Spiral bevel gears transmit torque between intersecting shafts, and their contact pattern is a direct indicator of how well the tooth surfaces, assembly, and load path agree. In aerospace and high-end equipment, a pattern that is centered at light load can move under torque, concentrate pressure near the heel, and shorten fatigue life. The reference case is a 29/12-tooth Gleason face-milled pair from a 2026 collaboration between Lanzhou University of Technology and Qinghai Huading Gearbox Co., Ltd. It uses a 90° shaft angle, a 35° spiral angle, a 46 mm face width, and a 9 mm transverse module. The case combined a cutter-based tooth-surface model, parametric CAD, transient finite-element contact analysis, fatigue post-processing, and a rolling-inspection check.
Gleason-generated tooth surfaces
The generating principle is a virtual generating gear in zero-clearance mesh with the workpiece. The cutter and the tooth surface keep a one-to-one spatial mapping: each point on the cutting cone corresponds to a point on the generated flank. For the wheel, a double-sided cutter is used. The straight cutting edge forms the working flank, while the circular arc edge forms the transition fillet. The cutter coordinate system is \(O_q(X_q,Y_q,Z_q)\), and the cutting edge is rotated by \(\theta_q\) about \(Z_q\) to form the cutter cone \(\Sigma_q\).
For the straight edge, the position vector and unit normal are
$$
\mathbf r_q(S_q,\theta_q)=
\begin{bmatrix}
(R_q-S_q\sin\alpha)\cos\theta_q\\
(R_q-S_q\sin\alpha)\sin\theta_q\\
-S_q\cos\alpha\\
1
\end{bmatrix},
$$
$$
\mathbf n_q(\theta_q)=
\begin{bmatrix}
\cos\alpha\cos\theta_q\\
\cos\alpha\sin\theta_q\\
-\sin\alpha
\end{bmatrix},
$$
where \(S_q\) and \(\theta_q\) are cutter-section coordinates, \(R_q\) is the cutter tip radius, and \(\alpha\) is the cutter profile angle. For the circular arc edge, the corresponding surface and normal are
$$
\mathbf r_q(\theta,\theta_q)=
\begin{bmatrix}
(C_x-\rho_q\sin\theta)\cos\theta_q\\
(C_x-\rho_q\sin\theta)\sin\theta_q\\
-(C_y-\rho_q\cos\theta)\\
1
\end{bmatrix},
$$
$$
\mathbf n_q(\theta,\theta_q)=
\begin{bmatrix}
\sin\theta\cos\theta_q\\
\sin\theta\sin\theta_q\\
-\cos\theta
\end{bmatrix},
$$
where \(C_x\) and \(C_y\) locate the arc center and \(\rho_q\) is the fillet radius.
The cutter surface is transformed into the wheel coordinate system through a chain of machine, cutter, blank, and auxiliary frames. The position and normal transform as
$$
\mathbf r_2=[M_2]\mathbf r_q,\qquad
\mathbf n_2=[L_2]\mathbf n_q,
$$
where \([M_2]\) is the homogeneous transformation matrix and \([L_2]\) is its \(3\times3\) rotation part. The meshing condition is
$$
f_2=\mathbf n_{m2}\cdot \mathbf v_{m2}^{(q2)}=0,
$$
which requires the common normal to be perpendicular to the relative velocity between the cutter cone and the wheel blank. Solving the tooth-surface and meshing equations gives discrete surface points. Interpolation, surface trimming, and lofting then produce a continuous tooth surface and a solid tooth model. Machine settings such as radial cutter position, angular cutter position, cradle angle, wheel mounting angle, vertical offset, and horizontal wheel position remain explicit parameters, so a change in the process settings regenerates the geometry rather than requiring manual reconstruction.
Parametric CAD and finite-element model
The gear pair is defined by the parameters in Table 1. The wheel is right-handed and the pinion is left-handed. The cone angles and tooth heights describe a hypoid-like Gleason geometry with a large wheel and a small pinion.
