Rectangular gears are non-circular gears whose pitch curve is rectangular rather than round, so the instantaneous gear ratio rises and falls four times per revolution. They are common in packaging and printing machinery where a constant input shaft must drive an output that speeds up and slows down on a fixed schedule. The shape forces a programmed angular-velocity profile without needing a cam, servo, or clutch. A typical pair drops output speed to roughly 0.4× input at the long-side mid-points and pushes it to about 2.5× at the short-side mid-points.
Rectangular Gears Interactive Calculator
Vary the input speed scale and rectangular-gear speed ratios to see the slow, mid, and fast output speeds from the 4-cycle non-circular gear action.
Equation Used
The rectangular gear pair uses the instantaneous non-circular gear relation omega_out / omega_in = r1 / r2. The calculator applies the worked-example slow, mid, and fast ratios to a normalized input speed, showing the programmed output speed variation over the four cycles per input revolution.
- Conjugate rectangular pitch curves remain correctly meshed.
- Driver and driven gears are timed 90 deg apart.
- Ratios represent the slow, mid, and fast pitch-radius cases from the worked example.
- Output speeds are normalized to the selected input speed scale.
How the Rectangular Gears (varying Speed) Works
A rectangular gear is a non-circular gear with a pitch curve traced as a rectangle with rounded corners. You mesh two of them so the contact point always sits on the line connecting the two centres — same conjugate-action rule as ordinary spur gears, just applied to a non-circular pitch line. As the driver rotates, the radius from its centre to the contact point changes continuously. Because the gear ratio at any instant equals the ratio of those two radii, the output shaft speeds up when the driver presents its short side and slows down when it presents its long side. Four full speed cycles happen per input revolution, which is what makes the rectangular pitch shape useful — it gives you a 4-per-rev variable angular velocity profile from a constant-RPM input.
The tooth profile is generated, not stamped from a standard cutter. Each tooth's pressure angle and root geometry shifts as you walk around the pitch curve, so the conjugate tooth profile must be CNC-cut against the actual pitch curve, usually by wire EDM or a 5-axis hob. If the centre distance is off by even 0.05 mm on a module-2 pair, the rounded corners bind on one quadrant and backlash opens up on the other — you will hear it as a four-times-per-rev knocking. Same symptom shows up if the two gears are timed wrong on assembly: the rectangles must be 90° offset between driver and driven, not 0° and not 45°. Get that wrong and the pair locks solid within a quarter turn.
Failure modes cluster around three things. First, corner wear — the rounded corners of the pitch curve carry the highest sliding velocity and pit first, especially in packaging applications running 24/7. Second, shaft deflection — the cyclic torque variation excites the input shaft at 4× running speed, and if your shaft's first bending mode sits near that frequency you get visible whip. Third, lubrication starvation at the long-side mid-points where surface speed peaks; an EP grease rated for the average load will fail at the instantaneous peak.
Key Components
- Driver Rectangular Gear: Carries the input shaft and presents a rectangular pitch curve, typically with corner radii of 15-25% of the long side. Tooth count is matched to the perimeter — a 200 mm × 100 mm pitch rectangle at module 2 carries roughly 95 teeth, every one with a slightly different profile.
- Driven Rectangular Gear: Identical pitch curve to the driver but mounted at 90° rotational offset. Centre distance is fixed and equal to half the sum of the long and short sides — for a 200 × 100 pair, centre distance must be 150 mm ± 0.05 mm or the corners bind.
- Conjugate Tooth Profile: Generated against the rectangular pitch curve rather than cut with a standard hob. Pressure angle varies from roughly 18° at the corners to 22° on the flat sides; surface finish on the flanks must hit Ra 0.8 µm or better to survive the cyclic Hertzian stress.
- Rigid Shaft and Bearing Pair: Both shafts must resist 4-per-rev torque pulses without deflecting more than 0.02 mm at the gear face. Tapered roller bearings or angular-contact pairs are standard; deep-groove ball bearings drift under cyclic axial loads inside 1000 hours.
- Timing Mark or Keyway: Establishes the mandatory 90° phase offset between driver and driven on assembly. Miss the mark by one tooth and the gears jam at the first corner-on-corner contact within 90° of input rotation.
Industries That Rely on the Rectangular Gears (varying Speed)
Rectangular gears live in machines where a constant-speed motor must produce a periodic speed-up-then-slow-down output without the cost or service overhead of a servo and cam profile. You see them most often in packaging, printing, and textile machinery where the cycle is locked to product geometry. The cyclic speed variation, the non-uniform rotation, and the 4-per-rev pattern are the signatures — if you need that pattern and you need it mechanically synchronised to a line shaft, a non-circular gear is cheaper and more reliable than the electronic equivalent.
- Flexible Packaging: Cross-seal jaw drive on horizontal flow-wrappers like the Bosch Pack 202 — the seal jaws must dwell during the seal stroke and accelerate during the gap, four times per product cycle on a multi-up tooling layout.
