From Gear Mesh to System: Advanced Prediction of Transmission Error

Eliminating common assumptions of single gear mesh calculations by considering the full system for higher-fidelity gear transmission error prediction and gearbox NVH analysis

The demand for lower emissions and greater electrification is intensifying in the realm of transmission applications, especially within the automotive industry. At the same time, there is a push to enhance performance and quality. This evolving landscape necessitates sophisticated structural design strategies,

focusing on lightweighting, durability, and the optimization of noise, vibration, and harshness (NVH). Fortunately, engineers now have access to ever-advancing tools that facilitate a CAE-led design approach, offering valuable insights at each stage of development with the appropriate level of detail. This approach minimizes design iterations and reduces the reliance on physical testing, ultimately saving both time and costs.

In a multi-fidelity modelling process, simpler and faster models are invaluable for early design exploration, such as assessing the impact of different tooth geometries on robustness. As development progresses, higher-fidelity models become crucial. Though they demand more detailed design inputs and are computationally intensive, their true value lies in the later stages of development, where they validate a select few, more refined design choices.

This paper delves into the analysis of gearbox transmission error and the resultant vibrations within a representative electric vehicle (EV) powertrain model. Transmission error (TE) quantifies the displacement of the gear mesh induced by the combination of varying tooth contact force, mesh stiffness, and flexibility of the system. Two levels of fidelity of TE analysis are considered in this paper: single mesh loaded tooth contact analysis (LTCA) and gearbox transmission error (GBTE) analysis. In addition, these approaches are combined in turn with two distinct tooth contact models—a Basic uncoupled 2D finite element model, and an Advanced coupled 3D finite element model. This gives us four different combinations to compare and contrast.

GBTE analysis, see Figure 1, is a quasi-static phased system analysis that evaluates the gearbox at multiple rotational angles. It performs an LTCA at each angle, accurately phasing all gear meshes. By considering system coupling and the phasing of all gears, GBTE analysis captures the complex interactions within the system, such as those occurring in planetary gearsets. It facilitates the calculation of the overall gearbox transmission error.

This analysis adopts less restrictive assumptions than single mesh calculations, allowing for variations in misalignment and torque through the gear meshes and system deflections during the meshing cycle. It considers changes in stiffness experienced by the gear mesh as the system rotates and recognizes that forces in one gear mesh can affect forces in others. Additionally, this approach allows the gear mesh to transfer moments in the misalignment direction, in addition to forces along the line of action, providing a fuller picture of the gear excitations, resulting in predictions of linear TE and tilt TE.

Linear or transverse TE is widely recognized as one of the main sources of gear whine noise. In contrast, tilt TE is less frequently discussed and is referred to by different terms, such as force axial shuttling, so it is worth clarifying, see Figure 2. Tilt TE results from the slight axial movement of the centroid of the gear mesh force as the gears rotate through the mesh. This axial shift arises from the varying load distribution along the lines of contact and is also influenced by the rest of the system, such as other gear meshes and bearing stiffnesses. By solving the entire system in the gearbox transmission error (GBTE) analysis at each rotational step, with all components accurately phased, these variations and interactions are effectively captured.

The GBTE analysis provides critical parameters for NVH analysis, including linear TE, tilt TE, and linear, tilt and cross-term mesh stiffness. These calculated values can be integrated into the NVH model. By coupling and solving the entire system with correct mesh phasing and accommodating variations and interactions, the analysis achieves much higher fidelity than single mesh LTCA.

Similarly, the Advanced tooth stiffness model captures more effects than the Basic model, see Figure 3. Specifically, the material coupling within a single tooth captures Poisson’s effect, which causes the tooth to assume an anticlastic curvature that alters the load distribution and can result in peaks towards the side edges if insufficient lead crowning or end relief is applied. Coupling between adjacent teeth can cause peaks near the tip and root due to the highest-loaded teeth deflecting more, increasing the load on the teeth that are just coming into or out of contact, especially at the corners where contact starts or ends.

Considering these effects not only impacts the load and stress distribution, but also influences the mesh stiffness, which is a crucial part of the TE calculation. The combined mesh bending stiffness from the Basic model is a relatively simple curve that is constant across the gear face width. In contrast, the combined mesh bending stiffness from the Advanced model is more complex and varies with face width, see Figure 4.

