Monte Carlo variety performance

The transfer of feature-based tolerances from a CAD system to a Monte Carlo simulator was the key to what is believed to be the first variation analysis of an automotive differential.

A groundbreaking use of tolerance analysis has allowed a US axle company to reduce the cost of developing automotive differentials. The project also enabled designers to build improvements into the product’s performance at the early stages.

Tolerance analysis has long been a valuable tool for reducing manufacturing costs. However, the use of analysis in the early stages of a design, when there is little information about part and assembly variation, has always been difficult. Frustrated with the absence of a realistic method of simulating the knock-on effect, or stack up, of all the tolerances in the final build of a differential assembly, engineers from the company approached Michigan-based Dimensional Control Solutions (DCS). The client, one of the world’s leading producers of differentials and driveline components, has asked not to be identified.

A differential presents an extremely challenging problem for tolerance simulation because of the complexities involved in dealing with the contacts of eight different teeth under the complicated geometrical conditions.

This problem was overcome largely thanks to 3DCS, software developed by DCS that has built advanced tolerance analysis and simulation capabilities into CatiaV5. This leading solid modeller can then seamlessly interface to a Monte Carlo simulator.Monte Carlo simulation involves constructing an analytical method that takes into account the tolerances of individual components. Then the construction and assembly of a number of individual components are simulated to predict final build manufacturing variation.

The simulator provided a statistical analysis of final build tolerances under a wide range of different designs and component tolerances that were defined by the analysts. This information made it possible to address quality issues in the concept design phase that previously were very hard to address before prototyping. To the best knowledge of DCS and the client, no one had ever successfully applied this technique to an automotive differential.

Analysing the tolerance stack-ups on a differential is extremely tricky. Firstly, there’s the fact that most differentials have four interlocking gears, all of whose interactions must be considered in simulating the final build stack-up. Then there’s the complexity of the gear corner, or area where the teeth from adjacent gears intersect.

This is far from a simple shaft-in-hole intersection. The tooth of each gear consists of a complicated 3D surface profile. The tolerance of each point on that surface must be considered to fully analyse the stack-up.

Backlash is usually the most critical final build tolerance because it involves a stack-up of all four gears. Excessive backlash can be experienced as a noise problem that often leads to warranty repairs, even though it does not normally affect the performance of the vehicle or the life of the differential.

Traditionally, preventing excessive backlash has largely been a trial-and-error process during the prototype phase. If backlash is a problem in prototype builds, engineers reduce manufacturing tolerances of components involved in the stack-up until the noise goes away.

There are two problems with this approach. Firstly, engineers usually have only a rough idea of the relative contribution of individual tolerances, so it’s possible to invest considerable amounts of money with minimal results. Then the considerable number of prototype assemblies that must be produced to ensure that final build tolerances are within specifications add to the expense.

Engineers from the axle company saw that the potential advantages of simulating final build tolerances of new differentials were enormous. Being able to simulate final builds based on various manufacturing assumptions makes it possible to investigate the effect of changing different tolerances or even altering the design without the expense of building prototypes.

In other applications where this analysis method has been applied engineers usually find that they can improve the performance of the product while reducing manufacturing costs. This is possible because they can determine the exact impact of each component tolerance on the final stack-up.

Engineers often find that they can loosen the tolerances of some components that have less impact on the final stack-up while tightening tolerances of the few critical dimensions that have the most impact.

In addition, the ability to simulate the construction and assembly of large numbers of components provides extremely valuable information that could make it possible to identify in the concept design phase less common problems that might not otherwise show up until a large number of prototypes had been tested.

The company was confident it could achieve this thanks to new capabilities of the Catia V5 solid modelling software used in the design process.

One particular advantage provided by V5 is the ability to attach tolerances not only at individual points but also along entire surfaces, meeting one of the critical requirements of this application. This makes it possible to quickly associate a tolerance to an entire geometric object.

Point-based tolerances were intermixed at key points to help make the results as robust as possible. The ability to view the entire simulation graphically within the Catia interface streamlined the modelling process, making it possible to complete the model in three weeks.

The engineers at DCS and the clients used the following build assumptions. After the differential is assembled the pinion and side-gears are readjusted to be nominally centred to one another starting in the lower left-hand corner and working anti-clockwise around to the upper right corner. The gears are then checked for interference and if necessary readjusted.

The model included an algorithm that simulates the engagement of the gears. The algorithm begins with equal spacing between the teeth of all four gears and then rotates the teeth until they touch.

The DCS engineers then used the 3DCS program to simulate 2,000 assembly builds and produce statistical information in the 74 measurements. Different reports showed how much variation occurred on each measurement and which tolerances and locating schemes affected backlash. The DCS engineers generated several types of graphical reports that provided a statistical distribution of final build tolerances. One report gave a graphical distribution of the most critical build tolerance including minimum, maximum, mean and six sigma values. Another displayed the contribution of each component tolerance and assembly operation to the final build tolerance.

The analysis results showed that the main contributor to the final build tolerances was the interface of the side gear to the pinion gear. The DCS engineers tried a number of different approaches to reduce final build tolerances by editing the geometry and tolerances of the Catia model and rerunning the simulated build analysis.

One approach that the analysis results recommended was eliminating the washers used to position the upper and lower pinion gears. This involved redesigning the case slightly to take up the extra space between the gears and wall, eliminating the contribution of the washers to the tolerance stack-up. Throughout the process the DCS engineers provided their modelling results to the client’s engineers, who in turn supplied information on the practical effects of changing various tolerances.

Examination of the analysis results proved beneficial in several ways. The axle company identified several promising avenues for significantly reducing final build tolerances without a major increase in manufacturing costs. At the same time the analysis showed that a number of tolerances that had little impact could be loosened to reduce costs without affecting the final build. The client’s analysis validated a new tool that is expected to be used heavily in the future to streamline the design process by providing predictions of manufacturing variation during the concept design.