In modern engineering applications, the demand for high-performance helical gears has increased significantly, particularly in electric vehicle powertrains where noise, vibration, and harshness (NVH) characteristics are critical. Helical gears are preferred due to their smooth operation and high load-carrying capacity. However, the reliability of simulation results for helical gears under varying rotational speeds is highly dependent on mesh quality and density in finite element analysis (FEA). This study investigates the impact of mesh discretization along the tooth width and involute directions on the simulation accuracy of maximum tooth contact stress (TCS), peak-to-peak transmission error (PPTE), and tooth contact pattern (TCP) distribution for helical gears operating at high and low rotational speeds. Using slice theory and dynamic modeling, we analyze how mesh quantity influences simulation outcomes, providing guidelines for optimal mesh settings to ensure reliable results in gear modification studies.
The importance of helical gears in automotive transmissions cannot be overstated, as they contribute to reduced noise levels and improved efficiency. In electric vehicles, the absence of internal combustion engines amplifies gear whine noise, which typically falls within the 700–4000 Hz range, making it perceptible and uncomfortable for occupants. Gear micro-modification, involving micron-level adjustments to tooth profiles, is a common technique to minimize transmission error and contact stress, thereby enhancing NVH performance. However, the accuracy of simulation-based evaluations of these modifications is contingent upon the mesh resolution used in computational models. Insufficient or excessive mesh elements can lead to erroneous predictions, affecting design decisions. This research addresses this issue by systematically varying mesh numbers along the tooth width and involute directions and assessing their effects on key performance indicators for helical gears.
To establish a theoretical foundation, we employ slice theory, which simplifies the complex contact behavior of helical gears by dividing the tooth into discrete slices along the face width. This approach allows for a detailed analysis of load distribution and stress concentrations. The dynamic model of a helical gear pair is based on an 8-degree-of-freedom system, incorporating factors such as mesh stiffness, damping, and manufacturing errors. The equations of motion are derived using lumped parameter methods, and Hertzian contact theory is applied to compute the maximum contact stress. Transmission error, defined as the deviation between the actual and theoretical positions of gear teeth during meshing, is calculated to assess vibrational excitations. The following sections detail the mathematical formulations, simulation methodology, and results, with an emphasis on the sensitivity of simulation outcomes to mesh density variations.
The general equation of motion for a helical gear system can be expressed in matrix form as:
$$ M \ddot{q}(t) + C \dot{q}(t) + K(t) [q(t) – e(t)] = F $$
where \( M \) is the mass matrix, \( C \) is the damping matrix, \( K(t) \) is the time-varying stiffness matrix, \( q(t) \) is the displacement vector, \( \dot{q}(t) \) is the velocity vector, \( \ddot{q}(t) \) is the acceleration vector, \( e(t) \) is the static transmission error matrix, and \( F \) is the load vector. For a helical gear pair, the displacement vector includes translational and rotational components along the X, Y, and Z axes. The relative displacement along the line of action, \( \mu \), is critical for calculating the mesh force, which incorporates damping \( c_m \) and stiffness \( k_m(t) \) terms, as well as the composite error \( e_m(t) \) due to modifications and misalignments.
The maximum tooth contact stress \( \sigma_H \) is derived from Hertzian theory and is given by:
$$ \sigma_H = \sqrt{ \frac{F_t}{\pi} \left( \frac{1 – \nu_1^2}{E_1} + \frac{1 – \nu_2^2}{E_2} \right) \frac{1}{\beta_{\Sigma}} } $$
where \( F_t \) is the tangential force, \( \nu_1 \) and \( \nu_2 \) are the Poisson’s ratios of the driven and driving gears, respectively, \( E_1 \) and \( E_2 \) are their elastic moduli, and \( \beta_{\Sigma} \) is the composite curvature radius at the point of maximum stress. The curvature radii \( \rho_1 \) and \( \rho_2 \) for the driven and driving gears are used to compute \( \beta_{\Sigma} \) as:
$$ \beta_{\Sigma} = \frac{1}{1/\rho_1 + 1/\rho_2} $$
Transmission error (TE) is a key metric for evaluating gear noise and vibration. It is defined as the difference between the actual and ideal angular positions of the gears, scaled by their base radii:
$$ TE = \phi_1 r_{b1} – \phi_2 r_{b2} $$
where \( \phi_1 \) and \( \phi_2 \) are the actual rotation angles of the driven and driving gears, and \( r_{b1} \) and \( r_{b2} \) are their base radii. The peak-to-peak transmission error (PPTE) is the range of TE over a meshing cycle, which directly influences dynamic responses.
