Strain-wave gearing uses a wave generator to deform a flexspline so its teeth engage a circular spline in moving zones. In the configuration shown, the circular spline is fixed, the wave generator is the input and the flexspline is the output. The tooth-count difference produces reduction and opposite-direction output rotation.
Strain Wave Gearing Interactive Calculator
Enter circular-spline and flexspline tooth counts for the standard fixed-circular configuration. The wave generator is the input and the flexspline is the output.
Equation Used
- Circular spline fixed; wave generator input; flexspline output.
- Tooth counts are positive integers and Nc must exceed Nf; invalid combinations return no motion result.
- The default 202/200 pair follows the cited manufacturer example. For other positive pairs, the calculator reports only the algebraic kinematic relation and does not claim a realizable two-lobe tooth mesh.
- The supporting cam/bearing, flex rim, ring ticks and engagement patches are schematic and do not reproduce the proprietary tooth profile or every entered tooth.
- Ideal kinematics only: elastic stress, coning, load distribution, torque, efficiency, torsional stiffness, life and product ratings are not predicted.
The 202/200 default gives a 100:1 reduction magnitude and −3.6° output per input revolution. Other accepted tooth-count pairs show the algebraic ratio only; the calculation does not establish whether their teeth can form a practical mesh.
How the three components interact
The rigid circular spline surrounds the flexspline. An elliptical wave generator and supporting bearing deform the flexspline so engagement occurs near opposite ends of the ellipse. As the generator turns, those engagement zones travel around the ring.
With fewer teeth on the flexspline than on the fixed circular spline, the flexspline turns slowly in the opposite direction. In the cited 202/200 example, one wave-generator revolution moves the flexspline backward by two of its own tooth pitches.
Reading the drawing
- Circular spline: the fixed outer ring.
- Wave generator: the rotating elliptical cam and bearing illustration.
- Flexspline: the deforming output member, identified by the green marker.
- Red arcs: schematic opposed engagement zones.
The tooth ticks are explanatory marks, not a drawing of every entered tooth or a manufacturable tooth profile. The ratio equation does not calculate flexspline stress, bearing contact or load distribution.
Exploring reduced rotary motion
This configuration demonstrates how a small difference between large tooth counts can give a large reduction ratio. It also makes the direction relationship visible: the fixed ring remains still while the flexspline marker moves opposite the input.
For an actual positioning drive, ratio is only one requirement. Torque, stiffness, transmission error, speed, duty cycle and bearings require data for the selected product. The calculator does not predict robot positioning accuracy or establish that a particular manufacturer’s unit suits an application.
Ratio and signed output travel
Let Nc be the circular-spline tooth count and Nf the flexspline count. For the displayed configuration, Δ = Nc − Nf must be positive. Reduction magnitude R = Nf/Δ, and output angle per input revolution is −360Δ/Nf degrees.
The signed speed relation is ωf/ωwg = −Δ/Nf. The negative sign means counter-rotation. The tooth-walk output is −Δ flexspline tooth pitches per input revolution.
The inputs are rounded to integer tooth counts and must be positive with Nc greater than Nf. Accepted counts define an algebraic result, not necessarily a realizable two-lobe tooth mesh.
Worked example: 202 and 200 teeth
- For Nc = 202 and Nf = 200, the tooth difference is 2.
- Reduction magnitude is 200/2 = 100:1.
- Output travel is −360 × 2/200 = −3.6° per input revolution, and tooth walk is −2.
Thus ten input revolutions correspond to −36° of ideal output travel. This verifies a motion ratio only. It does not calculate output torque, permissible speed, efficiency or positioning accuracy under load.
Ratio versus product performance
| Quantity | What this calculator supplies |
|---|---|
| Reduction and direction | Ideal algebraic relationship for a fixed circular spline. |
| Mesh feasibility | The default follows the cited manufacturer example; other pairs require a separate gear-design assessment. |
| Torque, speed and service life | No rating or prediction. |
| Backlash, stiffness and transmission error | No load-dependent performance calculation. |
A high ratio alone cannot establish the accuracy or durability of a reducer, or compare it fairly with a planetary or cycloidal unit.
Frequently asked questions
Which component is fixed?
The circular spline is fixed. The wave generator is the input and the flexspline is the output. The displayed formula applies to that configuration.
Why is output travel negative?
The flexspline rotates opposite the wave generator in this fixed-circular configuration. The reduction is shown as a positive magnitude, while angle and tooth walk retain the direction sign.
Does any positive tooth-count difference make a usable gear?
No. The equation gives a ratio for accepted counts, but it does not establish compatible tooth profiles, deformation or engagement. Only the cited 202/200 default is identified here as a documented manufacturer example.
Why do equal tooth counts return an invalid result?
They do not give the positive tooth difference required by this reducer model. The calculator also rejects a circular count below the flexspline count instead of assigning an output motion.
Can the ratio tell me available output torque?
No. Input torque, efficiency and component ratings are not inputs. Torque capacity and allowable speed must come from the actual design or product data.
Does the drawing prove zero backlash or positioning accuracy?
No. Tooth marks and engagement zones are schematic. The calculation does not model elastic deflection, lost motion, transmission error or wear.
References
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