Note: Descriptions are shown in the official language in which they were submitted.
1~4881Z
This invention relates to a balancing method for a
rotary machine, and more particularly to a balancing method
suitable for use in a large-capacity, large-sized steam
turbine and a generator.
In general, it is often t'nat a rotating portion
of a rotary machine is not completely or absolutely symmetric
with respect to its center axis, presenting a somewhat un-
balanced condition. Such an unbalanced condition results
in vibration in a shaft during the operation of a rotary
machine. Excessive vibrations lead to an abnormal condition
in bearing portions, thus failing to achieve the normal rotation
of a shaft. In addition, a force whicn causes the aforesaid
excessive vibrations is considerably large and acts on the
shaft, thereby presenting a possibility of the shaft being
damaged. An allowance is determined for the limit of vibra-
tions in a shaft, for preventing an accident arising from the
aforesaid shaft vibration. For this reason, a rotary machine
is subjected to a rotation test. In case the aforesaid
allowance is exceeded, then a correction weight is attached
to a rotary machine for reducing the degree of unbalance,
thereby reducing vibrations. Such an operation is referred
to as balancing.
A steam turbine and a generator provide a multiple
bearing system, wherein a plurality of shafts, tne opposite
ends of which are supported in bearings, are coupled to one
another. In general, it is customary that shafts are
respectively tested for balancing. A balancing procedure
is as follows;
(1) measure vibrations in shaft at bearing positions; -
(2) select balancing planes, i.e., positions on a -
shaft, to w'nich correction weights are to be attached;
- 2 -
1~)4881Z
(3) attach a trial weight to a balancing plane,
and then measure its influence on shaft vibrations. Determine
shaft vibrations caused by a unit weight, i.e., an influence
coefficient of the unit weight from the above result;
(4) determine a correction weight to nullify
vibrations by utilizing initial vibrations, i.e., shaft
vibrations in the initial condition and influence coefficients;
(S) attach the thus determined weight to the
balancing plane of a rotary machine. Then, operate it in this
condition, and confirm that shaft vibrations fall withln an
allowable level, thus completing the balancing operation.
When a steam turbine and a generator completed
in a factory are assembled in a power plant, then some degree
of unbalance would result. This dictates a balancing operation
by rotating a shaft system. Hitherto, in such a case, atten-
tion is drawn to a shaft which causes large shaft vibrations,
and then balancing is carried out for the shaft for reducing
the vibrations. Thereafter, balancing is given to a shaft
causing the next greatest vibrations.
In like manner, shafts are subjected to balancing
in turn, thereby reducing shaft vibrations to below an
allowable level, thus completing a balancing operation.
With a recent large-capacity, large-sized steam
turbine and a generator, a plurality of shafts are coupled
together, and these shaft systems affect ~a~h other,
thereby presenting a complex vibratory configuration. Thus,
difficulty is increased in reducing the shaft vibrations to
below a given level. According to the prior art balancing
operation, a plurality of cycles of balancing operation
are required for determining influence coefficients, with
the accompanying expenditure of much time and effort.
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881Z
Meanwhile, even if a correction weight is determined, it $s
difficult to estimate what kind of vibrations would take
place, until a correction weight is attached to a rotary
machine and then the rotary machine is acutally operated.
Thus, there exists some uncertainty in determination of a
correction weight. On the other hand, in case each shaft
of a plurality of shaft systems has to be~ subjected to
balancing as in the case of a power plant, many cycles of
operations are required until the balancing operation is
completely finished.
The method of measurement of shaft vibrations is
carried out, with a single vibration pick-up attached to one
place. For this reason, vibrations only in the direction
of a vibration pick-up to be attached are measured, i.e.,
vibrations in one direction. According to the prior art
balancing method, vibrations in one direction is only measured
in one measuring position, thus failing to accurately measure
vibrations within a plane, two-dimensionally. As a result,
poor accuracy results in vibrations which are utilized for
the computation of correction weight, and hence a correction
- weight obtained does not give an optimum value for reducing
an unbalance in a shaft. The prior art balancing still leaves
room for improvements in accuracy. In addition, a correction
weight is computed by using less accurate vibrations, so that
an optimum correction weight cannot be obtained, and thus
many cycles of balancing operations have been required, until
complete balancing is obtained.
