Page 267 - Machinery Component Maintenance

Balancing of Machinery Components 249

their supporting structure. If the structure is rigid, the force is larger than
if the structure is flexible (except at resonance). In practice, supporting
structures are neither entirely rigid nor entirely flexible but somewhere
in between. The rotor-bearing support offers some restraint, forming a
spring-mass system with damping, and having a single resonance fre-
quency. When the rotor speed is below this frequency, the principal iner-
tia axis of the rotor moves outward radially. This condition is illustrated
in Figure 6-8A.
If a soft pencil is held against the rotor, the so-called high spot is
marked at the same angular position as that of the unbalance. When the
rotor speed is increased, there is a small time lag between the instant at
which the unbalance passes the pencil and the instant at which the rotor
moves out enough to contact it. This is due to the damping in the system.
The angle between these two points is called the "angle of lag." (See Fig-
ure 6-8B.) As the rotor speed is increased further, resonance of the rotor
and its supporting structure will occur; at this speed the angle of lag is
90". (See Figure 6-8C.) As the rotor passes through resonance, there are
large vibration amplitudes and the angle of lag changes rapidly. As the
speed is increased further, vibration subsides again; when increased to
nearly twice resonance speed, the angle of lag approaches 180 degrees.
(See Figure 6-8D .) At speeds greater than approximately twice resonance
speed, the rotor tends to rotate about its principal inertia axis at constant
amplitude of vibration; the angle of lag (for all practical purposes) re-
mains 180".
In Figure 6-8 a soft pencil is held against an unbalanced rotor. In (A) a
high spot is marked. Angle of lag between unbalance and high spot in-
creases from 0" (A) to 180" in (D) as rotor speed increases. The axis of
rotation has moved from the shaft axis to the principal axis of inertia.
Figure 6-9 shows the interaction of rotational speed, angle of lag, and
vibration amplitude as a rotor is accelerated through the resonance fre-
quency of its suspension system.

Correlatlng CG Displacement with Unbalance

One of the most important fundamental aspects of balancing is the di-
rect relationship between the displacement of center-of-gravity of a rotor
from its journal axis, and the resulting unbalance. This relationship is a
prime consideration in tooling design, tolerance selection, and determi-
nation of balancing procedures.
For a disc-shaped rotor, conversion of CG displacement to unbalance,
and vice versa, is relatively simple. For longer workpieces it can be al-
most as simple, if certain approximations are made. First, consider a
disc-shaped rotor.