Variable rotary motion via an elliptical gear and slotted bar is a drive that pairs a non-circular (elliptical) gear with a follower carrying a slotted bar, producing an output shaft whose angular velocity rises and falls cyclically within each revolution. Unlike a constant-ratio spur pair, the instantaneous gear ratio changes with rotation angle because the pitch curve radius varies. We use it where a machine needs a slow working stroke and a fast return inside a single shaft revolution — common on shapers, textile take-up drives, and packaging cam-replacement retrofits running 60–300 RPM.
Variable Rotary Motion via Elliptical Gear and Slotted Bar Interactive Calculator
Vary ellipse size, eccentricity, input angle, speed, and crank radius to see changing output speed and slide position.
Equation Used
The calculator uses the article geometry constraint C = 2a and a focus-based elliptical gear phase model. Increasing eccentricity widens the cyclic speed swing, while the slotted bar converts the variable follower rotation into reciprocating slide motion.
- Matched elliptical pitch curves roll without slip.
- Shafts are located at the ellipse foci.
- Center distance is held at 2a.
- Slotted bar is modeled as an ideal Scotch-yoke slide.
How the Variable Rotary Motion via Elliptical Gear and Slotted Bar Actually Works
The driver is an elliptical gear cut on a true elliptical pitch curve, with the input shaft fixed at one focus of the ellipse — not at the geometric centre. As the driver rotates, the rolling-contact radius between driver and follower changes continuously, so the instantaneous gear ratio swings between a minimum and a maximum twice per revolution. The follower carries (or directly is) a slotted bar that rides on a crank pin, converting the non-uniform rotation into a reciprocating linear stroke with a built-in quick-return ratio. You get slow on the working stroke, fast on the return, all from one constant-speed input.
Geometry is unforgiving here. The two pitch curves must roll without sliding, which means the sum of the radii from each shaft centre to the contact point must equal the fixed centre distance at every angle — that's the design constraint that forces the follower to also be elliptical (or a matched non-circular shape) and forces the centre distance to equal 2a, twice the semi-major axis. Miss the centre distance by even 0.2 mm on a 100 mm gear and you get backlash on one side of the cycle and binding on the other. The teeth themselves are normal involute teeth, but each tooth is cut at the local pitch-curve normal, so a hobbed elliptical gear needs a CNC gear shaper or a wire-EDM finishing pass — you cannot cut these on a manual hob.
Failure modes cluster around three things. Loose centre distance causes tooth tip interference at the minor axis and skipping at the major axis. Slot wear in the slotted bar opens up the linear-stroke geometry and the quick-return ratio drifts — a 0.3 mm slot oversize on a 40 mm crank radius shifts the dwell timing by roughly 2°. And shaft alignment off the focus point by even 1% of the semi-major axis turns the smooth velocity curve into a lumpy one with a measurable second harmonic, which you'll hear as a beat at twice running speed.
Key Components
- Elliptical Driver Gear: Cut on a true elliptical pitch curve with the input shaft mounted through one focus, not the centre. The semi-major axis a and semi-minor axis b set the ratio swing — a 2:1 max/min ratio needs eccentricity e ≈ 0.6, which means b/a ≈ 0.8. Tooth count is typically 40–80 for smooth meshing.
- Mating Follower Gear: Identical elliptical pitch curve mounted at its own focus, with the centre distance fixed at 2a. The two ellipses roll on each other in pure rolling contact, so any centre-distance error directly converts to backlash on one half-cycle and binding on the other.
- Slotted Bar (Scotch-Yoke Style Output): Rides on a crank pin driven by the follower shaft. The slot must be hardened to 58 HRC minimum and ground to within 0.05 mm parallelism over its length, otherwise the variable-velocity output picks up secondary noise from slot slop. Slot length is sized to the crank radius plus 5 mm clearance.