It is an object of the present invention to provide
a balancing method which may avoid the shortcomings in the
prior art method and reduce the degree of shaft vibrations
in a rotary machine, efficiently.
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881Z
It is another ob~ect of the present invention to
provide a balancing system which may avoid the shortcomings
incurred in determination of correcting balancing, and enables
the determination of correction weight and measurements of
vibrations at a high accuracy. -
According to the present invention, there is provided
a balancing method for a multi-span rotor shaft comprising the
steps of measuring initial vibration amplitudes at at least
one desired vibration measuring point on the shaft; determining
10 . influence coefficients representative of vibration amplitudes
at the at least one vibration measuring point when a unit weight
is attached to at least one predetermined balancing plane on
said shaft; and determining at least one correction weight so
as to reduce the values of residual vibration amplitudes at the
at least one measuring point; the values of the residual vibration
amplitudes being dependent on the initial vibration amplitudes,
influence coefficients and the at least one correction weight
on the at least one balancing plane; wherein the step of .
determining influence coefficients includes determining the
influence coefficients for different measuring conditions; and
the step of determining the at least one correction weight
includes determining the at least one correction weight by the
method of least squares so as to minimize the sum of squares
of the residual vibration amplitudes.
- 5 -
' .' .. .. : ' ''''' ' ,' ', '
.. . . . . .
1~148812
FIG. 1 is a view illustrative of the balancing
method for use in a rotary maching according to the present
invention;
FIG. 2 is a view illustrative of the method for L
determining a correction weight;
FIG. 3 is a view illustrative of a transfer matrix;
FIG. 4 is a view illustrative o~f a vibrating mode;
FIG. 5 is a plot showing the relationship between
rotation speed and amplitude;
FIG. 6 is a view illustrative of a balancing plane;
FIG. 7 is a view illustrative of the correction -~
weight for the secondary vibration mode;
. FIGS. 8 and 9 are partial views of a rotary machine
having a groove adapted for use in attaching a correction
weight,
FIG. 10 is a view illustrative of the relationship
between the-rotation speed and the amplitude of vibration,
representing the advantages of the balancing according to
the invention9
FIG. 11 is a view illustrative of a balancing
method for particularly reducing a large residual vibration ~
amplitude; ~ .
FIG. 12 is a view illustrative of the balancing
method for reducing a correction weight at the sacrifice of
the residual vibrations, when the correction weight obtained
is extremely large;
FIG. 13 is a flow chart of one embodiment of the
invention;
FIG. 14 is a view showing detailed arrangement of
a vibration detecting pick-up;
FIG. 15 is an axial cross sectional view of a
~ C--
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1~)4881Z
vibration detecting pick-up;
FIG. 16 is a cross-sectional view taken at a right
angle to the shaft at a vibration detecting position, taken
along the line XIV - XIV of FIG. 15;
FIG. 17 is a view showing a locus of whirling of a
journal;
FIG. 18 is a flow chart of a ba~lancing system
according to the invention; and
FIG. 19 shows the results of balancing using a
balancing system according to the invention.
One embodiment of the present invention will be
described in more detail~by referring to FIG. 1. The method
according to~the present invention includes the steps of:
determining influence coefficients representing the relation-
ship between weight and vibrations, from the specification of
the rotor; measuring shaft vibrations occuring due to
unbalance; selecting balancing planes of the shaft, to which
correction weights are to be attached; and determining cor-
rection weights to be attached to the balancing planes, based
20 on the influence coefficients obtained from the results of -
measurements of shaft vibrations and the computation. Then,
correction weights thus obtained are attached to a rotary
shaft, vhich is then operated for confirming that vibrations
are reduced, thus completing balancing.
