Spiral bevel gears are the default choice for right-angle drives that must carry high torque at high speed: armored vehicle final drives, tank transmissions, aircraft engine gearboxes, and industrial reducers. Their teeth are curved and engage gradually, so several tooth pairs share load and motion is smoother than in straight bevel gears. The same geometry makes the tooth root a fatigue-critical region. Bending fatigue starts at the fillet where the curved tooth meets the hub; a crack that grows across the root can remove a tooth and jam or destroy the drive. Design against that failure requires a relationship between load, root stress, and cycles to failure.
Such relationships are difficult to obtain for spiral bevel gears. The tooth surface is generated by a complex machine-tool motion, and the contact pattern and root stress depend on mounting position, load, and alignment. A pulsator that bends a single tooth does not reproduce the meshing contact of a real gear pair. A rotating bench does, but a full-life test at design load still takes tens to hundreds of hours per pair, and a statistically meaningful test matrix multiplies that cost. The practical route is to combine a material S-N curve, a three-dimensional contact finite element model, and a fatigue post-processor, then use an accelerated bench test at higher stress to confirm the prediction.
Gear Pair and Material Data
The test pair was a tapered-tooth spiral bevel gear set with 16 pinion teeth and 27 wheel teeth, a 4.25 mm large-end transverse module, a 17 mm face width, a 90° shaft angle, a 20° normal pressure angle, and a 35° mean spiral angle. Table 1 lists the geometry. The pinion was left-hand and the wheel right-hand; height modification was +0.252 on the pinion and −0.252 on the wheel, with matching tangential modification of ±0.005. The addendum coefficient was 0.85 and the clearance coefficient 0.188. A cutter tip radius of 0.3 mm generated the root fillet.
| Parameter | Pinion | Wheel |
|---|---|---|
| Number of teeth | 16 | 27 |
| Hand | Left | Right |
| Large-end transverse module (mm) | 4.25 | 4.25 |
| Face width (mm) | 17 | 17 |
| Shaft angle (°) | 90 | 90 |
| Normal pressure angle (°) | 20 | 20 |
| Mean spiral angle (°) | 35 | 35 |
| Addendum coefficient | 0.85 | 0.85 |
| Clearance coefficient | 0.188 | 0.188 |
| Height modification coefficient | +0.252 | −0.252 |
| Tangential modification coefficient | +0.005 | −0.005 |
Machine-tool settings were calculated from tooth contact analysis and loaded tooth contact analysis, using contact pattern and transmission error as targets. The wheel was cut by the duplex method and the pinion by the single-side method. After grinding, the pair was measured on a CNC gear measuring center and met grade 6 of GB/T 11365—2019. The contact pattern from a rolling check agreed with the simulated pattern: contact started near mid-face, slightly toward the toe, and spread toward the heel as load increased.
The gears were made of 20CrNiMo, a low-carbon alloy carburizing steel. Specimens and gears came from the same melt and received the same heat treatment; they were produced together. Five tensile specimens failed in a brittle manner, with 6.5% elongation and less than 0.5% reduction of area. Mean ultimate tensile strength was 1664.7 MPa, close to the 1683.0 MPa value cited for AISI 9310, a widely used gear steel.
Eighteen specimens were run in load-controlled tension fatigue at a stress ratio of 0.05 on a GPS200 high-frequency machine; 16 produced valid failures. The fracture faces showed little necking, slight dimpling in the core, and intergranular cracking in the carburized case. The 50% survival S-N curve gave 883.90 MPa at 3×106 cycles. Because the pinion has 16 teeth, failure of one tooth corresponds to 15 of 16 teeth surviving, or 93.75%. The 50% curve was shifted downward to this 93.75% survival basis, and the lower curve was used as the material input for the gear fatigue simulation. Using the lower curve is conservative: it forces the simulated gear to reach a shorter life at a given stress than the average material data would predict.
Finite Element Model and Root Stress
Tooth flanks were generated numerically from the cutting principle, and the root transition surface was produced by the cutter tip fillet, so the three-dimensional geometry retained the actual root shape. The model included two teeth in mesh and three following teeth, five in total. The contact ratio is 2.06, so up to three pairs can share load; modeling only one pair would misstate the root stress. The pinion concave side and wheel convex side were taken as the contact pair.
The tooth surface mesh was 0.7 mm, refined to 0.1 mm at the root fillet. The material was modeled as linear elastic with density 7850 kg/m3, Young’s modulus 1.8×1011 Pa, Poisson’s ratio 0.3, and a tooth-surface friction coefficient of 0.03. The pinion ran at 900 r/min and the wheel carried 499 N·m. The transient dynamic solution gave the complete root stress history: as the pinion moved from heel-root engagement to toe-tip disengagement, the root stress passed from the heel through the mid-face to the toe. The root stress is therefore not a single static value; it is a cycle whose peak depends on load sharing among teeth.
Fatigue Life Simulation at Two Load Levels
The stress history was imported into nCode, the 93.75% S-N curve was assigned to the gear body, and mean stress was corrected with the Gerber method. Fatigue life was evaluated node by node; the node with the shortest life in the root region was taken as the critical point. The wheel torque was adjusted in 1 N·m steps until the target life was reached.
