Gear Geometry, Types, Rating and Selection: Gear Terminology

Gear Terminology: The Language You Must Speak

You cannot design, specify, or troubleshoot gears without mastering the terminology. Here is every critical term, defined precisely:

The Two Pitch Systems: Diametral Pitch vs. Module

The world uses two parallel systems for defining gear tooth size. Understanding both — and converting between them — is non-negotiable.

Diametral Pitch System (Inch-Based)

The diametral pitch system is dominant in the United States. Diametral pitch (P) is the number of teeth per inch of pitch diameter:

$$P = \frac{N}{D}$$

Where:

• $P$ = Diametral pitch
• $N$ = Number of teeth
• $D$ = Pitch diameter (inches)

The higher the diametral pitch number, the smaller the teeth. A 4-diametral-pitch gear has larger teeth than a 10-diametral-pitch gear. Each additional tooth increases the pitch diameter by $\frac{1}{P}$ inches.

Example: A gear with 20 teeth of 4 diametral pitch has a pitch diameter of $\frac{20}{4} = 5$ inches. Adding one tooth changes the diameter to $\frac{21}{4} = 5.25$ inches — an increase of 0.25 inches.

Module System (Metric-Based)

The module system is used globally wherever the metric system is standard. Module (m) is the pitch diameter in millimeters divided by the number of teeth:

$$m = \frac{D}{N}$$

Module is an actual dimension — it tells you how many millimeters of pitch diameter exist per tooth. A module of 2 means 2 mm of pitch diameter for each tooth.

Converting Between Systems

$$\text{Module} = \frac{25.4}{P}$$

$$P = \frac{25.4}{\text{Module}}$$

Spur Gears: The Workhorse of Power Transmission

the practitioner's stripped gear was a spur gear — the most common type in existence. External spur gears are cylindrical gears with straight teeth cut parallel to the axis. They transmit drive between parallel shafts rotating in opposite directions.

Strengths and Limitations

• Excellent at moderate speeds — simple, efficient, economical
• Tooth loads produce no axial thrust — simplifies bearing selection
• Tend to be noisy at high speeds — each tooth engages suddenly across its full face width
• Shafts rotate in opposite directions (external spur gears)

Master Formula Table: Standard Spur Gears

Tooth Proportions: 20° and 25° Involute Full-Depth Teeth (ANSI B6.1)

Practical Example: the practitioner Reverse-Engineers His Stripped Gear

the practitioner counted 40 teeth on the damaged gear. He measured the outside diameter at 10.5 inches. From the formula:

$$P = \frac{N + 2}{D_O} = \frac{40 + 2}{10.5} = 4 \text{ diametral pitch}$$

Pitch diameter: $D = \frac{40}{4} = 10.0$ inches

Addendum: $a = \frac{1}{4} = 0.250$ inches

Dedendum: $b = \frac{1.25}{4} = 0.3125$ inches

Whole depth: $h_t = \frac{2.25}{4} = 0.5625$ inches

Circular tooth thickness: $t = \frac{1.5708}{4} = 0.3927$ inches

With these numbers, the practitioner could now source a replacement — or machine one in-house.

Contact Ratio: Why Your Gears Must Exceed 1.0

The contact ratio tells you, on average, how many teeth are sharing the load at any given moment. For power transmission gears, the contact ratio should not be less than approximately 1.4 as a general rule. A ratio as low as 1.15 may be used in extreme cases, but only if tooth deflection, spacing errors, and manufacturing tolerances are carefully accounted for.

$$m_f = \frac{\sqrt{R_O^2 - R_B^2} + \sqrt{r_O^2 - r_B^2} - C\sin\phi}{p\cos\phi}$$

Where:

• $R_O$ = outside radius of first gear
• $R_B$ = base radius of first gear
• $r_O$ = outside radius of second gear
• $r_B$ = base radius of second gear
• $C$ = center distance
• $\phi$ = pressure angle
• $p$ = circular pitch

Undercut: The Hidden Killer of Small Pinions

When a pinion has a small number of teeth (below about 17 for 20° pressure angle), the hobbing process can cut into the base of the tooth below the involute profile. This undercut weakens the tooth dramatically and reduces the contact ratio.

