In modern helicopter designs, especially high-speed rotocraft, the coaxial reversal herringbone gear transmission system has been widely adopted due to its high efficiency, smooth transmission, and compact structure. However, unavoidable manufacturing and assembly errors inevitably lead to uneven load distribution among parallel power paths, which directly affects the reliability and service life of the entire drivetrain. This study focuses on the static load sharing characteristics of a coaxial reversal herringbone gear system under the combined effect of comprehensive errors, including manufacturing errors, alignment errors, and system stiffness variations. By establishing a static model that incorporates torque equilibrium, deformation compatibility, and bearing flexibility, we systematically investigate how errors from different gear stages influence the load sharing behavior. The results provide important guidance for the design and tolerance control of herringbone gears in helicopter main gearboxes.
The system under investigation consists of three stages: Stage I features a helical gear pair (input pinion and gear), Stage II includes two bevel gear pairs for power splitting and reversal, and Stage III contains two herringbone gear pairs that finally drive the coaxial counter-rotating rotors. A schematic representation of the typical coaxial reversal herringbone gear arrangement is shown below:
To conduct a static load sharing analysis, we treat all gear bodies as rigid, while the gear meshing, shafts, and bearings are modeled as linear springs. Friction, thermal expansion, and inertial effects are neglected. The torque transmitted from gear \(i\) to gear \(j\) is denoted as \(T_{i\_j}\). The entire power flow splits into two parallel paths: Path A (passing through gear Z3 → Z6 → Z8 → Z10) and Path B (passing through gear Z4 → Z5 → Z7 → Z9).
Static Equilibrium and Deformation Compatibility
Based on the torque balance of each rotating component, we derive the following set of independent equations:
$$
\begin{aligned}
T_{\text{in}} – T_{1\_2} &= 0 \\
– T_{1\_2} \cdot \eta_{1\_2} + T_{3\_6} + T_{4\_5} &= 0 \\
– T_{3\_6} \cdot \eta_{3\_6} + T_{8\_10} &= 0 \\
– T_{4\_5} \cdot \eta_{4\_5} + T_{7\_9} &= 0
\end{aligned}
$$
where \(\eta_{i\_j}\) represents the gear ratio between gears \(i\) and \(j\), and \(T_{\text{in}}\) is the input torque. The bearing elastic deformations in the x and y directions are incorporated through the following equilibrium conditions for each gear body:
$$
\begin{cases}
\sum \frac{T_{i\_j}}{r_{bi} \cos\alpha_n} \cos\lambda_{i\_j} – K_{xi} x_i = 0 \\[6pt]
\sum \frac{T_{i\_j}}{r_{bi} \cos\alpha_n} \sin\lambda_{i\_j} – K_{yi} y_i = 0
\end{cases}
$$
Here, \(r_{bi}\) is the base radius of gear \(i\), \(\alpha_n\) is the normal pressure angle, \(\lambda_{i\_j}\) is the angle between the line of action and the x-axis, and \(K_{xi}, K_{yi}\) are the support stiffness in the x and y directions respectively.
The angular displacement at the mesh point due to torsional deformation of the shaft is:
$$
\delta_{t\_i\_j}(T_{i\_j}) = \frac{T_{i\_j}}{K_{t\_i\_j}}
$$
where \(K_{t\_i\_j}\) denotes the torsional stiffness of the shaft between gears \(i\) and \(j\). The total angular displacement of a gear pair \(\delta_{S\_i\_j}\) is composed of the shaft deformation, the bearing-induced displacement \(\delta_{C\_i\_j}\), and the combined gear error \(\delta_{w\_i}\):
$$
\delta_{C\_i\_j} = \frac{ (x_i – x_j)\cos X_{i\_j} + (y_i – y_j)\sin X_{i\_j} }{r_{bi}}
$$
$$
\delta_{w\_i} = \delta_{e\_i} + \delta_{a\_i} = E_i \sin(\alpha_n + \beta_{ei} + \omega_i t) + A_i \sin(\alpha_n + \beta_{ai})
$$
Here, \(E_i\) and \(A_i\) are the amplitudes of manufacturing and alignment errors, \(\beta_{ei}\) and \(\beta_{ai}\) are their respective orientation angles, and \(\omega_i\) is the angular velocity of the driving gear. Finally, the mesh transmission error considering the mesh stiffness \(K_{i\_j}\) is:
$$
\delta_{i\_j} = \frac{T_{i\_j}}{K_{i\_j} r_i r_{bi} \cos\alpha_n} + \delta_{C\_i\_j} + \delta_{w\_i}
$$
The deformation compatibility conditions along the two power paths can be expressed as:
$$
\begin{aligned}
\delta_{1\_2} + \eta_{1\_2}\delta_{2\_3} + \eta_{1\_2}\delta_{3\_6} + \eta_{1\_2}\eta_{3\_6}\delta_{6\_8} + \eta_{1\_2}\eta_{3\_6}\delta_{8\_10} &= \delta_1 – \eta_{1\_2}\eta_{3\_6}\eta_{8\_10}\delta_{10} \\
\delta_{1\_2} + \eta_{1\_2}\delta_{2\_3} + \eta_{1\_2}\delta_{3\_4} + \eta_{1\_2}\delta_{4\_5} + \eta_{1\_2}\eta_{4\_5}\delta_{5\_7} + \eta_{1\_2}\eta_{4\_5}\delta_{7\_9} &= \delta_1 – \eta_{1\_2}\eta_{4\_5}\eta_{7\_9}\delta_{9}
\end{aligned}
$$
All the above equations are solved simultaneously to obtain the actual torques \(T_{7\_9}\) and \(T_{8\_10}\) in the two output herringbone gear pairs.