Description will be given ofthe method for deter-
mining a correction weight with reference to FIG. 2. Accor- -
ding to this method, a correction weight is computed by
utilizing a principle of the method of least squares so as
to minimize the sum of squares of residual vibration ampli-
tudes after attaching weights to the balancing planes,
while utilizing the results of measurements of shaft vibrations
~ - 7-
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~)4881Z
at many speeds, such as various critical speeds and rated speed.
Then, values of vibrations after a correction weightthusobtained
have been attached, i.e., residual vibration amplitudes are
determined by computation. In case the residual vibration
amplitudes fall within an allowable level, then the correction
weights thus obtained are used as final correction weights.
If the residual vibration amplitudes are out of the allouable
level, then the balancing condition is varied to carry out
the computation of correction weights again.
The present invention is characterized by the deter-
mination of correction weights, taking into consideration the
vibrating condition after correction weights have been attached.
A coefficient representing the relationship between
a weight and vibrations, i.e., an influence coefficient should
be determined by computation beforehand. To this end, a
vibration characteristic resulting, when an unbalance is
attached to a balancing plane of a rotary shaft, is computed,
and then the amplitude of vibrations in a vibrations-measuring
position, when a unit weight is attached to the balancing
plane, is determined, and the results thus obtained are to be
utilized as an influence coefficient for use in computing a
correction weight. The vibration characteristic of a shaft
system may be computed by using values dependent on the
specification of a rotor, such as dimensions of a shaft system,
weights of supporting ?ortions during the rotation, a
supporting condition such as an oil film characteristics.
The analysis of a shaft system may be carried out
by using a transfer matrix method. This method is carried
out as follows;
A shaft system is divided into elements such as
a beam portion, concentrated mass, and coupling portion by
~' .
i~
1¢~488~Z
means of a spring. Then, the vibration system is composed
as shown in FIG. 3 by using a matrix of state vector
lY] consisting of values such as deflection, slope, shearing
force, and moment in the dividing positions, and a transfer
matrix [A] of elements representing the relationship between
the values of the elements.
Assume deflections Vx, Vy, (cm)~, slopes ~ , a (rad),
shearing forces V , V (kg), and bending moments Mx,
My (kg cm), then state vector [Y] will be given as follows;
(wherein x represents values in the horizontal vibrations,
while y represents values in the vertical vibration).
i3~ lVY]
M
LV
[V] = ~V~ [M] = ~ ~
The transfer matrix [A] may be computed by the elements.
For instance, the transfer matrix [A] for the horizontal
vibrations of a beam element having a uniform cross section
is given below:
[A] = B4 aC2 aQC3-
Q 3 o Q Cl aC2
a C2 a C3 C QC
LaQCl a C2 Q4C3 CO -
CO = ~ (cosh ~ + cos ~), Cl = 2 ~ (sinh ~ + sin ~)
C2 = - 2 (cosh ~ - cos ~), C3 = 3(sinh ~ - sin ~)
Q , ~ ~2 Q4 ,
.
~,, .
, . . ' . ' ' . ,:
"- ' ' ' ' '
1~ 31Z
~herein Q: length (cm) ~: Young's modulus (kg/cm )
I: geometrical,moment of inertia (cm )
~: circular frequency (rad/s)
~: mass per unit length (kgs /cm )
The matrix of state vector [Yl before and after this
element, is given as below:
n+l ¦ - ~A] ~ 1-
n+l n
The transfer ~atLix [A] for a concentrated weight
is given below:
[A] = 1 0 0 0
0 1 0 0
0 2i2 1 0
m~ 0 0 1
wherein
m: mass (kg S /cm)
i: radius of gyration (cm)
~: circular frequency (rad/s)
In the position where an unbalance is attached,
an increase in shearing force results due to a centrifugal
force caused by thls unbalance.
In this case, the boundary condition at a shaft
end is free. In this case, a force matrix [P] consists of a
state vector such as force and moment is nullified:
[P] = O
A computing method for vibration-amplitude relating
to forced vibrations will be described hereinafter. Assume
that vibrations follow a sin wave having a circular frequency
~r-~
., ,
1~4881Z :
~(rad/s). The vibration-amplitude is a vector having a
magnitude and phase. Here now, a complex number is used
for representing values having such a magnitude and a phase.
