An Elliptical Spur Gear is a non-circular gear cut on an elliptical pitch curve, meshing with an identical mate so that one full input revolution produces two zones of fast rotation and two zones of slow rotation at the output. The Heidelberg KORD offset press uses this principle in its swinging gripper drive. The varying centre-to-pitch radius gives a cyclic instantaneous gear ratio, which lets a constant-speed motor drive an output that needs to dwell, snatch, or accelerate within each cycle. You get controlled irregular motion from a steady drive — without cams, without servos.
Elliptical Spur Gear Interactive Calculator
Vary eccentricity, cycle time, centre distance, and driver angle to see the changing speed ratio, output speed, and contact radii.
Equation Used
The calculator uses the article ratio relationship i = r2 / r1 and the stated elliptical-gear behavior of two fast and two slow output zones per revolution. Eccentricity e sets the fast/slow envelope; centre distance C splits into the instantaneous pitch radii r1 and r2.
- Identical elliptical spur gears are mounted at matched foci.
- Centre distance C is held equal to the major axis spacing, so r1 + r2 = C.
- The two-lobe ratio curve follows the article envelope, giving two fast and two slow zones per input revolution.
- Tooth compliance, backlash, friction, and manufacturing error are not included.
The Elliptical Spur Gear in Action
The Elliptical Spur Gear, also called the Elliptical Gear Wheel or Fast and Slow Motion Spur Gear in the packaging and printing trades, works by replacing the circular pitch line of a standard spur gear with an ellipse. Both gears in the pair are identical ellipses, mounted on shafts that sit one focal-distance apart. As the driver turns, the contact radius on the driver shrinks while the radius on the driven gear grows — and vice versa half a turn later. That radius swap is what produces Irregular circular motion by elliptical spur gear pairs: one input rev gives two fast peaks and two slow troughs at the output, with the ratio modulating continuously between roughly 1:e²/(1+e²) and (1+e²)/(1-e²) where e is the ellipse eccentricity.
For the gears to mesh without binding, the centre distance must equal the major axis of the ellipse — exactly. We're talking ±0.05 mm on a 100 mm gear pair. Get it wrong and the teeth either jam at the apex or skip at the equator. The teeth themselves are not standard involute either — they're laid out along the elliptical pitch curve with varying tooth thickness, so each tooth is technically unique. CNC profile grinding or wire EDM is the only practical way to cut them to the tolerances that matter.
If the eccentricity is too high (above roughly e = 0.5), the contact ratio drops below 1.0 at the minor-axis points and the pair will rattle through tooth disengagement. Too low (e below 0.1) and you barely get any speed variation — you may as well use a circular gear. Phasing matters too: if you mount the matched pair 90° out of phase, the output reverses its fast-slow timing, which is sometimes useful but often a build error. The classic failure mode on production lines is a pinned hub slipping by 30-40°, which throws the dwell timing off and tears product on the next station.
Key Components
- Driver Elliptical Gear: Mounted on the input shaft at one focus of the ellipse, not the geometric centre. The offset between focus and centre — equal to a × e where a is the semi-major axis — is what produces the radius modulation. Eccentricity typically sits between 0.2 and 0.4 for production work.
- Driven Elliptical Gear: An identical ellipse to the driver, mounted at one focus on the output shaft. The two gears must be cut from the same tooling pass to guarantee the pitch curves match within 0.02 mm — any deviation causes uneven mesh and audible knocking.
- Centre-Distance Frame: Holds the two shafts at a fixed centre distance equal to 2a (the full major axis). Frame deflection above 0.1 mm under load will cause backlash spikes at the apex points.
- Phasing Pin or Keyway: Locks the angular orientation of each gear to its shaft. The matched pair must be timed so that the major axis of one aligns with the minor axis of the other at the start position. A 5° phasing error visibly shifts the dwell window.
- Output Shaft and Bearing Pair: Carries the cyclic torque load, which peaks at the slow phase. Bearings should be sized for the peak instantaneous torque, not the average — a common sizing mistake gives bearings that fail at 30-40% of expected L10 life.
Who Uses the Elliptical Spur Gear
Elliptical spur gears show up wherever a designer needs a cheap, repeatable, mechanically-timed irregular motion off a constant-speed motor. They are common in packaging, printing, textile, and quick-return machine tool drives. Anywhere a cam-and-follower would work but you want a sealed, lubricated, lower-maintenance solution, an Elliptical Spur Gear pair earns its place.
- Offset Printing: Heidelberg KORD and GTO swinging gripper drives use elliptical spur gears to accelerate the gripper bar smoothly through the sheet transfer zone, then dwell while the sheet is clamped.
- Textile Machinery: Sulzer projectile looms use Fast and Slow Motion Spur Gear pairs in the picking-cam drive to give the projectile a hard launch acceleration followed by a controlled deceleration.
