A Uniform Speed of Sectional Spur Gear is a spur gear whose teeth occupy only a fraction of its circumference, paired with a full pinion so the pinion rotates at constant angular velocity while teeth are engaged and dwells stationary while the toothless arc passes by. The constant velocity ratio during engagement comes from standard involute tooth action — same as any spur pair. The mechanism converts continuous input rotation into a controlled drive-and-dwell output without clutches or cams, used widely on indexing tables, label feeders, and printing-press transfer drums to deliver clean, repeatable advances at speeds up to 300 cycles per minute.
Uniform Speed of Sectional Spur Gear Interactive Calculator
Vary the toothed sector angle and gear ratio to see drive angle, dwell angle, and output index motion.
Equation Used
The toothed sector drives the pinion only over its active angle. During that window the pinion advance is the sector angle multiplied by the spur gear velocity ratio i. The remaining part of the driver revolution is dwell.
- One driver revolution equals one drive-dwell cycle.
- Velocity ratio is constant only during tooth engagement.
- Output advance is reported as magnitude; actual rotation is opposite the driver.
- Backlash, end-tooth relief, and locking clearance are not included.
How the Uniform Speed of Sectional Spur Gear Works
The sectional spur gear is a normal spur gear with most of its teeth removed. What remains is an active arc — a sector — that meshes with a full pinion for a defined portion of the input revolution. While teeth engage, the pinion turns at the constant ratio i = Ndriver / Npinion, exactly as it would in a continuous pair. When the toothless arc rotates past the pinion centre, the pinion stops. A locking surface — usually a circular boss machined on the same blank as the sector — holds the pinion against rotation during the dwell. The result is a dwell-and-drive cycle from a single shaft, no clutch, no cam follower, no Geneva-style impact load.
The geometry has to be right or the mechanism eats itself. The first and last teeth of the sector see the worst loads because they enter and exit mesh under the full inertia of the driven train. We chamfer or relieve those end teeth by 0.05 to 0.10 mm of profile relief — not more, or the velocity ratio breaks down inside the engagement window. The pinion's locking flat must clear the boss diameter by 0.05 mm minimum, otherwise you get rubbing during dwell and the pinion creeps. If you notice the output index drifting cycle to cycle, that creep is almost always the cause.
Timing is the other failure mode. The sector must enter mesh exactly when the pinion's locking flat aligns with the boss release point. Off by 1° and you get a hammer-strike on the leading tooth — visible as pitting after a few thousand cycles. Off by 3° and the leading tooth shears. Constant velocity ratio during engagement is what separates this from a Geneva drive: no acceleration spike, no jerk, just clean uniform angular velocity for the full sector arc.
Key Components
- Sectional driver gear: A spur gear with teeth machined over only a defined arc — typically 90° to 270° depending on the dwell ratio required. The tooth profile is standard involute, module usually 1 to 4 mm for industrial applications, with end teeth relieved 0.05 to 0.10 mm to soften entry and exit shock.
- Full pinion: A standard spur pinion that meshes with the sector during the active arc. Tooth count sets the ratio — a 20-tooth pinion driving a 60-tooth sector gives 3:1 reduction during engagement. The pinion bore tolerance must be H7 to keep backlash under 0.03 mm at module 2.
- Locking boss: A circular surface on the driver, concentric with its axis, that fills the gap left by the missing teeth. It rides against a matching locking flat on the pinion to hold the output stationary during dwell. Clearance is held to 0.05 to 0.10 mm — tight enough to prevent creep, loose enough to avoid rubbing drag.
- Pinion locking flat: A flat or shallow concave surface ground into the pinion blank at the pitch line of the gap. Mates with the boss during dwell. The flat's angular extent must equal the dwell arc minus the engagement transition window, typically 2° to 5° shy on each side.
- Mounting plate and shaft bearings: Fixes the centre distance to within ±0.02 mm at module 2. Loose centre distance is the number-one cause of premature tooth wear in sectional gear pairs because backlash varies through the arc as the sector flexes under load.
Where the Uniform Speed of Sectional Spur Gear Is Used
You see sectional spur gears wherever a machine needs to index, dwell, then index again from a constantly running input shaft. Anywhere a Geneva drive would impose harsh acceleration, or a cam-and-follower would need its own dedicated drive, the sectional spur pair offers a quieter, lower-shock alternative with constant velocity ratio during the active stroke.
- Packaging machinery: Bottle-indexing star wheels on a Krones Sensometic filler — the sectional gear advances each pocket exactly 30° per cycle then dwells while the filling head descends.
- Printing: Sheet-transfer drums on a Heidelberg Speedmaster XL 106, where the gripper bar must accelerate to web speed, hold velocity through the transfer, then dwell for the next sheet pickup.
- Textile: Pattern-chain drive on a Stäubli dobby head where the heald frames step through the pattern sequence at constant angular velocity during the lift phase.
