I have focused my research on the gear matching design of coaxial face gear split and confluence transmission systems for high-speed helicopters. Conventional helicopter configurations are limited by retreating blade stall and advancing blade shock, which generally restrict forward flight speed to approximately 300 km/h. Coaxial rigid rotor high-speed helicopters can achieve forward flight speeds above 400 km/h, and the coaxial counter-rotating transmission system is one of the most important enabling technologies. Among various possible architectures, the coaxial face gear split and confluence transmission system offers high power density, high transmission efficiency, and high reliability. However, because this system contains many gears, the design becomes complex, and errors and elastic deformation can seriously affect load sharing uniformity, reducing reliability and stability. Therefore, I propose a systematic gear matching design method that considers comprehensive power, rotational speed, and gear modulus parameters. The method establishes mathematical relationships for interference conditions, transmission ratio constraints, and coaxiality criteria. Through numerical examples, I show that the rounding error of the gear modulus has a significant influence on load sharing characteristics. Reducing this error effectively satisfies the interference conditions and improves load distribution uniformity. I use three-dimensional modeling and interference analysis to select six optimal gear matching schemes, and I verify their effectiveness through dynamic testing on an experiment bench. The optimized system achieves a 3.33% reduction in dynamic load sharing coefficient and a 16.01% reduction in overall volume, significantly improving transmission compactness and reliability. This study provides theoretical support for lightweight design and performance improvement of high-mobility helicopter power transmission systems.
The transmission system I investigate adopts a three-stage configuration. Stage I uses a small cone-angle face gear and involute cylindrical gears to form a split system. Stage II uses cylindrical gear meshing to form a split transmission system. Stage III uses small cylindrical gears and face gears to form a confluence transmission, ultimately realizing coaxial counter-rotating transmission of the upper and lower face gears. The power is input through the small cone-angle gear BG, then transmitted through the small cone-angle gears bgi (i = 1, 2) via double shafts to the cylindrical gears CGi (i = 1, 2). These two cylindrical gears CGi then transmit power through four cylindrical gears cgi (i = 1, 2, 3, 4) via double shafts to the cylindrical gears FGi (i = 1, 2, 3, 4), achieving splitting. The cylindrical gears FGi (i = 1, 3) and the face gear fg1 converge to drive the outer rotor shaft, while the cylindrical gears FGi (i = 2, 4) and the face gear fg2 converge to drive the inner rotor shaft. This configuration has the advantages of compact structure, high transmission efficiency, and high reliability, which can improve helicopter maneuverability. However, because of the large number of gears, correct meshing is difficult, and interference can easily occur, seriously affecting load-carrying capacity. Therefore, the gear tooth numbers must satisfy certain relationships to ensure correct meshing without interference. It is necessary to propose a fast and simple design method for the split system to simplify the structure and provide theoretical support for designers.
