Spur gear design: Lewis and Buckingham equations

Spur gear forces, the Lewis beam strength with Barth velocity factor, Buckingham's dynamic load and wear strength, and the weaker-member check, with power-capacity and bending-plus-wear examples.

Drafted with Aria, reviewed by the AiCanCode.org team. Spotted an error? Use Give Feedback at the bottom of the page.

Why it matters

Gearbox, timing-gear, oil-pump and starter-motor gears must transmit torque for millions of tooth engagements without the teeth breaking at the root or pitting on the flanks. The Lewis equation sizes the tooth against bending, and Buckingham's equations add the dynamic load caused by tooth errors and the wear (surface) strength. Together they decide the module and face width.

Key ideas

Geometry: module m = d/z (mm), pitch circle diameter d = m·z, circular pitch p = π·m, face width b usually 9.5m to 12.5m (take 10m as a first guess), pressure angle φ = 20° for full-depth involute teeth. The pinion is the smaller gear; with the same material it is the weaker one in bending, because it has fewer teeth and a smaller form factor.

Forces: tangential force F_t = 2T/d = P/v does the work; radial force F_r = F_t·tan φ loads the bearings. Pitch-line velocity v = π·d·N/60.

Lewis bending equation: the tooth is treated as a cantilever loaded at its tip by F_t, with the critical section where a parabola of uniform strength touches the root fillet. The resulting beam strength is S_b = σ_b·b·m·Y, where the Lewis form factor Y (based on module) depends on tooth number and tooth system; for 20° full-depth teeth some data books give Y = 0.484 − 2.87/z, while tabulated values differ slightly between sources (for 20 teeth, about 0.32–0.34), so use the source your course follows. Some books use y = Y/π based on circular pitch.

Assumptions of Lewis: the full load acts at the tip of one tooth, radial force and stress concentration are ignored, and the load is static. Real gears share load between teeth and see dynamic loads, so the equation is used with corrections:

  • Barth velocity factor C_v (for ordinary cut teeth, C_v = 3/(3 + v) for v < 10 m/s; 6/(6 + v) for carefully cut teeth up to 20 m/s; 5.6/(5.6 + √v) for precision gears). The effective load is F_eff = C_s·F_t / C_v, with C_s a service factor.
  • Buckingham's dynamic load for accurate design: F_d = 21v·(bC + F_t) / (21v + √(bC + F_t)), added to the transmitted load. C is the deformation factor (N/mm), proportional to the tooth error e and depending on the materials and tooth form; take C (or C per unit error) from your data book.

Weaker member: when pinion and gear are different materials, compare σ_b·Y for each; the smaller product is the weaker gear and is designed first.

Wear (surface) strength: pitting depends on contact stress. Buckingham's wear load is S_w = b·Q·d_p·K, with ratio factor Q = 2z_g/(z_g + z_p) for external gears and load-stress factor K from the surface endurance strength and the elastic moduli (data book). For safety, both S_b and S_w must exceed the effective load F_eff = C_s·F_t + F_d.

Failure modes: tooth breakage (bending fatigue), pitting (surface fatigue), scoring (lubricant film breakdown) and abrasive wear.

Formulas

d = m·z, v = π·d·N / 60 F_t = 2T / d = P / v, F_r = F_t·tan φ Lewis: S_b = σ_b·b·m·Y Y ≈ 0.484 − 2.87 / z (20° full depth) Barth: C_v = 3 / (3 + v) (v < 10 m/s) F_eff = C_s·F_t / C_v (velocity-factor method) Buckingham dynamic load: F_d = 21v·(b·C + F_t) / (21v + √(b·C + F_t)) F_eff = C_s·F_t + F_d (Buckingham method) Wear strength: S_w = b·Q·d_p·K, Q = 2z_g / (z_g + z_p) Factor of safety: n = S_b / F_eff

Symbols: m = module (mm); z, z_p, z_g = number of teeth (pinion, gear); d, d_p = pitch diameter (mm); N = speed (rpm); v = pitch-line velocity (m/s); T = torque (N·mm); P = power (W); F_t, F_r = tangential and radial tooth forces (N); φ = pressure angle; σ_b = permissible bending stress (MPa); b = face width (mm); Y = Lewis form factor (–); C_v = velocity factor (–); C_s = service factor (–); C = deformation factor (N/mm); F_d = dynamic increment (N); Q = ratio factor (–); K = load-stress factor (MPa); S_b, S_w = beam and wear strengths (N). In Buckingham's equation, v must be in m/s, b in mm and C in N/mm.

