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What is robust design?
1.1 The importance of small variation
When mass production started in the dawn of the industrial revolution, variation came in the focal point of interest. An early example that illustrates this concerns mass production of guns. The French gunsmith Honoré le Blanc realized the importance for guns to have interchangeable parts. His solution was the invention of a system for making gun parts in a standardized way. The problem that challenged le Blanc is the same as in any modern day manufacturing, as, for example, in the production of bolts and nuts. It shall be possible to pick a bolt and a nut at random that fit together. This requires that the variation in diameter, in roundness, and in thread pitch is small from bolt to bolt and from nut to nut. Unless this is the case, there will be a substantial amount of scrapping, or even worse bolts that crack or fall off while they are in use.
Before the industrial revolution, this problem was handled by good craftsmen. In the industrial era, this was not an option anymore. The importance of managing the variation became obvious. Several approaches emerged. Specifying the tolerance limits was one of them and even if the gunsmith le Blanc did not get many immediate followers in France, some Americans saw the potential of his ideas and implemented them at the armoury in Springfield. This is sometimes considered as the birth of tolerance limits (which is not quite true as tolerance limits are much older than this).
To quote Edward Deming, a forefront figure in quality engineering, ‘Variation is the enemy of quality.’ The bolt and the nut is one example. Another one is thickness variation of the plastic film on the inside of a milk package–a plastic film preventing the beverage from coming in contact with the aluminium foil that is present in most milk (and juice) packages. If this thickness varies too much, it may occasionally happen that there is a point with direct contact between the beverage and the aluminium foil. However, it is not the fact that there is a contact point that should be the centre of interest. The focus should rather be the size of the film thickness variation. The contact point is just a symptom of this problem.
Investigations show that a substantial part of all failures observed on products in general are caused by variation. With this in mind it is obvious that variation needs to be addressed and reduced. The issue is just how.
This book is about random variation, or more specifically how to reduce random variation in a response variable y, but not just any way to reduce this variation. It will not be about tightening tolerances, not about feedback control systems, and not sorting units outside the tolerance limits. The focus is solely on preventing variation to propagate. It is this approach to variation that is called robust design.
1.2 Variance reduction
Example 1.1 Consider a bar that is attached to a wall. There is a support to the bar and a random variation in the insertion point in the wall, as sketched in Figure 1.1. We are interested in the position (x, y) of the end point of the bar and that its variation is small.
The variation of the end point position can be reduced in two fundamentally different ways. One is to reduce the variation of the insertion point in the wall. It may be costly. Typically it can involve investment in new and better equipment. However, it might be another way to reduce the variation of the end point position, namely to move the support. In that way the design becomes less sensitive to the variation in the insertion point (Figure 1.2). This is what we mean by robust design: to make the variation of the output insensitive (robust) against incoming variation.
Example 1.2 Two metal sheets are attached to each other. There are two holes in each one and they are attached to each other using two bolts (Figure 1.3). Assume that the maximum stress σmax in the metal sheets is of interest to us. A small variation in the position of the holes, or rather distance between them, affects this stress.
Suppose that this stress should be minimized. This can be achieved by increasing the precision of the positions of the holes. It can also be achieved by changing the design so that one hole is exchanged for a slot (Figure 1.4). In that way, the variation in the response, the maximum stress, is decreased without reducing the variation in the sources of variation, the hole positions. The stress is robust to the hole position.
We have seen two examples of robust design, the end point of the bar and the stress of the metal sheets. For both of them, ways to reduce the variation of the output without reducing or removing the original source of variation were pointed out.
In robust design, the original source of variation is called noise. This noise is typically sources of variation that the engineer cannot remove or even reduce, or something that can be reduced but at a considerable cost.
1.3 Variation propagation
The essence of robust design is to make use of nonlinearities in the transfer function y = g(x, z) in such a way that variation in the noise z is prevented from propagating. The key is in the derivative of the transfer function. We will study how this can be expressed mathematically.
Assume that Z is a random variable with mean μz and standard deviation σz and that x is a nonrandom variable. F...
