Gear Geometry, Types, Rating and Selection: Making the Pinion Harder Than the Gear

Making the Pinion Harder Than the Gear

This is one of the most important practical principles in gear engineering. The pinion has fewer teeth than the gear, so each pinion tooth does more work. Making the pinion harder than the gear:

• Equalizes the rate of wear between pinion and gear
• The harder pinion corrects gear tooth errors through initial wear
• The harder surface then burnishes the gear teeth, increasing their wear resistance through cold-working
• For high-ratio applications without severe shock: a casehardened pinion with an oil-treated gear (cut after treatment) is an excellent combination

Case-Hardening Steels Reference

Through-Hardening Steels Reference

Non-Ferrous Gear Materials

For spur and bevel gears: Hard cast bronze (ASTM B-10-18; SAE No. 62; 88-10-2 mixture):

• Copper: 86–89%
• Tin: 9–11%
• Zinc: 1–3%
• Lead: 0.20% max
• Iron: 0.06% max

For worm gears: S.A.E. nickel phosphor gear bronze (No. 65 + Ni):

• Copper: 87%
• Tin: 11%
• Nickel: 2%
• Phosphorus: 0.2%

Non-Metallic Gears

For moderate loads, non-metallic gears (phenolic laminates) offer advantages in noise reduction and shock absorption. Key design rules:

• The root diameter of a phenolic laminated pinion should provide a minimum distance from the keyway edge to the root diameter at least equal to the tooth depth
• Keyway stress should not exceed 3,000 psi on plain phenolic laminated gears
• Metal reinforcing end plates may be used when keyway stresses exceed 3,000 psi, but should not extend beyond the root diameter

Keyway stress formula:

$$S = \frac{33{,}000 \times \text{hp}}{V \times A}$$

Where:

• $S$ = unit stress (psi)
• hp = horsepower transmitted
• $V$ = peripheral speed of shaft (feet per minute)
• $A$ = area of keyway in pinion (length × height, in²)

Diametral Pitch Selection Guide for Non-Metallic Gears

The Module System: Metric Gear Design

German Standard Tooth Form (DIN 867)

The international standard for metric gears uses the module system with a 20° pressure angle and involute tooth flanks.

* When clearance = 0.157 × module (common American practice)

Module System Rules

To find the metric module:

$$\text{Module} = \frac{\text{Pitch Diameter (mm)}}{N}$$

To find pitch diameter:

$$D = N \times \text{module}$$

To find outside diameter:

$$D_O = (N + 2) \times \text{module}$$

Example: A gear has 40 teeth and module 8. Pitch diameter = $40 \times 8 = 320$ mm (12.598 inches). Outside diameter = $(40 + 2) \times 8 = 336$ mm.

Backlash: The Necessary Evil

What Is Backlash and Why Does It Matter?

Backlash is the play between mating tooth surfaces — specifically, the amount by which a tooth space exceeds the thickness of the engaging tooth, measured at the pitch circle. It serves critical functions:

• Prevents tooth binding due to thermal expansion
• Accommodates manufacturing tolerances in both gears and housings
• Allows lubricant film to form between mating surfaces
• Compensates for tooth deflection under load

How Backlash Is Provided

Backlash is created by reducing tooth thickness below the theoretical standard. The standard practice is to make half the allowance on each gear, with important exceptions:

• Pinions with very few teeth: Provide all backlash allowance on the mating gear to avoid weakening the pinion teeth
• Worm gearing: All allowance is typically provided on the worm (which is usually stronger than the wormgear)
• Bevel gearing: Backlash should never exceed $\frac{0.080}{P}$ where P is diametral pitch

Backlash Control Methods

1. Excess depth of cut — sinking the cutter deeper than standard depth (most common for spur and helical gears)
2. Thinner cutter — the cutter is designed to produce the desired backlash directly
3. Adjustable center distance — changing the distance between gear shafts at assembly (most common in bevel gearing)
4. Matching runout patterns — aligning high and low spots of mating gears to cancel each other

Excess Depth of Cut for Backlash

Where B = desired circular backlash.

Gear Inspection and Measurement

Checking Gear Size with Measurement Over Wires

The most accurate method of verifying gear tooth thickness is measurement over wires (or balls) placed in diametrically opposite tooth spaces. This technique eliminates the need for a gear tooth caliper and provides highly repeatable measurements.

For external gears: Each 0.001-inch reduction in pitch-line tooth thickness reduces the measurement over wires by an amount dependent on the number of teeth and pressure angle.

For internal gears: Each 0.001-inch reduction in pitch-line tooth thickness increases the measurement between wires by a corresponding amount.