Table 1. Basic parameters of the 29/12-tooth Gleason pair.
| Parameter | Wheel | Pinion |
|---|---|---|
| Number of teeth | 29 | 12 |
| Transverse module (mm) | 9 | 9 |
| Hand | Right | Left |
| Shaft angle (°) | 90 | |
| Spiral angle (°) | 35 | 35 |
| Face width (mm) | 46 | 46 |
| Addendum (mm) | 4.74 | 10.56 |
| Dedendum (mm) | 12.25 | 6.43 |
| Pitch cone angle (°) | 67.31 | 22.28 |
| Root cone angle (°) | 62.33 | 19.52 |
| Tip cone angle (°) | 70.7 | 27.26 |
The CAD workflow uses a parametric template. A two-dimensional sketch defines the blank, a spiral curve establishes the tooth trace, and tooth-slot point-cloud data are imported into the solid. A loft cut removes material along the fitted curve to form one slot; a circular pattern then produces the full wheel. The pinion is generated from the same template by changing the corresponding parameters. The pair is assembled with zero mounting error and checked for interference in motion simulation before the contact analysis, so assembly-induced interference does not contaminate the finite-element results.
The finite-element model is a seven-tooth segment of the pair. The material properties are \(E=202\ \mathrm{GPa}\), \(\mu=0.273\), and \(\rho=7.91\ \mathrm{g/cm^3}\). An adaptive tetrahedral mesh is used: the contact zone is refined to approximately \(1\ \mathrm{mm}\), while non-contact regions use a coarser size. The final mesh contains 228,332 nodes and 133,453 elements. The contact is surface-to-surface, frictional, with a coefficient of 0.2, and the solver uses the augmented Lagrange method. At the rated condition, the pinion speed is \(154\ \mathrm{rad/s}\) and the wheel reaction torque is \(8000\ \mathrm{N\cdot mm}\).
Contact response under torque
Two solution paths are compared at \(8000\ \mathrm{N\cdot mm}\). The steady analysis applies a step load, releases inertia, and reaches static equilibrium quickly. The transient analysis uses a ramp load: from 0 to 1 s, the load rises in 1000 steps of \(1\ \mathrm{ms}\), with an increment of \(8\ \mathrm{N\cdot mm}\) per step; from 1 to 3 s, the load is held constant with automatic time stepping. The transient peak contact stress reaches \(1.47\times10^{10}\ \mathrm{Pa}\), compared with \(9.90\times10^{9}\ \mathrm{Pa}\) in the steady analysis, about 1.5 times higher. The steady step load produces a higher initial stress because the full load is imposed instantly, while the transient ramp avoids that initial shock but develops a higher peak later through inertia and tooth-pair interaction.
At about \(0.6\ \mathrm{s}\), the stress fluctuation increases sharply. This timing corresponds to the handover between the next tooth pair entering mesh and the previous pair leaving. The transient model therefore captures a dynamic effect that a quasi-static step solution cannot represent: the contact stress peak is tied to the meshing transition, not only to the nominal torque.
Both analyses produce an elliptical contact footprint. The transient analysis also shows the footprint expanding as the load ramps, whereas the steady analysis gives only the equilibrium pattern. Under increasing torque, the contact area grows nonlinearly, and the pattern center shifts toward the heel, the large end of the tooth. The contact mode remains relatively stable, but the loaded region expands and redistributes. At \(400\), \(8000\), and \(15{,}000\ \mathrm{N\cdot mm}\), the pattern remains concentrated in the mid-face region, initially biased toward the toe, the small end. At light load, the pinion still shows a relatively large contact area; this is attributed to the soft tooth-flank design, which lowers contact stiffness and allows the pattern to spread before the load reaches the rated level.
Fatigue life ranking
The fatigue assessment uses the ANSYS fatigue tool with a strain-life \((\varepsilon\text{-}N)\) framework. The minimum computed life values are 333.69 at \(400\ \mathrm{N\cdot mm}\), 77.82 at \(8000\ \mathrm{N\cdot mm}\), and 18.78 at \(15{,}000\ \mathrm{N\cdot mm}\). The solver output does not carry a unit in the case data, so these values are best treated as relative rankings rather than absolute cycle counts. The reductions are about 4.3 times from 400 to \(8000\ \mathrm{N\cdot mm}\) and about 4.1 times from 8000 to \(15{,}000\ \mathrm{N\cdot mm}\). The life decrease is therefore strongly nonlinear with torque: a 20-fold torque increase from 400 to \(8000\ \mathrm{N\cdot mm}\) reduces life by only about 4.3 times, while a further 1.875-fold increase to \(15{,}000\ \mathrm{N\cdot mm}\) reduces it by about 4.1 times.
