Balancing Of Several Rotating Masses In The Single plane

          Let us Consider four masses of magnitude m₁, m₂, m₃ and m₄ are at a distance of r₁, r₂, r₃ and r₄ from the axis of rotating shaft. Let θ₁, θ₂, θ₃ and θ₄ be the angles of these masses with the horizontal line OX, as shown in Fig., Consider these masses rotate with a constant velocity of 'ω' rad/sec, about an axis through 'O' and perpendicular to the plane of paper.

          The magnitude and position of the balancing mass may be found out by using two methods, they are analytical method and graphical method. These methods are discussed below:

1. Analytical Method:

1. First we have to find out centrifugal force exerted by each mass on the rotating shaft.

2. We have to resolve the centrifugal forces horizontally and vertically, and find their sums, i.e., ΣH and ΣV. We know that 

Sum of horizontal components of the centrifugal forces, 

                      ΣH = m₁ . r₁. Cos θ1 + m₂ . r₂. Cos θ2  ………

and sum of vertical components of the centrifugal forces, 

                      ΣV = m₁ . r₁. Sin θ1 + m₂ . r₂. Sin θ2  ………

3. Magnitude of the resultant centrifugal force, 

4. If is the angle, which the resultant force makes with the horizontal, then 

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

6. Magnitude of balancing mass, such that 

                         F_c = m . r

       Where     m = Balancing Mass, 

                        r   =  It's radius of rotation

2. Graphical Method:

1. Draw space diagram of several masses based on their positions as shown in above Fig., 

2. We have to find out centrifugal force exerted by each mass on the rotating shaft.

3. Now draw the vector diagram with the centrifugal forces we obtained.

4. Here 'ab' represents the centrifugal force exerted by the mass m₁ (i.e., m₁ . r₁ ) in magnitude and direction to some suitable scale. 

5. Similarly for other masses m₂, m₃ and m₄, draw 'bc', 'cd', and 'de'.

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

6. Magnitude of balancing mass (m) at a given radius of rotation (r), such that 

                         F_c = m . ω². r = Resultant centrifugal force

                                          m . r = Resultant of  m₁ . r₁ , m₂ . r₂ , m₃ . r₃ , m₄ . r₄