In the first of this two-part series, we explored how new computer-based modeling of geared systems is allowing the design of gear microgeometry for durability and lower noise, vibration, and harshness, or *NVH*. No two gears are manufactured the same, but addressing variability with *optimized gear microgeometry* boosts durability and NVH.

Here we continue our explanation of an automated approach to optimize microgeometry, and *Monte Carlo* simulation for investigating microgeometry designs.

### Automated optimization

Two things are critical to good gear design: Consideration of misalignment variation with load, and knowledge of detailed gear contact behavior. Now, understand that no design transmits only one load — so additional data and variables must be considered when analyzing a range of loads for optimization. Here, automated optimization allows engineers to effectively process large amounts of data and drastically reduce the time required to create a design.

Some design software includes automated optimization tools to assist in microgeometry design. It resolves potentially conflicting demands of different targets such as durability, NVH, and so on. This is done through a designerselected *cost function* that defines the balance between NVH and durability requirements.

Know that optimization can be very sensitive to the relative weighting of these factors, so automated optimization should not be considered a black box that produces the perfect design at the click of a button. Instead, it is a tool that provides a starting-point design — one that requires further refinement.

Let us assume that for our example third gear pair, we increase cost function within the software with a low transmission error, or TE. Note that TE under higher load is given slightly lower weighting, as gear whine noise is often perceived to be less of a problem at high load. As for edge, tip, and root contact: Maximum contact load *close to the tooth perimeter* greatly reduces score.

The results of this optimization are shown in *Figs. 8* and *9*: TE is very low across the whole torque range, indicating that this design’s NVH quality is good. The contact load distributions show that the optimizer has avoided edge contact by introducing end relief, and the load is distributed over a wide area of the tooth — thus reducing overall peak load. In turn, contact damage for the full duty cycle is considerably reduced (pinion by 33% and wheel by 26%) compared with the manually optimized design. However, the optimizer converges on a solution with almost zero crowning, so contact is not well centered at the lowest and highest load. While this is not in itself a problem, it does mean that our design is likely to be sensitive to manufacturing variations in lead slope.

### Refined Optimization

Treating these results as a starting point, further re nements can be made to improve the design. The cost function can be adjusted by including a factor that biases the result so that maximum contact load is close to the tooth center. This is likely to force the optimizer to increase the amount of crowning which may in turn degrade TE. However, as TE is already very low for a transmission of this type, it is acceptable to sacri- ce some NVH performance for an overall improved design.

With this adjusted optimization, more crowning is included as predicted. The amount of crowning is still only 2 μm, which is small compared to crowning manufacturing tolerances possible in mass production. For the final design, crowning is thus manually increased to 5 μm.

The results of this refined optimization are shown in *Figs. 10* and *11*. Contact load distribution shows improvement over the previous design in that contact is more centralized across the load range, and peak load is reduced at the highest torques. On the other hand, there is slight increase in peak load at lower torques due to the reduction in contact area caused by additional crowning. The effect of this can be seen in the full duty-cycle contact damage, which increases to 48% and 38% for the pinion and wheel respectively.

A secondary effect of crowning is that it tends to increase TE at low loads due to the deviation of the gear surface from the ideal involute. At high loads the increase in TE is small as the higher contact forces deform the tooth surface back towards the involute shape. This is confirmed by TE results, which show a proportionally larger increase in TE below 100 Nm than at higher torques when compared with the previous optimization results. However, the TE is still low and falls well within what would is considered a quiet range.

### Manufacturing Variability

The optimization process described requires thousands of gear contact analyses. For this reason, the analysis algorithm must be very fast to make this a practical tool for the engineer: The computer code to perform this analysis is highly streamlined, and the optimization routines we describe here are performed in a couple hours. It is possible to leverage these fast solution times to carry out other parametric studies that require several evaluations of gear contact analysis.

One particularly useful application is the simulation of microgeometry manufacturing variability effects on gear performance. The optimization process produces a design, which is nominally the best, but it takes no account of the A design with very good performance, when made to exact optimization specifications, may give poor performance when even very small changes are made to microgeometry parameters.

Microgeometry parameters such as lead and involute slope, crowning, barreling, and tip and end relief cannot be varied individually to assess the sensitivity of the durability and NVH performance. There is too much interaction between these parameters for such an approach to provide any useful information on robustness. A better method is to vary each parameter simultaneously using Monte Carlo simulation — which randomly varies parameters within a normal distribution to examine the effect on some outcome. (We explore TE in the example that follows.) When sufficient random variations are analyzed, an outcome distribution can be generated and valuable information gained.

Let us consider some variation in the lead slope, involute slope, and crowning. The amount of variation is based on the tolerances for an *ISO Quality Grade 7* gear. The values are chosen randomly with a probability determined by a normal distribution; the mean is the nominal value from the optimization and the standard deviation is 1/6th of the tolerance range for that parameter. Crowning cannot have a negative value; therefore, the mean value must be great enough to ensure that the lower limit is greater than zero.

Let us explore Monte Carlo simulation on 200 variants of our automated and re ned microgeometry designs at low (14 Nm), medium (82 Nm), and high (177 Nm) loads. Histograms of the resulting TE distributions are shown in *Fig. 12*.

At 14 Nm, the automated design has a distribution with a lower mode and narrower peak than the refined design. This indicates that the automated design is better at this low torque level. However, at 82 and 177 Nm the refined design is better, as the mode is lower and the distribution widths are approximately the same. In this example it is difficult to say which design is ‘best’ overall, as no specific targets are defined — but this information is very useful in a real design situation.

Where targets are de ned, it is possible to set them based on a percentage failure rate rather than a nominal value. These distributions could be used to indicate where manufacturing tolerances need to be tightened or where they can be relaxed. The ability to relax tolerances with con dence can lead to signi - cant production cost savings through the use of cheaper manufacturing processes.

### Coming soon

The development of fast algorithms for analyzing gear contact behavior is allowing other parametric study types. Here, we examine Monte Carlo simulation to verify predictions of production variability on NVH performance. Beyond this, more sophisticated design-ofexperiments methods can be used to identify the sensitivity of individual parameters. The next development based on the approaches we explore here is to include manufacturing variability as a factor in the optimization cost function. Currently, run times for this (which grow by a factor of several hundred) make this impractical. That said, the continuous exponential improvement in computing performance means that a one-week task last year is an ‘overnight task’ this year, and will become a ‘coffee-break’ task next year.

By making use of such computer- based analysis techniques, gear designers can create robust designs faster and with greater confidence than before. Enabling engineers to carry out such detailed analysis upfront in the design process reduces the need for redesign, repeated prototyping, and later testing — and decreases development time and cost.

*For more information, visit www.romaxtech.com.*