A comparison is presented of the results of different fidelity methods to demonstrate how the latest advancement in GBTE analysis—incorporating the Advanced tooth stiffness model—stacks up against single mesh LTCA analysis and Basic tooth stiffness within a unified modeling environment.

The following case study was conducted using an EV powertrain simulation model, which includes representations of bearings, an electric motor, input and output gearsets, shaft assemblies, finite element (FE) internal components and housing, and housing mounts, see Figure 5. Gear excitations are calculated using both single mesh LTCA and GBTE analysis, each utilizing Basic or Advanced tooth stiffness. These excitations are then applied to vibration and acoustic analysis to assess their impact on NVH performance.

The initial results compare the first harmonic of linear TE from the output gear set across a range of torque levels, see Figure 6. These results highlight the difference in linear TE between the Basic and Advanced tooth stiffness models. At low torque, the differences are small, while at higher torque, the Advanced model predicts significantly lower linear TE. Further investigations, not shown here, indicate that this discrepancy largely stems from the variations in combined mesh bending stiffness mentioned earlier.

While the difference in linear TE between single mesh LTCA and GBTE analysis is relatively minor, the GBTE analysis uniquely predicts tilt TE, which the single mesh analysis does not account for and is therefore effectively zero. Similar to linear TE, tilt TE predicted by the Advanced tooth stiffness model is lower than that predicted by the Basic model, as illustrated in Figure 7.

To assess the significance of these differences and the impact of considering tilt TE, the vibration and acoustic response can be studied. The subsequent results compare the housing equivalent radiated power (ERP) at high and low torque levels, which indicates the acoustic radiation from the gearbox housing. In addition, a comparison is made between GBTE analysis, with and without tilt TE, and single mesh LTCA, see Figure 8.

GBTE analysis with only linear TE and single mesh LTCA yields very similar ERP at both low and high torque, using both the Basic and the Advanced tooth stiffness models. However, the full GBTE analysis, incorporating both linear and tilt TE, consistently shows higher ERP. This suggests that the main difference in ERP between single mesh LTCA and GBTE analysis is due to the inclusion of tilt TE excitation. In regions with larger discrepancies between the two methods, certain modes are more easily excited by tilt excitation, which can be analyzed further and mitigated if needed.

A comparison of ERP for the four combinations of types of analysis and tooth stiffness models is presented in Figure 9.

The single mesh LTCA with the Advanced tooth stiffness model yields the lowest ERP, while the GBTE with the Basic tooth stiffness model results in the highest ERP. The most accurate combination, the GBTE analysis using the Advanced tooth stiffness model, falls between these two extremes. Additionally, it is worth noting that at low torque levels, the difference between Basic and Advanced tooth stiffness is relatively minor, corresponding with earlier observations from the TE comparisons.

Structure-borne vibration can also be evaluated by analyzing the vibration at one of the housing mounts, as depicted in Figure 10. The trends mirror those observed in ERP. Comparisons including GBTE analysis without tilt TE excitation (not shown) show that the vibration when tilt TE is included is noticeably higher. When the four combinations of types of analysis and tooth stiffness models are examined, once again, the single mesh—Advanced case shows the lowest vibration, the GBTE analysis—Basic shows the highest vibration, and the highest fidelity case of GBTE analysis with Advanced tooth stiffness predicts vibration somewhere in between. Once again, at low torque, the differences between Basic and Advanced tooth stiffness are small, while much more pronounced at high torque.

Automotive customers have made similar observations in their analyses and specific applications. One customer noted that using the Advanced tooth stiffness model correctly predicted that a modification to the design resulted in improved NVH performance as observed in physical tests, while a competitor tool predicted the opposite. Another customer shared that, in some of their applications, the tilt TE has a relatively small impact, while in other projects, they have observed extreme sensitivity in sound power with changes to tilt TE.

Although reduced fidelity modelling lends itself to fast assessment of the design space for making early design decisions, the full system approach to TE calculation, i.e., the GBTE analysis, combined with Advanced LTCA, captures more effects and eliminates some underlying assumptions such as fixed load sharing and misalignment. Therefore, the TE predictions and subsequent NVH analysis from the higher-fidelity approach include more physics and produce results that allow more insight into the expected behavior of the final produced component. By taking a multi-fidelity CAE-led approach to design, the analyst can evaluate where additional detail makes a difference to the results and, as such, can judge how, when and where the highest fidelity is needed.

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