described motor is based on a physics phenomenon: if a rotor without bearings and windings is inserted into the stator that creates a rotating magnetic field, then such a rotor will be immediately attracted to one stator’s side and will roll without slipping along the inner surface of the stator, and the point of tangency will run across the circumference in synchronism with a magnetic field. Because the circumference of the rotor is shorter than the circumference of the stator [20-21], the rotor swing will be accompanied by its slow rotation in the opposite direction to the magnetic field rotation.

This phenomenon was used by Moskvitin to design a low-speed electric motor for gearless drive mechanism. He offered to insert a thin-walled flexible metal cylinder (rotor) into the stator of electric motor, and to energize the windings of two opposite poles of the electromagnets of the stator. The emerging magnetic field will stretch a circular section of the cylinder and turn it into an oval. The oval will abut its vertices in the inner surface of the stator. If we now get a magnetic field to rotate (similar to the rotating field in the stator of the asynchronous motor), the oval will rotate with the same speed as the field, and the cylinder itself will slowly rotate in the opposite direction (relatively to the rotating field).

Fig. 8. Patent SU № 68211, Moskvitin A.I. [20]

The invention of the Soviet engineer is essentially a kind of a SWG as it has three main components of the now famous Harmonic Drive: a wave generator, a flexible and a rigid (circular) elements. The basic difference of Moskvitin´s transmission is that it has no teeth and the flexible element (Flexspline) is not deformed by mechanical means: a cam or a disk, but as a result of interaction between the electromagnetic field of the stator and the flexible rotor [24-25]. Recognizing the advantages of the low-speed motor we can say that the absence of a rotating generator gave the mentioned invention another unique feature: low inertia and, consequently, high performance, i.e. the ability to instantly respond to the received command, and to begin to rotate almost simultaneously with the received signal. Moskvitin proposed to use his invention, the "Low-speed electric motor" [20-23] not only as a drive, but also as a generator to produce electric power (AC). Unfortunately, we do not know exactly what happened later: whether the invention was forgotten or it was closed for public because of its use in military or airspace industry - we hope so.

V. Responsin According to the Great Soviet Encyclopedia of 1979 [26] Responsin (Fig. 9) is a Russian name for a type of a wave transmission device that makes use of electromagnetic excitation of deformation waves. The concept of the Responsin [26-27] was put forth by the Soviet engineer Anatoly Ivanovich Moskvitin. The name came from combination of two words: Latin word

“responsum” – response, and Greek word “synchronismos” – simultaneity. The name explains the basic operating principle of the device. A rotating magnetic field is excited by the windings of the stator and is short-circuited through a flexible magnetic circuit located within a flexible wheel. This magnetic circuit functions as a generator of deformation waves. The windings of poles located in diametrically opposed positions are fed in pairs. This arrangement produces a given deformation of the flexible wheel.

Fig. 9. Responsin: (1) flexible wheel, (2) flexible magnetic circuit, (3) rigid wheel, (4) stator, (5) output shaft

When the kinematics of the interaction of the flexible wheel and the rigid wheel represent the conventional kinematics of wave transmissions, the rotational speed of the magnetic field is 100 to 200 times greater than that of the output shaft. The use of a magnetic field for direct deformation of the flexible wheel permits the elimination of all rapidly rotating components from the transmission; the Responsin consequently is characterized by inertia-free operation, fast response, and high kinematic accuracy. The absence of unbalanced rotating masses means that the mechanism does not vibrate during its operation. The efficiency of the Responsin is only 6–20 percent, associated with heavy losses in the magnetic circuit, which prevents the widespread use of the device. Fast response is very essential in industrial motion control and that is why this fact became the basis for the name. Responsin is unmatched in performance and exceeds all previously known devices: it “responds instantly”, for one second it can turn on and turn off about 1,500 times. The Responsin came into use in Russia in the 1960’s as a low-speed drive for servosystems and other precision mechanisms.

VI. Bauman Moscow State Technical UniversityIn mid 70-s of the 20th century scientists from the

Moscow Bauman State Technical University (BMSTU), Fig.10, designed and patented [31-33] wave gear systems with external wave generators that allowed better element configuration when designing electromechanical drives for automatic control systems (ACS) and antenna drives.