- Packaging: Bosch Doboy horizontal flow-wrappers use elliptical gear wheels in the cross-seal jaw drive, slowing the jaws during the seal-and-cut window so dwell time matches film thickness.
- Quick-Return Machine Tools: Older shaping machines like the Cincinnati Hypro shaper use elliptical spur gears as an alternative to the Whitworth linkage for the cutting-stroke drive — slow on the cut, fast on the return.
- Rotary Indexing: Some low-cost label-applicator indexers from Krones and Sacmi use Irregular circular motion by elliptical spur gear stages to time the application window without a separate cam shaft.
- Mechanical Watches and Clocks: Specialty clock movements occasionally use small elliptical gear pairs to drive equation-of-time displays, where the cyclic ratio matches the sun's apparent motion through the year.
The Formula Behind the Elliptical Spur Gear
The instantaneous gear ratio of a matched elliptical pair varies with the input angle θ. At the design sweet spot — eccentricity around e = 0.3 — you get a roughly 1.86:1 swing between fastest and slowest output speed, which is enough to drive most packaging dwell cycles cleanly. Push e to 0.45 and the ratio swing climbs above 2.6:1 but the contact ratio at the minor axis drops below 1.1, so you start hearing tooth knock under load. Drop e to 0.15 and the ratio swing falls under 1.35:1 — barely worth the cost of cutting non-circular teeth. Most production designs land in the 0.25-0.35 eccentricity band.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| i(θ) | Instantaneous gear ratio at input angle θ | dimensionless | dimensionless |
| ωin | Input angular velocity (constant) | rad/s | rev/min |
| ωout | Output angular velocity (varies with θ) | rad/s | rev/min |
| e | Eccentricity of the elliptical pitch curve | dimensionless | dimensionless |
| θ | Input shaft angle measured from major axis alignment | rad | deg |
Worked Example: Elliptical Spur Gear in a corrugated-box rotary die-cutter feed
You are sizing the elliptical spur gear pair for the sheet-feed drive on a Bobst Mastercut 106 PER rotary die-cutter retrofit. The input shaft runs at a steady 90 RPM off the main line drive. The feed needs to slow down during sheet pickup so the suction cups have 0.12 s of near-stationary contact, then accelerate to clear the sheet into the die-cut nip. You picked an eccentricity of e = 0.30 for the matched pair. You need to know the slow-phase output speed, the fast-phase output speed, and whether the dwell window is long enough.
Given
- ωin = 90 RPM
- e = 0.30 dimensionless
- Required dwell time = 0.12 s
Solution
Step 1 — at nominal eccentricity e = 0.30, calculate the slow-phase ratio at θ = 180° (cos θ = -1), where the output runs slowest:
Step 2 — calculate the fast-phase ratio at θ = 0° (cos θ = +1), where output runs fastest:
Step 3 — convert to actual output speeds at the nominal 90 RPM input:
ωout,fast = 1.857 × 90 = 167.1 RPM
Step 4 — at the low end of the typical operating range, e = 0.15, the slow-phase ratio climbs to 0.74 and fast-phase only reaches 1.36 — output sweeps from 66 RPM to 122 RPM. The dwell is too brisk; the suction cups will release the sheet before pickup completes. At the high end of the typical range, e = 0.45, slow-phase ratio drops to 0.38 (output 34 RPM) and fast-phase climbs to 2.51 (output 226 RPM). The dwell looks generous, but at 226 RPM peak the sheet whips into the die nip with enough overspeed that flag-out becomes likely above 18-point board.
Step 5 — the dwell window at e = 0.30 spans roughly 60° of input rotation around θ = 180° where output is within 10% of islow. At 90 RPM input, that's:
Result
The nominal output sweeps from 48. 4 RPM (slow phase) up to 167.1 RPM (fast phase) per input revolution, with a usable dwell window of 0.111 s. That dwell falls about 8% short of the 0.12 s the suction cups need — close enough that you would either bump eccentricity to 0.33 or slow the input to 85 RPM. At e = 0.15 the dwell barely exists and the cups would tear sheets; at e = 0.45 the dwell is comfortable but the fast-phase overspeed flags heavy board. If your measured slow-phase output reads 55 RPM instead of the predicted 48, suspect three things: (1) phasing pin slip on the driver hub letting the major axes drift toward parallel rather than perpendicular at start position, (2) centre distance set to 2a + 0.15 mm or more — a frame that has spread under load — which flattens the effective ratio swing, or (3) a mismatched gear pair where the two ellipses came from different cutting setups and the eccentricities don't agree to within 0.01.
