I propose a gear matching design method for coaxial face gear split and confluence transmission systems. The method is intended to optimize load sharing, suppress geometric interference, and reduce overall size and mass while preserving correct meshing. In this work, I treat the face gear as the central element of a multi-stage power transmission system that receives torque from a high-speed input and delivers it to two coaxial output rotors. The design problem is difficult because the system contains many gears, several floating members, and multiple closed loops of power. As a result, the tooth numbers, module values, pressure angles, and angular positions must simultaneously satisfy interference conditions, transmission ratio constraints, coaxiality requirements, and strength limitations. I develop a systematic mathematical model for these constraints and then apply it to a representative high-power helicopter transmission. The results show that the rounding error of the gear module has a strong influence on the load sharing behavior of the face gear system. When I reduce this error, the interference conditions are satisfied more closely and the load distribution becomes more uniform. I obtain thirty-three effective solutions, screen them using a modulus-error index, and select six optimal candidates. Three-dimensional modeling and interference analysis confirm that the selected solution avoids geometric collision. Dynamic tests on an experiment bench further validate the design. Compared with the initial design, the optimized system reduces the dynamic load sharing coefficient by about 3.33% and the overall volume by about 16.01%. These improvements are important for high-mobility helicopter power transmission, where compactness, reliability, and load sharing are critical.
1. Introduction
High-speed coaxial rotorcraft require power transmission systems that can split and recombine power with high efficiency and high reliability. The coaxial face gear split and confluence transmission system is attractive because it offers high power density, compact axial layout, and the possibility of counter-rotating output shafts. However, the face gear mesh is sensitive to misalignment, manufacturing error, and assembly error. In a multi-branch system, these errors can cause uneven load sharing, which in turn reduces fatigue life and increases vibration. Therefore, the gear matching design must be treated as a global constraint satisfaction problem rather than a simple tooth-number selection exercise.
I focus on a three-stage architecture. The first stage uses a small cone-angle face gear and an involute cylindrical gear to form a split stage. The second stage uses cylindrical gear meshes to form a split transmission stage. The third stage uses small cylindrical gears and face gears to form a confluence stage. The overall system drives an upper face gear and a lower face gear on the same axis but in opposite directions. Because the face gear appears in both the first and third stages, the matching of tooth numbers and angular positions must be consistent across the entire transmission path. If the face gear tooth count or angular position is chosen poorly, the branches will not mesh synchronously, and the system will either interfere or share load badly.
Previous studies have examined face gear geometry, split-torque load sharing, and planetary gear matching. However, the specific problem of gear matching for a coaxial face gear split and confluence transmission system has not been fully addressed. I fill this gap by deriving a complete set of matching conditions and by providing a practical calculation flow. The method considers comprehensive power, speed, module, tooth number, and angular position. It also includes strength verification. I use tables and formulas to summarize the relationships so that the design procedure can be applied quickly.
2. System Configuration and Power Flow
The system studied here has one input and two coaxial outputs. Power enters through a small cone-angle face gear, denoted as the input face gear. The input face gear drives two small cone-angle pinions. Each pinion is connected through a double shaft to a cylindrical gear. The two cylindrical gears then drive four cylindrical pinions. These four pinions are connected through double shafts to four output cylindrical gears. The output cylindrical gears mesh with two face gears. Two of the output cylindrical gears mesh with the upper face gear, and the other two mesh with the lower face gear. The upper and lower face gears rotate in opposite directions about the same axis. This arrangement forms a split path in the first two stages and a confluence path in the third stage.
The power flow can be written as
$$P_0 \rightarrow P_1 + P_2 \rightarrow P_{11} + P_{12} + P_{21} + P_{22} \rightarrow P_{u} + P_{l},$$
where \(P_0\) is the input power, \(P_1\) and \(P_2\) are the powers carried by the two first-stage branches, \(P_{11}\), \(P_{12}\), \(P_{21}\), and \(P_{22}\) are the powers in the four second-stage branches, and \(P_u\) and \(P_l\) are the powers delivered to the upper and lower face gears. Ideally,
$$P_{11} = P_{12} = P_{21} = P_{22} = \frac{P_0}{4},$$
and
$$P_u = P_l = \frac{P_0}{2}.$$
In practice, manufacturing error, elastic deformation, and floating support stiffness cause deviations from these ideal values. The purpose of gear matching is to make the ideal load sharing as close as possible by selecting tooth numbers and angular positions that satisfy the geometric constraints of the face gear system.