To avoid undercut, the following condition must be satisfied:

$$\frac{b - r_c}{\sin\phi} + r_c + 0.5 \leq n\sin\phi$$

Where:

• $b$ = dedendum constant
• $r_c$ = hob or rack tip radius constant
• $n$ = number of teeth in the gear
• $\phi$ = pressure angle

The practical fix: For pinions with 10–17 teeth (20° pressure angle) or 10–11 teeth (25° pressure angle), increase the pinion's addendum and outside diameter while decreasing the mating gear's addendum by the same amount. This avoids undercut, increases tooth strength, and maintains the same velocity ratio and center distance.

Enlarged Pinions: The Addendum Modification

When a pinion has fewer than about 17 teeth, standard tooth proportions create problems. The solution is addendum modification — enlarging the pinion by increasing its outside diameter and tooth thickness at the pitch circle, while correspondingly reducing the mating gear's dimensions.

The enlarged circular tooth thickness at the standard pitch diameter:

$$t = \frac{p}{2} + e\tan\phi$$

Where $e$ is the amount the outside diameter is increased over standard.

Example: A pinion with 10 teeth of 5 diametral pitch (14½° pressure angle) has its outside diameter increased by 0.2746 inches. The circular pitch is 0.6283 inches.

$$t = \frac{0.6283}{2} + 0.2746 \times \tan(14.5°) = 0.3142 + 0.0710 = 0.3852 \text{ inches}$$

The mating gear's tooth thickness is correspondingly reduced:

$$t = \frac{p}{2} - e\tan\phi = 0.3142 - 0.0710 = 0.2432 \text{ inches}$$

Helical Gears: When Spur Gears Aren't Quiet Enough

Six months after fixing his spur gear problem, the practitioner faced a new challenge. The plant installed a high-speed bottling line running at 3,000 RPM. The spur gears in the drive train screamed. The noise was unbearable on the production floor, and vibration was shaking sensors out of calibration.

His chief engineer said two words: "Helical gears."

What Makes Helical Gears Different

Helical gears are cylindrical gears with teeth cut at an angle (helix angle) to the axis. Instead of the entire tooth face engaging simultaneously (as with spur gears), engagement is gradual — starting at one end of the tooth and progressing to the other.

This produces:

• Superior load-carrying capacity compared to spur gears
• Significantly quieter operation — essential for high-speed applications
• Smooth, continuous power transmission due to gradual tooth engagement
• Axial thrust loads — the trade-off for all those benefits

Types of Helical Gears

• Parallel-shaft helical gears transmit drive between shafts rotating in opposite directions (like spur gears, but quieter)
• Crossed helical gears mesh together on non-parallel, non-intersecting axes
• Herringbone (double helical) gears cancel axial thrust by combining right-hand and left-hand helices on the same gear

Master Formula Table: Helical Gears

Where:

• $P_n$ = Normal diametral pitch of cutter
• $N$ = Number of teeth
• $\alpha$ = Helix angle
• $D$ = Pitch diameter
• $L$ = Lead of tooth helix

Axial Thrust: The Price of Quiet

Every helical gear produces an axial force along the shaft. The direction of thrust depends on three factors:

1. The hand of the helix (right or left)
2. The direction of rotation
3. Whether the gear is the driver or driven

Changing any one of these three factors reverses the thrust direction. Your bearing selection must account for this axial load.

Herringbone Gears: Canceling the Thrust

Double helical or herringbone gears combine right-hand and left-hand helices on a single gear body. The opposing thrust forces cancel each other out, eliminating the need for thrust bearings while retaining all the advantages of helical tooth engagement.

Herringbone gears are commonly used where pitch-line velocities range from 1,000 to 3,000 feet per minute in commercial gearing and up to 12,000 feet per minute or higher in specialized installations such as marine reduction gears, turbine drives, and high-speed motor drives.

The primary failure mode of herringbone gears is not tooth breakage but rather excessive wear or sub-surface failures such as pitting and spalling. Design is therefore based on durability — keeping tooth pressures within the allowable limits for wear. Tests have established that a critical surface pressure exists for any given set of material properties and friction coefficients. Above this critical value, wear is rapid and gear life is short. Below it, wear is negligible.

