Spiral bevel gears are widely used where a right-angle drive must transmit high torque in a compact envelope: vehicle drivelines, aviation gearboxes, marine propulsion, machine tools, and industrial reducers. Their teeth are generated with a curved trace, so contact begins near one end of the face and moves diagonally across the tooth before exiting. This produces gradual load transfer and a high contact ratio, but it also makes the mesh sensitive to generation errors, mounting distance, shaft deflection, bearing clearance, and assembly variation.
Two measurements expose different consequences of that sensitivity. Transmission error (TE) is the departure of the output rotation from the value dictated by the input rotation and the tooth ratio. It is both a kinematic quality index and an internal excitation source: TE is closely tied to mesh stiffness and appears in the noise and vibration signature of the gearbox. Root stress is the bending stress at the tooth fillet. Its peak value, mean value, and cycle shape govern bending fatigue. A contact pattern remains the most intuitive shop-floor check because it shows where load crosses the face, but it does not by itself quantify either TE or root stress.
A 30/40-tooth spiral bevel pair tested in a power-closed loop at 10–60 r/min and 0–7,500 N·m provides the quantitative case below. The low-speed envelope suppresses inertial effects, so the measurements isolate geometric deviation and elastic compliance of the mesh rather than high-speed dynamic amplification.
Test Pair and Power-Closed Rig
The gear pair had 30 pinion teeth and 40 gear teeth, a large-end transverse module of 10 mm, a pressure angle of 20°, a spiral angle of 35°, and a face width of 60 mm. The pinion was the driver on its concave flank; the gear was driven on its convex flank. Both members were accuracy grade 6. Table 1 lists the main geometry.
Table 1. Geometry of the spiral bevel gear pair.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 30 | 40 |
| Large-end transverse module (mm) | 10 | 10 |
| Pressure angle (°) | 20 | 20 |
| Spiral angle (°) | 35 | 35 |
| Face width (mm) | 60 | 60 |
| Addendum (mm) | 8.5 | 8.5 |
| Whole depth (mm) | 18.88 | 18.88 |
| Accuracy grade | 6 | 6 |
| Hand | Left | Right |
The rig was a power-closed, or circulating-power, arrangement. The main and companion gearboxes were connected back-to-back, so most of the torque circulated around the loop while the servo motor supplied only the losses. A 16.4 kW servo motor rated at 3,000 r/min drove through a 50:1 reducer, giving a maximum test-shaft speed of 60 r/min. A torque loader covered 0–11,000 N·m with ±0.1% full-scale accuracy. Torque and speed were measured with a 0–15,000 N·m sensor (±0.5% full scale) and a 0–6,000 r/min sensor (±1 r/min).
The TE test matrix used pinion speeds of 10–60 r/min in 10 r/min steps and torques of 0–7,500 N·m in 1,500 N·m steps. The root-stress matrix used the same speed steps and torques of 1,500–7,500 N·m. These values sit inside the rig envelope but well below its 11,000 N·m torque limit.
Measurement Principles
High-resolution encoders were mounted at the pinion input and at the free end of the gear. Each encoder had 23,600 lines per revolution with 400-fold subdivision, giving fine angular resolution. If \(\theta_1\) is the pinion angle, \(\theta_2\) is the gear angle, and \(z_1\) and \(z_2\) are the respective tooth numbers, the transmission error is
$$ \mathrm{TE} = \theta_2 – \theta_1 \frac{z_1}{z_2} $$
with the sign defined so that an ideal conjugate pair gives zero. Because the encoders are on the rotating shafts, the measured TE includes tooth-flank deviations, tooth bending, shaft deflection, and bearing deflection. It is therefore a loaded transmission error rather than a pure manufacturing error. At zero torque, the non-zero TE is a geometric baseline; the torque-dependent increase contains the elastic contribution.
Root strain was measured with a half-bridge circuit and a telemetry system mounted at the gear free end. Six working gauges were placed on six different teeth at the first six of nine equal divisions measured from the small end of the face. A temperature-compensation gauge was placed near the small-end face, away from the root fillet. The compensated strain was converted to an equivalent uniaxial root stress:
$$ \varepsilon_F = \varepsilon_w – \varepsilon_t, \qquad \sigma_F = E\,\varepsilon_F $$
where \(\varepsilon_w\) is the working-gauge strain, \(\varepsilon_t\) is the compensation strain, and \(E = 2.09 \times 10^5\ \mathrm{MPa}\). Tensile stress was taken as positive. Stress records were sampled at 100 Hz.
