Herringbone gears are widely used in high-power marine propulsion systems due to their high load-carrying capacity, smooth operation, and excellent vibration-damping characteristics. However, the complex geometry of double helical teeth introduces coupled bending, torsional, and axial vibrations, which significantly affect the noise and vibration behavior of the entire transmission system. To reduce the vibrational noise and improve the reliability of herringbone gear drives, it is essential to understand the dynamic response under realistic excitation conditions. In this study, I develop a comprehensive dynamic model of a herringbone gear pair that accounts for stiffness excitation, error excitation, and mesh impact excitation, and I investigate the influence of tooth profile modifications and lead crown modifications on the dynamic behavior. Both theoretical analysis and experimental tests are conducted to validate the findings.
1. Dynamic Model of Herringbone Gear System
I adopt a lumped-parameter method to establish a bending-torsional-axial coupled dynamic model of the herringbone gear transmission. The model includes two helical gear pairs (left and right ends) connected by a common shaft with axial, lateral, and torsional degrees of freedom. The motions of the pinion and gear at each end are described by translational displacements in the y-direction (lateral), z-direction (axial), and rotational displacements about the shaft axis (θ). The dynamic equations are derived using Newton’s second law. The complete set of differential equations is given below.
For the left pinion (index p1):
$$
\begin{aligned}
m_p \ddot{y}_{p1} + c_{p1y} \dot{y}_{p1} + k_{p1y} y_{p1} + c_{py} (\dot{y}_{p1} – \dot{y}_{p2}) + k_{py} (y_{p1} – y_{p2}) &= -F_{yp1}, \\
m_p \ddot{z}_{p1} + c_{pz} (\dot{z}_{p1} + \dot{z}_{p2}) + k_{pz} (z_{p1} + z_{p2}) &= -F_{z1}, \\
I_{p1} \ddot{\theta}_{p1} &= T_{p1} – F_{yp1} R_{p1}.
\end{aligned}
$$
For the left gear (index g1):
$$
\begin{aligned}
m_g \ddot{y}_{g1} + c_{g1y} \dot{y}_{g1} + k_{g1y} y_{g1} + c_{gy} (\dot{y}_{g1} – \dot{y}_{g2}) + k_{gy} (y_{g1} – y_{g2}) &= F_{yg1}, \\
m_g \ddot{z}_{g1} + c_{g1z} \dot{z}_{g1} + k_{g1z} z_{g1} + c_{gz} (\dot{z}_{g1} + \dot{z}_{g2}) + k_{gz} (z_{g1} + z_{g2}) &= F_{z1}, \\
I_{g1} \ddot{\theta}_{g1} &= -T_{g1} + F_{yg1} R_{g1}.
\end{aligned}
$$
Similarly, for the right pinion (p2) and right gear (g2), the equations are symmetric. In these equations, m represents mass, I represents moment of inertia, k and c denote stiffness and damping coefficients for various directions (subscripts y, z, etc.), R is the base circle radius, T is the applied torque, and F represents the dynamic mesh forces in the tangential (y) and axial (z) directions. The axial displacement excitation, which arises from manufacturing errors and leads to periodic axial motion at mesh frequency and shaft rotational frequency, is included as part of the error excitation in the Fz terms.
2. Internal Excitations
Three main internal excitations are considered: time-varying mesh stiffness, mesh impact, and axial displacement error.
2.1 Mesh Stiffness Excitation
The time-varying mesh stiffness is obtained from a loaded tooth contact analysis (LTCA) of the herringbone gear pair over one mesh cycle. The resulting stiffness variation is periodic and can be represented as a Fourier series. A typical stiffness curve is shown in the literature, with a mean value around 1.85 GN/m and a fluctuation of about ±7%. This excitation primarily affects the torsional and lateral vibrations.
2.2 Mesh Impact Excitation
Due to tooth profile errors and the finite stiffness of gear teeth, a sudden impact occurs at the beginning of each mesh cycle. This impact force is computed based on the relative velocity and the local contact stiffness. The impact force magnitude is typically around 4.7 kN for the studied gear pair.
2.3 Axial Displacement Excitation
In herringbone gears, manufacturing tolerances and assembly misalignments produce a small axial displacement that varies at both the mesh frequency and the shaft rotational frequency. The amplitude of this displacement is on the order of 5–10 μm. This excitation is the primary source of axial vibration in the system.
