In our study of enclosed differential herringbone gear transmission systems, we developed a comprehensive dynamics model based on lumped-parameter theory. The model accounts for bearing elastic deformation, time-varying mesh stiffness excitation, error excitations, and the influence of intermediate floating components. We introduced helical gear mesh stiffness formulas to compute the time-varying stiffness of herringbone gears in a parallel manner and solved the system dynamics using the Fourier series method to obtain dynamic load sharing coefficients. This paper presents our analysis of how eccentricity errors and tooth-frequency errors affect the load sharing characteristics of both the differential stage and the enclosed stage. Our findings reveal distinct sensitivities: the differential stage is highly sensitive to tooth-frequency errors, while the enclosed stage is more affected by eccentricity errors. The load sharing coefficient of the differential stage increases with tooth-frequency errors but remains nearly unaffected by eccentricity errors; conversely, the enclosed stage’s coefficient increases with eccentricity errors and is hardly influenced by tooth-frequency errors. Moreover, tooth-frequency errors have a stronger impact on the differential stage than eccentricity errors on the enclosed stage, leading to a larger load sharing coefficient for the differential stage overall.
1. System Configuration and Dynamics Model
The enclosed differential herringbone gear transmission system consists of a differential stage (sun gear Zs1, planet gears Zpi (i=1,…,N), internal gear Zr1, carrier H, intermediate floating component Zg1, and floating ring gear Zf1) and an enclosed stage (sun gear Zs2, star gears Zmj (j=1,…,M), internal gear Zr2, intermediate floating component Zg2, and floating ring gear Zf2). The input torque Td is split through the differential stage, passes through the floating components to the enclosed stage, and finally converges at the output shaft L. In our analysis, we used N=3 for the differential stage planets and M=5 for the enclosed stage stars to meet strength requirements. The system employs center gear floating to improve load sharing, but as we discovered, this measure is less effective when the number of planets or stars exceeds three.
The dynamics model has (18+N+M) degrees of freedom. The generalized displacement vector is:
\[
\mathbf{X} = \{ x_{s1}, H_{s1}, V_{s1}, x_{pi}, x_{r1}, H_{r1}, V_{r1}, x_{g1}, x_{f1}, x_{s2}, H_{s2}, V_{s2}, x_{mj}, x_{r2}, H_{r2}, V_{r2}, x_{g2}, x_{f2}, x_H, x_L \}^T
\]
Here, x represents linear displacements along the base circle (or pitch circle for floating components), and H, V represent horizontal and vertical displacements at the gear centers (in a rotating coordinate system for the differential stage and fixed for the enclosed stage). The elastic mesh forces between gear pairs are given by:
\[
P_{spi} = K_{spi}(t) \left[x_{s1} – x_{pi} + H_{s1}\cos\!\left(\frac{\pi}{2} – \alpha_1 + \frac{2\pi}{N}(i-1)\right) + V_{s1}\cos\!\left(\alpha_1 – \frac{2\pi}{N}(i-1)\right) – e_{spi}(t)\right]
\]
\[
P_{rpi} = K_{rpi}(t) \left[x_{pi} – x_{r1} – H_{r1}\cos\!\left(\frac{\pi}{2} – \alpha_2 – \frac{2\pi}{N}(i-1)\right) – V_{r1}\cos\!\left(\pi – \alpha_2 – \frac{2\pi}{N}(i-1)\right) – e_{rpi}(t)\right]
\]
\[
P_{smj} = K_{smj}(t) \left[x_{s2} – x_{mj} + H_{s2}\cos\!\left(\frac{\pi}{2} – \alpha_3 + \frac{2\pi}{M}(j-1)\right) + V_{s2}\cos\!\left(\alpha_3 – \frac{2\pi}{M}(j-1)\right) – e_{smj}(t)\right]
\]
\[
P_{rmj} = K_{rmj}(t) \left[x_{mj} – x_{r2} – H_{r2}\cos\!\left(\frac{\pi}{2} – \alpha_4 – \frac{2\pi}{M}(j-1)\right) – V_{r2}\cos\!\left(\pi – \alpha_4 – \frac{2\pi}{M}(j-1)\right) – e_{rmj}(t)\right]
\]
In these equations, K denote time-varying mesh stiffness, α are mesh angles, e(t) are error excitations (eccentricity and tooth-frequency combined). The damping forces D have similar forms with damping coefficients C.