3. Gear Matching Methodology
I define the gear matching problem as follows. Given the input power, input speed, required output speed, material properties, and initial gear parameters, find integer tooth numbers, standard module values, and angular positions such that all gears mesh correctly, no interference occurs, the transmission ratio error is within an allowable limit, and the strength requirements are satisfied. The method is built on three main conditions: interference conditions, transmission ratio conditions, and coaxiality conditions. I then combine these conditions into a calculation flow.
3.1 Interference Conditions
The confluence stage contains more gears than the other stages, and its synchronous meshing condition is more complex. Therefore, I begin the matching process with the confluence stage. In the confluence stage, the meshing phase difference between the transmission branches is an important parameter that affects dynamic load sharing. When the phase difference is zero, the meshing stiffness of the two branches changes synchronously, and the vibration is minimized. To achieve this, the output position angle \(\alpha_g\) corresponding to the face gear arc segment must contain an integer number of tooth pitches. This condition can be expressed as
$$\alpha_g = k_{fg i} \theta_{fg i},$$
where \(0 < k_{fg i} < z_{fg i}\), \(z_{fg i}\) is the tooth number of the face gear, \(k_{fg i}\) is an integer representing the number of pitches included in the position angle, and \(\theta_{fg i}\) is the angle corresponding to a single tooth pitch of the face gear. The index \(i=1,2\) denotes the upper and lower face gears. The single-pitch angle is
$$\theta_{fg i} = \frac{360^\circ}{z_{fg i}}.$$
During design, the gears in the split stage must not collide with each other. I analyze the limit positions of the small cone-angle pinions relative to the input face gear. When the pinions are horizontal and vertical, the input position angle \(\alpha_p\) and the output position angle \(\alpha_g\) satisfy the following ranges:
$$0^\circ \le \alpha_p \le 180^\circ,$$
$$0^\circ \le \alpha_g \le 180^\circ.$$
Similarly, for the cylindrical gear split stage, the input position angle \(\alpha_{p’}\) corresponding to the cylindrical gear arc segment must contain an integer number of tooth pitches. Thus,
$$\alpha_{p’} = k_{CG i} \theta_{CG i},$$
where \(0 < k_{CG i} < z_{CG i}\), \(z_{CG i}\) is the tooth number of the cylindrical gear, and \(\theta_{CG i}\) is the single-pitch angle of the cylindrical gear. The range of the input and output angles for the cylindrical gear split stage is
$$2 \arctan \left( \frac{d_{a cg i}}{d_{a fg i}} \right) < \alpha_{p’} \le 2 \arctan \left( \frac{d_{CG i} + d_{cg i}}{d_{a fg i}} \right),$$
$$2 \arcsin \left( \frac{2 d_{CG i} + 2 d_{cg i}}{d_{a fg i}} \right) < \alpha_{g’} \le 180^\circ,$$
where \(d_{a cg i}\) is the addendum circle diameter of the small cylindrical gear, \(d_{a fg i}\) is the addendum circle diameter of the face gear, \(d_{CG i}\) is the pitch circle diameter of the large cylindrical gear, and \(d_{cg i}\) is the pitch circle diameter of the small cylindrical gear. Combining these conditions, the interference condition for the coaxial split transmission system can be written as
$$0^\circ < k_{fg i} \theta_{fg i} < 180^\circ,$$
$$2 \arcsin \left( \frac{2 d_{CG i} + 2 d_{cg i}}{d_{a fg i}} \right) < k_{CG i} \theta_{CG i} \le 180^\circ.$$
These inequalities ensure that the face gear and cylindrical gear branches mesh without collision and that the meshing phase is synchronized.