Kinematic schemes of wave gear systems patented under [31] are shown on Fig.11. Wave gear systems of external deformation can be single-stage (Fig. 11, a, b) with reduction ratio U = 60…300; dual-stage (Fig. 11, d) with U = 2500…10000 or single-stage with wave coupling (Fig. 11, c) with U = 60…300. All of them can operate as a reducer or a multiplier. Flexible layout of crankshaft axes allows for the best possible configuration of the drive.

Fig. 10. Main Building of Bauman Moscow State Technical University

When used in the step-down mode, it can be powered by one or several motors. Non-driven shafts can be equipped with feedback converters without additional kinematic chains. However, when a wave gear drive is used under loads exceeding nominal ones, the rotational strength decreases due to the gear flexibility.

Fig. 11. Wave gear drives with external wave generator:

a – with a non-fixed flexible gear; b – with a fixed flexible gear (Flexspline); c – with a wave coupling; d – dual-stage wave gear; zf – flexible gear; zc – circular gear; zf2 – flexible gear of coupling; zc2 – circular gear of coupling

Another wave gear drive [32] that is shown on Fig.12 is interesting for its design and cinematic features. The invention was aimed at reducing the axial dimensions and lowering the bottom range of the reduction ratio to U = 20…25. The transmission consists of housing 1, flexible gear 2 with two sets of toothing located one under another (external 3 and internal 4), wave generator represented as

two internal tooth ring washers 5 and 6 that are fixed on the drive crankshaft 7 and the driven crankshaft 8 (the second driven shaft is not shown), rigid gear 9 with external toothing 10 that is connected to the output shaft 11. The drive works as follows. When the drive shaft 7 rotates, it makes the driven shaft 8 rotate too, while the toothed ring washers 5 and 6 of the wave generator deform the flexible gear 2 making it rotate elliptically. The motion is transferred to the flexible gear through the engagement of the washers 5 and 6 with the external toothing 3 of the flexible gear. From the flexible gear 2 it is further transferred to the output shaft 11 through the engagement of the internal toothing 4 with the external toothing 10 of the rigid gear 9. Decrease of the axial dimensions of the drive makes it lighter and allows for better configuration. As the input and output shafts are non-coaxial it is possible to have a central bore without increasing inertia of the drive. It is required, for example, in drives of radars to allow passage of waveguides.

Fig. 12. Wave gear drives with external wave generator

The problem of achieving high accuracy and high torsional stiffness in small size drives with high reduction ratios can be solved effectively through a patent-protected combined planetary-wave transmission. Its design rationally combines the benefits of the wave as well as the crank-planetary mechanisms. It is shown on Fig. 13. In this transmission the crank-planetary mechanism and the wave gear drive of external deformation are connected in parallel. The wave generator consists of two ring washers 4, 5. The internal surfaces of the washers deform the flexible gear ring 6 of the wave gear drive, while the external surfaces have toothing of the crank-planetary mechanism. Deforming ring washers are attached to three crankshafts 1, 2. External teeth of the washers 4, 5 engage with the internal teeth of the rigid gear 7a that is fixed on

the output shaft 7. The internal teeth of the flexible gear 6 engage with two gears, the rigid gear of the coupling 3 in the housing and with the rigid gear of the drive 7b on the output shaft 7. Axial dimensions and reduction ratios are the same both in the crank-planetary and the wave drives. The input energy in such a mechanism is divided into several streams (fours streams if there are two satellites) when the energy is transferred and transformed and then collected again at the output. It logically leads to higher rigidity and cinematic accuracy of the mechanism without considerably increasing its size and weight. When the wave and planetary mechanisms work in parallel, the former guarantees that there is no pitch play in the drive due to the flexible gear compliance. At low loads, when clearances in the planetary mechanism gearing have not been taken up, only the wave drive works.

