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John J. Yu, Ph.D. – Senior Engineer, Machinery Diagnostics – Bently Nevada Asset Condition Monitoring – GE Energy – john.yu@ge.com

of Influence Coefficients Between Static-Couple and Multiplane Methods on Two-Plane Balancing

This article was originally published in Vol. 131, Issue 1 of the Journal of Engineering for Gas Turbines and Power of the American Society of Mechanical Engineers (ASME) International. It is reprinted here with the permission of ASME, which retains all copyrights.

nbalance accounts for the majority of high vibration problems in rotating machines. High synchronous forces and vibration amplitudes due to mass unbalance produce excessive stresses

on the rotor and also affect bearings and casing, thus reducing the life span of the machine. The source of unbalance may be imperfect manu-facturing processes including assembly variation and material nonhomogeneity. Though rotors are typically balanced by manufacturers before they are installed for service, unbalance may still occur afterward for various reasons. These include deposits or erosion on (and shifting of) rotating parts, as well as thermal effects. Therefore, in many cases, field balancing is required to reduce

synchronous vibration levels. Topics on balancing have been of great interest to rotor dynamic researchers and engineers [1,2]. Typically a turbine, compressor, or generator section is supported by two bearings. This often requires two-plane balancing for most cases where cross-effects among different sections through couplings are trivial. There are a few papers discussing two-plane balancing with amplitude [3] or phase [4] only. These approaches would often require more runs in the field and may increase both the time and the cost for users of rotating machinery. The influence coefficient method is typically used for field trim balancing. There are basically two approaches to apply this method. The first one is to treat it as a multiplane balance problem involving

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a 2X2 matrix of complex influence coefficients, as Thearle [5] first presented in 1934. In this approach, two direct influence coefficients along with two cross-effect influence coefficients are generated so that correction weights at two balance planes can be determined. The second one is to treat it as two single-plane balance problems using static and couple components, respectively. The latter approach has been used extensively in the field [6,7]. Having valid influence coefficient data makes balancing much easier. Influence coefficient data can be employed to save trial runs for many machines of the same design or for future balanc-ing on the same machine. For two-plane balancing with influence coefficients, either static-couple or multiplane approaches can be used. However, no

relationship of influence coefficients was given between these two approaches. It was also some-times believed that static-couple balance could not reduce both static and couple vibration vectors successfully because static (couple) weights affect couple (static) response. In this paper, the multiplane approach with a 2X2 influence coef-ficient matrix is first presented, followed by the static-couple approach. In the latter approach, cross-effects between the static (couple) weights and the couple (static) component are introduced. Then, an analytical relationship of influence coef-ficients between these two approaches is derived for two-plane balancing. Real examples are given to verify the developed analytical conversion formulas as well as to show their application.

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Multiplane Method

As shown in Figure 1, synchronous 1X vibration vectors

are expressed as A1 and A2 measured by probes 1 and 2,

respectively.

Their orientations a1 and a2 are defined by phase

lagging relative to their probe orientation (Figure 1

shows the instant when the Keyphasor* pulse occurs).

Balance weights at weight planes 1 and 2 are expressed

as W1 and W2 with their orientations b1 and b2 refer-

enced to the probe orientation, respectively. Assuming

that the system is linear, changes in 1X vibration vectors

due to weight placement can be given by

where h11, h12, h21, and h22 form the 2x2 influence coeffi-

cient matrix. Superscripts “�(0)” and “�(1)” represent status

without and with weights W1 and W2, respectively.

Typically, the four influence coefficients, through two

trial runs, can be computed as follows:

Figure 1. Diagram of vibration and weight vectors when the Keyphasor pulse occurs.

(1)

(2)

where superscript (0) represents status without weights

and superscripts (1) and (2) denote status with the first

and second sets of weights. Note that the two sets of

weights must be chosen in a way that the weight matrix

is not singular.

(3)

Static-Couple Method

In the static-couple approach, as shown in Figure 2,

vibration vectors at both ends of the shaft are

expressed as the combination of static and couple

components as follows:

where S and C are defined as static and couple

components, respectively.