Definition of Load Sharing Coefficient
The static load sharing coefficient for the upper and lower output herringbone gears is defined as:
$$
\Omega_U = \max\left( \frac{T_{8\_10}}{T’_{8\_10}} \right), \quad \Omega_L = \max\left( \frac{T_{7\_9}}{T’_{7\_9}} \right)
$$
where \(T’_{i\_j}\) is the theoretical torque obtained from the ideal gear ratio. A load sharing coefficient closer to 1 indicates better load sharing performance.
System Parameters
The basic design parameters of the coaxial reversal herringbone gear system used in this study are summarized in the following tables.
| Gear | Number of Teeth | Module (mm) | Pressure Angle (°) | Helix Angle (°) |
|---|---|---|---|---|
| Z1 (Stage I input pinion) | 25 | 3 | 20 | 11 |
| Z2 (Stage I gear) | 70 | 3 | 20 | 11 |
| Z3, Z4 (Stage II small bevel) | 30 | 5 | 20 | 15 |
| Z5, Z6 (Stage II large bevel) | 70 | 5 | 20 | 30 |
| Z7, Z8 (Stage III small herringbone) | 22 | 5 | 20 | 30 |
| Z9, Z10 (Stage III large herringbone) | 120 | 5 | 20 | 30 |
| Component | Stiffness (N·m⁻¹) |
|---|---|
| Mesh stiffness \(K_{1\_2}\) | 1.85×10⁹ |
| Mesh stiffness \(K_{3\_6}, K_{4\_5}\) | 1.95×10⁹ |
| Mesh stiffness \(K_{7\_9}, K_{8\_10}\) | 2.35×10⁹ |
| Torsional stiffness \(K_{t2\_3}\) | 2.04×10⁷ |
| Torsional stiffness \(K_{t3\_4}\) | 1.145×10⁷ |
| Torsional stiffness \(K_{t5\_6}\) | 3.74×10⁵ |
| Torsional stiffness \(K_{t6\_8}\) | 3.237×10⁵ |
| Bearing Location | Stiffness (N·m⁻¹) |
|---|---|
| K1 | 1.45×10⁹ |
| K2 | 1.48×10⁹ |
| K3 | 1.34×10⁹ |
| K4 | 1.34×10⁹ |
| K5 | 5.14×10⁹ |
| K6 | 5.14×10⁹ |
| K7 | 1.98×10⁹ |
| K8 | 1.98×10⁹ |
| K9 | 7.01×10⁹ |
| K10 | 7.01×10⁹ |
Influence of Manufacturing Errors
We first examine the effect of manufacturing errors applied simultaneously to the Stage II bevel pinions Z3 and Z4, and then to the Stage III small herringbone gears Z7 and Z8. The error amplitude is varied from 0 to 50 μm. The resulting load sharing coefficients \(\Omega_U\) and \(\Omega_L\) are computed and presented below.
Case 1: Manufacturing errors on Z3 and Z4
When the manufacturing errors of Z3 and Z4 increase together, \(\Omega_L\) (lower herringbone gear) first decreases from 1.10 to 1.02 (at about 15 μm) and then rises to 1.23 at 50 μm, representing a net increase of 10.6%. Meanwhile, \(\Omega_U\) (upper herringbone gear) monotonically increases from 1.10 to 1.45, a 24.1% rise. The initial drop in \(\Omega_L\) at small error values is attributed to the fact that small manufacturing errors can partially compensate for the inherent unevenness caused by stiffness variations.