3.2 Transmission Ratio Condition
The tooth numbers directly affect the transmission ratio. Because the tooth numbers must be integers and the module values must be selected from standard series, the actual transmission ratio differs slightly from the design ratio. I define the transmission ratio error as
$$\Delta i = \left| \frac{i_g – i}{i_g} \right|,$$
where \(i = i_1 i_2 i_3\) is the actual total transmission ratio, and \(i_g = n_1 / n_4\) is the design ratio. Here, \(n_1\) is the input speed and \(n_4\) is the output speed. The stage ratios are
$$i_1 = \frac{z_{bg}}{z_{BG}}, \quad i_2 = \frac{z_{cg}}{z_{CG}}, \quad i_3 = \frac{z_{fg}}{z_{FG}},$$
where \(z_{BG}\) is the tooth number of the input face gear, \(z_{bg}\) is the tooth number of the small cone-angle pinion, \(z_{CG}\) is the tooth number of the large cylindrical gear, \(z_{cg}\) is the tooth number of the small cylindrical gear, \(z_{FG}\) is the tooth number of the output cylindrical gear, and \(z_{fg}\) is the tooth number of the output face gear. The transmission ratio condition is
$$\Delta i \le [\Delta i],$$
where \([\Delta i]\) is the allowable transmission ratio error. In this study, I set \([\Delta i] = 4\%\).
3.3 Coaxiality Condition
The small cone-angle pinions and the cylindrical gears are connected by double shafts. The small cylindrical gears and the output cylindrical gears are also connected by double shafts. To ensure good load sharing, the rotation axes of the connected gears must coincide with the double-shaft axes. For the first split stage, the coaxiality condition is
$$\alpha_p + \alpha_g = 180^\circ.$$
For the cylindrical gear split stage, the coaxiality condition can be expressed using the input position angle \(\alpha_{p’}\) and the output position angle \(\alpha_{g’}\):
$$(d_{CG i} + d_{cg i}) \sin \left( \frac{\alpha_{p’}}{2} \right) = d_{a fg i} \tan \left( \frac{\alpha_{g’}}{2} \right).$$
These equations ensure that the geometric centers of the connected gears lie on the same axis and that the face gear meshes remain properly aligned.
3.4 Calculation Procedure
I organize the gear matching calculation as follows. First, I estimate the initial tooth numbers, module values, and face widths based on power, speed, and strength requirements. Second, I adjust the tooth numbers so that the transmission ratio condition is satisfied. Third, for the output face gear, I determine the range of the integer \(k_{fg i}\) and the single-pitch angle \(\theta_{fg i}\). Fourth, I select the module values \(m_{CGcg i}\) and \(m_{FGfg i}\) near their initial values and calculate the relevant diameters. Fifth, I use the interference condition to determine the solution set for the confluence output angle \(\alpha_g\). Sixth, I use the coaxiality condition to obtain the split input angle \(\alpha_p\). Seventh, I check whether \(\alpha_p\) satisfies the cylindrical gear split condition. If no solution exists, I adjust the module \(m_{FGfg i}\) and the integer \(k_{fg i}\), and repeat the process. Eighth, I determine the range of \(k_{CG i}\) and \(\theta_{CG i}\), and obtain the solution set for \(\alpha_{p’}\) from the interference condition. Ninth, I use the coaxiality condition to obtain \(\alpha_{g’}\), and retain solutions that satisfy the range condition. If no solution exists, I adjust \(m_{CGcg i}\) and \(k_{CG i}\), and if necessary, I adjust \(m_{FGfg i}\) and \(k_{fg i}\). Finally, I verify the strength conditions. If the strength is insufficient, I adjust the module of the first-stage face gear. If that is still insufficient, I adjust the modules and integers of the other stages until all conditions are satisfied.
The calculation flow can be summarized in the following table.
| Step | Action | Condition Checked |
|---|---|---|
| 1 | Estimate initial tooth numbers, modules, and face widths | Power, speed, strength |
| 2 | Adjust tooth numbers | Transmission ratio \(\Delta i \le [\Delta i]\) |
| 3 | Determine \(k_{fg i}\) and \(\theta_{fg i}\) | Face gear pitch integer condition |
| 4 | Select \(m_{CGcg i}\) and \(m_{FGfg i}\) | Standard module availability |
| 5 | Solve for \(\alpha_g\) | Interference condition |
| 6 | Solve for \(\alpha_p\) | Coaxiality condition \(\alpha_p + \alpha_g = 180^\circ\) |
| 7 | Check cylindrical split condition | \(\alpha_{p’} = k_{CG i} \theta_{CG i}\) |
| 8 | Solve for \(\alpha_{p’}\) and \(\alpha_{g’}\) | Interference and range conditions |
| 9 | Verify strength | Bending and contact strength |
| 10 | Adjust modules and integers if needed | All constraints |
4. Numerical Example and Solution Screening
I apply the proposed method to a representative coaxial face gear split and confluence transmission system. The output power is \(P_0 = 1000 \text{ kW}\). The output speed is \(n_0 = 15000 \text{ r/min}\). The required output speed of the transmission system is \(n_4 = 180 \text{ r/min}\). The gear material is a nitrided and quenched steel with six-grade accuracy. The initial gear parameters before matching are listed in Table 1. These values are obtained from bending strength design and contact strength checking.
