Spiral bevel gears transmit rotation between intersecting shafts with high contact ratio, high load capacity, and good running smoothness. They are widely used in aerospace, marine, and automotive drivelines, where vibration and noise are increasingly important design constraints. The usual geometric design target is a controlled transmission error (TE) together with a contact pattern that remains inside the tooth. That target is established by tooth contact analysis (TCA), normally under light or nominal load. In service, the mesh deflects: teeth bend, shafts and bearings move, and the contact pattern migrates across the flank. The motion error that actually excites the system is then the loaded transmission error (LTE). Its peak-to-peak fluctuation over a mesh cycle, rather than its absolute level alone, is a direct source of mesh-frequency excitation.
The analysis here is based on a MASTA model of a 16/27-tooth spiral bevel gear pair, including its shafts and bearings. Five torque levels are applied to the output shaft: 5, 10, 20, 50, and 100 N·m. For each case, the LTE curve, the first 20 natural frequencies, the vibration acceleration waterfall, the acceleration spectrum at 600 r/min, and the noise curve from 500 to 1000 r/min are compared. The aim is to isolate how a change in LTE fluctuation, produced by a change in load, propagates into the vibration and noise signature of the gear system.
Gear Set and System Model
The main geometry of the gear pair is listed in Table 1. The pinion has 16 teeth and the gear has 27 teeth, with a module of 4.25 mm, a mean spiral angle of 35°, a normal pressure angle of 20°, and a 90° shaft angle. The face width is 17 mm. The pinion is left-hand and the gear is right-hand. The outer cone distance is 66.693 mm, and the pitch cone angles are 30.6507° for the pinion and 59.3493° for the gear.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 16 | 27 |
| Module (mm) | 4.25 | 4.25 |
| Mean spiral angle (°) | 35 | 35 |
| Normal pressure angle (°) | 20 | 20 |
| Shaft angle (°) | 90 | 90 |
| Face width (mm) | 17 | 17 |
| Hand | Left | Right |
| Outer cone distance (mm) | 66.693 | 66.693 |
| Pitch cone angle (°) | 30.6507 | 59.3493 |
| Face cone angle (°) | 35.3501 | 62.2108 |
| Root cone angle (°) | 27.7832 | 54.6499 |
| Addendum (mm) | 3.34 | 5.48 |
| Dedendum (mm) | 4.68 | 2.54 |
| Clearance (mm) | 0.8 | 0.8 |
TCA of the generated tooth surfaces produced a contact trace on the gear concave flank and a geometric TE amplitude of 10.313 arcsec. This is the unloaded design baseline. The loaded curves are assessed against the TCA interval rather than against a single TE value, because the position of the contact across the tooth determines whether the mesh remains fully inside the flank or reaches an edge.
The transmission is represented as a three-dimensional finite element system. Condensation nodes on the input and output shafts couple the gear mesh to the shaft–bearing structure, and modal superposition is used to calculate the system response. The condensation nodes also provide the output quantities used below: displacement, velocity, and acceleration. The pinion shaft carries the input speed; the gear shaft carries the applied torque. This arrangement allows the same gear pair to be evaluated at several loads without changing the mesh geometry.
Loaded Transmission Error Across Torque Levels
Table 2 gives the maximum, minimum, and peak-to-peak fluctuation of the LTE for the five torque levels. At 5, 10, and 20 N·m, the LTE curves remain inside the TCA interval, which indicates that the contact stays within the designed flank region. At 50 and 100 N·m, the curves cross the right endpoint of the interval, indicating edge contact. The absolute LTE maximum rises monotonically with torque, from 15.9 arcsec at 5 N·m to 75.73 arcsec at 100 N·m. The fluctuation, however, is not monotonic.
