Herringbone gear transmissions offer high load capacity, smooth operation, and minimal axial loads, making them critical components in heavy machinery for energy equipment, marine vessels, and aerospace applications. The increasing demand for stability, reliability, and low noise in large machinery drives the development of high-precision herringbone gears with extended service life. However, gear machining imperfections, particularly symmetry errors (misalignment between left and right helical halves), severely impact meshing stability. This study establishes a loaded tooth contact analysis (LTCA) model incorporating symmetry errors, analyzes their detrimental effects, and proposes a novel error coupling compensation method using intentional axial assembly offsets.
1. Loaded Contact Analysis Model with Symmetry Error
The herringbone gear pair is modeled as two parallel helical gear contacts. The elastic contact process is discretized along the major axis of the instantaneous contact ellipse into \(n\) points. The initial gap \(w_k\) at point \(k\) before loading is defined as:
$$ w_k = \delta + b_k $$
where \(\delta\) is the transmission error and \(b_k\) is the geometric spacing. Symmetry error \(\zeta\), defined as the vertical distance between the left and right tooth trace extensions on the gear axis (Figure 2), modifies the initial gap for the left (\(I\)) and right (\(II\)) helical halves:
$$
\begin{cases}
w_k^I = \delta + b_k^I + \zeta_s \\
w_k^{II} = \delta + b_k^{II} – \zeta_s
\end{cases}
$$
where \(\zeta_s = \zeta / 2\) is half the symmetry error in the meshing normal direction. Accounting for pinion axial float \(\sigma\) to balance axial forces, the displacement coordination equations incorporating symmetry error are:
$$
-[S]^t \{F\}^t – \{U\}^t + \{Z\} + \{d\}^t + \{\sigma_s\}^t = \{w\}^t, \quad t = I, II
$$
$$
\sum_{k=1}^{n} F_k^I = \sum_{k=1}^{n} F_k^{II}, \quad \left| \sum_{k=1}^{n} \sigma_{s,k}^I \right| + \left| \sum_{k=1}^{n} \sigma_{s,k}^{II} \right| = 0
$$
subject to the contact conditions:
$$
\begin{cases}
d_k^t = 0, & F_k^t > 0 \\
d_k^t > 0, & F_k^t = 0
\end{cases} \quad t = I, II
$$
Here, \([S]^t\) is the flexibility matrix, \(\{F\}^t\) is the load vector, \(\{U\}^t\) is the contact deformation vector, \(\{d\}^t\) is the residual gap vector, and \(\{\sigma_s\}^t\) is the axial float vector transformed to the meshing normal direction. The model is solved using a two-layer iterative algorithm combined with the golden section method.
2. Model Validation via Finite Element Analysis
A 3D model of a herringbone gear pair (Table 1) was created with a symmetry error \(\zeta = 0.02\) mm (introduced by rotating the left half of the driven gear by 0.01°). Nonlinear dynamic finite element analysis (FEA) in ANSYS used hexahedral elements (400,480 nodes, 134,864 elements). Boundary conditions included a revolute joint on the driven gear axis and a cylindrical joint on the pinion allowing axial float. Speed (2000 rpm) and torque (6000 N·m) were ramped from 0 to 0.1s.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of Teeth, \(z\) | 34 | 30 |
| Module (mm) | 5 | 5 |
| Pressure Angle, \(\alpha\) (°) | 20 | 20 |
| Helix Angle, \(\beta\) (°) | 35 | 35 |
| Face Width, \(b\) (mm) | 160 | 160 |
| Undercut Width, \(b_d\) (mm) | 24 | 24 |
| Undercut Depth, \(d\) (mm) | 5 | 4 |
| Symmetry Error, \(\zeta\) (mm) | 0 | 0.02 |
| Speed, \(n\) (rpm) | 2000 | – |
| Load Torque, \(T\) (N·m) | – | 6000 |
FEA results for axial displacement and loaded transmission error (LTE) showed good agreement with the LTCA model predictions (Figure 5). The symmetry error caused asymmetric contact forces, leading to pinion axial oscillation and LTE fluctuations, particularly during gear mesh entry/exit events due to stiffness variations. Axial displacement stabilized around 0.018 mm, and LTE fluctuated near -0.174 rad under steady speed.
3. Impact of Symmetry Error on Meshing Characteristics
FEA simulations analyzed symmetry errors (\(\zeta\)) of 0.05 mm, 0.10 mm, 0.15 mm, and 0.20 mm, representing typical tolerances achievable with conventional gear machining processes.
3.1 Tooth Contact Stress Distribution
Contact stress nephograms revealed significant asymmetric load distribution (Figure 6). The side with reduced initial gap (due to \(\zeta\)) experienced higher contact stress, shifting the contact pattern towards that side. Stress concentration increased with \(\zeta\), exceeding material yield limits during mesh entry for larger errors (\(\zeta \geq 0.15\) mm), risking premature failure and reduced gear machining component life.
3.2 Axial Displacement Dynamics
Axial displacement exhibited three distinct phases (Figure 7):
- Start-up Phase: Minimal displacement until initial tooth contact occurred. Duration decreased with increasing \(\zeta\) due to smaller initial gaps.