The static influence coefficient is computed based on

vectorial changes in S due to the static weights WS

(which can be sometimes placed as one weight in the

middle balance plane), as shown in Figure 2. The couple

influence coefficient is calculated based on vectorial

changes in C due to the couple weights WC (180 deg

apart at two ends). When the static (couple) component

is dominant, the static (couple) weight approach alone

may be adopted. In the case that both components are

high, up to four runs are often used to balance both

static and couple components.

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This article introduces the following static-couple

balance model to include these cross-effects:

where superscripts (0)and (1) represent status without

and with static weight(s) WS and/or couple weights WC.

Equation (4) also applies to the case where the static

weight WS is placed in the middle plane instead of two

end planes. The above four influence coefficients can be

computed by placing static/couple weights.

Having vibration data before and after static weight(s)

placement WS (without couple weights) yields

(4)

and

(5)

(6)

and

(7)

(8)

where ∆CC is the couple vibration component with

couple weight(s)—couple vibration component without

couple weights and ∆SC is the static vibration compo-

nent with weight(s)—static vibration component without

couple weights.

Equations (5) and (7) have been widely used to compute

the effect of static weight(s) to the static component and

the effect of couple weights to the couple component,

respectively. However, the cross-effect of static

weight(s) to the couple component or couple weights

to the static component has not been introduced so

far and has often been assumed to be zero. In a real

where ∆SS is the static vibration component with static

weight(s)—static vibration component without static

weight(s) and ∆CS is the couple vibration component

with static weight(s)—couple vibration component

without static weight(s).

Similarly, having vibration data before and after couple

weight placement WC (without static weights) yields

However, cross-effects of static weights to the couple

component or couple weights to the static component

have often been neglected when performing balancing.

A nonsymmetric rotor with respect to its two ends, or

strongly influenced by its adjacent section via coupling,

might have significant cross-effects.

Figure 2. Diagram of static/couple vibration and weight vectors when the Keyphasor pulse occurs.

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(13)

(14)

(15)

(16)

Using Equations (13)–(16), individual probe influence

vectors near plane 1 or 2 due to static or couple weights

can also be expressed in terms of static or couple influ-

ence vectors as follows:

(17)

(18)

(19)

(20)

Combining Equations (6), (9), and (10) yields (note that

∆CS=�(∆A1,S −∆A2,S)/2)

Combining Equations (7), (11), and (12) yields (note that

∆CC=�(∆A1,C −∆A2,C)/2)

Combining Equations (8), (11), and (12) yields (note that

∆SC=�(∆A1,C +∆A2,C)/2)

Note that all the above equations apply to cases where

static weights are placed either at the middle balance

weight plane only or at two end balance weight planes

with the same amount of weights in the same orienta-

tion. Couple weights are always defined throughout the

paper as placement at two end balance planes with the

same amount of weights in the opposite orientation (180

deg apart).

rotor where asymmetry exists due to rotor structure or

coupling effects, the cross-effect could be significant.

Equations (6) and (8) include these cross-effects. After

both static and couple balancing without considering

the cross-effects, residual unbalance response could

still be high. However, if these four influence coefficients

are obtained, both the static and couple vibration

components can be effectively reduced by applying

appropriate static and couple weights. Thus synchro-

nous vibration levels at plane 1 (A1=�S +C) and plane 2

(A2=�S −C) will be reduced accordingly.

Equation (4) shows that vibration can be effectively

reduced using the static-couple approach by including

cross-effects. There appears no need to reduce the

static (or the couple) component perfectly with the static

(or the couple) weights before making a trial run with

the couple (or the static) weights, if both the static and

couple weights are going to be tried. After trial runs with

static and couple weights, respectively, all direct and

cross-effects can be obtained, as shown in Equation 4.

where

∆A1,C =� A1 with couple weights − A1 without couple

weights

∆A2,C =� A2 with couple weights − A2 without couple

weights

Combining Equations (5), (9), and (10) yields (note that

∆SS=�∆A1,S+∆A2,S/2)

When the static or the couple component appears to

be larger, only static weight(s) or couple weights are

sometimes used. An optimized static or couple weight

solution can be obtained to include the cross-effect.

