Balancing of Several Masses Rotating in the Same Plane

Consider any number of masses (say four) of magnitude m1, m2, m3 and m4 at distances of r1, r2, r3 and r4 from the axis of the rotating shaft. Let ?1, ?2,?3 and ?4
be the angles of these masses with the horizontal line OX, as shown in Fig. 4 (a). Let these masses rotate about an axis through O and perpendicular to the plane of paper, with a constant angular velocity of ? rad/s. The magnitude and position of the balancing mass may be found out analytically or graphically as discussed below
a) Analytical method
The magnitude and direction of the balancing mass may be obtained, analytically,
as discussed below :
Resolve the centrifugal forces horizontally and vertically and find their sums, i.e.
?H = m1 . r1 cos?1 + m2 . r2 cos?2 + . . . . . .
?V = m1 . r1 sin?1 + m2 . r2 sin?2 + . . . . . .
Magnitude of the resultant centrifugal force,
FC = [(?H )2 + (?V )2]^.5
If ? is the angle, which the resultant force makes with the horizontal, then
tan ? = ?V/?H
The balancing force is then equal to the resultant force, but in opposite direction. Now find out the magnitude of the balancing mass,
FC = m. r
where m = Balancing mass, and
r = Its radius of rotation.
b) Graphical method
The magnitude and position of the balancing mass may also be obtained graphically using Table: 1 and then draw:
1. Find the centrifugal force (or product of the mass and radius of rotation)
exerted by each mass on the rotating shaft.
2. Draw the vector diagram with the obtained centrifugal forces to some suitable
scale.