Gears — General
The geometry and kinematics of gearing: tooth nomenclature, the involute profile and the law of conjugate action, contact ratio and interference, tooth systems, gear-train ratios, and the force analysis of spur, bevel, helical, and worm gears that feeds the strength chapters to follow.
Learning objectives
- Recognize spur, helical, bevel, and worm meshes from shaft geometry and operating priorities.
- Relate tooth count, module, pitch diameter, center distance, pressure angle, and speed ratio.
- Build a tooth-contact free-body diagram and resolve tangential, radial, and axial forces.
- Screen a proposed mesh for conjugate action, adequate contact ratio, and involute interference.
- Synthesize and audit simple or compound trains while respecting integer teeth and manufacturable sizes.
- State which geometry comes from calculation and which tolerances, ratings, and limits require current standards or vendor data.
Physical meaning
A gear pair is simultaneously a rolling-geometry model and a repeating contact problem. Pitch circles set the ideal velocity ratio, involute flanks preserve that ratio as contact moves, and the tooth-normal force creates the shaft and bearing loads. Geometry predicts motion; it does not by itself establish tooth strength, fatigue life, lubrication adequacy, noise, or reliability.
Problem-solving strategy
- Recognize shaft arrangement, ratio direction, duty, and whether the task is kinematics, force analysis, or selection.
- Sketch pitch circles and a separate free-body diagram; define positive rotation and axial directions before assigning signs.
- Use compatible module and pressure-angle systems, determine integer tooth counts, then check center distance and interference.
- Obtain torque from power and speed, calculate the tangential force, and resolve the tooth-normal force into bearing-load components.
- Iterate tooth counts, module, helix angle, and face arrangement; audit ratio error, contact ratio, packaging, thrust, and manufacturability.
13–1Types of Gears
Four Families
Spur gears have teeth parallel to the axis and transmit motion between parallel shafts — the simplest and most common. Helical gears have teeth at an angle (the helix angle), engaging gradually for quieter, higher-load operation, but producing axial thrust. Bevel gears have teeth on a conical surface and connect intersecting shafts (usually at 90°). Worm gearsets (a worm meshing with a worm wheel) give very high ratios between non-intersecting perpendicular shafts, often with self-locking action.
13–2Nomenclature
The Pitch Circle and Tooth Size
The pitch circle is the theoretical circle on which calculations are based; mating gears' pitch circles are tangent. The smaller gear is the pinion, the larger the gear. Tooth size is indexed by the diametral pitch \(P = N/d\) (teeth per inch, US) or its reciprocal the module \(m = d/N\) (mm, SI). The circular pitch \(p\) is the arc distance between corresponding points on adjacent teeth.
Tooth Depth Terms
The addendum \(a\) is the radial height of the tooth above the pitch circle; the dedendum \(b\) is the depth below it; their sum is the whole depth. Clearance is the excess of dedendum over the mating addendum, and backlash is the excess of tooth space over tooth thickness on the pitch circle — a small deliberate gap that prevents jamming.
Recognition strategy
The cues are compatible module, integer tooth counts, and an external simple pair. Pitch geometry gives dimensions, while equality of pitch-line velocity gives the inverse speed ratio.
Given information
- \(N_P=24\), \(N_G=72\), \(m=2.50\ \text{mm}\), and \(n_P=+1440\ \text{rev/min}\).
- Clockwise pinion rotation is defined positive; external meshing reverses the rotation sign.
- Ideal rigid pitch circles, no slip, and compatible tooth system are assumed.
Model and interpretation
Model the mesh as two tangent pitch circles. Use \(d=mN\), \(c=(d_P+d_G)/2\), \(p=\pi m\), and \(n_G/n_P=-N_P/N_G\); backlash and profile tolerances do not alter this ideal ratio.
Verification
Units: Module in millimetres multiplied by a dimensionless tooth count produces millimetres; center distance and circular pitch therefore also have length units.