Worked examples

Example 1 (standard). A steel pinion with 20 teeth, module 5 mm, face width 50 mm runs at 1000 rpm. Y = 0.32 (from the table), permissible bending stress 150 MPa, ordinary cut teeth, service factor 1. Find the power it can transmit by the Lewis–Barth method.

  1. d = 5 × 20 = 100 mm; v = π × 0.1 × 1000/60 = 5.236 m/s.
  2. C_v = 3/(3 + 5.236) = 0.3643.
  3. S_b = 150 × 50 × 5 × 0.32 = 12 000 N.
  4. Permissible F_t = S_b·C_v = 12 000 × 0.3643 = 4371 N.
  5. P = F_t·v = 4371 × 5.236 = 22.9 kW.

Example 2 (GATE level). The same pinion transmits F_t = 3000 N to a 60-tooth gear. From the data book the deformation factor is C = 228 N/mm for the expected tooth error, and K = 1.8 MPa. Service factor 1. Find the factors of safety in bending and wear by Buckingham's method.

  1. b·C + F_t = 50 × 228 + 3000 = 14 400 N; √14 400 = 120; 21v = 21 × 5.236 = 109.96.
  2. F_d = 109.96 × 14 400 / (109.96 + 120) = 6886 N.
  3. F_eff = 3000 + 6886 = 9886 N.
  4. Bending: n = 12 000 / 9886 = 1.21.
  5. Q = 2 × 60/(60 + 20) = 1.5; S_w = 50 × 1.5 × 100 × 1.8 = 13 500 N; n = 13 500/9886 = 1.37.
  6. Both exceed 1, but the margins are small; more accurate teeth (smaller C) would cut F_d sharply.

Common mistakes

  • Using σ = F_t/(b·m·Y) with F_t in N and b, m in metres, then reporting the answer in N/mm²; keep mm with N to get MPa.
  • Designing the gear instead of the pinion when both are the same material; the pinion is weaker.
  • Using the Lewis Y (module basis) in a formula written for y (circular-pitch basis).
  • Treating the dynamic load as a small correction; at moderate speed with ordinary cutting it can exceed the transmitted load.
  • Using rpm instead of m/s for v in Barth's or Buckingham's equation.
  • Forgetting the wear check; many gears fail by pitting, not breakage.

For GATE ME

Expect: tangential and radial forces from power and speed; Lewis beam strength and the power a gear can transmit; module or face width from a permissible stress; identifying the weaker of two gears by σ_b·Y; velocity factor effects; and Buckingham dynamic or wear load. Practise gear force questions together with the bearing reactions they cause.

Quick check

  1. A gear of 40 teeth has module 3 mm. What is its pitch diameter?
  2. F_t = 2 kN, φ = 20°. Find the radial force.
  3. Pinion: σ_b = 100 MPa, Y = 0.30; gear: σ_b = 80 MPa, Y = 0.40. Which is weaker?
  4. Barth factor at v = 6 m/s for ordinary cut gears?
  5. If the face width is doubled, how does the Lewis beam strength change?

Answers: 1. 120 mm. 2. 2000 × tan 20° = 728 N. 3. Pinion (σ_b·Y = 30 < 32). 4. 3/9 = 0.333. 5. It doubles.

Interview questions

All Design of Machine and Automotive Elements interview questions

Try answering each one aloud before you open it.

  1. 1.What is a spur gear and where is it commonly used?Concept

    A spur gear is a type of gear with straight teeth that are mounted on a parallel shaft. It is the simplest type of gear and is used to transmit motion and power between parallel shafts. Spur gears are commonly used in applications like clocks, washing machines, and conveyor systems due to their simplicity and efficiency.

