Crank-disk with Toothed Sector and Rack

Robbie Dickson
2026-04-29T01:07:58-07:00

Crank-disk with toothed sector and rack converts rotation into intermittent linear motion. See how it works, formula, worked example, and packaging-line uses.
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A crank-disk with toothed sector and rack is a rotary-to-linear mechanism where a rotating disk carries a partial gear (the toothed sector) that meshes with a straight rack only during part of each revolution. The geometry solves the problem of producing intermittent linear strokes with a built-in dwell from a continuously rotating shaft. While the sector teeth are engaged, the rack moves at a defined linear speed; when the sector clears, the rack stops and waits. You see this driving feed slides on packaging machines and indexing rams on assembly lines where one motor must produce stroke-then-pause cycles without a clutch.

How It Works Where It’s Used Formula Animated Calculator Worked Example Comparison FAQ

Crank-disk with Toothed Sector and Rack Interactive Calculator

Vary pitch diameter, sector angle, and gear module to see rack stroke, dwell timing, and approximate sector tooth count.

Pitch Diameter (D)

Sector Angle (theta)

Gear Module (m)

Rack Stroke

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Stroke Duty

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Dwell Angle

--

Sector Teeth

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Equation Used

s = pi * D * theta / 360; duty = theta / 360; dwell = 360 - theta; N = s / (pi * m)

The rack stroke is the pitch-circle arc length covered by the toothed sector. A larger pitch diameter or sector angle increases stroke. The same sector angle also sets the stroke duty cycle, while the bare arc sets dwell.

Rack is tangent to the pitch circle.

Sector teeth mesh only over the sector angle.

Stroke equals pitch-circle arc length during engagement.

Module is the same for sector and rack.

FIRGELLI Automations - Interactive Mechanism Calculators

Crank Disk with Toothed Sector and Rack.

How the Crank-disk with Toothed Sector and Rack Actually Works

The disk spins continuously. A toothed sector — typically covering anywhere from 60° to 270° of the disk's circumference — carries the gear teeth. A straight rack sits tangent to the disk's pitch circle on a linear slide. When the leading sector tooth enters the rack, the rack accelerates, traverses, and decelerates in lockstep with the disk's angular position. When the trailing tooth disengages, the rack dwells. The dwell length is set entirely by how much of the disk circumference is left bare. A 180° sector gives a 50% duty cycle — half stroke, half dwell. A 90° sector gives a 25% duty stroke and a long pause, which is what packaging cam-drives often need.

The geometry is unforgiving on two specifics. First, the entry tooth must engage the rack at zero relative velocity error or you will hear it — a metallic tick at every revolution and chipped tooth tips within weeks. We hold the centre distance to ±0.05 mm on a typical 80 mm pitch-circle sector and demand the first and last teeth be ground to a modified profile that ramps engagement gradually. Second, the rack needs a hard stop or a return spring to hold it during dwell, because nothing else stops it from drifting. Most production designs use a spring-loaded detent, a parallel cam, or a counter-rack on the opposite side of the disk to drive the return.

Failure modes are predictable. Tooth-tip fracture on the entry tooth means your engagement timing is off. Rack drift during dwell means your detent has worn or your return spring has lost preload. Backlash that grows over the first 100 hours usually means the rack guideway wasn't parallel to the sector's tangent line within 0.1 mm over the stroke length, and the teeth are wearing into a self-corrected fit — which works, but kills predictable timing.

Key Components

Crank disk: The driven disk that rotates continuously on the input shaft. Diameter is set by the required linear stroke and tooth count — a 100 mm pitch-circle disk with a 180° sector and module 2 teeth gives a stroke of roughly 157 mm. Disk runout must stay under 0.03 mm or the rack engagement chatters.

Toothed sector: The arc segment of gear teeth bonded or machined into the disk face. Sector arc angle directly sets the stroke duration; the bare arc sets the dwell. Entry and exit teeth are profile-modified — typically a 0.2 to 0.4 module tip relief — so the rack accelerates smoothly rather than colliding with the first tooth.

Linear rack: The straight gear bar that meshes with the sector. Module must match the sector exactly. Rack length must exceed the stroke by at least 2 tooth pitches on each end so the engagement and disengagement zones never reach the rack's last tooth, where stress concentrates and breaks teeth.