Sometimes one needs to know individual probe influ-

ence due to static or couple weights. The static weight

influence to probes near planes 1 and 2 can be given by

where

∆A1,S =� A1 with static weight(s) − A1 without static

weight(s)

∆A2,S =� A2 with static weight(s) − A2 without static

weight(s)

Similarly, the couple weight influence to probes near

planes 1 and 2 can be given by

and

(9)

(10)

and(11)

(12)

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Relationship of Influence Coefficients Between the Two Methods

When performing balancing in the field, sometimes the

number of weights or the amount of weights (heavy

metal weights may not be allowed due to high tempera-

ture on some rotors such as high pressure (HP) section)

is limited at balance planes. In this case, even if a 2X2

influence coefficient matrix is available that may lead

to placement of a large amount of weights at two end

planes, one would prefer to use less amounts of static

or couple weights only to reduce vibration to acceptable

levels. Using either static or couple weights would

depend on which component is dominant and which

weight placement is more efficient (having sensitivity of

static and couple influence vectors would help to deter-

mine). Having influence vectors for static and couple

weights with the same phase lag reference for weights

and vibration vectors (suggested to use for balancing,

preferably aligned to the probe orientation), one would

be able to see how the rotor is running before, after,

or close to the translational, pivotal, or other bending

modes based on phase lag angle of static and couple

influence vectors. The above-mentioned questions can

be answered by conversion of influence vectors from

the multiplane method to the static-couple method.

On the other hand, one would also need to know

influence vectors expressed in terms of the multiplane

method from known static and couple influence vectors

in some cases. Sometimes only one end balance plane

can be used due to unavailable empty holes or slot

section for weights, or difficult access on the other

end plane. In thermal bow/rub situations, calculating

additional unbalance (caused by thermal bow) using

vibration excursion vectors compensated by the normal

running condition vectors based on the multiplane

influence model would help to determine the thermal

bow/rub location (close to balance plane 1 or 2). Using

the 2X2 multiplane method would also directly lead

to weight placement at planes 1 and 2. Those would

require conversion of influence vectors from the static-

couple method to the multiplane method.

Applying arbitrary static weights WS only at planes 1

and 2, Equation (1) after replacing W1 and W2 each with

WS can be reformulated to

(21a)

(21b)

Applying the same static weights WS only at planes

1 and 2, Equation (4) after setting WC=�0 can be

reformulated to

(22a)

(22b)

Addition of Equations (21a) and (21b) followed by

subtraction of Equation (22) with application of Equation

(3) yields

(23)

Subtraction of Equation (21) from Equation (21) followed

by subtraction of Equation (22) with application of

Equation (3) yields

(24)

Similarly, applying arbitrary couple weights WC only at

planes 1 and 2, Equation (1) after replacing W1 with WC

and W2 with −WC can be reformulated to

(25a)

(25b)

Applying the same static weights WC at planes 1 and 2,

Equation (4) after setting WS=�0 can be reformulated to

(26a)

(26b)

Addition of Equations (25a) and (25b) followed by

subtraction of Equations (26a) with application of

Equation (3) yields

(27)

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Subtraction of Equation (25b) from Equation (25a)

followed by subtraction of Equation (26b) with

application of Equation (3) yields

(28)

Thus, conversion equations of influence vectors from

the multiplane method to the static-couple method are

given by Equation (23) (direct static influence vector),

Equation (24) (cross-effect of the couple component due

to static weights), Equation (27) (cross-effect of the static

component due to couple weights), and Equation (28)

(direct couple influence vector). Combining Equations

(23), (24), (27), and (28), conversion of influence vectors

from the static-couple method to the multiplane method

can also be given by

(29)

(30)

(31)

(32)

Combining Equations (13)–(16) and (29)–(32) yields

influence vectors with the multiplane method expressed

by individual probe influence vectors due to static and

couple weights as follows:

(33)

(34)

(35)

(36)

Combining Equations (33)–(36), individual probe

influence vectors due to static or couple weights can

also be expressed in terms of influence vectors with

the multiplane method as follows:

Note that all the above equations in this section apply

to cases where static weights are placed at two end

balance weight planes. In case the static weight is

defined as placement at the middle balance plane,

Equations (27), (28), (39), and (40) are still valid.

Figure 3. Rotor kit for balance calculations.

