Balancing Rotating Parts Machinery's Handbook, 31st Edition
201
a 2
a 1
W
M A
M A
M 2
A r A
M 1
r 1
M 2
r 2
a 3
M 1
2
1
Axis of Rotation
b 1
b 2
B
3
r B
b 3
r 3
M B
M B
M 3
M 3
c
Fig. 5. Counterbalancing Masses Located in Two or More Planes.— Unbalanced masses or weights rotating about a common axis in two separate planes of rotation form a couple, which must be counterbalanced by masses or weights, also located in two separate planes, say, planes A and B , and rotating about a common axis (see Couples on page 162 ). In ad- dition, they must be balanced in the direction perpendicular to the axis, as before. Since two counterbalancing masses are required, two separate equations are required to cal- culate the product of each mass or weight and its radius, and two additional equations are required to calculate the angular positions. The planes A and B selected as balancing planes may be any two planes separated by any convenient distance c , along the axis of rotation. In Fig. 5: For balancing plane A :
cos
sin
2
2
Mrb
Mrb
+ h ^
h
^
R
i
R
i
(3) (4)
M r
=
c
A A
cos sin
x y
Mrb Mrb
h
^
− −
R R
i i
tan
=
=
A i
h
^
For balancing plane B :
cos
sin
2
2
Mra
Mra
+ h ^
h
^
R
i
R
i
(5) (6)
M r
=
c
B B
cos sin
x y
Mra Mra
h
^
− −
R R
i i
tan
=
=
B i
h
^
where: M A and M B are the mass or weight of the counterbalancing masses in the balancing planes A and B , respectively; r A and r B are the radii; and θ A and θ B are the angular positions of the balancing masses in these planes. M , r , and θ are the mass or weight, radius, and angular positions of the unbalanced masses, with the subscripts defining the particular mass to which the values are assigned. The distance, c , between the balancing planes is always a positive value. The axial dimensions, a and b , may be either positive or negative, depending upon their position relative to the balancing plane; for example, in Fig. 5, the dimension b 2 would be negative. Example: Referring to Fig. 5, a set of values for the masses and dimensions has been selected and put into convenient table form below. The separation of balancing planes, c , is chosen as 15 inches. If in balancing plane A , the radius of the counterbalancing weight
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