A spur gear is a cylindrical gear with straight teeth cut parallel to its axis of rotation, used to transmit power and motion between parallel shafts. It solves the problem of changing speed, torque, or direction of rotation without slip, by meshing involute teeth that roll against each other at a fixed centre distance. The mesh produces a constant velocity ratio set by the tooth count, with efficiencies of 98-99% per stage. You see them everywhere from a Bridgeport mill's back-gear cluster to a 100:1 industrial reducer pulling 50 kW at 1,800 RPM.
Spur Gears Interactive Calculator
Vary the pinion and gear tooth counts to see the speed ratio, ideal torque multiplication, and animated gear mesh.
Equation Used
The spur gear ratio is set by tooth count: the driven gear teeth divided by the pinion teeth. A 20-tooth gear driven by a 12-tooth pinion gives i = 20 / 12 = 1.67, so the driven shaft turns at about 60% of input speed while ideal torque is multiplied by 1.67.
- External spur gears mesh without slip.
- The smaller gear is the driving pinion.
- Ideal torque multiplier ignores bearing, windage, and tooth friction losses.
- Pinion tooth shortfall is referenced to the common 17-tooth minimum noted for avoiding undercut.
Operating Principle of the Spur Gears
Two spur gears mesh by rolling their involute tooth flanks against each other along a line called the line of action. The involute tooth profile is the geometry that matters here — it guarantees that the velocity ratio stays constant even if the centre distance shifts slightly, which is why you can run a worn machine for years before the gears actually start hunting. The driving gear (usually called the pinion when it's the smaller one) pushes its tooth into the driven gear's tooth space, and the contact point sweeps along the flank from root to tip as the gears rotate. The pressure angle — almost always 20° on modern gears, sometimes 14.5° on older imperial stock — sets the angle of that line of action and the radial separating force between the two shafts.
The size of the teeth is set by the module (metric, in mm) or the diametral pitch (imperial, teeth per inch of pitch diameter). The pitch diameter is the imaginary circle where two meshing gears would behave like rolling cylinders, and the centre distance between two shafts is simply the sum of the two pitch radii. If you cut the centre distance 0.05 mm too tight, the teeth bind and you'll feel a hot spot on the gear case within an hour. If you cut it 0.10 mm too loose, backlash opens up and the drive will hammer every time the load reverses — a classic failure on retrofit CNC tables where the original ground gears were replaced with hobbed stock.
Contact ratio matters more than most builders realise. You want at least 1.4 teeth in mesh on average — below that you get whine, vibration, and uneven loading. Failures usually come from three places: pitting on the flank from Hertzian contact stress exceeding the AGMA allowable, root-fillet bending fatigue when the tooth count is too low for the torque, and scuffing when EP additives in the oil break down at flash temperatures above 150°C. The pinion and gear pair has to be designed as a system — not as two parts.
Key Components
- Pinion: The smaller of the two meshing gears, almost always the driver. Tooth count is typically 17 or higher to avoid undercutting at 20° pressure angle — go below 17 teeth on a standard cutter and you'll see the tooth root carved away during hobbing, weakening the bending strength by 20-30%.
- Gear (wheel): The larger driven member. Its tooth count divided by the pinion tooth count gives the gear ratio. For a single stage, sensible ratios run 1:1 up to about 7:1 — beyond that, contact ratio and tooth size become hard to balance and you should split into two stages.
- Involute tooth profile: The flank curve generated by unwrapping a string from a base circle. This profile is what allows constant velocity ratio under small centre-distance variation — typical tolerance on centre distance is ±0.025 mm for AGMA Q10 gears, ±0.008 mm for Q12.
- Pitch circle: The theoretical rolling circle. Pitch diameter D = m × z (metric) or D = z / P (imperial), where m is module, z is tooth count, P is diametral pitch. Centre distance C = (D₁ + D₂) / 2.
- Pressure angle: The angle between the line of action and the tangent to the pitch circles. 20° is the modern standard — it gives stronger teeth than the older 14.5° at the cost of slightly higher radial bearing loads.
- Backlash: The deliberate clearance between mating teeth, typically 0.03-0.10 mm for industrial gears. Too little and the gears bind under thermal expansion; too much and you get position error and impact loading on every reversal.
Real-World Applications of the Spur Gears
Spur gears go anywhere two parallel shafts need to transfer torque without slip — wherever helical gears would be overkill, expensive, or generate unwanted axial thrust. They dominate machine tools, conveyors, hand tools, kitchen appliances, automotive transmissions in lower gears, and almost every gearbox under 100 kW where noise isn't a primary concern. Below 10 m/s pitch line velocity they run quietly enough that nobody complains; above that you'd typically switch to helical.
- Machine tools: Back-gear cluster on a Bridgeport Series 1 milling machine, dropping spindle speed by roughly 8:1 for heavy roughing cuts in steel.
- Industrial gearboxes: SEW-Eurodrive R-series helical-bevel reducers use spur stages on the input side for ratios up to 7:1 per stage, handling 50-200 kW continuous duty.
- Automotive: Reverse gear in most manual transmissions, including the Tremec TR-6060, uses straight-cut spur teeth because the brief duty cycle doesn't justify the cost of helical.
- Power tools: DeWalt 20V cordless drill planetary stages — 3 spur planet gears around a sun pinion in each of two stages, giving roughly 30:1 reduction from a 20,000 RPM motor down to chuck speed.
- Wind turbines (auxiliary): Yaw drives on Vestas V90 turbines use spur gear final stages driving a large bull gear on the tower top, slewing the nacelle at fractions of an RPM.
- Conveyor systems: Drum motor drives on Interroll belt conveyors at parcel sortation centres like FedEx Memphis, using compact spur reductions inside the drum shell.
The Formula Behind the Spur Gears
The two formulas you reach for first are the gear ratio and the pitch diameter. Ratio sets your speed and torque trade — at the low end of typical reductions (around 2:1) you barely change torque but get a quiet, compact stage; at the nominal industrial sweet spot (3:1 to 5:1) you get useful torque multiplication with a contact ratio above 1.4 and reasonable pinion size; push beyond 7:1 in a single stage and the pinion gets so small relative to the gear that tooth count drops below 17, undercutting starts, and bending fatigue life collapses. Pitch diameter ties directly to centre distance, which is the dimension your gear case is bored to ��� get it wrong and the whole drive is scrap.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| i | Gear ratio (output torque multiplier, output speed divisor) | dimensionless | dimensionless |
| z1, z2 | Tooth count of pinion and gear | teeth | teeth |
| m | Module (metric tooth size) | mm | in (use diametral pitch P = 25.4 / m) |
| D | Pitch diameter | mm | in |
| C | Centre distance between shafts | mm | in |
Worked Example: Spur Gears in a corrugated box plant flexo printer drive
You are sizing the impression-cylinder drive train on a Bobst FFG 8.20 flexo folder-gluer at a corrugated packaging plant in Lyon. The cylinder must turn at 110 RPM during nominal production of 18,000 boxes per hour, fed by a 4 kW SEW geared motor running at 1,460 RPM at the motor shaft. You want a single spur stage using module 4 gears with a 20° pressure angle, and you need to know the tooth counts, pitch diameters, and centre distance the gear case must be bored to.
Given
- nmotor = 1460 RPM
- ncyl = 110 RPM
- m = 4 mm
- z1 (trial pinion) = 21 teeth
- Pressure angle = 20 °
Solution
Step 1 — calculate the required gear ratio at the nominal operating point of 110 RPM cylinder speed:
That ratio is far too high for a single spur stage — anything above 7:1 puts you into pinion undercut territory or oversized bull gears. The realistic single-stage design picks a sensible ratio around 5:1 and accepts that the rest comes from the SEW reducer upstream. Recompute assuming the SEW unit delivers a pre-reduced input of 550 RPM to this stage:
Step 2 — with z1 = 21 teeth on the pinion (above the 17-tooth undercut threshold), find the gear tooth count at nominal:
Step 3 — compute pitch diameters and centre distance:
At the low end of the plant's typical operating range — 60 RPM cylinder speed for short-run jobs — the same gear pair runs at i = 5.0 unchanged, but the SEW input drops proportionally and the gear teeth see lower Hertzian stress, extending pitting life. At the high end of nameplate, 130 RPM for thin liner board, tooth flash temperature climbs and you need to verify the ISO VG 220 oil hasn't dropped below 40 cSt at sump temperature — most Bobst service bulletins call for a synthetic PAO at this duty.
Result
The nominal design uses a 21-tooth pinion meshing with a 105-tooth gear at module 4, giving a 5:1 ratio and a centre distance of 252 mm bored into the gear case. At 60 RPM cylinder speed the drive runs cool and quiet with pitch line velocity around 0.26 m/s — well within the no-noise zone for spur gears. At nominal 110 RPM you're at 0.48 m/s pitch line velocity, still comfortably below the 10 m/s threshold where you'd need to consider helical. Push to 130 RPM and you start to hear the mesh in a quiet plant — not a fault, just spur-gear character at higher speed. If your measured cylinder speed differs from the predicted 110 RPM, the most common causes are: (1) the SEW reducer ratio tag was misread and the real input is 580 or 520 RPM not 550, (2) the centre distance was bored to 252.5 mm or larger, opening backlash beyond 0.10 mm and causing missed counts on the encoder during reversals, or (3) the pinion bore was reamed oversize and the key is rolling under load — pull the pinion and check for fretting marks on the keyway.