| Load (N·m) | LTE maximum (arcsec) | LTE minimum (arcsec) | Fluctuation (arcsec) | Contact state |
|---|---|---|---|---|
| 5 | 15.90 | 8.976 | 6.924 | Inside TCA interval |
| 10 | 19.76 | 14.80 | 4.96 | Inside TCA interval |
| 20 | 26.68 | 24.65 | 2.03 | Inside TCA interval |
| 50 | 49.19 | 42.44 | 6.75 | Beyond right endpoint |
| 100 | 75.73 | 65.37 | 10.36 | Beyond right endpoint |
The minimum fluctuation occurs at 20 N·m and equals 2.03 arcsec. The 100 N·m case has the largest fluctuation, 10.36 arcsec, about 5.1 times the minimum. The 5 N·m case, although it is the lightest load, has a fluctuation of 6.924 arcsec, larger than those at 10 and 20 N·m. The trend is therefore U-shaped: moderate torque gives the most stable LTE curve, while both low torque and high torque produce larger fluctuation. At high torque, edge contact produces a steep segment at the tooth boundary; at low torque, the contact is less evenly developed. For vibration analysis, this means that the absolute LTE level is not sufficient as a ranking metric. The peak-to-peak fluctuation is the quantity that tracks the excitation amplitude in the results below.
Load-Dependent Natural Frequencies
Mesh stiffness increases with load, so the natural frequencies of the transmission are load-dependent. For a simplified single-degree-of-freedom representation, the angular natural frequency and the corresponding cyclic frequency are
$$
\omega_n=\sqrt{\frac{k}{m}},\qquad f_n=\frac{1}{2\pi}\sqrt{\frac{k}{m}}
$$
where \(k\) is the mesh stiffness and \(m\) is the equivalent mass. For a fixed mass, an increase in mesh stiffness raises the natural frequency. The full model computes the first 20 modes. Table 3 shows selected orders for the five loads. Every selected order increases with torque, but the sensitivity differs: the first mode rises from 0.6239 to 1.0532 kHz, an increase of about 69%, while the 15th mode rises from 5.5366 to 6.3528 kHz, an increase of about 15%. The 12th mode, which dominates several of the waterfall peaks, rises from 4.0876 to 5.6128 kHz.
| Mode order | 5 N·m (kHz) | 10 N·m (kHz) | 20 N·m (kHz) | 50 N·m (kHz) | 100 N·m (kHz) |
|---|---|---|---|---|---|
| 1 | 0.6239 | 0.7079 | 0.8014 | 0.9395 | 1.0532 |
| 7 | 1.9322 | 2.0205 | 2.1204 | 2.2654 | 2.3785 |
| 10 | 3.311 | 3.4811 | 3.6453 | 3.9594 | 4.375 |
| 12 | 4.0876 | 4.5177 | 5.0017 | 5.412 | 5.6128 |
| 15 | 5.5366 | 5.5604 | 5.5911 | 5.8396 | 6.3528 |
| 16 | 5.8754 | 5.9963 | 6.1493 | 6.41 | 6.6584 |
This load dependence matters for resonance assessment. A natural frequency that is avoided at one torque may be crossed at another. A single nominal-load modal check is therefore not enough to describe the vibration behavior of a gear set that operates over a wide torque range.
Resonant Response in the Acceleration Waterfall
The vibration acceleration waterfall is built from the first mesh-frequency order as the input speed is swept. The horizontal axis is mesh frequency, the depth axis is input speed, and the vertical axis is acceleration amplitude. A ridge at a fixed frequency identifies a resonance: the mesh frequency coincides with a natural frequency, and the response at that speed rises sharply. Table 4 summarizes the largest acceleration amplitude and its dominant frequency for each load.
| Load (N·m) | LTE fluctuation (arcsec) | Maximum acceleration (m/s²) | Dominant frequency (kHz) | Mode order |
|---|---|---|---|---|
| 5 | 6.924 | 8,588 | ≈4.08 | 12 |
| 10 | 4.96 | 6,315 | ≈4.52 | 12 |
| 20 | 2.03 | 1,569 | ≈5.00 | 12 |
| 50 | 6.75 | 7,150 | ≈5.84 | 15 |
| 100 | 10.36 | 18,561 | ≈6.35 | 15 |
At 5, 10, and 20 N·m, the maximum acceleration occurs near the 12th natural frequency. At 50 and 100 N·m, the dominant peak moves to the 15th mode. The shift is a consequence of the load-dependent natural frequencies: as torque rises, the modes move upward, and the speed sweep reaches a different resonance. The peak frequency changes from about 4.08 kHz at 5 N·m to about 6.35 kHz at 100 N·m.