Signs and direction: One external contact reverses rotation, so a positive clockwise driver must produce a negative, counterclockwise driven speed as calculated.
Plausibility: The gear has three times as many teeth and three times the diameter, so one-third the speed, \(480\ \text{rev/min}\), preserves equal pitch-line velocity.
Engineering interpretation
The ideal pair provides exactly a \(3{:}1\) reduction. A released design must still specify pressure angle, tooth quality, profile modification, backlash, tolerances, lubrication, and an operating center-distance tolerance from the applicable gear standard and manufacturing process.
13–3Conjugate Action
Conjugate Profiles
Tooth profiles that satisfy this law are conjugate. While many curve pairs are theoretically conjugate, the involute is in near-universal use because it gives conjugate action, tolerates center-distance variation, and produces straight-sided rack teeth that are easy to manufacture.
13–4Involute Properties
The Involute of a Circle
An involute is traced by a point on a taut string unwound from a base circle. Its key property: the generating line (the string) is always normal to the involute and tangent to the base circle, so the line of action between two involute teeth is a straight line tangent to both base circles — guaranteeing the constant velocity ratio required by the law of gearing, even if the center distance changes slightly.
13–5Fundamentals
Pressure Angle, Base Circle, and Line of Action
The pressure angle \(\phi\) (standard values 20° or 25°, historically 14.5°) is the angle between the line of action and the common tangent to the pitch circles. The base circle from which the involute is generated relates to the pitch circle by \(r_b = r\cos\phi\). The base pitch \(p_b = p\cos\phi\) is the tooth spacing measured on the base circle and is the natural unit for contact-ratio calculations.
13–6Contact Ratio
How Many Teeth Share the Load
The contact ratio \(m_c\) is the average number of tooth pairs in contact — the length of the line of action \(L_{ab}\) divided by the base pitch. A contact ratio of 1.0 means one pair is always engaged; designers keep \(m_c \ge 1.2\) so that mounting inaccuracies cannot drop it below 1 (which would cause noise, vibration, and impact loading). Higher contact ratios share the load over more teeth and run more smoothly.
13–7Interference
Contact Below the Base Circle
Interference occurs when contact happens on the non-involute portion of the tooth (below the base circle, near the flank), as when the tip of the gear tooth digs into the flank of the pinion. It is most likely with small pinions and large gears. Eliminating it by undercutting weakens the tooth, so it is better avoided by limiting the minimum number of pinion teeth, which depends on the pressure angle and the gear ratio.
Practical Values
With \(k = 1\) (full-depth teeth) at \(\phi = 20°\): a 1:1 pair needs ≥ 13 teeth; a 4:1 pair needs a pinion of ≥ 16 teeth (meshing with 64). At \(\phi = 14.5°\) the 1:1 minimum jumps to 23 teeth — which is why the steeper 20°/25° pressure angles displaced the old 14.5° system. Stub teeth use \(k = 0.8\).
13–8The Forming of Gear Teeth
Generating and Forming Processes
Teeth are produced by forming (milling with a form cutter matching the tooth space, or casting/molding/stamping) or by generating (hobbing, shaping, and rack-cutting), where the cutter and blank roll together to generate the involute. Generation automatically avoids interference by undercutting if needed. Finishing operations — shaving, grinding, lapping, burnishing — improve accuracy and surface finish for quiet, durable high-speed gears.
13–9Straight Bevel Gears
Gears on Cones
Bevel gears transmit motion between intersecting shafts; their pitch surfaces are cones sharing a common apex. Tooth size varies along the face, so calculations use the large-end (back-cone) dimensions. The pitch-cone angles \(\gamma\) (pinion) and \(\Gamma\) (gear) satisfy \(\tan\gamma = N_P/N_G\) and \(\tan\Gamma = N_G/N_P\) for 90°-shaft sets. Tredgold's approximation uses an equivalent (virtual) spur gear on the back cone to apply spur-gear geometry.
