INTRODUCTION
An overhung rigid rotor is one that has its balance correction planes located outside the supporting bearings, as shown in Figure 1
^{
1
}
, and whose rotational speed is well below the first critical speed
^{
3
}
.
Cross effect, also known as the correction plane interference , can be defined as the change in the unbalance indication at one correction plane caused by a change in the unbalance at the other correction plane. The overhung rotors can be difficult to balance because of high levels of cross effect. To minimize cross effect, two trial weights, which generate a couple of forces, are added for the third run
^{
1
}
. Under identical conditions, the correction plane interference ratios for the new static/couple solution are comparable with the correction plane interference ratios for the effective static/couple solution
^{
2
}
.
ANALYSIS
Figure 1 illustrates an overhung rigid rotor as a shaft with a disk of width
l
and radius
r
for attaching trial and correction weights rotating in a clockwise direction with an angular speed ω as seen from the right side. The rotor is supported on the near bearing
N
close to the rotor and on the far bearing
F
far from the rotor, which are separated by a distance
h
. The center of gravity
G
of the rotor is farther away from the right correction plane
R
and closer to the left correction plane
L
, which is localized at a distance
c
from the near bearing
N
^{
1
}
. This is considered a perfectly balanced rotor.
Figure 1: Overhung rotor supported on bearings
A condition of dynamic unbalance is created by adding the weights
W
_{
L
}
and
W
_{
R
}
to the left and right correction planes, respectively. These two weights generate the two corresponding centrifugal forces, which are proportional to the factor
S = (r/g)ω
^{
2
}
, where
g
is the acceleration due to gravity
^{
1
}
. Assuming the radii to attach trial and correction weights are equal, that is
r
_{
L
}
=
r
_{
R
}
=
r
, then the factors
S
_{
L
}
=
S
_{
R
}
=
S
^{
2
}
. The unbalance weight in the left correction plane,
W
_{
L
}
=(W
_{
L
}
,a
_{
L
}
)
, as well as the unbalance weight in the right correction plane,
W
_{
R
}
=(W
_{
R
}
,a
_{
R
}
)
, expressed in polar coordinates, are complex vectors that are known as phasors, whose magnitudes WL and WR are the amounts of unbalance weight in the left and right correction planes, respectively, and whose directions are the corresponding angles,
a
_{
L
}
and
a
_{
R
}
of the unbalance weights
^{
1,2
}
.
If the two additional unbalance weights,
W
_{
R
}
and –
W
_{
R
}
, are added to the left correction plane, the state of unbalance is not altered. The vector combination of the left unbalance weight
W
_{
L
}
with the right unbalance weight
W
_{
R
}
in the left correction plane gives the quasistatic unbalance weight
W
_{
E
}
=(W
_{
E
}
,a
_{
E
}
) in the left correction plane, whose magnitude
W
_{
E
}
is the amount of unbalance weight in the left correction plane and whose direction is the angle aE of the unbalance weight in the left correction plane. This can be written as follows
^{
1
}
in Equation 1:
Therefore, the dynamic unbalance produced by the unbalance weights in the left and right correction planes is equivalent to the dynamic unbalance produced by the quasistatic unbalance weight in the left correction plane plus the couple unbalance created by the set of opposite unbalance weights in the left and right correction planes
^{
1
}
.
From the definition of the mobilities for the near and far bearings, M
^{
N
}
and M
^{
F
}
, respectively
^{
2
}
, the influence factors are given by the following expressions
^{
1
}
in Equation 2:
The general equations that relate the unbalance weights in the left and right correction planes of the rotor with the vibrations in the near and far bearings are the ones that follow at continuation
^{
1
}
in Equation 3:
Within the common balancing procedure, for the second run, the addition of a trial weight
W
_{
TL
}
in the left correction plane according to Equation 3 leads to Equation 4
^{
1
}
:
And, within the effective balancing procedure, for the third run, the addition of a trial weight
–W
_{
TR
}
in the left correction plane and of the opposite trial weight
W
_{
TR
}
in the right correction plane to form a couple according to Equation 3 leads to Equation 5
^{
1
}
:
The indicated couple unbalance weight in the right correction plane
W
_{
IR
}
caused by a trial weight
W
_{
TL
}
in the left correction plane is deduced from Equation 6:
The correction plane interference ratio, using Equation 2, is calculated as a percentage, as follows in Formula 7:
Therefore, the correction plane interference ratios for the effective static/couple solution
^{
1
}
are exactly the same as the correction plane interference ratios for the new static/couple solution
^{
2
}
.
It is observed in the second run of the balancing procedure that instead of the addition of the left trial weight
W
_{
TL
}
in the left correction plane to obtain the effective static/couple solution
^{
1
}
, that the quasistatic trial weight
W
_{
TE
}
is added directly in the left correction plane to obtain the new static/couple solution
^{
2
}
.
CONCLUSION
The correction plane interference ratios for the effective static/couple solution and the new static/couple solution are equal. The new solution is more direct than the effective solution, which is advantageous for practical applications.
REFERENCES

MéndezAdriani, José A., “Considerations on the Field Balancing of the Overhung Rigid Rotors”,
The Shock and Vibration Digest
, Vol. 37, No. 3, 2005, pp. 179187.

MéndezAdriani, José A., “A Detailed Equation Formulation for the Analysis of the Unbalance of an Overhung Rigid Rotor”,
Proceedings of the Vibration Institute Training Conference
, Jacksonville, Florida, June 1921, 2013.

Vance, John M.,
Rotordynamics of Turbomachinery
, John Wiley & Sons, New York, 1988, pp.116120.