  2. 2.Explain the Lewis equation in the context of spur gear design.Concept

    The Lewis equation is used to estimate the bending stress in the teeth of a spur gear. It is given by σ = Ft / (b·m·Y), where σ is the bending stress, Ft is the tangential force on the gear tooth, b is the face width, m is the module, and Y is the Lewis form factor. This equation helps in determining the strength of the gear tooth and ensuring it can withstand the applied loads.

  3. 3.What is the Buckingham equation and how does it differ from the Lewis equation?Concept

    Lewis gives the static beam strength of a tooth, S_b = σ_b·b·m·Y, treating it as a cantilever loaded at the tip. Dynamic effects are then allowed for crudely with a velocity factor C_v. Buckingham's method instead calculates the extra dynamic load caused by tooth errors and inertia, F_d = 21v(bC + F_t)/(21v + √(bC + F_t)), where C is a deformation factor that depends on the tooth error and the materials. The effective load F_t + F_d is then compared with both the Lewis beam strength and Buckingham's wear strength S_w = b·Q·d_p·K. So Buckingham's method is more accurate for precision and high-speed gears and also covers pitting.

  4. 4.Why is the Lewis form factor important in gear design?Application

    The Lewis form factor (Y) is important because it accounts for the shape and size of the gear tooth, influencing the distribution of stress along the tooth. A higher form factor indicates a more favorable tooth shape for stress distribution, reducing the likelihood of tooth failure. It is crucial for ensuring that the gear can handle the expected loads without excessive wear or damage.

  5. 5.What happens if the face width of a spur gear is increased?Application

    Increasing the face width of a spur gear generally increases its load-carrying capacity because it allows the load to be distributed over a larger area, reducing the stress on individual teeth. However, it also increases the weight and size of the gear, which may not be desirable in all applications. Designers must balance these factors to optimize gear performance.

  6. 6.How does gear module affect the design and performance of a spur gear?Application

    The gear module is a measure of the size of the gear teeth and is defined as the ratio of the pitch diameter to the number of teeth. A larger module results in larger teeth, which can carry more load but may also increase the size and weight of the gear. Selecting the appropriate module is crucial for balancing strength, size, and weight in gear design.

  7. 7.Why is it important to consider dynamic loads in gear design?Application

    Dynamic loads arise from gear inaccuracies, misalignments, and high-speed operations, leading to additional forces on the gear teeth. Ignoring these loads can result in premature gear failure due to fatigue or excessive wear. Considering dynamic loads ensures that the gear design is robust and reliable under real-world operating conditions.

  8. 8.Calculate the bending stress on a spur gear tooth using the Lewis equation given: Ft = 500 N, b = 10 mm, m = 5 mm, Y = 0.3.Numerical

    Using the Lewis equation: σ = Ft / (b·m·Y). Substituting the given values: σ = 500 N / (10 mm · 5 mm · 0.3) = 500 / 15 = 33.33 N/mm². Therefore, the bending stress on the gear tooth is 33.33 N/mm².

  9. 9.A spur gear has a module of 4 mm and a face width of 20 mm. If the Lewis form factor is 0.25 and the tangential force is 800 N, calculate the bending stress using the Lewis equation.Numerical

    Using the Lewis equation: σ = Ft / (b·m·Y). Substituting the given values: σ = 800 N / (20 mm · 4 mm · 0.25) = 800 / 20 = 40 N/mm². Therefore, the bending stress on the gear tooth is 40 N/mm².

  10. 10.Explain how gear inaccuracies can affect the performance of a spur gear system.Application

    Gear inaccuracies, such as deviations in tooth profile or spacing, can lead to uneven load distribution and increased dynamic loads. This can cause noise, vibration, and premature wear or failure of the gear teeth. Accurate manufacturing and alignment are essential to minimize these effects and ensure smooth and efficient gear operation.

Finished this topic? Mark it so your progress, study plan and readiness keep up.

Stuck on something here?

PreviousSliding contact bearings and hydrodynamic lubricationNext Helical, bevel and worm gear design