10' bz FIGURE 1-10. A trapezoid.
14" The Trapezoid FIGURE 1-11. Computing the area of a trapezoid.
A trapezoid (figure 1-10) is a quadrilateral having one pair of parallel sides. The area of a trapezoid is determined by using the formula:
1 = = (02 + 62) 12 where A is the area; ¿ is the given constant;
b and b are the lengths of the two parallel sides;
and h is the height.
EXAMPLE What is the area of a trapezoid whose bases are 14 inches and 10 inches, and whose altitude is 6 inches? (See fig. 1-11.)
FIRST: Substitute the known values in the formula.
#= (01+142) 2 1 = =(10 + 11)6.
NEXT: Solve the formula for the unknown value.
A = 2(21)% edge to trailing edge. If the wing is a tapered wing, the average width or chord, known as the mean chord, must be known in finding the area.
The formula for calculating wing area is:
A = SC where A is the area expressed in square feet, S is the wing span, and C is the average chord.
The process used in calculating wing area will depend upon the shape of the wing. In some instances it will be necessary to use the formula for finding wing area in conjunction with one of the formulas for the area of a quadrilateral or a
EXAMPLES
1. Find the area of the wing illustrated in To determine the area, it is necessary to decide what formula to use. It can be seen that the wing tips would form a 7-foot-diameter circle; the remainder of the wing planform is then in the shape of a rectangle. By combining the formulas for wing area and area of a circle, the area of a wing having circular tips can be calculated.
FIRST: Substitute the known value in the formula.
1 1 = 2X111 1 = 72 sq. in.
Wing Area To deseribe the planform of a wing (figure 1-12), several terms are required. Lo calculate wing area, it will be necessary to consider the meaning of the terms-span and chord. The wing span is the length of the wing from wing tip to wing tip.
The chord is the width of the wing from leading A = SC + TRE A = (25 - 7) (7) + (3.1416) (3.52).
The value for S is represented by the original wing span less the diameter of the circular tips.
NEXT: Solve the formula for the unknown value.
1 = (18 × 7) + (3.1416 × 12.25) 1 = 126 + 38.5 A = 161.5 sq. ft.
A = Wing Area, ft.* C = Average chord, ft.
S = Span, ft.
S - FIGURE 1-12. Wing planform.
2. Find the area of a tapered wing (figure 1-14) COMPUTING THE VOLUME OF SOLIDS whose structural span is 50 feet and whose mean Solids are objects with three dimensions-.
chord is 6'8". length, breadth, and thickness. They are of many FIRST: Substitute the known values in the shapes, the most common of which are prisms, formula. cylinders, pyramids, cones, and spheres. Occasionally, it is necessary to determine the volume A = SC of a rectangle, a cube, a cylinder, or a sphere.
A = 50' X 6'8". Since all volu les are not measured in the same NEXT: Solve the formula for the unknown value.
A = 50' × 6.67' A = 333.5 sq. ft.
3. Find the area of a trapezoidal wing (shown in figure 1-15) whose leading edge span measures 30 feet, whose trailing edge span measures 34 feet, and whose chord is 5 feet.
units, it is necessary to know all the common units of volume and how they are related to each other. For example, the mechanic may know the volume of a tank in cubic feet or cubic inches, but when the tank is full of gasoline, he will be interested in how many gallons it contains. The following table shows the relationship between some of the common units of volume.
UNITS OF SPACE MEASURE FIRST: Substitute the known values in the 1,728 cu. in. = 1 cu. ft.
formula. 27 cu. ft. = 1 cu. yd.
A = - (61 + b2) h 231 cu. in. = 1 gal.
7.5 gals. = 1 cu. ft.
1 = = (30 + 34)5. 2 pts. = 1 qt.
4 qts. = 1 gal.
NEXT: Solve the formula for the unknown value. Volume of a Rectangular Solid 1 A rectangular solid is a solid bounded by rec- 4 = 2(04)S tangles. In other words, it is a square-cornered 4 = 2(320) 4 = 160 sq. ft. 25' 6'8" 25' FIGURE 1-13. Wing with circular tips. FIGURE 1-14. Tapered wing with sweepback.