13–10Parallel Helical Gears
Helix Angle and Two Pressure Angles
Helical-gear teeth lie on a helix at the helix angle \(\psi\). They engage gradually (a tooth contact starts at one end and sweeps across), giving smoother, quieter, higher-capacity operation than spur gears, at the cost of an axial thrust component. Two pressure angles relate through the helix angle: the transverse \(\phi_t\) and the normal \(\phi_n\). The normal and transverse circular pitches relate by \(p_n = p_t\cos\psi\).
13–11Worm Gears
High-Ratio, Right-Angle Drives
A worm (a screw-like gear of one or a few threads) meshes with a worm wheel to give large speed reductions (often 10:1 to 100:1+) in a compact right-angle package. The ratio is \(N_W/N_G\) where \(N_W\) is the number of worm threads (starts). The lead angle \(\lambda\) governs efficiency and whether the set is self-locking (cannot be back-driven), valued in hoists. The lead is \(L = p_x N_W\) and \(\tan\lambda = L/(\pi d_W)\).
13–12Tooth Systems
Standardized Proportions
A tooth system standardizes addendum, dedendum, working depth, and tooth thickness as multiples of the module (or fractions of diametral pitch), so any two gears of the same system and pitch will mesh. The dominant standards are full-depth involute teeth at 20° and 25° pressure angles (e.g. addendum \(a = 1/P = m\), dedendum \(b = 1.25/P = 1.25m\)). Standard systems exist for spur, helical, bevel, and worm gearing.
13–13Gear Trains
Train Value
The train value \(e\) is the product of driving tooth numbers over the product of driven tooth numbers; the last gear's speed is \(n_L = e\,n_F\). Idler gears change direction but not the magnitude of the ratio (they cancel out of \(e\)). A simple train reaches about 10:1; a compound train (two gears on a shared shaft per stage) reaches 100:1 or more. Split the overall ratio as evenly as possible between stages to minimize size.
Recognition strategy
Two driven-to-driver ratios multiply because the intermediate gears share a shaft. Integer teeth make the ratio discrete, so the candidate must be checked against speed tolerance and package dimensions rather than rounded early.
Given information
- Stage tooth counts are \(20/88\) and \(22/98\), with \(m=3.00\ \text{mm}\).
- Input speed is \(1750\ \text{rev/min}\); target output band is \(89.1\) to \(90.9\ \text{rev/min}\).
- All gears are external and ideal; the two gears on the intermediate shaft have equal speed.
Model and interpretation
Take input clockwise as positive. Compute reduction \(i=(88/20)(98/22)\), output \(n_o=n_i/i\), and each fixed-axis center distance from \(c=m(N_p+N_g)/2\). Two external meshes produce no net direction reversal.
Verification
Units: Tooth ratios and percentage error are dimensionless, speed retains revolutions per minute, and module times tooth count correctly returns millimetres.
Signs and direction: Each external mesh reverses rotation; two meshes therefore restore the input direction, consistent with a positive clockwise output.
Plausibility: Two stages near \(4.4{:}1\) should yield roughly \(19.4{:}1\); \(1750/19.6\approx89.3\), independently confirming the detailed arithmetic.
Engineering interpretation
The integer candidate meets the kinematic requirement, but acceptance still depends on interference, contact ratio, shaft layout, bearing reactions, tooth strength, standard cutter availability, backlash, and cumulative tolerance. Similar stage ratios reduce extreme gear diameters but do not prove minimum cost.
13–14Force Analysis — Spur Gearing
Transmitted Load, Torque, and Power
The tooth force resolves into a tangential component \(W^t\) (the transmitted load that does the work) and a radial component \(W^r = W^t\tan\phi\) (which does no work but loads the shaft and bearings). Torque relates to the transmitted load by \(T = W^t d/2\), and power by \(H = T\omega\). Designers usually start from given power and speed and solve for \(W^t\) using the pitch-line velocity \(V\).
