Herringbone gear transmission systems are widely used in high‑power marine and aerospace applications due to their high load‑carrying capacity, smooth operation, and cancellation of axial forces. However, under high‑speed and heavy‑load conditions, gear tooth deformations, manufacturing errors, and support deflections lead to impact vibrations, increased noise, and reduced service life. Tooth surface modification has proven effective in alleviating these issues. In this study, we propose a quadratic parabolic three‑dimensional modification for the pinion tooth surface of high‑precision herringbone gears, and we use cubic B‑spline surfaces to represent the exact tooth geometry including both modification and manufacturing errors. A 12‑degree‑of‑freedom bending‑torsion‑axial coupled dynamic model is established, incorporating transmission error excitation and corner meshing impact excitation. The influence of modification on internal excitation factors is investigated. A multi‑objective optimization function comprising the fluctuation amplitude of loaded transmission error, the amplitude of corner meshing impact force, and the root‑mean‑square (RMS) value of relative vibration acceleration along the line of action is solved using an improved adaptive genetic algorithm. The results show a 20.42% reduction in the RMS of relative vibration acceleration after modification, indicating significant vibration and noise reduction. A closed‑power‑flow test rig is built, and Heidenhain circular gratings are employed to measure the vibration acceleration of herringbone gears along the line of action. The experimental results are in good agreement with theoretical predictions, with a maximum deviation of less than 14.5%.
1. Introduction
Herringbone gears are essential in heavy machinery such as ship propulsion systems, where reliability and low noise are critical. The complex geometry and high precision requirements of these gears make tooth surface modification a key technique for improving dynamic performance. Traditional modifications include profile crowning and lead crowning, but separate treatments often fail to achieve optimal results under combined loading. In our work, we adopt a four‑order parabolic three‑dimensional modification applied to the pinion of a herringbone gear pair. The modification surface is constructed using cubic B‑spline interpolation, which allows accurate representation of arbitrary deviation distributions. We then incorporate the modified tooth surface into a comprehensive dynamic model that captures the coupled bending, torsional, and axial vibrations of the herringbone gear system. The analysis focuses on how modification affects the two dominant internal excitations: transmission error fluctuation and off‑line meshing impact. A multi‑objective optimization framework is developed to determine the optimal modification parameters. Finally, experiments on a closed‑power‑flow test rig validate the theoretical findings, demonstrating the effectiveness of the proposed method.
2. Basic Theory of Tooth Surface Modification
Tooth surface modification removes a thin layer of material from the theoretical conjugate surface, creating a deviation that influences meshing behavior. For herringbone gears, we apply a four‑order parabolic modification along both the profile and lead directions. The modification amount at any point on the tooth surface is defined by eight parameters: four for the profile (two maximum amounts and two lengths) and four for the lead (two maximum amounts and two lengths). The modification shape is described by a quartic parabola, which provides a smooth transition near the tooth edges. The actual deviation surface is represented by a cubic B‑spline surface, which can fit complex geometries with high accuracy (within 1 μm). The B‑spline formulation is given by:
$$
\mathbf{r}(u,v) = \sum_{i=0}^{n} \sum_{j=0}^{m} \mathbf{V}_{ij} N_{i,3}(u) N_{j,3}(v)
$$
where \(u,v\) are parametric coordinates, \(\mathbf{V}_{ij}\) are control points, and \(N_{i,3}(u)\) are cubic B‑spline basis functions. This surface is then superimposed onto the standard involute tooth surface to obtain the actual modified geometry. The resulting tooth surface is used in tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) to predict transmission errors and contact patterns.
| Parameter | y₁ (μm) | y₂ (mm) | y₃ (μm) | y₄ (mm) | Parabola order |
|---|---|---|---|---|---|
| Profile | 14 | 1.8 | 15 | 3.0 | 4 |
| Parameter | y₅ (μm) | y₆ (mm) | y₇ (μm) | y₈ (mm) | Parabola order |
| Lead | 15 | 11 | 14 | 13 | 4 |
Similarly, another modification set B is used for comparison, with slightly different values (e.g., profile y₁=15 μm, y₂=2.0 mm, etc.). These parameters are defined at the tooth tip/root and at the two ends of the face width.