30 feet 5 feet 34 feet FIGURE 1-15. Trapezoid wing.
volume such as a box (figure 1-16). If the solid has equal dimensions, it is called a cube.
The formula for determining the volume of a rectangular solid may be expressed thus:
V = Iwh where: V = Volume.
1 = length.
w - width.
h = height.
EXAMPLE A rectangular-shaped baggage compartment measures 5 feet 6 inches in length, 3 feet 4 inches in width, and 2 feet 3 inches in height. How many cubic feet of baggage will it hold?
FIRST: Substitute the known values into the formula.
V = Loh V = 5'6" X 3'4" X 2'3".
NEXT: Solve the formula for the unknown value.
- S FIGURE 1-17. A cube.
If the rectangular solid is in the shape of a cube (figure 1-17), the formula can be expressed as the cube of the sides:
where Vis the volume and S is the side measurement of the cube.
Area and Volume of a Cylinder A solid having the shape of a can, length of pipe, or other such object is called a cylinder.
The ends of a cylinder are identical circles as shown in figure 1-18.
Surface Area The surface area of a cylinder is found by multiplying the circumference of the base by the altitude. The formula may be expressed as:
1 = Tah where A is the area, » is the given constant, d is the diameter, and h is the height of the cylinder.
V = A 105 = 41.25 cu. ft.
FIGURE 1-16. A rectangular solid. FIGURE 1-18. A cylinder.
Piston h = Stroke
- Cylinder
PISTON AT TOP CENTER PISTON AT BOTTOM CENTER FIGURE 1-19. Cylinder displacement.
EXAMPLE FIRST: Substitute the known values in the formula.
How many square feet of aluminum sheet would A - ndle be needed to fabricate a cylinder 12 feet long and 3 feet 6 inches in diameter? A = 3.1416 X 3'6" × 12'.
NEXT: Solve the formula for the unknown value.
A = 3.1416 X 3.5' X 12' A = 132.95, or 133 sq. ft.
Volume The volume of a cylinder may be found by multiplying the cross-sectional area by the height of the cylinder. The formula may be expressed as:
V = 7Th where V is the volume; » is the given constant;
r? is the square of the radius of the cylinder; and h is the height of the cylinder (figure 1-19).
EXAMPLE The cylinder of an aircraft engine has a bore (inside diameter) of 5.5 inches, and the engine has a stroke of 5.5 inches. What is the piston displacement of one cylinder? The stroke represents the height of the cylinder to be measured, cecause the volume displaced by the piston depends on the length of the stroke.
FIRST: Substitute the known values in the formula.
V = тґ°h V = (3.1416) (2.752) (5.5).
NEXT: Solve the formula for the unknown value.
V = 17.28 × 7.56 V = 130.64 cu. in.
GRAPHS AND CHARTS Graphs and charts are pictorial presentations of data, equations, and formulas. Through their use the relationship between two or more quantities may be more clearly understood. Also, a person can see certain conditions or relationships at a glance, while it would require considerable time to obtain the same information from a written description. Graphs may be used in a number of ways, such as representing a single equation or formula, or they may be used to solve two equations for a common value.
Graphs and charts take many forms. A few of the more common forms are called bar graphs, pictographs, broken-line curved-line graphs, and circle graphs. An example of each is shown in figure 1-20. The most useful of these graphs in technical work is the continuouscurved-line graph.
Interpreting or Reading Graphs and Charts It is more important, from the mechanic's viewpoint, to be able to read a graph properly than it is to draw one. The relationship between the horsepower of a certain engine at sea level and at any altitude up to 10,000 feet can be determined by use of the chart in figure 1-21. To use this type of chart, simply find the point on the horizontal axis that represents the desired altitude;
move upward along this line to the point where it intersects the curved line; then move to the left, reading the percent of sea level horsepower available on the vertical axis.
EXAMPLE What percent of the sea level horsepower is available at an altitude of 5,000 feet?
