11.
BALANCING OF ROTATING AND RECIPROCATING MASSES
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Inertia forces cause shaking of machine members and which may induce unwanted vibrations. Balancing is a technique of correcting or eliminating unwanted inertia forces, there by neutralizing or minimizing unpleasant and injurious vibratory effects. Balancing of inertia forces is effected by introducing additional masses or by removing some mass to counteract the unbalanced forces.
11.1. STATIC BALANCE •
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Static balance is a balance of forces due to the action of gravity.
Consider the disc and shaft combination shown in fig 1. The shaft, which is assumed to be perfectly straight, rests on hard and rigid rail and rolls without friction. Roll the disc gently by hand and allow it to come to rest.
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Then, mark the lowest point of the periphery of the disc, repeat this many times and observe the location of the marks.
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if the marks are placed randomly, then the disc is balanced. If the marks are concentrated in the same area, then the disc is statically unbalanced;
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i.e. the axis of the shaft and center of mass of the disc do not coincide.
The position of the marks indicates the position of the unbalance but not the magnitude. The correlation of the unbalance is effected by trial and error and this is done by either drilling out material at the mark or by adding mass to the periphery opposite to the location of the mark. 2
For a rotor with different masses as shown in fig 2, the requirement for static balance is that the center of gravity of the system be at the axis O-O of rotation.
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For this it can be concluded that moments about the x and y-axes must be zero. i.e.
Wr sin 0 Wr cos 0
(1) (2)
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11.2. STATIC BALANCE MACHINES
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11.3. DYNAMIC UNBALANCE •
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Consider the rotor, shown in the figure below with masses m 1 and m 2 placed at opposite ends of the rotor at distances r 1 and r 2 respectively. The rotor can be statically balanced in all angular positions if m 1 = m 2 and r 1 = r 2 . If the rotor is caused to rotate at rad/s , then centrifugal forces m1r 1 2 and m2r 2 2 act at masses m1 and m2 , respectively. These forces produce different reactions on the bearings at A and B. The entire system of forces rotates with the rotor at the speed of rad/s , thus causing the reaction forces to vary with it. The requirement for dynamic balance is the balance of forces due to the action of inertia forces. 5
11.4. BALANCING OF DIFFERENT MASSES LYING IN THE SAME TRANSVERSE PLANE •
Consider the rotor carrying masses m 1, m 2 , and m 3 at a radial distances r 1, r 2 , and r 3 , respectively, as shown below .
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For balance of the rotor, the vector sum of all forces, including the balancing mass, must be equal to zero; i.e.
F 0 F m r 0 c c
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2
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(4)
In scalar terms, equation (4) is written as
F c mi i r i 0 •
(3)
Keeping in mind that the term center O. Letting W i mi
g
(5) 2 m i i r iis along r i away from the
(6)
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and since 2
g
i
g
is constant, equation (5) can be re-written as
W r 0 i i
(7 ) 7
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Since
g
cannot be zero for some finite value of
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the above equation reduces to
W i r i 0
(8)
where W ir i is a vector in the direction of the inertia force. •
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For balance of the rotor, equation (8) must be satisfied. The value W ir i is tabulated and W er e of the balancing mass is determined so as to satisfy equation (8). If the mass me is set as the magnitude of the balancing mass, the radial position of the balancing mass r e can then be calculated, or vice-versa. Equation (8) can be solved either graphically or analytically.
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11.5. BALANCING OF DIFFERENT MASSES ROTATING IN DIFFERENT PLANES. •
If the rotating masses lie in different transverse planes as shown below, to achieve balance of the rotor, first, the equation
W i r i 0 •
(9)
must be satisfied. In addition balance of moments due to the inertia forces is also required.
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For balance of the rotor, the following conditions must be satisfied.
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The sum of the inertia forces must be equal to zero, including those due to the balancing masses; i.e.
W r 0 i i
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(10)
Moments of the inertia forces, including those of the balancing masses, about an arbitrary axis must be equal to zero; i.e.
W r a 0 i i
i
(11)
where a i s are the moment arms of each mass about the arbitrary axis. ’
In general, where unbalanced masses lie in different transverse and different axial planes, the resultant of the unbalanced forces and the resultant of the unbalanced moments are in different planes,
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In which case a single balancing mass would not satisfy both equation (10) and (11). 10
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The inertia force of the balancing mass, given by Re, can be obtained from
Re W i r i
(12)
where W i are the weights of the unbalanced masses and r i are their respective distances from the axis of rotation. •
The moment due to the inertia of the balancing mass is
M e W i r i ai •
The moment arm of the balancing mass from the arbitrary axis is obtained from
ae •
(13)
M e Re
W r a W r i i
i
(14)
i i
Equations (10) and (11) can be solved either graphically or analytically. 11