Wire Diameter Correction Factors

Values shown are for external gears per 0.001-inch reduction in tooth thickness.

Chordal Thickness Measurement

For gear tooth caliper measurements, the chordal thickness (straight-line thickness) differs from the circular thickness (arc thickness):

$$t_c = D \sin\left(\frac{90°}{N}\right)$$

Where:

• $t_c$ = chordal thickness
• $D$ = pitch diameter
• $N$ = number of teeth

Example: A pinion with 15 teeth of 3 diametral pitch (pitch diameter = 5 inches):

$$t_c = 5 \times \sin\left(\frac{90°}{15}\right) = 5 \times \sin(6°) = 5 \times 0.10453 = 0.5226 \text{ inches}$$

Selecting Milling Cutters for Gear Teeth

The Eight-Cutter System

When gear teeth are cut with formed milling cutters, the cutter must match both the pitch and the number of teeth. Tooth spaces vary in shape depending on the number of teeth — a small pinion has differently shaped spaces than a large gear of equal pitch.

Standard involute formed cutters come in series of eight cutters per diametral pitch:

Each cutter's tooth outline is correct only for the lowest number in its range. When used for higher numbers, slightly too much material is removed from the upper tooth surfaces — acceptable for ordinary work but not for high-accuracy applications.

For greater accuracy: An intermediate series with half-numbers is available for tooth counts between the standard ranges.

For metric module cutters: The same number ranges apply, but cutter numbers are designated in reverse order (No. 1 for 12–13 teeth, No. 2 for 14–16 teeth, etc.).

Replacement Gear Calculations: The Quick Reference

When you need to replace a damaged gear and have limited information, these formulas let you work backward from what you can measure.

Spur Gear Replacement Formulas

Bevel Gear Replacement Formulas (90° Shafts)

Your Next Steps

You now have the most comprehensive single-source gear reference available. Here's how to use it:

1. Bookmark this guide. You won't memorize these formulas — and you don't need to. What matters is knowing they exist and where to find them.

2. Practice the reverse-engineering method. Take a gear from your shop, count the teeth, measure the outside diameter, and work through the spur gear formulas. Verify your results against the gear's known specifications.

3. Build your material selection instinct. Every time you encounter a gear failure, identify the failure mode (wear, pitting, tooth breakage, spalling) and ask whether a different material or heat treatment would have prevented it.

4. Start with spur gears. Master the spur gear formulas first — they are the foundation for every other gear type. Helical gear calculations are spur gear calculations adjusted for the helix angle. Bevel gear calculations are spur gear calculations projected onto a cone.

5. When in doubt, design for durability. Tooth breakage is dramatic but rare. Surface fatigue (pitting, spalling) is the dominant failure mode in well-designed gear sets. Your material selection, heat treatment, and surface finish decisions matter more than most engineers realize.

Here's the question that separates good engineers from great ones:

When was the last time you actually calculated a gear dimension instead of just accepting what was on the drawing?

The drawings were made by someone. That someone used these formulas. When you understand the formulas, you understand the design intent — and that understanding is what lets you solve problems that no drawing can anticipate.

Every formula in this guide has been verified against ANSI, AGMA, DIN, and British Standard specifications. The data tables reflect the Machinery's Handbook reference standards for gear design and manufacturing.

Overview

This set of notes covers three interconnected areas of mechanical power transmission and drive system design:

• Geared motor units — pre-engineered combinations of electric motors and gearboxes classified by drive type, including selection tables for output speed, torque, power, and unit sizing
• Spur and helical gears — fundamental gear types used in mechanical power transmission, covering gear geometry, velocity ratios, tooth parameters, module selection, design principles, clearance, and force analysis
• Electric motors — selection and specification of three-phase and single-phase electric motors, including performance data, efficiency characteristics, mounting arrangements, synchronous speeds, overhung load calculations, and a step-by-step motor selection method

Together, these topics form the core knowledge required to design and specify mechanical drive systems from the prime mover (motor) through the transmission (gears/gearbox) to the driven load.