At 499 N·m, the simulation gave 3.009×106 cycles for a target of 3.0×106. The critical node lay at mid-face, slightly toward the heel. The maximum principal stress at that node was 745.47 MPa and the maximum equivalent stress 603.31 MPa. At 570 N·m, the target was 1.0×106 cycles; the critical region expanded and moved toward the heel, with maximum principal stress 826.87 MPa and maximum equivalent stress 669.48 MPa. Table 2 collects the results and the two stress estimates used for cross-checking.
| Wheel torque (N·m) | Target life (cycles) | Simulated life (cycles) | Max. principal stress (MPa) | Max. equivalent stress (MPa) | nCode stress (MPa) | ISO stress (MPa) |
|---|---|---|---|---|---|---|
| 570 | 1.0×106 | 1.0×106 | 826.87 | 669.48 | 692.00 | 727.95 |
| 499 | 3.0×106 | 3.009×106 | 745.47 | 603.31 | 623.10 | 637.27 |
nCode also reports an equivalent stress at the critical node; for the same load cases, these were 623.10 MPa at 499 N·m and 692.00 MPa at 570 N·m. The ISO 10300-3:2014 tooth-root strength method gave 637.27 MPa and 727.95 MPa, respectively. The nCode values were 2.3% and 5.2% below the ISO values, which is reasonable agreement for a contact model that resolves load sharing and fillet geometry. The four stress-life pairs produce straight lines on a log-log plot; the fitted slopes were \(k=-0.095465\) for the nCode stress and \(k=-0.121097\) for the ISO stress.
Accelerated Test Principle
On a log-log plot, the S-N relation is
$$
\lg \sigma = k \lg N + b
$$
For two stress levels \(\sigma_x > \sigma_y\) with lives \(N_x < N_y\), subtracting the two line equations gives
$$
\lg \sigma_x – \lg \sigma_y = k\left(\lg N_x – \lg N_y\right)
$$
$$
\frac{\sigma_x}{\sigma_y} = \left(\frac{N_x}{N_y}\right)^k
$$
The stress acceleration factor is defined as
$$
a = \frac{\sigma_x}{\sigma_y}
$$
Since \(k\) is negative, a shorter life \(N_x\) requires a higher stress \(\sigma_x\). For a threefold reduction in life, from \(3\times10^6\) to \(1\times10^6\) cycles, the simulation slopes give \(a=1.11058\) for the nCode stress and \(a=1.14229\) for the ISO stress. The corresponding wheel torques were 570 N·m and 499 N·m; their ratio is 1.142, matching the ISO-based factor. The test plan therefore used 570 N·m for the \(1\times10^6\) cycle step and 499 N·m for the \(3\times10^6\) cycle step. The accelerated step was run first, followed by the longer-life validation step.
Bench, Run-In, and Test Procedure
A rotating closed-power-flow rig was used rather than a pulsator. In a closed-power-flow rig, the test gear pair is connected through a mechanical loop; the torque is set once and circulates, so the load stays stable and the drive motor only supplies losses. The housing and shaft support stiffness is much greater than the tooth stiffness, so misalignment of the bevel pair is negligible. This matters because spiral bevel gears are sensitive to mounting error: a small offset can shift the contact pattern and change the root stress. A rotating rig also reproduces the multi-tooth meshing kinematics that a single-tooth pulsator cannot.
The lubricant was an API GL-5 85W-90 heavy-duty gear oil, changed once during the test. Before loading, the pair was run in at 1200 r/min in three steps: \(1.0\times10^5\) cycles at one-quarter torque, \(7.2\times10^4\) cycles at half torque, and \(3.6\times10^4\) cycles at three-quarter torque, for \(2.0\times10^5\) cycles in total. The speed was then set to 900 r/min and the torque raised to the test value. Oil temperature was held at 45–55 °C. After assembly, the rolling contact pattern was checked and matched the design pattern.
Failure was defined as a broken tooth. At 570 N·m, three pairs were run; the first developed pitting but no tooth break, and the next two failed. The two failures at 570 N·m corresponded to about 70 h of rig operation. At 499 N·m, two pairs were run for more than 130 h and both failed. The broken teeth were all at the pinion heel root. Post-test contact marks showed full-face load distribution without obvious edge loading, consistent with the preset contact pattern and with the contact zone spreading toward the heel under load.
Test Results and Correlation with Simulation
Table 3 lists the measured lives. At 570 N·m, the two failures occurred at \(6.46381\times10^5\) and \(19.03183\times10^5\) cycles, mean \(1.274782\times10^6\). At 499 N·m, the failures occurred at \(16.53261\times10^5\) and \(53.04999\times10^5\) cycles, mean \(3.47913\times10^6\). The scatter is large—about a factor of three at each load—which is typical of gear bending fatigue and is why a small sample gives only a rough estimate of the mean and slope.
| Specimen | Wheel torque (N·m) | Cycles to failure | Mean life (cycles) |
|---|---|---|---|
| 1 | 570 | 6.46381×105 | 1.274782×106 |
| 2 | 570 | 19.03183×105 | |
| 3 | 499 | 16.53261×105 | 3.47913×106 |
| 4 | 499 | 53.04999×105 |
At 570 N·m, the measured mean was 27.5% above the \(1.0\times10^6\) target. At 499 N·m, the measured mean was 16.0% above the \(3.0\times10^6\) target. On the log-life basis used for the S-N comparison, these are 1.7% and 1.0%, respectively. The test lives therefore sit slightly on the safe side of the simulation targets, not below them. Because the simulation used the 93.75% material curve, some conservatism is expected.