Choosing the Spur Gears: Pros and Cons
Spur gears compete with helical gears, bevel gears, and timing belts in the parallel-shaft power transmission space. Each has a place — the choice comes down to noise, axial thrust, cost, and how much torque per unit volume you need.
| Property | Spur gears | Helical gears | Timing belt drive |
|---|---|---|---|
| Pitch line speed limit (quiet operation) | ~10 m/s before whine becomes objectionable | ~25 m/s, runs noticeably quieter at all speeds | ~50 m/s for HTD/GT profiles |
| Efficiency per stage | 98-99% | 96-98% (axial thrust losses) | 95-98% (depends on tension) |
| Axial thrust load on bearings | Zero — purely radial | Significant, requires thrust bearings | Zero |
| Cost per stage (industrial Q10) | Lowest of the three | 20-40% more than spur | 30-60% less than spur for low torque |
| Single-stage ratio range (practical) | 1:1 to 7:1 | 1:1 to 10:1 | 1:1 to 8:1 |
| Lifetime at full load | 20,000+ hours with proper lube | 20,000+ hours, often longer due to load sharing | 5,000-15,000 hours, belt is the wear item |
| Backlash floor | 0.03 mm achievable, 0.01 mm with anti-backlash pair | Similar, 0.03 mm typical | Effectively zero in tension, but belt stretch adds compliance |
Frequently Asked Questions About Spur Gears
Whine at speed is almost never a tooth-quality issue if you're running Q10 or better — it's usually contact ratio. Below 1.4 average teeth in mesh you get a measurable transmission error spike every time the mesh transitions from two-tooth to one-tooth contact, and that spike radiates as airborne noise at gear-mesh frequency.
Check your contact ratio with the actual tip diameters as ground, not as drawn. If addendum was shortened during finishing to hit a backlash spec, the contact ratio drops fast. The fix is usually a profile shift on the pinion or a slight increase in addendum on the gear — not replacement.
Pinion size is the deciding factor. At 6:1 with module 3, a 17-tooth pinion gives you a 51 mm pitch diameter — fine for many drives but the bending stress at the root climbs because torque concentrates on a small pinion. Two 2.45:1 stages let you use larger pinions on both shafts, dropping root stress by roughly 40% and giving you headroom for shock loads.
The trade is package size and cost. Two stages need an extra shaft, two more bearings, and a second gear case bore. If the duty is steady-state and shock-free — say a fan drive — go single stage. If it's a winch, conveyor with start-stop, or anything with reversing loads, split it.
Three culprits, in order of frequency. First, surface hardness is below spec — if the pinion was through-hardened to 28 HRC instead of the case-hardened 58-62 HRC the calculation assumed, allowable contact stress drops by 60% and pitting starts almost immediately. Pull the gear and do a hardness check on a tooth flank.
Second, lubricant viscosity is wrong for the actual sump temperature. ISO VG 220 at 80°C drops to roughly 25 cSt — below the minimum film thickness for boundary lubrication on hardened gears. You need a higher-VI oil or a cooler.
Third, misalignment. Even 0.05 mm/m of shaft parallelism error concentrates load at one end of the tooth face and triples local Hertzian stress. Bluing the teeth and checking the contact pattern across the full face width tells you in five minutes.
Cold, but with a thermal-growth allowance built into the backlash spec. Steel gears in a steel case grow together so the centre distance change with temperature is small, but the gears themselves grow on the OD as well. A 252 mm centre distance running 30°C above ambient will see the pitch diameters grow by about 0.05 mm each, eating 0.05 mm of backlash.
Spec the cold backlash on the high side of normal — 0.08 to 0.10 mm for the case above — and you'll land at 0.03 to 0.05 mm hot, which is ideal. If you spec tight cold backlash and the machine binds at operating temperature, you'll cook the oil and gall the flanks within a shift.
No. The pressure angle defines the involute base circle, and two involutes generated from different base circles do not produce conjugate action. You'll get severe interference at the tip, rapid pitting, and audible knocking within minutes of running.
The visual check is easy — 14.5° teeth look noticeably more pointed at the tip and have rounder fillets at the root than 20° teeth. If the spares bin has both, sort them before mixing them up. Replace the pair, not just one gear.
Spring-loaded anti-backlash gears only remove backlash up to the preload force. The instant your cutting torque exceeds the spring preload, the secondary gear unloads and you get full backlash back, plus the position error from spring deflection.
Measure the preload spring force, multiply by the pitch radius to get the torque it can hold, and compare to your peak cutting torque. If peak is higher than preload, you need a stiffer spring, a larger gear pair so the same spring gives more torque, or a switch to a true zero-backlash design like a duplex worm or a harmonic drive.
References & Further Reading
- Wikipedia contributors. Spur gear (Gear article). Wikipedia
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