Use of Conversion Equations of Influence Coefficients on Two-Plane Balancing

John J.Yu Nicolas Pé ton

（1.Baker Hughes,Atlanta,USA,john.yu@baker hughes.com;2.Baker Hughes,Nantes,France,nicolas.peton@baker hughes.com）

Abstract：Typical rotors such as those on steam turbine or generator are often supported by two bearings with two balance planes at both ends.Vibrations are monitored by a pair of proximity probes at each bearing.There are two approaches to reduce 1X vibration due to unbalance at both ends via balancing with influence coefficient method.The first approach is to treat it as a multiple-plane balancing problem involving 2x2 matrix of complex influence coefficients.The second approach is to treat it as two single-plane balance problems using static(in-phase)and couple(180 degree out-of-phase)components,respectively.Conversion equations of influence coefficients between these two approaches have been found previously by the author.The corresponding spreadsheets that convert influence coefficients between these two formats are presented in the current paper.The paper shows effectiveness of these conversion equations in dealing with real balancing problems in the field.A detailed balance case is presented to demonstrate how the conversion equations are used to reduce vibration effectively.

Keywords：Two-plane Balancing；Field Balancing；Influence Coefficients；Static/couple Components

Nomenclature

synchronous vibration vector measured by probe 1

synchronous vibration vector measured by probe 2

couple vibration vector

couple vibration influence vector due to couple weight

couple vibration influence vector due to static weight(cross effect)

static vibration influence vector due to couple weight(cross effect)

static vibration influence vector due to static weight

probe 1 vibration influence vector due to plane 1 weight

probe 1 vibration influence vector due to plane 2 weight

probe 2 vibration influence vector due to plane 1 weight

probe 2 vibration influence vector due to plane 2 weight

probe 1 vibration influence vector due to couple weight

probe 2 vibration influence vector due to couple weight

probe 1 vibration influence vector due to static weight

probe 2 vibration influence vector due to static weight

static vibration vector

weight vector at balance plane 1weight vector at balance plane 2couple weight vectorstatic weight vector

Greek

α1=phase lag of

α2=phase lag of

β1=phase lag of

β2=phase lag of

Superscripts

(0)=initial status without weights

(1)=status with weights or with first trial weights

(2)=status with second trial weights

0 Introduction

High consistent vibration in rotating machines is usually caused by mass unbalance.The life span of the machine could be reduced as a result of excessive stresses on the rotor,bearings and casing if vibration level is not reduced via balancing.The source of unbalance includes assembly variation and material non-homogeneity.This can happen on new machines or machines after rotor repair or overhaul.Though rotors are often low-speed or sometimes high-speed balanced by manufacturers or workshops before they are installed for service,unbalance may still occur afterwards due to various reasons such as deposits on or erosion and shifting of rotating parts as well as thermal effects.Hence,balancing is often required in the field and has been of great interest to rotor dynamic researchers as well as practicing engineers.

A rotor such as that on a steam turbine or generator is often supported by two bearings with two balance planes available at both ends.Vibrations are monitored by a pair of proximity probes at each bearing.This often requires twoplane balancing of the rotor.Everett[1],and Foiles and Bently[2] discussed two-plane balancing with amplitude or phase only,which would often require more trial weight runs in the field.The influence coefficient method is typically used in the field for trim balancing with two approaches.The first method is to treat it as a multiple-plane balancing problem involving 2×2 matrix of complex influence coefficients,as indicated by Thearle[3].In this method,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 method is to treat it as two single-plane balance problems using static (in-phase)and couple (180 degrees out-of-phase) components,respectively,as shown by Wowk[4].

