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Machine Elements · Chapter 10 of 10 · Advanced

Spur and Helical Gears

Gears trade speed for torque at a fixed ratio. The force on a tooth comes from the power and the pitch line, and the Lewis equation treats that tooth as a small cantilever beam.

01

Readiness check

This closing chapter ties forces to gear geometry. Tick only what you can do closed-notes.

  • Relate power, torque, and angular speed.
  • Find a pitch-line velocity from diameter and speed.
  • Resolve a force into components with an angle.
  • Recall bending stress in a cantilever.
  • Read a form factor from a table.
0 or 1 weak itemsContinue with this chapter.
2 weak itemsRevisit bending stress in Chapter 2.
3 or more weak itemsReview torque and power before continuing.
02

The core idea

A gear transmits power as a tangential force at its pitch circle. That force bends the tooth like a tiny cantilever, and the Lewis equation sizes it against bending.

d = mN, V = πdn/60Wt = P/V, Wr = Wt tan φσ = Wt/(b·m·Y)

The module m sets the tooth size, and the pitch diameter is d = mN for N teeth. Power flowing through the mesh appears as a tangential force Wt = P/V at the pitch line, plus a radial component Wr = Wt tan φ that pushes the gears apart. That tangential force bends each engaged tooth, and the Lewis equation treats the tooth as a cantilever, giving a bending stress σ = Wt/(b·m·Y) with the form factor Y capturing the tooth shape. A velocity factor then accounts for dynamic load at speed.

The skill works when: you find the tangential force from power and pitch-line velocity, then bend the tooth with Lewis.
The skill breaks down when: the radial force is treated as load-carrying, or the velocity factor is ignored at high speed.
The concept. Two gears mesh at the pitch point, where the transmitted power becomes a tangential force Wt and a radial force Wr. The tangential force bends the teeth and drives the load.
03

The skills, taught in order

Five skills set the gear geometry, the forces at the mesh, the velocity ratio, the Lewis bending stress, and the dynamic correction.

10.1 Gear geometry

The module m (millimetres of pitch diameter per tooth) sets tooth size, so the pitch diameter is d = mN. Standard teeth use a 20° pressure angle and full-depth proportions. Two gears mesh only if they share a module and pressure angle.

10.2 Gear forces

The power transmitted equals the tangential force times the pitch-line velocity, so Wt = P/V = 2T/d. The normal tooth force also has a radial component Wr = Wt tan φ that separates the gears and loads the bearings but transmits no power.

10.3 Velocity ratio

Meshing gears have equal pitch-line velocity, so their speeds are inversely proportional to their tooth counts: n1/n2 = N2/N1. A small pinion driving a large gear trades speed for torque, the reason gearboxes exist.

10.4 Lewis bending stress

Treating the tooth as a cantilever loaded at its tip, the Lewis equation gives σ = Wt/(b·m·Y), with b the face width and Y the Lewis form factor from the tooth count. More teeth give a larger Y and a stronger tooth.

Number of teethLewis form factor Y
160.296
180.309
200.322
300.359

Lewis form factor for 20° full-depth teeth (Shigley, Table 14-2). Fewer teeth give a weaker, more pointed tooth.

10.5 Velocity factor and AGMA

At speed, meshing errors add a dynamic load, raised by a velocity factor Kv = (6.1 + V)/6.1 for cut teeth. The full AGMA method extends Lewis with factors for load distribution, size, and surface durability (pitting), but the bending core stays the Lewis form.

Engineering connection: the gear forces here set the bending moment on the shaft of Chapter 6 and the radial load on the bearings of Chapter 9, closing the drivetrain.

04

Worked example 1: forces on a spur gear

A 20-tooth spur pinion of module 4 mm runs at 1200 rev/min and transmits 5 kW. With a 20° pressure angle, find the pitch-line velocity, the tangential force, and the radial force.

Figure 1. The pitch diameter follows from module and tooth count; power over pitch-line velocity gives the tangential force, and the pressure angle gives the radial split.
  1. ProblemFind V, Wt, and Wr for the pinion in Figure 1.
  2. Given / findN = 20, m = 4 mm, n = 1200 rev/min, P = 5 kW, φ = 20°. Find V, Wt, Wr.
  3. AssumptionsPower transmitted entirely at the pitch line; standard 20° pressure angle.
  4. ModelPitch diameter d = mN; velocity V = πdn/60; Wt = P/V; Wr = Wt tan φ.
  5. Equationsd = mN, V = πdn/60Wt = P/V, Wr = Wt tan φ
  6. Solved = 4 × 20 = 80 mm = 0.080 m. V = π(0.080)(1200)/60 = 5.03 m/s. Wt = 5000/5.03 = 995 N. Wr = 995 × tan 20° = 995 × 0.364 = 362 N.
  7. CheckBy torque, T = P/ω = 5000/(2π·1200/60) = 39.8 N·m, and Wt = 2T/d = 2(39.8)/0.080 = 995 N, matching. The radial force is about a third of the tangential, as tan 20° gives.
  8. ConclusionThe tangential force does the work and bends the teeth; the radial force only loads the bearings. Both are needed for the shaft and bearing design.
Result. V = 5.03 m/s, Wt = 995 N, Wr = 362 N.
05

Worked example 2: Lewis bending stress

The same gear (Wt = 995 N, module 4 mm, 20 teeth so Y = 0.322) has a face width of 40 mm. Find the Lewis bending stress, then apply the velocity factor for V = 5.03 m/s.