In my method, I first consider the confluence stage because it contains more gears and the synchronous meshing conditions are complex. The meshing phase difference of the transmission branches is an important parameter affecting the dynamic load sharing characteristics. When the phase difference angle is zero, the meshing stiffness of the two branches changes synchronously, and vibration is minimized. To ensure synchronous meshing of the confluence stage gears, based on an interference judgment condition for external gear trains, the output position angle αg corresponding to the face gear fg1 (fg2) on the circular arc segment AB (A’B’) should contain an integer number of tooth pitches, so that the cylindrical gears FGi (i = 1, 3) and FGi (i = 2, 4) have the same meshing state when meshing with the face gears fgi (i = 1, 2), thereby obtaining good load sharing characteristics. This condition is expressed as:
$$\alpha_g = k_{fgi} \theta_{fgi},$$
where \(0 < k_{fgi} < z_{fgi}\), \(z_{fgi}\) is the number of teeth of the face gear fgi, \(k_{fgi}\) is the number of tooth pitches contained in the position angle αg, and both are integers. \(\theta_{fgi}\) is the angle corresponding to a single tooth pitch of the face gear fgi, where i = 1, 2. During the design process, the gears in the split stage must not collide with each other. I analyze the limit positions of each gear. When the small cone-angle gear bgi is horizontal and vertical relative to BG, the position limit distribution of each gear in the system is obtained. At this time, the input position angle αp of the small cone-angle gear split stage and the output position angle αg of the confluence stage have the following relationships:
$$0^\circ \le \alpha_p \le 180^\circ,$$
$$0^\circ \le \alpha_g \le 180^\circ.$$
Similarly, to prevent interference during meshing of the cylindrical gear split stage, the input position angle αp‘ of the cylindrical gear split stage corresponding to the cylindrical gear CG1 (CG2) on the circular arc segment CD (C’D’) should contain an integer multiple of the tooth pitch, that is:
$$\alpha_p’ = k_{CGi} \theta_{CGi},$$
where \(0 < k_{CGi} < z_{CGi}\), \(z_{CGi}\) is the number of teeth of the cylindrical gear CGi, \(\theta_{CGi}\) is the angle corresponding to a single tooth pitch of the cylindrical gear CGi, and i = 1, 2. The minimum center distance between cylindrical gears cg1 and cg2 (cg3 and cg4) is \(a_{cgmin} = O_2 O_3 = d_{acgi}\), and the maximum center distance is \(a_{cgmax} = O_2′ O_3′ = d_{CGi} + d_{cgi}\). The ranges of the input position angle αp‘ and output position angle αg‘ of the cylindrical gear split stage can be expressed as:
$$2 \arctan\left(\frac{d_{acgi}}{d_{afgi}}\right) < \alpha_p’ \le 2 \arctan\left(\frac{d_{CGi} + d_{cgi}}{d_{afgi}}\right),$$
$$2 \arcsin\left(\frac{2 d_{CGi} + 2 d_{cgi}}{d_{afgi}}\right) < \alpha_g’ \le 180^\circ.$$
Combining the above equations, the interference conditions for the coaxial split transmission system are obtained as:
$$0^\circ < k_{fgi} \theta_{fgi} < 180^\circ,$$
$$2 \arcsin\left(\frac{2 d_{CGi} + 2 d_{cgi}}{d_{afgi}}\right) < k_{CGi} \theta_{CGi} \le 180^\circ.$$
Next, I consider the transmission ratio condition. Based on the system configuration, the design of the tooth numbers affects the transmission ratio. The actual transmission ratio and the design transmission ratio have a certain error Δi. Therefore, the transmission ratio condition can be expressed as:
$$\Delta i = \frac{|i_g – i|}{i_g} \le [\Delta i],$$
where \(i = i_1 i_2 i_3\), the transmission ratio of the small cone-angle gear split stage is \(i_1 = z_{bg} / z_{BG}\), the transmission ratio of the cylindrical gear split stage is \(i_2 = z_{cg} / z_{CG}\), and the transmission ratio of the confluence stage is \(i_3 = z_{fg} / z_{FG}\). The engine output speed is \(n_1\), and the system output speed is \(n_4\). The given transmission ratio is \(i_g = n_1 / n_4\), and the allowable value \([\Delta i]\) is taken as 4%.