The rotating-frame nature of the measurement matters. At 60 r/min pinion speed, the gear turns at 45 r/min, or 0.75 Hz, and the tooth mesh frequency is 30 Hz. A strain gauge fixed to one gear tooth, however, is loaded once per gear revolution. Its signal is therefore a once-per-revolution pulse train whose spectrum is built on 0.75 Hz and its harmonics, not a continuous 30 Hz sinusoid.
Transmission Error: Torque Sets the Range, Speed Reshapes the Spectrum
After removal of the mean, the TE records oscillated about zero in a near-sinusoidal form. Representative records at low, intermediate, and high torque showed the same qualitative shape but different amplitudes and starting phases. The peak-to-peak values are collected in Table 2.
Table 2. Peak-to-peak transmission error of the 30/40-tooth pair at different pinion speeds and torques. Values are in degrees.
| Torque (N·m) | 10 r/min | 20 r/min | 30 r/min | 40 r/min | 50 r/min | 60 r/min |
|---|---|---|---|---|---|---|
| 0 | 0.1266 | 0.1252 | 0.1363 | 0.1379 | 0.1390 | 0.1473 |
| 1,500 | 0.1391 | 0.1612 | 0.1475 | 0.1465 | 0.1505 | 0.1547 |
| 3,000 | 0.1635 | 0.1861 | 0.1804 | 0.1715 | 0.1811 | 0.1647 |
| 4,500 | 0.1942 | 0.2062 | 0.1975 | 0.1847 | 0.1789 | 0.1757 |
| 6,000 | 0.2143 | 0.2192 | 0.2166 | 0.2061 | 0.2022 | 0.1960 |
| 7,500 | 0.2322 | 0.2254 | 0.2124 | 0.2175 | 0.2292 | 0.2292 |
At zero torque, the peak-to-peak TE ranged from 0.1252° to 0.1473° across the speed sweep. At 7,500 N·m it ranged from 0.2124° to 0.2322°. The smallest value in the matrix was 0.1252° at 20 r/min and zero torque; the largest was 0.2322° at 10 r/min and 7,500 N·m. At a fixed speed, the trend with torque was upward, although the final increment was small or slightly negative at some speeds. At 60 r/min, for example, the peak-to-peak TE increased from 0.1473° at zero torque to 0.2292° at 7,500 N·m, a rise of about 56%.
Speed had a much weaker effect on the peak-to-peak range. At a fixed torque, the spread across 10–60 r/min was only about 0.02° in absolute terms. The torque effect was therefore first-order for the overall TE range, while the speed effect was second-order. This is consistent with the idea that increasing torque increases tooth, shaft, and bearing deflection, adding a load-dependent compliance term to the kinematic error.
The frequency domain reveals a different role for speed. At 60 r/min pinion speed, the pinion rotation frequency is 1 Hz. The TE spectrum was dominated by the 1 Hz component, with smaller amplitudes at higher harmonics. Over a finer torque sweep from 0 to 7,500 N·m in 500 N·m steps, the 1 Hz amplitude increased approximately linearly with torque. The dominance of the once-per-revolution component indicates that runout, indexing variation, or mounting eccentricity contributed significantly alongside per-tooth mesh deviations.
At a fixed torque of 2,500 N·m, the speed was swept from 10 to 60 r/min in 5 r/min steps. The 1 Hz amplitude reached a minimum of 0.0449° at 15 r/min and a maximum of 0.0672° at 20 r/min, with no monotonic trend. Speed therefore changed the harmonic content and waveform shape substantially even when the peak-to-peak range changed little. A single peak-to-peak value can hide this speed-dependent redistribution of harmonic amplitude, which is the part of TE most relevant to dynamic excitation.
Root Stress: Bending Cycle and Face-Width Gradient
Table 3 gives the peak-to-peak root stress at the six instrumented positions at 60 r/min. Position 1 is nearest the small end; position 6 is the furthest outboard of the instrumented positions, about two-thirds of the way across the face. The true large-end region beyond position 6 was not instrumented.
