[V]ariety can destroy variety

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CEBON Engineering Design Centre, Engineering Department, Tmmpington Street, Cambridge CB2 IPZ, England ABSTRACT A novel materials-selection procedure has been developed and implemented in software. The procedure makes use of Materials Selection Charts: a new way of displaying material property data. A useful method of doing this is by plotting them as Material Property Charts, sometimes called ‘bubble’ or ‘Ashby’ charts, with one property on one axis and another property on the other. Each material has a range of values for each property, depending on the exact composition, grade, heat.

W. Ross Ashby

There are more things in heaven and earth, Horatio,
Than are dreamt of in your philosophy.

Hamlet (1.5.167-8)

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In his book An Introduction to Cybernetics, published in 1956, the English psychiatrist W. Ross Ashby proposed the Law of Requisite Variety. His original formulation isn’t easy to extract into a blog post, but the Principia Cybernetica website has a pretty good definition:

The larger the variety of actions available to a control system, the larger the variety of perturbations it is able to compensate.

Like many concepts in systems thinking, the Law of Requisite Variety is quite abstract, which makes it hard to get a handle on. Here’s a concrete example I find useful for thinking about it.

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Imagine you’re trying to balance a broomstick on your hand:

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This is an inherently unstable system, and so you have to keep moving your hand around to keep the broomstick balanced, but you can do it. You’re acting as a control system to keep the broomstick up.

If you constrain the broomstick to have only one degree of freedom, you have what’s called the inverted pendulum problem, which is a classic control systems problem. Here’s a diagram:

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The goal is to move the cart in order to keep the pendulum balanced. If you have sensor information that measures the tilt angle, θ, you can use that data to build a control system to push on the cart in order to keep the pendulum from falling over. Information about the tilt angle is part of the model that the control system has about the physical system it’s trying to control.

Now, imagine that the pendulum isn’t constrained to only one degree of freedom, but it now has two degrees of freedom: this is the situation when you’re balancing a broom on your hand. There are now two tilt angles to worry about: it can fall towards/away from your body, or it can fall left/right.

You can’t use the original inverted pendulum control system to solve this problem, because that only models one of the tilt angles. Imagine you can only move your hand forward and back, but not left or right. Because of this, the control system won’t be able to correct for the other angle: the pendulum will fall over.

The problem is that the new system can vary in ways that the control system wasn’t designed to handle: it can get into states that aren’t modeled by the original system.

This is what the Law of Requisite Variety is about: if you want to build a control system, the control system needs to be able to model every possible state that the system being controlled can get into: the state space of the control system has to be at least as large as the state space of the physical system. If it isn’t, then the physical system can get into states that the control system won’t be able to deal with.

Bringing this into the software world: when we build infrastructure software, we’re invariably building control systems. These control systems can only handle situations that it is designed for. We invariably run into trouble when the systems we build get into states that the designer never imagined happening. A fun example of this case is some pathological traffic pattern.

The fundamental problem with building software control systems is that we humans aren’t capable of imagining all possible states that the systems being controlled can get into. In particular, we can’t imagine the changes that people are going to make in the future that will create new states that we simply could not ever imagine needing to handle. And so, our control systems will invariably be inadequate, because they won’t be able to handle these situations. The variety of the world exceeds the variety our control systems are designed to handle.

Fortunately, we humans are capable of conceiving of a much wider variety of system states than the systems we build. That’s why, when our software-based control systems fail and the humans get paged in, the humans are eventually able to make sense of what state the system has gotten itself into and put things right.

Even we humans are not exempt from Ashby’s Law. But we can revise of our (mental) models of the system in ways that our software-based control systems cannot, and that’s why we can deal effectively with incidents. Because of how we can update our models, we can adapt where software cannot.

Using Material Selection Charts

Here is a materials selection chart for 2 common properties: Young's modulus (which describes how stiff a material is) and density.
On these charts, materials of each class (e.g. metals, polymers) form 'clusters' or 'bubbles' that are marked by the shaded regions. We can see immediately that:

• metals are the heaviest materials,
• foams are the lightest materials,
• ceramics are the stiffest materials.

But we could have found that out from tables given a bit of time, although by covering many materials at a glance, competing materials can be quickly identified.

Where selection charts are really useful is in showing the trade-off between 2 properties, because the charts plot combinations of properties. For instance if we want a light and stiff material we need to choose materials near the top left corner of the chart - so composites look good.

Note that the chart has logarithmic scales - each division is a multiple of 10; material properties often cover such huge ranges that log scales are essential.

There are a selection charts for many combinations of material properties, e.g. 'strength - toughness' and 'electrical resitivity - cost'. The next section shows how we cantake selection charts further.

Consider a design problem where the specification is for a component that is both light and stiff (e.g. the frame of a racing bicycle). The Young's modulus - density chart helps us to find the best materials - they lie towards the top left. The charts can be annotated to help reveal the 'best' materials, by placing a suitable selection box to show only stiff and light materials. What can we conclude? The values of Young's modulus for polymers are low, so most polymers are unlikely to be useful for stiffness-limited designs. Some metals, ceramics and woods could be considered - but composites appear best of all. This still leaves quite a lot of choices, so what might be considered next to narrow the choice further?

It is unlikely that only 2 material properties matter, so what other properties are important? Let's consider strength and cost - these properties are plotted on another selection chart. So, what else does this tell us about suitable materials classes? What can we conclude? The strength of ceramics is only sufficient for loading in compression - they would not be strong enough in tension, including loading in bending. Woods may not be strong enough, and composites might be too expensive. Metals appear to give good overall performance We should now be able to identify a promising class of materials, but how do we decide which members of this class are the best. For instance metals look promising, which particular metal should we select?

Selection charts can also be used to select between members of a given class by populating it with the main materials. For instance, we can do this for metals in the stiffness-density chart. What can we conclude? Some metals look very good for light, stiff components - e.g. magnesium, aluminium, titanium, while others are clearly eliminated - e.g. lead. Steels have rather a high density, but are also very stiff. Given their high strength and relatively low cost, they are likely to compete with the other metals. Let's summarise what we've learnt about materials selection.

Summary:

• By considering 2 (or more) charts, the properties needed to satisfy the main design requirements can be quickly assessed.
• The charts can be used to identify the best classes of materials, and then to look in more detail within these classes.
• There are many other factors still to be considered, particularly manufacturing methods. The selection made from the charts should be left quite broad to keep enough options open. A good way to approach the problem is to use the charts to eliminate materials which will definitely not be good enough, rather than to try and identify the single best material too soon in the design process.