As a result, elements of the matrix of state vector [Y] and
transfer matrix [A] will be represented by complex numbers.
The matrix of state vector [Y]l at one end is
expressed by an unknown matrix [X], by us~ing the boundary
condition in its position:
[1 1 [ 1]
wherein [I]: unit matrix
[X]: unknown matrix -
The term [1] in [1~ and [1] in the equation (1)
represents those which have no connection with the unknown
matrix [X] on the boundary, such as a vibratory force acting
on a vibration system, in computing a transfer matrix. ~-
Between the transfer matrix [A] and the matrixes of
[1~ ' [1~ 1 before and after the matrix [A], give
the following relation:
[ ~ = [A]n [ ] ................... (2)
The transfer matrix [A] may be computed for each
element, so that from the equations (1) and (2), the matrix of
state vector [1] at each position may be expressed as a
function of a matrix ~i] , as follows:
[Yl~n=[B]n M (3)
The following equation may be obtained from the
equations (2) and (3):
[1 +1 [1] ~1 n
,~ ,r ~
.~
4881Z
~X-
~
~ = [A] [B]
rYl Acgordingly, [B]n+l = [A]n [ ]n
¦ ¦ = [B]~ lJ Then, this cquation and the equation (1)determine [B]l.
[B]l = I O ¦ ............................. (5)
o o~
O IJ
[B]n may be computed by using the transfer matrix [A], and
equations (4) and (5). Accordingly, [B] may be obtained from
the equation (3), assuming the matrixes of state vector [Yl at
respective positions as a function of a unknown matrix ~ ¦ .
The matrix of state vector ~ at the other end is
given a jyl ~
On the other hand, the condition [P] = O may be
obtained from the boundary condition in this position, so that
an equation relating to an unknown matrix [X] may be obtained.
Thus, [X] may be obtained from the solution of the aforesaid
[ ] [ P] ~ 1~ Ll~ ¦C C22
[P]l~l [C21 C22] ~Y'l
IlJ
[X] = - [C21] [C22~
The equation (3) is substituted by the unknown matrix
[X] thus obtained for computing the value of matrix [Y] in the
respective positions. In this manner, the amplitude of
vibration may be obtained for the entire vibration system.
Balancing planes may be selected by using vibration
c'naracteristic of a shaft system. To this end, the vibration
,~ .
~
-- 1~41~812
characteristic of the shaft system, when an unbalance is
attached thereto, is computed for utilizing the results to be
obtained. FIG. 4 shows the vibrating mode at a natural fre- -
quency of the shaft supported at its opposite ends. The
vibrations mode at primary to third natural frequencies are
such that the directions of the primary vibrations remain
the same over the entire length of the shaft, the amplitude
of the secondary vibrations is at the minimum at the mid
point of the shaft, with the directions of vibrations at
the opposite ends being reversed, and the direction of the
third vibrations is one way at the mid way of the shaft, and
another at the opposite ends of the shaft. The rotary machine
tends to cause vibrations due to unbalance during operation.
The vibrations of this kind are apt to increase or decrease,
with an increase in rotation speed as shown in FIG. 5. The
rotary shaft has an inherent rotation speed which causes
severe vibration, and this rotation speed is referred to as
a critical speed. As shown in FIG. 5 three critical speeds
appear. These are referred to as a primary, secondary, and
third critical speeds, respectively. The vibrating mode at
these critical speeds correspond to the vibration character-
istics of a shaft system shown in FIG. 4.
By utilizing the vibratory characteristic of the
shaft system shown in FIG. 4, the balancing planes are
determined from the results of measurements of vibrations in
a shaft. Description will be given of the case where the
vibrations at the secondary critical speeds are large, and
the vibrations at the primary and third critical speeds
are small. Firstly, assume that the five balancing planes
are shown as at A to ~ in FIG. 6. The balancing planes which
are effective for the second critical speed are shown at
1~881Z
A and E which are close to the opposite ends of the shaft.