Table 1 summarizes the characteristics of each excitation and their dominant influence on the vibration components.
| Excitation type | Frequency content | Amplitude range | Dominant vibration direction |
|---|---|---|---|
| Mesh stiffness | Mesh frequency and harmonics | ±7% of mean stiffness | Torsional, lateral |
| Mesh impact | Broadband, centered at mesh frequency | ~4.7 kN peak | Torsional |
| Axial displacement | Mesh frequency + shaft frequency | 5–10 μm | Axial |
3. Effect of Tooth Modification on Dynamic Response
I investigate three modification scenarios: (1) no modification, (2) only profile modification (three-segment parabolic relief on the pinion), and (3) combined profile and lead crown modification (profile relief plus a parabolic crowning along the face width). The dynamic responses are computed for a specific operating condition: pinion speed 2881 rpm, gear torque 2000 N·m, and gear parameters as listed in Table 2.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 31 | 102 |
| Module (mm) | 4.5 | 4.5 |
| Pressure angle (°) | 20 | 20 |
| Helix angle (°) | 28.34 | 28.34 |
| Face width (mm) | 90 × 2 | 90 × 2 |
| Hand of helix | Right-left (double) | Left-right (double) |
| Gap width (mm) | 70 | 70 |
Table 3 presents the root-mean-square (RMS) values of the vibration accelerations in the circumferential direction (along the line of action of the gear pair) and in the axial direction for the pinion, for each modification case. The percentage reductions relative to the unmodified case are also shown.
| Modification type | Circumferential vibration (m/s²) | Axial vibration (m/s²) | Reduction in circumferential (%) | Reduction in axial (%) |
|---|---|---|---|---|
| No modification | 28.2856 | 20.9870 | — | — |
| Profile relief only | 20.5906 | 20.4207 | 27.2 | 2.0 |
| Profile relief + lead crown | 19.8035 | 11.4714 | 30.0 | 45.0 |
The results clearly show that profile modification alone significantly reduces the torsional/circumferential vibration (27.2% reduction) but has very little effect on axial vibration (only 2% reduction). In contrast, when both profile relief and lead crown modification are applied, the axial vibration is reduced by 45%, while the circumferential vibration is further reduced to 30%. This demonstrates that axial vibration in herringbone gears is primarily caused by axial displacement excitation, which can be effectively mitigated by lead crowning that improves load distribution along the face width and reduces the axial component of the mesh force.
4. Experimental Validation
To validate the theoretical findings, I performed vibration tests on a herringbone gearbox test rig. The test conditions are listed in Table 4. The torque direction was reversed for the modified gear test, while all other parameters remained identical.
| Parameter | Value |
|---|---|
| Motor speed (rpm) | 300 |
| Low-speed shaft speed (rpm) | 876 |
| High-speed shaft speed (rpm) | 2881 |
| Torque (N·m) | 2000 |
| Running time (min) | 30 |
| Torque direction (viewed from outside) | Counterclockwise |
Six accelerometers were mounted on the gearbox housing flange: four (positions 1#–4#) measured radial vibrations, one (5#) measured axial vibration, and one (6#) measured another radial direction. The average vibration amplitudes before and after tooth modification (profile relief + lead crown) are summarized in Table 5.
| Measurement point | Before modification (m/s²) | After modification (m/s²) | Reduction (%) |
|---|---|---|---|
| 1# (radial) | 3.33 | 3.01 | 10 |
| 2# (radial) | 7.23 | 5.70 | 21 |
| 3# (radial) | 3.62 | 2.86 | 21 |
| 4# (radial) | 4.64 | 3.20 | 31 |
| 5# (axial) | 2.05 | 2.38 | −14 (increase) |
| 6# (radial) | 4.98 | 2.96 | 41 |
The test results indicate that profile and lead crown modification effectively reduce radial vibrations by 10%–41%. However, the axial vibration at point 5# increased slightly (14%), which may be due to the reversal of torque direction or local resonance effects. Overall, the experimental trends are consistent with the theoretical analysis: circumferential vibration reductions are achieved through profile modification, while axial vibration requires careful lead crowning. The theoretical model predicted a 45% reduction in axial vibration with combined modifications, but the test showed a small increase. This discrepancy suggests that the axial excitation model may need refinement to account for additional factors such as shaft misalignment or housing flexibility.
5. Conclusions
In this work, I have presented a comprehensive dynamic analysis of herringbone gear transmissions. The key findings are:
- The bending-torsional-axial coupled lumped-parameter model accurately captures the essential vibration modes of herringbone gears.
- Mesh stiffness and mesh impact excitations dominate the torsional and lateral vibrations, while axial displacement excitation is the primary source of axial vibration.
- Profile modification alone reduces torsional vibration by about 27% but has negligible effect on axial vibration.
- Combined profile and lead crown modification reduces both torsional (30%) and axial (45%) vibrations, making it an effective strategy for noise and vibration control.
- Experimental tests confirm the beneficial effect of tooth modification on radial vibrations, although axial vibration behavior requires further investigation.
These insights provide a solid foundation for the design of low-noise herringbone gear drives in marine and industrial applications.