Enlarged Helical Pinions

Just as with spur gears, helical pinions with fewer than 24 teeth should be enlarged to avoid undercut and improve contact conditions. The enlargement factor $K_h$ for full-depth pinions of 20° normal pressure angle at 1 normal diametral pitch is calculated as:

$$K_h = 2.1 - (\sin\phi_t - \cos\phi_t \tan 5°)\sin\phi_t$$

Where the transverse pressure angle $\phi_t$ is found from:

$$\tan\phi_t = \frac{\tan\phi_n}{\cos\psi}$$

The enlarged outside diameter is then:

$$d_o = d + \frac{2 + K_h}{P_n}$$

Bevel Gears: Connecting Intersecting Shafts

the practitioner's third challenge came from the maintenance department. A right-angle drive in a conveyor system was failing every four months. The bevel gears showed heavy wear on one end of the teeth — classic misalignment. But before he could fix the problem, he needed to understand the five types of bevel gears and when each one is appropriate.

What Are Bevel Gears?

Bevel gears are conical gears — gears shaped like truncated cones — used to connect shafts with intersecting axes. The most common configuration is a 90° shaft angle, but bevel gears can be designed for virtually any angle between the shafts.

Types of Bevel Gears

Straight Bevel Gears are the most commonly used type. Their teeth are straight, with sides tapered so they would intersect the axis at a common point (the pitch cone apex) if extended inward. They are:

• The easiest to calculate and most economical to produce
• Best suited for peripheral speeds up to 1,000 feet per minute
• Recommended when maximum smoothness and quietness are not the primary concern
• Ideal for small production lots where fixed charges must be minimized
• Widely used in differentials because they allow compact, inexpensive designs with plain bearings

Zerol Bevel Gears have curved teeth lying in the same general direction as straight bevel gears. Think of them as spiral bevel gears with zero spiral angle. They are:

• Recommended when hardened gears of high accuracy are required (because Zerol gears can be ground)
• Preferred when only spiral-type cutting equipment is available
• Manufactured on the same machines as spiral bevel gears

Spiral Bevel Gears have curved oblique teeth that contact each other smoothly and gradually from one end to the other. They are:

• Recommended for peripheral speeds exceeding 1,000 feet per minute or 1,000 RPM
• Capable of producing smoother, quieter operation than straight or Zerol bevels
• Essential when extreme smoothness is desired, even at lower speeds
• Practical with smaller numbers of pinion teeth than straight bevels (due to continuous pitch line contact)
• Require grinding for speeds above 8,000 feet per minute

Hypoid Gears resemble spiral bevel gears but have non-intersecting, non-parallel axes — the pinion axis is offset from the gear axis. They are:

• Recommended when maximum smoothness of operation is desired
• Ideal for high reduction ratios where compactness, smoothness, and maximum pinion strength matter
• Essential for non-intersecting shafts
• The sliding action along the tooth length makes them even smoother than spiral bevels
• Widely used in automotive rear axles

Important: Bevel and hypoid gears may be used for both speed-reducing and speed-increasing drives. However, in speed-increasing drives, keep the ratio as low as possible and mount the pinion on anti-friction bearings — otherwise bearing friction will cause the drive to lock.

Bevel Gear Application Guide

Design of Bevel Gear Blanks: Critical Principles

The quality of a finished bevel gear depends heavily on the blank design:

• Sufficient metal under tooth roots — the amount of metal under the root should equal the whole depth of the tooth, maintained under both the large and small ends
• Webless ring gears need minimum stock between root line and tap drill holes of one-third tooth depth
• Rigidity for chucking — bores, hubs, and locating surfaces must be properly proportioned to gear diameter and pitch
• For heavily loaded gears, analyze the direction and magnitude of forces before designing the gear and its mounting

Bevel Gear Mounting: Deflection Limits

Bevel and hypoid gears in general should be mounted on anti-friction bearings in an oil-tight case. Critical guidelines:

• Bearing spacing should never be less than 70% of the pitch diameter
• On overhung mounted gears, the spread should be at least 2.5 times the overhang
• Shaft diameter should equal or exceed the overhang to provide sufficient stiffness
• Axial thrust should be taken at one place only — near the gear where the greatest thrust is developed
• Always provide for adjusting both gear and pinion axially during assembly

Worm Gears: High Ratios, Compact Packages

the practitioner's plant manager wanted to add a slow, precise indexing mechanism to a rotary table. The motor ran at 1,750 RPM. The table needed to turn at 35 RPM. That's a 50:1 ratio. A gear train with conventional gears would be enormous. The answer was a worm and wormgear.