Thus, it is recommended that the directions of weights to
be attached to the both balancing planes 13 be opposite to
each other as shown in FIG. 7. In this manner, the
vibrations at the second critical speed may be reduced, yet
exerting no adverse influence on the vibrations at the
primary critical speed other than the second critical speed.
According to the balancing method of the invention-
an attempt is made so as to determine weights which may reduce
the vibrations in these conditions, by utilizing the data
regarding many critical speeds and rated speed. In other
words, an attempt is made so as to reduce shaft vibrations
at all critical speeds, such as the primary, secondary and
third critical speeds. The method for computation herein
utilizes the method of least squares so as to minimize tne
sum of the square vibration amplitudes after attaching
weights, i.e., residual vibration amplitudes, in an attempt
to reduce shaft vibrations of the number more than the
number of balancing planes. The residual vibration amplitude
is defined as follows:
f = A + ~ a W ......................... (6)
m m n=l mn n
wherein
~ : residual vibration amplitude
Am: initial vibration amplitude
: influence coefficient
Wn: weights an respective balancing planes
n: number corresponding to the positions of
balancing planes 1 < n ~ N
N: number of balancing planes
m: number corresponding to the measuring positions
and condition of shaft vibrations 1< m <M
4881Z
~: product of the number of the measuring positions
and measuring condition.
For computing the correction weights, an evaluation
function is used as follows:
J = ~ 1 12 ................................. (7)
m=l
A correction weight Wn is determined under the
condition where the evaluation function J is minimized. To
this end, the evaluation function J is partially differen-
tiated by Wn, and then the term thus obtained is taken as
zero, as follows:
.
aJ = 0 ........................... .(8)
a wn
From the above equation, an equation for computing
Wn is obtained as follows:
[W] = - {[~] [a]} [~] [A] .................. .(9~
[W] = _ : column vector of correction weight
_WN -
[a] = ~ : matrix of influence
LaMl - ~MN~ coefficient
all ,,, aMl ..
[~] = : : : transposed matrix of
alN ., aMN influence coefficient
r 11
[A] = . : column vector of initial vibration
-AM- I amplitudes
The vibrations after the correction weight thus
obtained has been attached, i.e., a residual vibration ampli-
- 15-
:
.. . .
. - ,, - .
.. . - .. . . . . .. . .
.. , . . - . .
-- 1~4881Z
ude e may be obtained as follows:
= A + ~ ~ ~Jn ............... (10)
m m n=l mn
wherein WI : correction weight
: residual vibration amplitude
The residual vibration amplitudes are computed for
all of the vibration utilized for the computation of correc-
tion weights, and then if these residual vibration amplitudes
are below the vibration allowable value which is dependent on
the size of a shaft, then the weight is taken as a correction
weight required.
In case the residual vibration amplitudes are not
below the allowable value, then the condition for balancing,
such as balancing planes are changed, followed by a repeated
computation of a correction weight.
In this manner, the correction weight is determined
and then attached to a rotary shaft. For this purpose, part
of a wheel disc 32 is formed with a weight-attaching groove
33, and then the weight is fitted in the groove 33. In the
absence of the wheel disc 32, a shaft 31 is formed with a
weight-attaching groove 34, as shown in FIG. 9, and then the
weight is fitted therein.
After attaching a correction weight to a rotary
shaft, the rotary shaft is put into operation for conforming
that the shaft vibrations are reduced, thus completing the
balancing.
The adoption of this method permits a reduced number
of balancing operation for reducing shaft vibrations at all
critical speeds and rated speed in a shaft system including
many coupled shafts. This provides a rotary maching highly
~ I C~~ ,
~.
A ~ :
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1~4~381Z
conomical.
