Strain wave gearing is a high-ratio reducer that uses the controlled elastic deformation of a thin-walled flexspline to transmit torque between an elliptical wave generator and a rigid circular spline. It solves the problem of getting 50:1 to 320:1 reduction in a single stage with zero backlash and high torsional stiffness — something planetary or worm sets cannot match in the same package size. The flexspline has 2 fewer teeth than the circular spline, so each full input rotation advances the output by exactly that 2-tooth difference. Robots like the FANUC LR Mate and ABB IRB 1200 rely on it for repeatable joint positioning to within ±0.01°.
Strain Wave Gearing Interactive Calculator
Vary the circular spline and flexspline tooth counts to see the tooth difference, reduction ratio, and output motion per input revolution.
Equation Used
For a fixed circular spline, the reduction ratio is the flexspline tooth count divided by the tooth difference. With fewer flexspline teeth, the output moves in the opposite direction from the input by the tooth difference each input revolution.
- Circular spline is fixed.
- Flexspline is the output member.
- Circular spline tooth count is greater than flexspline tooth count.
- Ideal no-slip tooth engagement.
The Strain Wave Gearing in Action
Three parts do all the work — a rigid circular spline with internal teeth, a thin-walled flexspline cup with external teeth, and an elliptical wave generator bearing pressed inside the flexspline. The wave generator forces the flexspline into an oval shape, so its teeth only mesh with the circular spline at the two long ends of the ellipse. Because the flexspline has 2 teeth fewer than the circular spline (a typical pair is 200 vs 202 teeth for a 100:1 ratio), every full rotation of the input shaft walks the flexspline backwards by exactly 2 teeth relative to the circular spline. The output takes its rotation off the flexspline. That 2-tooth difference is what gives you the high single-stage ratio.
The design is built around tooth elastic deformation, not rolling contact. That is why torsional stiffness matters more than gear-strength numbers you would use on a spur set. If the flexspline wall is too thin, the cup deflects under torque and you lose positioning accuracy — the robot arm will overshoot under load and undershoot when unloading. If the wall is too thick, the cup cracks at the diaphragm fillet within a few thousand cycles. Manufacturers like Harmonic Drive SE and Nidec-Shimpo specify flexspline wall thicknesses in the 0.5-1.5 mm range depending on size, and the radial deformation under the wave generator is typically only 0.1-0.3% of the pitch diameter.
The most common failure mode you will see in the field is ratcheting — when peak torque exceeds the rated value, the flexspline teeth jump out of mesh at the major axis and the unit makes a hard knocking sound. Once a strain wave gearbox has ratcheted even once, the tooth flanks are damaged and positioning accuracy is gone. The second failure mode is fatigue cracking at the flexspline cup base, which shows up as gradually increasing backlash that was not there from new. Lubrication breakdown in the wave generator bearing is the third — the elliptical cam runs the bearing balls through cyclic radial loading that is brutal on grease.
Key Components
- Wave Generator: An elliptical steel cam fitted with a thin-race ball bearing on its outside. The cam profile typically deforms the flexspline by 0.1-0.3% of pitch diameter — for a 50 mm unit that is roughly 0.05-0.15 mm of radial deflection. This part is the input and runs at full motor speed.
- Flexspline: A thin-walled steel cup with external teeth cut on the open end. Wall thickness sits in the 0.5-1.5 mm range and the cup must flex elastically billions of cycles without fatigue. The diaphragm fillet radius is critical — Harmonic Drive AG specifies it tightly because that is where fatigue cracks initiate.
- Circular Spline: A rigid ring with internal teeth, 2 more teeth than the flexspline. This is usually the fixed reference, bolted to the gearbox housing. The 2-tooth difference is what produces the reduction ratio — Nflex / (Ncirc − Nflex).
- Cross-Roller Output Bearing: Many integrated units (the Harmonic Drive CSG and CSF series for example) include a cross-roller bearing on the output side rated for both axial and moment loads. Tilt stiffness on a Size 25 unit is around 1.5 × 104 Nm/rad, which is what lets you mount a robot wrist directly to the output flange without a secondary bearing.
- Dynamic Spline (component-set variant): On 'component set' units used in cobots like the Universal Robots UR series, a second internal-toothed ring with the same tooth count as the flexspline replaces the rigid output flange. This ring rotates with the flexspline and gives you a hollow-shaft configuration for routing cables through the joint.
Who Uses the Strain Wave Gearing
You see strain wave gearing wherever a designer needs high reduction in a small package with zero backlash — and is willing to pay for it. The cost per joint is roughly 5-10× a comparable planetary, so it never wins on price. It wins when positioning repeatability, hollow-shaft routing, or torque density is non-negotiable. Industrial robots, surgical robotics, satellite antenna pointing, and semiconductor wafer handlers are the natural fit. A typical 6-axis industrial robot uses 4-6 strain wave units, one per joint, supplied by Harmonic Drive SE, Nidec-Shimpo, or Leaderdrive.
