Enhancement of Field Balancing Methods in Rotating Machines

Enhancement of Field Balancing Methods in

Rotating Machines

A thesis submitted to The University of Manchester for the degree of

Doctor of Philosophy (PhD)

In the Faculty of Science and Engineering

2017

Sami Meshal F Ibn Shamsah

School of Mechanical, Aerospace and Civil Engineering

http://www.manchester.ac.uk/

http://www.se.manchester.ac.uk/

https://uk.linkedin.com/in/sami-ibn-shamsah-872b8426

https://uk.linkedin.com/in/sami-ibn-shamsah-872b8426

http://www.mace.manchester.ac.uk/

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Table of Contents

LIST OF TABLES ............................................................................................................ 7

LIST OF FIGURES .......................................................................................................... 8

LIST OF NOMENCLATURES ...................................................................................... 13

LIST OF ABBREVIATIONS ......................................................................................... 14

LIST OF PUBLICATIONS ............................................................................................ 15

ABSTRACT .................................................................................................................... 16

DECLARATION ............................................................................................................ 17

COPYRIGHT STATEMENT ......................................................................................... 18

ACKNOWLEDGMENTS .............................................................................................. 19

DEDICATION ................................................................................................................ 20

INTRODUCTION ................................................................................. 21 CHAPTER 1

General introduction ......................................................................................... 22 1.1

Motivations ....................................................................................................... 25 1.2

Aims and objectives ......................................................................................... 26 1.3

Research contributions ..................................................................................... 27 1.4

Layout of thesis ................................................................................................ 28 1.5

LITERATURE REVIEW ...................................................................... 31 CHAPTER 2

Introduction ...................................................................................................... 32 2.1

Modal balancing approach ............................................................................... 32 2.2

Influence coefficient balancing method ........................................................... 34 2.3

Unified balancing approach .............................................................................. 41 2.4

Mathematical model-based rotor balancing technique ..................................... 42 2.5

2.5.1 Model-based rotor balancing using full mathematical model ............... 43

2.5.2 Model-based rotor balancing using reduced mathematical model ........ 44

Summary and conclusion ................................................................................. 51 2.6

EXPERIMENTAL SETUP AND INSTRUMENTATION .................. 52 CHAPTER 3

Introduction ...................................................................................................... 53 3.1

Experimental rig ............................................................................................... 53 3.2

3.2.1 Main elements of the experimental rig .................................................. 55

Data acquisition system .................................................................................... 65 3.3

3.3.1 Sensors ................................................................................................... 65

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3.3.2 Signal conditioner .................................................................................. 67

3.3.3 DAQ device ........................................................................................... 67

3.3.4 DAQ software ........................................................................................ 68

Modal tests ....................................................................................................... 69 3.4

3.4.1 Modal test of assembled rig ................................................................... 69

3.4.2 Modal testing of free-free shaft ............................................................. 76

3.4.3 Discussion on the influence of supporting structure ............................. 79

Summary .......................................................................................................... 80 3.5

MATHEMATICAL MODELLING AND SIGNAL PROCESSING ... 81 CHAPTER 4

Introduction ...................................................................................................... 82 4.1

Mathematical modelling of a simple rotating machine using the FE method .. 82 4.2

4.2.1 Shaft element ......................................................................................... 83

4.2.2 Disc element .......................................................................................... 90

4.2.3 Foundation model .................................................................................. 91

4.2.4 Influence of damping on the rotor dynamic model ............................... 92

4.2.5 Modelling of the system ........................................................................ 95

Signal processing .............................................................................................. 98 4.3

4.3.1 3D waterfall plot .................................................................................... 98

4.3.2 Order tracking ........................................................................................ 99

4.3.3 Bode plot ............................................................................................. 100

Summary ........................................................................................................ 102 4.4

SENSITIVITY ANALYSIS OF IN-SITU ROTOR BALANCING ... 103 CHAPTER 5

Introduction .................................................................................................... 105 5.1

Experimental rig ............................................................................................. 106 5.2

Experiments .................................................................................................... 106 5.3

Unbalance estimation ..................................................................................... 109 5.4

Results and observations ................................................................................ 110 5.5

Conclusion ...................................................................................................... 115 5.6

MULTI-PLANES ROTOR UNBALANCE ESTIMATION USING CHAPTER 6

INFLUENCE COEFFICIENT METHOD .................................................................... 116

Introduction .................................................................................................... 118 6.1

Experimental rig ............................................................................................. 119 6.2

Machine runs and data acquisition ................................................................. 120 6.3

Application of IC method ............................................................................... 123 6.4

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Unbalance estimation and discussion ............................................................. 124 6.5

Concluding remarks ....................................................................................... 128 6.6

SENSITIVITY ANALYSIS OF THE INFLUENCE COEFFICIENT CHAPTER 7

BALANCING METHOD FOR MULTIPLE PLANES ROTOR BALANCING WITH

REDUCED NUMBER OF SENSORS ......................................................................... 129

Introduction .................................................................................................... 131 7.1

Theory of IC balancing method ...................................................................... 133 7.2

Example-1: rig with one balancing plane ....................................................... 136 7.3

7.3.1 Experimental setup .............................................................................. 136

7.3.2 Instrumentation .................................................................................... 137

7.3.3 Modal tests .......................................................................................... 138

7.3.4 Experiments carried out ....................................................................... 140

Sensitivity analysis of unbalance estimation .................................................. 141 7.4

7.4.1 Using single speed ............................................................................... 141

7.4.2 Using speed range ............................................................................... 144

Example-2: rig with two balancing planes ..................................................... 147 7.5

7.5.1 Experiments carried out ....................................................................... 149

Sensitivity analysis of unbalance estimation .................................................. 152 7.6

7.6.1 Using vertical and horizontal responses .............................................. 152

7.6.2 Using radial responses only ................................................................. 161

Comparison of results ..................................................................................... 171 7.7

Overall observations ....................................................................................... 172 7.8


MATHEMATICAL MODEL-BASED ROTOR UNBALANCE CHAPTER 8

ESTIMATION USING A SINGLE MACHINE RUNDOWN WITH REDUCED

NUMBER OF SENSORS ............................................................................................. 175

Introduction .................................................................................................... 176 8.1

Earlier method ................................................................................................ 177 8.2

Proposed method ............................................................................................ 179 8.3

8.3.1 Theory ................................................................................................. 180

8.3.2 Parameter estimation ........................................................................... 182

Simulated example ......................................................................................... 185 8.4

Results and discussion .................................................................................... 185 8.5

8.5.1 Vertical response only ......................................................................... 186

8.5.2 Horizontal response only ..................................................................... 187

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8.5.3 Radial responses only .......................................................................... 188

Conclusion ...................................................................................................... 191 8.6

MULTI-PLANES ROTOR UNBALANCE IDENTIFICATION USING CHAPTER 9

DATA FROM A SINGLE MACHINE RUN-UP WITH REDUCED NUMBER OF

SENSORS ............................................................................................................. 193

Introduction .................................................................................................... 195 9.1

Earlier method ................................................................................................ 196 9.2

Proposed method ............................................................................................ 204 9.3

Experimental rig with one balancing plane .................................................... 205 9.4

Modal tests ..................................................................................................... 206 9.5

Experiments conducted .................................................................................. 209 9.6

Unbalance estimation ..................................................................................... 211 9.7

9.7.1 Part 1: application of the earlier method ............................................. 211

9.7.2 Part 2: application of the proposed method ......................................... 212

9.7.3 Comparison between the earlier and proposed methods ..................... 213

Modified test rig with two balancing planes .................................................. 214 9.8

9.8.1 Experiments and unbalance estimation ............................................... 218


CONCLUSIONS AND FUTURE WORK ...................................... 223 CHAPTER 10

Summary of research context ..................................................................... 224 10.1

Main achievements ..................................................................................... 225 10.2

Overall conclusion ...................................................................................... 227 10.3

Future work ................................................................................................. 227 10.4

REFERENCES .............................................................................................................. 228

APPENDICES .............................................................................................................. 239

Word Count: 43,197

LIST OF TABLES

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LIST OF TABLES

Table 3.1 Natural frequencies of the test rig ................................................................... 74

Table 5.1 Unbalance and phase of 8 runs ..................................................................... 108

Table 5.2 Sensitivity using different runs ..................................................................... 110

Table 6.1 Mass unbalances and phase angles of 6 machine runs.................................. 122

Table 7.1 List of 6 machine runs with different added unbalances (mass and phase

angles) ........................................................................................................................... 141

Table 7.2 Different scenarios used for the estimation of the added unbalance ............. 141

Table 7.3 List of 13 machine run-ups with different added unbalances (mass and phase

angles) ........................................................................................................................... 150

Table 7.4 Different scenarios used for the added unbalance estimation ....................... 151

Table 7.5 List of 10 scenarios used for the added unbalance estimation ...................... 162

Table 8.1 The unbalance and foundation stiffness configurations for the simulated

examples ........................................................................................................................ 186

Table 8.2 The estimated unbalance for the simulated examples using vertical and

horizontal directions separately..................................................................................... 188

Table 8.3 The estimated unbalance for the simulated examples using 𝑟𝑘1 and 𝑟𝑘2 directions separately ...................................................................................................... 189

Table 9.1 Experimentally identified natural frequencies of test rig with one balancing

disc at zero RPM ........................................................................................................... 207


angles) ........................................................................................................................... 210


Table 9.4 Estimated unbalance for the different scenarios using pair of orthogonal

sensors (at vertical and horizontal directions) at a bearing pedestal ............................. 212

Table 9.5 Estimated unbalance using only one sensor at a bearing pedestal (at radial

direction) ....................................................................................................................... 213

Table 9.6 Experimentally identified natural frequencies of test rig with two balancing

discs at zero RPM ......................................................................................................... 215


angles) ........................................................................................................................... 219


Table 9.9 Estimated unbalance for the different scenarios using 1 sensor (at radial

direction) per bearing pedestal ...................................................................................... 220

LIST OF FIGURES

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LIST OF FIGURES

Figure 1.1 (a) Photograph of a 9HA gas turbine rotor on the half shell casing [2], (b)

block diagram of the turbine in part a ............................................................................. 22

Figure 1.2 Graphical abstract of the thesis layout ........................................................... 30

Figure 3.1 3D drawings of the experimental rig, (a) first configuration, (b) second

configuration ................................................................................................................... 54

Figure 3.2 Presentation of the different elements of the assembled test rig (first

configuration) .................................................................................................................. 55

Figure 3.3 Steps of constructing the experimental rig (first configuration) .................... 56

Figure 3.4 Layout of the (a) first and (b) second configurations of the experimental

rig .................................................................................................................................... 57

Figure 3.5 (a) Photograph and (b) dimensions of typical balancing disc........................ 58

Figure 3.6 (a) Photograph of typical ball bearing (model: SY20TF) and (b) its

dimensions in mm ........................................................................................................... 59

Figure 3.7 Steps of attaching the double-sided adhesive tape and shim to the horizontal

beam ................................................................................................................................ 60

Figure 3.8 Photographs of the electrical motor used in the experiments ........................ 60

Figure 3.9 Delta USB-RS485 converter.......................................................................... 61

Figure 3.10 Flexible coupling (a) before and (b) after assembling ................................. 62

Figure 3.11 Steps of constructing the foundation ........................................................... 63

Figure 3.12 Photograph of the foundation ...................................................................... 63

Figure 3.13 Machine guard (a) close position, (b) open position ................................... 64

Figure 3.14 Hinge operated safety switch ....................................................................... 64

Figure 3.15 Functional diagram of typical DAQ system ................................................ 65

Figure 3.16 Schematic cross-sectional view of compression mode ICP acceleration

sensor .............................................................................................................................. 66

Figure 3.17 (a) Front and (b) back sides of the signal conditioner used in the modal

test ................................................................................................................................... 67

Figure 3.18 16-bit 16-channel DAQ hardware ............................................................... 68

Figure 3.19 DAQ driver software ................................................................................... 69

LIST OF FIGURES

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Figure 3.20 Setup of the modal test of the (a) first and (b) second test rig

configurations .................................................................................................................. 70

Figure 3.21 Schematic of the setup and instrumentations used for the modal test (first

configuration) .................................................................................................................. 71

Figure 3.22 Typical FRF plots in (a) vertical and (b) horizontal directions at location 4

(first configuration) ......................................................................................................... 73

Figure 3.23 Mode shapes of the (a)1st, (b) 2

nd, (c) 3

rd and (d) 4

th modes of the system (1

st

configuration) .................................................................................................................. 74

Figure 3.24 Typical FRF plots in (a) vertical and (b) horizontal directions at location 4

(second configuration) .................................................................................................... 75

Figure 3.25 Mode shapes of the (a)1st, (b) 2

nd, (c) 3

rd and (d) 4

th modes of the system

(2nd

configuration) ........................................................................................................... 76

Figure 3.26 Shaft dimensions .......................................................................................... 76

Figure 3.27 Setup of the modal test of a free-free steel shaft ......................................... 78

Figure 3.28 Typical FRF plot of free-free shaft obtained during modal testing ............. 79

Figure 4.1 The local coordinates of a beam element ...................................................... 83

Figure 4.2 Condition of Euler-Bernoulli beam theory .................................................... 84

Figure 4.3 A rotor element with degrees of freedom (a) horizontal plane, (b) vertical

plane, (c) combination of horizontal and vertical ........................................................... 89

Figure ‎4.4 FRF plot indicating the natural frequency and half-power amplitudes (1st rig

configuration) .................................................................................................................. 94

Figure ‎4.5 FRF plot marking the natural frequency and half-power amplitudes (2nd

rig

configuration) .................................................................................................................. 95

Figure ‎4.6 A simple schematic representation of the rotor system used to demonstrate

the matrix assembly......................................................................................................... 98

Figure ‎4.7 Typical 3D waterfall plot ............................................................................... 99

Figure ‎4.8 (a) Time waveform, (b) vibration spectrum of the time waveform ............. 100

Figure ‎4.9 Typical Bode plot of the 1× shaft displacement for machine coast-up ....... 101

Figure ‎4.10 Typical Bode plot of the 2× shaft displacement for machine coast-up ..... 101

Figure 5.1 Photographic representation of the experimental rig ................................... 107

Figure 5.2 Demonstration of the added unbalance (mass and phase angle) ................. 108

LIST OF FIGURES

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Figure 5.3 Typical measured 1× displacement responses in vertical direction for the

experimental runs 1, 4 and 6 at (a) B1 and (b) B2 ........................................................ 108

Figure 5.4 The actual and estimated unbalances and phase angles for (a) case I, (b) case

II, (c) case III, (d) case IV, (e) case V and (f) case VI .................................................. 113

Figure 5.5 A typical comparison between the actual and estimated rotor unbalance at

different single speeds for Case II ................................................................................. 114

Figure 6.1 Mechanical layout of the test rig ................................................................. 120

Figure 6.2 Typical measured 1× displacement responses in horizontal direction for the

experimental runs 3 and 5 at bearings (a) B1 and (b) B2.............................................. 121

Figure 6.3 Actual and estimated unbalances (amplitude and phase) of (𝑟𝑢𝑛3 − 𝑟𝑢𝑛0) at

(a) disc d1 and (b) disc d2; * : estimated unbalance, : actual added unbalance .... 125





Figure 7.1 Photographs of the rig (a) assembled rig, (b) balancing disc, (c) flexible

coupling ......................................................................................................................... 137

Figure 7.2 Laser tachometer .......................................................................................... 138

Figure 7.3 Typical measured FRF plots of the rotor at distance of 75cm from bearing

B1, (a) vertical direction, (b) horizontal direction ........................................................ 139

Figure 7.4 Typical arrangement of accelerometers on bearing pedestal ....................... 140

Figure 7.5 Comparison between the actual and estimated rotor unbalance for (a,b) case

III, (c,d) case IV; ○: estimated unbalance using different single speeds, ☆: error, :

actual added unbalance ................................................................................................. 143

Figure 7.6 Comparison between the actual and estimated rotor unbalance for (a,b) case

III, (c,d) case IV; ○: estimated unbalance using speed range, ☆: error, : actual added

unbalance....................................................................................................................... 146

Figure 7.7 Photograph of the test rig with two balancing discs .................................... 147

Figure 7.8 Typical measured FRF plots of the rotor at distance of 75cm from bearing B1

in (a) vertical and (b) horizontal directions ................................................................... 148

