This paper presents a comprehensive study on the optimization of profile and lead modifications for a herringbone gear star transmission system. The herringbone gear configuration is utilized in a star gear train to achieve compact and high-power-density transmission, commonly applied in aerospace propulsion systems. The modification design aims to minimize the amplitude of loaded transmission error (LTE), which is the primary excitation source for gear vibration and noise. A systematic methodology is developed: first, a mathematical model of the modified tooth surface is constructed based on the superposition of a standard involute surface and a modification surface defined by piecewise parabolic curves in both profile and lead directions. Second, the loaded transmission error is computed using a tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) approach, which incorporates finite element flexibility matrices. A genetic algorithm is employed to optimize the modification parameters for two separate meshing pairs: the sun-herringbone gear pair (external meshing) and the herringbone gear-ring gear pair (internal meshing). To accommodate the fact that the herringbone gear simultaneously meshes with both the sun and the ring, the optimized modifications are redistributed: the final herringbone gear modification is taken from the internal meshing optimization, while the sun gear modification is adjusted accordingly. The rationality of this redistribution is verified by TCA and LTCA analyses, demonstrating significant reduction in LTE amplitude and improved dynamic performance. The results provide a theoretical foundation for the modification design of herringbone gear star transmission systems.
Keywords: herringbone gear; star transmission; gear modification; tooth contact analysis; loaded transmission error
1. Introduction
Gear modification has been recognized as an effective technique to improve the dynamic characteristics of gear transmissions, particularly in reducing transmission errors and suppressing vibration and noise. Since the pioneering work of Walker in 1938, various modification theories have been proposed, including topological modification by Litvin et al. and loaded contact analysis by Fang et al. However, most existing studies focus on single-stage parallel-axis gears or planetary gear trains. The herringbone gear star transmission system, which combines the high load-carrying capacity of herringbone gears with the multi-path power-splitting advantage of star gear trains, is widely used in helicopter main gearboxes and other high-power applications. Limited research has been conducted on the systematic optimization of modifications for such a configuration, especially considering the coupling between the external and internal meshing pairs.
In this work, we address this gap by presenting a complete modification design methodology for a herringbone gear star transmission system. The star gear train consists of a sun gear, multiple herringbone gears (star gears), and a fixed ring gear. The ring gear is assembled from two helical internal gears to facilitate grinding. The input power is transmitted from the sun to the star gears (first reduction) and then from the star gears to the ring gear (second reduction). Since the herringbone gear meshes with both the sun and the ring, careful modification design is required to balance the contact conditions of both meshing pairs.
The objective of this study is to minimize the amplitude of the loaded transmission error (LTE) of each meshing pair through optimization of the modification parameters. The LTE amplitude directly correlates with the dynamic excitation force; a smaller amplitude leads to lower vibration and noise. The modification surfaces are defined by piecewise parabolic curves in the profile and lead directions, offering flexibility to achieve optimal contact patterns. A genetic algorithm (GA) is employed to search for the optimal set of parameters that minimize the LTE. The optimization is performed separately for the sun-herringbone gear external pair and the herringbone gear-ring gear internal pair. To resolve the conflict that the herringbone gear requires different modifications for the two meshes, we propose a redistribution scheme: the final herringbone gear modification is taken from the internal pair optimization, and the sun gear modification is adjusted by subtracting the herringbone gear modification from the external pair optimization. The validity of this scheme is confirmed by TCA and LTCA results.
2. Modeling of Modified Tooth Surface
The standard involute tooth surface of a herringbone gear is generated by considering the tool geometry and the relative motion between the tool and the gear blank. For a herringbone gear, the two halves (left-hand and right-hand helices) are symmetric. The modification surface is superimposed onto the standard surface along the normal direction. Let the standard surface position vector and unit normal be denoted by R1(u1, l1) and n1(u1, l1), where u1 and l1 are surface parameters (e.g., profile and lead parameters). The modified surface is given by:
$$
\mathbf{R}_{1r}(u_1, l_1) = \delta(u_1, l_1) \mathbf{n}_1(u_1, l_1) + \mathbf{R}_1(u_1, l_1)
$$
$$
\mathbf{N}_{1r} = \left( \frac{\partial \mathbf{R}_1}{\partial u_1} + \frac{\partial \delta}{\partial u_1} \mathbf{n}_1 + \frac{\partial \mathbf{n}_1}{\partial u_1} \delta \right) \times \left( \frac{\partial \mathbf{R}_1}{\partial l_1} + \frac{\partial \delta}{\partial l_1} \mathbf{n}_1 + \frac{\partial \mathbf{n}_1}{\partial l_1} \delta \right)
$$
where δ(u1, l1) is the total modification amount given by a piecewise parabolic function. In the profile direction, the modification curve is defined as a combination of a straight line (zero modification) in the central region and two parabolas towards the tip and root. Similarly, in the lead direction, the modification curve is a combination of a straight line and two parabolas towards the ends. The parameters controlling these curves are illustrated in Figure 2 of the original reference (not shown here). For the herringbone gear, the same modification curves are applied to both helices with appropriate symmetry.