2. Error Representation and Time-Varying Stiffness
We represented manufacturing and assembly errors through gear eccentricity errors and tooth-frequency errors. The equivalent error displacement along the line of action for each gear pair is the superposition of eccentricity and tooth-frequency components. For example, for the sun-planet pair (differential stage):
\[
e_{spi}(t) = E_{spi}\sin(\omega_1 t + \phi_{spi}) – E_{pi}\sin(\omega_{pH}t + \phi_{pi} – \alpha_1) – E_{s1}\sin\!\left(\omega_{sH}t – \frac{2\pi}{N}(i-1) – \phi_{s1} + \alpha_1\right)
\]
Similar expressions were used for other pairs. Here E denote error amplitudes, ω are angular frequencies (mesh frequencies, relative rotational speeds, etc.), and φ are initial phases.
For the time-varying mesh stiffness of herringbone gears, we adopted the helical gear stiffness formula developed by Maatar and Velez (1996) and applied it in parallel to account for the two helices. The stiffness is expressed as:
\[
k(t) = k_0 \left[ 1 + \sum_{k=1}^{\infty} A_k \cos(2\pi k \tau – \psi) + B_k \sin(2\pi k \tau – \psi) \right] L_m
\]
where τ = t/Tm, Tm is the mesh period, ψ is the mesh phase difference between planet/star gears, and the coefficients are defined in the literature. We accounted for phase differences between planets/stars and between internal/external meshes following Parker and Lin (2004).
3. Solution Method and Load Sharing Coefficient Definition
The system dynamics equations in matrix form are:
\[
[M]\{\ddot{x}\} + [C]\{\dot{x}\} + [K]\{x\} = \{P\}
\]
We solved this nonlinear system using the Fourier series method. By decomposing the stiffness into average and fluctuating parts, \([K] = [\bar{K}] + [\Delta K]\), and the displacement into static and dynamic parts, \(\{x\} = \{\bar{x}\} + \{\Delta x\}\), we obtained a linear time-invariant system for the dynamic response:
\[
[M]\{\Delta\ddot{x}\} + [C]\{\Delta\dot{x}\} + [\bar{K}]\{\Delta x\} = \{F\}
\]
The excitation vector \(\{F\}\) includes mesh stiffness fluctuation, error, gravity, and load excitations. We expanded the fluctuating stiffness into Fourier series up to the 5th harmonic of the mesh frequency, plus rotational frequencies, giving 18 excitation frequencies. The response was obtained by superposition of frequency-domain solutions.
For load sharing evaluation, we defined coefficients at each mesh cycle and then over the entire system period. For the differential stage external meshes, the cycle coefficient is:
\[
b_{spi,k1} = \frac{N (P_{spi,k1})_{\max}}{\sum_{i=1}^N (P_{spi,k1})_{\max}}
\]
and similarly for internal meshes and for the enclosed stage. The system-period load sharing coefficients are:
\[
B_{spi} = \max_{k1} |b_{spi,k1} – 1| + 1,\quad B_{rpi} = \max_{k2} |b_{rpi,k2} – 1| + 1
\]
\[
B_{smj} = \max_{k3} |b_{smj,k3} – 1| + 1,\quad B_{rmj} = \max_{k4} |b_{rmj,k4} – 1| + 1
\]
Finally, the differential stage load sharing coefficient Bp is the maximum of Bspi and Brpi, and similarly Bm for the enclosed stage.
4. Analysis and Results
We studied a specific enclosed differential herringbone gear reducer with the following parameters: module 7 mm, differential stage sun 37 teeth, planet 56, ring 149, helix angle 22°, face width factor 1.2; enclosed stage sun 71 teeth, star 39, ring 149, face width factor 0.6. Input speed 3000 rpm, power 20,000 kW. Bearing support stiffness 3×108 N/m. Baseline eccentricity error 20 μm, tooth-frequency error 8 μm. The interior and exterior meshes showed nearly identical load sharing coefficients, so we used the external mesh (sun-planet or sun-star) to characterize each stage.
4.1 Isolated Effects of Eccentricity and Tooth-Frequency Errors
First, we considered only eccentricity errors (tooth-frequency set to zero). The load sharing coefficients are shown in the table below:
| Stage | B (load sharing coefficient) |
|---|---|
| Differential (planet-sun) | 1.0233 |
| Enclosed (star-sun) | 1.1457 |
The enclosed stage coefficient (1.1457) is larger than the differential stage (1.0233). This indicates that eccentricity errors cause more uneven load sharing in the enclosed stage. The reason is that while center gear floating can reduce the effect of eccentricity, it is less effective when the number of planets/stars exceeds three. The differential stage with three planets benefits more from floating, whereas the enclosed stage with five stars does not.