The waterfall also contains secondary ridges near 1.932, 3.311, and 5.875 kHz at 5 N·m; near 2.020, 3.481, and 5.996 kHz at 10 N·m; near 2.120, 3.645, 5.538, and 6.149 kHz at 20 N·m; near 2.265, 3.874, 5.500, and 6.410 kHz at 50 N·m; and near 2.379, 4.060, and 6.658 kHz at 100 N·m. These correspond to the 7th, 10th, 16th and, in some cases, 8th, 9th, 13th, and 14th modes. Not every natural frequency produces a strong visible peak. A mode becomes dangerous only when it is sufficiently excited by the mesh force, so the mode list should not be read as a list of equally severe resonances.
The maximum acceleration follows the LTE fluctuation ranking: 20 N·m is lowest, followed by 10, 50, 5, and 100 N·m. The 100 N·m peak, 18,561 m/s², is about 11.8 times the 20 N·m peak of 1,569 m/s². This is the central coupling in the results. The absolute LTE maximum rises monotonically with load, but the peak acceleration does not. The acceleration ranking follows the fluctuation instead. Even though the resonant frequency shifts with load, reducing the LTE fluctuation reduces the resonant acceleration amplitude.
Forced Response at Constant Speed
The waterfall describes the response when the mesh frequency crosses a resonance. A constant-speed spectrum shows how the same system behaves during ordinary running. At an input speed of 600 r/min, the mesh frequency is 0.16 kHz. In the acceleration amplitude spectra of both the pinion and the gear, the 20 N·m case has the smallest amplitude. The amplitudes then increase in the order 10, 50, 5, and 100 N·m.
This is not a resonance sweep, so the ranking cannot be explained by a shift in natural frequency alone. It reflects the amplitude of the mesh excitation at the operating speed. The fact that the spectrum order matches the LTE fluctuation order indicates that the fluctuation controls the forced response as well as the resonant peak. A load case with a smooth LTE curve produces lower acceleration at the mesh frequency; a load case with a steep or edge-contacted LTE curve produces higher acceleration even when the system is not at resonance.
Predicted Noise Trend
For input speeds from 500 to 1000 r/min, the predicted noise curves rank from highest to lowest as 100, 5, 50, 10, and 20 N·m. This is the same order as the acceleration response and the inverse order of the LTE fluctuation. The 20 N·m case, which has the smallest fluctuation, also has the lowest predicted noise. The 100 N·m case, which has the largest fluctuation, has the highest predicted noise.
The noise result is a simulation prediction of the structural acoustic response, not a measured sound-pressure level. Absolute values depend on damping, housing radiation, boundary conditions, and the measurement environment. The relative trend is nevertheless consistent across the waterfall, the constant-speed spectrum, and the noise curve: lower LTE fluctuation produces lower vibration and noise in this gear system.
Design Implications and Scope
The practical target is not simply a small unloaded TE. A gear set can have a low geometric TE and still develop a steep LTE curve once the contact reaches an edge. Loaded TCA should therefore track both contact position and LTE fluctuation across the operating torque range, with particular attention to the torque at which edge contact begins. In this case, the transition occurs between 20 and 50 N·m. The 20 N·m condition gives the lowest fluctuation and the lowest vibration, but the useful design window is the range of loads over which that smooth contact condition can be maintained.
Because natural frequencies shift with load, resonance checks should be repeated at several torque levels rather than at a single nominal load. The dominant resonance moved from the 12th to the 15th mode as torque increased, and the peak frequency rose by more than 2 kHz across the load range. A design that is quiet at one load can therefore encounter a different resonance at another. The waterfall is a more informative tool than a single modal or steady-state calculation for this type of gear set.
The limits of the analysis should be kept in view. The results come from one 16/27-tooth spiral bevel gear pair and one shaft–bearing representation, using modal superposition in MASTA. They are simulation predictions rather than test data, and absolute acceleration and noise levels depend on damping, housing, and boundary conditions that are not part of the gear pair itself. Within those limits, the transferable result is clear: reducing the peak-to-peak fluctuation of the loaded transmission error reduces mesh vibration and noise, and the effect is visible both at resonance and during constant-speed operation.