Example 1 – Rotor Kit Verification

The first real example presented here is mainly to verify

the above-developed equations of influence coefficient

conversion between multiplane and static-couple

methods. In this example, a Bently Nevada* RK-4 rotor

kit was used, as shown in Figure 3. A shaft with the

diameter and length of 0.01 m and 0.56 m, respectively,

was supported by two brass bushing bearings and

driven by a 75 W motor. Three 0.8 kg disks were

attached to the shaft with one close to bearing No. 1

and two close to bearing No. 2, thus having asymmetri-

cal mass distribution with respect to the two bearings.

The rotor was also supported by a midspan spring to

prevent excessive bow in the middle of the shaft. The

data acquisition and processing system consisted of

two pairs of X-Y displacement proximity probes, one

speed probe, and one Keyphasor* probe for speed and

phase measurement. Two balance weight planes 1 and

2 are located adjacent to bearing Nos. 1 and 2 as well

as their corresponding proximity probes. The shaft was

rotated in the counterclockwise direction when viewed

from the motor to bearing #2.

In this example, the running speed for balance was set

at 4800 rpm for demonstration. Since higher amplitudes

occurred in the horizontal direction at the running speed,

influence coefficient calculations were carried out in

terms of vibration readings measured by the two hori-

zontal probes located 90 deg right of top, as shown in

Figure 3. From an initial run without any balance weight

placement, synchronous vibration vectors at bearing

Nos. 1 and 2 in the horizontal direction were as follows:

(37)

(38)

(39)

(40)

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Figure 4. Polar plots and vibration vectors at approximately 4800 rpm for initial run, and first and second trial runs with weight placements.

With the following two 0.4 g weights placed at planes 1

and 2 (see Figure 4):

the corresponding vibration vectors became

Placing the following two 0.8 g weights at planes 1 and 2

(see Figure 4) after removing the above two 0.4 g weights

corresponded to the following vibration vectors:

Figure 4 shows polar plots and vibration vectors at

approximately 4800 rpm for three different runs as well

as two sets of weight placement. Using Equation (2), the

influence coefficient matrix for the multiplane method

is computed as

Assuming that synchronous vibration vectors are lin-

early proportional to applied balance weights, arbitrary

two weight placement sets (as long as its weight matrix

is not ill conditioned or singular) should yield the same

influence coefficient matrix for this multiplane method at

this running speed. Actually, the other two sets of weight

placement (placing only one weight at one time at one

plane followed by the other plane) were tried, which

produced the results very close to the above ones.

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and the second trial run with static weights

can be computed, respectively, as follows:

Note that

and

The influence vectors due to static and couple weights

placed at two ends are computed directly from their

definition, as shown in the right column of Table 1.

The left column of Table 1 shows calculated results,

using Equations (23), (24), (27), (28), and (37)–(40),

based on known h11, h12, h21, and h22 values from the

multiplane method. It is found that the results in the left

column are the same as those in the right column; small

differences appear just due to rounding errors during

computations.

It is shown from this real example that influence vectors

for the static-couple method can be calculated from

known influence coefficients h11, h12, h21, and h22 in a

2X2 matrix for the multiplane method, without having to

place trial static or couple weights. Since Eqs. (29)–(32)

are equivalent to Equations (23), (24), (27), and (28), and

Equations (33)–(36) are equivalent to Equations (37)–(40),

Equations (29)–(36) also hold true in this example.

Therefore, influence coefficients h11, h12, h21, and h22

can also be obtained from influence vectors for the

static-couple method without having to place two sets

of trial weights.

In this example, it is found that static weights affect the

couple vibration vector (HCS is about 2.8 mils pp/g < 24

deg) and that couple weights affect the static vibration

vectors HSC is about 4.0 mils pp/g < 39 deg). These

cross-effects are even higher than the direct static

influence vector HSS about 1.1 mils pp/ g < 161 deg).

The high influence vector HCC about 16.6 mils pp/g < 44

deg) indicates a very sensitive couple weight effect. The

phase readings in HSS and HCC indicate that the rotor kit

runs after the first bending resonance speed and before

the second bending resonance speed. This is in good

agreement with the polar plots of Figure 4.