Next, we examined only tooth-frequency errors (eccentricity set to zero). Results:
| Stage | B |
|---|---|
| Differential | 1.2905 |
| Enclosed | 1.0072 |
Now the differential stage coefficient (1.2905) is much larger than the enclosed (1.0072). This is because load sharing improves with higher load levels; the enclosed stage sun gear carries a torque Zr1/Zs1 times that of the differential stage (about 4 times in our design), making it much less sensitive to tooth-frequency errors. Thus, tooth-frequency errors strongly affect the differential stage but barely influence the enclosed stage.
4.2 Combined Effects
When both errors are present, the combined load sharing coefficients are:
| Stage | B |
|---|---|
| Differential | 1.2918 |
| Enclosed | 1.1442 |
The differential stage coefficient (1.2918) is very close to the value under tooth-frequency errors alone (1.2905), confirming that eccentricity errors have negligible effect on the differential stage. The enclosed stage coefficient (1.1442) is close to the value under eccentricity errors alone (1.1457), confirming that tooth-frequency errors hardly affect the enclosed stage.
4.3 Parametric Sensitivity
We varied the eccentricity error while keeping tooth-frequency error fixed at 8 μm, and vice versa. The trends are summarized in figures (represented here by tables).
| Eccentricity error (μm) | Bdifferential | Benclosed |
|---|---|---|
| 0 | 1.2905 | 1.0072 |
| 10 | 1.2910 | 1.0715 |
| 20 | 1.2918 | 1.1442 |
| 30 | 1.2925 | 1.2158 |
The differential stage coefficient remains nearly constant (around 1.291–1.293), while the enclosed stage increases almost linearly with eccentricity.
| Tooth-frequency error (μm) | Bdifferential | Benclosed |
|---|---|---|
| 0 | 1.0233 | 1.1457 |
| 4 | 1.1531 | 1.1449 |
| 8 | 1.2918 | 1.1442 |
| 12 | 1.4306 | 1.1436 |
The enclosed stage coefficient is nearly immune to tooth-frequency variations, while the differential stage coefficient increases significantly with tooth-frequency error.
5. Key Conclusions
Our study on the enclosed differential herringbone gear transmission system reveals the following important conclusions regarding the impact of eccentricity and tooth-frequency errors on dynamic load sharing:
- Sensitivity to error type: The differential stage load sharing is governed primarily by tooth-frequency errors and is almost unaffected by eccentricity errors. Conversely, the enclosed stage load sharing is governed by eccentricity errors and is immune to tooth-frequency errors.
- Reason for differential stage sensitivity: The differential stage carries a much lower torque than the enclosed stage (due to the torque amplification factor of the planetary ratio). Lower torque makes the system more sensitive to periodic excitations like tooth-frequency errors. The enclosed stage, with higher torque, is more robust to such errors.
- Reason for enclosed stage sensitivity: The enclosed stage uses five star gears (M=5). Center gear floating, which is the floating mechanism employed, is less effective at reducing the effect of eccentricity errors when the number of planets exceeds three. This is why eccentricity errors cause significant load imbalance in the enclosed stage.
- Dominant error: Under combined errors, the differential stage load sharing coefficient (1.2918) is larger than that of the enclosed stage (1.1442), indicating that the differential stage experiences more severe load imbalance. This is because tooth-frequency errors have a stronger impact on the differential stage than eccentricity errors do on the enclosed stage.
- Design implications: To improve overall load sharing, designers should:
- For the differential stage: control tooth-frequency accuracy (manufacturing quality of tooth profile and spacing) more tightly.
- For the enclosed stage: minimize eccentricity errors (e.g., precision mounting and balancing of gears) and consider additional floating mechanisms or compliant supports to compensate for the limited effectiveness of center gear floating when M > 3.
In summary, our analysis of herringbone gear dynamics for enclosed differential gear trains shows that the two stages respond very differently to errors. Recognizing these differences is essential for achieving balanced load distribution and reliable performance in high-power aerospace and marine applications. The use of herringbone gears in such systems offers smooth and quiet operation, but careful attention must be paid to the sensitivity of each stage to distinct error types. We hope our findings provide a useful guide for the design and optimization of herringbone gear planetary transmissions.