- Industrial Robotics: FANUC LR Mate 200iD/7L 6-axis robots use Harmonic Drive CSG-series units in the wrist joints J4-J6 to hit the rated ±0.02 mm repeatability.
- Collaborative Robotics: Universal Robots UR5e uses a hollow-shaft strain wave drive at every joint so the actuator wiring can pass straight through the centre of the joint, eliminating external cable harnesses.
- Aerospace: NASA's Curiosity and Perseverance Mars rovers use Harmonic Drive units in the robotic arm joints because the zero-backlash spec survives the launch shock and 7-month transit.
- Semiconductor Manufacturing: Brooks Automation and Yaskawa wafer-handling robots in 300 mm fabs use strain wave reducers to position wafers to ±0.05 mm repeatability across millions of cycles.
- Medical Robotics: Intuitive Surgical's da Vinci Xi system uses miniature strain wave gearboxes in the EndoWrist instrument joints — the zero backlash is what allows the surgeon's hand motion to be scaled 5:1 without dead band.
- Satellite Antenna Pointing: Geostationary comms satellites built by Airbus Defence & Space use radiation-hardened strain wave drives for solar array drive assemblies and antenna gimbals, where any backlash would smear the beam pointing.
The Formula Behind the Strain Wave Gearing
The reduction ratio depends only on the tooth counts of the flexspline and circular spline. What changes across the operating range is not the ratio but the torsional stiffness and the allowable peak torque — at the low end of rated speed you can pull near peak (momentary) torque without ratcheting, but at the high end you are constrained by the wave generator bearing's DN limit and grease shear-rate. The sweet spot for most industrial sizes sits around 30-50% of rated input speed and 50-70% of rated continuous torque. Push past 80% rated torque and you start consuming fatigue life on the flexspline cup at a rate that will not give you the 10,000-hour service life the catalogue claims.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| i | Reduction ratio (input revs per output rev, with circular spline fixed) | dimensionless | dimensionless |
| Nflex | Tooth count on the flexspline (external teeth) | teeth | teeth |
| Ncirc | Tooth count on the circular spline (internal teeth, always 2 more than flexspline for standard units) | teeth | teeth |
| Tout | Output torque, equals input torque × i × η (η typically 0.65-0.85 depending on size and speed) | Nm | lb-ft |
Worked Example: Strain Wave Gearing in a stratospheric balloon gondola pointing system
You are sizing the azimuth drive on a stratospheric balloon-borne telescope gondola flown out of the Esrange Space Center in Kiruna, Sweden, that needs to point a 60 kg payload to within 0.01° of a target star while the balloon rotates underneath at up to 3 RPM. You have selected a Harmonic Drive CSF-25-100-2UH unit — Size 25, ratio 100:1, with cross-roller output bearing — driven by a Kollmorgen AKM52K servo. Flexspline tooth count is 200, circular spline is 202. You need to verify the output speed and torque across the full operating range the balloon will see during a 12-hour flight.
Given
- Nflex = 200 teeth
- Ncirc = 202 teeth
- nin,nom = 3000 RPM
- Tin,nom = 1.2 Nm
- η = 0.78 dimensionless
Solution
Step 1 — confirm the reduction ratio from tooth counts:
Step 2 — compute output speed at the nominal servo speed of 3000 RPM, which is the ground-test condition where you are slewing the gondola to the next target:
That gives you 180°/s of azimuth slew — fast enough to retarget across the sky in under 2 seconds. Compute output torque at nominal:
Step 3 — at the low end of the operating range, holding station on a target with a 6 RPM input dither to fight balloon rotation, the output crawls at 0.06 RPM or 0.36°/s. Torque is what matters here, not speed — and at 6 RPM input the wave generator bearing is barely turning, so peak (momentary) torque can hit 192 Nm per the CSF-25-100 datasheet without ratcheting. That headroom is what lets the gondola hold pointing through wind gusts.
Step 4 — at the high end, the AKM52K can hit 6000 RPM during emergency stow, which would give:
But the CSF-25 is rated for 3500 RPM continuous input — running at 6000 RPM exceeds the average input speed limit and the wave generator grease (typically Harmonic Grease SK-1A) will shear-thin and migrate within a few hundred hours. So 6000 RPM is only acceptable for short stow events, not steady operation.
Result
Nominal output is 30 RPM at 93. 6 Nm with 100:1 reduction — well inside the CSF-25-100's continuous rating of 137 Nm. At the low-speed pointing condition of 0.06 RPM output the unit is essentially holding torque, and you have 192 Nm of momentary peak available for gust rejection; at the high-speed stow condition of 60 RPM output you are bumping the input speed limit and grease life becomes the binding constraint, not torque. If your measured pointing accuracy drifts beyond the 0.01° spec, suspect (1) flexspline torsional wind-up because you are running closer to peak torque than calculated — recheck the actual cable-wrap drag on the gondola, (2) thermal contraction of the wave generator at stratospheric −60 °C ambient causing preload loss in the elliptical bearing, or (3) servo encoder resolution mismatch where the 100:1 ratio amplifies any input-side count error by 100× at the output.