Key Concepts

• Geared motor drive classifications define standard combinations of motor power, gear ratio, output speed, output torque, and unit frame size for pre-engineered gearmotor assemblies
• Velocity ratio (VR) is the fundamental relationship between driver and driven gear, determined by the ratio of teeth or pitch circle diameters
• Module (M) is the key sizing parameter for gear teeth, linking pitch circle diameter to the number of teeth and governing tooth strength
• Hunting teeth ensure even wear distribution by requiring no common factor between the number of teeth in the pinion and wheel
• Gear tooth forces consist of tangential, separating (radial), and (for helical gears) axial components — all of which must be accounted for in shaft and bearing design
• Gear efficiency is typically 95–96% per pair for well-machined, lubricated gears on rolling-element bearings; overall efficiency compounds across multiple stages
• Squirrel cage induction motors are the most common type in engineering applications, self-adjusting to load via changes in current draw and slip
• Motor selection follows a structured method: determine mechanical requirements → choose motor from performance tables → verify speed at design load → check overhung and thrust loads → extract dimensions

Geared Motor Units

Drive Classification System

• Geared motor units are pre-engineered assemblies combining an electric motor with an integrated gearbox
• Units are classified into drive classifications (e.g., Classification 2, 3, 4) which represent different ranges of output capability
• Each classification provides a selection table cross-referencing:
  • Nominal output speed (rev/min) — ranging from approximately 20 to 288 rev/min
  • Nominal gear ratio — typically from 5:1 up to 70:1
  • Motor power (kW) — ranging from 0.12 kW to 4.0 kW (varies by classification)
• For each combination, the table specifies:
  • Output power (kW)
  • Output torque (Nm)
  • Unit frame size (e.g., JPM11, JPM17, JPM22, JPM26, JPM30)

Selection Considerations

• As gear ratio increases, the available motor power range narrows (higher ratios support fewer high-power options)
• As output speed decreases (higher ratio), output torque increases proportionally
• Actual output speeds depend on the full-load speed of the motor and the exact gear ratio, and may differ from nominal speeds listed
• Higher drive classifications generally support higher output torques and powers for equivalent speed ranges
• Frame size increases with increasing power and torque requirements

Geared Motor Dimensions

• Geared motors are available with plug-in and solid output shafts
• Standard mounting is foot mounting (Type 2), with dimensions specified for each frame size
• Key dimensional parameters include:
  • Overall envelope (height, width, length)
  • Shaft dimensions (diameter, keyway, length)
  • Mounting bolt patterns (foot bolt spacing, flange PCD)
  • Centre height and shaft centreline offsets
• Dimensions scale with frame size — larger frames (e.g., JPM30) have significantly larger envelopes and shaft diameters than smaller frames (e.g., JPM11)

Spur and Helical Gears

Types of Gears

• Common gear types in engineering include: spur, helical, double helical (herringbone), bevel, hypoid, and worm
• Two gears in mesh are called a gear pair
  • Mesh is normally external, but may be internal (one gear has teeth cut internally)
  • The smaller gear is the pinion; the larger is the wheel
  • The gear transmitting input torque/power is the driver; the output gear is the driven
• In standard mechanical power transmission, the driver is typically the pinion — the wheel rotates slower, providing speed reduction
• When more than two gears are in continuous mesh, this forms a gear train
  • Simple gear train — gears in series on separate shafts; intermediate gears are called idler gears (they do not change the overall velocity ratio)
  • Compound gear train — multiple gear pairs where intermediate shafts carry both a wheel and a pinion; the overall VR is the product of individual pair VRs
  • Planetary (epicyclic) gear train — compact arrangement with a sun gear, planet gears, and a ring gear

Spur vs Helical Gears

• Spur gears have teeth cut parallel to the shaft axis
• Helical gears have teeth cut at an angle (the helix angle, α) to the shaft axis
  • Typical helix angle: ~20° for single helical, ~30–35° for double helical (herringbone)
  • In a helical gear pair, one helix must be right-hand and the other left-hand
  • Helical gears are inherently stronger than spur gears of the same module, allowing a smaller module selection (one standard size down)
  • Helical gears produce an axial force component not present in spur gears

Velocity Ratio (VR)

• For all gears except worm-and-wheel: VR = number of teeth in wheel ÷ number of teeth in pinion
• For a compound gear train: overall VR = product of individual pair VRs
• For a worm and wheel: VR = number of teeth in wheel ÷ number of starts in worm

• Very high velocity ratios are undesirable due to the large number of teeth needed on the wheel, making accurate machining difficult and requiring large centre distances

Number of Teeth

• It is impractical to have gears with too few teeth (below ~3 teeth causes profile issues)
• Rule of thumb minimums:
  • Spur gears: ≥17 teeth on the pinion
  • Helical gears (20° helix angle): ≥14 teeth on the pinion
• Hunting teeth — for maximum life with meshing gears, it is desirable to distribute wear uniformly among all teeth
  • The ideal condition (all teeth hunting) requires no common factor between the number of teeth in the pinion and the wheel
  • This ensures that the same teeth re-mesh only after the pinion has completed a number of revolutions equal to the number of teeth in the wheel
  • The velocity ratio in this case cannot be reduced to a simpler ratio

