## Engineering workshop #1: Units and conversions

10 February 2010

Bob Johnson, Technical Director of DAMT, highlights the advantages of the metric system and provides examples of how one set of units can be converted to any other consistent set without the need for a crib sheet

Welcome to a new series of basic training articles written by Bob Johnson. These brief pieces seek to provide practical down-to-earth guidance on a range of engineering topics. The subject matter will vary from basic engineering principles all the way through to the Finite Element Analysis (FEA) of assemblies.

Why is Bob presenting this material? Briefly, he is one half of the consultancy and training company, DAMT, and specialises in the reliable analysis of complex assemblies such as offshore connectors.

Bob provides the NAFEMS training course on basic FEA and considers that well-thought-out FEA can accurately mimic the real world and add tangible value to the design process.

Outside of engineering he has completed the last nine London marathons (four of which in under three hours) and this year was third Vet50 in the Ben Nevis fell race (2hrs 2mins) and third Vet50 in the 3-Peaks Cyclo-Cross (3hrs 51mins). To top all this, in 2006 he completed the London Marathon whilst dressed as a 6ft (32lb) Dalek!

## Down to business

All engineering calculations need to be carried out in a sensible and consistent set of units. Don’t think that errors with units are a thing of the past: the $125 million Mars Climate Orbiter (launched 1998) was sent 60 miles closer to the Martian surface because thruster data was supplied in pounds-force instead of Newtons! This article seeks to show the advantage of the metric system of units (as used by NASA and not its supplier in this case!) and provides examples of how one set of units can be converted to any other consistent set without the need for a crib sheet.

Any calculation that we care to undertake will need to respect a consistent system of units. Any answer we quote must contain (a) the numerical calculation and (b) the outcome of a study of the units involved. Mistakes in either part will invalidate the overall result but that said mistakes in the units are more embarrassing!

{Fig #1} shows the three main systems of units in use today. On the left, two “absolute” systems and on the right one “gravitational” system of units.

Most engineers agree that an “absolute” system of units is best because the base quantities (in the S.I. system: mass, length and time) are always the same wherever you are. (Note that the base quantities in the Imperial (US) system are length, time and force – the force varies depending on the local gravitational field so would not be applicable on the moon). Even US text books consider that we will all use an absolute system such as S.I. one day!

Concentrating on absolute systems, then Figure #1 shows the standard S.I. system (kg, m, s, N) and the so-called “modified” S.I. system (tonnes, mm, s, N). The “modified” system is in common use today because calculations will render displacements in millimetres and stresses in Newtons-per-square millimetre (easily stated as Mega-Pascals or MPa). The stresses have a sensible magnitude in that, for instance, the yield stress of a piece of steel might be 400MPa in the modified S.I. system (as opposed to 400 million Newtons per square metre in the standard S.I. system).

{Fig. #2} introduces the concept of “Unity Brackets” and these allow us to convert one unit for (say) distance into any other unit for distance.

For example, we can easily make a unity bracket from the relationship that 25.400mm is exactly equal to one inch. We can convert the above “equation” into the quotient form of [25.4mm / 1 inch] or [1 inch / 25.4mm]. In Figure 2 (and here in the text) the quotient is shown within square brackets to remind us that the value is one – thus the name “unity bracket”.

Figure #2 illustrates how unity brackets can be formed from a number of simple expressions. All we are doing is making a quotient such that the top equals the bottom!

{fig.3} demonstrates how these unity brackets can be used to convert one system of units to any other. The first example shows how the 3.528 litre capacity of a Rover V8 engine can be converted to cubic-inches (say for consumption of an interested party in the ‘States!).

In this example three unity brackets are used – note that two of which are cubed. Remember that one to any power is still one so all we’re doing here is multiplying the capacity in litres by one, three times. If we arrange our string of unity brackets such that cancelling can occur then we’re left with the capacity in cubic inches.

Figure #3 shows a further example which uses five unity brackets to convert from a motor torque of 140Nm to the Imperial equivalent of 103 lbsf-ft. Please study the examples – it really is an excellent system and endorsed by the Institute of Mechanical Engineers so it must be good! More next time – don’t have nightmares!

Coming up in Bob’s Engineering Workshop next month: forces, reactions and free-body diagrams.

Bob can be contacted at .(JavaScript must be enabled to view this email address)

### Comments on this article:

hi,

very cool site!

thanks

**Posted by** falkon on Sunday 28 2010 at 10:01 AM

i want 3D human head (skull and brain)geometry file in abaqus format.

**Posted by** mostafa on Tuesday 24 2010 at 04:45 PM

Engineers who are concerned about accuracy with units and unit conversions should be aware that PTC’s Mathcad 15.0 and Mathcad Prime 1.0 perform unit conversions automatically in Mathcad’s live mathematics interface. Here’s a brief illustration (http://blogs.ptc.com/bid/30820/Why-is-Mathcad-so-Keen-on-Units). As a former mathematics teacher, I am sure that performing unit calculations using Mathcad will be easier and more accurate than the Unity Brackets method.

**Posted by** Chris Hartmann on Friday 06 2011 at 09:02 PM

What is the font you used to draw up the unity brackets? It is some of the neatest presentation I have ever seen. What program was it? The guide was very helpful too too. I’m studying thermodynamics at university this term and this helped clear some things up that we’re really brief in the lecture. We have to do it by hand so it’s a handy guide!

**Posted by** Vin on Thursday 16 2012 at 08:20 AM

Great, helpful, practical article. I’ve been using this method since my uni days and it has always worked out and saved me embarrassment.

I did however have an occasion when starting to work with data from a different CAD package to usual, when I was under considerable time pressure and decided to try and bypass the pre-simulation hand-calcs stage. It wasn’t pretty and it wasn’t useful. The only good thing that came out of that simulation was a good story to scare students with!

Here’s to next month’s article being equally useful.

Posted byAlthea on Monday 22 2010 at 04:18 PM