Variable circular motion is rotary motion in which the output shaft's angular velocity changes through each revolution while the input shaft turns at a constant speed. The principle relies on a non-constant instantaneous gear ratio — usually generated by non-circular gears, eccentric mountings, or elliptical pitch curves — so the same input rev produces fast and slow output phases within one cycle. Designers use it to create dwell, rapid-return, or shock-free indexing without clutches or cams. You see it inside packaging cutoff drums, printing press inkers, and quick-return shapers.
Variable Circular Motion Interactive Calculator
Vary input speed and instantaneous pitch radii to see the output speed, ratio, pitch-line velocity, and fast/slow phase of an elliptical gear pair.
Equation Used
The mesh point has the same tangential pitch-line velocity on both gears. When the input pitch radius is larger than the output pitch radius, the output speeds up; when it is smaller, the output slows down.
- Rolling pitch contact with no slip at the mesh point.
- Instantaneous radii are measured from each shaft center to the contact point.
- Angular speeds use the same rotational units, so rpm can be used directly in the ratio.
- Positive speed ratio magnitude is shown; meshing gears rotate in opposite directions physically.
Inside the Variable Circular Motion
Take a pair of meshing gears. If both pitch curves are circles concentric with their shafts, the ratio is fixed and the output turns at a steady speed. Now swap one or both for an elliptical pitch curve, or shift a circular gear's mounting hole off-centre, and the rolling contact point moves closer to and further from each rotation axis as the pair turns. The instantaneous gear ratio becomes a function of input angle, so the output shaft accelerates and decelerates inside every single revolution — that is variable circular motion in one sentence.
The physics is conservation of pitch-line velocity at the mesh point. At any instant, ωin × rin(θ) = ωout × rout(θ), where the radii vary with rotation angle θ. For a true 1:1 elliptical pair the centre distance must equal the sum of the semi-major axes, and the gears must be mounted with foci aligned to the shaft centres — get this wrong by even 0.2 mm on a 100 mm pitch diameter pair and the teeth either bind at the close phase or separate at the open phase. We have seen rebuilt corrugator wipe-roll drives chatter audibly because someone shimmed the centre distance to a nominal circular spec instead of the elliptical-pair spec.
Tooth profile is the second place builders trip up. On non-circular gears you cannot just stamp involute teeth from a standard hob — the pressure angle changes around the pitch curve, so each tooth needs its profile generated against the local instantaneous radius. Modern wire-EDM and 5-axis cutting handle this fine, but vintage rebuilds where someone re-cut a damaged elliptical with a stock involute cutter will run hot, lose contact ratio at the major-axis position, and pit the flanks within a few hundred hours.
Key Components
- Non-Circular Pitch Gears: A matched pair (often elliptical, lobed, or eggshell-curve) whose rolling pitch curves replace the circles of a normal gear set. The pitch curve is mathematically derived so that the pair stays in continuous rolling contact — typical pitch-curve dimensional tolerance is ±0.05 mm to keep contact stress uniform around the cycle.
- Eccentric Hub or Bushing: On simpler builds, a circular gear is mounted with its bore offset from the geometric centre by a few millimetres. The eccentricity e sets the velocity ratio swing — a 5 mm offset on a 60 mm pitch radius gives roughly ±8% velocity variation at the output.
- Aligned Centre Distance Fixture: Centre-to-centre spacing must match the pitch-curve sum at every angle, not just nominal. For an elliptical 1:1 pair with semi-major a and semi-minor b, centre distance C = 2a exactly. Drift more than 0.5% and you get backlash spikes at the minor-axis crossover.
- Output Shaft and Bearing Pair: The output bearings see a cyclic radial load that swings with the variable torque, so size them for the peak instantaneous load not the average. We typically spec deep-groove ball bearings one size up from a constant-speed equivalent drive.
- Phasing Key or Timing Mark: Both gears must be assembled with their major axes in the correct relative orientation — usually 90° offset for a 1:1 elliptical pair so one gear's fast phase matches the other's slow phase. A keyway or scribed timing mark on the hub face prevents 90° assembly errors that swap the dwell and rise phases.
Real-World Applications of the Variable Circular Motion
Variable circular motion shows up wherever a process needs the output to slow down or dwell during a working stroke and snap through during the return — without the cost and noise of a cam or the lockup of a Geneva drive. The mechanism is purely rolling, so it runs quieter and at higher RPM than equivalent intermittent drives, and it never has to engage and disengage. That is why packaging, printing, and textile machinery have used it for over a century.
- Web Converting: Cutoff knife drums on Paper Converting Machine Company (PCMC) tissue rewinders, where elliptical gears slow the knife at the cut point so the blade matches paper-web speed only during the slice.
- Printing: Oscillating ink-distributor rollers on Heidelberg Speedmaster offset presses, driven by a small eccentric gear pair to vary roller traverse speed across the form.