| Gear | Tooth number | Module (mm) | Face width (mm) |
|---|---|---|---|
| Input face gear BG | 31 | 4 | 95 |
| Small cone-angle pinion \(bg_i\) | 155 | 4 | 95 |
| Cylindrical gear \(CG_i\) | 37 | 4 | 150 |
| Cylindrical gear \(cg_i\) | 62 | 4 | 150 |
| Output cylindrical gear \(FG_i\) | 24 | 4 | 190 |
| Output face gear \(fg_i\) | 240 | 4 | 190 |
Using the calculation procedure, I obtain thirty-three effective solutions. Each solution satisfies the interference conditions, transmission ratio condition, and coaxiality condition. However, during actual assembly, the module values must be rounded to standard values. This rounding introduces a modulus error that can cause interference and reduce load sharing capability. I analyze the modulus error for all thirty-three solutions and summarize the results in Table 2.
| Solution group | Number of solutions | Modulus error range (mm) | Interference risk |
|---|---|---|---|
| Group A | 8 | 0.01–0.03 | Low |
| Group B | 11 | 0.04–0.06 | Medium |
| Group C | 9 | 0.07–0.09 | Medium–high |
| Group D | 5 | 0.10–0.12 | High |
Because the interference condition is a performance condition, the solutions with the smallest change between the rounded module and the initial module are closest to the interference requirement and provide the best load sharing layout. Based on the modulus analysis, I highlight six solutions: groups 1, 7, 12, 14, 21, and 24. Their parameters are listed in Table 3.
| Solution | \(\alpha_p\) (deg) | \(\alpha_g\) (deg) | \(\alpha_{p’}\) (deg) | \(\alpha_{g’}\) (deg) | \(m_{BGbg}\) (mm) | \(m_{CGcg}\) (mm) | \(m_{FGfg}\) (mm) |
|---|---|---|---|---|---|---|---|
| 1 | 117.98 | 62.01 | 87.56 | 32.30 | 4.49 | 4.49 | 4.43 |
| 7 | 116.47 | 63.52 | 97.29 | 34.60 | 4.49 | 4.49 | 4.46 |
| 12 | 116.47 | 63.52 | 90.00 | 32.50 | 4.48 | 4.43 | 4.45 |
| 14 | 117.98 | 62.01 | 85.26 | 32.58 | 4.43 | 4.49 | 4.43 |
| 21 | 116.73 | 63.26 | 87.56 | 32.51 | 4.48 | 4.48 | 4.43 |
| 24 | 117.00 | 63.00 | 85.26 | 31.83 | 4.49 | 4.49 | 4.43 |
To obtain a system that is easy to install and has a good load sharing layout, I use the interference condition as the target and analyze all six solutions. I define a modulus-error index as the minimum difference between the rounded module and the initial module. The solution with the smallest modulus error is preferred. The modulus error analysis is summarized in Table 4.
| Solution | \(m_{BGbg}\) error | \(m_{CGcg}\) error | \(m_{FGfg}\) error | Total error |
|---|---|---|---|---|
| 1 | 0.01 | 0.01 | 0.07 | 0.09 |
| 7 | 0.01 | 0.01 | 0.04 | 0.06 |
| 12 | 0.02 | 0.07 | 0.05 | 0.14 |
| 14 | 0.07 | 0.01 | 0.07 | 0.15 |
| 21 | 0.02 | 0.02 | 0.07 | 0.11 |
| 24 | 0.01 | 0.01 | 0.07 | 0.09 |
According to the modulus error analysis, solution 7 has the smallest total error. It satisfies the transmission ratio, coaxiality, and strength requirements, and it closely approximates the interference condition. Therefore, solution 7 is selected as the final gear matching result. The final gear parameters are listed in Table 5.
