Gear Tooth Contact Stress and Bending Analysis

The problem

EV gears run at high speed with high torque from zero rpm, so tooth bending and pitting resistance are central to gearbox life. Hand formulas include many correction factors, while FEA shows the real stress field but is very sensitive to tooth geometry, contact settings and mesh. A careful comparison teaches students how to use each method and where each can mislead.

Objectives

• Generate accurate involute tooth profiles with equation-driven curves.
• Set up a contact study for a tooth pair at worst-case mesh positions.
• Converge root bending stress and contact pressure with focused mesh controls.
• Compare FEA with Lewis bending, Hertz contact and a simplified ISO 6336 or AGMA-style calculation.
• Study the effect of profile modification or root fillet radius on stress.

What you need

Key concepts

Involute profile

The tooth flank shape that keeps the velocity ratio constant through mesh. Its equation in parametric form is x = rb(cos t + t sin t), y = rb(sin t − t cos t).

Highest point of single tooth contact

The load position on the flank where one tooth carries the full load and the bending moment at the root is near its maximum.

Hertz contact stress

Peak pressure between two curved surfaces in contact, governing pitting. For gears it is evaluated near the pitch point from equivalent radii of curvature.

Mesh sensitivity at contact

Contact pressure is concentrated in a narrow band, so elements smaller than the contact half-width are needed for credible peaks.

Step-by-step guide

1. 1

   Choose the gear pair

   Take an example first-stage pair: module 2 mm, 20-degree pressure angle, pinion 19 teeth, wheel 61 teeth, face width 20 mm, pinion torque 150 N·m and a case-carburised steel. Compute base, pitch, tip and root radii, the transverse contact ratio and the positions of the single-tooth contact limits along the line of action. Label all values as examples and keep them in a parameter sheet.

   Checkpoint: A geometry table with contact ratio between 1.2 and 2 and the highest point of single tooth contact located.

2. 2

   Model the involute

   In a sketch, add an Equation Driven Curve with the parametric involute equations based on the base radius, limited between the base and tip circles. Mirror it for the other flank at the correct tooth thickness angle. Add the root fillet as a trochoid generated by the cutter rack, or as a circular fillet of known radius for a first model, then pattern the tooth around the gear.

   Checkpoint: Measured tooth thickness at the pitch circle matches the theoretical value within 0.01 mm.

3. 3

   Prepare an analysis model

   Cut each gear down to three or five teeth around the contact zone to save elements, keeping enough rim depth to represent its stiffness realistically. Position the pair at the highest point of single tooth contact using a driven angle in the assembly. Create configurations for other mesh positions along the line of action so you can sweep the roll angle later without rebuilding.

   Checkpoint: In the assembly, Interference Detection shows a touching but not overlapping tooth pair at each position.

4. 4

   Set contacts and boundary conditions

   Define a no-penetration contact set between the meshing flanks and bonded contact elsewhere if needed. Fix the wheel bore and apply torque to the pinion through a remote load or a torque on its bore face, with the pinion restrained radially but free to rotate. Use soft springs or the stabilisation option if the solver reports rigid-body motion before contact closes.

   Checkpoint: The study solves without instability warnings and the pinion rotates slightly to close the contact.

5. 5

   Control the mesh

   Add mesh controls at the pinion root fillet and along both flanks near the contact line, with elements several times smaller than the expected Hertz contact half-width you calculated. Run three refinement levels and track maximum principal stress at the root and peak contact pressure. Use a coarse draft-quality mesh only for early setup checks, never for reported results. Save the converged mesh settings so every later position uses them.

   Checkpoint: Root stress changes by under 5 per cent and contact pressure by under 10 per cent between the last two meshes.

6. 6

   Calculate by hand

   Compute Lewis bending stress with the form factor for the tooth count, then Hertz contact stress from the equivalent radius of curvature at the pitch point. Add a simplified ISO 6336 or AGMA-style calculation with the main factors (application, dynamic, face load and transverse load) taken from your source, stating each assumption. Present all three methods side by side in one table for the same load.

   Checkpoint: A table of root and contact stresses by hand, simplified standard and FEA for the same load.

7. 7

   Sweep mesh positions

   Run the study across five or more positions along the line of action using the configurations. Plot root stress and contact pressure against roll angle. Expect root stress to peak near the highest point of single tooth contact and contact pressure to peak near the lowest point of single tooth contact or the pitch point, and check whether your results agree and why.

   Checkpoint: Plots of root stress and contact pressure versus roll angle with the peak positions identified and explained.

8. 8

   Study the root fillet

   Create configurations with root fillet radius values from small up to the largest the cutter can generate, and rerun the worst mesh position for each. Plot root stress against fillet radius. Explain how a larger fillet lowers the stress concentration, and confirm with Interference Detection through a full mesh cycle that the enlarged fillet does not clash with the mating tooth tip.

   Checkpoint: A stress-versus-fillet plot and a recommended radius that clears the mating tooth.

9. 9

   Try a profile modification

   Add a small linear or parabolic tip relief to the pinion profile by modifying the equation near the tip, and rerun the positions near contact entry. Compare contact pressure at entry and root stress with the unmodified gear. Discuss the link between tip relief, load sharing between tooth pairs and transmission error, which drives gear whine in quiet electric drivetrains.

   Checkpoint: The contact pressure peak at entry is reduced by the tip relief, quantified as a percentage.

10. 10

    Interpret and report

    Compare the three methods and explain the differences: standard factors for dynamics and load distribution, boundary effects and rim stiffness in FEA, and linear material behaviour. State which result you would use for design and why. Report geometry, the mesh study, the sweeps and the fillet and relief studies with plots and units, and note that real gears also need fatigue rating and testing.

    Checkpoint: A report with method comparison, converged results and a clear statement of limitations.

Testing & validation

• Tooth thickness and contact ratio match theory.
• Mesh convergence meets the stated thresholds for root and contact stress.
• FEA and hand methods are within a band that is explained by known factors.
• Peak stress positions agree with gear theory.

Expected results

An accurate involute gear pair model, converged FEA root and contact stresses across mesh positions, a comparison with Lewis, Hertz and a simplified standards approach, and quantified effects of root fillet radius and tip relief.

Take it further

• Model a helical pair and study face load distribution.
• Run a Fatigue study on the root stress history across a duty cycle.
• Include shaft deflection and misalignment in the contact model.
• Compare with a dedicated gear analysis tool if available.

Viva and review questions

Learn it at DIYguru

These DIYguru courses and programmes teach what this project needs.