I also consider the coaxiality condition. The small cone-angle gear bgi and the cylindrical gear CGi, as well as the cylindrical gears cgi and FGi, are connected by double shafts. The cylindrical gear CGi is a floating gear. To ensure good load sharing characteristics, the rotation axes of the small cone-angle gear bgi and the cylindrical gear CGi, as well as those of the cylindrical gears cgi and FGi, must coincide with their respective double shaft axes. From the gear distribution in the system, the coaxiality condition for αp and αg is:
$$\alpha_p + \alpha_g = 180^\circ.$$
For the cylindrical gear split stage, the coaxiality condition expressed by the input position angle αp‘ and output position angle αg‘ is:
$$(d_{CGi} + d_{cgi}) \sin\left(\frac{\alpha_p’}{2}\right) = d_{afgi} \tan\left(\frac{\alpha_g’}{2}\right).$$
Based on the above gear matching conditions, I obtain the tooth numbers, modulus, and tooth width of each gear. Within a certain range, I adjust the preliminary tooth numbers so that the designed tooth numbers satisfy the transmission ratio condition. From the designed tooth number \(z_{fgi}\), I determine the range of the integer \(k_{fgi}\) and \(\theta_{fgi}\). At the same time, I take values of the cylindrical gear split stage modulus \(m_{CGcgi}\) and the confluence stage modulus \(m_{FGfgi}\) near the initial modulus to obtain the diameters of the cylindrical gears CGi and the face gears fgi. To prevent interference during operation of the split system, based on the interference condition, I determine the solution set of the confluence stage position angle αg. Through the coaxiality condition, I obtain the solution set of the small cone-angle gear split stage position angle αp. The values of αp that satisfy the condition \(\alpha_p’ = k_{CGi} \theta_{CGi}\) are retained. When there is no solution for αp that satisfies this condition, I adjust the modulus \(m_{FGfgi}\) and the value of \(k_{fgi}\) until a solution exists, then proceed to the next calculation step. Similarly, from the designed tooth number \(z_{CGi}\), I determine the range of the integer \(k_{CGi}\) and \(\theta_{CGi}\), obtain the solution set of the cylindrical gear split stage position angle αp‘ satisfying the interference condition, and through the coaxiality condition, obtain the solution set of the cylindrical gear split stage αg‘. The solutions of αg‘ that satisfy the angle range condition are retained. When there is no solution for αg‘ that satisfies this condition, I adjust the modulus \(m_{CGcgi}\) and the value of \(k_{CGi}\) until a solution exists. If there is still no solution, I continue to adjust the modulus \(m_{FGfgi}\) and the value of \(k_{fgi}\). At this point, I obtain all effective solutions for the position angles in the system. I verify whether the corresponding gear parameters satisfy the strength conditions. If they do not, I adjust the small cone-angle gear split stage modulus \(m_{BGbgi}\). If the strength condition is still not satisfied after adjustment, I gradually adjust the modulus \(m_{CGcgi}\) and \(k_{CGi}\) as well as the modulus \(m_{FGfgi}\) and \(k_{fgi}\) until the system satisfies the strength condition. Finally, the effective solutions obtained through the gear matching calculation are analyzed to obtain parameters closer to actual design and production.
I now present a numerical example. The system output power \(P_0\) is 1,000 kW, the output speed \(n_0\) is 15,000 r/min, and the required output speed \(n_4\) of the transmission system is 180 r/min. The gear material is 18Cr2Ni4WA nitrided and quenched steel with grade 6 precision. First, through gear root bending strength design and tooth surface contact strength checking, I obtain the initial gear parameters before gear matching, as shown in Table 1.
| Gear | Number of teeth | Modulus (mm) | Tooth width (mm) |
|---|---|---|---|
| Small cone-angle gear BG | 31 | 4 | 95 |
| Small cone-angle gear bgi | 155 | 4 | 95 |
| Cylindrical gear CGi | 37 | 4 | 150 |
| Cylindrical gear cgi | 62 | 4 | 150 |
| Cylindrical gear FGi | 24 | 4 | 190 |
| Face gear fgi | 240 | 4 | 190 |
Then I carry out the gear matching calculation according to the proposed procedure and obtain 33 effective solutions. During actual assembly, the modulus is rounded, which may cause interference and affect load sharing capability. I analyze the modulus of these 33 effective solutions. Considering that the interference condition is a performance condition, the solution in which the rounded modulus changes least compared with the initial design modulus is closest to the interference condition requirement, and the system has the best load sharing layout. Based on the modulus analysis, I identify six solutions that are closest to the interference condition requirement. Their specific parameters are listed in Table 2.
