**5.7 Units**

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 (lb_{f})
for force, slugs for mass, and feet for displacement. The
complexity in the units arises because pounds mass (lb_{m})
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 lb_{f},
lb_{m}, 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 lb_{f} will accelerate 1.0 slug
at 1.0 ft/sec^{2}. Therefore:

1.0 lb_{f} = k * 1.0
slug * 1.0 ft/sec^{2}

and:

k = 1.0 lb_{f}-sec^{2}/(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
lb_{f} 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/sec^{2}. Using units of lb_{f},
lb_{m}, and feet in Newton's equation produces.

1.0 lbf = k * 1.0 lb_{m}
* 32.2 ft/sec^{2}

and:

k = 1.0/32.2 lb_{f}-sec^{2}/(lb_{m}-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 lb_{m}
* 32.2 ft/sec^{2} * 12 in/ft

and:

k = 1.0/386.4 lb_{f}-sec^{2}/(lb_{m}-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 lb_{m},
lb_{f}, and inches are being used instead of the
consistent units of slugs, lb_{f}, 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 lb_{f} = 1.0 slug *
1.0 ft/sec^{2}

and

1.0 lb_{f} = 1.0 lb_{m}
* 32.2 ft/sec^{2}

Rearranging the equation containing slugs gives:

1.0 slug = 1.0 (lb_{f}-sec^{2}/ft)

Rearranging the equation containing lb_{m}
gives:

1.0 lb_{m} = 1.0/32.2
(lb_{f}-sec^{2}/ft)

or

32.2 lb_{m} = 1.0 (lb_{f}-sec^{2}/ft)

It is clear that 1.0 slug = 32.2 lb_{m}.

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 (kg_{f}),
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.)