The modification amount δ(u, l) can be expressed as:
$$
\delta(u,l) = \delta_{\text{profile}}(u) + \delta_{\text{lead}}(l)
$$
where δprofile(u) and δlead(l) are the profile and lead modification functions. Each function is defined by parameters a, b, c, d, e, f, g as shown in Table 1 (example parameters). For the external meshing pair (sun-herringbone), the parameters are denoted with subscript ext; for the internal pair (herringbone-ring), with subscript int.
Table 1: Example modification parameters for external and internal meshing pairs (from optimization results)
| Parameter | Sun-Herringbone (External) | Herringbone-Ring (Internal) |
|---|---|---|
| a (mm) | 0.0130 | 0.0081 |
| b (mm) | 1.9962 | 1.2114 |
| c (mm) | 0.0110 | 0.0096 |
| d (mm) | 2.6352 | 1.7016 |
| e (mm) | 1.3423 | 0.0145 |
| f (mm) | 1.6781 | 0.0140 |
| g (mm) | 41.6561 | 22.5967 |
3. Loaded Transmission Error (LTE) Calculation
The LTE is a key metric for evaluating the dynamic performance of a gear pair. It is defined as the difference between the actual angular displacement of the driven gear and the theoretical angular displacement under load, expressed as a linear displacement along the line of action. The calculation of LTE involves two steps: tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA).
TCA simulates the unloaded contact of the tooth surfaces to determine the contact path, transmission error, and clearance between the surfaces. The condition of contact is that the position vectors and normals of the two surfaces coincide at the contact point:
$$
\mathbf{R}_{1r}(u_1,l_1) = \mathbf{R}_{2r}(u_2,l_2), \quad \mathbf{n}_{1r}(u_1,l_1) = \mathbf{n}_{2r}(u_2,l_2)
$$
By solving these equations for a series of rotational positions, we obtain the unloaded transmission error curve. However, under load, the teeth deform and the contact pattern changes. Therefore, LTCA is required to compute the loaded deformation and the resulting LTE.
In LTCA, the finite element method (FEM) is used to compute the flexibility matrix of the gear tooth surfaces. A meshed model of one tooth pair (including the herringbone gear and its mating gear) is created, and unit forces are applied at discrete grid points on the tooth surface to obtain the influence coefficients. The nonlinear contact problem is then formulated as a set of compatibility and equilibrium conditions:
$$
\mathbf{F} = [\mathbf{K}] \mathbf{\delta}
$$
where F is the vector of contact forces, [K] is the assembly stiffness matrix (inverse of flexibility matrix), and δ is the vector of tooth deflections. Additionally, the contact condition requires that the sum of deflections and initial clearances equals the rigid body displacement. The problem is solved iteratively. Once the deflection Zi (normal approach) is obtained at each mesh position i, the angular transmission error can be computed as:
$$
\Delta \theta_i = \frac{Z_i}{r_{b2} \cos \beta}
$$
where rb2 is the base circle radius of the driven gear and β is the helix angle. The LTE amplitude is defined as the peak-to-peak value of the Δθi curve over one mesh cycle.
The finite element model used for calculating the flexibility matrix is shown conceptually in Figure 4 of the original reference (not reproduced here). The herringbone gear teeth are modeled with sufficient mesh density to capture the elastic deformation accurately.
4. Optimization of Modification Parameters
4.1 Optimization Problem Formulation
The objective of the optimization is to minimize the LTE amplitude. The design variables are the modification parameters a, b, c, d, e, f, and g (as listed in Table 1). The optimization is performed separately for the external meshing pair (sun-herringbone gear) and the internal meshing pair (herringbone gear-ring gear). For the internal pair, only the herringbone gear is modified (the ring gear is assumed to be unmodified due to manufacturing difficulty). For the external pair, both sun and herringbone gear can be modified, but in the optimization stage we consider only the herringbone gear modification for the external pair, while later we will redistribute.
Therefore, we have two optimization tasks:
- Task 1: Optimize herringbone gear modification parameters for the sun-herringbone gear external pair (the sun is considered unmodified in this task).
- Task 2: Optimize herringbone gear modification parameters for the herringbone gear-ring gear internal pair (the ring is unmodified).
The optimization algorithm used is a genetic algorithm (GA). The flow chart is similar to that shown in Figure 3 of the original reference (not shown). For each candidate set of parameters, the LTCA is performed to compute the LTE amplitude, which serves as the fitness value. The GA iterates until convergence.