Figure 7.9 A typical accelerometer installation at a bearing in 3 directions ................. 149

Figure 7.10 Case I (a) disc D1 (𝑒1,3), (b) error of unbalance in disc D1, (c) disc

D2(𝑒2,3), (d) error of unbalance in disc D2; ○: estimated unbalance, ☆: error, :


LIST OF FIGURES

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Figure 7.11 Case II (a) disc D1 (𝑒1,4), (b) error of unbalance in disc D1, (c) disc



Figure 7.12 Case IV (a) disc D1 (𝑒1,6), (b) error of unbalance in disc D1, (c) disc



Figure 7.13 Case IX (a) disc D1 (𝑒1,11), (b) error of unbalance in disc D1, (c) disc



Figure 7.14 Case I (a) disc D1 (𝑒1,3), (b) error of unbalance in disc D1, (c) disc



Figure 7.15 Case II (a) disc D1 (𝑒1,4), (b) error of unbalance in disc D1, (c) disc D2

(𝑒2,4), (d) error of unbalance in disc D2; ○: estimated unbalance, ☆: error, : actual

added unbalance ............................................................................................................ 166

Figure 7.16 Case IV (a) disc D1 (𝑒1,6), (b) error of unbalance in disc D1, (c) disc D2


added unbalance ............................................................................................................ 168

Figure 7.17 Case IX (a) disc D1 (𝑒1,11), (b) error of unbalance in disc D1, (c) disc D2


added unbalance ............................................................................................................ 170

Figure 7.18 Grouped bar chart of the comparison between actual and estimated

unbalances for (a) disc D1, and (b) disc D2 .................................................................. 172

Figure 8.1 (a) Schematic representation of the rig, (b) measurements are taken in the

radial direction .............................................................................................................. 178


plane, (c) combination of horizontal and vertical ......................................................... 179

Figure 8.3 Schematic representation of the rotor .......................................................... 179

Figure 8.4 Coordinates of a point in two systems ......................................................... 181

Figure 8.5 Phase angles with respect to vertical and horizontal axes in cases of (a)

response taken at normal directions and (b) response taken at radial directions .......... 187

Figure 8.6 Comparison between the actual and estimated responses at (a) bearing 1 𝑟𝑘1 and (b) bearing 2 𝑟𝑘1, for run number 5: , actual ; , estimated ............................. 190

Figure 8.7 Comparison between the actual and estimated responses at (a) bearing 1 𝑟𝑘2 and (b) bearing 2 𝑟𝑘2, for run number 5: , actual; , estimated .............................. 191

LIST OF FIGURES

12


plane, (c) combination of horizontal, vertical and radial planes ................................... 197

Figure 9.2 (a) A simple schematic representation of the rig, (b) vibration measurement

directions of bearing pedestal at node 𝑘........................................................................ 199

Figure 9.3 Typical rotor mass unbalance distribution along the rotor length; (a)

continuous form of rotor mass unbalance, (b) discretized form of rotor mass

unbalance....................................................................................................................... 200

Figure 9.4 Photographs of the test rig with one balancing disc .................................... 206

Figure 9.5 Typical measured FRF plots of the rotor at distance of 42cm from bearing B1

in (a) vertical, (b) horizontal directions......................................................................... 207

Figure 9.6 Measured mode shapes of the rig, (a) mode 1, (b) mode 2, (c) mode 3 and (d)

mode 4 ........................................................................................................................... 208

Figure 9.7 A typical accelerometer installation at a bearing in 3 directions ................. 209

Figure 9.8 Demonstration of the added unbalance (mass and phase angle) ................. 210

Figure 9.9 Typical measured 1× displacement responses in vertical direction for the

machine runs 4 and 5 at bearings (a) B1 and (b) B2 ..................................................... 211

Figure 9.10 Grouped bar chart of the comparison between actual and estimated

unbalances by both the earlier and proposed methods, (a) mass and (b) phase angle .. 214

Figure 9.11 Photograph of the test rig with two balancing discs .................................. 215

Figure 9.12 Typical measured FRF plots of the rotor at distance of 53cm from bearing

B1 in (a) vertical and (b) horizontal directions ............................................................. 216

Figure 9.13 Measured mode shapes of the rig, (a) mode 1, (b) mode 2, (c) mode 3 and

(d) mode 4 ..................................................................................................................... 217

Figure 9.14 Typical measured 1× displacement responses in horizontal direction for the

machine runs 3 and 5 at bearings (a) B1 and (b) B2 ..................................................... 218

LIST OF NOMENCLATURES

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LIST OF NOMENCLATURES

Notation Description

A : Area of shaft element

𝐴𝑛 : The shape of beam factor

CF : Foundation damping

𝐃𝑚 : Displacement vector during 𝑚𝑡ℎ machine run-up

𝐝𝑝 : Displacement vector during 𝑝𝑡ℎ machine run-up

E :‎Young’s‎modulus

𝑒0 : Residual rotor unbalance

𝑒𝑝 : Unbalance added to rig at 𝑝𝑡ℎ machine run-up

𝑒𝑞 : Unbalance added to rig at 𝑞𝑡ℎ machine run-up

funb : Unbalance force vector

𝑓𝑛 : Natural frequency in Hz

𝐺 : Gyroscopic vector

𝐺𝑑 : Gyroscopic matrix for the disc

𝐼 : The second moment of area

𝐼𝑑 : The diametral moment of inertia

𝐼𝑝 : The polar moment of inertia

𝐾𝐹 : Foundation stiffness

𝑀𝑑 : Balancing disc mass

𝑀𝐹 : Foundation mass

𝑚𝑠ℎ𝑎𝑓𝑡 : Shaft mass

𝐒 : Machine sensitivity

s : Cross-sectional area

𝐓 : Transformation matrix

𝑇 : Kinetic energy

U : Strain energy

𝑉𝑠ℎ𝑎𝑓𝑡 : Shaft volume

Z : Dynamic stiffness matrix

ρ : Material density

ω : Rotational speed in rad/sec

𝜁 : Damping ratio

LIST OF ABBREVIATIONS

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LIST OF ABBREVIATIONS

Abbreviation Description

ADC : Analog to Digital Converter

BNC : Bayonet Neill–Concelman

DAQ : Data Acquisition

DOF : Degree of Freedom

EOM : Equation of Motion

FE : Finite Element

FRF : Frequency Response Function

HOS : Higher Order Spectra

IC : Influence Coefficient

ICP : Integrated Circuit Piezo-electric

MB : Modal Balancing

PCA : Principal Component Analysis

PC : Personal Computer

RPM : Revolution per Minute

SVD : Singular-Value Decomposition

STFT : Short-Time Fourier Transform

TG : Turbogenerator

USB : Universal Serial Bus

VCM : Vibration based Condition Monitoring

LIST OF PUBLICATIONS

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LIST OF PUBLICATIONS

Journal publications

1. Ibn Shamsah, S., Sinha, J.K., Mandal, P. Precaution during the field balancing of

rotating machines. Journal of Maintenance Engineering 2016. 1(1), p.59-66.

2. Ibn Shamsah, S., Sinha, J.K. Rotor unbalance estimation with reduced number

of sensors. Machines, 2016. 4(19).

3. Ibn Shamsah, S., Sinha, J.K., Mandal, P. Reliable Machine Balancing for

Efficient Maintenance. Journal of Quality in Maintenance Engineering, Under review.

4. Ibn Shamsah, S., Sinha, J.K., Mandal, P. Rotor unbalance identification using

reduced sensors and data from single machine run-up. Journal of Sound and Vibration,

Under review.

5. Ibn Shamsah, M., Ibn Shamsah, S., Traditional In-Situ Gas Compressor Rotor

Balancing: A Case Study. Journal of Maintenance Engineering 2016. 1(1), p.297-304.

Conference publications

1. Ibn Shamsah, S., Sinha, J.K. Rotor unbalance estimation using a single machine

rundown with reduced number of sensors, Proceeding of the International Conference

on Engineering Vibration (ICoEV), Ljubljana, Slovenia, 2015.

2. Ibn Shamsah, S., Sinha, J.K., Mandal, P. Sensitivity analysis of in-situ rotor

balancing, Proceeding of the Vibration in Rotating Machinery (VIRM 11), Manchester,

United Kingdom, 2016.

3. Ibn Shamsah, S., Sinha, J.K., Mandal, P. Multi-planes rotor unbalance

estimation using influence coefficient method, Proceeding of the Twelfth International

Conference on Vibration Engineering and Technology of Machinery (VETOMAC XII),

Warsaw, Poland, 2016.

4. Ibn Shamsah, S., Sinha, J.K., Mandal, P. Application of model-based rotor

unbalance estimation using reduced sensors and data from a single run-up, Proceeding

of the 2nd

International Conference on Maintenance Engineering (IncoME II),

Manchester, UK, 2017.

https://www.linkedin.com/in/sami-ibn-shamsah-872b8426/

https://www.linkedin.com/in/jyoti-sinha-phd-ceng-fimeche-18654935/

https://www.linkedin.com/in/partha-mandal-8a359b35/



http://www.mdpi.com/2075-1702/4/4/19

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ABSTRACT

16

ABSTRACT

The University of Manchester

Sami Meshal F Ibn Shamsah

PhD in Mechanical Engineering

"Enhancement of Field Balancing Methods in Rotating Machines"

2017

The influence coefficient (IC) method is an acceptable field balancing approach for

rotating machines. However, it is generally observed that the IC method often uses

vibration response acquired at single machine speed at bearing pedestals for the rotor

unbalance estimation for industrial applications. The estimated rotor unbalance may not

be accurate at a single speed either due to noise in the measured signal or measurement

at single speed not reflecting the machine dynamics accuratly or both. Therefore, an

improved unbalance estimation is proposed by using the IC method, but using vibration

measurements at multiple rotor speeds together in a single band to estimate rotor

unbalance accuratly. Sensitivity analysis of the proposed method is also carried out to

understand the dependency of adding more speeds in a single band on the accuracy of

unbalance estimation.

In the recent past, with the support of the advanced computer technology, the model-

based rotor fault identification approach has been introduced earlier. This method

requires vibration measurements of a single machine transient operation and reasonably

accurate numerical model of the rotating machine. Despite all the significant research

contributions towards the enhancement of the aforementioned two balancing methods

(i.e. IC and model-based approaches), they are currently applied using two orthogonal

vibration sensors per bearing pedestal. Therefore, this study proposes that the two

balancing methods can be enhanced by applying them with using only one sensor at a

bearing pedestal. The proposed balancing techniques are applied on experimental rigs

with single as well as multiple balancing planes. Also, several added unbalance

scenarios are used for both methods. The proposed rotor mass unbalance estimation

methods can estimate the rotor unbalance of different unbalance configurations

accurately for all cases. This indicates that the proposed unbalance estimation

approaches have the potential for future industrial application.



DECLARATION

17

DECLARATION

"I, Sami Ibn Shamsah, declare that no portion of the work referred to in the thesis has

been submitted in support of an application for another degree or qualification of this or

other universities or institutes of learning."

https://www.researchgate.net/profile/Sami_Ibn_Shamsah


COPYRIGHT STATEMENT

18

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http://www.library.manchester.ac.uk/about/regulations/

ACKNOWLEDGMENTS

19

ACKNOWLEDGMENTS

First and foremost, the author would like to praise God, the almighty for providing him

this opportunity and granting him the capability to proceed successfully. The author

wishes to sincerely thank the Government of the Kingdom of Saudi Arabia for granting

him a full scholarship under The Custodian of The Two Holy Mosques’ overseas

scholarship program. He would also like to thank most sincerely, Saudi Arabia’s‎

Ministry of Education and the Saudi Arabian Cultural Bureau in the UK for their

continued support throughout pursuing his PhD. Also, the author would like to extend

special thanks to his supervisor Professor Jyoti Kumar Sinha and co-supervisor Dr

Parthasarathi Mandal, who made this dissertation come into being. They are not only his

advisers, they are also his mentors who gave their endless support and helped him to go

through all the difficulties he came across during his PhD study. He would also like to

acknowledge Dr Akilu Kaltungo and Dr Adrian Nembhard who helped him during his

research. And last but not least, the author would like to thank all technicians who

worked in Pariser Building workshop, especially Mr David Jones and Mr Anthony

Williams, for their support while conducting the experiments.

https://www.moe.gov.sa/en/HigherEducation/ExternalEducation/Pages/TheCustodianofTheTwoHolyMosquesOverseasScholarshipProgram.aspx

https://www.moe.gov.sa/en/HigherEducation/ExternalEducation/Pages/TheCustodianofTheTwoHolyMosquesOverseasScholarshipProgram.aspx

https://www.moe.gov.sa/en

https://www.moe.gov.sa/en

http://uksacb.org/uk-en1313/


https://www.research.manchester.ac.uk/portal/Jyoti.Sinha.html



http://www.manchester.ac.uk/research/partha.mandal/

https://www.linkedin.com/in/akilu-yunusa-kaltungo-phd-msc-mimeche-ceng-33785318/

http://www.mace.manchester.ac.uk/people/staff/profile/?ea=akilu.kaltungo

https://www.linkedin.com/in/dradriannembhard/

http://www.mace.manchester.ac.uk/people/staff/profile/?ea=david.j.jones

http://www.mace.manchester.ac.uk/people/staff/profile/?ea=anthony.williams

http://www.mace.manchester.ac.uk/people/staff/profile/?ea=anthony.williams

DEDICATION

20

DEDICATION

This thesis is dedicated to my mum Afra, dad Meshal, uncle Mamdouh, wife Amal,

daughter Al Jawhara, sisters, brothers and friends.

For their endless love, support and encouragement

https://www.linkedin.com/in/mamdouh-ibn-shamsah-671a5535/

‎CHAPTER 1 INTRODUCTION

21

CHAPTER 1

INTRODUCTION


22

General introduction 1.1

In the construction of typical rotating machinery such as gas turbine, rotor, bearings and

foundation are often considered as the major components (Figure ‎1.1). The rotating

shaft and all other rotating parts connected to it such as gears, couplings and impellers

are commonly known as the rotor. Fluid filled journal bearings or rolling-element

bearings are typically used to support the rotor. The term foundation, also known as

machine supporting structure, is used to describe all components that are located

underneath the bearings but linked with them [1]. Rotating machines are considered as

the bedrock of most industries such as oil and gas power plants, mining, aerospace, and

chemical factories as they play a vital role in many activities. Therefore, the reliability

of this class of machines is essential to these industries. During the last few decades, a

growing body of research has emerged with the aim to enhance the reliability of rotating

machines.

Figure ‎1.1 (a) Photograph of a 9HA gas turbine rotor on the half shell casing [2], (b) block

diagram of the turbine in part a

The achievement of the desired reliability targets of industrial rotating machines is often

hampered by the existence of various rotor-related faults [3-5]. Most of these faults are

unpreventable due to a number of reasons such as manufacturing/installation

imperfections, inappropriate commissioning and wears and tears owing to day-to-day

operations [6-9]. Malfunctions in rotating machines may lead to damage in critical parts

of the machine or even worse, cause catastrophic damages to the entire machine, which

Bearing Bearing …

𝐟𝑅

𝐟𝐹

(a) (b)


23

has safety implications as well as economic considerations. It is to this end that the early

detection and reliable diagnosis of rotor faults in their initial stages have become

essential in industries to enhance machine reliability and maintenance cost

effectiveness. Recently, significant efforts have been made by manufacturing companies

to implement effective machinery maintenance programs that can detect and diagnose

rotor faults at their initial stage.

Although there are various commonly encountered rotor-related faults, rotor mass

unbalance is one of the most common malfunctions in rotating machines which

repeatedly occurs throughout their operations [10]. This stimulus develops when the

mass is asymmetrically distributed around the axis of rotation [11]. If the machine

vibration due to rotor mass unbalance exceeds the allowable limit, it may lead to

machine failure. As a consequence, machine downtime and unscheduled maintenance

actions will be required, which in turn influence the operating cost negatively. Also,

high levels of machine vibrations due to rotor mass unbalance may result in a

significant cutback of the machine fatigue life [12]. Therefore, the regular field

balancing is essential to keep machine vibration within an acceptable level, and hence

ensure safe machine operation and long service life [13].