Note that the static and couple components are referred to as in-phase and 180 degrees out-of-phase components,respectively.The static component is usually due to first and/or third modes that are very symmetric to the rotor mass center while the couple component is typically due the second mode that is nearly anti-symmetric.Static weight is defined as a weight vector at the first balance plane,always accompanied by the same weight vector at the second balance plane.And couple weight is defined as a weight vector at the first balance plane,always accompanied by the opposite or 180 degrees out-of-phase weight vector at the second balance plane.Therefore,the current“static”and“couple”vibration or weight vectors are really referred to as“in-phase”and“180 degrees out-of-phase”,respectively.“Static”and couple”terms have their origin from rigid rotor balancing and have been extended to flexible rotors for balancing the first,the second,and/or even the third modes.These terms are well defined in the standards of balancing and have been widely accepted and used in the field.Therefore,these terms are used throughout this paper so that practicing engineers can apply the solutions from the paper with the above definitions.

Yu indicated the relationship of influence coefficients between static-couple and multiplane methods on two-plane balancing[5].Thus,static or couple influence coefficients due to static or couple weights can be obtained directly without having to place static or couple trial weights if influence coefficients used in the multi-plane approach are known.From static and couple influence data as well as cross effects,influence data for the multi-plane method can be obtained directly as well without having to place any trial weights at either plane.Yu applied the relationship into twoplane balancing of symmetric rotors[6].Yu,and Yu and Ashar used these conversion equations to deal with two-plane rotor balancing problems in the field effectively[7-8].

The current paper will demonstrate a case where previously obtained static and couple pair weight influence data could be converted into one-plane influence data that made one-shot balancing successfully.Such a case has not been presented before.Moreover,corresponding spreadsheets have been developed that can easily convert influence coefficients between these different expressions,and will be presented here for reference.

1 Theory

Note that each influence coefficient is a complex number that contains not only sensitivity of synchronous 1×vibration amplitude from a particular probe versus balance weight but also 1×vibration phase lag relative to the weight.When influence data is concerned,a common reference of orientation should be used for both weight and vibration vectors.Since phase lag of the synchronous 1×vibration vector is always referenced to the probe against shaft rotation from vibration test equipment such as ADRE?,phase lag of the weight vector should also be referenced to the probe against shaft rotation in calculation.As a result,developed influence vectors such as static and couple influence coefficients would be meaningful by looking vibration phase lags relative to the weight vector.

1.1 Multiplane Balance Model

As shown in Figure 1,synchronous 1×vibration vectors are expressed asmeasured by probes 1 and 2,respectively.Their orientationsα1andα2are defined by phase lag relative to their probe orientation (Fig.1 shows the instance when Keyphasor?pulse occurs).Balance weights at weight planes 1 and 2 are expressed aswith their orientationsβ1andβ2referenced to the probe orientation,respectively.Assuming the system is linear,changes in 1×vibration vectors due to weight placement can be given by

Fig.1 Diagram of vibration and weight vectors when Keyphasor?pulse occurs

1.2 Static-Couple Balance Model

In the static-couple method,as shown in Fig.2,vibration vectors at both ends of the shaft are expressed as combinations of static and couple components as follows:

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

whereare defined as static and couple components,respectively.Similarly,static weightis defined as being in-phase each at two ends with the same amount while couple weightis defined as the weight vector at plane 1 accompanied with the,180 degrees out-of-phase with the same amount.The static-couple balance method fits to both rigid and flexible rotors.

The following static-couple balance model was introduced by Yu[9]:

where superscripts“(0)”and“(1)”represent status without and with weights.

Sometimes one needs to know individual probe influence data due to static or couple pair weight.Yu [5] yielded the relationship between individual probe influence vectors near plane 1 or 2 and static or couple influence vectors,due to static or couple pair weight.Individual probe influence vectors can be given in terms of static or couple vibration component influence vectors as below:

while static or couple vibration component influence vectors can be given in terms of individual probe influence vectors as below:

1.3 Conversion Equations

As proved by Yu[5],the relationship on two-plane balancing between static-couple influence coefficients

and multiplane influence coefficients

is as follows:

2 Real example

In this example,high synchronous 1×vibration due to unbalance was observed via proximity probes on an air-cooled generator driven by a gas turbine,as shown in Fig.3.The base load full power output is approximately 100 MW.The gas turbine was composed of low pressure booster,high pressure core,and power turbine.The 2-pole generator was directly coupled to a power turbine with a spool at rated speed of 3600 r/min.The machine was viewed from the gas turbine to the generator and therefore the generator was considered to be rotating in the counter-clockwise direction.The 9-meter long generator rotor weighted about 26500 kg and was supported by two elliptical journal bearings.

