Balancing Rotating Parts Machinery's Handbook, 31st Edition
200
where: M 1 , M 2 , M 3 , . . . , M n = any unbalanced mass or weight, kg or lb M B = counterbalancing mass or weight, kg or lb
r = radius to center of gravity of any unbalanced mass or weight, mm or inch r B = radius to center of gravity of counterbalancing mass or weight, mm or inch θ = angular position of r of any unbalanced mass or weight, degrees θ B = angular position of r B of counterbalancing mass or weight, degrees x and y = position with sign according to Table 1 Table 1 is helpful in finding the angular position of the counterbalancing mass or weight. It indicates the range of the angles within which this angular position occurs by noting the plus and minus signs of the numerator and the denominator of the terms in Equation (2). In a like manner, Table 1 is useful in determining the sign of the sine or cosine functions for angles ranging from 0 to 360 degrees. Balancing problems are usually solved most conve niently by arranging the arithmetical calculations in a tabular form. Table 1. Relationship of Angle Function Signs to Quadrant in Which They Occur Angle θ 0 ° to 90 ° 90 ° to 180 ° 180 ° to 270 ° 270 ° to 360 ° Signs of the Functions tan II I + r
– x + r + r
+ y – y
+ x + y – y
x y r y r x
x y r y r x
x y r y r x
x y r y r x
+ + + + + +
− + + + + −
− − + − + −
+ − + − + +
IV + r
sine
III
cosine
Example: Referring to Fig. 4, particular values of unbalanced weights M 1 , M 2 , M 3 are entered in the table below. Calculate the magnitude of the counterbalancing weight M B if its radius r is to be 10 inches. Solution: The table is arranged with the given masses M , their radii r , and their an- gular positions θ . The angle functions sin theta and cos theta are determined from the angles using a calculator or trig table. Finally, the quantities are used to fill in the last two columns. M r in. θ deg. cos θ sin θ Mr cos θ Mr sin θ No. lb. 1 10 10 30 0.8660 0.5000 86.6 50.0 2 5 20 120 − 0.5000 0.8660 − 50.0 86.6 3 15 15 200 − 0.9397 − 0.3420 − 211.4 − 77.0 − 174.8 = ∑ Mr cos θ 59.6 = ∑ Mr sin θ . . . . . ; cos sin tan cos sin M r Mr Mr M lbs Mr Mr x y 10 1748 596 2 + ^ h 185 1748 596 h 341 10 B B B B B 2 2 2 ° ′ i i i i i i R R R R = = = + = − − − = − − − ^ ^ = + − = ^ ^ ^ ^ ^ h h h h h h
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