Numerous vibration based rotor balancing techniques have been proposed in the

literature [14-17]. For different reasons, only a few of these balancing methods are

acknowledged by practising balancing engineers. Some of these methods are relatively

involved and mandate judgment from highly skilled engineers with thorough knowledge

of rotor dynamics. One of the most popular rotor balancing techniques in industries is

the influence coefficient (IC) balancing method. This balancing approach has some

advantages such as simplicity and high efficiency, which make it suitable for a broad

range of industrial rotating machines. In addition, the IC balancing method does not

require any prior knowledge of the underlying dynamics of the machine. It only requires

the vibration response of the machine at different trial masses to define the correction

weights. One machine’s transient operation with residual unbalance plus one machine’s

transient operation per balancing plane are needed to balance the rotor using the IC

balancing approach. A survey of the literature of the IC balancing method was presented

by some authors [18, 19].


24

With the development of the modern computer technology, the model-based rotor

balancing method has been introduced. This balancing approach relies on accurate

numerical models of some parts of the rotating machine as well as measured vibration

response from a single machine’s run-up/run-down [20]. The finite element (FE)

method has been found to be the most appropriate tool for the numerical modelling in

structural engineering today [21]. Often, an accurate mathematical model of the rotor

and approximate numerical model of the bearings can be constructed using the FE

method. Considerable research has been done on the applications of the model-based

rotor balancing method [22-24].


25

Motivations 1.2

Rotor mass unbalance is one of the most commonly encountered malfunctions in

rotating machines which occurs due to several reasons including corrosion, deposition

of dirt on rotors, manufacturing imperfections, cracked fans, incorrect keyways, etc.

Therefore, regular balancing of rotating machines is imperative to ensure safe machine

operation. Several rotor balancing techniques have emerged over the years, and a review

of these techniques indicates that their applications are often time-consuming due to the

requirement of multiple measurement locations. Consequently, the development of an

approach that simplifies rotor balancing through the rationalisation of measurement

locations could be useful to industries.


26

Aims and objectives 1.3

The ultimate aim of this research project is to enhance the field balancing methods in

rotating machines. To achieve this goal, the following research objectives are needed to

be fulfilled:

1. To experimentally investigate the effectiveness of the application of the

influence coefficient balancing method using vibration measurements acquired

at multiple speeds in a single band.

2. To propose and experimentally examine the effectiveness of the application of

the influence coefficient balancing method using vibration measurements at

multiple machine speeds from only one vibration sensor per bearing pedestal.

3. To develop a model-based method for identifying rotor mass unbalance using

single vibration sensor per bearing pedestal and single machine’s transient

operation (run-up/run-down).


27

Research contributions 1.4

The current industrial applications of IC method generally use vibration measurements

at a single machine speed for the rotor unbalance estimation. The measured vibrations at

a single machine speed do not adequately reflect the dynamics of the rotor and might

include a high level of noise. As a consequence, applying the IC method using vibration

measurements at a single machine speed might not provide an accurate estimation of the

rotor mass unbalance and hence results in a bad balancing. Therefore, as mentioned in

the first aim of the thesis, the first contribution of the current research project is the

enhancement of the rotor mass unbalance estimation by applying the IC method using

vibration measurements acquired at multiple rotor speeds in a single band instead of

single rotor speed.

Both IC and model-based balancing techniques are currently applied using pair of

vibration sensors mounted orthogonally at each bearing pedestal [25, 26]. Therefore,

applying these methods on vast and complex rotating machines with a large number of

vibration sensors could be overwhelming as well as significantly relying on a highly

skilled engineer with sound knowledge of rotor-dynamics during data analysis.

Moreover, high cost related to the maintenance of the monitoring system is required.

Thus, as mentioned in the second and third aims of the thesis, the second contribution of

the current research project is to enhance the existing IC and model-based unbalance

identification approaches by using significantly reduced number of sensors.


28

Layout of thesis 1.5

The thesis is written in the alternative format with its main content presented in the form

of‎ published/submitted‎ research‎ papers‎ of‎ the‎ candidate’s‎ own work. Same as the

traditional format, Chapter 2 provides a review of literature related to this research

project, Chapter 3 describes the experimental setup and instrumentation, and Chapter 4

explains the mathematical modelling and signal processing. However, Chapters 5-9

present the published/submitted research papers. The last Chapter includes the

concluding remarks and possible future work. Further details of the outline are

provided:

Chapter 2 gives a review of literature related to this research project. It starts by

reviewing literature pertaining to the modal balancing method. Then, the IC unbalance

estimation method is discussed and some research papers on this topic are reviewed.

The unified balancing approach which combines both modal balancing and IC

balancing approaches is, then, reviewed. After that, a brief literature review on the

model-based rotor unbalance identification method is given.

Chapter 3 describes the experimental setup. The main elements of the laboratory rig as

well as the Data Acquisition (DAQ) system are described. Then, the modal testing

procedure and results are provided.

Chapter 4 explains the detailed steps of modelling a typical rotating machine. Then, it

gives a brief idea about signal processing and briefly presents the concepts of the 3D

waterfall plot, order tracking and Bode plot.

Chapter 5 presents the application of the IC unbalance estimation approach using

vibration data acquired from a test rig with a single balancing plane at a single rotor

speed as well as wide range of speeds during machine run-up.

Chapter 6 applies the same approach explained in Chapter 5 but using vibration data

acquired from an experimental rig with multiple balancing planes during machine’s‎

transient operation.


29

Chapter 7 introduces the application of the IC unbalance estimation technique using

reduced number of vibration sensors (i.e. 1 sensor at a bearing pedestal instead of pair

of sensors arranged in orthogonal directions). The concept is applied experimentally on

laboratory rigs with single as well as multiple balancing planes.

Chapter 8 applies the model-based unbalance estimation approach on a numerically

simulated rotating machine with reduced number of sensors and single machine’s‎


Chapter 9 provides an experimental validation to the proposed model-based unbalance

estimation method. The method is applied on test rigs with single as well as multiple

balancing planes.

Chapter 10 addresses the possible conclusions of the present research, summarising the

main findings and contributions. A closure is given, while possibilities for future

research are provided.

A graphical abstract of the layout of the thesis is illustrated in Figure ‎1.2.

30

‎CH

AP

TE

R 1

INT

RO

DU

CT

ION

Concluding remarks and future research 10

IC balancing (measurements at

multiple speeds in a single band,

2 directions/pedestal, 1 disc)

5 IC balancing (measurements at

multiple speeds in a single band,

2 directions/ pedestal, 2 discs)

6 IC balancing (measurements

at multiple speeds in a single

band, 1 direction/ pedestal)

7

Model-based balancing of

simulated rotor (measurements

at 1 direction, 1 disc)

8

Experimental validation of the

proposed model-based method

9

Single

disc

Double

discs

Modal

tests Test rigs

Modal

balancing

IC

balancing

Unified

approach

Model-based

approach

Signal

processing Rotor FE

modelling

Enhancement of Field Balancing Methods in Rotating

Machines

Introduction 1 Review of relevant

literature

2 Numerical modelling and

signal processing

4 Experiments

3

Instrumentation Rig layout

Natural

frequencies

Mode

shapes

Figure ‎1.2 Graphical abstract of the thesis layout

‎CHAPTER 2 LITERATURE REVIEW

31

CHAPTER 2

LITERATURE REVIEW


32

Introduction 2.1

A review of background literature and research papers relevant to the current research project

is given in this chapter. The chapter starts by reviewing the literature concerning the modal

balancing method (Section 2.2). This is followed by a review of research into IC balancing

method (Section 2.3). Special attention is, then, devoted to papers dealing with the unified

balancing approach which combines modal balancing and IC balancing approaches (Section

2.4). Finally, a comprehensive review of literature pertaining to the mathematical model-

based rotor unbalance identification techniques is presented in Section 2.5.

Modal balancing approach 2.2

In the modal balancing approach, each mode is balanced individually starting with the lowest

mode. The number of machine runs required for rotor balancing mainly depends on the

number of modes that are needed to be balanced. To balance the first mode, two machine

runs are required; one run with residual unbalance and one run with added mass unbalance.

Then, extra run is required for balancing of each of the higher modes [27]. The balancing

mass of each mode is carefully chosen to avoid upsetting the previously balanced modes. The

combined effect of the added masses should not have any effect on the previously corrected

modes [27].

Literature reviews on the modal balancing approach were presented by some authors [28-30].

In the 1960s, Bishop, Gladwell and Parkinson presented a number of papers [31-34] that have

outstanding contributions to the theory and application of the modal balancing approach. The

theoretical background of the flexible rotor balancing, mode by mode, was discussed by

Bishop and Gladwell [31]. First, they investigated the balancing of rigid rotating machines at

low speeds mathematically. Then, they considered the balancing of flexible rotating

machines. They provided an example of balancing a uniform shaft in two modes. They also

investigated the influence of a slightly bent shaft as well as the influence of‎the‎shaft’s‎weight‎

on the balancing process.

The procedure of isolating whirl modes for accurate balancing was discussed by Bishop and

Parkinson [33]. In this approach, the rotor runs at a speed that is close to the critical speed in


33

order to amplify modal distortion in a certain mode. The proposed balancing approach was

applied on small size test rigs as well as real rotating machines. The authors highlighted some

limitations of this rotor balancing approach. To overcome the shortcomings of the proposed

balancing approach, they adapted the technique of resonance testing which was firstly

published by Kennedy and Pancu [35].

Parkinson and Bishop [34] presented the problem of the vibration due to residual unbalance

in the rotor after applying the modal balancing. They proposed that the residual vibration of

rigid rotors (i.e. run below their first critical speed) can be balanced by adding a single mass.

The method was illustrated on a rotor of boiler feed pump. The application of the modal

balancing technique on large rotating machines with flexible foundation was discussed by

Lindley and Bishop [36]. Demonstration of the application of the method on various rotating

machines such as pumps and turbomachines was presented by Moore and Dodd in three

different papers [37-39].

Some researchers proposed modal balancing approaches that do not require trial masses or

test runs [40-44]. Gnielka [40] extended a modal balancing technique which was originally

proposed by Gasch and Drechsler [45] to balance a rotor that was initially bowed. This

method does not require any test runs; it only requires a pre-knowledge of the flexural mode

shapes of the rotor as well as the generalised masses. First, the rotor is run in the vicinity of

the first critical speed, and the machine vibration response is acquired. Then, the differential

equation of motion of the bowed shaft is solved using the frequency response function (FRF).

As the system is non-linear, trial and error process was used and the unbalance was identified

by using the least square technique. Also, Morton [41] proposed a modal balancing method

that does not require trial masses. He was able to identify the bearing model by calculating

the shaft response function. This technique is applicable for all types of bearings and can be

used on rotors with multiple bearings.

Some researchers [46, 47] developed a modal balancing approach that can perform balancing

during the operation of the rotating machine. Lee and Kim [46] have used balancing head

which consists of single/multiple discs that are assembled to the shaft and carry correction

masses. First, the machine vibration response is measured, and then the correction masses

move while the rotor vibrates to balance the machine. The rotor unbalance is controlled

manually by a controller that monitors the whirl data of the shaft on an oscilloscope. The


34

oscilloscope, then, transmits the magnitude and direction of the signals to the balancing head.

The magnitude of the signal is changed by rotating the balancing discs in opposite directions.

After adjusting the magnitude, the discs are rotated together to change the direction. The

method was applied to a test rig, and the balancing head was successfully able to balance the

rotor during operation at various speeds. Later, Lee et al. [47] improved this balancing

approach by automatically controlling the balancing head through software on a personal

computer. The experimental results prove the effectiveness of the modified modal balancing

approach.

Deepthikumar et al. [48] used a method that was initially developed by Yang and Lin [49] for

modal balancing of a flexible rotor with single balancing plane that has distributed mass

unbalance and bow. First, they applied the method to a numerical model. Then, they validated

the simulation results experimentally. They were able to balance the rotor with measured

machine vibration below the first critical speed. They also introduced the concept of

quantifying‎ the‎distributed‎unbalance‎using‎‘Norm’‎of‎eccentricity‎polynomial‎ function.‎To

apply this method efficiently, a fairly accurate numerical model of the rotor is required.

Notwithstanding the fact that the modal balancing method requires fewer trial runs than the

influence coefficient balancing method and is not very complicated in principle, applying it

on large and complex rotating machines such as turbogenerator (TG) set is not straight

forward and requires an engineer with a sound knowledge of rotor dynamics. A full list of the

shortcomings of the modal balancing approach was presented by Darlow [50].

Influence coefficient balancing method 2.3

Over the years, many rotor balancing techniques have been proposed in the literature [29, 51,

52]. Amongst all vibration based rotor balancing techniques, the most predominant approach

is the influence coefficient (IC) balancing method [21, 53]. IC balancing method is also

known as field balancing, because the balancing is performed at the site without

disassembling rotor from the machine. In this balancing approach, the rotor system is

assumed to be linear, and the influence of the individual unbalances can be superposed to

give the influence of a set of unbalances [29]. In addition, the IC method does not require any

prior knowledge of the dynamics of the rotor [17]; it requires only the vibration response of

the rotor at different trial masses.


35

The number of test runs required to perform balancing using the IC method depends mainly

on the number of balancing planes [21]. In the case of a rotor with one correction plane, two

transient machine operations are needed to obtain the influence coefficients and hence

identify the rotor mass unbalance. The first machine’s transient operation is carried out with

residual unbalance (known as reference run) and the second machine’s transient operation is

carried out with added correction weight. An extra machine’s transient operation is required

for each additional correction plane. Therefore, due to its simplicity and high efficiency, the

IC balancing method has become the conventional field balancing approach in most

industries around the world.

To enhance understanding of the IC unbalance estimation approach, assume a simple rotating

machine which consists of shaft that is supported on flexible foundation through two bearings

at its ends and carries a single balancing plane at the midspan between the bearings. Assume

that the machine runs at 𝑓𝑘 Hz and the displacement is measured at both bearing pedestals in

the vertical and horizontal orthogonal directions. In order to evaluate the sensitivity of the

machine, the displacements are measured in two different run-ups or run-downs, i.e. the first

run with residual unbalance only and the second run with residual unbalance plus added mass

unbalance. Thus, the sensitivity matrix is written as follows,

𝐒 = [𝐝1(𝑓𝑘)−𝐝0(𝑓𝑘)

𝑒1] (2.1)

where 𝑒1 is added unbalance to the balancing plane at the 2𝑛𝑑 machine run-up, and

displacement vectors 𝐝0(𝑓𝑘) as well as 𝐝1(𝑓𝑘) can be written as:

𝐝0(𝑓𝑘) = [y1,0(𝑓𝑘) x1,0(𝑓𝑘) y2,0(𝑓𝑘) x2,0(𝑓𝑘)]T

𝐝1(𝑓𝑘) = [y1,1(𝑓𝑘) x1,1(𝑓𝑘) y2,1(𝑓𝑘) x2,1(𝑓𝑘)]T

(2.2)

where y1,0 and x1,0 are the displacements at the 1st bearing pedestal in the vertical and

horizontal directions respectively for the 1st machine run, similarly the displacements y2,0 and

x2,0 at 2nd

bearing. Same thing applies to y1,1, x1,1, y2,1 and x2,1. The sensitivity matrix

together with the vibration measurements acquired at the‎first‎machine’s‎transient‎operation

are used to obtain the residual rotor mass unbalance 𝑒0 as shown in the following equation


36

[𝐒]4×1[𝑒0]1×1 = [𝐝0]4×1 (2.3)

Thus,

𝑒0 = [( 𝐒T𝐒)

−1𝐒T]𝐝0 (2.4)

When adding more balancing planes to the rotating machine, then more machine run-ups/run-

downs with adding mass unbalance to each run are required for balancing.

Comprehensive reviews of literature pertaining to the IC balancing method were conducted

by Darlow [29] and Zhou and Shi [19]. The IC balancing approach was firstly proposed in

the 1930s by some researchers [54, 55]. Thearle [54] has explained the application of the IC

method with referring to the balancing of a turbogenerator, which consists of two rotors that

are supported by multiple bearings. Sinha [17] has explained the IC method for single/

multiple balancing planes in details in his book. He also presented graphical and

mathematical approaches and provided some examples to enhance understanding. Friswell et

al. [21] provided a detailed explanation of the IC balancing method and introduced the

theoretical background of the method. Moreover, they gave few examples on the application

of the IC balancing method. They also compared the IC balancing approach with the modal

balancing approach, and discussed the advantages and limitations of each balancing method.