FIRST: Locate the point on the horizontal axis that represents 5,000 feet. Move upward to the point where the line intersects the curved line.
NEXT: Move to the left, reading the percent of sea level horsepower available at 5,000 feet. The available horsepower is 80%.
Nomograms It is often necessary to make calculations using the same formula, but using different sets of values for the variables. It is possible to obtain a solution by use of a slide rule or by preparing a table giving the solution of the formula resulting from successive changes of each variable. However, in the case of formulas involving several mathematical operations, the labor entailed would usually be very great.
It is possible to avoid all this labor by using a diagram representing the formula, in which each variable is represented by one or more graduated lines. From this diagram, the solution of the formula for any given variable may be read by means of an index line. A diagram of this type is known as a nomogram.
Much of the information needed to solve aeronautical problems will be presented in nomogram form. Instruction manuals for the various aircraft contain numerous nomograms, many of which 23
1955 ·→ 1960 +→→ 1965
(a) Bar graph (b) Pictograph
(c) Broken line graph (d) Continuous curved-line graph
(e) Circle graph FIGURE 1-20. Types of graphs.
SL 1 2 3 4 5 7 9 10 Altitude × 1000 ft:
FIGURE 1-21. Horsepower vs. altitude chart.
appear quite complex. Many of the presentations will possess several curves on the same coordinate axis, each curve drawn for different constants in the equation. In the latter case, it is essential to select the proper curve for the desired conditions.
Again, as with the simpler graphs, it is more important for the mechanic to be able to read nomograms than it is to draw them.
The following example is taken from the maintenance manual for the Allison 501-D13 turboprop engine. A nomogram (figure 1-22) is used to determine the power requirements when the engine is operating at minimum torque. The OAT (outside air temperature), station barometric pressure, and engine r.p.m. are three factors that must be known to use this particular nomogram.
EXAMPLE Determine the calculated horsepower of a certain engine, using the nomogram in figure 1-22.
Assume that the OAT is 10° C., the barometric pressure is 28.5 in. Hg, and the engine is operating at 10,000 r.p.m.
FIRST: Locate the reference points on the OAT scale and on the barometric pressure scale that correspond to the given temperature and pressure readings. These are identified as I and 2, respectively, on the chart. With the aid of a straightedge, connect these two points and establish point 3 on the pivot line.
NEXT: Locate the engine speed, identified as 1, on the engine speed r.p.m.
scale. Using a straightedge, connect points ® and Q and establish point $ on the calculated horsepower scale. The calculated horsepower is read at point 5. The calculated horsepower is 98%.
MODEL 501-D13 ENGINE AND MODEL 606 PROPELLER
CALCULATED TORQUEMETER
OUTSIDE AIR (READING) HORSEPOWER
•C TEMP •F PIVOT BAROMETRIC ENGINE SPEED RPM LINE PRESSURE - IN. HG
- 140 - 10500
32.0 -20 - 130 31.0 10400 NOTE: VALID FOR WINDS - 120 • 30.0 10 BELOW 11 M. P. H. 10300 Ễ110 29.0
- 10200 20 - 28.0 100 30 - 27.0 - 10100
• 26.0 · 10000 90 10 ~ 50 25.0 15-80 — 80 - 24.0 111 9800 20-70 23.0 25 - - 70
- 80
22.0 30 -
- 90 EXAMPLE 35 - - 21.0
- 100 - 80 40 -
- 110 - 55 - 20.0 FIGURE 1-22. Power requirements at minimum torque.
MEASUREMENT SYSTEMS Our customary system of measurement (figure 1-23) is part of our cultural heritage from the days when the thirteen colonies were under British rule.
It started as a hodge-podge of Anglo-Saxon, Roman, and Norman-French weights and measures.
medieval times, commissions appointed by various English monarchs had reduced the chaos of measurement by setting specific standards for some of the most important units. Early records, for instance, indicate that an inch was defined as the length of "three barleycorns, round and dry" when laid together; a pennyweight, or one-twentieth of a Tower ounce, was equal to 32 wheatcorns from the "midst of the ear."
about 20 percent smaller than the Imperial gallon that the British adopted in 1824 and have since used to measure most liquids).