Fig. 13. Combined planetary-wave drive with a flexible gear wheel

As the loads increase, the planetary drive gets engaged. As the planetary drive has a bigger module and a bigger gear diameter, its rigidity is higher than that of the wave drive. That is why when the loads are high, they are mostly absorbed by the planetary mechanism and less so by the wave one. This step-down design (Fig.13) guarantees large reduction ratios with small axial dimensions, high torque capacity, rotational rigidity and kinematic precision. It is achieved through multi-field and multi-toothed meshing (two fields in the planetary drive and four in the wave drive and the wave coupling). The presence of a deformable flexible gear in a closed space means that loads can be distributed evenly within the fields of gear engagement.

VII. Design Practice of Wave Gearing DriveThere are several ways to calculate geometrical

parameters of Wave Gearing Drives. The current practice was developed by the department “Theory of Mechanisms and Machines” at BMSTU (www.bmstu.ru) and is based on a supposition that the design of wave generators for the

drives under discussion provides constant curvature of the median layer of the deformed flexible gear within the engagement field limited by central angles (Fig.14 a, b). Beyond these limits the flexible gear has a free deformation shape. In the segment where the curvature is constant, the engagement in the wave gear is considered as an internal involute engagement of a rigid gear having жznumber of teeth and a conventional gear that has parameters of a flexible gear and уz number of teeth.

Fig. 14, a, b. Geometrics of wave gearing

A. Geometric Features of Wave Gearing When the flexible gear of the harmonic drive is under load, it changes its initial shape. It is caused by the presence of clearances and the elasticity of the parts interacting with the flexible gear. The deformation of the flexible gear 1 is limited from the outside by the rigid gear 2 and from the inside by the wave generator h. The flexible gear bearing on the wave generator within the limits of the constant curvature (Fig. 14, a) tends to take the shape of the rigid gear. As the torque applied to the flexible gear increases, the area of taken up clearances increases too, which leads to a higher number of tooth pairs in engagement. Thanks to the multi-toothed contact, loading capacity of the harmonic drive is higher than that of the planetary one (load can be transferred by up to 40% of all

the tooth pairs). Wave gear drive efficiency is also higher because the engaged teeth are hardly moving when the flexible gear is fitting against the rigid one. When steel flexible gears are used in single-stage wave gear drives, the reduction ratio can be as high as 60 – 320 with the efficiency ranging between 0.85 – 0.80. In dual-stage wave gear drives, the reduction ratio of 2500 to 10000 and the efficiency of 0.7 to 0.1 can be achieved. Manufacturing and assembly errors can be significantly neutralized owing to the multi-tooth and multi-field contact in the wave gearing, which results in its high cinematic accuracy. Since the flexible gear has a relatively small radial deformation, it can be made as a bell-shaped shell to be used in airtight wave gear drives where rotation is transferred through an airtight partition without using movable seals. Flexible bearing and flexible gear are crucial parts of the wave gear drive. Flexible gear (Fig. 14, a) has a thin-walled end cap that allows axial movements of the cylindrical shell when it is deformed from the other end. The length of the flexible gear is set between 0.4 and 1.2dсг where dсг is the diameter of the non-deformed median surface of the flexible gear. Reduction ratio of the drive, its design, nominal torque at the output shaft, wave generator frequency, life of the drive and strength of the flexible gear are the initial parameters for the calculation. Design calculations aim at determining the median surface diameter of the flexible gear based on the bending resistance, fatigue calculation or calculation of the set ratio of torsional rigidity. The bigger of the calculated diameters is taken as a base to determine the engagement module

ггс / zdm =′ that is rounded to the nearest standard

value. Reference diameters and thickness сh of the flexible

gear rim under the toothing ring are calculated through the following formulae:

.10)5/60(

;;4

ггс

жжгг

−⋅+=

==

mzzh

mzdmzd(1)

The main variable parameter is the relative radial deformation of the flexible gear along the big axis:

γ

−=г

гж

сг

0

z

zz

r

w, (2)

where 2,19,0 K=γ is the ratio of relative radial

deformation. The calculated number of teeth of the conventional gear

is:

)/(1 гс0

гу rwK

zz

β±= (3)

In formula (3) as well as in all the consecutive ones that contain double signs of arithmetic actions, the upper sign refers to the internal deformation of the flexible gear by disk or cam wave generator and the lower one refers to the external deformation by circular generator (Fig. 14, b):

ββ−β−πβ−βπ−πβ=β cossin2/

sin2cos)/4()/4(K (4)

whereβ is the angular coordinate of the constant curvature

segment ( °<β<° 6540 ).