Fig.3 Machine train diagram of generator rotor balance example

Vibration was monitored byXYpairs of non-contacting proximity probes mounted at 45-degree right (X-probe) and 45-degree left(Y-probe)relative to the 0 degree vertical reference.The alert alarm was set as 3.0 mil pp (76.2 μm pp) at rated speed of 3600 r/min.As usual,for this counterclockwise shaft rotation machine,X-directional amplitude was higher than its correspondingY-directional amplitude.Therefore,X-directional vibration was used to calculate its amplitude and phase lag as well as the corresponding influence data.Generator drive end (DE) and non-drive end (NDE) are referred to as bearing 1 and bearing 2,respectively,for vibration readings and influence data fromX-probes.

The initial run without weight placement at rated speed of 3600 r/min as indicated in Fig.4 showed the following stabilized 1Xvibration vectors fromX-probes as below

Fig.4 Direct and 1×trend plots(a)and 1×polar plots(b)at DE and NDE X-probes during the initial run without weight placement

This type of machine had been balanced previously at two different sites.At one site,a static pair weight (same amount of weights placed in the same orientation at both plane 1 and plane 2) was used at that time to reduce vibration level at both bearing 1 and bearing 2 satisfactorily.And its static pair influence data at rated speed of 3600 rpm was known for probe 1 (DE-Xprobe) and probe 2 (NDE-Xprobe),respectively as below:

At another site,a couple pair weight (same amount of weights placed in the opposite orientation at plane 1 and plane 2)was used at that time to reduce vibration level at the second critical speed of around 3000 r/min.And its couple pair influence data at rated speed of 3600 r/min was also computed for probe 1 (DE-Xprobe) and probe 2 (NDE-Xprobe),respectively as below:

It can be noticed that vibration was higher than 3 mil pp only at generator DE.Was it possible to place weights only at DE fan ring,i.e.,plane 1?

To answer this question,influence coefficientsandneed to be obtained first.Based on the conversion equations described in the early part of this paper,all influence coefficients in any format,including,can be computed easily using a developed Excel spreadsheet,as shown in Fig.5.

Fig.5 Conversion of influence data between different formats using Excel spreadsheet for the example

Influence vectorscan also be computed directly as shown in Yu[5]as below

The corresponding plane1 weight only solution by using the above obtained influence coefficientsbased onwith

would yield predicted 1X vibration vectors at bearing1 and bearing 2 as below

Therefore,the above calculated weights were added at Plane 1 only,as shown in Fig.6.

Fig.6 One-shot balance weights at generator DE fan ring(Plane 1)

The one-shot balancing with weights at Plane 1 only reduced vibration at both ends effectively.Fig.7 shows direct and 1× trend plots as well as 1× polar plots at DE and NDE X-probes after placement of 5 weights at DE fan ring plane.Stabilized 1× vibration vectors measured byx-probes as shown in Fig.7 after balance became

Fig.7 Direct and 1×trend plots(a)and 1×polar plots(b)at DE and NDE X-probes after the one-shot balancing

which are very close to the predicted 1×vibration vectors.

Note that the static pair influence vectorsas well as the couple influence vectorsat rated speed were not directly obtained from the same machine.These influence vectors were generated from other machines with the same design.However,they appeared to be accurate enough for predicted vibration response.Certainly,if these influence vectors generated from the identical machine,the results might be more accurate.

3 Conclusions

Conversion equations of influence coefficients can be very effective to deal with two-plane balancing problems.Previously obtained influence data due to static and couple pair weights can be used for future one-shot balancing with one-plane weights only.This can be applied not only for the identical machine but also for the other machines with the same design,in terms of validity of influence data.Application of the conversion equations can avoid unnecessary trial weight runs in the process of balancing,thus saving resources and fuels and releasing the machine for safe operation sooner than later.