In this study, we developed a dynamic model of a two-stage reducer for an electric vehicle using specialized software. The helical gear parameters are summarized in Table 1, which includes tooth numbers, module, pressure angle, helix angle, profile shift coefficients, face width, and operating conditions at high and low rotational speeds. The gears are made of standard alloy steel with appropriate heat treatment to ensure durability. Micro-modification was applied to both the drive and coast sides of the teeth, incorporating lead crowning, tip relief, profile crowning, and pressure angle corrections. The modification values were optimized to centralize the contact pattern and reduce stress concentrations, as illustrated in the subsequent results.
| Parameter | Driving Gear Value | Driven Gear Value |
|---|---|---|
| Number of Teeth | 23 | 93 |
| Normal Module | 1.93 | |
| Pressure Angle (°) | 16.0 | |
| Helix Angle (°) | 25.6 | |
| Profile Shift Coefficient | 0.4300 | 0.0387 |
| Face Width (mm) | 42.0 | 46.5 |
| Center Distance (mm) | 125 | |
| Accuracy Grade | 8 | |
| Input Speed 1 (r/min) | 20,000 | |
| Input Speed 2 (r/min) | 3,500 | |
| Input Torque 1 (N·m) | 19.5 | |
| Input Torque 2 (N·m) | 111.4 | |
The mesh generation process involved discretizing the tooth surface into elements along the face width and involute directions. We varied the number of mesh elements from 20 to 160 in increments of 10, monitoring the convergence of TCS, PPTE, and TCP. The initial mesh density was set low to evaluate the impact of coarseness on simulation accuracy, and it was gradually increased until the results stabilized, indicating mesh independence. The slice theory was implemented by dividing the gear tooth into multiple segments along the face width, each treated as a spur gear slice, to account for the helical angle effects. This approach simplifies the contact analysis while maintaining accuracy. The mesh model was constructed using hexahedral elements to ensure precise stress calculations and reduce computational time.
The simulation results for maximum TCS under high and low rotational speeds are depicted in Figure 1, which shows the variation with mesh numbers along the face width and involute directions. At high rotational speeds (20,000 r/min), the maximum TCS decreased initially as the face width mesh number increased, eventually stabilizing beyond approximately 120 elements. The difference between the maximum and minimum TCS values was about 4.1 MPa over the range of 20 to 160 elements. In contrast, at low rotational speeds (3,500 r/min), the TCS variation was more pronounced, with a difference of 24.0 MPa, and stabilization occurred at around 110 elements. Changes in the involute direction mesh number had minimal impact on TCS, indicating that the face width discretization is more critical for accurate stress prediction. This behavior can be attributed to the larger deformations along the face width due to elastic deflections of shafts and bearings, which dominate the contact behavior. The Hertzian theory supports this, as the curvature radius at the contact point increases with finer meshing, reducing the calculated stress.
For PPTE, the simulations revealed an inverse trend compared to TCS. As the face width mesh number increased, PPTE values rose initially and then plateaued. At high speeds, the PPTE ranged from 0.1320 μm to 0.1385 μm, with a maximum variation of 0.0065 μm. The results stabilized when the face width mesh number exceeded 100. At low speeds, the PPTE variation was larger, from 0.3260 μm to 0.4200 μm, with a difference of 0.0940 μm, and stabilization occurred at mesh numbers above 100. The involute direction mesh number again showed negligible influence. This increase in PPTE with finer meshing can be explained by the reduction in contact stress peaks, which enlarges the gaps between mating teeth, thereby amplifying the transmission error. The dynamic model equations account for this through the stiffness and error terms, where finer meshing better captures the micro-geometry variations.
