TYPICAL EXAMPLES OF ALIGNMENT CHARTS.
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By Raoul J. Hoffman
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Reprinted from the October 1942, issue of Flying Magazine
t is often necessary, for estimating
Iof calculations purposes, to make a large number with the same formula.
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A slide rule is very convenient for ready use if carefully operated. However, to minimize the chance of error, usually by a misplaced decimal point, graphical computations with the aid of alignment charts - or "nomograms" - are great timesavers.
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Nomograms are generally used when the degree of accuracy of the results is not expected to be too close. This is the case in preliminary designs. Also the size of the nomogram has a great deal to do with its limitations.
There are nomograms of different construction principles. In the following, however, only the construction of alignment charts having parallel scales will be shown. Nomograms for formulas containing two variables are constructed by laying out the values of one variable on one side of the line and the values of the other variable on the other side. A single-line nomogram is shown in Fig. 1 for finding the area SPORT AVIATION
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of a circle if the diameter is given (or vice versa). Any convenient scale can be used. This simplest form of graphical
chart is frequently used for conversion of a system of units of the same
character to units of another system. For instance, the temperature measured in centigrades to that of the
Fahrenheit scale.
Simple additions may be done by the use of two scales having inch
divisions.
Connect the zero point
with the first figure and read at the second figure the sum on the lower
scale as sketched in the bottom layout in Fig. 1. This process is basic for slide rule operations.
Most formulas generally contain
three or four variables. The diagrams shown in Fig. 2 give the work-
ings of many nomograms.
Three
equidistant parallel lines - A, B and C - are drawn. A line is drawn
5L on paper and place a center point at a five-unit distance; for example, five inches. Then draw a parallel line at the distance the size of the scale is wanted. If the scale required is 2.5L the distance will be 2.5 from
out for a nomogram conforming to the equation Y = X2Z or X2 = Y/Z is shown. Three parallel equidistant lines are drawn. Then the directions toward increasing values indicated by arrows are found by visual checking
the center point. Connect all divisions on the scale 5L with the center point. Then their intersections on
with the equation. Then the scale lengths are selected. If the scale for
scale. A single-line nomogram is shown in Fig. 1.
line has been selected for the Z scale, therefore the scale should be 2L/2 or IL. All lines will use IL scale or any other scale of the same length.
the parallel line will be the 2.5L
One does not have to understand the derivation of logarithmic values in order to use a slide rule or to construct nomograms with log-scales. The characteristics of logarithms
of figures is that the sum of two logarithms represent the product of the
two figures. The sum of log. 2 and
log. 3 is not log. 5 but log. 6.
All
we have to do now is to take two rules with logarithmic divisions, place them above each other and we have a regular slide rule.
The use of a
Y and Z is IL, then the scale for X2
should be 2L
However, the center
If a constant is in the equation the
layout will be the same except the location will be different. The last scale must be laid out by making a
few samples. The exponents indicate the size of the scale. If the exponent is 4 then the scale should be twice the length of the variable having an exponent 2. The center scale should be half the length indicated. Another nomogram is constructed
for the above equation in Fig. 8 by
Length H is measured on the B line
slide rule is simple. Place mark 1 over one figure and read the product on the second figure as shown in
placing the variable X on the outer line. The scales are marked on the layout.
length or 2H. Now if the C line is
es beyond the lower rule use the mark 1 on the other end of the scale.
In Fig. 9 a nomogram is shown for the equation giving the relative horizontal flying speed if the wing loading and the absolute lift coefficient are known. The equation is squared in order to simplify the selection of the scales. The scales are of the same length. Placing the velocity on the outer line another nomo-
across the three lines from point 0.
and connected with O. This line will intersect the C line at twice the
graduated into double length divisions, then any number of units added to the C scale will add the same number of units on the B scale, as
shown in the lower diagram in Fig. 2.
This simple form of adding or subtracting is applied to the construction of the nomogram in Fig. 3. Three equidistant lines have the same divisions on the outer lines and twice as many graduations on the center line. The length of the scales limits the range of computation. Two examples are indicated by dotted lines. Two plus 4 and 8 minus 4. The results are found on the center scale. One does not have to draw a line across the scales, as placing a straight edge on the nomogram will keep the chart in perfect condition. Logarithmic scales are of greatest aid in constructing nomograms. Five
logarithmic scales are given with their basic lengths noted. The scale
2L is twice the length of IL. In case another length is required draw scale NEW
AMATEUR
Fig. 4.
If the second figure pass-
Two examples are noted in the sketch-
es.
The same setting is used when
dividing 6 by 3 or 12 by 3. There is no zero point on a slide rule and 2,000 or .002 is located on the same
mark 2. One has to be careful in reading between 1 and 2 shown in Fig. 4. The second mark 1 means 11 or 1100.
Substituting the regular scales in
however, will be 2L, IL and 4L, as illustrated in Fig. 10. i
outer scales are laid out with scales
The same construction is used as for the three-scale charts. Two pairs of scales are connected by a common
the nomogram of Fig. 3 with logarithmic scales, the process of addition will convert to multiplication. The two having the same length and the cen-
ter scale with half of the length of the outer scales. The center scale is located by a few trial calculations. The accuracy depends on the size and the exactness of the scales and of the layout. The three-scale multiplication nomogram can be applied to many equations having three variables with
various exponents.
BUILDER'S
MANUAL
In Fig. 6 a lay-
READY
First Volume 1958 of the AMATEUR BUILDER'S MANUAL now available. FREE to members . . . or send $2.50 for an EXTRA copy. (Limited supply of First Volume and Second Volume 1957 MANUALS still available $1.25 per copy). If you have not yet received your 1957 MEMBERSHIP DIRECTORY. . .write for your copy t o d a y . . . F R E E to EAA members . . . non-members send $1.00. 16
gram may be constructed. The scales,
Many equations have four variables.
line called the reference, or turning
line. Usually the direction of connecting scales is from left to right. Therefore, the second and third scale will be half of the outer scales. A nomogram layout for an equation
having four variables is shown in Fig. 11.
For the equation given in Fig. 9 a
four-scale nomogram may be constructed as illustrated in Fig. 12. In
this nomogram the center line gives the wing loading, which is useful in preliminary investigations of performance calculations.
Nomograms having five or more
variables are constructed by using
for each additional variable another reference line as shown in a layout in Fig. 13. £ SEPTEMBER
1958