- Textile Machinery: Picking-stick drives on Dornier rapier looms, which use non-circular gears to accelerate the rapier through the shed and decelerate it before the catch.
- Packaging: Variable-speed wrap-feed drums on Bosch Sigpack horizontal flow-wrappers, which slow the film during the seal phase and speed up during register pickup.
- Machine Tools: Quick-return shaper rams on classic Cincinnati and Atlas shapers — the cutting stroke runs slow for chip load, the return stroke runs fast to save cycle time.
- Assembly Automation: Pick-and-place rotary indexers using elliptical-gear inputs to dwell at each station for part transfer, then accelerate between stations without a Geneva's tooth shock.
The Formula Behind the Variable Circular Motion
What you actually want to know is the instantaneous output speed at any input angle, because that tells you whether the slow phase is slow enough to match a process (a cut, a seal, a pick) and whether the fast phase is fast enough to recover the cycle. At the low end of the typical eccentricity range — say e/r = 0.05 — the velocity ratio swings only ±5%, which is barely noticeable and probably not worth the part complexity. At the high end — e/r approaching 0.3 — the ratio swings nearly 2:1 between fast and slow phase, which gives you real dwell behaviour but also drives up tooth contact stress at the close-phase. The sweet spot for most packaging and printing work sits around e/r = 0.10 to 0.15.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| ωout(θ) | Instantaneous output angular velocity at input angle θ | rad/s | RPM |
| ωin | Input angular velocity (constant) | rad/s | RPM |
| rin(θ) | Input pitch radius at angle θ measured from input centre | mm | in |
| rout(θ) | Output pitch radius at the conjugate angle on the output gear | mm | in |
| e | Eccentricity (offset of bore from geometric centre, for eccentric-mounted variant) | mm | in |
Worked Example: Variable Circular Motion in a glass-bottle labeller wrap drum
You are sizing the elliptical gear pair driving the wrap drum on a Krones Autocol roll-fed labeller running 24,000 bottles per hour onto 330 ml long-neck beer bottles. Input shaft turns at 200 RPM constant from the main line shaft. The drum must slow during the 90° label-application arc to match bottle surface speed within ±2%, then speed up through the 270° non-contact arc to keep the cycle. You have specified a 1:1 elliptical pair with semi-major a = 60 mm and semi-minor b = 48 mm, mounted with major axes 90° offset.
Given
- ωin = 200 RPM
- a (semi-major) = 60 mm
- b (semi-minor) = 48 mm
- C (centre distance) = 120 mm
- θ at label-apply midpoint = 0 ° (major axis aligned)
Solution
Step 1 — at the label-application midpoint, the input gear's major axis points along the centre line, so rin = a = 60 mm and rout = b = 48 mm (output gear major axis is perpendicular). Compute output speed at this slow-phase point:
Step 2 — at the fast-phase point, 90° later, the geometry inverts: rin = 60, rout = 48:
Step 3 — nominal mean output speed, averaged over a full revolution, equals the input by conservation (1:1 pair):
At the low end of the typical eccentricity ratio (b/a = 0.95, mild ellipse) the swing would be only 200 ±5 RPM — not enough velocity contrast to do useful work in a labeller. At the high end (b/a = 0.7, aggressive ellipse) you would get 140 RPM slow phase and 286 RPM fast phase, but tooth contact stress at the close-phase position rises sharply and you start chewing flanks within a few thousand cycles. Our specified b/a = 0.8 sits in the sweet spot — 25% velocity drop during apply, manageable contact stress, and the tooth count works out to a clean integer at module 2.
Result
Output swings from 160 RPM during the label-application arc up to 250 RPM during the recovery arc, with a mean of 200 RPM matching the input. That 160 RPM slow phase is what gives you the bottle-matching surface speed during apply — fast enough to keep up with line throughput, slow enough that the label glue tack window stays in spec. Compared to the mild b/a = 0.95 case (only ±5% swing, useless here) and the aggressive b/a = 0.7 case (massive swing but flank pitting in months), the b/a = 0.8 design is the right balance for this duty. If you measure 170 RPM slow phase instead of 160, check the timing mark phasing first — a 5° assembly error on the major-axis offset shifts the velocity profile and shows up exactly like that. If the slow phase reads correctly but you see 240 RPM at the fast-phase peak instead of 250, suspect centre-distance creep from a worn bearing housing letting C drift above 120 mm; a 0.3 mm gap there flattens the ratio swing measurably. And if the drum rumbles audibly at the major-axis crossover, your tooth profile was probably cut with a standard involute hob instead of the conjugate profile the elliptical pair requires.