How Worm Gears Work

Worm gears transmit motion between shafts at right angles (typically) that do not lie in a common plane. The worm resembles a screw thread, and the wormgear (wheel) has teeth that mesh with the worm's threads.

Worm gearing is divided into two general classes:

Fine-Pitch Worm Gearing is used primarily to transmit motion rather than power. Tooth strength is rarely an important factor — durability and accuracy are what matter. Housing constructions and lubrication methods differ significantly from coarse-pitch gearing.

Coarse-Pitch Worm Gearing is designed for power transmission. Here, tooth strength, load capacity, and efficiency are primary concerns.

The Efficiency Trade-Off

Worm gears have line tooth contact and can achieve very high ratios in a single stage. But there's a fundamental trade-off: the higher the ratio, the lower the efficiency. This is because higher ratios mean lower lead angles, which increase the sliding friction between the worm and wormgear.

Single-Thread vs. Multi-Thread Worms

Single-thread worms are used to obtain high ratios (up to 50:1 as a general rule, though ratios up to 100:1 are possible). They are comparatively inefficient due to the low lead angle. Single-thread worms are employed:

• When a large speed reduction is necessary in one stage
• As a means of adjustment
• When mechanical advantage or self-locking are important factors

Multi-thread worms are designed for efficient power transmission. The lead angle should be as high as possible — preferably between 25° and 45° — which requires multiple threads. Common thread counts range from 1 to 6 or 8, depending on the ratio.

Example combinations for a 6:1 ratio:

Pro tip: The number of wormgear teeth may deliberately not be an exact multiple of the worm threads to obtain "hunting tooth" action, which promotes even wear across all teeth.

Standard Axial Pitches for Fine-Pitch Worm Gearing (ANSI B6.9)

Eight standard axial pitches cover the normal range: 0.030, 0.040, 0.050, 0.065, 0.080, 0.100, 0.130, and 0.160 inch.

Axial pitch is the design basis because:

1. It establishes the lead — a basic dimension for worm production and inspection
2. The axial pitch of the worm equals the circular pitch of the gear in the central plane
3. Only one set of change gears or master lead cam is required for a given lead, regardless of lead angle

Worm Gear Material Selection

Worm gears demand careful material pairing to manage the sliding contact:

The S.A.E. nickel phosphor gear bronze (No. 65 + Ni) contains 87% copper, 11% tin, 2% nickel, and 0.2% phosphorus — an excellent combination for worm gears.

Planetary (Epicyclic) Gearing: Compact Power, Inline Shafts

When the practitioner was asked to redesign a conveyor drive to fit within an impossibly tight space — with driving and driven shafts that had to be perfectly inline — he discovered the elegant world of planetary gearing.

What Makes Planetary Gears Special

Planetary (epicyclic) gearing provides:

• Compact design — far smaller than equivalent conventional gear trains
• Driving and driven shafts in line — critical for space-constrained applications
• Large speed reductions available in a single stage
• Load sharing across multiple planet gears, increasing power density

How It Works

A basic planetary gear set consists of three elements:

1. Sun gear — the central gear
2. Planet gears — typically 3–5 gears that orbit the sun gear while meshing with both the sun and the ring
3. Ring gear (annulus) — the outer internal gear

By fixing one element and driving another, you control the speed and direction of the output:

Speed Ratio Formulas

For the simplest planetary arrangement (sun gear + planet + ring gear):

Speed reduction (ring fixed, sun drives carrier):

$$F = \frac{1}{1 + \frac{C}{B}}$$

Speed increase (ring fixed, carrier drives sun):

$$D = 1 + \frac{C}{B}$$

Where:

• $F$ = rotation of follower per revolution of driver
• $D$ = rotation of driver per revolution of follower
• $B$ = size of driven gear (teeth or pitch diameter)
• $C$ = size of fixed gear (teeth or pitch diameter)

Direction of Rotation

If the final result of the speed formula is negative, the driver and follower rotate in opposite directions. If positive, both rotate in the same direction.