Hopkirk [55] formulated the two-plane, two-sensor, single-speed balancing procedure using

influence coefficients in the manner as most modern methods. He presented an analytical

solution to solve for the required balancing mass using only the amplitude information from

the measured vibration response. In 1964, Goodman [56] presented a least square approach to

extend the technique to multi-plane balancing using data from different machine speeds and

measurement locations. His method was refined in 1972 by Lund and Tonnesen [16] and

verified in the same year by Tessarzik et al. [57].

In his book, Den Hartog [58] briefly discussed the IC technique for two balancing planes.

Grobel [59] was able to balance a large rotating machine mode by mode using the IC

approach. Church and Plunkett [60] presented a technique that can generate the influence

coefficients without trial masses. The method relies on using a shaker to excite the rotor at

zero revolutions per minute (RPM). The theory was tested on a flexible rotor that was


37

supported by stiff rolling-element bearings. However, research done by Tonnesen [61]

showed that applying this method on a real rotating machine is not straight forward and does

not provide reliable results. Therefore, this approach would not be practical for field rotor

balancing.

Rieger [62] developed a computer program for balancing rotors. In his program, he studied

the effectiveness of the IC balancing method analytically. He examined three practical rotor-

bearing systems in his study (i.e. rigid rotor supported by fluid film bearings, supercritical

flexible three-disc overhung rotor supported by fluid film bearings and supercritical three-

bearing rotor with one disc overhung supported by fluid-film bearings). The balance

improvement with two, three and four balancing planes was also studied. He showed that the

number of bearing supports does not have any influence on the quality of balancing. Rieger

also examined the impact of measurement errors and installation of correction weight on the

quality of the obtained balance.

LeGrow [63] used a numerical model to get the influence coefficients of an actual rotating

machine. Despite the fact that this approach can save time and money, it could not balance

the tested rotor adequately. The author also presented a method for balancing rotors which is

similar to the IC method. He first mentioned the purpose of rotor balancing and difficulties

introduced by flexible rotors. LeGrow concluded that the weighted least square IC procedure

is not very practical because it is time-consuming and requires many machine runs.

The different approaches for multiple-speeds and multiple-planes balancing using the IC

method were presented by Badgley [64]. He described the IC method and gave some notes

regarding the rotor balancing process in practice. Moreover, the author mentioned that the

machine downtime is the main source of the high cost involved in the in-situ rotor balancing.

Therefore, he proposed that the best way for reducing the machine downtime is to determine

the calibration weight while the machine is running, and then shut it down only long enough

to install the calibration weight. However, the author did not provide an explanation

regarding the way of calculating the calibration weight without shutting the rotating machine

down. Moreover, he discussed the error caused by the use of the electronic equipment;

Badgley advised using the mean value of every parameter for the sake of increasing the

effectiveness when determining the correction weight.


38

An extended test program for the evaluation of the validity of the IC technique was carried

out by Tessarzik, Fleming, Badgley and Anderson [65-67]. The outcomes of seven

experimental tests concerning the balancing of an experimental rig were discussed in the last

paper [67]. The rig was operating over the full speed range and covering four critical speeds.

Two of the four critical speeds did not require balancing as they were heavily damped. The

other two critical speeds were lightly damped, and hence required balancing. Generally, the

method was found to be effective for all critical speeds.

In one of the experimental tests, the authors attempted to balance the fourth critical speed

without including the measured vibration data from the first critical speed. They found that

the fourth critical speed was improved, but only at the cost of the vibration level at the first

critical speed. When they used all vibration data, including data from the first critical speed,

to balance the fourth critical speed, no significant increase in the vibration level was detected

at the first critical speed.

Ling and Cao [68] presented frequency response functions (FRFs) for analysing the

relationship between the FRFs and ICs theoretically as well as deriving the corresponding

mathematical equations for balancing high-speed rotors. Furthermore, they have analysed the

relationships between the mass unbalances and FRFs. The analyses were based on the modal

balancing (MB) technique along with the equations related to dynamic and static unbalance

masses. They have conducted some experiments on a high-speed rotor to validate the theory.

It was found that the experimental results were in good agreement with the analytical

solution.

Kang et al. [69] derived a formulation of IC matrices from the motion equations of nonlinear

rotors by using the finite element (FE) method and complex coordinate representation. An

algorithm of plane separation was formulated based on the exact point IC technique. Using

the inference from two and three-plane separation, they introduced a generalised technique

for multiple-plane separation for balancing rigid rotors. The authors provided some examples

to validate their work.

Yu [70] as well as Lee et al. [71] used the IC method to estimate the rotor unbalance at a

constant rotor speed. Zhou et al. [72] were successfully able to balance a rotor-bearing

system using the IC method during varying speed period. Dyer and Ni [73] have extended the


39

IC technique to the active control and on-line estimation. In their study, they have

successfully implemented an adaptive control scheme that combines flexible rotor balancing

method and the on-line estimation of the IC using an active balancing system. Recently, Xu

et al. [74] have used the IC method to balance a rigid rotor in two balancing planes. They

have used cross-correlation method to extract the fundamental frequency signal. They were

successfully able to reduce the vibration level by around 90% with a total of four machine’s

transient operations.

It can be observed from the reviewed literature that the IC balancing method is applied using

vibration measurements acquired at a single rotor speed. The measured vibrations at a single

rotor speed possibly do not fully reflect the dynamics of the rotating machine and might

include a high level of noise. Therefore, applying the IC method using vibration

measurements at a single rotor speed may not provide an accurate estimation of the rotor

mass unbalance, and hence could result in a bad balancing. Therefore, an opportunity exists

to enhance the efficiency of the conventional IC balancing method by using vibration

measurements acquired at multiple rotor speeds in a single band, instead of individual single

speeds.

The previously proposed influence coefficient methods were applied with using multiple

sensors mounted orthogonally at a bearing pedestal. In the case of large and complex rotating

machines with multiple shafts like the industrial steam turbine, the rotor is usually supported

through a substantial number of bearing pedestals. Accordingly, a large number of sensors is

used to collect the vibration response of the machine. Hence, huge amounts of data sets are

generated during rotor unbalance diagnosis, which could be overwhelming as well as

significantly relying on experience and engineering judgment during data analysis.

Furthermore, considerable time and effort are required to identify the severity of the acquired

information. Moreover, an additional cost related to the maintenance of the monitoring

system is needed.

Notwithstanding that the IC balancing method is almost a mature and well suited to be

applied in most industries, the author found an opportunity of developing the currently used

IC method by reducing the number of sensors without necessarily compromising the valuable

information required for the diagnosis and prognosis of rotor mass unbalance. This

improvement is certainly of great benefit to any industry as it reduces the time needed for the


40

complex signal processing and also minimises the maintenance cost effectively. In addition,

the significant reduction in the number of sensors will considerably reduce the probability of

tripping the machine because of false alarms, which occur as a result of faulty sensors. The

idea of reducing number of sensors is not new in the field of condition monitoring as it was

applied by several researchers for the identification and diagnosis of several rotor faults [75-

82]. Some of the proposed condition monitoring methods that use a reduced number of

vibration sensors are reviewed in the next three paragraphs.

Yunusa-Kaltungo et al. [76] have explored the use of the combined bispectrum and

trispectrum with one accelerometer mounted at 45-degree to vertical/horizontal directions at

each bearing location for diagnosing different rotor faults. The rotor faults analysed are

misalignment, crack and rotor/stator rub. The method has been applied on an experimental rig

with artificial rotor faults. They have been able to distinguish between the healthy and faulty

conditions. Furthermore, it has been found that the rotor malfunctions may be identified by

merging the bispectrum and trispectrum components.

Nembhard et al. [77] have presented multiple rotor faults diagnosis method that uses one

accelerometer and one K-type thermocouple at a bearing pedestal. In order to get accurate

temperature readings, the thermocouples were mounted between the bearing casing and

outside of the outer race of each bearing. The method has been applied on a test rig which

consists of two rigidly coupled shafts supported by four rolling-element bearings. The faults

discussed were coupling misalignment, cracked rotor and rotor rub. After comparing the

vibration response spectrums of the healthy and faulty conditions, it has been found that the

1× (i.e. one multiplied by the rotating frequency) component of the faulty spectrum was

significantly higher than the healthy one. Therefore, they have been successfully able to

indicate the presence of fault conditions. Moreover, by combining the principal component

analysis (PCA) with vibration and temperature measurements in the analysis, they have been

able to classify the different faults successfully and also have got useful information on the

faults severity.

Sinha et al. [79] have proposed a vibration-based method that uses higher order spectra

(HOS), namely bi-spectrum, to identify two faults (i.e. shaft rub and shaft misalignment).

They have used the same laboratory rig that was used by Yunusa-Kaltungo et al. [76] and

Nembhard et al. [81]. The method is based on the fusion of vibration response from all


41

vibration sensors in the frequency domain, in order to have a composite spectrum for a

rotating machine and then the computation of HOS. In their method, they have reduced the

number of vibration sensors to a single sensor at a bearing pedestal. The theoretical and

experimental results have shown good agreement.

Unified balancing approach 2.4

Both of the main balancing approaches (i.e. modal balancing and IC balancing methods) have

advantages as well as shortcomings. For example, the modal balancing approach has the

advantage of using less sensitivity runs at high rotor speed. However, applying this method

on complex rotating machines effectively mandates a highly skilled engineer with a good

knowledge of machine dynamics. Moreover, modal balancing approach depends on the

assumption of planar modes which may not be valid for systems with significant damping or

bearing cross-coupling effect. Similarly, the IC balancing approach has the advantage of

being linear and not requiring much pre-knowledge of the physics of the rotor system, and

disadvantage of suffering from the reliance on a significant number of sensitivity runs at high

machine speeds.

On the light of the aforementioned discussion, some researchers [83-87] proposed a unified

balancing approach that combines both modal balancing and influence coefficient balancing

methods. The unified balancing approach combines the advantages and avoids the limitations

of both modal balancing and IC balancing methods. In this balancing approach, the modal

trial mass sets are calculated such that they do not disturb the previously balanced critical

modes. The trial and balancing weights are obtained from influence coefficients that are

determined from experimental tests. Basically, the unified balancing approach does not rely

on the assumption of planar modes [29]. Also, this balancing method usually can be

automated.

Foils et al. [88] presented a literature review on the unified balancing method. Darlow [29]

has specified a chapter in his book for the unified balancing approach; he explored the

method and discussed the theoretical background and explained the procedure for the

application of the method. Parkinson et al. [89] provided a theoretical introduction to the

unified balancing approach and listed the advantages of the proposed balancing method.

Later, Darlow [50] presented the analytical basis and detailed procedure of implementing the


42

unified balancing approach. In addition, he reported an experimental validation of the

proposed balancing approach. Zorzi et al. [90] proposed an optimised unified balancing

approach that avoids the restrictions imposed by former unified balancing methods. Unlike

previous approaches, this approach allows the application of weight constraints at any

balancing speed including the critical speeds. The experimental results showed that the

proposed method is feasible for high-speed rotor balancing.

Tan and Wang [91] presented a mathematical approach that unifies the two most popular

balancing methods (i.e. modal balancing and influence coefficient balancing methods). They

applied the proposed approach to the low-speed balancing of flexible rotating machines. They

have discussed the conditions of balancing flexible rotors at low speeds without the need of

high-speed rotor balancing. However, no attempts have been made for validating the

proposed method experimentally. Kang et al. [92] presented a modified unified balancing

approach for unsymmetrical rotor bearing systems. They formulated the modified unified

approach from FE equations of a rotor, and they took into account the unequal properties of

the rotating parts of the machine as well as the asymmetry of bearings. Furthermore, they

provided some examples to verify the validity of the proposed balancing approach.

Mathematical model-based rotor balancing technique 2.5

In the recent past, with the support of the advanced computer technology, the model-based

rotor fault identification approach has been introduced [93-95]. In this approach, a priori

information about the system is analytically included in the identification process. Therefore,

unlike the signal-based approaches, the mathematical model-based rotor malfunction

identification approaches utilise all information concerning dynamic and health of the

machine parts [96]. Thus, the rotor faults can be identified faster and more accurately and

reliably than the traditional methods. The model-based method requires an accurate

numerical model of the rotating machine for the identification and quantification of the rotor

faults [97]. The FE method has been found to be the most appropriate tool for the numerical

modelling in structural engineering today [27]. This section is divided into two sub-sections,

where the first sub-section presents the model-based rotor balancing approaches that use the

complete mathematical model of the rotating machine, and the second subsection presents the

model-based rotor balancing approaches that use reduced mathematical model of the rotating

machine.


43

2.5.1 Model-based rotor balancing using full mathematical model

Some researchers presented model-based unbalance estimation approaches that use the full

numerical model of the machine (i.e. including rotor, bearings and foundation) [98-100].

Comprehensive surveys of literature on model-based rotor faults diagnosis methods,

including rotor unbalance, with rich bibliography are given by Parkinson [101], Foiles et al.

[88] and Edwards et al. [28]. Recently, Lees et al. [102] presented an updated review of

literature on the model-based balancing approach and stated the possible future trends.

Bachschmid and Pennacchi [103] proposed a model-based rotor fault diagnosis method that

requires the full mathematical model of the rotating machine. They have applied the proposed

method on small size laboratory rigs as well as real rotating machines. The faults analysed in

the proposed balancing method were rotor mass unbalance, radial/angular misalignment,

crack and thermal bow. They were able to distinguish between the various faults that generate

alike symptoms. Furthermore, the identification and location of the rotor unbalance in a 125

MW gas turbogenerator set that has a length of around 20 meters and supported by four oil

film bearings was reasonable. The mathematical modelling of the foundation was difficult

owing to the existence of some local resonances. This shortcoming contributes to the

weakness of the numerical model which eventually affects the accuracy of the estimated rotor

unbalance.

Markert et al. [104] and Bachschmid et al. [105] presented a multiple fault estimation method

(including rotor unbalance) that uses the full numerical model of the rotating machine and

permits the online identification of rotor faults. According to their proposed method, the

models of the rotor faults are defined as equivalent loads. The equivalent loads are virtual

force or moment systems that generate similar behaviour to the damaged rotating machine.

The least squares fitting algorithm in the frequency domain was used with the model-based

diagnostic method to obtain information about the location and extent of the different rotor

fault types. The proposed method was validated experimentally.

Jain and Kundra [106] presented a method that uses the full mathematical model of a small

test apparatus with two discs and supported by two bearings for the identification of multiple

faults, namely rotor mass unbalance and transverse fatigue crack. To calculate the equivalent

loads from the mathematical model, it is required to have the measured vibrations of all


44

degrees of freedom (DOFs) of both the damaged and undamaged systems. As the vibration

response is acquired at a limited number of DOFs in practice, they have applied the modal

expansion method to estimate the full vibrational state of the system. Similar to the

aforementioned references (i.e. [104] and [105]) they have applied the least square approach

in order to fit the theoretical equivalent loads from the fault models into the equivalent loads

from the acquired data. The error involved in the identified rotor malfunctions is probably

caused by the estimation of the non-measured data.

Bachschmid et al. [107] presented a model-based unbalance identification approach. The

proposed method was applied to a 28 meters long turbogenerator set which consists of two

turbines (i.e. high-intermediate pressure and low pressure) and one generator that are rigidly

coupled. The supporting structure of the steam turbine was represented by a modal model

which is coupled to the rotor via the mixed co-ordinate technique. In a different publication

[108], they applied the same model-based identification approach on a different industrial

turbogenerator set which was affected by a rub in the sealing. Although the models of the

rotating machines were not fine-tuned, the proposed method was able to identify both faults

efficiently.

2.5.2 Model-based rotor balancing using reduced mathematical model

In the last few decades, as the turbo-machinery size has increased significantly, most

industries have replaced the massive and costly concrete foundations with lightweight

flexible supporting structures such as the fabricated steel structures. The reason is that the

fabricated steel structures are generally cheaper, easier to build and increase the space for

auxiliary equipment below the main machine [102, 109]. It has been found that the

lightweight machine’s‎foundation considerably influence the dynamic behaviour of the entire

rotating machine system [26, 102, 109-113]. Thus, the effect of the flexible supporting

structure of the rotating machine should be taken into consideration in the routine condition

monitoring and the associated maintenance strategy. Neto et al. [114] have highlighted the

effects of the flexible foundation on the rotating machine by presenting some real case

studies.

