Topics of discussion
- Durability, NVH with load variation
- Microgeometry optimization
- Manufacturing variability simulation
- Case study of a speed gear pair
No two gears are manufactured the same, but addressing manufacturing variability with optimized gear microgeometry boosts durability and lowers noise, vibration, and harshness, or NVH.
New developments in computer-based modeling of geared systems are now allowing the design of gear microgeometry for durability and NVH — for realistic in-service duty cycles in a virtual environment. However, ideal designs can rarely be manufactured, as mass production always exhibits geometric variability. Here we review an automated approach to optimize microgeometry, and how Monte Carlo simulation methods can be used to investigate microgeometry designs. (Here, Monte Carlo method refers to a class of algorithms that model systems with repeated random sampling.) We'll then analyze the example automotive gearbox pictured above — not only for nominal NVH and durability performance, but also for immunity to manufacturing variability.
Importance of microgeometry
Microgeometry design is fundamental to gear design, affecting both durability and NVH quality. However, it is often considered late in the process. A lack of suitable and accurate analysis tools leads to low confidence in predictive design — which in turn leads to over-engineered gears which are necessarily biased towards durability. As a result, NVH quality tends to end up compromised, and may not even be considered at all.
To illustrate: In automotive applications, it is common for gearboxes to be designed to meet some duty cycle, derived either from a company standard or from measured road load data for a particular market. The gear designer is then presented with a condensed version of the duty cycle, which typically comprises a single loading condition (usually maximum rated load) over a duration, which is supposed to represent the equivalent damage of the whole duty cycle. Using this “worst-case” approach to gear design may be quicker and easier to execute, but variation of durability and NVH performance with load is not taken into consideration.
A typical traditional approach to microgeometry design is as follows:
Design gear macrogeometry for maximum load — possibly with some consideration for NVH, but often not.
Determine gear durability based on maximum load and duration only.
Apply some default microgeometry in the hope of avoiding tooth contact durability problems.
Hope the design is not too noisy.
Refine microgeometry by experience/experiment and repeated prototyping.
An improvement on this process is to add an analysis, to enable the calculation of system deflections under load, thus giving accurate predictions of gear misalignments. This has two main benefits. First, it gives a more accurate estimate of gear durability, allowing safety factors to be reduced with greater confidence. Second, it gives clues to the microgeometry modifications that may be required to correct for misalignment. However, it still does not take into consideration the load variation and misalignment, and can only give a vague qualitative indication of the tooth contact behavior of the gears, which must subsequently be validated by a gear tooth-marking test on a prototype gearbox.
This implies that further enhancements to the gear design process — namely detailed analysis of tooth contact behavior across the entire range of operating loads — would provide an engineer with valuable information which could then be used to design better microgeometry as well as give a more accurate prediction of gear durability.
In fact, there is a potential problem with this information: There is too much of it. Consider a final drive gear set in an automotive transaxle. If there are five speed gears, there are five potential power flows in the forward-drive condition where the final gears carry load. Using say, seven load conditions to cover the entire gearbox torque range leaves an engineer with 35 load conditions to analyze and interpret. Even with a speed gear pair where there are seven loading conditions, this is still no small task when the number of possible microgeometry parameters that can be varied is considered.
Automated optimization tools can help engineers to cope with this glut of data. Optimization requires multiple calculations of gear contact behavior, so analysis must be fast and accurate. Once developed, such a high-speed approach can then be used to investigate other problems requiring repeated analysis, such as manufacturing variability.
A nominal gear design produced by an engineer is never manufactured. Variability within tolerances that can be achieved by the manufacturing process leads to variation in gear behavior during operation. Considering the durability and NVH performance of only the nominal gear at the design stage could lead to problems in production if the design is sensitive to manufacturing variation. One way of assessing this design robustness is through Monte-Carlo simulations of microgeometry parameter variations.