Linear guide or slide: Constrains the rack to pure translation. Parallelism to the sector tangent must be held within 0.1 mm over the stroke. Use a profiled rail like a THK SR or HSR series for any production duty — plain bushings allow the rack to lift off the sector under load and skip teeth.

Return mechanism: Either a counter-sector on the same disk driving a second rack, a spring return, or a parallel cam that drives the rack back during the dwell phase of the primary sector. Without one, the rack just sits where it stopped.

Detent or hard stop: Holds the rack steady during dwell. A spring-loaded ball detent into a notch on the rack works well for forces under 50 N. Above that, use a positive mechanical stop or a parallel locking cam timed to the disk.

Who Uses the Crank-disk with Toothed Sector and Rack

You find this mechanism wherever a continuously running shaft must produce a stroke-pause-stroke cycle without electronic control, clutches, or pneumatics. Packaging, textile feeding, and small-part assembly are the obvious homes, but it shows up in surprising places — automatic firearm loaders, ticket printers, and bookbinding machinery all use variants. The appeal is simple: one motor, one shaft, perfect timing, no controls.

Packaging machinery: Bosch Pack 102 cartoning machines use crank-disk-with-sector drives for the carton-feed pusher, giving a hard 180° feed stroke and a 180° dwell that synchronises with the glue applicator.

Textile machinery: Picanol air-jet looms historically used sector-and-rack drives on the weft-feeder advance, where a 90° engagement window matched the shed-open phase of the loom cycle.

Bookbinding: Müller Martini Presto perfect binders use a sector-rack drive on the book-block transfer arm to advance a block by a precise pitch and dwell during the spine-glue application.

Ticket and label printing: Boca Systems thermal ticket printers use a small sector-and-rack mechanism for paper advance, giving exact pitch with no servo.

Small-arms feeding: Browning M2 belt-fed mechanisms use a related toothed-sector geometry to advance the ammunition belt one round per bolt cycle with a clean dwell during firing.

Pharmaceutical filling: IMA Group capsule filling machines use sector-rack drives on the powder dosator advance, where the dwell phase allows the powder to settle before the next compression stroke.

The Formula Behind the Crank-disk with Toothed Sector and Rack

The headline calculation is the linear stroke produced by one engagement of the sector. At the low end of practical sector arcs — around 60° — you get a short stroke and a long dwell, useful for indexing applications where the dwell does most of the work. At the high end — 270° or so — you approach a near-continuous rack drive with a brief dwell, useful when you need most of the cycle to be active stroke. The sweet spot for packaging and feed-advance work sits between 120° and 180°, balancing stroke length against the time the rack needs to settle.

S = (θsector / 360°) × π × Dpitch

Variables

Symbol | Meaning | Unit (SI) | Unit (Imperial)
S | Linear stroke produced per disk revolution | mm | in
θsector | Arc angle of the toothed sector | degrees | degrees
Dpitch | Pitch-circle diameter of the sector | mm | in
vrack | Linear rack speed during engagement | mm/s | in/s
Ndisk | Disk rotational speed | RPM | RPM

Worked Example: Crank-disk with Toothed Sector and Rack in a confectionery wrapping line

Sizing the sector-rack drive for the foil-feed pusher on a Carle & Montanari Sigma 600 chocolate wrapping machine. The pusher must advance a foil sheet by 120 mm, then dwell while the wrap-fold tooling closes. Disk pitch-circle diameter is 80 mm. The line runs nominally at 200 RPM with operating excursions from 100 RPM (start-up) to 280 RPM (peak production).

Given

Dpitch = 80 mm

Srequired = 120 mm

Nnom = 200 RPM

Nlow = 100 RPM

Nhigh = 280 RPM

Solution

Step 1 — solve for the required sector arc angle from the stroke equation:

θsector = (S / (π × Dpitch)) × 360° = (120 / (π × 80)) × 360° = 171.9°

Round to 172° of toothed arc, leaving 188° of bare disk for the dwell phase. That gives a 47.8% stroke duty cycle and a 52.2% dwell — a clean match for foil-feed timing.