Based on the above discussion, it could be asserted that the availability of a sufficiently

accurate foundation model is extremely useful for efficient operation and rotor fault

diagnosis. However, using the theory to construct a reliable and accurate complete


45

mathematical model of the industrial rotating machines is an elusive task. This is due to many

uncertainties such as the huge number of joints [115]. Therefore, the most reliable approach

is to perform in-situ testing. One possible modelling approach is the technique of

experimental modal analysis. This modelling approach relies on the data from the frequency

response function (FRF) curves where vibration modal parameters are extracted from the

measured data, and hence the dynamical behaviour can be described adequately [116, 117].

Such an in-situ modal testing is possibly the best method to stay away from all uncertainties

and updating of the numerical modelling approach. However, applying it to a large industrial

rotating machine such as gas turbine, which has numerous joints and fluid bearings, is

complicated due to the limited time constraints.

An alternative modelling approach is the use of a merged theoretical-experimental model for

the rotor-bearings-foundation system using modal coordinates [118]. In this approach, the

rotor and bearings are modelled numerically, and the foundation is modelled through modal

testing. Cavalca et al. [117] have used this approach to model their laboratory scaled test rig.

They have analysed the effect of the foundation sub-structure on a rotor-bearing system.

Generally, this technique is useful when the rotor can be disassembled easily from the system

[119]. Therefore, the main drawback of such modelling approach is that it requires the rotor

to be disassembled from the supporting structure to perform the modal tests, which is not

practical for most of the existing power stations.

Due to numerous practical difficulties, a reliable FE model of the foundation of the rotating

machine is hard to construct [120]. It is often found that two machines with exactly same

configuration, at the same location, constructed to the same drawings, exhibit noticeably

different vibrational behaviour. A possible reason for this different vibrational manners is the

nonlinear behaviour of the keyways and the huge number of connections between

components such as press fits, bolted joints and welds which combine to change the stiffness

of the structure significantly [121].

Considering these difficulties, using measured vibration response at the bearing locations

together with an acceptable numerical model of the rotor and a reasonably accurate model of

the bearings seems to be the most promising approach to identify the foundation model [122].

This approach was proposed firstly by Lees and Friswell [123]. The proposed method uses a

reduced mathematical model of the machine (i.e. good model of the rotor and acceptable


46

model of the bearings) as well as measured pedestal vibration from a single machine’s

transient operation to estimate the foundation model as well as the state of rotor mass

unbalance. For the sake of simplicity, no damping has been included in the calculation.

Although the estimation of the amplitude of the mass unbalance was of good accuracy, no

attempts for the estimation of the phase have been made as this relies strongly on the quality

of the bearing model.

Edwards et al. [124] have developed a model-based method that can identify both the

foundation model as well as amplitude and phase of rotor mass unbalance using measured

response of a single machine’s transient operation. The identification method has shown

robustness with different unbalance configurations under many conditions such as poor

selection of balancing planes, and with different rotor and foundation configurations. To

validate the method, they have applied it experimentally on a laboratory size test rig.

Lees et al. [125] have applied this method to a turbogenerator set with turbine blades loss. An

accurate numerical model of the rotor and fairly accurate model of the journal bearings have

been used. In order to check the accuracy of the obtained foundation model, they have

estimated the machine response using rotor-bearing model together with estimated unbalance

and foundation parameters. A good fit between measured and estimated responses has been

shown, which proves that the estimated foundation model is of a good accuracy. Smart et al.

[109] proposed that for large rotating machines such as gas turbine, the identification of

foundation model using measured vibration at the entire speed range of the machine’s‎ run-

up/run-down in a single band might be inaccurate. Therefore, they suggested that the

frequency range should be split into smaller bands.

As suggested by Smart et al. [109, 126], Sinha et al. [25, 127] have presented a method that

estimates both rotor mass unbalance and foundation model of a rotor-bearing-foundation

system. The method was similar in concept to references [123-125]. They have estimated the

parameters numerically and experimentally using three approaches. In the first approach, they

have used the measured machine response at the whole speed range at once. This approach

has estimated the rotor unbalance accurately, but has shown poor results for the foundation

parameters (i.e. mass, stiffness and damping). The reason was that the number of critical

speeds was probably more than the number of the measured degrees of freedom (DOFs). In

the second approach, they have split the speed range into smaller segments, but have


47

identified a different unbalance vector within each frequency band. This method has shown

poor estimations of the rotor unbalance. In the final approach, they have split the frequency

range into bands and have estimated a global unbalance vector, but different foundation

models in each frequency band. This method has provided an excellent estimation of both

state of unbalance and foundation model. Despite that this approach estimated rotor

unbalance accurately, it can be computationally involved when applied on large rotating

machines that have several bearings such as the turbogenerator set. Therefore, enhancing this

method by applying it with reduced number of sensors without affecting the estimation

process could be welcomed by power plants.

Jalan et al. [128] have presented a model-based technique for diagnosing multiple faults,

namely rotor mass unbalance and misalignment, in rotor-bearing systems. They have been

successfully able to identify both rotor malfunctions. The numerical results have been

validated by experimental measurements on a small laboratory rig. After running the test rig

with mass unbalance and misalignment, 1× (one multiplied by rotating frequency) and 2×

(two multiplied by rotating frequency) running speed components have been displayed. This

can be expected as it is well known that the rotor unbalance affects the 1× running speed, and

the misalignment mostly affects the 2× running speed [53].

Tiwari et al. [129, 130] have proposed a method for simultaneous identification of rotor mass

unbalance and bearing dynamic coefficients of a flexible rotor-bearing system by using run-

down vibration measurements. The identification procedure has been found to be very ill-

conditioned. After applying column scaling, they have found that the estimation of the

bearing coefficients still ill-conditioned. Thus, to solve this problem, they have implemented

the Tikhonov regularisation [131]. The results of the actual and estimated residual unbalances

and bearing coefficients were in good agreement. Also, their method has shown robustness

against noise. The proposed rotor fault identification technique has been validated

experimentally by Tiwari and Chakravarthy [132].

To enhance understanding of the theory behind the mathematical model-based unbalance

estimation approach consider a rotating machinery with rotor, journal bearings and machine

supporting structure, where the rotor is connected to the foundation through the bearings. The

equation of motion of such structure can be written as shown in Equation (2.5) below:


48

[

𝐙𝑅,𝑖𝑖 𝐙𝑅,𝑖𝑏 0

𝐙𝑅,𝑏𝑖 𝐙𝑅,𝑏𝑏 + 𝐙𝐵 −𝐙𝐵0 −𝐙𝐵 𝐙𝐵 + �̅�𝐹

] {

𝐝𝑅,𝑖𝐝𝑅,𝑏𝐝𝐹,𝑏

} = {𝐟𝑢00} (2.5)

where 𝐙, 𝐝 and 𝐟𝑢 are the dynamic stiffness matrix (also called impedance matrix),

displacement vector and unbalance forces respectively. The subscripts 𝐹, 𝑅, 𝐵, 𝑖 and 𝑏

denote the foundation, the rotor, the bearings, the internal degrees of freedom and the bearing

(connection) degrees of freedom, respectively.

The first two rows of Equation (2.5) are used to eliminate the non-measured degrees of

freedom of the rotor (i.e. 𝐝𝑅,𝑖 and 𝐝𝑅,𝑏) as shown in the equation below:

�̅�𝐹𝐝𝐹,𝑏 = 𝐙𝐵(𝑃−1𝐙𝐵 − 𝐼)𝐝𝐹,𝑏 − 𝐙𝐵𝑃

−1𝐙𝑅,𝑏𝑖𝐙𝑅,𝑖𝑖−1 𝐟𝑢 (2.6)

where 𝑃 = 𝐙𝑅,𝑏𝑏 + 𝐙𝐵 − 𝐙𝑅,𝑏𝑖𝐙𝑅,𝑖𝑖−1 𝐙𝑅,𝑖𝑏. It is assumed that reasonably accurate analytical

models of the rotor 𝐙𝑅 and bearings 𝐙𝐵 are available, and the vibration response at bearing

pedestals 𝐝𝐹,𝑏 is measured. Thus, the unbalance forces 𝐟𝑢 and the reduced foundation model

�̅�𝐹 are the only unknowns in Equation (2.6).

Assume that the unbalance planes are positioned at nodes 𝑛1 to 𝑛𝑝. The amplitude of

unbalance and phase angles associated to the unbalance planes can be written as

[𝑢𝑛1 , 𝑢𝑛2 , 𝑢𝑛3 , … , 𝑢𝑛𝑝]T

and [α𝑛1 , α𝑛2 , α𝑛3 , … , α𝑛𝑝]T

. Therefore, the complex quantity of the

rotor unbalance for the 𝑖th balancing plane may be evaluated as:

𝑢𝑛𝑖 exp(jα𝑛𝑖) = 𝑎𝑛𝑖 + j𝑏𝑛𝑖 (2.7)

Hence, the unbalance forces in the vertical and horizontal directions can be written as shown

in Equation (2.8).


49

𝐟𝑢 = ω2

{

0⋮0

𝑎𝑛1 + 𝑗𝑏𝑛1−𝑗𝑎𝑛1 + 𝑏𝑛1

0⋮0

𝑎𝑛𝑝 + 𝑗𝑏𝑛𝑝−𝑗𝑎𝑛𝑝 + 𝑏𝑛𝑝

0⋮0 }

(2.8)

The expression for the unbalance forces may be further simplified as shown in Equation (2.9)

𝐟𝑢 = ω2𝐓𝐠 (2.9)

where 𝐠 = [𝑎𝑛1 𝑏𝑛1 𝑎𝑛2 𝑏𝑛2 ⋯ 𝑎𝑛𝑝 𝑏𝑛𝑝]T

and 𝐓 is the transformation matrix which is

defined such that Equations (2.8) and (2.9) are equivalent. Substituting Equation (2.9) into

Equation (2.6) gives

�̅�𝐹𝐝𝐹,𝑏 = 𝐙𝐵(𝑃−1𝐙𝐵 − 𝐼)𝐝𝐹,𝑏 −ω

2(𝐙𝐵𝑃−1𝐙𝑅,𝑏𝑖𝐙𝑅,𝑖𝑖

−1 𝐓)𝐠 (2.10)

In order to identify the rotor unbalance in a least-square sense, the foundation parameters are

grouped into a vector 𝐯. Assume that the impedance matrix of the reduced model of the

foundation is written in terms of mass, damping and stiffness matrices. If there are 𝑡

measured degrees of freedom at the foundation-bearing interface, then the vector 𝐯 can be

written as

𝐯 = [�̅�𝐹,11 �̅�𝐹,12 … �̅�𝐹,𝑡𝑡 𝑐�̅�,11 𝑐�̅�,12 … 𝑐�̅�,𝑡𝑡 �̅�𝐹,11 �̅�𝐹,12 … �̅�𝐹,𝑡𝑡]T (2.11)

With this expression of 𝐯, there is a linear transformation such that

[�̅�𝐹]{𝐝𝐹,𝑏} = [𝐖]{𝐯} (2.12)

where 𝐖 contains the measured response terms at each frequency. For the N𝑡ℎ measured

frequency:


50

𝐖(ω𝑁) = [W0(ω𝑁) W1(ω𝑁) W2(ω𝑁)]T (2.13)

If all elements of the foundation mass, damping and stiffness matrices are identified, then

W𝑘(ω𝑁) = (𝑗ω𝑁)𝑘

[ 𝐝𝐹,𝑏T (ω𝑁) 0 … 0

0 𝐝𝐹,𝑏T (ω𝑁) 0

⋮ ⋮ ⋱ ⋮0 0 … 𝐝𝐹,𝑏

T (ω𝑁)]

(2.14)

where 𝑘 = 0, 1, 2. Hence, Equation (2.10) becomes

𝐙𝐵(𝑃−1𝐙𝐵 − 𝐼)𝐝𝐹,𝑏 = ω𝑁

2 (𝐙𝐵𝑃−1𝐙𝑅,𝑏𝑖𝐙𝑅,𝑖𝑖

−1 𝐓)𝐠 + [𝐖]{𝐯} (2.15)

Let

R(ω𝑁) = ω𝑁2 𝐙𝐵𝑃

−1(ω𝑁)𝐙𝑅,𝑏𝑖(ω𝑁)𝐙𝑅,𝑖𝑖−1 (ω𝑁)𝐓 (2.16)

and

H(ω𝑁) = 𝐙𝐵(ω𝑁) (𝑃−1(ω𝑁)𝐙𝐵(ω𝑁) − 𝐼)𝐝𝐹,𝑏(ω𝑁) (2.17)

Substitution of Equations (2.16) and (2.17) into Equation (2.15) gives

[𝐖(ω𝑁)]{𝐯} + [R(ω𝑁)]{𝐠} = H(ω𝑁) (2.18)

Equation (2.18) can be rearranged as shown in Equation (2.19)

[𝐖(ω𝑁) R(ω𝑁)] {𝐯𝐠} = [H(ω𝑁)] (2.19)

The system of equations in Equation (2.19) is overdetermined. Therefore, a solution can be

obtained by applying the least squares technique as shown in Equation (2.20)

{𝐯𝐠} = [[[𝐖 R]

T[𝐖 R]]−1[𝐖 R]T] [H] (2.20)


51

Summary and conclusion 2.6

This Chapter started by reviewing the modal balancing technique. Then, review on papers

pertaining to the IC balancing approach was given. After that, the unified balancing method

that combines both modal balancing and IC balancing approaches was reviewed. Finally, a

review of literature regarding the model-based rotor mass unbalance identification approach

was presented.

The IC balancing approach is a powerful and almost mature method for balancing of

industrial rotating machines. However, this balancing approach is currently applied using

vibration measurements acquired at a single machine speed which is mostly the machine

operating speed. The use of acquired vibration measurements at a single machine speed might

introduce error in the estimated rotor unbalance, hence results in poor rotor balancing.

Therefore, the influence of applying the IC balancing method using vibration measurements

at multiple speeds in a single band will be investigated in this research study and compared

with the estimation using measurements at single speeds.

It has been found that almost no researchers (according to the best knowledge of the author)

have identified the rotor mass unbalance using a single sensor per bearing location together

with vibration measurements from a single machine run-up/down. Therefore, this research

project attempts to enhance the current IC and model-based balancing approaches by using a

single vibration sensor per bearing pedestal.

It is comprehensible that since rotating machines represent the largest and most important

class of machinery used for many industrial applications, a large amount of papers, books,

patents, reports and various texts has been published during the last few decades. Therefore,

it‎ is‎not‎an‎easy‎matter‎ and‎surely‎out‎of‎ the‎author’s‎ability‎ to‎ recognise and describe the

significance of each work and include each contribution in the entire field. On the other hand,

a general description of the direction in which various researchers made their contributions

has been achieved in this brief literature review.

‎CHAPTER 3 EXPERIMENTAL SETUP AND INSTRUMENTATION

52

CHAPTER 3

EXPERIMENTAL SETUP AND

INSTRUMENTATION


53

Introduction 3.1

This chapter presents the laboratory rigs and instrumentations used for the experiments

in details. It starts by showing the steps of constructing and assembling the test rigs, and

then explains the different mechanical and electrical elements of the experimental

apparatus (Section 3.2). After that, the different parts of the data acquisition (DAQ)

system are described in details in Section 3.3. The last part of this chapter shows the

modal testing of the system (Section 3.4).

Experimental rig 3.2

The rotor mass unbalance estimation methods proposed in the current study are needed

to be validated experimentally in order to show their practicality in real machines.

Therefore, two configurations of a simple rotating machine were first designed using

professional commercial engineering drawing software SolidWorks [133] (Figure ‎3.1).

The first configuration is with one balancing disc and the second configuration is with

two balancing discs. Each part of the test rig was manufactured individually according

to the drawings. Then, all parts were assembled to form the complete test apparatus.