Compound Drives

When two driving members rotate at different speeds, compound planetary arrangements become possible. The central shaft with its attached link is one driver, and the internal gear (instead of being fixed) is rotated as the second driver. This opens up an enormous range of possible ratios and configurations.

Planetary Bevel Gears

Two forms of planetary gears use the bevel type:

1. Simple planetary bevel — the planet gear rotates about a fixed bevel gear, with the driven shaft at the center
2. Humpage reduction gear — sometimes called cone-pulley back-gearing, used within the cone pulleys of certain machine tools

Ratchet Gearing: Controlled Intermittent Motion

Types and Applications

Ratchet gearing serves two primary functions:

1. Transmitting intermittent motion — converting oscillating input into one-direction rotary output
2. Preventing backward rotation — holding loads in hoisting mechanisms, jacks, and similar equipment

Key Variants

Simple ratchet and pawl — A toothed ratchet wheel and a pivoted pawl. When the lever oscillates, the pawl drives the wheel in one direction while slipping over the teeth on the return stroke.

Stationary pawl — The pawl is fixed to a stationary member and prevents the ratchet wheel from rotating backward (used in hoisting drums and winches).

Locking ratchet — The pawl prevents rotation in either direction as long as it engages the wheel.

Multiple-pawl ratchet — Two pawls of different lengths (differing by half the tooth pitch) effectively halve the practical pitch, providing finer feed from relatively coarse teeth. This approach is preferable to using a single fine-pitch ratchet, which would have weaker teeth.

Reversing ratchet — Teeth shaped so either side can be used for driving, with a double-ended pawl that can be repositioned.

Frictional ratchet — Uses rollers or balls between the ratchet wheel and an outer ring. Rotation in one direction causes the rollers to wedge (driving), while the opposite direction allows free rotation. No positive tooth engagement — motion is transmitted through friction.

Designing Ratchet Wheel Teeth

Critical design rule: The faces of the engaging teeth must be angled so that a line perpendicular to the tooth face passes between the center of the ratchet wheel and the center of the pawl pivot. This ensures the pawl stays engaged under load.

Calculating Ratchet Tooth Pitch

For ratchet wheels used in holding suspended loads:

$$P = \sqrt{\frac{M}{L \times S \times N \times F}}$$

Where:

• $P$ = circular pitch (inches), measured at the outside circumference
• $M$ = turning moment on the ratchet wheel shaft (inch-pounds)
• $L$ = length of tooth face (inches) — the thickness of the ratchet gear
• $S$ = safe stress (2,500 psi with shock; 4,000 psi without shock, for steel)
• $N$ = number of teeth in ratchet wheel
• $F$ = factor: 50 for ≤12 teeth; 35 for 12–20 teeth; 20 for >20 teeth

Gear Materials: Choosing the Right Steel for the Job

The fourth lesson the practitioner learned the hard way. He had the right gear geometry but specified the wrong material. Within three months, the replacement gear showed pitting on the tooth flanks — classic surface fatigue failure from insufficient hardness.

Classification of Gear Steels

Gear steels fall into three fundamental categories:

1. Casehardening steels — extremely hard, fine-grained case with a comparatively soft, ductile core; used when maximum wear resistance is the priority
2. Through-hardening steels — uniform hardness from surface to core; used when strength, toughness, and shock resistance are paramount
3. Machined-after-treatment steels — hardened then tempered to a machinable hardness; used when grinding is impractical and high accuracy is required

When to Use Casehardening Steels

Casehardening steels provide:

• Maximum resistance to wear
• A fine-grained, extremely hard surface (when properly treated)
• A relatively soft, ductile core that absorbs shock

The critical requirement: Casehardened steels should be double-quenched to realize the full benefits of the core properties. This is especially true for alloy steels — the benefits rarely justify the cost without refining the core through a second quench. The penalty is increased distortion.

When to Use Through-Hardening Steels

Through-hardening steels provide:

• Great strength and high endurance limits
• Toughness and resistance to shock
• Fairly high surface hardness (though not as high as casehardened steels)

The limitation: These steels become distorted when hardened. They are unsuitable for high-speed gearing where noise matters or where accuracy is critical — unless the teeth can be ground after hardening.