FIG. 10 shows one example of test results of
balancing for a model rotor. Two curves shown therein represent
variations in vibration-amplitude due to rotation speed at
a certain position, before and after balancing. In this
figure, critical speeds appear in two positions. By utilizing
this method, a correction weight is deter~mined from vibrations
before balancing, and then the weight is attached to the shaft
for measuring shaft vibrations. The result is given as curve
like one after balancing. According to this method of balan-
cing, shaft vibrations may be reduced to 1/5 according to one
balancing operation. In this example, the shaft vibrations
are reduced to 1/10. Accordingly, even if the shaft vibrations
are 10 times as large as an allowable level, the shaft vibra-
tions may be reduced within an allowable level according to
a single balancing operation. According to the application
of the present invention, the shaft vibrations may be
suEficiently reduced according to a single balancing operation.
In the above example, if the residual vibration
amplitudes remain out of an allowable range, after attaching
correction weights, then the condition for computation should
be varied for carrying out the computation again. FIG. 11
shows a method of balancing in such a case. Firstly, a correc-
tion weight is computed according to a principle of the
method of least squares, so as to minimize the sum of squares
of residual vibration amplitudes by utilizing many vibration
data, such as critical speeds and rated speed, and then the
residual vibration amplitudes are computed, when the weight
thus obtained is attached. In case the residual vibration
amplitudes are less than the allowable value, then the weight
thus determined is used as a correction weight.
If the residual vibration amplitudes are larger
_ Iq_
.~
.
': . ' '
: . ' '' ,
.
1~4881Z
~han the allowable level, then the computation is repeated,
with attention paid to the large residual vibration so as to
reduce same.
The following equation is used as an evaluation
fun-ction in place of the equation (7):
J ~ ~ml Am ................................ (11)
m=l
wherein Fm: residual vibration amplitude
Am: factor
The factor Am in the equation (11) is taken as being 1 for
the first time.
The result of this computation accords with the
result obtained from the equation (7).
Computation is carried out for a correction weight
at the first cycle, and if the residual vibratlons are no
less than the allowable level, then the factor Am is computed
according to the following equation:
S ~ 1 12 A i - 1 ¦
R = J~ r .................................. (12)
~ (i) = I F l/R
wherein ~m: residual vibration amplitude
Am : a factor used in the preceding computation for
a correction weight
A( ): factor for computation of a correction weight
in the next balancing
M: product of the number of measuring positions
and number of measuring conditions
A correction weight is computed by utilizing A thus obtained.
By repeating the aforesaid computation, a correction weight
may be determined so as to particularly reduce a large
residual vibration amplitude.
Meanwhile, in case the correction weight obtained
- 18--
.
' ' , ' ' ' ~ .
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881Z
s extremely large, and yet in case a some allowance is
given to residual vibration amplitudes, with accompanying
some increase in residual vibration, then computation is
repeated for a correction weight so as to reduce the
correction weight, at the sacrifice of residual vibration.
FIG. 12 shows a method to be used in this case. Firstly,
many vibration data such as many critica~ speeds and rated
speeds are utilized to compute a correction weight according
to a principle of the method of least squares which minimizes
the sum of squares of residual vibration amplitudes, and then
the residual vibration amplitudes, when this weight is attached,
are computed. In case the correction weight obtained is
extremely large and yet some allowance is given thereto,
then a correction weight is computed again so as to reduce the
correction weight, at the sacrifice of residual vibration to
some extent.
An evaluation function corresponding to the equation
(7) is given as follows:
M 2 N
20 J = ~ l~ml ~m~ ~ lwn¦ ~n .................. (13)
m=l n=l
wherein ~m: residual vibration amplitude
W : weights for respective balancing planes
: factor given as data
N: number of balancing planes
: factor
A correction weight is determined in this manner
and thus a desired balancing may be achieved with the result
of a reduced correction weight.
Here is another method for determining balancing
planes by determining a correction weight. In other words,
~9_ ~
B
fi
.' :
1~48812
several kinds of conditions are determined, and then a
correctionweight for these conditions is determined~ after
which the best correction weight is adopted as a final
correction weight. For instance, assume many balancing -
planes, and then select a plurality of balancing planes
among these, so that a correction weight may be determined
according to the foregoing method for a ~ase where a correc-
tion weight is attached to this balancing plane. The best
correction weight is selected among these. The conditions
for determining as being the best correction weights are
that a residual vibration is small, a correction weight is
small, and the number of the balancing plane is less.
