Balancing of Rotating Masses
Balancing A Single Rotating Masses
If a mass of M kg is fastened to a shaft rotating at w rad/s at radius r meter, the centrifugal force, producing out of balance effect acting radially outwards on the shaft will be equal Mw2rNewton. This out of balance in any one of the following two ways:
a. By introducing single revolving mass in the same transverse. Introduce a second mass B kg, called the balance mass, diametrically opposite to M at radius R rotating with sameangular speed of w rad/s fig
For complete balance, the centrifugal force of the two masses must be equal an opposite in the plane of rotation.
Mw2r = Bw2R
Mr = BR
Or hence for such balance the product of mass and its radius must be equal to the product of balance mass and its radius. The product BR or Mr is very often called the mass moment.
b. By introducing two masses one in each in two parallel transverse planes.
Sometimes it is not possible to introduce balance mass in the same transverse plane in which disturbing mass M is placed .in that case two masses can be placed one each in two parallel transverse planes to affect a complete balance. it may be remembered that one revolving mass in one plane cannot be balanced by another mass revolving in another parallel plane, as, no doubt balancing mass can be adjusted such that centrifugal forces may be equal and opposite indirection but at the same time will give rise to a couple which will remain unbalanced.
So let M be the distributing mass and B1, B2 be the balance masses placed at radius of r, b1 and b2 respectively from the axis of rotating , let the distances of planes of revolution ofB1 and B2 from that of M be a and c respectively and between B1 and B2d.
Balancing of Several Coplanar Rotating Masses
If several masses are connected to s shaft at different radii in one plane perpendicular to the shaft and the shaft is made to rotate, each mass will set up out of balance centrifugal force on the shaft. In such a case complete balance can be obtained by placing only one balance mass in the same plane whose magnitude and relative angular position can be determined by means of a force diagram. Since all the masses are connected to the shaft, all will have the same angular velocity w, we need not calculate the actual magnitude of centrifugal force of any, but deal only with mass moments.
If the three masses (M1, M2 and M3 are fastened to shaft at radiir1, r2 and r3 resp.
In order to determine the magnitude of balance mass B to be placed at radius b we proceed as follows.
1. Find out mass moment of each weight i.e. M1r1, M2r2 etc.
2. Draw vector diagram for these mass moments at a suitable scale. Commencing at p draw pq to represent M1r1 from q to draw qr to represent M 2r2. and from r draw rs to represent M3r3
3. The closing side sp (from s to p and not from p to s represents the magnitude and direction of balancing mass moment Bb.
4. Measure sp on the scale considered and divided by b, the quotient will be themagnitude of balance mass B.
Balancing of Several Masses in Different Parallel Planes
The technique of tackling this problem is to transfer the centrifugal force acting in each plane to a single parallel plane which is usually termed as reference plane and thereafter the procedure for balancing is almost the same as for different forces acting in the same plane.
Balancing of Reciprocating Masses
Acceleration and force of reciprocating parts. To find acceleration of reciprocating parts such as crosshead or piston, consider asimple crank and connecting rod arrangement Inwhich P is the piston or crosshead whose acceleration is to be determined.
r = length of crank ;
L = length of connecting rod;
x = movement of piston or cross head at any instant from its outermost position when revolves θ radian from its inner dead center position.