Using either Equation (1) for multiplane method or

Equation (4) for static-couple method, the required

balance weights to offset the initial vibration at two

planes can be determined. The former approach yields

the following balance weights:

Thus, the above four values within the matrix are the

influence coefficients for the multiplane method at

this running speed. It is noted that the above two sets

of weight placement were also just for couple and

static weights, respectively. Therefore, the influence

coefficients for the static-couple method can be directly

computed. Using Equation (3), static and couple vibra-

tion vectors for the initial run without weight placement,

the first trial run with couple weights

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The latter approach yields the following weights:

Note that

The above two sets of weights are identical. Among

available weights and holes, the final weights and their

orientations were chosen as follows:

Figure 5 shows synchronous orbits before and after

the balance with the above weights. The synchronous

vibration level has been reduced from around 6 mils to

less than 1 mil after placing the above weights.

Table 1. Verification of influence coefficient conversion between multiplane and static-couplemethods on a real example.

Figure 5. Synchronous orbits before and after balance at bearing Nos. 1 and 2.

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Figure 6. Polar plots and vibration vectors at 3600 rpm before and after balance.

Figure 7. Synchronous orbits at 3600 rpm before and after balance.

Table 2. Calculated influence vectors in static and couple methods from the known influence vectors in the multiplane method, without placing static or couple trial weights.

Example 2 – Steam Turbine Generator Application

The second example is to demonstrate how to apply

the developed conversion between the two methods

when an influence coefficient matrix for the multiplane

method is known. In this example, high synchronous

vibration due to unbalance was observed via proximity

probes on a 66 MW hydrogen-cooled generator driven

by a steam turbine. The machine is a two-pole genera-

tor and was run at 3600 rpm. It rotates clockwise when

viewed from the turbine towards the generator. The

two generator bearings were named as bearing Nos.

5 (drive-end) and 6 (nondrive-end). A pair of X-Y probes

was installed at 45 deg left and right at bearing No. 5

while another pair of X-Y probes was installed at 60 deg

left and 30 deg right at bearing No. 6.

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Synchronous vibration amplitudes were higher on

Y-probes than on X-probes at the two bearings on

the generator. Balance calculations were therefore

conducted on Y-probes only. In order to use the

same nomenclature and subscripts for the equations

developed earlier, probes and weight plane at bearing

No. 5 are denoted as 1 while those at bearing No. 6 are

denoted as 2. As shown in Figure 6, Y-probe readings at

bearing Nos. 5 and 6 were

The previous influence coefficients used for the

multiplane method were given by

where h11, h12, h21, and h22 were applied to Equation (1)

in which synchronous vibration vectors were defined as

original ones from the two Y-probes 1 was referenced

to 45 deg left and 2 was referenced to 60 deg left, while

weights at both ends were all referenced to 45 deg left.

The balance plane radius where weights were placed

was about 0.254 m (10 in.) with the one at bearing No.

5 slightly larger than that at bearing No. 6 (about 1%

difference). Note that the radius difference between the

two weight planes would not affect the validity of all

the equations developed in the paper. Weight planes

at bearing Nos. 5 and 6 had 44 and 36 holes for weight

placement, respectively. Their weight sizes were also

different between two planes.

Using Eq. (1), the required balance weights at two planes

appeared to be

These large amounts of weight at two planes were

unable to be placed into empty holes or achieved by

adjustment of existing weights. An alternative needed

to be found. The study of influence data was then

performed. Influence coefficients for static and couple

weights were calculated based on known h11, h12, h21,

and h22 values without placing static or couple trial

weights. Note that the Y-probe at bearing No. 6 was

not parallel to the Y-probe at bearing No. 5. In order to

evaluate static and couple effects better, the synchro-

nous vector at bearing No. 6, as though it was measured

by a proximity probe at 45 deg left, needed to be known,

and h11, h12, h21, and h22 needed to be applicable to this

change. Although the above-mentioned synchronous

vector at bearing No. 6 could be determined by using

vectors from both X and Y probes, h11, h12, h21, and h22

might not fit the new defined vector. Therefore, the origi-

nal vector was used as the new vector except its phase

was lagged an addition 15 deg. Thus, the two vibration

vectors referenced to 45 deg left became

and the influence matrix with both vibration and weight

vectors referenced to 45 deg left became

Table 2 shows calculated influence vectors for static

and couple weights from known influence vectors

h11, h12, h21, and h22 used for the multiplane method,

without having to place static or couple trial weights.