- Watchmaking and timing: Date-disc advance on automatic calendar movements similar to the ETA 2824-2 calibre — a sector pinion drives the date wheel one tooth per 24-hour cycle.
- Automated assembly: Rotary index tables on a Mikron NRG-50 transfer machine, where 8 stations dwell for 1.2 seconds of machining then index 45° in 0.4 seconds.
- Food processing: Wrapper-feed roll on a Theegarten-Pactec EK4 chocolate wrapping machine, advancing the foil one pitch per cycle and holding it dead still while the fold heads close.
The Formula Behind the Uniform Speed of Sectional Spur Gear
The core calculation tells you how far the output pinion rotates during one engagement, and at what angular velocity. The driver runs continuously at ωin, but the pinion only sees rotation while sector teeth are in mesh. At the low end of the typical sector arc — around 60° of driver rotation — you get short crisp indexes with long dwells, good for slow assembly stations. At the high end — 300° of arc — the dwell becomes the minority of the cycle and the mechanism behaves almost like a continuous pair with brief stops. The sweet spot for most packaging and printing applications sits between 120° and 180° of active arc, balancing index speed against available dwell time for the secondary process.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| θout | Pinion rotation per engagement cycle | rad or ° | ° or rev |
| θarc | Angular extent of the toothed sector on the driver | rad or ° | ° or rev |
| Ndriver | Equivalent full tooth count of the driver gear (as if fully toothed) | teeth | teeth |
| Npinion | Tooth count of the mating pinion | teeth | teeth |
| ωin | Driver angular velocity (constant) | rad/s | RPM |
| ωout | Pinion angular velocity during engagement | rad/s | RPM |
Worked Example: Uniform Speed of Sectional Spur Gear in a pharmaceutical blister-pack indexer
Sizing the sectional spur drive on the forming-station indexer of an Uhlmann UPS4 MT blister-pack line. The driver shaft runs continuously at 90 RPM off the main camshaft. The forming station needs to advance the foil web by exactly 60° of pinion rotation per cycle, then dwell for the heat-seal stroke. The driver is a 72-tooth equivalent gear with a toothed arc of θarc = 120°, mating with a 24-tooth pinion on the foil-feed shaft.
Given
- Ndriver = 72 teeth
- Npinion = 24 teeth
- θarc = 120 °
- ωin = 90 RPM
Solution
Step 1 — compute the gear ratio during engagement:
Step 2 — at nominal 120° sector arc, compute pinion rotation per cycle:
That is one full revolution of the pinion per cycle — too much. The web would over-feed by 6×. Reset: the design intent is θout = 60°, which means we need a 1:3 step-up at the pinion or the sector arc has to shrink. With i fixed at 3.0 (driver larger than pinion is wrong for this application — flip the pair so pinion drives sector, or invert the ratio). Treating the sector as the driven side instead, iused = 24 / 72 = 0.333, so:
Step 3 — compute pinion angular velocity during engagement at nominal 90 RPM input:
At the low end of the typical line speed — 45 RPM input — the pinion sees ωout,low = 0.333 × 45 = 15 RPM during engagement. The web creeps cleanly through the forming station with plenty of dwell margin, ideal for thicker laminates that need extended seal time. At the high end of the rated range — 180 RPM input — ωout,high = 60 RPM during engagement. In theory fine, but in practice the foil web develops snap-back oscillation above roughly 150 RPM input on this size of indexer because the engagement transition imposes a 0.04 second rise time the web cannot follow without slack. Most Uhlmann UPS4 lines settle at 90 to 120 RPM input for that reason.
Result
Nominal pinion rotation per cycle is 60° at an engagement velocity of 30 RPM, with the sector arc set to 180°. The 60° advance feels like a crisp, snappy index when you watch the line — the foil jumps one pitch then sits dead still for the seal stroke. Across the typical operating range, 45 RPM input gives a slow 15 RPM engagement that handles thick laminates well, while 180 RPM input pushes engagement to 60 RPM and the web starts to oscillate at the transition. If you measure pinion rotation at 55° instead of the predicted 60°, check first for centre-distance error above 0.05 mm allowing the sector to skip the leading tooth on entry, then for a worn locking boss letting the pinion creep backward 2° to 5° during dwell, and finally for an out-of-time keyway on the driver shaft shifting engagement onset by a fraction of a degree per cycle.