The damage mechanism also changes with load. At \(400\ \mathrm{N\cdot mm}\), surface fretting wear dominates the life result. At \(15{,}000\ \mathrm{N\cdot mm}\), the stress-strain hysteresis loop produces a non-proportional loading path, which accelerates fatigue damage and causes the minimum life to fall steeply. These results are model-based; the rolling inspection described below validates the contact pattern, not the absolute fatigue life.
Rolling-inspection validation
The contact pattern of the ground gear pair is checked on a universal rolling inspection machine. The pinion position is adjusted with handwheels to control offset displacement error \(\Delta E\), axial displacements \(\Delta G\) and \(\Delta P\), and shaft-angle error \(\Sigma\). Dial indicators monitor the linear displacements and an electronic angle gauge monitors the shaft angle. The adjustment limits are given in Table 2.
Table 2. Installation-error adjustment and monitoring.
| Parameter | Adjustment object | Step | Resolution | Allowed error | Monitoring |
|---|---|---|---|---|---|
| \(\Delta E\) (mm) | Offset displacement error | ±0.01 | 0.01 | ±0.03 | Dial indicator |
| \(\Delta G\), \(\Delta P\) (mm) | Axial displacement errors | ±0.02 | 0.01 | ±0.05 | Dial indicator |
| \(\Sigma\) (″) | Shaft-angle error | ±10 | 10 | ±40 | Electronic angle gauge |
The test starts at \(20\ \mathrm{rad/s}\) with no load for 1 min to close the assembly clearance, then accelerates at \(0.08\ \mathrm{rad/s^2}\) to \(154\ \mathrm{rad/s}\). A closed-loop servo motor applies the load, and a torque sensor monitors the wheel torque. A PID controller raises the torque at \(200\ \mathrm{N\cdot mm/s}\) to the target values of \(400\), \(8000\), and \(15{,}000\ \mathrm{N\cdot mm}\). The pair runs for 5 min at the rated speed, then stops for inspection of the marking compound on the tooth surface.
The measured contact patterns agree with the finite-element predictions in position and orientation for all three torque levels. The pattern remains within the intended contact region, and the loaded pattern shifts and expands in the same direction as the simulation. This supports the contact model, the assembly setup, and the manufacturing parameters for this gear pair. The rolling check is a pattern-validation test; it does not supply a fatigue-life curve.
What the results mean for spiral bevel gear design
The main technical point is that the contact pattern of a spiral bevel gear is load-dependent. A single light-load check can show a large pinion contact area without revealing how the pattern will move under torque. In this pair, the area grows nonlinearly, and the center shifts toward the heel as load rises. A design or acceptance check that uses only one torque level can therefore miss the loaded contact state.
Loading path also matters. The steady step load gives a higher initial stress, while the transient ramp develops a higher peak near the tooth-pair handover. When peak stress, inertia, or contact-pattern evolution is of interest, a transient analysis with a realistic load ramp is more informative than a quasi-static step solution. The parametric surface model makes this practical: cutter settings and machine settings remain explicit inputs, so the tooth geometry can be regenerated and re-analyzed without manual remodeling.
The fatigue ranking shows that life is not proportional to torque. The transition from surface fretting at light load to a non-proportional stress-strain path at high load produces a steep drop in the computed life. That result is useful for ranking load cases and for identifying which torque range deserves attention, but it remains a numerical prediction. The evidence covers one 29/12-tooth pair and three torque levels; broader use requires matching cutter data, machine settings, and experimental fatigue data. Within that scope, the combination of Gleason-based parametric modeling, transient contact analysis, and rolling inspection gives a coherent chain from tooth-surface generation to loaded contact behavior and design assessment.