A Scroll Gear is a non-circular gear cut on a spiral pitch curve so that the radius — and therefore the gear ratio — changes continuously as the gear rotates through its working arc. Franz Reuleaux documented the form in his 19th-century kinematic catalogues, classifying it among the variable-ratio rolling-contact pairs. As the mating pinion rolls along the spiral pitch line, output speed either increases or decreases progressively over the stroke. The result is a single gear that delivers a programmed speed or torque profile without electronics — used on stamping presses, looms, and watch fusees.
Scroll Gear Interactive Calculator
Vary the scroll radius range, pinion radius, and working arc to see the changing gear ratio and spiral pitch curve.
Equation Used
The calculator models the scroll pitch radius as a linear Archimedean spiral from r_min to r_max across the working arc. The instantaneous ratio is the scroll pitch radius divided by the fixed pinion pitch radius.
- Scroll pitch curve is treated as an Archimedean spiral over the working arc.
- Pinion pitch radius is constant.
- Rolling contact is ideal, with no backlash, slip, or tooth compliance.
- Ratios are pitch-radius ratios, not including losses.
The Scroll Gear in Action
A Scroll Gear works by replacing the constant pitch radius of an ordinary spur gear with a spiral pitch curve — usually a section of an Archimedean or logarithmic spiral. As the gear turns, the contact point with its mating pinion walks outward (or inward) along that spiral, so the instantaneous gear ratio changes every degree of rotation. If the pinion has a fixed pitch radius of 20 mm and the scroll's radius sweeps from 40 mm to 80 mm across its working arc, the ratio walks from 2:1 to 4:1 smoothly. That is what people mean by Scroll-gears (gradually increasing speed) — a single gear that gives you a controlled velocity ramp from one input revolution.
The geometry is unforgiving. Both gears must be cut so the rolling pitch curves stay tangent at every point of mesh — that is the conjugate condition. Miss the spiral profile by even 0.1 mm of radial error and you get either backlash that opens up mid-stroke or interference that locks the teeth. Tooth count and tooth spacing also vary along the spiral because each tooth sits at a different radius, so most Scroll Gear pairs are CNC-machined from a parametric tooth-by-tooth model rather than hobbed. Common failure modes are tooth-tip chipping at the small-radius end (where pressure angle gets aggressive) and pitting at the large-radius end (where sliding velocity peaks).
Real-world Scroll Gearing usually runs through a partial revolution — typically 180° to 300° of working arc — then either resets via a return spring, a complementary scroll, or a dwell sector with circular teeth. Run past the spiral's defined arc and the teeth simply stop meshing. That is by design: the mechanism is meant to deliver a programmed motion stroke, not continuous rotation.
Key Components
- Spiral pitch curve: The mathematical backbone of the gear — usually an Archimedean spiral r = a + bθ or a logarithmic spiral r = a·e^(bθ). The choice sets whether the ratio changes linearly with angle or geometrically. Tolerances on the pitch curve are tight — radial error must stay under roughly 0.05 mm on a 100 mm-radius gear or backlash becomes visible as torque ripple.
- Variable-pitch teeth: Each tooth is generated individually because pitch radius, pressure angle, and tooth thickness all change along the spiral. Teeth at the small-radius end are thinner and stronger in shear; teeth at the large-radius end are wider and see higher sliding speeds. AGMA quality 8 or better is typical for industrial work.
- Mating pinion: Usually a standard circular spur or helical pinion sized to a fixed pitch radius matching the scroll's tooth module. The pinion does the work of constant rotation; the scroll does the work of variable ratio. Pinion bore and shaft fit must hold ±0.015 mm runout — any wobble walks straight into the conjugate mesh.
- Working arc limits: Hard stops or dwell sectors that define where the spiral begins and ends. Most Scroll Gear designs use 180° to 300° of active arc. Beyond that arc the teeth either disengage or transition into a constant-radius section for return strokes.
- Return mechanism: Either a torsion spring, a counter-rotating second scroll, or a Geneva-style index that resets the gear to its starting angle. On watch fusees, a chain wound around the scroll body provides both the return path and the working force.
Where the Scroll Gear Is Used
Scroll Gearing shows up wherever a designer needs a programmed speed or torque profile from a single rotating shaft, without resorting to a servo and encoder loop. The Scroll gearing (form) is mechanical signal-shaping — you bake the velocity curve into the metal. That is exactly why old high-end machines used it long before electronics existed, and why it still appears in cost-sensitive automation today.
- Horology: The fusee in marine chronometers and early pocket watches — a conical Scroll Gear that compensates for declining mainspring torque, used by John Harrison in the H1 sea clock and still made today by manufacturers like A. Lange & Söhne in the Pour le Mérite series.
- Mechanical presses: Toggle-press knee-joint drives where a Scroll Gear gives the ram fast approach and slow squeeze — used historically on Bliss and Minster eccentric presses for cold-heading and coining work.
- Textile machinery: Pickup-cam drives on Jacquard and dobby looms where the warp shed must open quickly then dwell — Scroll Gears running 240° of arc replace cam-and-follower stacks on machines like the Vamatex Leonardo.