When to Use a Elliptical Spur Gear and When Not To
Elliptical spur gears compete with cam-and-follower drives, Geneva drives, and modern servo-driven axes whenever cyclic non-uniform motion is needed. The Elliptical Gear Wheel approach has a specific niche — sealed, lubricated, mid-speed, repeatable — that the alternatives don't fully cover. Here's how the trade-offs land on the dimensions designers actually compare.
| Property | Elliptical Spur Gear | Cam and Follower | Servo-Driven Axis |
|---|---|---|---|
| Max practical input speed | 1500 RPM | 800 RPM (follower bounce) | 6000 RPM |
| Output motion programmability | Fixed by gear geometry | Fixed by cam profile | Fully programmable in software |
| Typical lifespan (hours) | 20,000-40,000 | 8,000-15,000 (cam wear) | 30,000+ (motor bearings) |
| Initial cost (matched pair / drive) | $400-1,200 (CNC ground pair) | $200-600 (cam + follower) | $2,500-8,000 (servo + drive + controller) |
| Maintenance interval | Lube every 2,000 hr | Inspect/replace every 1,000 hr | Encoder check yearly |
| Repeatability cycle-to-cycle | ±0.1° (mechanical) | ±0.3° (follower lift) | ±0.01° (closed loop) |
| Best application fit | Mid-speed cyclic dwell drives | Complex non-symmetric profiles | Variable recipe production |
Frequently Asked Questions About Elliptical Spur Gear
The formula assumes the gears are mounted at the focal points of their ellipses, not the geometric centres. A common build error is to bore the hub at the centre of the blank rather than offset by a×e. If your hub is centred geometrically instead of at the focus, you'll get smooth meshing but almost no ratio variation — the ellipses just rotate about their centres like circles with extra teeth.
Check by measuring the distance from the bore centre to the major-axis tip. It should equal a + a×e, not a. On a gear with semi-major axis 50 mm and e = 0.3, that's 65 mm, not 50 mm.
Yes — Elliptical Gear Wheel, Elliptical Spur Gear, Fast and Slow Motion Spur Gear, and Irregular circular motion by elliptical spur gear all describe the same mechanism. The naming varies by industry: print and packaging trades favour Elliptical Gear Wheel, machine-tool catalogues use Fast and Slow Motion Spur Gear, and academic kinematics texts use the longer descriptive form.
Geneva gives you true zero-velocity dwell — the output literally stops between index events. Elliptical gears give you a slow phase, not a stop. If your process tolerates the output creeping at 30-50% of average speed during the dwell window (most adhesive-application, suction-pickup, and seal-jaw closures do), the elliptical pair runs smoother and lasts longer.
If you need genuine motionless dwell — registration printing, hot-foil stamping, precision pick-and-place — Geneva or a servo wins. Rule of thumb: process tolerance window above 5% of cycle time, elliptical works fine. Below 2%, you need true dwell.
That's contact-ratio collapse at the minor-axis mesh point. At the minor axis, the local pitch radii are smallest and tooth engagement count is lowest. Under dynamic load, tooth deflection lets the leading tooth disengage before the next one picks up — a millisecond gap, then a hard re-engagement that rings through the frame.
Three fixes in order of cost: reduce eccentricity (recut at e = 0.25 instead of 0.35), increase face width by 30-40% to share load across more tooth length, or add a flywheel on the output shaft to absorb the inertia spike. The flywheel is the cheap field fix.
No. The pitch curves must satisfy the rolling-without-slipping condition at every angle, which only works when both pitch curves are conjugate ellipses with shared focal geometry. A circular driver against an elliptical driven gear would either bind or skip teeth — there's no centre distance that keeps them in contact through a full revolution.
If you need asymmetric speed profiles, look at lobed non-circular gears or eccentric circular gears (which are circular gears mounted off-centre — different mechanism, similar effect, but limited in how much variation you can achieve before contact ratio collapses).
Yes, and this catches people out. Power is conserved across the mesh, so when output speed drops to 0.54× input, output torque rises to 1.86× input torque. If your motor delivers 10 N·m steady, the output shaft sees 18.6 N·m peak at the slow phase. Bearings, keyways, and downstream couplings must be sized for that peak, not the average.
A common failure: a designer sizes the output coupling for nominal torque and it shears within the first week of production. Always spec the output drivetrain for ifast/islow × nominal torque, with a 1.5× safety factor on top.
Tighter than a circular gear pair of equivalent module. For a 100 mm major-axis pair at module 2, hold centre distance to ±0.03 mm. The reason: centre-distance error doesn't just shift backlash uniformly the way it does on circular gears — on elliptical gears it modulates backlash through the cycle, with peaks at the apex angles. A 0.1 mm offset that would be invisible on a circular pair becomes audible knock at the major-axis crossings on an elliptical pair.
Use a precision-bored frame, not slotted holes. If you need adjustability for setup, use eccentric bushings on one shaft and lock them after dial-indicating the mesh.
References & Further Reading
- Wikipedia contributors. Non-circular gear. Wikipedia
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