The distribution of TCP was also analyzed to assess the effect of mesh density on contact pattern realism. The TCP size and location are vital for evaluating gear meshing quality, as per standard guidelines. Figure 2 illustrates the TCP changes with mesh numbers. Along the involute direction, TCP distribution remained consistent regardless of mesh count. However, along the face width, the TCP area expanded as the mesh number increased up to 80 elements, after which it remained stable. This stabilization indicates that beyond a certain mesh density, the contact pattern is accurately represented, and further refinement does not yield significant improvements. The contact pattern requirements, such as those specified in gear accuracy standards, emphasize the need for a centralized and evenly distributed pattern to avoid edge loading and ensure longevity.
To quantify the mesh sensitivity, we performed a convergence analysis for TCS and PPTE at both speed conditions. The results are summarized in Table 2, which lists the critical mesh numbers where simulation outcomes become reliable. For high-speed operations, a face width mesh number of 120 or more ensures stable TCS and PPTE values, whereas for low-speed conditions, 110 elements suffice. The involute direction requires fewer elements, typically around 20–30, due to the lower sensitivity. This disparity highlights the importance of tailoring mesh settings to the operational regime of helical gears. The slice theory facilitates this by allowing independent control over mesh density in different directions, optimizing computational resources while maintaining accuracy.
| Parameter | High Speed (20,000 r/min) | Low Speed (3,500 r/min) |
|---|---|---|
| Face Width Mesh Number for Stable TCS | ≥120 | ≥110 |
| Face Width Mesh Number for Stable PPTE | ≥100 | ≥100 |
| Involute Direction Mesh Number for Stability | ≥20 | ≥20 |
| TCP Stabilization Mesh Number | ≥80 | ≥80 |
The implications of these findings are significant for the design and simulation of helical gears in automotive applications. In high-speed scenarios, such as electric vehicle reducers, the mesh requirements are more stringent due to the heightened sensitivity to dynamic excitations. Underestimation of mesh density can lead to non-conservative stress predictions and inadequate gear modifications, exacerbating noise issues. Conversely, in low-speed, high-torque conditions, the mesh can be coarser, but still requires careful consideration to avoid inaccuracies in transmission error estimation. The slice theory-based approach provides a balanced method for mesh optimization, as it accounts for the helical geometry without excessive computational cost.
Further analysis using the dynamic equations reveals that the time-varying mesh stiffness \( k_m(t) \) plays a crucial role in the simulation outcomes. The stiffness can be expressed as a function of the mesh density, where finer meshing captures the stiffness variations more accurately. For helical gears, the mesh stiffness is influenced by the contact ratio and helix angle, which are incorporated into the slice model. The composite error \( e_m(t) \) includes contributions from micro-modifications, and its representation improves with higher mesh resolution, particularly along the face width. The damping term \( c_m \) also affects the dynamic response, but its dependence on mesh density is less pronounced compared to stiffness and error terms.
In conclusion, this study demonstrates that the simulation reliability of helical gears is highly dependent on mesh discretization, especially along the face width direction. The maximum tooth contact stress and peak-to-peak transmission error are sensitive to face width mesh numbers, with high-speed conditions requiring finer meshing for accurate results. The tooth contact pattern distribution stabilizes at moderate mesh densities, and the involute direction has minimal impact. Based on slice theory, we recommend a face width mesh number of at least 110–120 for reliable simulations of helical gears under varying rotational speeds. These insights aid in optimizing finite element models for gear design, ensuring that micro-modification strategies are evaluated with high fidelity. Future work could explore the effects of mesh type and element shape on simulation accuracy, as well as experimental validation under real-world operating conditions.
The mathematical models and simulation techniques presented here provide a foundation for advancing the design of helical gears in electric vehicles, contributing to reduced NVH and improved performance. By adhering to the recommended mesh guidelines, engineers can achieve more dependable simulation outcomes, facilitating the development of quieter and more efficient powertrains. The ongoing evolution of helical gear technology will benefit from such refined computational approaches, aligning with the industry’s push towards sustainability and enhanced user experience.