Step 2 — at nominal 200 RPM, compute rack speed during engagement:

vnom = (π × Dpitch × Nnom) / 60 = (π × 80 × 200) / 60 = 837.8 mm/s

That is the peak rack velocity — fast enough that the foil sheet must be held flat on the slide or it will buckle. At 200 RPM the dwell lasts (188/360) × (60/200) = 0.157 s, which is the window the fold tooling has to operate.

Step 3 — at the low end of operating range, 100 RPM:

vlow = (π × 80 × 100) / 60 = 418.9 mm/s

At start-up speed the rack moves at half-pace and the dwell stretches to 0.313 s. The foil settles cleanly and you can hand-feed the first few sheets without jamming. This is the speed an operator should run at when threading new foil stock.

Step 4 — at the high end, 280 RPM:

vhigh = (π × 80 × 280) / 60 = 1172.9 mm/s

At peak production rack speed climbs past 1.17 m/s and dwell drops to 0.112 s. Above roughly 250 RPM you start seeing foil flutter at the leading edge during the high-speed traverse, and the fold tooling barely has time to clear before the next stroke begins. The mechanism still works mechanically, but the wrapping quality starts to drop — a real production line would set the upper limit at 240 RPM and accept the throughput cap.

Result

A 172° toothed sector on an 80 mm pitch-circle disk gives the required 120 mm stroke with a 188° dwell. At 200 RPM nominal the rack travels at 838 mm/s with a 0.157 s dwell, which is the timing the wrap-fold tooling is designed around. At 100 RPM start-up the rack creeps at 419 mm/s and the dwell doubles — comfortable for threading and hand-feeding. At 280 RPM peak the rack hits 1.17 m/s and the dwell collapses to 0.112 s, where foil flutter and tooling clearance become the limiting factors well before the gear teeth do. If you measure rack stroke shorter than the predicted 120 mm, suspect three causes: the rack guide rail has lost parallelism with the sector tangent (check with a dial indicator over the stroke — must be within 0.1 mm), the entry-tooth profile relief has worn flat and the rack is skipping the first half-tooth of engagement, or the sector retaining bolts have backed off and the sector has rotated 1-2° on the disk face.

When to Use a Crank-disk with Toothed Sector and Rack and When Not To

The sector-and-rack drive competes with a Geneva mechanism, a slider-crank, and a servo-driven rack-and-pinion. Each owns a different region of the design space. The decision usually comes down to whether you need a built-in dwell, how fast you want to cycle, and whether your plant has the controls budget for a servo.

Property | Crank-disk with toothed sector and rack | Geneva mechanism | Servo-driven rack and pinion
Maximum practical speed | Up to 300 RPM in production, limited by tooth engagement shock | Up to 600 RPM with proper indexing | Up to 3000 RPM, limited by motor only
Stroke length range | 20-500 mm typical, set by disk diameter and arc | Fixed angular index, not linear | Unlimited, set by rack length
Dwell-to-stroke ratio control | Continuously variable from 25% to 75% by sector arc design | Fixed by slot count (4 slots = 75% dwell) | Fully programmable
Cost per unit (small machine) | $80-300 in machined steel | $150-500 for a quality indexer | $1500-4000 with servo and driver
Repeatability per cycle | ±0.05 mm with ground teeth | ±0.02 mm at the index position | ±0.01 mm with encoder feedback
Lifespan at rated load | 10-50 million cycles before sector tooth refurbishment | 100+ million cycles with proper lubrication | Limited by servo bearings, typically 20,000+ hours
Control complexity | Zero — purely mechanical | Zero — purely mechanical | PLC or motion controller required

Frequently Asked Questions About Crank-disk with Toothed Sector and Rack
Why does my sector-rack drive lose 1-2 mm of stroke after the first 50 hours of running, even though I never changed the geometry?

This is run-in wear on the entry tooth, and it is almost universal on a fresh build that didn't ship with profile-modified teeth. The first tooth on the sector takes the full impact of engagement, and on a stock involute profile that impact plastically deforms the tooth tip in the first few thousand cycles. The deformation effectively delays engagement by a fraction of a tooth pitch, which translates to lost stroke.

The fix is to specify tip relief — typically 0.2 to 0.4 module of material removed from the leading and trailing teeth — when the sector is cut. If the damage is already done, you can re-grind the entry tooth and the stroke will stabilise. Don't just keep running it. The deformed tooth eventually fractures and takes the rack tooth with it.