3. Dynamic Model of Herringbone Gear System
We develop a 12‑degree‑of‑freedom (DOF) lumped‑parameter model for a herringbone gear pair supported by rolling element bearings. The model accounts for bending (y‑direction), axial (z‑direction), and torsional (θ) motions of the left and right helical sections of both pinion and gear. The generalized displacement vector is:
$$
\mathbf{\delta} = \left[ y_{p1}, z_{p1}, \theta_{p1}, y_{g1}, z_{g1}, \theta_{g1}, y_{p2}, z_{p2}, \theta_{p2}, y_{g2}, z_{g2}, \theta_{g2} \right]^T
$$
where subscripts p and g denote pinion and gear, and 1,2 denote the left and right helical sections. The equations of motion are derived using Newton’s second law. For example, the left pinion section satisfies:
$$
\begin{aligned}
m_{p1} \ddot{y}_{p1} + c_{p1y} \dot{y}_{p1} + k_{p1y} y_{p1} &= -F_{y1} + m_{p1} g \\
m_{p1} \ddot{z}_{p1} + c_{p12z} (\dot{z}_{p1}-\dot{z}_{p2}) + k_{p12z} (z_{p1}-z_{p2}) &= -F_{z1} \\
I_{p1} \ddot{\theta}_{p1} &= -F_{y1} R_p + T_{p1} – F_{s1} R_p
\end{aligned}
$$
Similar equations hold for the other three masses. Here, \(m\) and \(I\) are mass and moment of inertia, \(R_p\) and \(R_g\) are base circle radii, \(c\) and \(k\) are damping and stiffness of bearings and connecting shafts, \(F_{y1}, F_{z1}\) are dynamic mesh forces in the transverse and axial directions, and \(F_{s1}\) is the off‑line impact force. The dynamic mesh forces are computed using time‑varying mesh stiffness and transmission error obtained from LTCA. The system is solved numerically using a Runge‑Kutta method, and the steady‑state periodic response is analyzed.
| Parameter | Pinion | Gear |
|---|---|---|
| Normal module (mm) | 6 | 6 |
| Pressure angle (°) | 20 | 20 |
| Helix angle (°) | 24.43 | -24.43 |
| Number of teeth | 17 | 44 |
| Face width (mm) | 55 | 55 |
| Load torque (N·m) | — | 828 |
| Input speed (r/min) | 2000 | — |
4. Influence of Modification on Internal Excitation
4.1 Transmission Error
Transmission error is a primary internal excitation in gear dynamics. Using LTCA, we compute the loaded transmission error for the unmodified and modified herringbone gear surfaces. For the unmodified case, the fluctuation amplitude is about 2.9 arc‑seconds. After applying modification set A, the amplitude reduces to 2.2 arc‑seconds, and for set B it is 2.3 arc‑seconds. However, the waveform of set A contains additional inflection points corresponding to a second harmonic component in the frequency spectrum. This extra harmonic can excite resonances at sub‑harmonic speeds. Therefore, in addition to the fluctuation amplitude, the RMS of the vibration acceleration is also considered as an optimization objective. Figure 6 in the original study (here we omit figure numbers) shows that the vibration response of set A at 1/3 and 1/2 resonance speeds is significantly higher than that of set B, even though their peak‑to‑peak transmission errors are similar. Hence, a comprehensive optimization must account for both amplitude and spectral content.
4.2 Off‑line Meshing Impact
Off‑line meshing impact occurs when teeth enter contact earlier than the theoretical line of action due to deformations. The impact velocity can be computed from the geometry of the actual contact point. After modification, the load distribution at the entry point can be reduced or even eliminated. For unmodified herringbone gears under 828 N·m and 2000 r/min, the load sharing coefficient at the mesh entry is 0.032 and the impact force amplitude is 1160 N. With a light modification (set C), the entry load coefficient drops to 0.016 and the impact force to 786 N. A heavier modification (set D) that brings the entry load coefficient to zero reduces the impact force further to 435 N. However, excessive modification reduces the effective contact ratio and increases tooth deformations. Thus, a trade‑off is required, which is incorporated into our multi‑objective optimization.
| Parameter | y₁ (μm) | y₂ (mm) | y₃ (μm) | y₄ (mm) | Parabola order |
|---|---|---|---|---|---|
| Profile (Set C) | 8 | 1.8 | 10 | 2.8 | 4 |
| Profile (Set D) | 16 | 1.9 | 16 | 2.9 | 4 |
| Parameter | y₅ (μm) | y₆ (mm) | y₇ (μm) | y₈ (mm) | Parabola order |
| Lead (Set C) | 9 | 10 | 9 | 10 | 4 |
| Lead (Set D) | 11 | 10 | 13 | 10 | 4 |
5. Multi‑Objective Optimization of Tooth Surface Modification
We formulate a multi‑objective optimization problem that minimizes three functions simultaneously: (1) the peak‑to‑peak amplitude of the loaded transmission error \(f_e\), (2) the maximum off‑line meshing impact force \(f_I\), and (3) the RMS of the relative vibration acceleration along the line of action \(f_a\). The overall objective is a weighted sum:
$$
f(\mathbf{y}) = w_1 f_e(\mathbf{y}) + w_2 f_I(\mathbf{y}) + w_3 f_a(\mathbf{y})
$$
where \(\mathbf{y} = [y_1, y_2, \dots, y_8]^T\) are the eight modification parameters. The weights are chosen based on three typical torque levels: 621, 828, and 1035 N·m, with duty factors 0.2, 0.5, and 0.3 respectively. Constraints include bounds on modification amounts and lengths, as well as symmetry conditions (e.g., \(|y_1 – y_3| \leq Q_{y0}\)). The optimization is solved using an improved adaptive genetic algorithm that employs dynamic crossover and mutation probabilities to avoid premature convergence. The algorithm flowchart (omitted here) begins with population initialization, evaluates the three objective functions via LTCA and dynamic simulation, and evolves the population until convergence.
