Loaded transmission error as the link between geometry and vibration
Spiral bevel gears transmit motion and torque between intersecting shafts in automotive, marine, and aerospace drivetrains. Their curved teeth provide gradual engagement, high load capacity, and relatively smooth motion. Under torque, however, tooth bending, contact deformation, and body deflection introduce an elastic angular lag between the pinion and the gear. This lag is described by the loaded transmission error (LTE), which acts as a displacement excitation at the mesh and is a major source of gear whine and vibration.
The LTE is not a purely geometric quantity. It combines the geometric transmission error designed into the tooth flanks with the angular deformation produced by the load:
$$
E_L(\varphi)=E_T(\varphi)+\delta_\varphi,
$$
where \(E_T(\varphi)\) is the geometric transmission error at pinion angle \(\varphi\), and \(\delta_\varphi\) is the load-induced angular deformation obtained from the contact solution. Because the geometric term is sensitive to tooth-flank topology, a change of only a few micrometres can alter the contact pattern and the resulting vibration. Conventional CAD-to-CAE workflows that build a solid model and then remesh the flanks may lose this level of detail and may also introduce assembly error when the pinion and gear are positioned manually.
Tooth-surface mesh and finite element assembly
The example gear pair has a 16-tooth pinion and a 27-tooth gear with a module of 4.25 mm, a midpoint spiral angle of 35°, a normal pressure angle of 20°, and a shaft angle of 90°. The pinion is left-handed and the gear is right-handed. Table 1 lists the basic data used for the model.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 16 | 27 |
| Module (mm) | 4.25 | 4.25 |
| Midpoint spiral angle (°) | 35 | 35 |
| Normal pressure angle (°) | 20 | 20 |
| Shaft angle (°) | 90 | 90 |
| Spiral hand | Left | Right |
| Outer cone distance (mm) | 66.693 | 66.693 |
| Pitch cone angle (°) | 30.650 7 | 59.349 3 |
| Face cone angle (°) | 35.350 1 | 62.216 8 |
| Root cone angle (°) | 27.783 2 | 54.649 9 |
| Addendum (mm) | 4.682 | 2.542 |
| Dedendum (mm) | 3.342 | 5.482 |
| Clearance (mm) | 0.8 | 0.8 |
The gear tooth surface is generated from machine-tool settings rather than from an approximate CAD tooth. For the gear, the nodes are arranged on a rotation-projection plane. The coordinates of node \((i,j)\) satisfy
$$
x_2(u_{ij},v_{ij})=L_{ij},\qquad
\sqrt{y_2^2(u_{ij},v_{ij})+z_2^2(u_{ij},v_{ij})}=R_{ij},
$$
for \(i=1,\dots,n\) and \(j=1,\dots,m\). Here \(L_{ij}\) and \(R_{ij}\) are coordinates on the projection plane, \(u\) and \(v\) are tooth-surface parameters, \(n\) is the number of nodes along the tooth height, and \(m\) is the number along the face-width direction. The position vector and normal at each node are obtained from the tooth-surface equations. These nodes become the vertices of the hexahedral mesh, so the flank geometry is carried directly into the finite element model instead of being approximated by a remeshed CAD solid.
Hexahedral elements are used because contact is a geometrically nonlinear problem with boundary nonlinearity, and hexahedral meshes resist distortion while giving efficient and accurate contact convergence. The root transition is refined with six nodes across the fillet, where the tooth shape changes abruptly and stress concentration is expected.
Gear tooth contact analysis supplies the contact path and the instantaneous angular positions of the pinion and gear. A coordinate transformation places a five-tooth sector in the calculated meshing position, and the node and element data are written as an input file for ABAQUS. This avoids manual adjustment of the assembly position and the associated risk of tooth interference. The material is alloy steel with Young’s modulus \(2.1\times10^5\) MPa, Poisson ratio 0.3, and density \(7.9\times10^{-9}\) t/mm³. Contact pairs cover the convex and concave flanks and the root and tip transition regions. The gear flank is the slave surface and the pinion flank is the master surface to reduce interpenetration. Reference points on the two axes are coupled to the inner rim and hub section. The implicit static analysis uses two steps: 1 s to remove the backlash and establish contact, followed by 3 s under torque, with the pinion fully constrained and the gear free to rotate about its axis.
Contact stress and load-dependent transmission error
At a torque of 30 N·m, the contact-stress fields show the expected load sharing. Contact stress is highest during single-tooth contact and lower during double-tooth contact, when two tooth pairs share the load. The maximum principal stress occurs near the centre of the contact ellipse. As the mesh progresses, the contact ellipse moves from the heel at the tooth tip toward the toe at the tooth root. When the middle gear tooth leaves mesh, the maximum contact stress lies below the addendum line. No edge contact was observed in this load case, and the contact region remained in the middle of the working flank.
