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PAGE: "Reference | Operating Principles :: Harmonic Drive LLC"
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Operating Principles

The operating principles of strain wave gearing are both elegant and unique.
The high performance attributes of this gearing technology including zero backlash, high torque, compact size, and excellent positional accuracy are a direct result of the unique operating principles.

Form Follows Function...
Component Assembly
Click to View Video Clip of Component Assembly

The mechanism is comprised of three components: Wave Generator, Flexspline, and Circular Spline. Each of these will be explained below.

Wave Generator
The Wave Generator is actually an assembly of a bearing and a steel disk which is called a Wave Generator plug. The outer surface of the Wave Generator plug has an elliptical shape that is carefully machined to a precise specification. A specially designed ball bearing is pressed around this bearing plug causing this bearing to conform to the same elliptical shape of the Wave Generator plug.

The Wave Generator is typically used as the input member, usually attached to a servo motor.

The Flexspline is a thin-walled steel cup. This geometry allows the walls of the cup to be radically compliant, yet remain torsionally stiff since the cup has a large diameter. Gear teeth are machined into the outer surface near the open end of the cup (near the "brim").

The cup has a rigid boss at one end to provide a rugged mounting surface. During assembly, the Wave Generator is inserted inside the Flexspline such that the bearing is at the same axial location as the Flexspline teeth. The Flexspline wall near the brim of the cup conforms to the same elliptical shape of the bearing. This causes the teeth on the outer surface of the Flexspline to also conform to this elliptical shape. Effectively, the Flexspline now has an elliptical gear pitch diameter on its outer surface.

The Flexspline is usually the output member of the mechanism.

Important: Although the steel Flexspline flexes during normal operation, there is no concern about fatigue failure. The stresses developed are far below the endurance limit of the material. Thus the Flexspline will achieve infinite life when used according to catalog ratings. This is explained better in the figure below.


Circular Spline
The Circular Spline is a rigid circular steel ring with teeth on the inside diameter. The Circular Spline is usually attached to the housing and does not rotate.

The Circular Spline is located such that its teeth mesh with those of the Flexspline. The tooth pattern of the Flexspline (which is now elliptical -as a result of conforming to the Wave Generator's elliptical shape) engages the tooth profile of the Circular Spline (circular) along the major axis of the ellipse. This engagement is like an ellipse inscribed concentrically within a circle. Mathematically, an inscribed ellipse will contact a circle at two points. However, the gear teeth have a finite height. So there are actually two regions (instead of two points) of tooth engagement.

In fact, up to roughly 30% of the teeth are engaged at all times. The load torque is distributed as shown in the following diagram.


The pressure angle of the gear teeth transforms the output torque's tangential force into a radial force acting on the Wave Generator bearing. The teeth of the Flexspline and Circular Spline are engaged near the major axis of the ellipse, and disengaged at the minor axis of the ellipse.

Here's the trick...
The Flexspline has two less teeth than the Circular Spline. So every time the Wave Generator rotates one revolution, the Flexspline and Circular Spline shift by two teeth. The gear ratio is calculated by {#Flexspline Teeth} / {#Flexspline Teeth - #Circular Spline Teeth}. Since the Flexspline has two less teeth than the Circular Spline, the term {#Flexspline Teeth - #Circular Spline Teeth} = -2. For example, if the Flexspline has 200 teeth, the Circular Spline has 202 teeth, the gear ratio will be calculated as 200/(200-202) = -100. The negative sign indicates that the input and output rotate in opposite directions.

Click to View Video Clip of Gear Operation
Click to View Video Clip of Animated Gear Operation

Kinematics and Performance

Zero Backlash
The unique design and operating principles yield some very convenient benefits. The tooth engagement motion (kinematics) of the strain wave gear is very different than that of planetary or spur gearing. The teeth engage in a manner that allows up to 30% of the teeth (= 60 teeth for a 100:1 gear ratio) to be engaged at all times. This contrasts with maybe 6 teeth for a planetary gear, and 1 or 2 teeth for a spur gear.

In addition, the unique kinematics allow the teeth of a gear to be engaged on both sides of the tooth flank (yes really!). Since backlash is defined as the difference between the tooth space and tooth width. This difference is zero strain wave gearing.

Consistent Performance
As part of the design, the gearteeth of the Flexspline are preloaded against those of the Circular Spline at the major axis of the ellipse. They are preloaded such that the stresses are well below the material's endurance limit. This has an important benefit.

In conventional gearing, wear results in an increase in backlash over time.

In the gear as the gear teeth wear, this elastic radial deformation acts like a very stiff spring to compensate for space between the teeth that would otherwise cause an increase in backlash. This allows the performance to remain constant over the life of the gear.

High Positional Accuracy
The combination of strain wave gearing principles and our manufacturing technology allow positional accuracy of 30 arc-sec (=0.008 degrees!). All three gearing components (Wave Generator, Flexspline, and Circular Spline) are held concentric at all times. In addition, the tooth height, pitch circle, and tolerances are controlled to millionths of an inch. These factors, when combined with the 30% tooth engagement allows for sustained accuracy far better than any other gearing technology.

Highest Torque -to- Weight Ratio
Harmonic Drive™ strain wave gearing offers the highest torque/weight and torque/volume ratios of any gearing technologies. The lightweight construction and single stage gear ratios of up to 160:1 allows the gears to be used in applications requiring minimum weight or volume. Small motors can exploit the large mechanical advantage of a 160:1 gear ratio to create a compact, lightweight, and low cost package.

Affordable Precision
Harmonic Drive™ strain wave gearing offers many performance advantages as compared to conventional gearing technologies. Yet, its simple and elegant design allows manufacturing costs to be roughly equal to that of other precision motion control technologies. This provides an attractive cost/benefit proposition for most motion control applications.

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