Figure 2. The Lewis model treats the tooth as a cantilever loaded by the tangential force; the velocity factor inflates the static stress for the dynamic load at running speed.
  1. ProblemFind the Lewis bending stress, static and velocity-corrected, for the tooth in Figure 2.
  2. Given / findWt = 995 N, m = 4 mm, Y = 0.322, b = 40 mm, V = 5.03 m/s. Find σ and the corrected σ.
  3. AssumptionsTooth as a cantilever loaded at the tip; single tooth carries the full load; cut teeth (Barth velocity factor).
  4. ModelLewis σ = Wt/(b·m·Y), then multiply by Kv = (6.1 + V)/6.1.
  5. Equationsσ = Wt/(b·m·Y)Kv = (6.1 + V)/6.1
  6. Solveσ = 995/(40 × 4 × 0.322) = 995/51.5 = 19.3 MPa. Kv = (6.1 + 5.03)/6.1 = 1.82. Corrected σ = 1.82 × 19.3 = 35.2 MPa.
  7. CheckThe static stress is modest, but the velocity factor nearly doubles it, showing why speed matters. A steel gear with this stress has a large bending margin; pitting durability would likely govern instead.
  8. ConclusionLewis sizes the tooth against bending, and the velocity factor brings in dynamic load. The full AGMA method refines this, but the cantilever idea is the core.
Result. Static σ = 19.3 MPa; with Kv = 1.82, σ = 35.2 MPa.
06

Misconceptions and diagnostics

MistakeSymptomDiagnostic questionCorrection
Using the radial force for powerWrong transmitted load"Which force is tangent to the pitch circle?"Only Wt transmits power; Wr just loads bearings.
Ignoring the velocity factorStress under-predicted at speed"Is the pitch-line velocity high?"Apply Kv = (6.1 + V)/6.1 for cut teeth.
Wrong Y for the tooth countStress off for the pinion"Did I read Y at this number of teeth?"Use the form factor for the actual tooth count.
Checking only bendingGear pits before it breaks"Is surface durability checked too?"Also check contact (pitting) stress in the AGMA method.
07

Practice ladder

Level 1 · Direct skill

A gear of module 5 mm has 30 teeth. Find its pitch diameter.

Show answer

d = mN = 5 × 30 = 150 mm. The module times the tooth count is the pitch diameter directly.

Level 2 · Mixed concept

The pinion of Worked Example 1 drives a 60-tooth gear. Find the output speed and the velocity ratio.

Show answer

n2 = n1(N1/N2) = 1200(20/60) = 400 rev/min. The ratio is 3:1, trading speed for three times the torque.

Level 3 · Independent problem

A spur gear transmits Wt = 1500 N with module 5 mm, face width 50 mm, and 18 teeth (Y = 0.309). Find the static Lewis stress.

Show answer

σ = Wt/(b·m·Y) = 1500/(50 × 5 × 0.309) = 1500/77.25 = 19.4 MPa.

Level 4 · Transfer to real engineering

Find a real geartrain (a drill, a bicycle, a clock). Identify the pinion and gear, estimate the ratio, and explain what it trades and why.

What good work looks like

A tooth-count ratio that matches the speed or torque change, and a clear statement of what the gearset is for: speed reduction, torque multiplication, or direction change.

08

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I used the tangential force, not the radial, for the transmitted power."
"Give me three gear pairs; I will find the velocity ratio for each."
"What is the tooth stress?" Treating the tooth as a cantilever yourself is the skill.
"Design this gearset." Finding the forces and bending stress is the point.

Portfolio task

Analyse a real gear pair: find the module and ratio, compute the tangential and radial forces from the power, and find the Lewis bending stress with the velocity factor.

Must include: the geometry and ratio, the gear forces, and a velocity-corrected Lewis stress.
09

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Relate pitch diameter, module, and tooth count.

d = mN, the module times the number of teeth.

2. Write the tangential and radial gear forces.

Wt = P/V = 2T/d, and Wr = Wt tan φ.

3. State the velocity ratio.

n1/n2 = N2/N1: speeds are inverse to tooth counts.

4. Write the Lewis bending stress.

σ = Wt/(b·m·Y), with Y the Lewis form factor.

5. Why apply a velocity factor?

Meshing errors add a dynamic load at speed, raised by Kv = (6.1 + V)/6.1.

TodayFinish this quiz and Levels 1 and 2 of the ladder.
+1 dayRe-derive the gear forces and Lewis stress from a blank page.
+3 daysAnalyse a second gear pair.
+7 daysCombine shaft, bearing, and gear into one drivetrain design.
+30 daysRevisit the whole course through the Machine Elements hub.
10

Textbook mapping

ItemMapping
Primary sourceBudynas and Nisbett, Shigley's Mechanical Engineering Design, Chapters 13 and 14 (Gears and Spur/Helical Gears)
Cross-referenceNorton, Ch. 12 · Manufacturing Processes (gear cutting)
Core topics10.1 Gear geometry · 10.2 Gear forces · 10.3 Velocity ratio · 10.4 Lewis bending stress · 10.5 Velocity factor and AGMA
Engineering connectionGear forces set the loads on the shaft and bearings, completing the drivetrain.
Read nextReturn to the Machine Elements hub and integrate the elements.