The LTE is calculated from the contact solution using the angular deformation at each meshing position. The geometric term comes from the tooth contact analysis, while the elastic term comes from the finite element contact result. The LTE is expressed in microradians. Table 2 summarises the response for the three torque levels considered.
| Torque (N·m) | LTE fluctuation (μrad) | Maximum LTE (μrad) | Measured first-order mesh acceleration (m/s²) |
|---|---|---|---|
| 10 | 18.64 | — | — |
| 30 | 11.18 | — | 0.109 7 |
| 50 | 31.24 | 259.15 | 0.180 2 |
The fluctuation is therefore non-monotonic. It is 18.64 μrad at 10 N·m, falls to 11.18 μrad at 30 N·m, and rises to 31.24 μrad at 50 N·m. At 50 N·m, the peak LTE reaches 259.15 μrad and the curve lies above the right-hand endpoint of the geometric transmission error curve, which is the critical condition for edge contact. Severe edge contact concentrates stress at the tooth tip or root, reduces tooth strength, makes transmission unsteady, and further amplifies the LTE fluctuation. At 30 N·m, the fluctuation is the smallest of the three cases, so this load gives the smoothest transmission among the tested conditions.
The non-monotonic trend reflects the interaction between load sharing and contact migration. At moderate torque, the contact pattern remains favourably placed on the flank and elastic deformation partially compensates the geometric transmission error. At higher torque, the contact pattern moves toward the edge and the increased compliance amplifies the fluctuation.
Vibration response and the resonance condition
The time-varying LTE generates a dynamic mesh force. Using the LTE curve as the excitation, a system-level vibration model in MASTA evaluates the acceleration waterfall for the first three meshing harmonics. The first harmonic produces the largest acceleration and dominates the overall vibration response.
For the 30 N·m case, the first-harmonic response has prominent peaks at 1.66 kHz and 2.19 kHz, corresponding to the first and third natural frequencies, with acceleration amplitudes of 131.91 m/s² and 128.93 m/s², respectively. The second natural frequency at 2.10 kHz does not produce a comparable peak. The maximum first-harmonic acceleration is smallest at 30 N·m; it is higher at 10 N·m and highest at 50 N·m, following the same ranking as the LTE fluctuation.
Frequency coincidence alone is not sufficient to produce severe vibration. At 1.66 kHz and 2.19 kHz, the two compliance curves have equal absolute values and opposite phase, so the dynamic mesh force reaches a peak. At 2.10 kHz, the compliance curves intersect, but their absolute values are unequal and the phase difference is not 180°, so the mesh force does not peak. The phase response spans 0°–180° over the relevant range. The practical resonance criterion therefore combines frequency proximity with the compliance and phase condition at the mesh.
Experimental check at 540 r/min
Vibration tests were run on a closed-power-flow bevel gear fatigue rig using an M+P dynamic signal analyzer. An accelerometer was mounted on the pinion shaft close to the mesh. The input shaft speed was 540 r/min, and the driven gear was loaded at 10 N·m, 30 N·m, and 50 N·m. The measured axial acceleration spectra contain the mesh frequency and its harmonics, with the first harmonic dominant, consistent with the simulation.
At 540 r/min, the first mesh harmonic of the 16-tooth pinion is 144 Hz, well below the 1.66–2.19 kHz range of the calculated resonance peaks. The test therefore checks the load ranking of the first mesh harmonic rather than the full frequency-response map. The measured first-order mesh acceleration is 0.109 7 m/s² at 30 N·m and 0.180 2 m/s² at 50 N·m; the 10 N·m case lies between them. This ranking follows the LTE fluctuation: the 30 N·m case has the lowest vibration, and the 50 N·m case has the highest. Absolute acceleration levels are not directly comparable with the system-level simulation because the test measures axial shaft response at one location.
Design implications
For spiral bevel gears, the design target should be a low LTE fluctuation at the dominant operating torque, not merely a prescribed geometric transmission error amplitude. The geometric flank design and the load-induced contact migration must be evaluated together. A high-fidelity tooth-surface mesh and a contact-analysis-based assembly are important because micron-scale flank changes and small mounting errors can shift the contact pattern and alter the vibration excitation.
High torque requires particular attention to edge contact. An LTE curve that rises above the endpoint of the geometric transmission error curve is a warning that the contact is approaching the edge; if edge contact becomes severe, stress concentrates at the tip or root and the transmission becomes less stable. For resonance screening, a natural frequency close to a mesh harmonic is not enough by itself. The compliance magnitudes and phase relation at the mesh should also be checked. In this example, the 30 N·m condition gives the smallest LTE fluctuation and the lowest first-harmonic acceleration, but that optimum is specific to the flank design, mounting, and load range; it should not be treated as a universal recommendation for other spiral bevel gear pairs.