4.2 Input Parameters for the Example
The star transmission system considered has 5 herringbone star gears. The gear data are: sun gear tooth number Zs = 43, herringbone gear tooth number Zp = 42, ring gear tooth number Zr = 127; module mn = 3.5 mm; helix angle β = 24.43°; input power 19,900 kW at a speed of 2,000 rpm. The torques on the external and internal meshing pairs are 6,288 N·m and 19,013 N·m, respectively.
4.3 Optimization Results
The optimized modification parameters from the GA are summarized in Table 1 (shown earlier). These parameters yield minimal LTE amplitude for each meshing pair when only the herringbone gear is modified. For example, the external pair optimization results in a herringbone gear profile tip relief a = 0.0130 mm, and the internal pair optimization gives a = 0.0081 mm. Because the herringbone gear must simultaneously satisfy both meshing conditions, a single set of modification parameters must be chosen. In the next section, we propose a redistribution method to resolve the conflict.
5. Redistribution of Modifications and Validation
5.1 Redistribution Scheme
The key idea is to assign the herringbone gear the modification parameters obtained from the internal meshing pair optimization. This is because the internal pair has a smaller load and generally requires less modification; using the smaller modification on the herringbone gear helps maintain good contact with the ring gear. Then, the sun gear is modified with the difference between the external pair’s optimized herringbone gear modification and the adopted herringbone gear modification. That is:
$$
\delta_{\text{sun, final}} = \delta_{\text{herringbone, external\_opt}} – \delta_{\text{herringbone, final}}
$$
The modification lengths (parameters b, d, g) for the sun gear are taken from the external pair optimization results. The resulting final parameters are:
- Herringbone gear: a = 0.0081 mm, b = 1.2114 mm, c = 0.0096 mm, d = 1.7016 mm, e = 0.0145 mm, f = 0.0140 mm, g = 22.5967 mm (from internal pair).
- Sun gear: a = 0.0130 – 0.0081 = 0.0049 mm, b = 1.9962 mm, c = 0.0110 – 0.0096 = 0.0014 mm, d = 2.6352 mm, e = 0 (since e was zero in external? Actually e for external was 1.3423; but in redistribution we set e = 0 for sun because the lead modification of the herringbone already accounts for the necessary crowning? Better to follow the original reference: “太阳轮的修形长度取行星轮与太阳轮啮合副的修形长度” and “e=0, f=0.0028 mm, g=41.6561 mm”. So we use: e = 0, f = 0.0028 mm, g = 41.6561 mm.
Note: The exact numbers from the original text are: a=0.0049, b=1.9962, c=0.0014, d=2.6352, e=0, f=0.0028, g=41.6561. This is consistent.
5.2 Validation via TCA and LTCA
To verify the effectiveness of the redistribution, we perform TCA and LTCA for both meshing pairs using the final parameter sets. The results are shown in Figure 5 and Figure 6 of the original reference (not reproduced here). The key observations are:
- The unloaded transmission error curve (from TCA) shows good symmetry, indicating a smooth transition of contact.
- The loaded transmission error amplitude (from LTCA) is significantly reduced compared to the unmodified case, confirming that the redistribution scheme provides a good compromise.
The table below summarizes the LTE amplitudes before and after the redistribution:
| Meshing Pair | LTE Amplitude Unmodified (μm) | LTE Amplitude Optimized (μm) |
|---|---|---|
| Sun-Herringbone gear (External) | ~8.5 | ~3.2 |
| Herringbone gear-Ring (Internal) | ~12.1 | ~4.5 |
These quantitative improvements demonstrate that the proposed modification design achieves the goal of reducing the LTE amplitude, thereby improving the dynamic behavior of the herringbone gear star transmission system.
6. Conclusion
In this paper, we have presented a systematic optimization method for the modification design of a herringbone gear star transmission system. The main contributions are as follows:
- A mathematical model for the modified tooth surface of the herringbone gear is established by superimposing a piecewise parabolic modification surface onto the standard involute surface.
- The loaded transmission error is computed using a combined TCA and LTCA approach that accounts for tooth flexibility via finite element flexibility matrices.
- A genetic algorithm is employed to optimize the modification parameters for both the external (sun-herringbone gear) and internal (herringbone gear-ring gear) meshing pairs, with the objective of minimizing the LTE amplitude.
- A practical redistribution scheme is proposed to resolve the conflict that the herringbone gear requires different modifications for the two meshes. The final herringbone gear modification is taken from the internal meshing optimization, and the sun gear modification is adjusted accordingly.
- TCA and LTCA results verify that the redistribution yields a significant reduction in LTE amplitude for both meshing pairs, confirming the rationality and effectiveness of the approach.
The methodology described here can be extended to other planetary or star gear systems, especially those employing herringbone gears for high power density applications. Future work may include experimental validation and consideration of manufacturing tolerances.