The issue of units, along with coordinate systems, is a common problem in defining and using loads. English units are commonly used on the Space Shuttle. The standard consistent set of English units is pounds force (lbf) for force, slugs for mass, and feet for displacement. The complexity in the units arises because pounds mass (lbm) and inches are generally used on the Space Shuttle and its payloads rather than slugs and feet. This section covers some of the ramifications of these choices for units.
Sir Isaac Newton's famous equation states that:
F = M * a
However, the equation assumes that consistent units are used. The equation should actually read:
F = k * M * a
where k is a proportionality constant to deal with units.
Some dimensional analysis can demonstrate the difficulty with using the non-consistent units of lbf, lbm, and inches. The units and value of k for three cases are shown below.
Using consistent English units in Newton's equation means that 1.0 lbf will accelerate 1.0 slug at 1.0 ft/sec2. Therefore:
1.0 lbf = k * 1.0 slug * 1.0 ft/sec2
k = 1.0 lbf-sec2/(slug-ft)
Since k has a value of 1.0, the term k can be ignored which is one purpose of consistent units.
Now, engineers like to have the case where 1.0 lbf accelerates a 1.0 unit mass at 1.0 gravity (G). 1.0 G is set to the standard value for the acceleration of gravity of 32.2 ft/sec2. Using units of lbf, lbm, and feet in Newton's equation produces.
1.0 lbf = k * 1.0 lbm * 32.2 ft/sec2
k = 1.0/32.2 lbf-sec2/(lbm-ft)
The proportionality constant is now 1.0/32.2 or 0.03106.
Finally, using inches rather than feet produces:
1.0 lbf = k * 1.0 lbm * 32.2 ft/sec2 * 12 in/ft
k = 1.0/386.4 lbf-sec2/(lbm-in)
The proportionality constant in this case, which is the typical case for the Space Shuttle and its payloads, is now 1.0/386.4 or 0.002588. Users of NASTRAN will recognize the value 0.002588 because this value is entered as the parameter WTMASS. The WTMASS parameter is the proportionality constant which is used by NASTRAN to allow for the fact that lbm, lbf, and inches are being used instead of the consistent units of slugs, lbf, and feet.
To a further aid in understanding the situation, it is also useful to understand the mass units. Ignoring the factor k for the moment, two of Newton's equations shown earlier give:
1.0 lbf = 1.0 slug * 1.0 ft/sec2
1.0 lbf = 1.0 lbm * 32.2 ft/sec2
Rearranging the equation containing slugs gives:
1.0 slug = 1.0 (lbf-sec2/ft)
Rearranging the equation containing lbm gives:
1.0 lbm = 1.0/32.2 (lbf-sec2/ft)
32.2 lbm = 1.0 (lbf-sec2/ft)
It is clear that 1.0 slug = 32.2 lbm.
The basic problem in the English units being used on the Space Shuttle and its payloads is that units of pound mass instead of slugs are being used for mass. The use of inches instead of feet is an additional difficulty. However, before users of metric units start to laugh at this problem in using English units, it must be pointed out that they can also fall into a similar trap. Units of kilogram-force (kgf), rather than Newtons, have appeared in documents. The convenience of having one mass unit accelerated at one G by one force unit is often irresistible no matter what system of units is used.
(This page is taken from The Payload Loads Design Guide which is sponsored by the Lyndon B. Johnson Space Center (JSC) located in Houston, Texas, USA. It is meant to provide helpful structural loads and dynamics information to the developers of payloads that fly on the Space Shuttle.)