Abstract
This article investigates the simulation reliability of helical gear operating under various rotational speeds. Gear modification, a micro-level structural design, is significantly influenced by both subjective modification amounts and the mesh accuracy of simulation calculations. Focusing on a case study of a secondary reducer development for an electric vehicle, this research delves into the mesh model partitioning of modified helical gear and studies the simulation methods. By analyzing the effect of mesh count variations along the tooth width and involute direction on maximum tooth contact stress (TCS), peak-to-peak transmission error (PPTE), and tooth contact pattern (TCP) distribution, insights into optimal meshing strategies are gained. The results reveal that mesh refinement primarily impacts simulation accuracy at low speeds, whereas it has a lesser effect at high speeds. A comprehensive understanding of these relationships facilitates the optimization of gear design and simulation processes.
1. Introduction
With the tightening of global emission regulations, electric vehicles (EVs) have garnered significant attention from automakers worldwide. Compared to traditional internal combustion engine vehicles, EVs offer better fuel economy and environmental friendliness. However, the absence of engine noise shielding has accentuated wind, tire, and mechanical noises, particularly those emanating from gear reducers. These noises, predominantly in the 700–4000 Hz range, are highly audible and detrimental to ride comfort . The trend towards higher rotational speeds in EV gear reducers further exacerbates these high-frequency noise issues.
The primary source of these noises is the dynamic excitation within the gearbox, resulting from transmission errors during gear meshing. Transmission errors arise from deviations between actual and theoretical gear engagements, generating excitation forces that induce vibrations and, subsequently, radiated noise . Among various strategies to mitigate these errors, optimizing the micro-geometry of gear teeth is both cost-effective and highly efficient. Gear modifications can significantly improve load distribution, reduce internal excitations, and minimize gear whine .
Simulation methods are widely employed to optimize gear transmission errors and tooth contact stresses, offering time and cost savings . However, the accuracy of these simulations is inherently tied to mesh refinement, which, despite theoretical improvements with increased mesh density, can negatively impact simulation outcomes if not appropriately managed . This study aims to explore the influence of mesh counts along the tooth width and involute directions on simulation reliability for helical gear operating at various rotational speeds.
2. Modeling and Gear Modification Scheme
2.1 Gear Transmission System Model
The study utilizes Ricardo Software’s SABR module to establish a dynamic model of a two-stage gear reducer for an EV. Based on the load spectrum and gear parameters outlined in Table 1, a standard helical gear pair model is constructed, considering gear materials, manufacturing precision, and heat treatment processes.
Table 1: Load Spectrum and Gear Parameters
| Parameter | Drive Gear | Driven Gear |
|---|---|---|
| Number of Teeth | 23 | 93 |
| Normal Module | 1.93 | 1.93 |
| Pressure Angle (°) | 16.0 | 16.0 |
| Helix Angle (°) | 25.6 | 25.6 |
| Addendum Modification | 0.4300 | 0.0387 |
| Face Width (mm) | 42.0 | 46.5 |
| Center Distance (mm) | 125 | 125 |
| Accuracy Grade | 8 | 8 |
| Input Speed 1 (rpm) | 20,000 | – |
| Input Speed 2 (rpm) | 3,500 | – |
| Input Torque 1 (Nm) | 19.5 | – |
| Input Torque 2 (Nm) | 111.4 | – |
2.2 Gear Micro-Modification Scheme
A comprehensive modification approach encompassing tooth flank, end relief, and profile modifications is adopted, with equal modifications applied to both the drive and coast flanks. The specific modification parameters for high and low speeds are detailed in Table 2, resulting in optimized contact patterns as depicted in Figure 1.
Table 2: Helical Gear Modification Parameters
| Modification Parameter | High-Speed Modification (μm) | Low-Speed Modification (μm) |
|---|---|---|
| Crowning (Flank) | 2, 6 | 6, 6 |
| Tip Relief | -6, -4 | -6, -6 |
| Addendum Modification | 2, 7 | 3, 8 |
| Profile Crowning | 3, 3 | 8, 8 |
| Pressure Angle Mod. | 2, -2 | 7, -7 |
Figure 1: Optimized Contact Patterns after Modification
2.3 Tooth Surface Mesh Model
Mesh refinement significantly impacts simulation accuracy and efficiency. The study employs a step-wise refinement approach, varying mesh counts along the tooth width and involute directions, from an initial 20 meshes until TCS, PPTE, and TCP stabilize (Figure 2).
Figure 2: Helical Gear Tooth Surface Mesh Model
3. Theoretical Calculations
3.1 Dynamics Model of the Gear Pair
The 8-degree-of-freedom parallel-axis helical gear system dynamics are modeled using the lumped parameter method (Figure 3). The equations of motion are derived considering gear meshing stiffness, damping, and errors.