Consider our model in Fig. 1 at this article's opening, a five-speed manual transaxle gearbox. The analysis that follows is solely of third-gear-pair microgeometry design, a detail of which is shown in Fig. 2. A duty cycle histogram for 3rd gear has been calculated from road load data and is shown in Fig. 3; the nominal torque capacity for this transmission is 170 Nm. The duty cycle includes both drive and coast conditions and a number of shock loads which exceed the nominal capacity for short periods of time. A condensed design load duty cycle which comprises a single load case (at the maximum nominal load) is derived to represent the equivalent damage of the full duty cycle — 170 Nm for 17 hours.
Analysis method: Basic rating
A basic ISO 6336 rating of a gear by a designer accounts for basic macrogeometry properties of the gear and the load applied to it. No misalignment of the gear is taken into consideration and the peak load factor KHβ is assumed to be one. For the full duty cycle, this yields a contact damage of 6% for the pinion gear (counter shaft) and 5% for the wheel gear (input shaft). For the condensed duty cycle, damage is 7% on the pinion and 6% on the wheel. This shows that the condensed duty cycle does indeed yield the same result as for the full duty cycle when the most basic rating is used.
To calculate a more accurate and reliable rating for the gears, the misalignment of the gear pair must be taken into consideration. To calculate gear mesh misalignment, the deflection of the gears under load must be calculated. The most accurate way to do this is to apply the torque load to the simulation-software model, and solve it iteratively to determine the entire system's deflections. The resulting displacements of the gear mesh can then be resolved into the line of action of the gear pair to obtain the misalignment.
This static analysis must be performed separately for each loading condition. Why? Components such as bearings and clearance connections do not behave linearly. Solving the entire system is necessary as the flexibility of all components, including the housing, have an influence on the displacement at the gear mesh. The system deflections at the design load of 170 Nm are shown in an exaggerated form in Fig. 4. Also shown is a detail of misalignment at the third speed gear mesh due to the entire system's flexibility. The third-gear misalignments for the entire duty cycle are shown in Fig. 5. There is a fairly linear relationship between torque and misalignment for the drive loading conditions.
The ISO 6336 rating (including the effects of misalignment) can be calculated by assuming a simplified function for gear-mesh stiffness. For the condensed duty cycle, damage is 51% and 40% for the pinion and wheel respectively. When the complete duty cycle with variation in misalignment is used to calculate contact durability, damage increases still further to 68% and 53% for the pinion and wheel respectively.
Note how including misalignment has a major impact on gear durability. In addition, the difference between the condensed and full duty cycle results shows that the effect of misalignment variation also ensures that the condensed duty cycle is no longer representative of the full duty cycle from a damage point of view.
The aim of microgeometry optimization is to ensure good distribution of load across the tooth face while simultaneously keeping the transmission error (or TE) low across as much of the operating range as possible. TE is the source of gear whine. It is caused by the non-conjugacy of motion in the gear pair due to a combination of misalignment and gear teeth deflection under load. Unfortunately, the modifications often required to maintain a good contact for durability are not always conducive to low TE (and hence good NVH performance) and so require a design compromise. Furthermore, as misalignment varies with load, there is the additional problem that this balance must be maintained across a range of misalignments. In effect, the gear must be designed so that it is insensitive to misalignment variation.
Tune in next month for the second half of this two-part series, when we'll explore automated optimization, the variation of gear transmission error, and biasing within software for more accurate predictions of gear performance. For more information, visit www.romaxtech.com.
Model with housing cutaway
Our example five-speed front-wheel-drive manual transaxle gearbox above is modeled in RomaxDesigner software; the detailed static and dynamic model includes shafts, bearings, gears, synchronizers, differential, and housing. • Simple shafts are modeled as one-dimensional Timoshenko beams. • Complex shafts are modeled as three-dimensional FE components. • Rolling-element bearings are modeled as six-degree-of-freedom nonlinear components. • Gears are modeled with several macro and microgeometry parameters. • Radial and axial clearances are modeled using non-linear contact elements, and the housing is modeled as a three-dimensional FE component.