When to Use a Uniform Speed of Sectional Spur Gear and When Not To
The sectional spur gear competes with several other intermittent-motion mechanisms. Each has a different sweet spot in terms of speed, shock load, and accuracy. Pick based on cycle rate and how much abuse the driven load can absorb at the index transitions.
| Property | Sectional Spur Gear | Geneva Drive | Cam-and-Follower Indexer |
|---|---|---|---|
| Max practical cycle rate | Up to ~300 cpm | Up to ~500 cpm in 4-slot, lower with more slots | Up to ~1500 cpm with rolling-element follower |
| Output velocity profile during index | Constant angular velocity (uniform) | Sinusoidal — peaks mid-stroke | Fully programmable via cam profile |
| Peak acceleration / shock at transitions | Low — softened by tooth relief, ~2 to 5 g | High — finite at slot entry, 10 to 30 g | Lowest — modified-sine cams hold under 5 g |
| Indexing accuracy | ±0.05° with H7 bores and tight centre distance | ±0.1° limited by slot clearance | ±0.01° on precision globoidal cams |
| Cost to manufacture (single set) | Low — standard gear cutting | Medium — slotted plate plus driver pin | High — ground cam plus follower assembly |
| Lifespan at rated load | 20,000 to 50,000 hours with proper relief | 10,000 to 20,000 hours — pin-slot wear dominant | 30,000 to 80,000 hours on hardened cams |
| Best application fit | Continuous indexing under moderate load with constant velocity needed | Simple low-cost intermittent drives, station counts of 3 to 8 | High-speed precision indexing where dwell and motion profiles must be tuned |
Frequently Asked Questions About Uniform Speed of Sectional Spur Gear
Chatter on entry almost always points to insufficient end-tooth relief, not the main profile. The first tooth of the sector enters mesh at finite angular velocity while the pinion is stationary — that velocity step has to be absorbed somewhere. Without 0.05 to 0.10 mm of profile relief on the leading tooth, the impact transfers straight into the bearings and you hear it as chatter.
Check the leading tooth tip with a profile tracer. If you see a sharp corner instead of a gentle taper over the top 30% of the tooth height, that is your problem. A second cause is timing of the locking-boss release — if the boss releases the pinion 1° to 2° after the leading tooth lands, the tooth carries the entire load alone for that interval and rings.
Put the sector on whichever shaft runs continuously. The full pinion needs to start, run at constant velocity, then dwell — so it goes on the intermittent output. The toothed-and-toothless sector goes on the input because it can spin through the toothless arc without consequence.
If you put the sector on the output you create a problem: when the input pinion spins past the toothless gap, nothing supports the output and external loads can back-drive it freely. Always sector on input, full pinion on output. The locking boss-and-flat then does its job constraining the output during dwell.
Drift over many cycles, not single-cycle error, is almost always the locking interface — not the gear teeth. The boss-to-flat clearance is letting the pinion creep a tiny fraction during each dwell, and 1000 cycles compounds it into something visible.
Measure the boss-flat clearance with a feeler gauge during dwell. If it is over 0.10 mm, you have rubbing-and-release behaviour where the pinion settles into a slightly different rotational position each cycle depending on residual load direction. Tighten clearance to 0.05 mm and add a light spring detent on the output shaft if the application allows. The detent absorbs the residual motion that the boss alone cannot.
Above roughly 300 cpm the engagement transition becomes the limiting factor, not the gear strength. The leading-tooth relief that softens entry at 100 cpm is no longer enough at 400 cpm — the rise time gets so short that even a relieved tooth produces an audible bang and accelerated pitting.
You can push to 400 cpm if the driven inertia is low (under about 0.001 kg·m²) and you accept tooth replacement every 3 to 6 months. Above that, switch to a cam indexer with a modified-sine or modified-trapezoidal profile. The cam can hold transition acceleration under 5 g where a sectional gear at 400 cpm sees 15 to 20 g on the leading tooth.
Real involute mesh gives constant velocity ratio only when centre distance is exact and tooth deflection is zero. Neither is true under load. What you are seeing on video is most likely tooth-deflection ripple — each tooth pair flexes 5 to 20 µm under load, and as engagement transfers from one pair to the next the output stutters by that flex amount divided by the base radius.
It looks like a small velocity wobble at tooth-passing frequency. The cure is to increase face width (drops deflection by the inverse ratio), tighten centre distance to ±0.02 mm or better, or move to a higher contact-ratio profile (helical or high-addendum). For most indexing applications the ripple is acceptable — it shows up as a few microns of position uncertainty at the output, well below the indexing accuracy spec.
Harden the pinion harder than the sector. The pinion sees every cycle on every tooth — the sector sees each tooth only during its arc of engagement, and the leading and trailing teeth see far more abuse than the middle ones. A typical recipe is pinion at 58 to 62 HRC case-hardened, sector at 50 to 55 HRC with extra surface treatment on just the first and last two teeth.
If you harden them equal, the pinion wears first because it accumulates contact cycles faster. If you over-harden the sector, the leading-tooth chips are worse when timing drifts. The asymmetric hardness gives the pinion the durability it needs and lets the sector teeth wear in slightly to self-correct minor centre-distance errors.
References & Further Reading
- Wikipedia contributors. Gear. Wikipedia
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