• Note: 20/38 and 20/40 have very low cycle repeats (2) because they share a common factor — these combinations lead to uneven wear

Gear Parameters and Geometry

• Pitch Circle Diameter (PCD) — the theoretical circle on which the gear teeth are considered to mesh; denoted as d for pinion and D for wheel
• The relationship between VR and PCD: VR = N/n = D/d (where N = teeth in wheel, n = teeth in pinion)
• Nominal centre distance: C = 0.5 × (d + D) — actual centre distance is usually slightly greater
• Addendum (A) — height of tooth above the PCD line
• Dedendum (B) — height of tooth below the PCD line
• Pressure angle (θ) — angle made by the tangent to the gears at the point of contact; usually 20° (assume unless stated otherwise)
• Pitch point (P) — the point of contact on the PCD; must remain fixed as gears mesh to maintain constant velocity ratio
• Involute profile — the standard tooth profile that keeps the pitch point fixed; can be visualised as the curve traced by unwinding a cord from a cylinder
• For a rack and pinion, the mating profile on the pinion is involute while the rack profile is a straight-sided form at the pressure angle

Clearance

• To minimise friction, teeth should contact only along the front face of the driver and back face of the driven
• Two types of clearance are required:
  • Radial clearance (bottom clearance) — obtained by making dedendum > addendum; usually B = 1.25A
  • Circumferential clearance — very small when gears are new; increases with wear; obtained by making centre distance slightly larger than nominal
• Circumferential clearance causes backlash — the back-and-forth play when one gear is held fixed and the other is rocked

Module (M)

• Module is one of the most important parameters in gear design, defined as: M = d/n = D/N (PCD divided by number of teeth)
• The module must be the same for both pinion and wheel in a gear pair
• Standard modules (first choice, in mm): 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40, 50
• As module increases, tooth size increases → stronger teeth capable of transmitting more torque and power
• Standard proportions based on module:
  • Addendum: A = M
  • Dedendum: B = 1.25 M
  • Tooth depth: A + B = 2.25 M
• Face width (W) rules of thumb based on loading:
  • Light loads: W = 8M
  • Moderate loads: W = 10M
  • Heavy loads: W = 12M
• The face width of the pinion is typically 5–10% larger than the wheel (depending on assembly tolerances)
• Module selection can be done using a module selection chart (log-log plot of power vs pinion speed with module lines)
  • For spur gears with face width = 10M and 18 teeth on pinion
  • Can also be used for face widths 8–12M and pinions with 17–19 teeth
  • Can be used for helical gears with helix angles up to 20° by choosing one standard size smaller module

Gear Design Approach

• A comprehensive gear design procedure references detailed engineering standards with dozens of variables and charts
• The most critical factor: the greater the loading, the larger the module (and therefore the larger the teeth)
• A simplified approach uses the module selection chart to determine appropriate module based on power and pinion speed

Gear Tooth Forces

• Forces act at the pitch point (point of contact on the PCD)
• The resultant transverse force F acts perpendicular to the tooth at the pitch point and represents the total load on the gear shaft at the gear location
• This resultant decomposes into:
  • Tangential force (Fₜ) — the useful force that transmits torque
  • Separating force (Fₛ) — the radial force pushing gears apart along the line of centres; keeps gears in mesh
  • θ is the pressure angle between F and Fₜ (typically 20°)

Spur Gear Forces

• Tangential force: Fₜ = 2T / d
  • Where T = torque (Nm), d = PCD (m)
• Separating force: Fₛ = Fₜ × tan θ
• Resultant transverse force: F = √(Fₜ² + Fₛ²)

Helical Gear Forces

• Tangential force is the same as for spur gears: Fₜ = 2T / d
• Separating force (modified): Fₛ = (Fₜ × tan θ) / cos α
  • Where α is the helix angle
• Axial force (additional): Fₐ = Fₜ × tan α
• Resultant transverse force: F = √(Fₜ² + Fₛ²)
• The resultant transverse force for a helical gear is only slightly larger than for a spur gear, but the axial force is an important additional load that must be carried by the bearings

Worked Examples Summary

Example — Spur Gear Pair Design:

• Given: VR in range 2.5–2.7, pinion teeth = 18, module = 5 mm
• Approach: Tabulate possible wheel teeth (45, 46, 47, 48) and check VR and hunting condition
• Result: 18:47 ratio selected (VR = 2.556, all teeth hunting — no common factors)
• PCD of pinion = 5 × 18 = 90 mm; PCD of wheel = 5 × 47 = 235 mm
• Centre distance = 0.5 × (90 + 235) = 162.5 mm
• Addendum = 5 mm, Dedendum = 6.25 mm, Tooth depth = 11.25 mm
• Face widths (moderate load): Wheel = 50 mm, Pinion = 53.5 mm

