When several masses revolve in different planes, they may be transferred to a reference plane (briefly written as R.P.), which may be defined as the plane passing through a point on the axis of rotation and perpendicular to it. The effect of transferring a revolving mass (in one plane) to a reference plane is to cause a force of magnitude equal to the centrifugal force of the revolving mass to act in the reference plane, together with a couple of magnitude equal to the product of the force and the distance between the plane of rotation and the reference plane.
In order to have a complete balance of the several revolving masses in different planes, the following two conditions must be satisfied:
1. The forces in the reference plane must balance, i.e. the resultant force must be zero.
2. The couples about the reference plane must balance, i.e. the resultant couple must be zero.
Let us now consider four masses m1, m2, m3 and m 4 revolving in planes 1, 2, 3 and 4 respectively as shown in Fig. The relative angular positions of these masses are shown in the end view. The magnitude of the balancing masses mL and mM in planes L and M may be obtained as discussed below:
1. Take one of the planes, say L as the reference plane (R.P.). The distances of all the other planes to the left of the reference plane may be regarded as negative, and those to the right as positive.
2. Tabulate the data as shown in Table, The planes are tabulated in the same order in which they occur, reading from left to right.
| Plane(1) |
Mass (m) (2) |
Radius(r) (3) |
Cent.force /ω2 (4) |
Distance from plane L (l) (5) |
Couple /ω2(m.r.l.) (6) |
1L(R.P.)
2
3
M
4
|
m1
mL
m2
m3
mm
m4
|
r1
rL
r2
r3
rM
r4 |
m1.r1
mL.rL
m2. r2
m3.r3
mm.rM
m4.r4 |
-l1
0
l2
l3
lM
l4 |
-m1.r1.l1
0
m2. r2.l2
m3.r3. l3
mm.rM.lM
m4.r4.l4 |
3. A couple may be represented by a vector drawn perpendicular to the plane of the couple.
The couple C1 introduced by transferring m1 to the reference plane through 0 is proportional to ml or 1.1 1 and acts in a plane through Om1 and perpendicular to the paper. The vector representing this couple is drawn in the plane of the paper and perpendicular to Om 1 as shown by OC1 in Fig.
(c) Similarly, the vectors OC2, OC3 and OC4 are drawn perpendicular to Om2, Om3 and Om4 respectively and in the plane of the paper.
4. The couple vectors as discussed above, are turned counter clockwise through a right angle for convenience of drawing as shown in Fig.
(d). We see that their relative positions remains unaffected. Now the vectors OC2, OC3 and OC4 are parallel and in the same direction as Om2, Om3 and Om4, while the vector OC1 is parallel to Om1 but in opposite direction.
Hence the couple vectors are drawn radially outwards for the masses on one side of the reference plane and radially inward for the masses on the other side of the reference plane.
5. 6. Now draw the couple polygon as shown in Fig.
(e) The vector d0 represents the balanced couple. Since the balanced couple CM is proportional to mM.rMIM therefore
From this expression, the value of the balancing mass-m M~n the-plane M may, becbtained. and the angle of inclination of this mass may be measured from Fig.
(b) Now draw the force polygon as shown in Fig.
(f) (The vector eo(in the direction from e to 0 ) represents the balanced force. Since the balanced force is proportional to M L.rL, therefore,
From this expression, the value of the balancing mass mL in the plane L may be obtained and the angle of inclination of this mass with the horizontal may be measured from Fig