- Printing: Sheet-feed registration drive on a Heidelberg XL 75 offset press where the gripper bar must decelerate before pickup and accelerate after release, indexed off a constant-speed line shaft.
- Textile Machinery: Picking-stick acceleration drive on Sulzer P7100 projectile looms — the projectile launch needs a precise velocity profile that a rectangular gear pair generates from the main camshaft.
- Food Processing: Dough divider piston drive on a Werner & Pfleiderer scale-and-divide line, where the piston must dwell briefly at top and bottom of stroke to allow dough flow.
- Industrial Sewing: Bobbin-thread tensioning shaft on heavy-duty Juki LU-2810 walking-foot machines, providing the cyclic tension peaks that match the needle-bar cycle without an electronic tensioner.
- Pharmaceutical: Tablet ejector drive on rotary tablet presses similar to the Fette FE55, where the lower punch must accelerate the tablet clear of the die at a specific point in the turret rotation.
The Formula Behind the Rectangular Gears (varying Speed)
The instantaneous gear ratio at any input angle θ governs the output speed at that moment. At the short-side mid-points the ratio is at its minimum and the output speed peaks — for a 2:1 long-to-short ratio that gives roughly 2.5× input speed. At the long-side mid-points the ratio is at its maximum and the output crawls at about 0.4× input. Halfway between, on the corner radii, the ratio sweeps through unity. The sweet spot for designing a rectangular pair is a long-to-short ratio between 1.5 and 2.5 — below 1.5 the speed variation is barely worth the manufacturing cost, above 2.5 the corner pressure angles get extreme and pitting accelerates.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| ωout | Instantaneous output angular velocity | rad/s | deg/s |
| ωin | Constant input angular velocity | rad/s | deg/s |
| r1(θ) | Driver pitch radius at input angle θ, measured from driver centre to contact point | mm | in |
| r2(θ) | Driven pitch radius at the same instant, equal to (centre distance − r1) | mm | in |
| θ | Input shaft angle measured from a reference flat | rad | deg |
Worked Example: Rectangular Gears (varying Speed) in a glass ampoule filling line
You are sizing a rectangular gear pair for the carrier-chain indexing drive on a Bosch ALF 5000 ampoule filling line running 18,000 ampoules per hour. The line shaft turns at a constant 75 RPM. The carrier chain must dwell briefly while each ampoule receives 1 mL of product, then accelerate to the next station — four ampoules indexed per input revolution. You picked a pitch rectangle of 200 mm long side and 100 mm short side, giving a long-to-short ratio of 2.0. Centre distance fixes at 150 mm.
Given
- ωin = 75 RPM
- Long side a = 200 mm
- Short side b = 100 mm
- Centre distance C = 150 mm
- Cycles per input rev = 4 —
Solution
Step 1 — at the long-side mid-point (slowest output, dwell phase), the driver pitch radius is half the long side, so r1 = 100 mm and r2 = C − r1 = 50 mm:
So at the dwell point the output crawls at 0.50 × 75 = 37.5 RPM. This is the operating point where the ampoule actually receives the fill — the carrier chain is moving slowly enough that the fill nozzle tracks cleanly without splash.
Step 2 — at the corner mid-point, both radii cross through equality at r1 = r2 = 75 mm, giving the nominal ratio:
Output speed equals input — 75 RPM. This is the transition phase between dwell and rapid index, and it's where the chain tension peaks because angular acceleration is at its highest.
Step 3 — at the short-side mid-point (fastest output, rapid index), r1 = 50 mm and r2 = 100 mm:
Output speed peaks at 2.00 × 75 = 150 RPM. The chain is whipping the next ampoule into station here. In theory you can push the long-to-short ratio higher to get a more aggressive index, but above a 2.5:1 ratio the corner pressure angle exceeds 28° and tooth pitting starts inside 5,000 hours on a 24/7 line.
Result
Output speed swings from 37. 5 RPM at the dwell phase to 150 RPM at the index phase, with 75 RPM at the corner crossings. That 4:1 instantaneous speed swing is what gives you the dwell-and-fire motion the filler needs without a servo. At 1.5:1 long-to-short you would only get a 2.25:1 swing — too gentle for clean ampoule transfer — and at 2.5:1 you get a 6.25:1 swing which is great kinematically but eats teeth at the corners. If you measure an actual swing narrower than 4:1, the most likely causes are: (1) a worn corner radius on the driver pitch curve flattening the peak ratio, (2) shaft deflection of more than 0.05 mm at the gear face under cyclic torque, robbing the geometry of its theoretical reach, or (3) backlash opened up by an incorrect 150 mm centre distance — even a 0.1 mm error on centre distance shifts the peak ratio by roughly 8%.
