Engineering Mechanics

(^590) „„„„„ A Textbook of Engineering Mechanics

• The balancing of masses rotating in different planes is beyond the scope of this book.
  † Sometimes for simplicity, an assumed force is found out, such that
  2
  22
  Centrifugal force
  Assumed force
  mr
  mr
  ω
  ===×
  ωω
  29.7. BALANCING OF SEVERAL ROTATING MASSES
  Consider any number of masses (say three) A, B and C
  attached to a shaft rotating in *one plane. In order to balance
  these masses, let us attach another body D to the same shaft
  as shown in Fig. 29.3.
  Let m 1 = Mass of the body A
  θ 1 = Angle which the body A makes with
  the horizontal,
  m 2 ,θ 2 = Corresponding values for the body
  B, and
  m 3 ,θ 3 = Corresponding values for the body
  C.
  The magnitude and position of the balancing mass,
  may be found out by any one of the following two methods :

1. Analytical method and 2. Graphical method.

29.8.ANALYTICAL METHOD FOR THE BALANCING OF SEVERAL ROTATING

MASSES IN ONE PLANE BY ANOTHER MASS IN THE SAME PLANE

The magnitude and position of the balancing body may be obtained, analytically as discussed

below :

1. First of all, find out the †centrifugal force exerted by each body on the rotating shaft.


2. Resolve the centrifugal forces (as found above) horizontally, and find out the resultant of
   the horizontal components (i.e. ∑H).


3. Now resolve all the centrifugal forces vertically, and find out the resultant of the vertical
   components (i.e. ∑V).


4. Magnitude of the resultant force is given by the relation :

=Σ +Σ() ()HV^22

5. If θ be the angle, which the resultant force makes with the horizontal, then

tan

V

H

Σ

θ=

Σ

6. The balancing force is, then equal to the resultant force, but in the opposite direction.


7. Now find out the magnitude of the balancing mass. This can be done by calculating the
   mass of a body, which can produce a centrifugal force equal to the resultant force [as per
   item (4) above].

Fig. 29.3. Balancing of several rotating

masses