- Loading Phase: Sudden axial jump upon contact collision, followed by high-amplitude oscillation. Peak displacement magnitude increased with \(\zeta\).
- Steady-Speed Phase: Periodic oscillation with reduced amplitude. Mean axial displacement closely approximated \(\zeta\) (Table 2).
| \(\zeta\) (mm) | Start-up Time (s) | Max. Displacement (mm) | Mean Steady-State Disp. (mm) |
|---|---|---|---|
| 0.05 | 0.0034 | 0.091 | 0.051 |
| 0.10 | 0.0023 | 0.138 | 0.099 |
| 0.15 | 0.0009 | 0.176 | 0.149 |
| 0.20 | 0.0002 | 0.208 | 0.186 |
3.3 Axial Force Generation
Axial force dynamics mirrored displacement behavior (Figure 8):
- Start-up Phase: Negligible axial force before contact.
- Loading Phase: Force spikes during collisions, increasing in magnitude with \(\zeta\). Multiple collisions occurred before stabilization.
- Steady-Speed Phase: Periodic force fluctuation synchronized with gear mesh frequency (\(f_m = 999\) Hz). Force amplitude and variability increased significantly with \(\zeta\) (Table 3), causing instability, noise, and potential tooth separation.
| \(\zeta\) (mm) | Max. Force (N) | Mean Force (N) | Std. Dev. (N) | Dominant Freq. (Hz) |
|---|---|---|---|---|
| 0.05 | 970.27 | 99.38 | 322.31 | 994.02 |
| 0.10 | 1527.58 | 130.78 | 516.44 | 1002.14 |
| 0.15 | 1989.74 | 168.35 | 845.26 | 1014.57 |
| 0.20 | 2172.76 | 231.28 | 989.76 | 999.74 |
4. Symmetry Error Compensation via Axial Assembly Offset
Precision gear machining to minimize \(\zeta\) is costly. We propose compensating \(\zeta\) by introducing a controlled axial assembly offset \(\lambda\) during installation. This modifies the initial gap equations:
$$
\begin{cases}
w_k^I = \delta + b_k^I + \zeta_s – \lambda \\
w_k^{II} = \delta + b_k^{II} – \zeta_s + \lambda
\end{cases}
$$
The optimal \(\lambda\) counteracts \(\zeta_s\), equalizing the initial gaps. Compensation values (\(\lambda\)) of 0.05 mm, 0.10 mm, 0.15 mm, and 0.18 mm were analyzed for the respective \(\zeta\) values.
4.1 Compensated Contact Stress
Compensation dramatically improved load distribution symmetry (Figure 9). Contact patterns became balanced across both helical halves. The effectiveness was more pronounced for larger initial symmetry errors, demonstrating the method’s capability to mitigate gear machining-induced inaccuracies.
4.2 Compensated Axial Displacement
Compensation eliminated the large initial axial jump and drastically reduced oscillation amplitudes in all phases (Figure 10). Peak displacements during steady operation were minimized to below 7.83 µm, representing reductions exceeding 96% compared to the uncompensated case for \(\zeta = 0.20\) mm. Motion became predominantly periodic with minimal random fluctuation.
4.3 Compensated Axial Force
Axial forces were confined within a narrow band (±300 N) during steady operation (Figure 11). The violent force spikes during loading vanished. Mean axial forces were reduced to very low levels (56.14 N – 64.68 N), representing only 1.31% – 1.82% of the peak forces observed on a single helix side without compensation. This confirms near-perfect axial force balance was achieved.
| \(\zeta\) (mm) | Comp. \(\lambda\) (mm) | Mean Axial Force (N) | % of Uncompensated Peak Force |
|---|---|---|---|
| 0.05 | 0.05 | 56.14 | 1.48% |
| 0.10 | 0.10 | 55.37 | 1.31% |
| 0.15 | 0.15 | 62.98 | 1.78% |
| 0.20 | 0.18 | 64.68 | 1.82% |
5. Conclusions
- An LTCA model incorporating symmetry error (\(\zeta\)) was developed and validated against FEA. A novel compensation method using an axial assembly offset (\(\lambda\)) was proposed, defined by the modified gap equation: \(w_k^I = \delta + b_k^I + \zeta_s – \lambda; w_k^{II} = \delta + b_k^{II} – \zeta_s + \lambda\).
- Symmetry error, inherent in conventional gear machining, causes severe asymmetric loading. Contact stress increases and shifts towards the side with the smaller initial gap, escalating with \(\zeta\) and risking gear failure.
- \(\zeta\) induces significant axial vibration characterized by a large initial displacement jump, high-amplitude oscillation during loading, and a mean steady-state offset approximately equal to \(\zeta\). Axial force fluctuations, synchronized with mesh frequency, increase dramatically in magnitude and variability with \(\zeta\), destabilizing the transmission.
- The error coupling compensation method (\(\lambda \approx \zeta\)) effectively neutralizes the impact of symmetry error. It restores symmetric contact patterns, reduces peak axial displacement by over 96%, and lowers mean axial force to less than 2% of the uncompensated peak force. This method offers a practical and cost-effective solution to enhance stability without requiring ultra-high precision gear machining.