When to Use a Rectangular Gears (varying Speed) and When Not To
Rectangular gears are one of three honest ways to get programmed cyclic speed variation off a constant-speed input. The other two are elliptical gears and servo-driven cam profiles. Pick based on what your cycle actually needs: how many speed cycles per input rev, what peak-to-trough ratio, and whether the duty is 24/7 or intermittent.
| Property | Rectangular Gears | Elliptical Gears | Servo + Electronic Cam |
|---|---|---|---|
| Speed cycles per input revolution | 4 | 2 | Programmable, any integer |
| Typical peak-to-trough speed ratio | 2.25:1 to 6.25:1 | 1.5:1 to 4:1 | Unlimited (motor-limited) |
| Manufacturing cost (single pair, module 2) | $1,800-3,500 (wire EDM) | $900-1,800 | $4,000-12,000 servo + drive + controller |
| Maintenance interval (24/7 duty) | 8,000-15,000 hr (corner pitting) | 12,000-20,000 hr | Servo bearings 30,000+ hr, but encoder & drive electronics 5-7 yr |
| Profile flexibility | Fixed by gear geometry — no field changes | Fixed by gear geometry | Reprogrammable in software |
| Best application fit | 4-per-rev cyclic packaging, printing indexing | 2-per-rev seal-jaw and cross-seal drives | Variable-format lines, recipe changes, R&D rigs |
| Failure mode signature | 4-per-rev knock from corner wear | 2-per-rev vibration from major-axis pitting | Following-error faults, encoder drift |
Frequently Asked Questions About Rectangular Gears (varying Speed)
Conjugate action requires that at every instant, the sum of the two pitch radii equals the centre distance. With a rectangle, the long-side radius plus the short-side radius equals (a+b)/2, which is exactly the centre distance C. At 0° offset both gears present their long sides simultaneously, so the radii sum to a, not C — the gears physically cannot mesh and will jam at the first quarter turn.
At 45° offset the rounded corners try to mesh corner-to-corner, and you get a brief moment of correct meshing followed by interference as the geometry walks out of conjugacy. Only the 90° offset keeps r1(θ) + r2(θ) = C continuously through every input angle.
That delayed knock pattern is almost always thermal growth on the centre distance. The shafts and housing warm up under load, the centre distance drifts by 0.05-0.15 mm depending on housing material, and the rounded corners start binding once the geometry walks outside its 0.05 mm tolerance window.
Quick diagnostic: shut down hot, measure centre distance with the gears removed, then measure again cold the next morning. If the delta exceeds 0.05 mm, you need either a steel housing instead of aluminium, or a thermally compensated bearing arrangement. A common production fix is to bore the housing 0.03 mm oversize on the cold spec so the operating-temperature centre distance lands on the nominal.
Pick rectangular gears when you need a smooth speed variation with finite dwell — the output never actually stops, it just slows to roughly 0.4× input. Pick Geneva when you need a true zero-velocity dwell with the output locked stationary. The fill-on-the-fly ampoule line works with rectangular gears because the fill nozzle tracks the slow phase. A pick-and-place head needing absolute station accuracy needs Geneva because any residual motion during placement smears the position.
Cost-wise the Geneva is cheaper to build (standard milling, no wire EDM) but the rectangular pair runs quieter and accelerates the load more gently — Geneva indexing produces a hard cosine acceleration spike that a rectangular gear smooths out.
Instantaneous input torque equals output torque divided by the instantaneous gear ratio, ignoring losses. If your output load is constant at 50 Nm, then at the dwell phase (ratio 0.5) the input sees 50 × 0.5 = 25 Nm, and at the index phase (ratio 2.0) the input sees 50 × 2.0 = 100 Nm. That's a 4:1 torque pulse on the input shaft happening four times per revolution.
This is why you cannot size the input motor and shaft for the average torque — you must size for the peak. A common mistake is picking a motor on RMS torque and finding it stalls at the index phase under high-friction loads. Size the motor's continuous rating to handle the peak instantaneous torque, not the mean.
You can change the rectangle dimensions but you must hold two relationships rigidly. First, centre distance C must equal (a+b)/2 exactly — that's not a design choice, it's a meshing requirement. Second, the long-to-short ratio a/b sets your peak-to-trough output speed swing as (a/b)2, so a 180×120 blank gives you (1.5)2 = 2.25:1 swing instead of (2.0)2 = 4:1.
If your application needed the 4:1 swing for clean dwell-and-index motion, a 2.25:1 swing will give you a sluggish index that may not clear the station before the next cycle. Verify the kinematic requirement before substituting — the gear blank size is not the design variable, the speed-swing requirement is.
Those four pitting bands sit exactly at the corner regions of the pitch rectangle, where pressure angle peaks (typically 25-28° versus 20° on the flats) and sliding velocity is highest. Hertzian contact stress at the corners can run 30-40% above the flat-flank value, so fatigue life there is roughly half what the rest of the gear sees.
Three things help: increase the corner radius from a typical 15% of the short side up to 22-25%, which spreads the pressure-angle excursion over more teeth; specify a harder surface treatment at the corners (case depth 0.8-1.2 mm with 58-62 HRC); and switch from a general-purpose EP grease to a synthetic with a higher film strength rating, because the corners run at the highest sliding velocity and starve first.
References & Further Reading
- Wikipedia contributors. Non-circular gear. Wikipedia
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