The next step is to determine the radius of the median circumference of the deformed flexible gear:

)2/2/( гсгус xmhchzmr ++++= ∗∗α (5)

where ∗αh , ∗c are the initial profile parameters andгx is

the addendum modification coefficient:

δ++= ∗∗α )2/( сг mhсhx (6)

with 4,10,1 K=δ being the displacement modificationcoefficient.

As the values β , γ , δ change within their possible

range, engagement optimization can be performed with theoretical contact ratio being the objective function. Median circumference radius of the non-deformed flexible gear is:

усу

ггс r

z

zr = (7)

Centre distance of the drive equals eccentricity of the deforming disks and is calculated as follows:

усгс0гс )/1( rrwreaw ++±== (8)

Then the engagement angle of the harmonic drive is

α−=α

ww a

mzz

2

cos)(arccos уж

(9)

Rigid internal teeth gear in the drives with disk or cam internal wave generators is machined by shaper cutter with

0z number of teeth. Cutter tool pressure angle of the rigid

gear and the shaper cutter is

−α−−α−α

−α=α0ж

гужж

tg2))(inv(invinvinv

0 zz

xzzww

(10)

and the rigid gear displacement coefficient is:

21)cos/(cos 0ж

жж 0

zzx w

−−αα= (11)

The other parameters and execution dimensions of the wave gear drive parts are calculated in the same way as those of the internal involute gearing.

VIII. Wave Gear Applications From the very beginning it was obvious that applications of the wave gear (Harmonic Drive) are unlimited. A new class of constant-ratio mechanical drive systems is used for power transmission, angular positioning, or motional conversation. As Musser [17] said: “They can change speed with high reduction ratio to 1000000:1 and transmit motion through sealed enclosures”.

The main advantages include compactness and light

weight, simple construction, extremely low backlash and long life, reconfigurable ratios within a standard housing, good resolution and excellent repeatability (linear representation) when repositioning inertial loads, high torque capability, and coaxial input and output shafts. It is important that high gear reduction ratios are possible in a single stage and in a small volume (a ratio from 30:1 up to 320:1 is possible in the same space in which planetary gears typically only produce a 10:1 ratio). Drive efficiency of some models can be over 90 per cent. Disadvantages include a tendency for 'wind-up' (a torsional spring rate) and potential degradation over time from mechanical shocks and the environment. A unique and useful characteristic of wave gear is its ability to transmit mechanical motion through sealed walls (a hermetic seal). The wall can be a part of a welded steel enclosure which is capable of maintaining complete isolation of two environments. As it was mentioned before wave gears are successfully used in many areas and aerospace in particular. There was a time in USA that all things in space were Harmonic Drive driven: Antenna Pointing Mechanisms, Solar Array Drives for Satellites, Flight Controls, Valve Actuators, Lunar and Interplanetary Rovers (Fig.15), etc.

Fig.15. Nomad Planetary Rover [18] (photograph courtesy of NASA Ames Research Centre)

The performance requirements of space vehicles are increasing steadily. This would not be possible without continuous improvements from the applied gear and actuator technology. Harmonic Drive systems fulfill a complex set of requirements, providing high positioning accuracy and repeatability, high torque capacity and high torsional stiffness, a compact and light design at a competitive price. The first major space application in USA was in 1971 as the mechanical transmission element within the individual wheel drives of the Lunar Roving Vehicle on the Apollo 15 mission. This application brought this unique gear principle into the public eye for the first time. Soon after harmonic drive gears were used as part of the telescope drive actuator for the imaging photopolarimeter that flew on NASA‘s Pioneer 10 planetary probe launched in March 1972. In 1984 this actuator was still working perfectly as Pioneer left the Solar System. At that time the hermetic sealing possible with the harmonic drive gear was an important additional attribute, given the high vapor pressure and migrating tendencies of the silicone-based lubricants then used. Also, the winches used on Skylab to deploy the solar