Can I run two sectors on the same disk to drive two racks alternately, or does that cause problems?

Yes, and it is a common configuration — one sector on each face of the disk, or two sectors at 180° on the same face driving racks on opposite sides. This gives you two independent stroke-dwell cycles per revolution from one motor, which is useful when you have a feed-and-cut or push-and-return arrangement.

The catch is timing overlap. If both sectors are engaged simultaneously you double the torque demand on the input shaft, which usually means upsizing the motor and the shaft. Most designs deliberately offset the sectors so engagement windows don't overlap. Check the torque profile across one full revolution before committing to dual sectors.

How do I decide between a 90° sector with a long dwell versus a 270° sector with a short dwell for the same stroke?

The decision is driven by what happens during the dwell, not what happens during the stroke. If your downstream process needs time — glue setting, sensor reading, tooling clearance — pick the longer dwell and the larger disk. A 90° sector on a bigger disk gives the same stroke as a 270° sector on a smaller disk, but with three times the dwell time at the same RPM.

The trade-off is rack speed. A 90° sector compresses the same stroke into a quarter of the revolution, so the rack moves four times faster than the equivalent 270° design. Above about 1 m/s rack velocity you start fighting inertia, especially if the rack is carrying any payload. The practical sweet spot for most packaging work is 150°-180°, which keeps rack speed reasonable while still giving you a useful dwell.

My rack drifts about 0.5 mm during the dwell phase even though the sector is fully disengaged. What is causing this?

Drift during dwell means there is no positive lock holding the rack still, and something is pushing it. The usual suspect is the linear guide itself — if the guide rail is not perfectly horizontal, gravity will roll the rack downhill during the dwell. A 0.1° tilt over a 200 mm rack is enough to produce visible drift.

The other common cause is a worn detent. A spring-loaded ball detent into a rack notch is the standard solution, and the spring loses preload over time. Check the detent force with a push-pull gauge — if it's below about 70% of the original spec, replace the spring. For higher-load applications, abandon the detent and use a positive mechanical stop or a parallel locking cam timed to the disk dwell phase.

Is there a way to get variable stroke length without re-cutting the sector?

Not directly — the sector arc fixes the stroke. But you can get effective stroke variability by mounting the rack on a secondary slide with an adjustable hard stop. The sector still drives the full geometric stroke, but the rack only delivers usable motion until it hits the stop, and the remaining engagement just slips at the stop face. This is rough on the teeth and only works at low loads.

The cleaner solution is to design the disk with multiple sector positions — bolt-on sector segments that you can swap to change the arc. We have done this on machines that need to handle different product sizes; the operator unbolts a 120° sector and bolts on a 180° sector in about 15 minutes. If you need true on-the-fly variable stroke, you are in servo territory and the sector-rack mechanism is the wrong choice.

How tight does the centre distance between the disk and the rack need to be?

Tighter than most people expect. Standard rack-and-pinion runs happily with ±0.1 mm centre distance variation because the engagement is continuous and self-correcting. A sector-and-rack does not have that luxury — the entry tooth has to find the right rack tooth on every revolution, and any centre distance error shows up as either a hard tooth strike (too close) or a partial engagement skip (too far).

For a typical module 2 sector on an 80 mm pitch circle, hold the centre distance to ±0.05 mm and the parallelism of the rack to the sector tangent within 0.1 mm over the stroke. Use a shimmed mounting block, not slotted holes, so the alignment doesn't drift after the first big load event. If you can hear a tick at the start of every engagement, your centre distance is wrong by at least 0.1 mm.

References & Further Reading

Wikipedia contributors. Rack and pinion. Wikipedia

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About the Author

Robbie Dickson — Chief Engineer & Founder, FIRGELLI Automations

Robbie Dickson is the Chief Engineer and Founder of FIRGELLI Automations. He has spent over 20 years designing motion-control hardware for industrial automation, packaging machinery, and custom mechanism work. His background covers gear-train design, linear actuator development, and the kind of unforgiving production-line geometry where a 0.05 mm centre-distance error costs a customer a shift's worth of output. He writes the FIRGELLI Mechanisms library to give working engineers and serious DIYers the practical numbers and decisions that get glossed over in textbooks.

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