Using the simulation S-N slope and the measured 570 N·m life, the 499 N·m life was inferred as \(3.824346\times10^6\) cycles, 1.63% above the \(3.0\times10^6\) target on the log-life basis. The measured 499 N·m mean was \(3.47913\times10^6\), which is 0.63% below the inferred value on the same basis. The inference and the measurement therefore agree closely, even though the absolute cycle counts differ by about 10%.
| Life basis | Test strength, nCode basis (MPa) | Simulation, nCode stress (MPa) | Deviation | Test strength, ISO basis (MPa) | Simulation, ISO stress (MPa) | Deviation |
|---|---|---|---|---|---|---|
| 1×106 | 709.77 | 692.00 | −2.5% | 751.75 | 727.95 | −3.2% |
| 3×106 | 632.82 | 623.10 | −1.5% | 649.91 | 637.27 | −1.9% |
Table 4 compares the test-evaluated strengths with the simulation. At \(1\times10^6\) cycles, the nCode-evaluated test strength was 709.77 MPa and the simulation 692.00 MPa, a deviation of −2.5%; the ISO-evaluated test strength was 751.75 MPa and the simulation 727.95 MPa, a deviation of −3.2%. At \(3\times10^6\) cycles, the corresponding values were 632.82 MPa versus 623.10 MPa (−1.5%) and 649.91 MPa versus 637.27 MPa (−1.9%). The largest deviation in the whole comparison was 3.2%.
| Quantity | nCode basis (MPa) | ISO basis (MPa) |
|---|---|---|
| 1×106 test strength | 709.77 | 751.75 |
| 3×106 strength inferred from 1×106 | 639.10 | 658.11 |
| 3×106 test strength | 632.82 | 649.91 |
| Deviation of inference from test | +1.0% | +1.3% |
Table 5 checks the extrapolation. From the \(1\times10^6\) test strength, the S-N line predicts 639.10 MPa on the nCode basis and 658.11 MPa on the ISO basis at \(3\times10^6\) cycles. The measured values were 632.82 MPa and 649.91 MPa; the predictions were 1.0% and 1.3% high. The ISO-evaluated \(3\times10^6\) strength of 649.91 MPa is the strength value supported by the bench data.
Fitting the four test points gave S-N slopes of \(k=-0.104454\) for the nCode stress and \(k=-0.132503\) for the ISO stress. The corresponding threefold-life acceleration factors were 1.12156 and 1.15670. These are slightly higher than the simulation-based factors of 1.11058 and 1.14229, because the test-fitted lines are steeper. The direction of the difference is consistent: a steeper S-N line means a larger stress increment is needed to shorten life by the same factor. Using the ISO-fitted slope, a tenfold reduction in life would require a stress factor of 1.32159. That value is an extrapolation beyond the tested range and should not be used for design without independent verification.
Engineering Significance and Limits
The chain—material S-N curve, three-dimensional contact finite element analysis, fatigue post-processing, and accelerated bench test—produced a consistent picture for this gear pair. Simulation and test agreed within 3.2% in stress at the two life levels; the extrapolated \(3\times10^6\) strength from the \(1\times10^6\) test was within 1.3% of the measured value; and the failure location fell within the predicted critical root region at the pinion heel. The threefold acceleration factor was about 1.12–1.16, so a \(3\times10^6\) cycle test could be replaced by a \(1\times10^6\) cycle test at 12–16% higher stress. This is a practical reduction in test time, but not a dramatic one: for this material and heat treatment, the S-N line is shallow, so the stress increase must be modest.
The evaluation also has clear limits. The data set contains four valid failures at two load levels, enough to fit a line but not enough to characterize scatter or a reliability curve. The tests used one geometry, one material heat, one lubricant, and one run-in schedule. The first pair at the higher load pitted before any tooth broke; pitting can become the limiting failure mode and is not predicted by the bending-only simulation. The 93.75% survival curve is a component-level assumption based on 16 pinion teeth, not a statistical estimate from a large gear sample. The test-fitted line is a 50% survival line from four points, while the simulation used the 93.75% material curve, so the comparison tests the whole chain rather than the material curve alone. The acceleration factors above threefold life, including the tenfold value of 1.32159, lie outside the tested range. The nCode and ISO stress estimates differ by 2–5%, so the choice of stress measure matters when setting a test load.
Within those limits, the result is useful: for a complex bevel gear where full-life testing is expensive and material data are scarce, a validated simulation plus a short high-stress bench test can give a defensible estimate of bending strength and life. The next step is to repeat the chain on more geometries and heats and to extend the test matrix so that scatter and high-stress extrapolation can be checked directly.