panels were powered using Harmonic Drive gears. Current USA aerospace applications include the wheel drives, antenna and robotic arm of the Mars Exploration Rovers and 26 joints on Robonaut 2 [15], the first humanoid robot to work in space. Today’s largest project for aerospace applications is to develop dry lubricated gears. The aim is to increase the operational temperature range of space gears for future missions, including planetary exploration. The 1990s have seen a rapid increase in SWG applications as requirements for increased accuracy and improved dynamic performance have necessitated the use of high quality gears and actuators, in fields as diverse as surgical robotics, measuring machines, silicon wafer processing equipment, and more [37?]. The Harmonic Drive gear (SWG) is the subject of continuous development, due to new market requirements from each of the major application areas. Common requirements are the need for higher power density, higher torsional stiffness and lower ratios. In the last few years in Japan the new "S" series tooth profile has been developed through theoretical modeling [37-38]. The CAD graphic shows the results of the mathematical and kinematic studies that were conducted. The Fig.16 of the engaging teeth shows how the new S (Fig.16) tooth profile was designed to permit multiple tooth engagement [38].

Fig. 16. new "S" series tooth profile [25]

The original toothform has an involute curve profile that is common to most spur gears. The new "S" series toothform does not use the involute curve, instead it uses a series of pure convex and concave circular arcs created to match the loci of engagement points dictated by theoretical and CAD analysis. The new tooth profile provides an even higher packaging ratio of torque per cubic inch, and with lower weight for those applications where space and weight are at a premium. The advantages of the "S" Series drive units include double the torsional stiffness, double the peak torque ratings, double the life, and high efficiency, in the range of 60% to 90% depending upon the load, speed and lubrication.

IX. Evolution of Wave Gear (Harmonic Drive) Design It has been more than 50 years since the first wave gears were invented. They are successfully used in the sky, on the sea and on the ground. But why were the wave gear invented only in the 20th century, after hundreds of years of traditional gear use? Hadn’t there been a person that was able to understand that flexibility could give gears unique and remarkable properties?

History of discoveries and inventions at a glance seems

to be a random chain of revelations, a result of the efforts of a lone genius who is driven by his own internal motivation. But it is only at a glance. In addition to the internal motivation of talented inventors, there are the needs of social development of the society. These needs ultimately determine the fate of a technical invention. The History of Science and Technology proves that the idea of SWG was “in the air”. Still, the invention took place only in the mid of the 20th century when it became possible to cut fine-grained wheels and to use new high strength materials. The 20th century did not only provide an opportunity for the wave gears to be invented, it encouraged their invention through the development of the high-speed rotation machines that required a gearing with high reduction ratios; space rockets and sputniks that needed extremely lightweight mechanisms; special devices for which high accuracy and possibility to transfer motion through the sealed wall were of most importance. Wave transmission, and SWG in particular, were born in due time, when science and technology allowed them to boldly invade the areas that are traditionally defined as “rigid-body mechanics”: lifting and transportation machinery, space vehicles and sputniks, highly precise machine tools and work tables, actuators and chemical systems where motion can be generated within a completely sealed vessel and more.

X. Conclusions The Conclusion can be drawn as follows: (1) The wave gears (Harmonic Drives) have no equal in

many applications. But the traditional gears are “allies” of the new ones, organically complementing the possible use of mechanical transmissions. The further development of the scientific and technological progress without using mechanical transmission is unthinkable.

(2) Walton Musser patented the Strain Wave Gearing (SWG) in 1957 but the main idea of his invention as a continuous deflection wave generated in a flexible element was known before and was claimed in a patent of the Soviet (Russian) engineer A.I. Moskvitin in 1944.

(3) In mid 1960s the Responsin, a type of wave transmission device, was already used in Russia as a low-speed drive for servo actuators and other precision mechanisms.

(4) A range of application possibilities will be further extended and become almost limitless when SWG stages are combined to form multiple-stage systems, including combination of internal and external wave generators, flexible splines with multiple sets of teeth, and internal or external circular splines.

(5) The SWG drive is the subject of continuous development. Common requirements for new models are the need for higher power density, higher torsional stiffness and lower ratios. More attention is to be paid to the development of dry lubricated gears to increase their operational temperature range for future missions, including planetary exploration.

Acknowledgements This research is sponsored by Bauman Moscow State

Technical University (MGTU).

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