Machinery's Handbook, 31st Edition
Balancing Rotating Parts 199 Such dynamically unbalanced conditions can occur even though static balance (bal- ance at zero speed) exists. Dynamic balance can be achieved by the addition of one or two masses rotating about the same axis and at the same speed as the unbalanced masses. A single unbalanced mass can be balanced by one counterbalancing mass lo- cated 180 degrees opposite and in the same plane of rotation as the unbalanced mass if the product of their respective radii and masses are equal; i.e., M 1 r 1 = M 2 r 2 . Two or more unbalanced masses rotating in the same plane can be balanced by a single mass rotating in the same plane, or by two masses rotating about the same axis in two separate planes. Likewise, two or more unbalanced masses rotating in different planes about a common axis can be balanced by two masses rotating about the same axis in separate planes. When the unbalanced masses are in separate planes they may be in static balance but not in dynamic balance; i.e., they may be balanced when not rotating but unbalanced when rotating. If a system is in dynamic balance, it will remain in balance at all speeds, although this is not strictly true at the critical speed of the system. (See Critical Speeds on page 204 .) In all the equations that follow, the symbol M denotes either mass in kilograms or in slugs, or weight in pounds. Either mass or weight units may be used; the equations may be used with metric or with customary English units without change; however, in a given problem the units must be all metric or all customary English. Counterbalancing Several Masses Located in a Single Plane.— In all balancing prob lems, the product of the counterbalancing mass (or weight) and its radius are calculated; it is thus necessary to select either the mass or the radius and then calculate the other value from the product of the two quantities. Design considerations usually make this decision self-evident. The angular position of the counterbalancing mass must also be calculated. Referring to Fig. 4:
cos
sin
2
2
M r
Mr
Mr
(1) (2)
+ h ^
h
^
=
R
i
R
i
B B
cos sin
x y
Mr Mr
h
^
− −
R R
i i
tan
=
=
B i
h
^
M 2
r 2
2
r 1
M 1
1
r B
r 3 3
B
M 3
M
B
Fig. 4.
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