Figure 3: Dynamics Model of the Helical Gear Pair
The dynamics equations are formulated as:
begin{align*} m_1 \ddot{x}_1 – (c_{m}\mu + k_{m}(t)[\mu – e_{m}(t)])\cos\beta_b\sin\delta &= 0, \\ I_{z1} \ddot{\theta}_1 + (c_{m}\mu + k_{m}(t)[\mu – e_{m}(t)])r_1\cos\beta_b &= T_1, \\ vdots & \\ m_2 \ddot{x}_2 – (c_{m}\mu + k_{m}(t)[\mu – e_{m}(t)])\cos\beta_b\sin\delta &= 0, \\ I_{z2} \ddot{\theta}_2 + (c_{m}\mu + k_{m}(t)[\mu – e_{m}(t)])r_2\cos\beta_b &= T_2, end{align*}
where m1,m2 are the masses, Iz1,Iz2 are the moments of inertia, r1,r2 are the base circle radii, βb is the helix angle, δ is the tilt angle, θ1,θ2 are the rotational angles, T1,T2 are the torques, cm is the meshing damping, km(t) is the time-varying meshing stiffness, em(t) is the composite meshing error, and μ is the relative displacement along the line of action.
3.2 Tooth Contact Stress Calculations
The maximum tooth contact stress ((\sigma_H)) is computed using Hertzian contact theory:
sigmaH=π(1−ν12)E1+(1−ν22)E2Ft(ρΣ1)1/2,
where Ft is the circumferential force, ν1,ν2 are the Poisson’s ratios, E1,E2 are the Young’s moduli, and ρΣ is the combined curvature radius at the contact point.
3.3 Transmission Error Calculations
Transmission error (TE) is defined as the deviation between the actual and ideal positions of the driven gear tooth profile along the line of action:
textTE=φ1rb1−φ2rb2,
where φ1,φ2 are the actual rotational angles, rb1,rb2 are the pitch circle radii, and Δφ is the angular deviation due to transmission error.
4. Results and Discussion
4.1 Influence of Mesh Density on TCS Simulation Accuracy
The impact of mesh density variations on TCS simulation accuracy is analyzed for both high and low rotational speeds (Figure 4).
Figure 4: Variation of Maximum TCS with Mesh Counts
At both speeds, TCS sensitivity to mesh density along the involute direction is low. Conversely, mesh refinement along the tooth width significantly influences TCS, especially at low speeds. This is attributed to higher elastic deformations along the tooth width during operation. The TCS initially decreases with increasing tooth width meshes before stabilizing beyond a threshold (approximately 110 meshes for low speeds and 120 meshes for high speeds).
Table 3: TCS Sensitivity to Mesh Density
| Rotational Speed (rpm) | Mesh Count Range | TCS Variation (MPa) |
|---|---|---|
| 20,000 | 20-70 | 2.1 |
| 70-85 | 0.8 (61.9% reduction) | |
| 85-120 | 0.4 (81.0% reduction) | |
| >120 | Stable | |
| 3,500 | 20-40 | 10.0 |
| 40-55 | 5.0 (50.0% reduction) | |
| 55-110 | 2.0 (80.0% reduction) | |
| >110 | Stable |
4.2 Influence of Mesh Density on PPTE Simulation Accuracy
The PPTE simulation accuracy exhibits an opposing trend to TCS, increasing with tooth width mesh refinement before stabilizing (Figure 5).
Figure 5: Variation of PPTE with Mesh Counts
At 20,000 rpm, PPTE stabilizes beyond 100 tooth width meshes, while at 3,500 rpm, it stabilizes around 100 meshes. This suggests that PPTE is less sensitive to mesh density variations, particularly at high speeds.
4.3 Influence of Mesh Density on TCP Distribution
TCP distribution stabilizes beyond 80 tooth width meshes (Figure 6), indicating that further mesh refinement beyond this point has negligible impact on TCP simulation outcomes.
Figure 6: Variation of TCP Distribution with Mesh Counts
5. Conclusion
This study comprehensively analyzed the impact of mesh density variations along the tooth width and involute directions on the simulation reliability of helical gear operating at different rotational speeds. Key findings include:
- Insensitivity to Involute Mesh Density: Simulation accuracy for TCS, PPTE, and TCP distribution is relatively insensitive to mesh density changes along the involute direction.
- Sensitivity to Tooth Width Mesh Density: Mesh refinement along the tooth width significantly influences TCS and PPTE, with TCS decreasing and PPTE increasing before stabilizing beyond a certain threshold (approximately 110 meshes).
- TCP Stabilization: TCP distribution stabilizes beyond 80 tooth width meshes, suggesting that further mesh refinement has minimal impact on TCP outcomes.
- Speed-dependent Mesh Density Requirements: While both high and low speeds exhibit similar trends, low speeds demand a lower optimal mesh count, highlighting the importance of rotational speed in determining simulation reliability.
These insights facilitate the optimization of gear design and simulation processes, ensuring reliable outcomes for helical gear performance analysis across various operational conditions.