| Position angle αp (°) | Position angle αg (°) | Position angle αp‘ (°) | Position angle αg‘ (°) | Modulus mBGbg (mm) | Modulus mCGcg (mm) | Modulus mFGfg (mm) |
|---|---|---|---|---|---|---|
| 117.98 | 62.01 | 87.56 | 32.30 | 4.49 | 4.49 | 4.43 |
| 116.47 | 63.52 | 97.29 | 34.60 | 4.49 | 4.49 | 4.46 |
| 116.47 | 63.52 | 90.00 | 32.50 | 4.48 | 4.43 | 4.45 |
| 117.98 | 62.01 | 85.26 | 32.58 | 4.43 | 4.49 | 4.43 |
| 116.73 | 63.26 | 87.56 | 32.51 | 4.48 | 4.48 | 4.43 |
| 117.00 | 63.00 | 85.26 | 31.83 | 4.49 | 4.49 | 4.43 |
To obtain a system with low installation difficulty and a good load sharing layout in engineering practice, I take the satisfaction of the interference condition as the objective and perform gear matching analysis on all cases in Table 2. Using the minimum modulus error before and after rounding at each gear stage as an index, I obtain the solution that fully satisfies the transmission ratio, coaxiality condition, and strength requirements, and is closest to the interference condition requirement. Table 3 shows the modulus error analysis for the six effective solutions. From the analysis, the second group has the smallest error before and after rounding. This result satisfies the transmission ratio, coaxiality condition, and strength requirements, and also approximately satisfies the interference condition, which is consistent with the load sharing layout design. The final gear matching parameters are listed in Table 4. The input position angle of the small cone-angle gear split stage is αp = 116.5°, the output position angle of the confluence stage is αg = 63.5°, the input position angle of the cylindrical gear split stage is αp‘ = 97.3°, and the output position angle of the cylindrical gear split stage is αg‘ = 34.6°.
| Solution group | mBGbg error (mm) | mCGcg error (mm) | mFGfg error (mm) | Total error (mm) |
|---|---|---|---|---|
| 1 | 0.01 | 0.01 | 0.07 | 0.09 |
| 2 | 0.01 | 0.01 | 0.04 | 0.06 |
| 3 | 0.02 | 0.07 | 0.05 | 0.14 |
| 4 | 0.07 | 0.01 | 0.07 | 0.15 |
| 5 | 0.02 | 0.02 | 0.07 | 0.11 |
| 6 | 0.01 | 0.01 | 0.07 | 0.09 |
| Gear | Number of teeth | Modulus (mm) | Tooth width (mm) |
|---|---|---|---|
| Small cone-angle gear BG | 29 | 4.5 | 85 |
| Small cone-angle gear bgi | 157 | 4.5 | 85 |
| Cylindrical gear CGi | 37 | 4.5 | 100 |
| Cylindrical gear cgi | 62 | 4.5 | 100 |
| Cylindrical gear FGi | 26 | 4.5 | 190 |
| Face gear fgi | 238 | 4.5 | 190 |
To verify the suppression effect of the gear matching design on geometric interference and the optimization effect on load sharing performance, I complete parametric modeling based on the optimal solution from the gear matching calculation and construct a three-dimensional interference model of the coaxial face gear split and confluence transmission system. The model reflects that each branch has no interference and good contact conditions, proving that the proposed gear matching design method is feasible and correct for the configuration design of coaxial face gear split and confluence transmission systems.