The aforesaid procedure is shown in FIG. 13. In
other words, in case the initial vibration amplitude A is
larger than an allowable level, a weight is obtained so as
to reduce the residual vibration ~m by minimizing
~ I E ml
m=l
In case the residual vibration amplitude em is
larger than an allowable level, then ~ ¦F~¦ ~m is
m=l
minimized so as to obtain the weight which reduces the
residual vibration.
In case the magnitude of a weight is limited, then
M N
~ ¦~ 12~ + ~ IW 12~ is minimized to obtain a
m=l n=l
compensated weight.
FIGS. 14, 15, 16 show a vibration detecting
method for a balancing system embodying the present invention.
--~o_
,: :
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., , , ~,
~0~8812
A shaft-vibration measuring rod 51 is provided
with (i) a hollow cylindrical outer case 53, (ii) a vibration
pick-up 54 and tip 55 attached to the upper and lower ends
of a contact shaft 52 movable along the length of the outer
case 53 therein, (iii) a coil spring 56 and a spring-hold-
down member 57 between the outer case 53 and the contact
shaft 52 for imparting a suitable pressure to the tip 55
beforehand, and (iv) a stopper 58 in the vicinity of the
vibration pick-up on the contact shaft 52, the aforesaid
stopper 58 being adapted to adjust the tension of the coil
spring 56.
FIG. 15 shows the measurement of vibrations in
the shaft by using the shaft-vibration-measuring rod 51
of the aforesaid arrangement. The outer case 53 for the
rod 51 is so positioned as to pierce through a rod hole 67
in bearing cover 66, then through a rod hole 64 in a bearing
cover 63 and then a rod hole 62 in a bearing bushing 61,
then the tip 55 at the lower end of the contact shaft 52 is
brought into press contact with a rotary shaft 60. Then,
the top end of the outer case 53 is secured to the bearing
cover 66 by means of a metal piece 71. The metal piece 71
is secured to a spherical seat 70 supported through the
medium of upper and lower spherical bearing surfaces 68, 69,
which are secured to the bearing cover 66 by bolts 72.
Two of the aforesaid rods 51 are attached in the
circumferential same plane as shown in FIG. 16 at a right
angle to each other. Vibrations at two point A ar.d B may
be measured by means of the rods 51 intersecting at a right
angle with each other, the aforesaid points A and B being
angularly spaced 90 in the circumferential direction of the
rotary shaft 60. Vibrations x and y at the points A, B at
~f ,,~
':
,
1~48812
a certain rotation speed ~ may be expressed as follows:
x = ¦x¦ sin (~t + x) .~................... (14)
Y = ¦Y¦ sin (~t + ~y)
wherein ¦x¦, ¦Y¦, ~x' ~y represent the amplitudes and phases
at the points .~ and B, and are the values to be measured. In
general, with a horizontal rotary machine of a turbine-generator,
the vibration characteristics at the points A and B are varied
due to the anisotropy of bearings, with the result that the
amplitudes ¦x¦ and IYI in the equation are varied. For this
reason, the rotary shaft 60 at a given rotation speed describes
an elliptic locus, when (x,y) is plotted, taking ~t as a
parameter. This elliptic locus varies its shape depending
on rotation speed. In other words, an angle ~ formed by
the x axis and a major axis varies. FIG. 17 shows a locus
of a whirling shaft 60 at a critical speed and a rated speed
wnich have been actually measured. Shown at 100, 101 are
bearings. Since the whirling of the shaft assumes the
shape of an ellipse, the direction of measurement should not
be limited to one direction. Otherwise, it may happen that
vibrations in the vicinity of a minor axis of an ellipse
are measured as a peak vibration due to a variation in the
direction of the major axis of an ellipse, despite the fact
that the measurement should have been given to the vibration
in a position affording the largest amplitude, i.e., the ~¦
position in the vicinity of a major axis of the ellipse.