When to Use a Strain Wave Gearing and When Not To
Strain wave gearing competes against planetary reducers and cycloidal drives in the precision-reducer market. Each one wins on different axes — pick based on what you actually need to optimise for, not on what looks best in a brochure.
| Property | Strain Wave Gearing | Planetary Gearbox | Cycloidal Drive (RV) |
|---|---|---|---|
| Single-stage ratio range | 30:1 to 320:1 | 3:1 to 10:1 per stage | 30:1 to 200:1 |
| Backlash (arc-min) | < 0.5 (effectively zero) | 3-15 typical, 1 for precision grades | < 1 |
| Torsional stiffness | Moderate (limited by flexspline elasticity) | High | Very high (best of the three) |
| Peak torque vs continuous | ~2× rated, hard ratcheting limit | ~2.5× rated, no hard cliff | ~5× rated, very tolerant of shock |
| Efficiency at rated load | 65-85% | 94-97% per stage | 85-93% |
| Max input speed | 3500-6500 RPM (size dependent) | 6000-10000 RPM | 4000 RPM |
| Service life (typical) | 10,000-15,000 hours | 20,000-30,000 hours | 20,000+ hours |
| Relative cost (per Nm) | High (5-10° planetary) | Low (baseline) | High (4-8× planetary) |
| Hollow-shaft availability | Standard option | Rare and expensive | Available, larger packages |
| Best application fit | Robot joints, satellite gimbals | General industrial drives, mobile equipment | Heavy industrial robot bases, shock loads |
Frequently Asked Questions About Strain Wave Gearing
Real strain wave units are zero-backlash at the tooth mesh, but the total angular transmission error (sometimes called lost motion) is the sum of tooth mesh, torsional wind-up of the flexspline, and any wear on the wave generator bearing. After ~5,000 hours the wave generator bearing balls develop microspalling on the inner race because the elliptical cam loads them cyclically — this shows up as 1-2 arc-min of slack that was not there from new.
Check the unit by locking the input and applying ±50% rated torque at the output while measuring rotation with a dial indicator. If you see more than 2 arc-min of hysteresis, the wave generator bearing is the usual culprit, not the gear teeth.
Pick on torsional stiffness, not on torque rating. The Size 20 will be roughly 2× the torsional stiffness of the Size 17 in Nm/arc-min, which directly determines your positioning accuracy under load. If your application has any moment load on the output flange — like a robot wrist with a long end effector — the Size 20 also gives you a meaningful jump in tilt stiffness through its larger cross-roller bearing.
Rule of thumb: size on torsional stiffness so the wind-up under your worst-case load stays below 25% of your required positioning resolution. If you need 0.01° at the output, target less than 0.0025° of wind-up.
Efficiency drops sharply at low input speeds and at low load fractions. The 0.78 efficiency in most catalogues is quoted at rated speed and rated torque — at 10% of rated load and 10% of rated speed, real efficiency can fall to 0.45-0.55 because the no-load running torque (mostly wave generator bearing drag and grease churn) becomes a significant fraction of total torque.
Check the catalogue efficiency curve at your actual operating point before assuming the gearbox is faulty. Cold ambient also kills efficiency — Harmonic Grease SK-1A roughly doubles in viscosity below 0 °C, and at −20 °C you can lose another 10-15 efficiency points until the unit warms up.
You can, but it is generally a bad idea. Most strain wave units are not designed to back-drive — efficiency in the reverse direction drops to 30-50% on small sizes and the unit may be self-locking entirely on high-ratio units (above 100:1). The wave generator bearing was sized assuming it sees input-side speed and load, so back-driving puts the high-speed stress on a part that is now seeing high torque instead.
If you genuinely need a step-up, use a planetary or a belt drive. Strain wave gearing is a one-direction tool.
That is the early warning sign of imminent ratcheting. When peak torque approaches the unit's ratcheting torque limit, the flexspline cup deflects enough that the teeth at the major axis of the ellipse momentarily disengage and re-engage out of phase. You hear it as a knock at output frequency because the heavy-load tooth pair only crosses the major axis twice per input rev, but accumulated phase error makes the audible event repeat at output frequency.
Stop the machine immediately and reduce peak torque. Once a unit has ratcheted even once, the tooth tip flanks are deformed and positioning accuracy degrades permanently — typical loss is 3-8 arc-min of new transmission error.
Only if your application actually consumes the extra accuracy. The CSG series from Harmonic Drive SE has roughly 50% lower transmission error than the CSF (around 1 arc-min vs 1.5 arc-min on a Size 25) and tighter starting-torque variation. For a pick-and-place doing 0.05 mm placement at a 300 mm reach, the CSF is fine — geometric error from the arm structure swamps the gearbox error.
The CSG earns its premium on semiconductor wafer handlers, optical metrology stages, and surgical robotics where the encoder is on the motor side and you cannot close a position loop on the output. There the gearbox transmission error becomes the dominant accuracy term and the CSG pays back.
References & Further Reading
- Wikipedia contributors. Strain wave gearing. Wikipedia
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