Figure ‎3.2 presents the different elements of the assembled test rig. The steps of

assembling the various parts to create the complete small scale laboratory rig are shown

in Figure ‎3.3. Figure ‎3.4 displays photographs of the constructed experimental rigs. The

rigs were designed to simulate the real rotating machines in order to detect rotor mass

unbalance using vibration measurements acquired during the transient period (i.e.

machine run-up and run-down). The test rig is located in the Dynamics Laboratory at

Pariser Building at the University of Manchester.


54

Figure ‎3.1 3D drawings of the experimental rig, (a) first configuration, (b) second configuration

(a)

(b)


55

Figure ‎3.2 Presentation of the different elements of the assembled test rig (first configuration)

3.2.1 Main elements of the experimental rig

The essential mechanical and electrical parts of the test apparatus including the shaft,

disc, motor, etc. are described in the following seven sub-sections.

3.2.1.1 Rotating shaft

The rotating mild steel solid shaft was manufactured by SKF Company. The length of

the shaft is 1000mm with a circular cross section of 20mm diameter. The hardness of

the steel shaft is 60 HRC and‎its‎mass‎is‎2.46‎kg.‎The‎density‎and‎Young’s‎modulus‎of‎

the shaft are calculated in Sub-section 3.4.2.

Ball bearing Balancing disc Flexible coupling

Motor support

Rig support Bearing support


56

Figure ‎3.3 Steps of constructing the experimental rig (first configuration)

(1) (2)

(6)

(3) (4)

(5)


57

Figure ‎3.4 Layout of the (a) first and (b) second configurations of the experimental rig

(b)

(a)


58

3.2.1.2 Balancing disc

In the first configuration, the shaft carries a single balancing disc positioned at midspan

between the two bearings. In the second configuration, the shaft carries two identical

balancing discs at distances of 240 and 665mm from the bearing closer to the motor.

The balancing discs were manufactured to simulate the rotating parts that are attached to

the rotating shaft in real rotating machines. Each disc is made of mild steel and has a

density of 7828 kg/m3. The diameter and thickness of both discs are 130 and 20mm

respectively. Two flanges with diameter of 40mm and thickness of 20mm each are

added to the disc to ensure that the disc will not slip when the shaft rotates at high

speed. Each disc contains staggered tapped holes (M5) in two different pitch diameters

which are 70 and 120mm. The angle between two adjacent holes for a particular pitch

diameter is 30 degree. The staggered tapped holes were made for the purpose of

inserting the mass unbalance in them, hence simulating the unbalance in real rotating

machines. Typical balancing disc is shown in Figure ‎3.5.

Figure ‎3.5 (a) Photograph and (b) dimensions of typical balancing disc

3.2.1.3 Ball bearings and supporting structure

The assembled rotor (i.e. shaft and balancing disc(s)) is supported by two identical

greased lubricated pedestal ball bearings as illustrated in Figure ‎3.3. The bearings,

which were manufactured by SKF (model: SY20TF), can accommodate speeds up to

8500 RPM. The bearing housings are made of grey cast iron and can be re-lubricated

𝟑𝟎°

40 mm 20 mm

Side view

Front view

130

mm

60 mm

20 mm (a) (b)


59

through a grease nipple in the housing. Figure ‎3.6 shows a photograph of conventional

pedestal ball bearing.

Figure ‎3.6 (a) Photograph of typical ball bearing (model: SY20TF) and (b) its dimensions in mm

In order to include more than one critical speed within the‎ machine’s‎ run-up speed

range, i.e. 0 to 50 Hz, two thin and flexible steel horizontal beams are used to support

the bearings. The dimensions of the horizontal beams for the first and second

configurations are 530mm × 25mm × 3mm and 530mm × 25mm × 8mm respectively.

The horizontal beams were sandwiched between two layers of double-sided adhesive

tape (model: 3M Acrylic Foam Tape, 25mm, white) in order to increase the damping

effect when the rotor pass through the critical speeds. The outer side of the tape is

attached to a very thin shim to make the damping effect stay for a long time

(Figure ‎3.7). Each horizontal beam is screwed to two rectangular steel blocks (107mm

× 25mm × 25mm) that are screwed to a thick horizontal base plate (580mm × 150mm

× 15mm).

33.3

20.5

65

32

127

97

20

(a) (b)


60

Figure ‎3.7 Steps of attaching the double-sided adhesive tape and shim to the horizontal beam

3.2.1.4 Electrical motor

The rotor-bearing system was mated to a three-phase DC motor (Figure ‎3.3). The

electrical motor was manufactured by Beatson Fans & Motors Ltd, and its model is

GF6382. A photograph of the motor is shown in Figure ‎3.8. The power of the motor is

0.75 kW and its maximum speed is approximately 3000 RPM. The motor was

connected to a control panel to run the motor’s shaft at different speeds.

Figure ‎3.8 Photographs of the electrical motor used in the experiments

(1)

(4)

(2)

(3)


61

To run-up/run-down the motor linearly at a given period of time with a specified range

of speeds, the control panel was controlled by software developed by Softstart

Company. The laptop was connected to the control panel through universal serial bus

(USB) portable device which is shown in Figure ‎3.9. The USB device was

manufactured by Delta Electronics, Inc. (model: IFD6500).

Figure ‎3.9 Delta USB-RS485 converter

For the purpose of aligning the motor’s shaft centre with the rotor’s shaft centre, two

blocks (150mm × 25mm × 55mm) were positioned under the motor as shown in

Figure ‎3.8. The motor was fixed to a rigid thick horizontal plate by four long screws

passing through the two blocks. The thick horizontal plate was mounted on a massive

steel platform by means of two screws.

3.2.1.5 Flexible coupling

The rotor-bearing system was mated to the electrical motor through a flexible coupling

made of 7075 Aluminium‎ to‎ transmit‎ rotational‎ power‎ from‎ the‎motor’s‎ shaft‎ to‎ the‎

rotor’s‎ shaft.‎ The‎ flexible‎ coupling‎ was made by Ruland Manufacturing Co., Inc.

(model: FCMR38-20-16-A) (Figure ‎3.10). The weight, length, outside diameter and

bore diameters of the coupling are 236g, 57.2mm, 38.1mm, 20mm, and 16mm

respectively. The temperature range of the coupling is between -40 and +107C, and its

speed capability ranges up to 6000 RPM. The biggest advantage of this type of

couplings is that they can accommodate misalignment and compensate for end

movement while transmitting power [134]. This coupling permits angular misalignment

of up to 3 degrees. In addition, it allows some parallel misalignment between the motor


62

and rotor shafts. The coupling hubs were first aligned on the shafts. Then, the M5

screws of both coupling hubs were tightened using 4mm hex key.

Figure ‎3.10 Flexible coupling (a) before and (b) after assembling

3.2.1.6 Foundation

The three base plates are mounted on a massive rectangular platform made of mild steel.

The length, width and thickness of the steel platform are 1743mm, 1136mm and 30mm

respectively. The steel platform lies atop a huge steel plate with similar length and

width but has a thickness of 50mm. For the purpose of mitigating noise and vibration, a

12mm thick anti-vibration (TICO) pad was sandwiched between the two steel plates.

The anti-vibration pad, which was manufactured by Tiflex Company, consists of

synthetic rubber and filled with cork particles. The massive steel plate is placed atop

hollow rectangular foundation table (1743mm × 1136mm × 690mm) which was built

of bricks. The steps of constructing the foundation are shown in Figure ‎3.11. A

photograph of the foundation table, steel plates and TICO pad is provided in

Figure ‎3.12.

(a) (b)


63

Figure ‎3.11 Steps of constructing the foundation

Figure ‎3.12 Photograph of the foundation

3.2.1.7 Machine guard and safety switch

The experimental rig was guarded by a rectangular metal steel casing for safety

precaution. The top, front and right sides of the guard are transparent windows. The

length, width and height of the guard are 1155mm, 600mm and 280mm respectively.

Figure ‎3.13 shows a photograph of the guard of the test rig. An electrical switch was

attached to one of the hinges on the back of the guard (Figure ‎3.14). It was

(1)

(4) (3)

(2)

Steel platform

Anti-vibration pad

Steel plate

30mm

12mm

50mm


64

manufactured by Steute (material number: 95554002); it works in a way such that when

the guard is rotated to the open position by more than 10 degrees, the normally closed

contacts in the safety switch are mechanically forced open turning the control power off,

hence disabling the machine. As the switch contacts can be closed only when the shaft

of the switch is rotated to the closed position, the rotating machine cannot be restarted

until the guard is closed.

Figure ‎3.13 Machine guard (a) close position, (b) open position

Figure ‎3.14 Hinge operated safety switch

(a) (b)


65

Data acquisition system 3.3

The term data acquisition, abbreviated by the acronyms DAQ, is often used to describe

the sampling of signals that measure real world physical phenomena including motion,

temperature, pressure, etc. and converting them into digital values that can be stored and

analysed on a personal computer (PC). A typical DAQ system consists of sensors,

signal conditioners, DAQ hardware, DAQ software, and PC. Figure ‎3.15 shows a

functional diagram of typical DAQ system. The main elements of the DAQ system are

explained in the following four sub-sections.

Figure ‎3.15 Functional diagram of typical DAQ system

3.3.1 Sensors

Although the terms transducer and sensor are used interchangeably, they have quite

different meanings. The term transducer is usually used to describe the device that

converts any type of energy into another. However, the sensor is a device that is

responsible for converting a physical phenomenon into electrical charge which can be

measured by the DAQ hardware [135]. Sensors are commonly used in industries for

making in-situ measurements. The type of sensor is chosen depending on the physical

phenomenon that is needed to be measured. For example, thermocouples are used to

convert the temperature which is a physical phenomenon into a voltage which is an

electrical signal [136].

Physical

Phenomenon Sensor Field wiring

Signal

conditioner

Field wiring DAQ device DAQ

Software


66

The type of the sensor used in the current research project is Integrated Circuit

Piezoelectric (ICP) acceleration sensor. It is manufactured by PCB Piezotronics Inc. and

its model is 352C33. The ICP acceleration sensor is an electromechanical sensor which

consists of a piezo-electric material that is sandwiched between a seismic mass and the

base of the sensor, as illustrated in Figure ‎3.16.

Figure ‎3.16 Schematic cross-sectional view of compression mode ICP acceleration sensor

The term piezoelectric is used to describe the material that produces positive/negative

electric charges when it is compressed or subjected to shear forces [137]. The most

popular piezoelectric materials are quartz (naturally piezoelectric) and polycrystalline

ceramics [135]. The latter is human-made crystals which are forced to be piezoelectric.

The piezoelectric material used in the above accelerometer model is polycrystalline

ceramics.

There are three primary configurations utilised in the ICP accelerometers design. These

are compression mode design, shear mode design and flexural mode design [138]. The

accelerometers used in the current research are compression mode accelerometers. In

this type of accelerometers, when the sensor is subjected to acceleration, the seismic

mass imposes a force on the piezoelectric crystal and hence compresses and stretches it.

As a result of the compression, the piezoelectric crystal produces a charge which is

proportional‎to‎the‎acceleration‎(according‎to‎Newton’s‎second‎law‎of‎motion:‎𝐹𝑜𝑟𝑐𝑒 =

𝑚𝑎𝑠𝑠 × 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛). The charge output is then converted to a low impedance

voltage [139]. This conversion is made by built-in electronics (Figure ‎3.16). The 5.8g

Seismic mass

Pre-load bolt

Case

Piezo-electric crystal

Base

Mounting stud

Built-in electronics


67

accelerometer has a sensitivity of 100 mV/g and works in a frequency range of 0.5 to

10,000 Hz.

3.3.2 Signal conditioner

The analogue signals produced by the sensors often need some conditioning before

reaching the analogue to digital converter (ADC) in the DAQ hardware. Signal

conditioning devices are capable of removing the unwanted components of the analogue

signal and prepare it for the DAQ device. Signal conditioning usually includes

attenuation, amplification, filtration, linearization, isolation, multiplexing, etc. [140].

Furthermore, some types of signal conditioners can provide an adjustable current source

to drive some models of sensors [141].

As a matter of fact, almost all signals from sensors have a certain amount of noise.

Therefore, it is not easy for the very small signals to survive without conditioning.

Some types of sensors often generate very tiny signals, which cannot survive without

conditioning. Therefore, signal conditioners often contain low pass filter (commonly

known as an antialiasing filter) which is capable of blocking frequencies higher than the

desired frequency range and hence increasing the measurement accuracy [142]. The

type of the signal conditioner used in the current research study is PCB 482C05

(Figure ‎3.17).

Figure ‎3.17 (a) Front and (b) back sides of the signal conditioner used in the modal test

3.3.3 DAQ device

The next step after conditioning the analogue signal is to translate it into a digital form.

This is done through the ADC. The ADC is the heart of the DAQ hardware as it makes

(a) (b)


68

the signal readable and can be analysed by the computer [142]. The DAQ card is

required to be connected to PC in case of using PC based data acquisition systems, and

the driver software performs the data acquisition. The DAQ works on the principle of

converting the input voltage into a number of the nearest binary bins, depending on the

sensitivity of the DAQ hardware [17]. The DAQ device used in the current research

project was manufactured by National Instruments company (model: NI USB-6229

BNC). Figure ‎3.18 shows a photograph of the DAQ hardware employed in the modal

test.

Figure ‎3.18 16-bit 16-channel DAQ hardware

3.3.4 DAQ software

DAQ software is the final component of the DAQ system [142]. It is required for the

DAQ device to be able to communicate with the PC. Without proper driver software,

the whole DAQ system will not function properly. The driver software can be defined

as a set of instructions that are written by a programmer to perform a particular task

when executed by the computer [143]. The driver software can be written in any

programming language that computer can understand. Examples of the popular

programming languages used to construct the drive software are C, C++, Java and

Fortran [136]. The driver software utilised in the current research project was developed

by Austin consultants (Figure ‎3.19).


69

Figure ‎3.19 DAQ driver software

Modal tests 3.4

The modal test is carried out to know the modal properties of the entire system

including natural frequencies, mode shapes, etc. It is necessary to conduct a modal test

before running the machine due to many reasons such as avoiding running the machine

at or close to the critical speeds. Also, these parameters are required for the FE

modelling in order to predict the structural response to dynamic loading. There are four

main stages in the modal testing of any structure, namely structure exciting stage,

sensing stage, signal conditioning stage, and processing stage.

3.4.1 Modal test of assembled rig

The set up for the modal test is shown in Figure ‎3.20. Figure ‎3.21 shows a schematic of

the setup and instrumentations used for the modal test.

ur


70

Figure ‎3.20 Setup of the modal test of the (a) first and (b) second test rig configurations

Sensors Impact locations (b)

(a) Sensors Impact locations


71

Figure ‎3.21 Schematic of the setup and instrumentations used for the modal test (first

configuration)

An instrumented hammer with a sensitivity of 1.1 mV/g (model: PCB 086C04) has been

used to excite the rig. This has been done by impacting different locations on the shaft.

The instrumented hammer has been connected to the four-channel signal conditioner via

BNC (Bayonet Neill–Concelman) to BNC cable. Seven ICP accelerometers have been

used to measure the dynamic response of the excited structure and transmit the analogue

signal to the signal conditioners via micro-to-BNC cables. Then, the conditioned

analogue signal has been transmitted to the 16-bit 16-channel DAQ hardware using

BNC-to-BNC cables. After that, the digitised signal has been transmitted to data

logging software on a Laptop computer via USB cable for further processing and

analysis.

The accelerometers have been mounted using super glue in mutually perpendicular

directions. Super glue is an excellent choice for vibration testing as this type of adhesive

dries quickly. Furthermore, as the glue has a very good stiffness compared to other

adhesives, it provides an excellent frequency response. To get a good indication of the

𝒙

𝒚

𝒛 a b

Accelerometer , Impact hammer , Micro to BNC signal cable

BNC to BNC signal cable (Sensor) BNC to BNC signal cable (Hammer) , Note: All dimensions are in cm

Front view Side view

𝒙 𝒛

𝒚

19

9 58 42 42 7

100

Motor


72

mode shapes, locations of the accelerometers have been chosen carefully as displayed in

Figure ‎3.21. For each impact location, the hammer has impacted the shaft ten times,

where the response decayed to zero before each new impact. This is done to reduce the

noise level in the signal.

Although the impact hammer test is designed to reproduce the ideal impact of structure,

which has infinitely small impact duration, it is almost impossible to perform it in real

as there is always a known contact time. Therefore, every effort has been made to

ensure that the contact time is very small and no double hits have been done. It should

be mentioned that acquiring the vibration response has been started few seconds before

the first impact and stopped few seconds after the last impact. The impact test has been

done in both vertical and horizontal radial directions.

