Balancing of rotating masses

In this chapter we shall discuss the balancing of unbalanced forces caused by rotating masses, in order to minimize the loads on bearings and stresses in the various members, which causes dangerous vibrations when a machine is running.
The following cases are important from the subject point of view:
1. Balancing of a single rotating mass by a single mass rotating in the same plane.
2. Balancing of a single rotating mass by two masses rotating in different planes.
3. Balancing of different masses rotating in the same plane.
4. Balancing of different masses rotating in different planes.

1. Balancing of a Single Rotating Mass By a Single Mass Rotating in the Same Plane

Consider a disturbing mass m1 attached to a shaft rotating at ? rad/s as shown in
Fig .1. Let r1 be the radius of rotation of the mass m1. We know that the centrifugal
force exerted by the mass m1 on the shaft,
FCl = m1. ?2. r1
This centrifugal force produces bending moment on the shaft. In order to counteract the effect of this force, a balancing mass (m2) may be attached in the same plane of rotation as that of disturbing mass (m1) such that the centrifugal forces due to the two masses are equal and opposite.
Let r2 = Radius of rotation of the balancing mass m2, centrifugal force due to mass m2,
FC2 = m2. ?2. r2 …………….(2)
Equating equations (1) and (2),
m1. ?2. r1 = m2. ?2. r2 or m1. r1 = m2. r2
2. Balancing of a Single Rotating Mass By Two Masses Rotating in Different Planes
In order to put the system in complete balance, two balancing masses are placed in
two different planes, parallel to the plane of rotation of the disturbing mass, There
are two possibilities may arise while attaching the two balancing masses: