AREA OF IRREGULAR PLANE FIGURE Machinery's Handbook, 31st Edition
73
W 3 -- V ( [
A = Example: The area of the accompanying figure was divided into 8 strips on a full-size drawing and the following data obtained. Calculate the area using Simpson’s rule. 0 V N + ) 4 V 1 V 3 ( + … V N – 1 + + + ) 2 V 2 V 4 ( + … V N – 2 + + + ) ]
W = 1 cm V 0 = 0 cm V 1 = 1.91 cm V 2 = 3.18 cm V 3 = 3.81 cm V 4 = 4.13 cm V 5 = 5.27 cm V 6 = 6.35 cm V 7 = 4.45 cm V 8 = 1.27 cm
W
v 0
v 1 v 2 v 3 v 4 v 5 v 6
v 7
v 8
Substituting the given data in the Simpson’s formula, A 1 3 -- 0+1.27 ( ) 4 1.91 + 3.81 + 5.27 + 4.45 ( ) 2 3.18 + 4.13 + 6.35 ( ) + + [ ] = 1 3 -- 1.27 4 15.44 ( ) 2 13.66 ( ) + + [ ] = 30.12 cm 2 = In applying Simpson’s rule, it should be noted that the larger the number of strips into which the area is divided the more accurate the results obtained. Areas Enclosed by Cycloidal Curves.— The area between a cycloid and the line upon which the generating circle rolls equals three times the area of the generating circle (see diagram, page 79). The areas between epicycloidal and hypocycloidal curves and the “fixed circle” upon which the generating circle is rolled may be determined by the follow ing formulas, in which a = radius of the fixed circle upon which the generating circle rolls, b = radius of the generating circle, A = the area for the epicycloidal curve, and A 1 = the area for the hypocycloidal curve. A b 2 3 a 2 b + ( ) a = ----------------- A 1 b 2 3 a 2 b – ( ) a = ----------------- Contents of Cylindrical Tanks at Different Levels.— In conjunction with the table Seg- ments of Circles for Radius = 1 starting on page 84, the following relations can give a close approximation of the liquid contents, at any level, in a cylindrical tank.
Measuring Stick
L
x
More Than Half-Filled Less Than Half-Filled
r
d
y
r
x
y
Less Than Half-Filled
More Than Half-Filled
A long measuring rule calibrated in length units or a plain stick can be used for measuring contents at a particular level. In turn, the rule or stick can be graduated to serve as a volume gauge for the tank in question. The requirements are that the tank must have a circular cross section; the dimensions of the tank must be known; the gauge rod has to be inserted verti- cally through the top center of the tank so that it rests precisely in the center at the bottom of the tank; and the calculations must be done using consistent metric or US customary (also called English ) units. The formulas and parameters are:
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