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Friday, 3 March 2017

EXERCISES -SOLUTIONS - Balancing Of Reciprocating masses

Question.5. Explain the method of balancing a single rotating mass by another rotating mass in same plane.

Answer.
When a certain mass is connected to a rotating shaft, it will exert some centrifugal force. The tendency of this centrifugal force is to bend the shaft and to produce vibrations in it. To prevent the effect of the unbalanced centrifugal force, another mass is connected to the opposite side of the shaft at such a position so that the effect of the centrifugal force due to first mass can be balanced.
Fig. Balancing of a Single Rotating Mass by Another Rotating Mass in the Same Plane
Let us consider a mass connected to a shaft as shown in the Fig.
Let


The force acts radially outwards and continuously changes its direction. To counteract this force, a balance mass is introduced at a distance from the axis of the shaft in the same plane of the disturbing force such that centrifugal force of the two masses are equal and opposite.
Letbe the centrifugal force due to balance mass.
For complete balance,
 The product may be split in any convenient way. Normally, is kept large to reduce .

Question.6. Explain the method of balancing a number of masses rotating in one plane by another mass rotating in the same plane.

Answer.
Let us consider four masses attached rigidly to the shaft rotating at angular speed  as shown in Fig. (a). 
Let  are the distances of from the axis of rotating shaft. are the angles of masses  with the X-axis.
The centrifugal force for each mass is proportional to the product of mass and its radius of rotation as  is same for each mass. All these forces are acting radially outwards and form a system of concurrent coplanar forces at point O i.e. on the axis of rotation. The resultant of all such forces can be found by the following two methods:
1. Analytical method,
2. Graphical Method.


(a) Space Diagram                                                                                                  (b) Vector Diagram
Fig. Balancing of a Number of Masses Rotating in One Plane by Another Mass Rotating in the Same Plane
1.            Analytical method :
(i) Resolve each force horizontally and vertically.
(ii) Find the resultant horizontal component () of the centrifugal forces which is
(iii) Similarly, find the resultant vertical () of the centrifugal forces which is
(iv) Find the magnitude of the resultant centrifugal force.
(v) Find the direction of the resultant centrifugal force.
Where is the angle made by the resultant force with X-axis.
(vi) The balancing and the resultant forces are equal and opposite to each other.
(vii) Find the magnitude of the balancing mass from the relation,

2. Graphical Method : The analytical method is lengthy procedure and also there is possibility of mistake in the calculations. Therefore, graphical is generally preferred. The steps followed in the graphical method are as follows:
(i) Draw the space diagram as shown in fig. (a).
(ii) Draw the vector diagram as shown in fig. (b) in such a way that ‘ab’ represents the centrifugal force due to mass which is equal to. Magnitude and direction is drawn on some suitable scale. Similarly bc, cd and de are drawn to represent centrifugal forces due to masses.
(iii) The resultant force is shown by the closing side ‘ae'(dotted line).
(iv) The balancing force is equal and opposite to the resultant force.
(v) Find the magnitude of the balancing mass m such that
 

Question.7. Explain the method of balancing a number of masses rotating in different planes.

Answer.
Let us consider four masses  rotating in four different planes at different radii  respectively as shown in fig. Let us consider that all the above four masses are attached to a single rotor rotating with angular velocity . Let the distance of planes from the plane are  respectively.

Fig. Balancing of a Number of Masses Rotating in Different Planes
Let us consider the plane P as reference plane. Now, if we transfer the masses  rotating in planes  to the reference plane , then there will occur four unbalanced forcesand three unbalanced couples  in the reference plane. For complete balancing of the system, the following two conditions must be satisfied:

It should be noted that the distances of planes to the left side of reference plane are taken as negative, while to the right side as positive.

Question.8. Five masses A,B,C,D and E rotate in the same plane at equal radii. The masses A, B and C are 10 kg, 5 kg, and 8 kg respectively. The angular position of masses B, C, D and E measured in the same direction from A are 
respectively. Find the masses D and E for complete balance.

Answer.
Graphical Method:

Component
Mass(m) (kg)
Radius(r)(m)
mr(kgm)
A
10
r
10r
B
5
r
5r
C
8
r
8r
D
E
 mdr
mcr
r
r
 mcr
mcr

From the tabular data and statement of the problem, it is clear that the system is in complete balance. Therefore, the force polygon should be a closed one. Hence it is drawn with the help of data in column 4 of the table as under: 
1. Draw a line parallel to and equal to  or some suitable scale.
2. From point a , draw line ab parallel to OB and equal to 5r.
3. Similarly, draw line bc parallel to OC and equal to 8r.
4 From point c, draw line cd parallel to OD.
5. From point O, draw line ed parallel to OE to cut line cd at point d.
6. Measure cd and equate it to mdr to find the value of md.
7. Similarly, measure de and equate it to mer . Find the value of me .

Let us now solve the above problem by analytical method.
Analytical Method: {see fig. a}
Component
Mass (m)(kg)
Radius (r) (m)
mr (kg m)
mr sin θ
A
0
r10r10r
0
B
5
r5r2.5r4.33
C
8
r8r-5.657r5.657r
Dmdrmdr-0.8666mdr-0.5
Emermer
0
-mer

For complete balance,


From equation (i),

From equation (ii),






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