- Crank Pin: A hardened ground pin pressed into the follower hub, typically 12–25 mm diameter for industrial sizes. Surface finish must be Ra 0.4 µm or better — anything rougher accelerates slot wear and the quick-return ratio drifts within a few hundred hours of running.
- Centre-Distance Mounting Plate: Holds both shafts at exactly 2a apart. Use dowelled mounting, not slotted holes — the constraint is geometric, not adjustable. We spec a hardened steel plate ground flat to 0.02 mm to keep the two pitch curves rolling cleanly through the whole revolution.
Industries That Rely on the Variable Rotary Motion via Elliptical Gear and Slotted Bar
You see this mechanism wherever a designer needs cyclic speed variation from a constant-speed input shaft, without resorting to a servo or a cam. The cost trade is favourable when the cycle pattern is fixed and the machine runs millions of cycles — the gear pair will outlive the rest of the machine. It shows up in textile, packaging, printing, and metalworking machinery, usually as a quick-return drive or as a non-uniform feed drive where the working stroke needs more dwell time than the return.
- Metalworking: Quick-return ram drive on a Cincinnati 24-inch shaper, where the cutting stroke runs at roughly 60% of the cycle and the return stroke at 40%, giving the tool more time under chip load.
- Textile machinery: Non-uniform take-up drive on a Saurer Allma TC2 cabling machine, where yarn package build needs slow traverse at the package ends and fast traverse across the middle.
- Printing: Sheet-feeder cam-replacement retrofit on Heidelberg Speedmaster SM 74 grippers, used to soften the gripper-open velocity profile at sheet release.
- Packaging: Variable-pitch indexing on Bosch SVE 2520 horizontal cartoners, where the pusher needs a long dwell at insertion and a fast retract.
- Agricultural equipment: Non-uniform feed roll drive on a Krone BiG Pack 1290 baler tucker arm, providing a slower compression phase and a faster recovery phase.
- Pumps: Variable-flow piston metering pumps where the elliptical pair gives a near-flat dispense stroke and a fast suction stroke from a single-speed motor.
The Formula Behind the Variable Rotary Motion via Elliptical Gear and Slotted Bar
The instantaneous gear ratio is what you actually design around — it tells you the output shaft's angular velocity at any input angle, and it sets the quick-return ratio that determines how skewed the working stroke is versus the return. At low eccentricity (e ≈ 0.2) the ratio swing is mild and the output is barely distinguishable from a constant-speed shaft. At nominal eccentricity (e ≈ 0.5) you get a usable 3:1 max-to-min ratio swing, which covers most quick-return applications. Push the eccentricity above 0.7 and the ratio swing exceeds 5:1 but tooth pressure angle at the minor axis climbs past 30° and the gears start undercutting — you lose tooth strength fast.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| ωin | Constant input angular velocity at the driver focus | rad/s | rev/min (RPM) |
| ωout | Instantaneous output angular velocity at the follower focus | rad/s | rev/min (RPM) |
| e | Eccentricity of the elliptical pitch curve, e = √(1 − b²/a²) | dimensionless (0 to <1) | dimensionless |
| θ | Input shaft angle measured from the major axis | rad | degrees |
| a, b | Semi-major and semi-minor axes of the elliptical pitch curve | mm | in |
Worked Example: Variable Rotary Motion via Elliptical Gear and Slotted Bar in a corrugator glue-applicator drive retrofit
Sizing the elliptical gear and slotted-bar drive on the glue-applicator wipe roll of a BHS Corrugated single-facer rebuild, where the wipe roll must dwell slowly across the flute crests and snap back fast between contact strokes. The driver runs at 180 RPM steady from the line shaft, semi-major axis a = 50 mm so centre distance is 100 mm, and we want to evaluate three eccentricity choices to land the right quick-return feel.
