In my research, I focused on the traction transmission system of high-speed electric multiple units (EMUs), where the gear transmission system is a core transmission unit on the bogie. The herringbone gear, due to its unique symmetrical structure, offers advantages such as low axial force, high load-carrying capacity, small deformation, high overlap ratio, and smoother transmission. These characteristics make it widely applicable in high-speed transmission devices. I took a certain EMU traction transmission device as the research object and explored the application of herringbone gears in EMU gear transmission. Through simulation analysis and exploratory research, I aimed to provide insights into the dynamic behavior, gear modification, and fatigue life of the system.
Model Establishment and Static Analysis
I first established a simulation model of the EMU traction herringbone gear transmission system using Romax Designer software. The model included gears, shafts, and bearings. Based on the relevant parameters of the driving device, I defined the gear characteristics as shown in Table 1.
| Parameter | Symbol | Pinion | Gear | Unit |
|---|---|---|---|---|
| Number of teeth | z | 35 | 85 | — |
| Helix angle | β | 25 | deg | |
| Normal pressure angle | αn | 20 | deg | |
| Normal module | mn | 6 | mm | |
| Center distance | a | 380 | mm | |
| Face width | b | 68 | 66 | mm |
| Addendum coefficient | ha* | 1.05 | — | |
| Clearance coefficient | cn* | 0.35 | — | |
| Gear ratio | u | 2.429 | — | |
The material of the gears was 18CrNiMo7-6 with carburized and hardened treatment, while the shaft material was 20CrNiMo. I selected the bearing types according to the speed and load requirements. The complete system model is shown in the figure below.
I performed static analysis under five operating conditions: startup, sustained-1, sustained-2, high-speed-1, and high-speed-2. The contact stress and bending stress on the driven gear were obtained, as summarized in Table 2.
| Condition | Startup | Sustained-1 | Sustained-2 | High-speed-1 | High-speed-2 |
|---|---|---|---|---|---|
| Contact stress of gear set 1 (MPa) | 769 | 634 | 597 | 545 | 511 |
| Bending stress of gear set 1 (MPa) | 283 | 195 | 176 | 141 | 129 |
| Contact stress of gear set 2 (MPa) | 711 | 586 | 553 | 519 | 479 |
| Bending stress of gear set 2 (MPa) | 262 | 178 | 163 | 128 | 117 |
The maximum contact stress occurred in the startup condition, reaching 769 MPa, while the maximum bending stress was 283 MPa. The total contact ratio of the herringbone gear was calculated as 2.2540, which is greater than 2, satisfying the requirement for smooth transmission. The maximum mesh misalignment was 26.88 μm, caused by shaft deflection and deformation. The shaft deflection amplitude was 1.141 mm under the combined effects of weight and gear meshing forces.
Gear Modification Using Genetic Algorithm
To improve the meshing performance of the herringbone gears, I applied tooth profile modification and helix modification. The fundamental reason for gear modification is to reduce mesh impact and transmission error caused by elastic deformation and manufacturing errors. The optimal modification parameters were obtained using the second-generation genetic algorithm (GA) implemented in Romax. The optimization process considered all five operating conditions as constraints, with the objectives of minimizing transmission error and surface contact load.
The design variables included lead crown, lead slope, involute crown, and involute slope. The optimal values are listed in Table 3.
| Tooth flank | Lead crown (μm) | Lead slope (μm) | Involute crown (μm) | Involute slope (μm) |
|---|---|---|---|---|
| Right flank | 0.39 | 14.33 | 6.72 | 12.71 |
| Left flank | 0.17 | 14.63 | 5.98 | 11.39 |
The profile modification curve followed the general equation:
$$ \Delta = \Delta_{\max} \left(\frac{x}{L}\right)^e $$
where \( \Delta_{\max} \) is the maximum modification amount, \(L\) is the modification length, and \(e\) is the exponent determining the curve shape. For the herringbone gear, I used a parabolic curve with \(e=1.5\). For helix modification, the drum shape amount was determined by:
$$ C_c = \frac{F_m}{b C_\gamma} F_{\beta y} $$
where \(F_m\) is the tangential force, \(b\) is the face width, \(C_\gamma\) is the mesh stiffness, and \(F_{\beta y}\) is the misalignment factor.
After applying the optimal modification parameters, I compared the contact stress, unit load, and transmission error before and after modification. The results are shown in Table 4 and Table 5.
| Condition | Startup | Sustained-1 | Sustained-2 | High-speed-1 | High-speed-2 |
|---|---|---|---|---|---|
| Right flank before (MPa) | 772 | 634 | 597 | 547 | 511 |
| Right flank after (MPa) | 716 | 582 | 540 | 513 | 474 |
| Reduction (%) | 7.3 | 8.2 | 9.5 | 6.2 | 7.2 |
| Left flank before (MPa) | 711 | 586 | 553 | 519 | 479 |
| Left flank after (MPa) | 652 | 563 | 526 | 498 | 461 |
| Reduction (%) | 8.3 | 3.9 | 4.9 | 4.0 | 3.8 |
| Condition | Startup | Sustained-2 | High-speed-1 |
|---|---|---|---|
| Before (μm) | 64.4 | 36.3 | 30.0 |
| After (μm) | 41.3 | 27.5 | 23.6 |
| Reduction (%) | 35.9 | 24.2 | 21.3 |
The unit length load on the right flank was reduced from 531 N/mm to 461 N/mm in the startup condition, a 13.2% reduction. On the left flank, the unit load decreased from 394 N/mm to 333 N/mm, a 15.5% reduction. These results demonstrate that the modification of herringbone gears significantly improves the contact pattern and reduces stress concentrations, thereby increasing the load-carrying capacity and transmission smoothness.