In short, as some of the founders of this country realized, the customary system was a makeshift based largely on folkways.
Metric System The metric system is the dominant language of measurement in use today. Most of the world countries used the metric system prior to World War II. Since the war, more countries have converted or are in the process of converting to the metric system. Only the United States and 13 smaller countries have not made the conversion.
Congress has the power to define the standard of weights and measures. Repeatedly the metric system has been proposed and each time the question has been voted down.
The metric system was developed by a French statesman, Talleyrand, Bishop of Autum, using a "meter" as a standard; the meter being a specific portion of the circumference of the earth at the From this base measurement the meter was developed and accepted as the standard. Divisions and multiples of the meter are based on the decimal system.
ELECTRIC
LENGTH MASS VOLUME TEMPERATURE CURRENT TIME METRIC Celsius Meter Kilogram Liter (Centigrade) Ampere Second
CUSTOMARY inch fluid ounce Fahrenheit ampere second foot teaspoon year tablespoon fathom cup rod pint mile quart gallon barrell peck bushel FIGURE 1-23. Some common units.
The Logic of Metric No other system of measurement that has been actually used can match the inherent simplicity of International Metric. It was designed deliberately to fill all the needs of scientists and engineers.
Laymen need only know and use a few simple parts of it. It is logically streamlined, whereas other systems developed more or less haphazardly.
At this time there are only six base units in the International Metric System. The unit of length is the meter. The unit of mass is the kilogram.
The unit of time is the second. The unit of electric the unit, whatever it may be. For example, kilo stands for 1,000: one kilometer equals 1,000 meters, and one kilogram equals 1,000 grams. Micro is the prefix for one millionth: one meter equals one million micrometers, and one gram equals one million micrograms (figure 1-24).
current is the ampere. The unit of temperature is the kelvin (which in common use is translated into giga (10°) One billion times the degree Celsius, formerly called degree centi- mega (10°) One million times grade). The unit of luminous intensity is the candela. kilo (103) One thousand times hecto (102) One hundred times All the other units of measurement in the Inter- deca (10) Ten times national Metric System are derived from these six base units. Area is measured in square meters; deci (10-') One tenth of speed in meters per second; density in kilograms per cubic meter. The newton, the unit of force, centi (10-3) One hundredth of is a simple relationship involving meters, kilograms, milli (10-3) One thousandth of and seconds; and the pascal, unit of pressure, is micro (10-*) One millionth of defined as one newton per square meter. In some other cases, the relationship between the derived nano (10-*) One billionth of and base units must be expressed by rather more pico (10- ) One trillionth of complicated formulas-which is inevitable in any measurement system, owing to the innate complexity of some of the things we measure. Similar relationships among mass, area, time and other quantities in the customary system usually require similar formulas, made all the more complicated because they can contain arbitrary constants. For example, one horsepower is defined as 550 footpounds per second.
The third intrinsic advantage is that metric is based on the decimal system. Multiples and subnultiples of any given unit are always related by powers of 10. For instance, there are 10 millimeters in one centimeter; 100 centimeters in one meter; and 1,000 meters in one kilometer. This greatly simplifies converting larger to smaller measurements. For example, in order to calculate the number of meters in 3.794 kilometers, multiply by 1,000 (move the decimal point three places to the right) and the answer is 3,794. For comparison, in order to find the number of inches in 3.794 FIGUrE 1-21. Names and symbols for metric prefixes.
Conversion: Metric To Conventional People tend to resist changes, usually because they do not understand cither the purpose of the change or the new order. Terminology for customary units and metric units have been discussed. A conversion table also has been included. Examples of its use follow:
To convert inches to millimeters, multiply the number of inches by 25. (Ex. 25 into mm =25 X25=625 mm) To convert millimeters to inches multiply millimeters by .01. (Ex. 625 mmX.04=25 inches.)
miles, it is necessary to multiply first by 5,280 and To convert square inches to square centimeters then by 12. multiply by 6.5. (Ex. 100 sq. in.X6.5=650 sq. cm.)