The direct couple influence vector HCC was the most

sensitive one (0.0111 mil pp/g 131 deg), indicating that

appropriate couple weights would effectively reduce the

current synchronous vibration level, especially to bear-

ing No. 5 (h1,C=�0.0135 mil pp/g 131 deg). Static weights

appeared not to be sensitive to synchronous vibration

vectors at the running speed for this generator, as

shown in Table 2. The current static and couple vibration

vectors were as follows:

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Using Equation (4) by setting WS=�0 and neglecting HSC

effect, the required couple weights were calculated as

follows:

or

Based on available weights and holes on the two

balance planes as well as the above estimation, the

following chosen weights

would yield synchronous vibration vectors of about 0.2

mil pp and 0.7 mil pp at bearing Nos. 5 and 6, predicted

from the original multiplane influence coefficient matrix.

After placing the above weights, synchronous vibrations

at bearing Nos. 5 and 6 were reduced to 0.2 mil pp and

0.4 mil pp, respectively, as shown in Figures 6 and 7.

Conclusions

Based on both analytical and real case studies pre-

sented in this article, the following five conclusions are

stated regarding influence vectors using static-couple

and multiplane methods for two-plane balancing:

1. For the static-couple method, cross-effects between

static weights and couple response as well as

between couple weights and static response can be

included so that a good combination of static and

couple weights can be applied to offset synchronous

vibration more effectively.

2. Conversion equations of influence vectors between

the static-couple and multiplane methods are

given in this paper. Equations (23), (24), (27), and (28)

are used for conversion from multiplane format to

static-couple format and Equations (29)–(32) are

used for conversion from static-couple format to

multiplane format.

3. Individual probe influence vectors due to static or

couple weights can be vital information. Static and

couple influence vectors as well as cross-effects can

be evaluated from them by using Equations (13)–(16)

and multiplane influence vectors can be evaluated

from them by using Equations (33)–(36).

4. The above analytical findings have been confirmed

by experimental results.

5. The analytical findings can be applied to real rotat-

ing machinery balancing as shown in this article.

Effective balance weights can be best evaluated

by using conversion equations of influence vectors

between multiplane and static-couple formats.

Knowing influence vectors in both formats can also

help troubleshoot unbalance changes as well as

running modes.

* denotes a trademark of Bently Nevada, LLC, a wholly owned subsidiary of General Electric Company.

Acknowledgment

The author is grateful to Robert C. Eisenmann, Sr. of

GE Energy for his support and comments on the current work.

Vol .29 No.1 2009 ORBIT 85

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Greek

Superscripts

References

[1] Ehrich, F. F., 1999, Handbook of Rotordynamics,

Krieger, Malabar, FL.

[2] Foiles, W. C., Allaire, P. E., and Gunter, E. J., 1998,

“�Review: Rotor Balancing,” Shock Vib., 5, pp. 325–336.

[3] Everett, L. J., 1987, “�Two-Pane Balancing of a Rotor

System Without Phase Response Measurements,”

Trans. ASME, J. Vib., Acoust., Stress, Reliab. Des., 109,

pp. 162–167.

[4] Foiles, W. C., and Bently, D. E., 1988, “�Balancing With

Phase Only (Single-Plane and Multiplane),” Trans.

ASME, J. Vib., Acoust., Stress, Reliab. Des., 110, pp.

151–157.

[5] Thearle, E. L., 1934, “�Dynamic Balancing of Rotating

Machinery in the Field,” Trans. ASME, 56, pp. 745–753.

[6] Wowk, V., 1995, Machinery Vibration: Balancing,

McGraw-Hill, New York.

[7] Eisenmann, R. C., Sr., and Eisenmann, R. C., Jr., 1997,

Machinery Malfunction Diagnosis and Correction:

Vibration Analysis and Troubleshooting for the

Process Industries, Prentice-Hall, Englewood

Cliffs, NJ.

Nomenclature