13–15Force Analysis — Bevel Gearing
Three Force Components
Bevel-gear tooth forces have three components evaluated at the mean radius \(r_{av}\): tangential \(W^t\), radial \(W^r\), and axial (thrust) \(W^a\). With pressure angle \(\phi\) and pitch angle \(\gamma\), the radial and axial parts are \(W^r = W^t\tan\phi\cos\gamma\) and \(W^a = W^t\tan\phi\sin\gamma\). The axial component pushes the gears apart, so bevel-gear shafts need thrust bearings positioned to react it.
Recognition strategy
Torque first determines the work-producing tangential force at the mean radius. Pressure and pitch-cone angles then divide the separating force into radial and axial components for the bevel-pinion free body.
Given information
- \(T=180\ \text{N}\!\cdot\!\text{m}\), \(d_{\mathrm{mean}}=0.120\ \text{m}\), \(\phi=20.0^\circ\), and \(\gamma=35.0^\circ\).
- Positive tangential force opposes pinion motion; radial force points toward the axis and axial force acts along it.
- Steady torque, rigid alignment, negligible friction, and a single ideal contact force are assumed.
Model and interpretation
Isolate the pinion and place \(W^t\) tangent to the pitch circle, \(W^r\) toward the pinion center, and \(W^a\) along its axis. Resolve with \(W^r=W^t\tan\phi\cos\gamma\) and \(W^a=W^t\tan\phi\sin\gamma\).
Verification
Units: Torque divided by radius gives newtons, while trigonometric ratios are dimensionless; all three components and their resultant therefore remain forces.
Signs and direction: Magnitudes are reported because the mating-gear location and viewing direction are not specified; the free-body arrows set actual signs, and the mating gear receives equal and opposite contact-force components.
Plausibility: The radial and axial components must reconstruct \(1091.9\ \text{N}\), and the resultant must exceed \(3000\ \text{N}\); both geometric tests pass.
Engineering interpretation
The axial force is absent from a spur-gear model and must be reacted deliberately. Bearing arrangement, shaft deflection, housing stiffness, pitch-cone assembly, dynamic loading, tooth strength, lubrication, and alignment must be checked with current manufacturer and applicable standard data before release.
13–16Force Analysis — Helical Gearing
Tangential, Radial, and Axial from Two Angles
Helical-gear tooth force has three components set by the normal pressure angle \(\phi_n\) and the helix angle \(\psi\): tangential \(W^t\) (transmits power), radial \(W^r = W^t\tan\phi_t\) (separates the gears), and axial \(W^a = W^t\tan\psi\) (the thrust inherent to helical gearing). The thrust must be carried by the shaft bearings — a key drawback addressed by double-helical (herringbone) gears, whose opposing helices cancel the net thrust.
13–17Force Analysis — Worm Gearing
Forces and the Role of Friction
In a worm drive, the worm and wheel forces are related through the lead angle \(\lambda\), the normal pressure angle \(\phi_n\), and — uniquely important here — the coefficient of friction \(f\), because worm gears slide heavily. The worm's tangential force equals the wheel's axial force and vice versa (the right-angle geometry). Friction strongly governs efficiency, which falls sharply at small lead angles (where self-locking occurs). Heat dissipation often limits worm-gear capacity more than tooth strength.
Chapter recap
- Pitch geometry controls ideal ratio and packaging; involute contact preserves conjugate motion over modest center-distance variation.
- Contact ratio and interference are geometry screens, not substitutes for bending, pitting, scuffing, or fatigue checks.
- Tangential force transmits torque, while radial and axial components determine shaft, bearing, and housing reactions.
- Integer tooth selection is iterative because ratio accuracy, size, thrust, manufacturability, dynamics, and reliability interact.
Before moving on: be able to sketch the mesh and free body, compute pitch dimensions and train value, resolve gear forces with a declared sign convention, and list the standard or vendor data still needed before a geometry becomes a rated design.