Code written in commercial software package MATLAB [144] has been used to

generate FRF of the acquired vibration measurements of each location. The sampling

rate was 5000 Hz, and the frequency resolution was 0.61 Hz. The first four natural

frequencies of the first configuration by appearance are listed in Table ‎3.1. Figure ‎3.22

shows typical FRF plot from the modal test on the first test rig configuration. It has been

observed that the 17.09 and 29.91 Hz natural frequencies are dominant in the horizontal

direction while the 31.13 and 58.59 Hz are dominant in the vertical direction. The mode

shapes of the first four modes are shown in Figure ‎3.23.


73

Figure ‎3.22 Typical FRF plots in (a) vertical and (b) horizontal directions at location 4 (first

configuration)

(a)

(b)


74

Figure ‎3.23 Mode shapes of the (a)1st, (b) 2

nd, (c) 3

rd and (d) 4


st

configuration)

In the second configuration, the test was done by impacting three different locations

along the shaft at both vertical and horizontal locations. The first four natural

frequencies of the second configuration by appearance are listed in Table ‎3.1.

Figure ‎3.24 shows typical FRF plot from the modal test on the second test rig

configuration. The mode shapes of the first four modes are shown in Figure ‎3.25.

Table ‎3.1 Natural frequencies of the test rig

Mode 1

st configuration 2

nd configuration

Frequency (Hz) Frequency (Hz)

1 17.09 24.41

2 29.91 31.13

3 31.13 53.1

4 58.59 84.23

(d) (c)

(b) (a)


75

Figure ‎3.24 Typical FRF plots in (a) vertical and (b) horizontal directions at location 4 (second

configuration)

(a)

(b)


76

Figure ‎3.25 Mode shapes of the (a)1st, (b) 2

nd, (c) 3

rd and (d) 4


nd

configuration)

3.4.2 Modal testing of free-free shaft

The modal test of the free-free shaft is carried out in order to obtain the exact value of

the Young’s modulus, which is needed in the FE modelling. First, the mass of the shaft

was measured on a digital scale and found to be 2.46 kg. The dimensions of the shaft

are shown in Figure ‎3.26.

Figure ‎3.26 Shaft dimensions

(d) (c)

(b) (a)

Front view

1000 mm

20 mm

Side view


77

Therefore, the volume of the shaft can be calculated as follows,

𝑉𝑠ℎ𝑎𝑓𝑡 = (𝜋(0.02)2

4) 1.000 = 3.142𝑒−4 𝑚3 (3.1)

Hence, the density is determined as shown in the following formula

𝜌 =𝑚𝑠ℎ𝑎𝑓𝑡

𝑉𝑠ℎ𝑎𝑓𝑡=

2.46

3.142𝑒−4= 7829 𝑘𝑔/𝑚3 (3.2)

The main formula for calculating the natural frequency [145] is

𝑓𝑛 =𝐴𝑛

2𝜋𝑙2√𝐸𝐼

𝜌𝑠 (3.3)

where 𝐴𝑛 is the shape of beam factor, 𝑙 is the shaft length, √𝐸

𝜌 is material dependent

factor with

𝐸:‎Young’s‎modulus

𝜌: Material density

and √𝐼

𝑠 is section geometry factor with

𝐼: Second moment of area

𝑠: Area of the cross section

The second moment of area of the shaft can be calculated as

𝐼 =𝜋𝑑4

64=𝜋(0.02)4

64= 7.85𝑒−9𝑚4 (3.4)

The shape of beam factor can be calculated as shown below

𝐴𝑛 = [(2𝑛 + 1)𝜋

2]2

(3.5)

Hence, the first shape of beam factor for free-free shaft is 𝐴1 = 22.2. Now, the only two

unknowns in Equation (3.3) are‎the‎natural‎frequency‎and‎Young’s‎modulus.‎In‎order‎to‎


78

obtain‎the‎Young’s‎modulus,‎the‎natural‎frequency‎should‎be‎found‎experimentally,‎and‎

then substituted it in Equation (3.3). The shaft was hanged to a beam via two bungee

cords with steel hooks (one at each end) as shown in Figure ‎3.27. The first natural

frequency by appearance has been observed at 84.84 Hz. Typical FRF plot is shown in

Figure ‎3.28.

Figure ‎3.27 Setup of the modal test of a free-free steel shaft


79

Figure ‎3.28 Typical FRF plot of free-free shaft obtained during modal testing

Thus, Young’s modulus can be calculated by rearranging Equation (3.3) as shown in the

following equation

𝐸 =4𝜋2𝑓𝑛

2(𝜌𝑠𝑙4)

𝐴𝑛2 𝐼

=4𝜋2(84.84)2(7829∗0.000314∗1.0004)

22.22∗7.85𝑒−9= 180.5𝑒9𝑃𝑎 (3.6)

Therefore,‎ the‎ value‎ of‎ Young’s‎ modulus‎ that‎ will‎ be used in the FE model is

180.5𝑒9 𝑃𝑎.

3.4.3 Discussion on the influence of supporting structure

It can be seen in Figure ‎3.28 that the natural frequency of the rotor with added balancing

disc is 84.84 Hz. However, it can be seen in Figure ‎3.22 that the natural frequency of

the assembled rig with one balancing disc is 17.09 Hz. This change in the natural

frequency is considered as the result of combined effect of the entire rotor-bearing-

foundation assembly.


80

Summary 3.5

This chapter presented two configurations of the laboratory rig used for the

experiments. The first configuration consists of a solid shaft which is supported on

relatively flexible foundation through two greased lubricated pedestal ball bearings and

carries a steel balancing disc at midspan between the two bearings. A flexible coupling

made of Aluminium connects the shaft to an electrical motor. Each of the two ball

bearings is mounted on flexible steel horizontal beam. Each horizontal beam is screwed

to two rectangular steel blocks that are screwed atop thick base plate as shown in

Figure ‎3.3. Both base plates of the two bearing pedestals are mounted on a massive steel

platform. The second configuration is similar to the first configuration, but with an extra

balancing disc and thicker horizontal beams. Both configurations are shown in

Figure ‎3.4.

The DAQ system was also discussed in this chapter. The main elements of the DAQ

system were explained. The data acquisition begins by converting a physical

phenomenon into electrical charges which are measurable by the sensor. Then, the

signal is conditioned by the signal conditioning unit and prepared for the DAQ

hardware. The primary function of the DAQ hardware is to convert the analogue signal

into digital signal. Finally, with the help of driver software, the digital signal is

displayed and stored in PC for further processing. In the last part of the current chapter,

modal tests were carried out on both configurations, and the natural frequencies as well

as mode shapes were obtained.

‎CHAPTER 4 MATHEMATICAL MODELLING AND SIGNAL PROCESSING

81

CHAPTER 4

MATHEMATICAL MODELLING AND

SIGNAL PROCESSING


82

Introduction 4.1

There are several approaches for modelling rotating machines such as the experimental

modal tests modelling technique. This modelling approach has the advantage of

avoiding all uncertainties in the mathematical modelling technique. However, the

application of this modelling approach on rotating machines with fluid bearings is

complicated, because the dynamics of the fluid bearings during the rotor rotation

influence the dynamic behaviour of the whole machine. Some researchers [115, 166,

174] presented an overview on various experimental modal tests modelling approach

and discussed the limitations of this method.

Another alternative is the application the finite element (FE) modelling approach [175].

Some researchers such as Edwards et al. [28] and Lees et al. [102] provided literature

reviews pretaing to the FE modelling of rotating machines. This modelling approach has

been found to be the most suitable tool for the numerical modelling in structural

engineering nowadays [27]. Although the FE modelling approach is powerful, its

derivation is logical and simple. The texts written by Zienkiewicz et al. [176] and Cook

et al. [177] give details of the formulation of element matrices for different structural

element types such as shells, beams, continua and plates. The steps of the numerical

modelling of rotating machines using the finite element technique was presented by

some authors such as Friswell et al. [21] as well as Sinha [17]. Some researchers

proposed rotor faults identification approaches that rely mainly on an accurate

numerical model of the rotating machine [98-100].

The current research project includes a rotor mass unbalance estimation approach that

requires an accurate mathematical model of the rotating machine. Therefore, this

chapter explains the detailed steps of modelling a simple rotating machine using the FE

method. The second part of this chapter gives a brief idea about signal processing, and

briefly presents the concepts of the 3D waterfall plot, order tracking and Bode plot.

Mathematical modelling of a simple rotating machine using the FE method 4.2

This sub-section includes modelling the subsystems of simple rotating machinery, i.e.

rotor and foundation. Eventually, the subsystems are assembled to form the complete


83

system. The modelled rotor is assumed to be symmetric (i.e. stiffness and mass

properties do not change along the rotor).

4.2.1 Shaft element

One-dimensional numerical models of structural beams are typically created using beam

theories. Due to the fact that the beam is a three-dimensional structural member, all

models unavoidably approximate the underlying physics. Consider a beam element with

a certain length and has two nodes, one at each end. Each one of these nodes has two

DOFs (i.e. lateral displacement and slope) as shown in Figure ‎4.1. The beam

deformation should have continuous deflection and slope at any two adjacent elements.

In order to fulfil this continuity condition, each node will have two nodal variables (i.e.

deflection 𝑢𝑖 and slope Ψ𝑖). Hence, any two adjacent beam elements have the same slope

and deflection at the shared node.

Figure ‎4.1 The local coordinates of a beam element

The Euler-Bernoulli beam theory, which dates back to the 18th

century [146], states that

the plane that is perpendicular to the neutral axis before deformation stays plane and

perpendicular to the neutral axis after bending deformation [147]. This is illustrated in

Figure ‎4.2.

element 1 element 4 element 3 element 2 element 5

𝑦

𝑧

𝑥 Ψ1

𝑢1 𝑢2 Ψ2

𝜉

𝑙

element 3


84

Figure ‎4.2 Condition of Euler-Bernoulli beam theory

This assumption indicates that the slope is the 1st derivative of the deflection (i. e.Ψ𝑖 =

(𝝏𝒖

𝝏𝝃)𝑖). The deflection within the element can be approximated as shown in Equation

(4.1), below:

𝑢(𝜉) = [𝑁1(𝜉) 𝑁2(𝜉) 𝑁3(𝜉) 𝑁4(𝜉)] {

𝑢1Ψ1𝑢2Ψ2

} (4.1)

where 𝑁1, 𝑁2, 𝑁3 and 𝑁4 are the shape functions (also called Hermitian functions

[148]). As the beam element has four nodal variables, a cubic expansion for the

deflection can be identified as

𝑢(𝜉) = 𝑎𝑖 𝜉3 + 𝑏𝑖 𝜉

2 + 𝑐𝑖 𝜉 + 𝑑𝑖 (4.2)

According to the assumption of the Euler-Bernoulli beam theory, slope is computed

from Equation (4.2) as follows

𝜕𝑢

𝜕𝜉= 3𝑎𝑖 𝜉

2 + 2𝑏𝑖 𝜉 + 𝑐𝑖 (4.3)

The shape function 𝑁1 is equal to the displacement of the beam when 𝑢1 = 1 and Ψ1 =

𝑢2 = Ψ2 = 0. Hence, Equation (4.2) can be rewritten as

𝑢(𝜉) = 𝑁1(𝜉) = 𝑎1 𝜉3 + 𝑏1 𝜉

2 + 𝑐1 𝜉 + 𝑑1 (4.4)

𝑧

𝑥

Neutral axis

Ψ = 𝝏𝒙

𝝏𝒛


85

The four boundary conditions are

𝑁1(𝜉)|𝜉=0 = 1

𝜕𝑁1(𝜉)

𝜕𝜉|𝜉=0

= 0

𝑁1(𝜉)|𝜉=𝑙 = 0

𝜕𝑁1(𝜉)

𝜕𝜉|𝜉=𝑙

= 0

The four coefficients 𝑎1, 𝑏1, 𝑐1 and 𝑑1 can be determined from the four boundaries as

follows

𝑁1(𝜉)|𝜉=0 = 1 𝑦𝑖𝑒𝑙𝑑𝑠→ 𝑑1 = 1

𝜕𝑁1(𝜉)

𝜕𝜉|𝜉=0

= 0 𝑦𝑖𝑒𝑙𝑑𝑠→ 𝑐1 = 0

𝜕𝑁1(𝜉)

𝜕𝜉|𝜉=𝑙

= 0 𝑦𝑖𝑒𝑙𝑑𝑠→ 3𝑎1𝑙 + 2𝑏1 = 0

𝑦𝑖𝑒𝑙𝑑𝑠→ 𝑎1 =

−2𝑏13𝑙

𝑁1(𝜉)|𝜉=𝑙 = 0 𝑦𝑖𝑒𝑙𝑑𝑠→

−2𝑏1 𝑙2

3+ 𝑏1 𝑙

2 = −1 𝑦𝑖𝑒𝑙𝑑𝑠→ 𝑏1 =

−3

𝑙2 , hence 𝑎1 =

2

𝑙3

Substitution of the four coefficients in Equation (4.4) gives

𝑁1(𝜉) = 2

𝑙3 𝜉3 +

−3

𝑙2 𝜉2 + 1 (4.5)

The shape function 𝑁2 is equal to the displacement of the beam when Ψ1 = 1 and u1 =

𝑢2 = Ψ2 = 0. The same procedure is followed to calculate 𝑁2. A similar conclusion can

be drawn for the functions N3(𝜉) and N4(𝜉) (i.e. for each N𝑖(𝜉) : 4 equations with 4

unknowns 𝑎𝑖, 𝑏𝑖 , 𝑐𝑖 and 𝑑𝑖). Thus, N2, N3 and N4 can be written as


86

𝑁2(𝜉) = 𝜉3

𝑙2−2

𝑙 𝜉2 + 𝜉 (4.6)

𝑁3(𝜉) = 3

𝑙2 𝜉2 −

2

𝑙3 𝜉3 (4.7)

𝑁4(𝜉) = 𝜉3

𝑙2−𝜉2

𝑙 (4.8)

The material is assumed to be linearly elastic,‎ obeying‎ Hook’s‎ law‎ with‎ Young’s‎

modulus 𝐸, which is assumed to be constant within the element. The elastic strain

energy stored in a beam element which has a flexural rigidity 𝐸𝐼 is given by

𝑈 =1

2 ∫ 𝐸 𝐼(𝜉) (

𝜕2𝑢

𝜕𝜉2)2

𝑑𝜉𝑙

0 (4.9)

Assume that the cross section does not vary within the element. Then, Equation (4.9)

becomes

𝑈 =1

2𝐸 𝐼 ∫ (

𝜕2𝑢

𝜕𝜉2)2

𝑑𝜉𝑙

0 (4.10)

The approximation to the strain energy along with the approximation to the lateral

displacement of the centre line of the beam, which is given by Equation (4.1), can be

used to obtain the following equation

𝑈 =1

2 {

𝑢1Ψ1𝑢2Ψ2

}

T

[

𝑘11 𝑘12 𝑘13 𝑘14𝑘21𝑘31𝑘41

𝑘22𝑘32𝑘42

𝑘23𝑘33𝑘43

𝑘24𝑘34𝑘44

] {

𝑢1Ψ1𝑢2Ψ2

} (4.11)

where the elements of the stiffness matrix are

𝑘𝑖𝑗 = 𝐸 𝐼 ∫ 𝑁𝑖′′(𝜉) 𝑁𝑗

′′(𝜉)𝑙

0 𝑑𝜉 (4.12)

The second derivatives of the Hermitian functions are


87

𝑁1′′(𝜉) = −

6

𝑙2(1 −

2𝜉

𝑙) , 𝑁2

′′(𝜉) = 2

𝑙(−2 +

3𝜉

𝑙)

𝑁3′′(𝜉) =

6

𝑙2(1 −

2𝜉

𝑙) , 𝑁4

′′(𝜉) = 2

𝑙(−1 +

3𝜉

𝑙)

(4.13)

As an example, 𝑘12 can be expressed as shown in Equation (4.14)