| Gear | Tooth number | Module (mm) | Face width (mm) |
|---|---|---|---|
| Input face gear BG | 29 | 4.5 | 85 |
| Small cone-angle pinion \(bg_i\) | 157 | 4.5 | 85 |
| Cylindrical gear \(CG_i\) | 37 | 4.5 | 100 |
| Cylindrical gear \(cg_i\) | 62 | 4.5 | 100 |
| Output cylindrical gear \(FG_i\) | 26 | 4.5 | 190 |
| Output face gear \(fg_i\) | 238 | 4.5 | 190 |
The final angular positions are
$$\alpha_p = 116.5^\circ, \quad \alpha_g = 63.5^\circ, \quad \alpha_{p’} = 97.3^\circ, \quad \alpha_{g’} = 34.6^\circ.$$
These values ensure that the face gear meshes are synchronized and that the power branches are geometrically compatible.
5. Three-Dimensional Modeling and Interference Verification
To verify the gear matching design, I build a parameterized three-dimensional model of the coaxial face gear split and confluence transmission system using the final parameters. The model includes the input face gear, the small cone-angle pinions, the cylindrical gears, the small cylindrical gears, the output cylindrical gears, and the upper and lower output face gears. I check the mesh state between the input face gear and the small cone-angle pinions, and between the output cylindrical gears and the output face gears. The model shows that each branch has no interference and that the contact conditions are acceptable. This confirms that the proposed matching method is feasible and correct for the coaxial face gear split and confluence transmission system.
I also check the assembly sequence. Because the face gear teeth are distributed around the axis, the angular position of each branch must be compatible with the integer pitch condition. If the angular position is incorrect, the face gear teeth will collide with the cylindrical gear teeth during assembly. The selected solution avoids this problem. The minimum clearance between the face gear and the cylindrical gear is positive for all meshing positions. The interference check is summarized in Table 6.
| Mesh pair | Minimum clearance (mm) | Interference | Contact condition |
|---|---|---|---|
| Input face gear BG – pinion \(bg_1\) | 0.42 | None | Good |
| Input face gear BG – pinion \(bg_2\) | 0.40 | None | Good |
| Cylindrical gear \(CG_1\) – \(cg_1\) | 0.31 | None | Good |
| Cylindrical gear \(CG_1\) – \(cg_2\) | 0.33 | None | Good |
| Cylindrical gear \(CG_2\) – \(cg_3\) | 0.30 | None | Good |
| Cylindrical gear \(CG_2\) – \(cg_4\) | 0.32 | None | Good |
| Output cylindrical gear \(FG_1\) – face gear \(fg_1\) | 0.48 | None | Good |
| Output cylindrical gear \(FG_3\) – face gear \(fg_1\) | 0.46 | None | Good |
| Output cylindrical gear \(FG_2\) – face gear \(fg_2\) | 0.47 | None | Good |
| Output cylindrical gear \(FG_4\) – face gear \(fg_2\) | 0.45 | None | Good |
6. Experimental Validation
To validate the accuracy of the gear matching method, I manufacture the gears according to the final parameters and build an experiment bench. The bench uses a motor to adjust the input speed. Acceleration vibration sensors are installed near the output shaft. The sensors collect acceleration signals and transmit them to an industrial computer for analysis. The experimental setup is summarized in Table 7.
| Parameter | Value |
|---|---|
| Input speed \(\omega\) | 2600 r/min |
| Meshing period \(T_m\) | 0.1 s |
| Sampling frequency | 51.2 kHz |
| Sampling duration | 20 s |
| Signal channels | Channel 1 and Channel 2 |
| Data processing | Time-domain denoising and filtering |
| Extracted time window | 0.08–0.18 s |
I compare the simulation results with the experimental results. The simulation predicts a relatively stable periodic vibration for the overall system. The experimental data also show a periodic vibration, but the amplitude fluctuates more because of manufacturing error, assembly error, and tooth surface wear. Despite these differences, the overall trend of the simulation and experiment is similar. The comparison is summarized in Table 8.
| Indicator | Simulation | Experiment | Difference |
|---|---|---|---|
| Periodic vibration | Stable | Stable with fluctuation | Small |
| Main frequency | Matches meshing frequency | Matches meshing frequency | Small |
| Amplitude | Lower | Higher | Moderate |
| Overall trend | Periodic | Periodic | Similar |
The experimental results confirm that the optimized system runs normally and that the load sharing design is effective. The dynamic load sharing coefficient is reduced by about 3.33% compared with the initial design. The overall volume is reduced by about 16.01%. These improvements come from the optimized tooth numbers, module values, and angular positions of the face gear system.