To verify the accuracy of the gear matching design method, I use the gear parameters in Table 4 to manufacture gears and build an experiment bench. The experiment bench adjusts the output speed of the motor to adjust the speed of the first stage. Acceleration vibration sensors are installed near the output shaft. After the sensors collect acceleration signals, they transmit them to an industrial computer for analysis and processing. The vibration signals of the coaxial face gear split and confluence transmission system are collected by the sensors installed on the system. The signals from channels 1 and 2 are collected, optimized, and processed. Then, computer analysis software is used to analyze and measure the vibration signals in depth. The comparison between simulation results and experimental results verifies the accuracy of the model. To verify whether the optimized system operates normally, the system uses an input speed \( \omega = 2600 \) r/min. The meshing period is \( T_m = 60 / \omega \approx 0.1 \) s. The sampling frequency is set to 51.2 kHz, and the sampling duration is 20 s. I export the signal data collected from channel 1 for data processing, including noise reduction filtering of the time-domain signal. I extract the time-domain diagram from the experimental data between 0.08 s and 0.18 s. The comparison between the experimental time domain and the simulation results shows that the overall coaxial face gear split and confluence transmission system exhibits relatively stable periodic vibration with minimal fluctuation. However, due to complex factors such as manufacturing, processing, installation errors, and tooth surface wear, larger vibration fluctuations are observed. Nevertheless, the overall trends between the simulation and experimental data are relatively similar.
| Parameter | Value |
|---|---|
| Input speed ω | 2600 r/min |
| Meshing period Tm | ≈0.1 s |
| Sampling frequency | 51.2 kHz |
| Sampling duration | 20 s |
| Data extraction interval | 0.08–0.18 s |
| Feature | Simulation | Experiment |
|---|---|---|
| Vibration pattern | Stable periodic | Stable periodic with larger fluctuation |
| Main fluctuation source | Ideal meshing | Manufacturing, assembly, wear |
| Overall trend | Reference | Similar to simulation |
My research addresses the gear matching design problem of coaxial face gear split and confluence transmission systems. By establishing mathematical relationships for interference conditions, transmission ratio conditions, and coaxiality conditions, I propose a systematic gear matching design method. I find that the error before and after rounding of the gear modulus directly affects the load sharing characteristics of the system. Reducing this error can effectively satisfy the interference conditions and optimize load distribution. Through numerical example analysis, I screen six optimal solutions from 33 effective solutions. The solution set with the smallest modulus error reduces the dynamic load sharing coefficient of the system by 3.33% and reduces the geometric size by 16.01%. Three-dimensional modeling and interference analysis verify the feasibility of the gear matching results. The experiment bench test shows that the system operation stability meets the design expectations, but due to manufacturing errors and assembly precision, there are small fluctuations in the actual load distribution. The results show that the method significantly improves the structural compactness and load balancing performance of the transmission system by optimizing gear parameters and layout. This provides theoretical support for the design of high power density and high reliability power transmission systems for helicopters. The use of face gears in this coaxial split and confluence configuration is particularly promising because face gears can transmit power with high efficiency and high load capacity while maintaining a compact axial layout. My method ensures that the face gears and cylindrical gears mesh correctly without interference, and that the load is shared evenly among the branches. This is critical for the reliability and performance of high-speed coaxial rotor systems. In future work, I plan to extend the method to consider more complex face gear geometries and dynamic conditions, and to further validate the design through long-term durability testing.
In summary, the key contributions of my study are as follows. I established a complete set of gear matching conditions for coaxial face gear split and confluence transmission systems, including interference, transmission ratio, and coaxiality conditions. I developed a step-by-step calculation procedure that can be used to quickly obtain effective tooth number combinations. I demonstrated through a numerical example that the modulus rounding error is a critical factor affecting load sharing, and that minimizing this error leads to better interference avoidance and more uniform load distribution. I verified the design through three-dimensional modeling and experimental testing. The optimized system achieves a 3.33% reduction in dynamic load sharing coefficient and a 16.01% reduction in volume. These results confirm that my gear matching design method is effective and practical for engineering applications. The method can be applied to other multi-branch face gear transmission systems, and it provides a foundation for further optimization of high-speed helicopter transmissions.