The optimized modification parameters are listed in Table 4. The resulting RMS of relative vibration acceleration along the line of action is reduced from 29.38 m/s² (unmodified) to 23.38 m/s², a reduction of 20.42%. The frequency spectrum after modification shows a significant reduction in higher harmonics, confirming the mitigation of off‑line impacts.
| Parameter | y₁ (μm) | y₂ (mm) | y₃ (μm) | y₄ (mm) | Parabola order |
|---|---|---|---|---|---|
| Profile | 16 | 1.6 | 18 | 3.2 | 4 |
| Parameter | y₅ (μm) | y₆ (mm) | y₇ (μm) | y₈ (mm) | Parabola order |
| Lead | 14 | 11.2 | 14 | 11.2 | 4 |
6. Experimental Validation
To verify the theoretical predictions, we constructed a closed‑power‑flow test rig for herringbone gears. The rig includes a driving motor, a slave gearbox, torsional shafts, and a loading device that allows a constant circulating power. Two Heidenhain ROD280 angle encoders with 18,000 lines per revolution (resolution ±5 arc‑seconds) are mounted on the pinion and gear shafts via flexible couplings. The signals are acquired by a PCI‑8502 data acquisition card with a sampling frequency of 40 MHz. By simultaneously measuring the rotation angles of both shafts, the dynamic transmission error is obtained. The relative vibration acceleration along the line of action is derived by double differentiation of the angular transmission error after converting to linear displacement using base circle radii.
Time‑domain synchronous averaging (TSA) is applied to suppress random noise. The signal is divided into segments corresponding to one shaft revolution, and these segments are averaged 100 times. This reduces the RMS of the noise by a factor of 10, revealing the periodic components clearly. The frequency response of TSA shows that the desired tooth‑mesh frequency and its harmonics are preserved while other frequencies are attenuated.
The experiment was performed at 2000 r/min and a load torque of 828 N·m, using both unmodified and optimized modified herringbone gear sets. The measured RMS of vibration acceleration for the unmodified case was 32.13 m/s², and for the modified case it dropped to 25.13 m/s², a reduction of 21.8%. The theoretical prediction gave 23.38 m/s², yielding an error of approximately 7%. Figure 20 in the original reference (omitted here) shows that the dominant frequency components in both simulation and experiment are at the first two harmonics of the mesh frequency, confirming the theory.
| Load (N·m) | Unmodified (Sim.) | Modified (Sim.) | Unmodified (Exp.) | Modified (Exp.) |
|---|---|---|---|---|
| 414 | 18.2 | 19.5 | 20.1 | 21.3 |
| 621 | 24.5 | 21.0 | 27.0 | 23.5 |
| 828 | 29.4 | 23.4 | 32.1 | 25.1 |
| 1035 | 33.8 | 27.6 | 36.5 | 30.0 |
The experimental results show the same trend as simulations, with the largest reduction (about 18.7% in experiment vs. 20.4% in simulation) occurring at the design load of 828 N·m. At the light load of 414 N·m, the modified gear actually produced slightly higher vibration than the unmodified one, because the modification was optimized for higher loads; this highlights that a single modification cannot be optimal for all operating conditions.
7. Conclusion
In this study, we presented a systematic approach for the three‑dimensional modification of herringbone gears to reduce vibration and noise. The key contributions are:
- Proposal of a four‑order parabolic modification combined with cubic B‑spline surface modeling for high‑precision herringbone gear tooth surfaces.
- Development of a 12‑DOF coupled dynamic model for herringbone gears that includes transmission error and off‑line impact excitations.
- Quantitative analysis of how tooth modification reduces the fluctuation of loaded transmission error and the amplitude of off‑line meshing impact.
- Multi‑objective optimization using an improved adaptive genetic algorithm, leading to a 20.42% reduction in the RMS of vibration acceleration at the design load.
- Experimental validation on a closed‑power‑flow test rig using high‑resolution encoders, with good agreement (maximum error 14.5%) between theory and measurement.
The results demonstrate that the proposed three‑dimensional modification method is effective for improving the dynamic performance of herringbone gear transmission systems. Future work will extend the optimization to variable‑speed and variable‑load conditions.