Case 2: Manufacturing errors on Z7 and Z8
For the Stage III herringbone gears, the load sharing coefficients increase significantly. \(\Omega_U\) rises from 1.10 to 1.65 (50% increase), and \(\Omega_L\) increases from 1.10 to 1.58 (43.3% increase). This indicates that errors in the herringbone gears themselves have a stronger effect on load sharing than those in the upstream bevel gears.
The following table summarizes the sensitivity of load sharing coefficients to manufacturing error magnitudes for the two cases.
| Error Source | Error Range (μm) | \(\Omega_U\) variation | \(\Omega_L\) variation |
|---|---|---|---|
| Z3 & Z4 | 0 → 50 | +24.1% | +10.6% (with dip) |
| Z7 & Z8 | 0 → 50 | +50% | +43.3% |
Influence of Assembly Errors
Assembly errors (misalignment) of Z3 and Z4, and of Z7 and Z8, are studied in the same manner. The results show a similar trend: assembly errors in Stage II lead to a moderate increase in load sharing coefficients, while errors in Stage III herringbone gears cause a more drastic deterioration. For instance, when assembly errors of Z7 and Z8 increase from 0 to 50 μm, \(\Omega_U\) grows from 1.10 to 1.46 (24.7% increase). In contrast, for Z3 and Z4, \(\Omega_U\) first decreases slightly (due to compensation) and then increases.
Combined Effect of Comprehensive Errors
Next, we consider the comprehensive error (sum of manufacturing and assembly errors) applied individually to gears Z3, Z8, Z9, and Z10. Figure 2 (omitted in text) in the original paper shows that as the comprehensive error of Z8 increases from 0 to 50 μm, \(\Omega_U\) rises from 1.10 to 1.80 (a 38.9% increase), and \(\Omega_L\) rises from 1.10 to 1.42 (22.5% increase). The comprehensive error of Z3 produces the smallest effect: \(\Omega_U\) increases only from 1.10 to 1.15, and \(\Omega_L\) to 1.23. Errors on Z9 and Z10 also result in significant increases, but the most critical gear for load sharing performance is the Stage III small herringbone gear Z8.
To quantify the influence more systematically, we tabulate the maximum load sharing coefficients at the 50 μm error level for each gear:
| Gear with Error | \(\Omega_U\) at 50 μm | \(\Omega_L\) at 50 μm |
|---|---|---|
| Z3 | 1.15 | 1.23 |
| Z8 | 1.80 | 1.42 |
| Z9 | 1.56 | 1.38 |
| Z10 | 1.50 | 1.35 |
These results clearly show that the herringbone gear Z8 (the pinion in the third stage) is the most sensitive element. Therefore, during design and manufacturing of coaxial reversal herringbone gear systems, special attention should be paid to controlling the comprehensive error of Stage III herringbone gear pinions to ensure satisfactory load sharing.
Effect of Operating Speed Combined with Manufacturing Errors
We also investigate the combined influence of rotational speed (from 7,000 to 15,000 rpm) and manufacturing errors of Z3 and Z8. As speed increases, the dynamic effects (though not directly modeled in static analysis) cause additional deformations that appear as effective changes in mesh stiffness and error compensation. For both gears, the load sharing coefficients generally increase with speed. For example, when the manufacturing error of Z3 is 25 μm, increasing the speed from 7,000 to 15,000 rpm raises \(\Omega_U\) from 1.17 to 1.25, and \(\Omega_L\) from 1.03 to 1.05. For Z8, the increase is more pronounced. This indicates that higher speeds degrade load sharing performance further, especially when combined with larger manufacturing errors.
Conclusion
In this work, we have established a comprehensive static model of a coaxial reversal herringbone gear transmission system, accounting for mesh stiffness, torsional stiffness, support stiffness, elastic deformations, and both manufacturing and alignment errors. The key findings are:
- Manufacturing errors on Stage II bevel gears increase the load sharing coefficient of the upper herringbone gear by up to 24.1%, while errors on Stage III herringbone gears cause a 50% increase. Assembly errors show similar but slightly weaker effects.
- Among all gears, the Stage III small herringbone gear Z8 is the most critical: a comprehensive error of 50 μm raises the load sharing coefficient by 38.9% for the upper output and 22.5% for the lower output.
- Increasing operating speed magnifies the unfavorable influence of manufacturing errors, leading to higher load sharing coefficients and poorer load distribution.
- To achieve a load sharing coefficient within ±5% of the ideal value, the manufacturing errors of herringbone gear Z7 and Z8 should be controlled within a range of 5 ± 2 μm, and the comprehensive error of Z3 should be limited to 5–25 μm.
These conclusions provide quantitative guidelines for the tolerance design of coaxial reversal herringbone gear systems in helicopter main transmissions, emphasizing the importance of precise manufacturing and assembly of the herringbone gear stages.