Given
- ωin = 180 RPM
- a = 50 mm
- e (low / nominal / high) = 0.3 / 0.5 / 0.7 dimensionless
- θ at major axis (slow stroke) = 0 degrees
- θ at minor axis (fast stroke) = 180 degrees
Solution
Step 1 — at the nominal eccentricity e = 0.5, calculate the slow-stroke output speed when θ = 0° (driver at major axis, output at slowest):
Wait — that's the inverted case. With the input shaft on the focus closest to the contact at θ = 0°, the output runs fast. Flip the convention: at θ = 0° the output is slow, ratio is 1/3:
Step 2 — at θ = 180°, the fast stroke (return):
So at nominal e = 0.5 the wipe roll dwells at 60 RPM during the working stroke and snaps back at 540 RPM — a clean 9:1 swing, with the working stroke covering roughly 75% of the cycle time. That matches the corrugator wipe pattern well: long contact, fast lift.
Step 3 — at the low end of the range, e = 0.3:
At e = 0.3 the swing is only 3.4:1 and the dwell-vs-snap difference is mild — operators on the corrugator floor would describe it as "barely a quick-return," and you'd see glue smearing during the lift phase because the wipe roll doesn't pull away fast enough.
Step 4 — at the high end, e = 0.7:
Theoretically gorgeous — a 32:1 swing — but at e = 0.7 the pressure angle at the minor axis approaches 32° and the teeth start undercutting on a 40-tooth pinion. We'd see tooth tip chipping inside 200 hours on a hardened 8620 gear running 24/7, and the 1020 RPM peak on the slotted bar would slam the crank pin into slot endplay hard enough to ring the machine frame.
Result
At nominal e = 0. 5, the output cycles between 60 RPM (slow stroke) and 540 RPM (fast stroke) from a steady 180 RPM input — a 9:1 quick-return ratio that matches the corrugator wipe pattern. The low-end e = 0.3 case (97 / 334 RPM) feels too uniform and risks glue smearing; the high-end e = 0.7 case (32 / 1020 RPM) is mathematically clean but the teeth undercut and slot impact loads spike — pick e = 0.5 as the design point. If your measured slow-stroke speed is 80 RPM instead of 60, check three things: (1) centre-distance error — 0.3 mm long on a 100 mm centre flattens the ratio swing measurably, (2) the input shaft drilled through the geometric centre instead of the focus, which kills the asymmetry entirely, or (3) the follower gear cut from the wrong pitch-curve master — symmetric ovals are not ellipses and the math above does not apply.
Choosing the Variable Rotary Motion via Elliptical Gear and Slotted Bar: Pros and Cons
Variable rotary motion via elliptical gears competes with three other mechanisms that produce non-uniform output: scotch yokes with offset cranks, four-bar quick-return linkages (Whitworth), and servo-driven cam profiles. Each has a different sweet spot for speed, cost, and adjustability.
| Property | Elliptical Gear + Slotted Bar | Whitworth Quick-Return Linkage | Servo + Electronic Cam |
|---|---|---|---|
| Typical operating speed | 60–600 RPM | 30–200 RPM | 0–3000 RPM |
| Quick-return ratio range | 1.5:1 to 5:1 (limited by tooth strength) | 1.5:1 to 4:1 | Arbitrary, software-defined |
| Profile adjustability | Fixed at manufacture | Fixed at manufacture | Reprogrammable in seconds |
| Capital cost (typical 100 mm centre size) | $800–$2,500 for the gear pair | $300–$700 for the linkage | $4,000–$15,000 with drive and feedback |
| Lifespan at rated load | 20,000+ hours, gear-tooth limited | 10,000–15,000 hours, pin-joint wear | Servo motor 30,000 hours, drive electronics 50,000+ |
| Maintenance interval | Re-grease every 2,000 hours, inspect slot wear annually | Re-grease pin joints every 500 hours | Effectively zero on the mechanism, drive firmware updates only |
| Best application fit | High-cycle fixed-profile machines (textile, corrugator, shaper) | Low-cost low-speed shapers and slotters | Variable-recipe packaging and printing where profile changes per job |
| Mechanical complexity | Moderate — two custom gears, one slot | Low — four pin-jointed links | High — motor, drive, encoder, controller |
Frequently Asked Questions About Variable Rotary Motion via Elliptical Gear and Slotted Bar
The minor-axis crossing is where the instantaneous ratio swings fastest and the tooth-loading direction reverses. If your centre distance is even slightly long, backlash opens up at the minor axis but stays closed at the major axis — so you only hear the knock on that half-cycle.