Variable Circular Motion vs Alternatives
Variable circular motion is one of three common ways to give a rotating shaft non-uniform output. The other two are Geneva drives (intermittent, fully indexing) and cam-driven oscillators (full motion-profile freedom). Each has a different sweet spot, and the choice depends on what you actually need from the output curve.
| Property | Variable Circular Motion (non-circular gears) | Geneva Drive | Cam-Driven Oscillator |
|---|---|---|---|
| Maximum continuous RPM | 800-1500 RPM (smooth rolling contact) | 60-200 RPM (limited by tooth-shock) | 300-600 RPM (cam follower bounce) |
| Output motion type | Continuous rotation, varying speed | Intermittent index-then-dwell | Any programmed profile (rise-dwell-fall) |
| Velocity ratio swing | Typically 1.2:1 to 2:1 | 0 to ∞ (full stop dwell) | Unlimited (cam-defined) |
| Manufacturing cost | High — wire-EDM or 5-axis cut required | Low — standard cylindrical machining | Medium — cam grinding plus follower assembly |
| Backlash and shock at transition | Smooth — pure rolling, no impact | High — engagement shock at every index | Low — depends on follower preload |
| Service life at rated load | 10,000+ hours with conjugate teeth | 5,000-8,000 hours, pin wear limited | 8,000-12,000 hours, follower-roller limited |
| Best application fit | Continuous-cycle packaging, printing, textile | Rotary turret indexers, dial-feed assembly | Engine valves, custom motion profiles |
Frequently Asked Questions About Variable Circular Motion
You are almost certainly running a centre distance set for the nominal pitch curve average rather than the elliptical sum. For a 1:1 elliptical pair the centre distance C must equal 2a exactly, where a is the semi-major axis. If you set C using the average pitch radius (a+b)/2 by mistake, the gears will mesh fine at the 45° crossover position but bind savagely at the major-axis aligned position, because the actual pitch curves want more clearance there.
Quick diagnostic: rotate the pair by hand with no load. If torque to turn spikes every 180°, your centre distance is short. If the gears slap and lose contact every 180°, it is long. Re-machine the housing or shim the bearing pedestal until the spike disappears.
Yes, and it is a common shortcut for low-end velocity swing — up to about ±10% — but with two real penalties. First, the centre distance varies through the rotation because the eccentric gear's effective pitch radius shifts, so you must allow either a spring-loaded idler or a slotted mounting that tracks the change, otherwise tooth contact alternates between binding and separating. Second, the output velocity profile is not a clean sinusoid — it has a small but real second-harmonic content that excites resonance in long output shafts.
For anything above 10% swing or above 300 RPM, cut proper conjugate non-circular gears. The eccentric trick is fine for slow agitator drives and demonstration models.
Count the velocity peaks per revolution that your process needs. An elliptical 1:1 pair gives you two slow phases and two fast phases per output revolution — perfect for a two-station operation like a dual-head labeller or a twin-knife cutoff. A three-lobe set gives three slow-fast cycles per rev, suited for a three-up packaging drum or a triple-pocket indexer.
Rule of thumb: match the lobe count to the number of working positions on your output drum. Mismatch and you either get processes happening during the fast phase (bad) or you have to add a reduction stage that defeats the point of the variable drive.
Three common causes, in order of likelihood. First, torsional windup in the output shaft — at the fast phase the output is accelerating hardest and a long or undersized shaft twists, smearing the peak. Check shaft polar moment of inertia and length; if J×L is high you will see exactly this signature. Second, drag from a downstream load that is itself a varying torque (web tension, label gum, etc.) — the load fights the acceleration. Third, encoder sampling rate — if you are measuring with a 1024-line encoder at 1 kHz update on a 250 RPM shaft, you are aliasing the peak.
Fix the measurement first by upping encoder resolution or sample rate, then look at shaft stiffness, then load.
Worse than a standard gear pair would. The instantaneous tooth contact in a non-circular gear changes pressure angle continuously, and the tooth in mesh at the moment of jam might be loaded near its weakest geometry — close to the major-axis crossover where the local pressure angle is steepest. We have seen elliptical gears shear two or three teeth from a single jam event that a standard spur pair would have shrugged off.
Always fit a torque limiter or shear pin upstream of the non-circular pair. A Mayr EAS-Compact slip clutch sized for 1.5× peak running torque is the standard fix for packaging applications.
Extremely sensitive — that is the single most common build error on rebuilds. For a 1:1 elliptical pair the major axes must be 90° offset. A 5° phasing error shifts the slow-phase angular position by 5° and skews the velocity curve asymmetrically, so the deceleration ramp into the slow phase no longer matches the acceleration ramp out of it. In a labeller this shows up as the front edge of the label landing fine but the trailing edge wrinkling.
Always assemble with the timing marks both pointing radially outward at the same instant, then rotate one gear exactly 90° before engaging the mesh. Mark with a punch, not a Sharpie — Sharpie wears off the first time someone wipes the gearbox down.
References & Further Reading
- Wikipedia contributors. Non-circular gear. Wikipedia
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