Less Is More, Part VI: Budgeting for Sustainability
First, we need a formalism for addressing sustainable design of precision machines. This follows from the formalism used for basic machine design. This is referred to as the “sustainability budget.” Let me explain.
In the design of machines, specially precision ones (that is, machines that can operate reliably and repeatably positioning workpieces or tooling to great accuracy and with very high resolution – for example, repeatably positioning something within a couple of microns (or nanometers).) This is often accomplished using a technique called “error mapping” and developing an “error budget.” These are methods for accounting for the magnitude and eventual impact of the numerous potential sources of error in a machine’s performance – relative to dimensional accuracy, form error, or surface finish.
One does this by determining the likelihood of errors due, for example, to thermal distortion (remember, things expand when heated and contract when cooled so if a machine component is subjected to either of these due to operation or environment) the component will change shape and that will affect the accuracy of the machine. Seems small but, over long machine components, it adds up. Or, for high accuracy, small temperature changes can have a big effect. Steel, for example, has a coefficient of thermal expansion of 11.7 microns/meter/degree C. So, a steel component 10 cm (or about 4 inches; one tenth of a meter) long that experiences a temperature rise of 5 degrees C during operation will “grow” almost 6 microns due to the thermal expansion. That’s a lot in the precision manufacturing community! Larger structures can grow more. And, 5 degrees C is easy to experience in most conventional manufacturing facilities.
We can put the machine in a conditioned environment where the temperature is maintained constant but that cost money and, importantly, uses a lot of energy. Or we can put circulating oil systems on machines with temperature controlled oil to maintain a constant temperature but that adds to the machine’s energy footprint also. And, since the circulating system usually runs even when the machine is not producing work, this makes the idle state of the machine almost as bad as the production state.
There are materials with almost zero coefficients of thermal expansion – but they are costly in terms of money and energy to create. So, we’d like to design the machine to have as little sensitivity to thermal distortion while using materials that have lower environmental impact.
One of the concepts in precision machine design relates to identifying, first, the “sensitive direction” in the machine. This is the direction in which an error impacts the part quality: dimension, form, roughness. For example, if you are trying to create a surface with a certain dimension by machining, then you want to control the position of the cutting tool relative to that surface with great accuracy. Any error in the position of the tool relative to the surface will result in an error. So, for this operation, the axis of tool motion towards and away from the surface during machining would be the sensitive direction.
The way we can keep track of all the contributions to the errors in the machine can be referrer to as an “error budget.” This budget allowed us to include all sources of error and an estimate of their relative magnitudes and then determine which of these sources actually impacted a sensitive direction resulting in a part error. The term budget is chosen exactly to represent what, like a budget for household expenditures, is available to be distributed over all the requirements for operation. Just as in a household budget where some of the monthly funds must cover groceries, insurance, transportation, etc., in an error budget, we allocate the elements making up the total error in such a way that, when we are done with designing the machine, the cumulative error, in the sensitive direction, do not exceed our requirements.
So, errors in the machine due to thermal effects, loads due to moving workpieces or forces generated in machining (which cause another type of distortion, elastic distortion, due to the elasticity of the material the component is made of), gravity loads, or vibrational excitation due to rotating spindles or tooling, are estimated. From these estimates it is determined by modeling the machine structure kinematically, in what way these errors affect the machine tool operation and accuracy and, then, to what extent they affect the sensitive direction.
Now, an important concept in this method is that an error that exists but does not affect the sensitive direction is not of concern. Meaning, something could be going on in the machine but as long as it does not affect the location of the tool relative to the workpiece in the example we’ve been discussing, we don’t need to worry about it.
This would be like going to a restaurant which serves a fixed price buffet. You could eat a lot, or a little, and, from the point of view of your budget, it wouldn’t matter. With graduate students, that means you can eat a lot!
So, in the case of an error source not impacting the sensitive direction, we have a lot more design freedom with no apparent penalty in terms of performance.
So by now you are asking what the heck this has to do with the subject of the blog!
Consider if we would add constraints on the environmental performance of a machine while insisting that the other quality metrics are met as well as the manufacturing performance (throughput, lead time, cost/piece to operate, etc.) This could be included in our budget analysis but, in this case, we’d call it a “sustainability budget”. A sustainability budget would operate similarly to an error budget except we would be looking for the impact, from environmental metric point of view, of the design and operation of the precision machine, process or system.
Then, using the idea of sensitive directions (and the complementary concept of non-sensitive directions – that is, those directions for which any error from a specific source has no effect) we can imagine an analysis which measures the impact of materials, designs, or operating conditions on the overall environmental behavior. Then we look for instances of materials, design features or operating conditions that give the largest range of variability, from the point of view of design, with the least environmental impact. That is, those instances for which little or no sensitivity is displayed.
Following a methodology based on this would allow us to design the machine, or system of machines, in such a way that the basic performance, precision and accuracy, would meet the core error budget constraints but, in addition, we could do so in a way that was more sustainable.
Great idea but how do we do this?! Let’s get started.
In the design of a precision machine the first requirement is to derive an error budget. Now it gets a bit complicated. Creating an error budget relies on two sets of rules — connectivity and combinational. Connectivity rules define the behavior of machine components and interfaces in the presence of errors. That is, how does the error in one component affect the position (for example) of another component. This is sort of like trying to level a table in a restaurant by sticking little bags of sugar under one of the legs. Sometimes you are lucky and it works the first time. Other times changing one leg makes another lose contact with the floor and the table still wobbles. That’s is a simple example but that is connectivity.
Then, the combinational rules define how the errors are to be combined to determine the impact on the accuracy of the workpiece. That means, how all these connected components, experiencing the various sources of error, combine to affect the sensitive direction. Not surprisingly, this is done with mathematics.
The procedure is comprised of the following three steps:
Step 1 — make the model of connectivity. This is called the error map. We do this by determining a kinematic model of the machine and its principal components in the form of a series of matrices,
Step 2 — analyze systematically each type of error in the system and use the mold to determine the relative tool-work errors. This is determining a relationship defining how the errors affect the sensitive direction, and finally
Step 3 — combine the errors to yield maximum and minimum estimates of the total error of the machine. Sort of like specifying tolerances on a part length – the error of importance will likely be within this range.
If we revise this approach for a sustainability budget, we’d follow the same three basic steps but with some different objectives. For example, we would add some elements to the three steps, or, actually, develop a parallel set of “models” and analysis.
Parallel to Step 1 would be:
– determine an energy, material and resources (consumables, etc.) model of the machine and its principal components in the form of a series of relationships defining the energy consumption, materials use as a function of machine design or operation. (This might be referred to as energy or materials mapping.)
Parallel to Step 2 would be:
– analyze systematically each type of energy and material use in the system and determine the relative performance-energy/material impact (for example, from Ashby charts).
Parallel to Step 3 would be:
– combine the energy/materials impacts to yield upper and lower bound estimates of the total energy/material impact of the machine.
Importantly, in this parallel analysis, the embedded energy and materials must be counted. That is, we cannot only look at the energy to move an axis of the machine (for example in a precision machine tool) but we’d need to consider also the energy associated with the earlier material processing and conversion, any subcomponents or subsystems, etc. Also, some measure of global warming gas generation and any other environmental impact effects must be included.
We are, essentially, estimating the footprint of this device we are designing. This makes the analysis rather complex and, unfortunately, not as analytical as the construction of the conventional error budget first described.
But, it makes sense. And, next time, we’ll add details and apply this to an example.
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