- Aerospace test rigs: Variable-ratio loading drives on fatigue test stands at Boeing and Airbus suppliers where a sinusoidal force profile must come out of a constant-RPM motor — Scroll Gears generate the profile mechanically without servo tuning.
- Stage and theatre automation: Variable-speed curtain and scrim drives where the load must accelerate slowly then run flat-out — a Scroll Gear paired with a constant-RPM gearmotor delivers the soft-start without a VFD.
- Packaging machinery: Web-tension dancer drives on labelers and shrink-sleeve applicators — Scroll Gearing maintains constant linear web speed as roll diameter shrinks from 300 mm down to 80 mm.
The Formula Behind the Scroll Gear
The core calculation for a Scroll Gear is the instantaneous gear ratio as a function of rotation angle, because that is what tells you what the output speed actually does over the working stroke. At the low end of the typical operating arc — say the first 30° of a 240° stroke — the ratio is near its minimum and output speed is at its slowest. At the high end the ratio has walked to its maximum and output speed is highest. The sweet spot for design sits where you place the part of your load profile that needs the most speed: usually the middle 60% of the arc, because tooth stress is most uniform there. Push the velocity peak too close to either end of the spiral and you stack pressure-angle problems on top of speed problems.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| i(θ) | Instantaneous gear ratio at scroll rotation angle θ | dimensionless | dimensionless |
| rscroll(θ) | Pitch radius of the scroll gear at angle θ | mm | in |
| rpinion | Constant pitch radius of the mating pinion | mm | in |
| a | Starting radius of the spiral at θ = 0 | mm | in |
| b | Spiral growth coefficient (radius gained per radian) | mm/rad | in/rad |
| θ | Scroll rotation angle measured from the start of the working arc | rad | rad |
Worked Example: Scroll Gear in a label-applicator dancer drive
You are sizing the Scroll Gear that drives the rewind shaft on a Herma 152C pressure-sensitive label applicator running 80 m/min linear web speed. The supply roll starts at 300 mm diameter and shrinks to 80 mm by end of run. The drive motor turns a constant 200 RPM and you want the rewind shaft RPM to ramp from 85 RPM (full roll) up to 318 RPM (empty core) so linear web speed stays constant. The pinion has a fixed pitch radius of 25 mm. The working arc is 270° (4.712 rad).
Given
- rpinion = 25 mm
- nstart = 85 RPM (full roll)
- nend = 318 RPM (empty core)
- ninput = 200 RPM (constant)
- θarc = 4.712 rad (270°)
Solution
Step 1 — work out the required ratio at the start and end of the arc. Ratio is input speed over output speed:
iend = 200 / 318 = 0.629
Step 2 — convert ratios to scroll pitch radii using rscroll = i × rpinion:
rend = 0.629 × 25 = 15.7 mm
Step 3 — solve for the spiral coefficients a and b across the 4.712 rad working arc:
b = (rend − rstart) / θarc = (15.7 − 58.8) / 4.712 = −9.15 mm/rad
So the scroll spirals inward as it turns. Now check three operating points along the arc.
Step 4 — at the low-speed end of the typical operating range (θ = 0, full roll), output RPM is 85. The web feels solid, the dancer arm sits high, and the rewind torque is at its maximum because the roll is heavy and the moment arm is long. This is where tooth stress peaks on the scroll — design margin must live here.
Step 5 — at the nominal mid-arc point (θ = 2.356 rad, half-spent roll), the radius is r = 58.8 + (−9.15)(2.356) = 37.2 mm, ratio i = 1.49, output speed:
This is the sweet spot — sliding velocity is moderate, pressure angle is well-behaved, and tooth load is balanced. If you have a single point to optimise tooth profile around, this is it.
Step 6 — at the high-speed end (θ = 4.712 rad, empty core), output RPM hits 318. Sliding velocity at the mesh peaks at roughly 0.52 m/s — fine for case-hardened steel running in a light oil mist, but borderline for plastic gears, which is why most Herma-class machines use steel scrolls.
Result
The Scroll Gear must run from a 58. 8 mm starting radius down to a 15.7 mm ending radius across 270° of arc, giving a continuously variable output from 85 RPM at full roll to 318 RPM at empty core. At 85 RPM the rewind feels almost stalled to a hand on the shaft — the roll is heavy and the gear is loafing. At 134 RPM (mid-arc) the system is in its design sweet spot and that is where tooth profile should be optimised. At 318 RPM the empty core spins fast enough that any imbalance in the spent core becomes audible. If the measured web speed drifts off the 80 m/min target, the three most common causes are: (1) spiral pitch error above 0.05 mm radial — usually from CNC backlash during gear cutting — which puts a step in the ratio curve and shows up as a periodic web-tension wobble, (2) pinion shaft runout above 0.015 mm causing once-per-rev tension pulses, or (3) the scroll gear assembled 1 or 2 teeth out of phase with the roll-diameter sensor, which offsets the entire ratio curve and gives the right shape but the wrong absolute speeds.