Example — Spur Gear Force Calculation:

• Given: PCD = 100 mm, Torque = 800 Nm
• Fₜ = 2 × 800 / 0.1 = 16 kN
• Fₛ = 16 × tan 20° = 5.82 kN
• F = √(16² + 5.82²) = 17 kN

Example — Helical Gear Force Calculation (20° helix angle):

• Fₜ = 16 kN (same as spur)
• Fₛ = (16 × tan 20°) / cos 20° = 6.2 kN
• F = √(16² + 6.2²) = 17.2 kN (only marginally greater than spur)
• Fₐ = 16 × tan 20° = 5.82 kN (additional axial load on bearings)

Electric Motors

Motor Types

• The two most common motor types used in engineering are:
  • Three-phase squirrel cage induction motor — the workhorse of industrial applications
  • Single-phase squirrel cage induction motor — used for domestic and light commercial applications where three-phase supply is unavailable

Three-Phase Motors

• Available configurations include: totally enclosed fan cooled, dust ignition proof, non-sparking, flameproof, two-speed, brake motors, geared motors, and slip ring motors
• The standard off-the-shelf configuration is the totally enclosed fan cooled type with protection designation IP55 or higher
• Data in standard references typically covers sizes from 0.18 to 110 kW (frame sizes 63–280)
• Frame size = distance in mm between the base of the motor feet and the centreline of the rotor — a common designation used by all manufacturers; as frame size increases, motor power increases

Synchronous Speeds (Three-Phase, 50 Hz)

Single-Phase Motors

• Available in three starting methods:
  • Permanent capacitor type
  • Capacitor start / induction run type
  • Capacitor start / capacitor run type
• Three mounting arrangements: standard foot mount, flange mounted, and "C" type face mount
• Only two synchronous speeds available: 2-pole (3000 rev/min) and 4-pole (1500 rev/min)

Motor Operating Characteristics

• Under no-load conditions, the actual motor speed equals the synchronous speed (approximately)
• Full-load speed is less than synchronous speed (the difference is called slip)
• Between no load and full load, the speed-load relationship is very close to linear — linear interpolation can be used for intermediate loads with little error
• Squirrel cage motors self-adjust to load: as load increases, current draw increases to provide the required torque
  • It is poor practice to overload the motor (excess current causes overheating and failure)
  • It is also poor practice to significantly undersize the load relative to motor capacity (motor runs at lower efficiency and wastes space/cost)
• Electric motors, like most prime movers, have lower efficiency at part load than at full load

Performance Data

• Performance tables list for each motor type:
  • Output power (kW), full-load speed (RPM)
  • No-load and full-load current (A), locked rotor current
  • Efficiency at 100% FL, 75% FL, and 50% FL
  • Power factor at 100% FL, 75% FL, and 50% FL
  • Full-load torque (Nm)
  • Starting torque, pull-up torque, maximum torque (multiples of FL torque)
  • Moment of inertia (J), net weight

Motor Mounting Arrangements

• B3 Footmount — standard arrangement with feet for floor mounting
• B5 Flangemount — flange mounted to driven equipment
• B14A / B14B Facemount — face-mounted configuration
• Additional variants exist for combined foot/flange mounting (Type F 160/280 series)
• Dimensions are standardised and tabulated by frame size for all mounting types

Overhung Load and Thrust

• When a gear, pulley, chain-wheel, or flywheel is directly attached to the motor shaft, it creates an overhung (radial) load

• This load must not exceed the motor's allowable value, or bearing life will be reduced

• Overhung load formula:

  F = (2fT) / d = (60fP) / (π × d × N)

  Where:

  • F = overhung load (N)
  • T = motor torque at design load (Nm) — not maximum/full-load torque
  • P = motor power at design load (W) — not maximum/full-load power
  • d = PCD of pulley, sprocket, or gear (m)
  • N = speed at design load (rev/min)
  • f = drive application factor:
    • Chain drive or toothed belt: f = 1.0
    • Gear drive: f = 1.25
    • Vee belt: f = 1.5
    • Flat friction belt: f = 2.0

• If overhung load is excessive, options include:

  • Use a larger pulley or gear (increases d, reducing F)
  • Use a larger motor (higher allowable load)
  • Use an intermediate (lay) shaft with its own bearings coupled to the motor via a flexible coupling — this prevents the overhung load from reaching the motor bearings