CA1048812A - Balancing method for use in multiple-span rotor shaft system and balancing system using same - Google Patents

Abstract

BALANCING METHOD FOR USE IN MULTI-SPAN ROTOR
SHAFT SYSTEM AND BALANCING SYSTEM USING SAME

ABSTRACT OF THE DISCLOSURE
A balancing method for use in a multi-span rotor shaft system for a prime mover, such as a large-capacity, large-sized steam turbine and generator. A balancing influence coefficient representing the relationship between a unit weight and vibration, when the aforesaid unit weight is attached to a given balancing plane of a multi-span rotor, is first determined, and then a balancing plane, to which is to be attached a correction weight, is selected, then a correction weight is determined so as to minimize values of residual vibration amplitude, by using the method of least squares, which values are dependent on the values of the initial vibration amplitude, influence coefficients and correction weights. Thus, a weight corresponding to the aforesaid correction weight is attached to the aforesaid balancing plane to reduce vibrations in a shaft.

Description

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;

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.

.il 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.

., ."~

, 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--B

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-~. , - : , , . : , ~)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 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 ~

.~

~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

~: 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-:

.. . .
. - ,, - .
.. . - .. . . . . .. . .
.. , . . - . .

-- 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 ~ :

- . . .

- `
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 .
' ' , ' ' ' ~ .

-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:

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_ ,: :
.
., , , ~, ~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 ~, .

~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. -~

,.~,.

Claims (9)

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:

1. 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 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.

2. A balancing method as claimed in claim 1, further comprising the steps of:
determining the sums of products of squares of the at least one correction weight and factors corresponding to respective correction weights, for respective balancing planes;
determining compensated correction weights for making the total sum of said sums of products and sums of squares of said residual vibration amplitudes a minimum; and making all of said compensated correction weights smaller than an allowable value.

3. A balancing method as claimed in claim 1, characterized in that said different measuring conditions include a plurality of vibration measuring points.

4. A balancing method as claimed in claim 1, characterized in that said different measuring conditions include a plurality of revolutionary speeds.

5. A balancing method as claimed in claim 1, characterized in that said different measuring conditions include a plurality of revolutionary speeds and a plurality of vibration measuring points.

6. A balancing method as claimed in claim 1, characterized in that said various measuring conditions include a plurality of revolutionary speeds and biaxial measurements in two sub-stantially orthogonal directions.

7. A balancing method as claimed in claim 1, characterized in that said different measuring conditions include a plurality of revolutionary speeds, a plurality of vibration measuring points and biaxial measurements in two substantially orthogonal directions.

8. 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 influence coefficients representative of vibration amplitudes at the at least one vibration measuring point when a unit weight is attached to respective ones of predetermined balancing planes on the shaft; and determining at least one correction weight so as to reduce the values of residual vibration amplitudes at the at least one vibration measuring point, the values of residual vibration amplitudes being dependent on the initial vibration amplitudes, influence coefficients and the at least one correction weight on the balancing planes;

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 by the method of least squares, at least one compensated correction weight for minimizing the sum of squares of respective residual amplitudes of respective vibration measuring positions multiplied by a weight factor of an i th step given by the equation:

where;
.epsilon.m is the residual vibration amplitude, .lambda.m(i) denotes' a weight factor used in the step of number i computation for a compensating correction weight, and M denotes a product of the number of measuring positions and number of measuring conditions, S and R are parameters utilized for simplification of the equation; and repeating the above stated steps until all residual vibration amplitudes with the at least one compensated correction weight attached to the rotor shaft becomes lower than an allowable value.

9. A balancing method as claimed in claim 8, further comprising the steps of:
determining the sums of products of squares of said compensated correction weights and values of said factor corresponding to respective correction weights for respective balancing planes; and making all of recompensated correction weights smaller than an allowable value, said recompensated and sums of squares of residual vibration amplitudes at respective vibration measuring points multiplied by said weight fators a minimum.