Check centre distance with a feeler gauge against a hardened spacer block. Spec is 2a ± 0.05 mm on a 100 mm centre. If centre distance is correct, look at the tooth contact pattern with marking blue — knock at the minor axis with a centred contact pattern usually means the follower gear was cut from a slightly oversize pitch-curve master, and the only fix is replacement.
The decision comes down to how skewed you actually need the working stroke versus the return. e = 0.5 gives a 9:1 ratio swing and the working stroke takes about 75% of cycle time; e = 0.6 gives roughly 16:1 and pushes the working stroke to about 80% of cycle time, but doubles the peak velocity on the slotted bar.
Rule of thumb: if your slot-and-pin assembly is rated above 1000 RPM peak and the working stroke duty fraction matters more than peak load, go to e = 0.6. If you're at the limit of crank pin Hertzian contact stress, stay at e = 0.5 — the marginal duty-fraction gain is not worth the 40% peak velocity bump.
Three causes account for almost every case. First, slot oversize: a 0.5 mm slot wear on a 40 mm crank radius adds compliance that softens the velocity peaks and flattens the ratio. Measure slot width with a pin gauge — anything more than 0.1 mm over crank pin diameter and you've found it.
Second, drive-side coupling slop. A loose key or a worn jaw coupling between the gearmotor and the input shaft lets the input speed itself fluctuate, smearing the output curve. Put a tachometer on the input shaft and confirm it's actually constant.
Third — and most often missed — the input shaft was bored through the gear's geometric centre, not its focus. This is a manufacturing error that produces a perfectly running gear with the wrong velocity curve. There is no field fix; the gear has to be re-bored or replaced.
You can, and you should — but the oil level setting is different. With a constant centre distance and a varying contact radius, the meshing point sweeps up and down vertically through the cycle. Set the oil level to cover the lowest meshing position by at least 5 mm, not the lowest tooth tip of either gear at rest.
If you set the oil level by the resting tooth tip the way you would for a spur pair, the contact zone will run dry for half the cycle and you'll see micropitting on the major-axis flank within the first 1,000 hours.
That second harmonic is almost always shaft-eccentricity error. The formula assumes the input shaft passes exactly through one focus of the ellipse. If the bore is offset from the focus by even 1% of the semi-major axis — 0.5 mm on a 50 mm a — the velocity curve picks up a 2× harmonic with measurable amplitude.
Check focus location with a CMM or a precision indicator running on the gear's outer ellipse. If the focus offset is the cause, the bump amplitude scales linearly with the offset, so 0.2 mm of correction kills 40% of the harmonic content. There's no software fix on a mechanical drive — it's a remachining job.
Below about 150 RPM and at quick-return ratios under 3:1, a Whitworth linkage wins on cost — you're looking at $300–$700 in pin-jointed parts versus $1,500+ for a custom elliptical gear pair. The Whitworth also tolerates dirty environments better because it has no continuous tooth contact to protect.
The elliptical gear wins above 200 RPM, above 4:1 quick-return ratios, or in cycle counts above 10 million where pin-joint wear in the Whitworth becomes the limiting factor. We'd also pick the gear pair if vibration matters — pin-jointed linkages produce hammering at the dead-centre transitions that elliptical gears do not.
References & Further Reading
- Wikipedia contributors. Non-circular gear. Wikipedia
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