Dynamic Analysis of the Transmission System
In the dynamic analysis, I investigated the modal flexibility, natural frequencies, modal shapes, dynamic mesh stiffness, and vibration responses of the herringbone gear transmission system. The dynamic excitation sources include meshing impact, transmission error, and time-varying mesh stiffness. I calculated the system’s first 40 natural frequencies and modal flexibilities. The maximum modal flexibility was \(1.33 \times 10^{-4}\) μm/N at the 16th order with a frequency of 1873.1 Hz. The corresponding frequency–flexibility relationship is shown in Figure 4.2 in the original study, but I summarize the key values in Table 6.
| Order | Frequency (Hz) | Flexibility (μm/N) |
|---|---|---|
| 3 | 191.7 | 4.77e-5 |
| 6 | 470.8 | 2.35e-5 |
| 9 | 716.5 | 1.09e-4 |
| 16 | 1873.1 | 1.33e-4 |
| 21 | 2624.3 | 3.76e-5 |
| 27 | 3436.2 | 1.81e-5 |
The dynamic mesh stiffness of the herringbone gear changed with the rotating frequency, exhibiting peaks at certain frequencies. The maximum stiffness reached \(1.5 \times 10^6\) N/mm at about 1824 Hz, with a corresponding phase angle of 62.8 degrees. After this frequency, the stiffness stabilized around \(4.78 \times 10^5\) N/mm. The modal shapes indicated that at lower orders, the shaft bending and gear misalignment were severe, while at the 16th order (normal operation), the deformation was relatively small, indicating better transmission stability.
I also analyzed the dynamic contact forces and displacements of the four bearings. The results are presented in Table 7.
| Bearing | Max contact force (N) | Frequency at max force (Hz) | Max displacement (μm) |
|---|---|---|---|
| Bearing 1 | 1.33e+3 | 1000 | 1.98 |
| Bearing 2 | 5.5e+2 | 300 | 0.36 |
| Bearing 3 | 1.06e+3 | 300 | 0.37 |
| Bearing 4 | 1.62e+3 | 830 | 1.70 |
The harmonic excitation responses were computed for the first three harmonics (35th, 70th, and 105th orders). The maximum response acceleration appeared at the bearing between the input shaft and pinion, reaching \(17.18\, \text{m/s}^2\). The response waterfall diagram indicated that the vibration acceleration increased with the response frequency and input shaft speed.
Fatigue Life and Scuffing Analysis
Finally, I evaluated the fatigue damage and life of the herringbone gear transmission system using the Miner linear cumulative damage rule, rain-flow counting method, and local stress-strain approach. The gear material had a contact fatigue limit of 1500 MPa and a bending fatigue limit of 500 MPa. The calculated safety factors in all combined conditions were greater than 1.5, satisfying the strength requirements.
For scuffing analysis, I applied the flash temperature method based on the Blok equation:
$$ \theta_{fl} = \frac{C_m \mu_m \Gamma_X X_B X_M}{a^{1/4}} \left( \frac{v_w}{v_r} \right)^{1/4} \left( \frac{v_r}{v_w} \right)^{1/2} $$
The maximum flash temperature was 55.391 °C, leading to a maximum contact temperature of 117.936 °C, while the scuffing temperature was 328.707 °C. The corresponding scuffing safety factor was 4.638 for the right flank and 4.785 for the left flank, indicating a low risk of scuffing under normal lubrication.
I also assessed the bearing fatigue life using ISO 281:2007. The damage and life values for the four bearings are summarized in Table 8.
| Bearing | Damage ISO (h) | L10M ISO (h) | L10M TS16281 basic (h) |
|---|---|---|---|
| Bearing 1 | 6.137e-3 | 16294 | 22719 |
| Bearing 2 | 4.239e-5 | 2.359e6 | 8.099e6 |
| Bearing 3 | 5.999e-6 | 1.667e7 | 1.396e7 |
| Bearing 4 | 4.439e-3 | 22527 | 30973 |
The most damaged bearing (bearing 1) had a rated life of 16294 hours, which is longer than the required 1.1×104 hours for the EMU application, thus confirming the suitability of the selected bearing. I further analyzed the contact load distribution on the rolling elements of bearing 1. The maximum raceway load was 23102.4 N at a contact angle of 139.741 degrees, and the maximum raceway contact stress was 2411 MPa. Only the left seven rolling elements were loaded, indicating the load zone was limited to a certain angular range.
In conclusion, my study on the herringbone gear transmission system of EMUs, based on Romax simulation, demonstrated that herringbone gears can effectively serve in high-speed traction drives. The gear modification process significantly reduced transmission error and contact stress, thereby enhancing the transmission stability and load capacity. The dynamic analysis revealed critical frequencies and response characteristics that are essential for vibration control. Fatigue and scuffing assessments confirmed that the designed herringbone gear system meets the durability requirements for EMU applications. This research provides a valuable reference for the application of herringbone gears in high-speed railway traction systems.