On the other hand, in case the measurement of vibrations is ~-
given in two directions, then the aforesaid shortcoming
may be avoided, and at least one point will detect the
vibrations in the vicinity of a major axis of the ellipse.
This results in a higllly accurate measurement of vibrating
configuration of the rotary shaft 60 according to the bi- ;
~J
~4881Z
~irectional vibration measurement.
In general, the rotary shaft 60 includes an unbal-
anced mass, and thus the above unbalanced mass causes a
centrifugal force, which in turn causes unbalanced vibrations
in the rotary shaft 60. When the unbalanced vibrations become
excessive, then the rotary shaft 60 will contact the bearing
bushing 61, thereby leading to a damage in the rotary shaft 60.
For this reason, it is mandatory to reduce unbalanced vi-
brations occuring in the rotary shaft 60. Thus, the balancing
is required for reducing the unbalanced vibration. For
achieving highly efficient, highly accurate balancing, vibration
data should be measured at high accuracy.
The aforesaid bi-direction vibration measurement
enables the highly accurate determination of correction weights
(weight having a magnitude and an angle).
Description will be given of the method for
determining the correction weights. Assume the influence
coefficients a n( ), amn( ) (n = 1, 2, .....
N...... number of balancing planes; m = 1, 2,
20 M........ number of measurements; V...... vertical direction;
H...... horizontal direction), and initial unbalanced
vibrations Am( ), Am( ~. The residual vibration amplitudes
m(V), ~(H) are given as follows:
(V) (V) ~ (V)W
(H) = A (H) + ~ ~ (H)W
m m n=l mn n
The following evaluation function J is so defined as to
reduce the vibrations m(V), Em( ):
, . .
, . . .
1~4881Z
J = ~ {~ m( ) + Im( )I ~ ( ) } ............... (16)
wherein A (V), ~ (H) represent weight factors. Assume that
the weight factor is 1, then an optimum correction welght
so as to minimize the evaluation function J is givan as below:
W t = ~ {~ a} ~ A ............ (17)
wherein,
~ (n-l, ..... , N)
Ol (V)X - C! (V) :~
mn mn y
a (V) a (V)
a = mn mn x (m=l, ... ,M) .. (18)
,a~n(H)x~ C~mn(H)y
(H)y a (H)
mx (m=l, .... M) ~ nx
[A]= Amy(H) ¦ (n=l, ........ N)
AmY(H)_ ......... (19)
FIG. 18 shows a flow chart of the balancing system
according to the present invention. -~
Vibrations at the point A and B, which have been
detected by means of vibration pick-up 54 are fed to a
tracking filter. The vibrations are analyzed into relation-
synchronous-unbalanced vibration components by this tracking
filter. This unbalanced vibration components are printed as
outputs by a typewriter in a data-processing device for the
utilization of monitoring the vibration data. Meanwhile,
the influence coefficients which have been obtained at the
points A and B in two directions beforehand are fed into a
small-sized electric computer. In addition, excessive
~, .
348812
~ibrations generating rotation speed, i.e., vibrations at
the point A and B at critical speed and rated speed are fed
to the small-sized electric computer, thereby determining the
correction weights by using the equation (18). Then, the
residual vibration at the points A and B, when the aforesiad
correction weights are attached, are computed, followed by
confirming that the vibrations remain within an allowable
level, then balancing planes are changed, and then computation
in the preceding manner is continued, until the residual
vibration fall within the allowable level. Meanwhile,
the aforesaid small-sized electric computer may be built in
the data processing device.
Description will now be given of the results of
the actual balancing by using the aforesaid vibration measuring
system. When the rotation speed of a rotor was increased,
then there took place excessive unbalance in the bearing No.
1 and bearing No. 2 at the opposite ends of the rotor. For
reducing the aforesaid excessive unbalanced vibratlons, the
following two types of balancing tests were given;
(i) measurement of vibrations in one direction
(vertical direction)
(ii) measurement of vibrations in two directions
(vertical and horizontal directions).
FIG. 19 shows the results of balancing of the
bearing No. 2 in the horizontal and vertical directions. -~
,.~,.