Moreover, multiples and submultiples of all the International Metric units follow a consistent nam- To convert square centimeters to square inches ing scheme, which consists of attaching a prefix to multiply by .16. (Ex. 100X.16=16 sq. in.)
IF YOU WHEN YOU KNOW: YOU CAN FIND: MULTIPLY BY:
LENGTH inches millimeters 25 feet centimeters 30 yards meters 0.9 miles kilometers 1.6 millimeters inches 0.04 centimeters inches 0.4 meters yards 1.1 kilometer miles 0.6 AREA square inches square centimeters 6.5 square feet square meters 0.09 square yards square meters 0.8 square miles square kilometers 2.6 acres square hectometers (hectares) 0.4 square centimeters square inches 0.16 square meters square yards 1.2 square kilometers square miles 0.4 square hectometers (hectares) acres 2.5 MASS ounces grams 28 pounds kilograms 0.45 short tons megagrams (metric tons) 0.9 grams ounces 0.035 kilograms pounds 2.2 megagrams (metric tons) short tons 1.1 LIQUID ounces milliliters 30 VOLUME pints liters 0.47 quarts liters 0.95 gallons liters 3.8 milliliters ounces 0.034 liters pints 2.1 liters quarts 1.06 liters gallons 0.26 TEMPER- degrees Fahrenheit degrees Celsius 5/9 ( after subtracting 32) ATURE degrees Celsius degrees Fahrenheit 9/5 (then add 32) FIGURE 1-25. Converting customary to metric.
Figure 1-26 is practically self explanatory. Meas- been converted to decimal divisions of inches and urements starting at 64 inch up to 20 inches have to millimeters.
Inches Inches Inches Inche Fractions Decimals MM Fractions Decimals MM Fractions Decimals M M F'ractions Decimals | M M 0004 .01 25/32 .781 19.844 2.165 55. 3-11/16 3.6875 93,663 004 7874 20. 2-3/16 2.1875 55.563 3.7008 54.
51/64 .797 20.241 2.2047 56. 3-23/32 3.719 94.456 1/64 0156 .397 3/16 8125 20.638 2-7/32 2:219 56.356 3.7401 95.
0197 .0295 -50 .75 53/64 .8268 828 21.034 21. 2-1/4 2.244 57. 57.150 3-3/4 3.750 3.7795 05.250 96.
1/32 03125 794 27/32 844 21.431 2-9/32 2.281 57.944 3-25/32 96.044 3/64 .0394 0469 55/64 859 8661 21.828 22. 2-5/16 2,2835 2012 58. 58.738 3-13/16 3.8125 3.8189 96.838 97.
.059 1.5 875 22,225 2.3228 59. 3-27/32 3.844 97,631 1/16 .062 1.588 8906 22.622 2-11/32 2.344 59.531 3.8583 5/64 .0781 .0787 1.981 2. 29/32 9055 9062 23. 23.019 2.3ł8 2.5622 60. 60.325 3-7/8 3.875 0.8976 08.425 3/32 094 2.381 59/64 .922 23.416 2.4016 61. 3-29/32 3.9062 99.219 7/64 .0984 2.778 15/16 .0449 9375 24. 23.813 2-13/32 2-7/16 2.406 2.438 61.119 61.913 3-15/16 5.9370 3.9375 100. 100.013
24.606 35005 24.209 2-15/165 2.4803 2.469 2,4409 53 5m 62. 1-1/16 3-31/32 4.000 3.969 3.9764 100.806 101. 101.600- 2-1/2 2.500 4.002 103.188
1.000 25.400 2.5197 64. 4-1/8 4.125 104.775
1.0236 -17 2.531 64.294 4.1338 105.
1.062 L.0312 26.988 194 2-9/16 2.559 2.562 65. 65.088 1-3/16 4-1/4 4.250 4.1875 106.363 107.950
27. 2-19/32 65,881 1-5/16 4.312 109.538
27.781 4.3507 110.