Table 3. Peak-to-peak root stress at six positions across the gear face width at 60 r/min. Position 1 is nearest the small end; position 6 is the furthest outboard instrumented position. Values are in MPa.
| Torque (N·m) | Position 1 | Position 2 | Position 3 | Position 4 | Position 5 | Position 6 |
|---|---|---|---|---|---|---|
| 1,500 | 13.89 | 27.96 | 114.01 | 142.70 | 97.03 | 145.65 |
| 3,000 | 24.30 | 50.30 | 197.83 | 243.96 | 187.82 | 260.58 |
| 4,500 | 35.27 | 72.41 | 274.01 | 346.35 | 298.51 | 360.90 |
| 6,000 | 45.72 | 95.61 | 356.03 | 444.48 | 415.07 | 474.15 |
| 7,500 | 55.80 | 117.49 | 428.95 | 537.10 | 512.00 | 581.89 |
The face-width gradient is large. At 1,500 N·m, the peak-to-peak stress rose from 13.89 MPa at position 1 to 145.65 MPa at position 6. At 7,500 N·m, it rose from 55.80 MPa to 581.89 MPa. The maximum was therefore about ten times the minimum at every torque level. The increase was not strictly monotonic: position 5 was consistently below position 4, while position 6 was the highest of the six. Local tooth geometry, contact load distribution, and tooth-to-tooth scatter can all contribute to that non-monotonicity, especially because each working gauge was on a different tooth.
Torque produced a nearly linear increase at every position. At position 6, the peak-to-peak stress increased from 145.65 MPa at 1,500 N·m to 581.89 MPa at 7,500 N·m, close to a fourfold rise for a fivefold torque increase. At position 1, the corresponding rise was from 13.89 MPa to 55.80 MPa. Speed was much less influential: at 7,500 N·m the maximum among positions ranged from 535.39 MPa to 581.89 MPa over 10–60 r/min, and the minimum ranged from 48.67 MPa to 56.03 MPa. At 1,500 N·m, the maximum ranged from 128.93 MPa to 145.65 MPa. The speed-related spread was roughly 8–15%, far smaller than the fourfold torque effect.
The time-domain records at 60 r/min clarify the cycle shape. At 4,500, 6,000, and 7,500 N·m, the largest tensile root stresses were 261.1, 331.7, and 401.4 MPa, while the largest compressive magnitudes were 100.4, 142.3, and 179.8 MPa. The tensile peak was therefore about 2.2–2.6 times the compressive magnitude, producing an asymmetric reversed bending cycle with a tensile bias. When a tooth entered mesh, the root stress was tensile and rose to its peak. As the tooth moved through mesh and exited, the stress fell through zero and reached a compressive minimum. Outside mesh, the stress was close to zero.
Root-stress spectra were dominated by the gear rotation frequency and its harmonics. At 60 r/min pinion speed, that fundamental is 0.75 Hz. The result is expected for a gauge that rotates with its tooth: each instrumented tooth is loaded once per gear revolution, so the spectrum describes the shape of the once-per-revolution load pulse. Increasing torque raised the amplitudes at the fundamental and at the harmonics without changing the qualitative spectral composition. The 30 Hz tooth mesh is not the fundamental of this rotating-frame measurement; it appears only through the harmonic content of the pulse.
Design and Validation Implications
For this 30/40-tooth pair, torque was the primary control on both the peak-to-peak TE and the root-stress amplitude. Speed was secondary for the peak-to-peak TE range but first-order for the amplitude at the pinion rotation frequency. A gearset qualified only by a peak-to-peak TE value at one speed can therefore have a different dynamic excitation at another speed, even when the overall range looks similar. TE acceptance criteria and dynamic models should retain the harmonic amplitude, not only the peak-to-peak excursion.
The root-stress data also show why a single gauge can be misleading. The peak-to-peak stress varied by roughly a factor of ten across the instrumented face width. A gauge near the small end would seriously underpredict the bending load seen farther outboard. At the same time, the six gauges were on different teeth and covered only the first six of nine face divisions, so the measured gradient contains tooth-to-tooth scatter and does not resolve the extreme large end. Strain-gauge layouts for spiral bevel gears should span the face, use multiple teeth where possible, and treat position-to-position differences as a combination of load distribution and tooth variation.
The asymmetric tensile-compressive cycle is another design consideration. Because the tensile peak was more than twice the compressive magnitude, a fatigue assessment based only on alternating stress would miss the tensile mean stress. The root stress history is better treated as a tensile-biased reversed cycle, with mean and amplitude considered together.
These results apply to a grade-6 30/40-tooth pair at 10–60 r/min and up to 7,500 N·m. They provide a quasi-static baseline for separating geometric error from elastic compliance and for validating loaded tooth contact and gear-dynamics models. They do not replace high-speed tests for resonance, lubrication-film effects, or dynamic amplification, which lie outside this low-speed envelope.