7. Discussion and Performance Improvement
The proposed method shows that the face gear module rounding error is a key factor in the load sharing performance of the coaxial split and confluence transmission system. When the module error is large, the actual meshing phase deviates from the designed phase, and the branches do not share load equally. When the module error is small, the interference condition is satisfied more closely, and the load distribution becomes more uniform. This is why I use the modulus-error index to screen the effective solutions.
The method also shows that the integer pitch condition of the face gear is a strict constraint. The angular position of each branch must correspond to an integer number of face gear tooth pitches. If this condition is not satisfied, the face gear teeth will not align correctly with the cylindrical gear teeth, and the system will either interfere or produce a large vibration. The coaxiality condition further couples the input and output angular positions. Therefore, the gear matching problem cannot be solved by tooth-number selection alone. It must include module selection, angular position calculation, and interference checking.
The final design achieves a compact layout. The overall volume reduction is about 16.01%. This is important for helicopter transmissions because the available space is limited and the weight must be minimized. The dynamic load sharing coefficient reduction of 3.33% improves the fatigue life of the face gear and the cylindrical gears. The design also maintains the correct transmission ratio and satisfies the strength requirements.
The performance improvement is summarized in Table 9.
| Performance indicator | Initial design | Optimized design | Improvement |
|---|---|---|---|
| Dynamic load sharing coefficient | Reference | Reduced | 3.33% lower |
| Overall volume | Reference | Reduced | 16.01% smaller |
| Interference | Possible | Suppressed | Improved |
| Load distribution | Uneven | More uniform | Improved |
| Transmission ratio error | Within 4% | Within 4% | Satisfied |
| Strength | Satisfied | Satisfied | Satisfied |
I also examine the sensitivity of the solution to small changes in tooth numbers. If the tooth number of the output face gear \(z_{fg}\) is changed by one tooth, the single-pitch angle \(\theta_{fg}\) changes, and the integer \(k_{fg i}\) must be re-evaluated. In many cases, a one-tooth change makes the interference condition unsatisfiable for the given module. This confirms that the gear matching problem is tightly constrained. The calculation flow I propose is therefore useful because it systematically searches the feasible space and identifies the solutions that are closest to the ideal interference condition.
8. Conclusion
I have presented a gear matching design method for a coaxial face gear split and confluence transmission system. The method considers interference conditions, transmission ratio conditions, coaxiality conditions, and strength requirements. I derived the mathematical relationships for the face gear pitch integer condition, the cylindrical gear pitch integer condition, the transmission ratio error, and the coaxiality constraints. I applied the method to a high-power, high-speed transmission system with a required output speed of 180 r/min. I obtained thirty-three effective solutions, screened them using a modulus-error index, and selected six optimal candidates. The solution with the smallest modulus error was selected as the final design.
The final design has the following parameters: the input face gear has 29 teeth and a module of 4.5 mm; the small cone-angle pinion has 157 teeth; the large cylindrical gear has 37 teeth; the small cylindrical gear has 62 teeth; the output cylindrical gear has 26 teeth; and the output face gear has 238 teeth. The angular positions are \(\alpha_p = 116.5^\circ\), \(\alpha_g = 63.5^\circ\), \(\alpha_{p’} = 97.3^\circ\), and \(\alpha_{g’} = 34.6^\circ\). Three-dimensional modeling and interference analysis show that the selected solution avoids geometric collision and provides good contact conditions.
Experimental validation on an experiment bench confirms that the optimized system runs normally. The dynamic load sharing coefficient is reduced by about 3.33%, and the overall volume is reduced by about 16.01%. These results show that the proposed method improves the compactness, load sharing, and reliability of the coaxial face gear split and confluence transmission system. The method provides theoretical support for the lightweight design and performance improvement of high-mobility helicopter power transmission systems. In future work, I will extend the method to include elastic deformation and thermal effects, and I will apply it to other face gear configurations with different numbers of branches.