Scroll Gear vs Alternatives
A Scroll Gear is one of several ways to deliver a programmed speed or torque profile. The choice between Scroll Gearing, a servo with electronic cam, and a mechanical cam-and-follower depends on whether you can afford electronics, how often the profile changes, and how clean the motion needs to be at the start and end of the stroke.
| Property | Scroll Gear | Servo with electronic cam | Cam-and-follower |
|---|---|---|---|
| Typical operating speed | 50–500 RPM input | 0–6000 RPM, software-limited | 30–800 RPM input |
| Profile accuracy | ±1% if cut to AGMA 8 | ±0.01% with closed-loop encoder | ±0.5% if cam ground to ±0.02 mm |
| Cost (single unit, industrial scale) | $800–$3000 CNC-cut steel | $4000–$12000 servo + drive + controller | $400–$1500 ground steel cam |
| Reprogrammability | None — profile is in the metal | Full — change in software | None — must regrind cam |
| Maintenance interval | ~5000 hours, oil mist lubed | ~20000 hours servo bearings | ~2000 hours follower roller |
| Working arc per cycle | 180°–300° typical | Unlimited | 0°–360° depending on cam |
| Failure mode | Tooth pitting at large-radius end | Encoder fault, drive trip | Follower roller flat-spotting |
| Best application fit | Fixed profile, no electronics | Variable profile, frequent changeover | Fixed profile, high cycle count |
Frequently Asked Questions About Scroll Gear
That symptom is almost always a tooth-spacing error introduced during CNC tooth-by-tooth generation, not a pitch-curve error. Because each tooth on a scroll sits at a unique radius and unique pressure angle, a 0.5° indexing error on the rotary table during cutting puts one tooth slightly out of step with its neighbours. You will see it as a once-per-revolution torque spike at exactly the same θ every cycle.
Diagnostic check: mount a dial indicator on the pinion shaft and rotate the scroll slowly by hand through the working arc. Any tooth that gives a sudden indicator deflection is the culprit. Fix is usually to recut that tooth pair or live with the error if it falls outside the load-critical part of the arc.
Choose Scroll Gearing when the profile never needs to change, the duty cycle is high, and EMI or controller crashes would stop production. A servo can replicate any scroll curve in software, but every servo profile depends on a controller, a drive, an encoder, and clean power. A Scroll Gear is mechanical signal-shaping — it cannot lose its program and it does not care about voltage sags.
Rule of thumb: if you would still need a mechanical backup for the profile in case the servo drops out, the Scroll Gear is doing the real work and the servo is redundant. Skip the servo.
A uniform speed offset across the whole arc points to the pinion pitch radius, not the scroll. If the pinion was cut on the high side of its tolerance band — say 25.4 mm instead of 25.0 mm — every ratio along the arc shrinks by the same factor and output speed rises proportionally. Eight percent is roughly a 2 mm pinion oversize on a 25 mm part, which is too much to be machining error and usually means the wrong pinion module was installed.
Quick check: measure pinion pitch diameter with a gear-tooth caliper or over pins, compare to the print, and verify the module matches the scroll's tooth module. A mismatched module also causes audible meshing noise that builds with arc angle.
You can, and watchmakers have done it for centuries with the fusee-and-barrel pair. The key constraint is that the two scrolls must be conjugate — as one winds out, the other winds in, and the sum of their working radii at every angle must give a consistent centre distance. That makes the design a coupled-spiral problem, not two independent scrolls bolted together.
For most industrial work it is cheaper to add a return-stroke dwell sector with circular teeth and reset the scroll mechanically. True coupled-scroll pairs are worth the design effort only when you need the variable ratio in both directions of rotation.
Wear concentration tells you where the load profile actually lives on the spiral, regardless of what your design intent was. If the tooth pitting is all at the large-radius end, you are running the high-speed/low-torque portion of your duty cycle for most of every shift, and sliding velocity there is grinding the tooth flanks. If wear is at the small-radius end, you are stalling against high torque in that region.
The fix is rarely a metallurgy upgrade — it is rebalancing the duty cycle. Either change the input motor RPM so the load-heavy portion of your cycle moves toward the middle of the arc, or redesign the spiral so the radii at the worn end shift to better-suited values. A Scroll Gear assumes the load profile matches the geometry; if they drift apart, the metal tells you immediately.
Tighter than a normal spur gear pair, because backlash compounds across the spiral. On a typical industrial scroll with module 2 teeth, hold centre distance to ±0.025 mm. Open it to ±0.10 mm and you will see backlash open up by the time you reach the small-radius end of the arc, which shows up as torque reversal slap during the high-speed portion of the cycle.
Use a precision-bored housing rather than slotted mounting, and check centre distance after thermal soak — a 40°C temperature swing on a 150 mm centre distance steel housing moves the bores about 0.07 mm, which is enough to push a tight design out of spec.
References & Further Reading
- Wikipedia contributors. Non-circular gear. Wikipedia
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