28. 2-5/8 66.675 1-3/8 4.375 111.125
28.575 1-7/16 4.438 112.713 2-21/32 655.450 1-1/2 4.500 114.300
1.156 29.369 1.5275 115.
2-11/16 4-9/16 1.562 115,888
1.1675 30.163 4-5/8 4.625 117.475 30,956 2-23/32 4-11/16 4.6875 119.063/
31. 2-3/4 1.7244 120.
31.750 70. 1-3/4 4.750 120.650
32. 2-25/32 70,6439 4-13/16 4.8125 122.238
32.544 71. 1-7/8 4.875 123.825
33. 33.338 2-13/16 71.4376 4-15/16 4.9212 125 125.413
34. 2-27/32 5.000 127.000 17326 2-7/8 5.1181 2-29/32 5.500
5.518 4173 2-15/16 2.0375 5.750 438 36.513 TS. 5.9055 2-31/32 2.969 6.000 469 37.306 2.9921 6.250 .4961 3.000 6.2092 S-1/32 2.0312 6.500
3.0315 77. 6.6929 I.5354 3-1/16 3.062 6.750
1.562 39.688 3.0709 78. 7.000
1.5748 3-5/32 3.094 7.0866
1.594 3.1102 79. 7.4803
1.6142 3.125 7.500
1.625 3.1496 80. T.8740
1.6535 3-5/32 3.156 8.000
1.6562 42.069 3-3/16 3.1875 8.2677
1.6875 42.863 3.1890 81. 8.500 5118 13. 1.6929 43. 3-7/32 3.219 8.6614 33/64 5156 13.097 1-23/32 1.719 43.656 3.2283 82. 9.000 35/64 17/35 531 .547 13.891 10.494 1-5/4 1.7323 1.750 44. 44.450 :3-1/4 3.2677 3.250 83. 0.0551 .5512 14. 1.25/32 1.7717 45. 3-9/32 3.281 9.500 9/16 14.288 1.781 45.244 3.3071 84. 9.8425
14.5 1.8110 3-5/16 3.312 10.000 37/64 .578 14.684 1-15/16 1.8125 46.038 3-11/32 3.344 10.2362 .5906 15. 1-27/32 1.844 46.831 3.3464 85.
19/32 15.081 1.8504 47. 3-3/8 3.375 85.725 11 11.000 39/64 10.910 1-7/8 1.875 47.625 3.3858 86. 11.0236 5/8 15,875 1.8898 48. 3-13/32 3.406 86.519 11.4173 41/64 16. 16.272 1-29/32 1.9062 1.929I 48.419 49. 3-7/16 3.4252 3.438 87. 87.313 † 12 12.000 21/32 .6496 .056 16.669 16.5 1-15/16 1.9685 1.9375 49.213 50. 3-13/32 3.469 3.4646 88. 85.106 1 1.3 13.000 13.7795 .6693 1-31/32 1.969 50.006 3-1/2 3.500 88.000 14 11.000 .672 17.066 2.000 50,800 3.5039 89.
11/16 .6875 17.463 2.0079 51. 3-17/32 3.531 89.694 15.7480 45/64 .703 17.859 2:1/32 2.03125 51.594 3.5433 90. 16 16.000 23/32 .7087 .719 18.256 2-1/16 2.0179 2.062 52. 52.388 3-9/16 3,562 3.5827 91. 90.4877 17 17.000 17.7165 .7283 18.5 2.0868 53. 3-19/32 3.594 91.281 18 18.000 47/64 .734 18.653 2-3/32 2.094 53.181 3.632 92. 19 .7480 19. 0-1/8 2.125 53.975 3-5/8 3.625 02.075 19.6350 49/64 3/4 .750 .7656 19.050 19.447 2-5/32 2.156 2.12€ 54. 51.769 3-21/32 3.656 5.6614 93. 92.869 20 20.000 FIcURE 1-26. Fractions, decimals, and millimeters.