𝑘12 = 𝐸 𝐼 ∫ 𝑁1′′(𝜉) 𝑁2

′′(𝜉)𝑙

0 𝑑𝜉

= 𝐸 𝐼 ∫ −6

𝑙2(1 −

2𝜉

𝑙) 2

𝑙(−2 +

3𝜉

𝑙) 𝑑𝜉

𝑙

0

=12𝐸 𝐼

𝑙3∫ (2 − 7

𝜉

𝑙+ 6

𝜉2

𝑙2) 𝑑𝜉

𝑙

0

=12𝐸 𝐼

𝑙3 [2𝜉 −

7

2

𝜉2

𝑙+ 2

𝜉3

𝑙2]0

𝑙

= 12𝐸 𝐼

𝑙2 [2 −

7

2+ 2] =

6𝐸𝐼

𝑙2

(4.14)

The rest of the stiffness matrix terms are calculated same way as in Equation (4.14). The

integrand in Equation (4.12) is symmetric, thereby reducing the amount of computing,

because 𝑘𝑖𝑗 = 𝑘𝑗𝑖. Thus, the element stiffness matrix can be written as

𝐾 = [

𝑘11 𝑘12 𝑘13 𝑘14𝑘12𝑘13𝑘14

𝑘22𝑘23𝑘24

𝑘23𝑘33𝑘34

𝑘24𝑘34𝑘44

] = 𝐸 𝐼

𝑙3[

12 6𝑙 −12 6𝑙6𝑙−126𝑙

4𝑙2

−6𝑙2𝑙2

−6𝑙12−6𝑙

2𝑙2

−6𝑙4𝑙2

] (4.15)

The mass matrix can be calculated using the kinetic energy. Ignoring the rotational

effects, the kinetic energy of the beam may be written as

𝑇 = 1

2 ∫ 𝜌 𝐴(𝜉)�̇�2𝑑𝜉𝑙

0 (4.16)

where 𝜌 and A are the density of the material and the cross sectional area of the beam

element respectively. The approximation to the kinetic energy along with the

approximation to the displacement of the beam centreline can be used to obtain the

following equation


88

𝑇 =1

2

{

�̇�1Ψ̇1�̇�2Ψ̇2}

T

[

𝑚11 𝑚12 𝑚13 𝑚14𝑚21𝑚31𝑚41

𝑚22𝑚32𝑚42

𝑚23𝑚33𝑚43

𝑚24𝑚34𝑚44

]

{

�̇�1Ψ̇1�̇�2Ψ̇2}

(4.17)

Assuming uniform cross-sectional beam, the elements of the mass matrix can be written

as

𝑚𝑖𝑗 = 𝜌 𝐴 ∫ 𝑁𝑖(𝜉) 𝑁𝑗(𝜉)𝑙

0 𝑑𝜉 (4.18)

As an example, 𝑚32 can be expressed as shown in the following equation

𝑚32 = 𝜌 𝐴 ∫ 𝑁3(𝜉) 𝑁2(𝜉)𝑙

0 𝑑𝜉

= 𝜌 𝐴 ∫ (3

𝑙2 𝜉2 −

2

𝑙3 𝜉3) (

𝜉3

𝑙2−2

𝑙 𝜉2 + 𝜉)

𝑙

0 𝑑𝜉

= 𝜌 𝐴∫ (−2

𝑙5 𝜉6 +

7

𝑙4 𝜉5 −

8

𝑙3 𝜉4 +

3

𝑙2 𝜉3)

𝑙

0

𝑑𝜉

= 𝜌 𝐴 [−2

7𝑙5 𝜉7 +

7

6𝑙4 𝜉6 −

8

5𝑙3 𝜉5 +

3

4𝑙2 𝜉4]

0

𝑙

= 𝜌 𝐴 𝑙2 [−2

7+7

6 −8

5 +3

4] =

13

420 𝜌 𝐴 𝑙2

(4.19)

Calculating the other integrals gives the element mass matrix as shown below

𝑀 = [

𝑚11 𝑚12 𝑚13 𝑚14𝑚21𝑚31𝑚41

𝑚22𝑚32𝑚42

𝑚23𝑚33𝑚43

𝑚24𝑚34𝑚44

] = 𝜌 𝐴 𝑙

420 [

156 22𝑙 54 −13𝑙22𝑙54−13𝑙

4 𝑙2

13𝑙−3 𝑙2

13𝑙156−22𝑙

−3 𝑙2

−22𝑙4 𝑙2

] (4.20)

Figure ‎4.3 shows the coordinates defining bending in both vertical and horizontal

planes.


89

Figure ‎4.3 A rotor element with degrees of freedom (a) horizontal plane, (b) vertical plane, (c)

combination of horizontal and vertical

Assume the shaft is symmetric; the element stiffness and mass matrices of both planes

for Euler-Bernoulli beam can be obtained based on the local coordinate vector

[𝑢𝑘, 𝑣𝑘, 𝜃𝑘 , Ψ𝑘 , 𝑢𝑙 , 𝑣𝑙 , 𝜃𝑙 , Ψ𝑙]T as shown in Equation (4.21)

𝐾𝑒𝑙 =𝐸𝐼

𝑙3

[ 12 0 0 6𝑙 −12 0 0 6𝑙0 12 −6𝑙 0 0 −12 −6𝑙 00 −6𝑙 4𝑙2 0 0 6𝑙 2𝑙2 06𝑙 0 0 4𝑙2 −6𝑙 0 0 2𝑙2

−12 0 0 −6𝑙 12 0 0 −6𝑙0 −12 6𝑙 0 0 12 6𝑙 00 −6𝑙 2𝑙2 0 0 6𝑙 4𝑙2 06𝑙 0 0 2𝑙2 −6𝑙 0 0 4𝑙2 ]

and

𝑀𝑒𝑙 =𝜌𝐴𝑙

420

[ 156 0 0 22𝑙 54 0 0 −13𝑙0 156 −22𝑙 0 0 54 13𝑙 00 −22𝑙 4𝑙2 0 0 −13𝑙 −3𝑙2 022𝑙 0 0 4𝑙2 13𝑙 0 0 −3𝑙2

54 0 0 13𝑙 156 0 0 −22𝑙0 54 −13𝑙 0 0 156 22𝑙 00 13𝑙 −3𝑙2 0 0 22𝑙 4𝑙2 0

−13𝑙 0 0 −3𝑙2 −22𝑙 0 0 4𝑙2 ]

(4.21)

where the subscript 𝑒𝑙 denotes the element.

𝒛

𝒚

𝒛− 𝒙 plane

(a)

𝒛− 𝒚 plane

(b)

𝚿𝒌 𝛉𝒌

𝒖𝒌 𝒖𝒍

𝚿𝒍

𝒗𝒌 𝒗𝒍 𝛉𝒍

𝒛

𝒙

(c) O

𝒛

𝒚

𝒖𝒍

𝒙 𝒖𝒌

𝚿𝒌 𝒗𝒌

𝛉𝒌 𝒗𝒍

𝛉𝒍

𝚿𝒍


90

The gyroscopic matrix of the beam element may be developed as shown in the equation

below [21]

𝐺𝑒𝑙 =𝜌𝐼

15𝑙

[ 0 36 −3𝑙 0 0 −36 −3𝑙 0−36 0 0 −3𝑙 36 0 0 −3𝑙3𝑙 0 0 4𝑙2 −3𝑙 0 0 −𝑙2

0 3𝑙 −4𝑙2 0 0 −3𝑙 𝑙2 00 −36 3𝑙 0 0 36 3𝑙 036 0 0 3𝑙 −36 0 0 3𝑙3𝑙 0 0 −𝑙2 −3𝑙 0 0 4𝑙2

0 3𝑙 𝑙2 0 0 −3𝑙 −4𝑙2 0 ]

(4.22)

The shaft can be divided into a finite number of elements where each element has two

nodes. Each node has two translational and two rotational degrees of freedom. It is

assumed that the entire shaft elements are identical (i.e. the cross section, dimensions,

and material properties are same for each element). Thus, the equation of motion

(EOM) may be written as

[𝑀]𝑒𝑙�̈�𝑛𝑒 + ([𝐶]𝑒𝑙 + Ω[𝐺]𝑒𝑙)�̇�𝑛𝑒 + [𝐾]𝑒𝑙𝐝𝑛𝑒 = 𝐟𝑛𝑒 (4.23)

where 𝐝𝑛𝑒 and 𝐟𝑛𝑒 are the elemental nodal displacement and force vectors, respectively,

and matrices [𝑀]𝑒𝑙, [𝐶]𝑒𝑙, [𝐺]𝑒𝑙 and [𝐾]𝑒𝑙 are the elemental mass, damping, gyroscopic

and stiffness matrices, respectively.

4.2.2 Disc element

After assembling the matrices of the shaft elements, the constraints, including the disc,

are applied. The element mass matrix for the disc 𝑀𝑑 may be written as shown in

Equation (4.24) [21]

𝑀𝑑 = [

𝑚𝑑 0 0 00 𝑚𝑑 0 00 0 𝐼𝑑 00 0 0 𝐼𝑑

] (4.24)

where md and 𝐼𝑑 are the mass and diametral moment of inertia of the disc, respectively.

The element gyroscopic matrix for the disc 𝐺𝑑 can be written as [21]


91

𝐺𝑑 = [

0 0 0 00 0 0 00 0 0 𝐼𝑝0 0 −𝐼𝑝 0

] (4.25)

where 𝐼𝑝 is the polar moment of inertia. In case of disc with thickness 𝑡, inner diameter

𝐷𝑖 and outer diameter 𝐷𝑜, the polar and diametral moments of inertia can be calculated

as shown in the equation below:

𝐼𝑝 =1

32𝜌𝜋(𝐷𝑜

4 − 𝐷𝑖4)𝑡

and

𝐼𝑑 = 𝐼𝑝

2+𝑚𝑑𝑡

2

12= (

1

64𝜌𝜋(𝐷𝑜

4 − 𝐷𝑖4)𝑡) +

𝑚𝑑𝑡2

12

(4.26)

The EOM of the disc can be expressed as

[𝑀𝑑]�̈�𝑑 − Ω[𝐺𝑑]�̇�𝑑 = 𝐟𝑑 (4.27)

where the vectors 𝐝𝑑 and 𝐟𝑑 are the disc displacement and force vectors, respectively.

4.2.3 Foundation model

The effect of the foundation can be taken into consideration by adding additional terms

to the DOFs of the nodes where the bearings are located. Foundation can be modelled as

follows

[ 𝑚𝑓1𝑣 0 0 0 0

0 𝑚𝑓1ℎ 0 0 0

0 0 ⋱ 0 00 0 0 𝑚𝑓𝑡𝑣 0

0 0 0 0 𝑚𝑓𝑡ℎ]

{�̈�𝑓} +

[ 𝑐𝑓1𝑣 0 0 0 0

0 𝑐𝑓1ℎ 0 0 0

0 0 ⋱ 0 00 0 0 𝑐𝑓𝑡𝑣 0

0 0 0 0 𝑐𝑓𝑡ℎ]

{�̇�𝑓}

+

[ 𝑘𝑓1𝑣 0 0 0 0

0 𝑘𝑓1ℎ 0 0 0

0 0 ⋱ 0 00 0 0 𝑘𝑓𝑡𝑣 0

0 0 0 0 𝑘𝑓𝑡ℎ]

{𝐝𝑓} = {𝐟𝑓}

(4.28)

where the subscripts 𝑓 and 𝑡 denote the foundation and total number of bearings

respectively.


92

4.2.4 Influence of damping on the rotor dynamic model

Damping generally causes the dissipation of energy in the vibration system, which

depends on many factors such as joints, surrounding medium, frictions, material

intermolecular force and strength of the applied loads. Unlike mass and stiffness

matrices, the damping matrix cannot be constructed using material properties and

geometrical dimensions. However, the damping coefficient should be calculated in

order to understand the influence of the damping on the rotor dynamic model.

Some researchers [149, 151] discussed the theory of using the logarithmic decrement

method to calculate the damping ratio. It is defined as the natural logarithm of the ratio

of two consecutive amplitudes. This method is generally useful when dealing with

single-degree-of-freedom systems or in case of a system vibrating at a single frequency.

However, most of the systems in practise are multiple-degree-of-freedom and have

several natural frequencies; hence the logarithmic decrement method may not be only

related to a single natural frequency. In addition, the free oscillation could have

influence from various natural frequencies. Therefore, a different approach is generally

employed for the estimation of damping, namely the half-power spectrum method (also

called half-power point) [17].

The half-power spectrum method [150] is a simple and very good alternative for most

multiple-degree-of-freedom systems. Therefore, this method is employed in the current

research project. In order to understand the half power spectrum method, consider a

spring-damper-mass system that vibrates under the influence of an external force. The

equation of displacement of such structure in the frequency domain can be written as

follows

𝑥(𝑓)

𝐹(𝑓)=

1

(𝐾−𝑀𝜔2)+𝑗𝐶𝜔=

1𝐾⁄

(1−(𝑓

𝑓𝑛)2)+𝑗(2𝜁

𝑓

𝑓𝑛) (4.29)

where 𝜁 is the damping ratio and 𝑓𝑛 is the natural frequency. The amplitude of the

frequency response function (FRF) of the displacement to the force in terms of the non-

dimensional form can be written as shown in Equation (4.30).


93

𝑋(𝑓) =𝐾𝑥(𝑓)

𝐹(𝑓)=

1

√(1−(𝑓

𝑓𝑛)2)2

+(2𝜁𝑓

𝑓𝑛)2

(4.30)

where 𝑋(𝑓) is the non-dimensional amplitude of the FRF. At the natural frequency,

𝑋(𝑓) becomes:

𝑋(𝑓𝑛) =1

2𝜁 (4.31)

In terms of power:

𝑋2(𝑓𝑛) = (1

2𝜁)2

(4.32)

Hence, the half-power point will be 1

2(1

2𝜁)2

, and the non-dimensional amplitude is

written as 1

√2(1

2𝜁). Inserting the half-power point value in Equation (4.30) provides

1

2(1

2𝜁)2

=1

(1−(𝑓

𝑓𝑛)2)2

+(2𝜁𝑓

𝑓𝑛)2 (4.33)

When simplifying and solving Equation (4.33), two roots for the frequency 𝑓 are

produced:

𝑓1 = (1 − 𝜁)𝑓𝑛 and 𝑓2 = (1 + 𝜁)𝑓𝑛 (4.34)

Thus, the damping ratio can be estimated as shown in the equation below;

𝜁 =𝑓2−𝑓1

2𝑓𝑛 (4.35)

The half-power point is used in this research project to estimate the system damping

using the measured FRF plot. Figure ‎4.4 shows the FRF plot of the rotating machine

with only one balancing disc. The frequencies associated with the half-power amplitude

at the natural frequency, i.e. 17.09 Hz, are found to be:


94

𝑓1 = 16.817 Hz and 𝑓2 = 17.328 Hz

The frequencies 𝑓1 and 𝑓2 are also highlighted in Figure ‎4.4.

Figure ‎4.4 FRF plot indicating the natural frequency and half-power amplitudes (1st rig

configuration)

Therefore, the damping ratio for the system can be evaluated by applying Equation

(4.35) as follows:

𝜁 =𝑓2−𝑓1

2𝑓𝑛=17.328−16.817

2(17.09)= 0.0150 (1.5%) (4.36)

The half-power spectrum method is applied again to obtain the damping ratio for the

second configuration of the experimental rig (i.e. rotor with two balancing discs).

Figure ‎4.5 shows the FRF plot of the rotating machine with two balancing planes.

16.817 17.328

𝟓.𝟓𝟓

√𝟐= 𝟑.𝟗𝟐𝟒

𝒇𝒏:𝟏𝟕.𝟎𝟗

𝒂:𝟓.𝟓𝟓


95

Figure ‎4.5 FRF plot marking the natural frequency and half-power amplitudes (2nd

rig

configuration)

Thus, the damping ratio for the system can be evaluated by applying Equation (4.35) as

follows:

𝜁 =𝑓2−𝑓1

2𝑓𝑛=31.5116−30.406

2(31.13)= 0.0178 (1.78%) (4.37)

It is clear that the damping ratios for both rig configurations used in the current research

project are relatively small. This indicates that the influence of damping on the rotor

dynamic model is small. Hence, damping coefficient is not included in the model.

4.2.5 Modelling of the system

Assuming that rotating machine is vibrating because of unbalance-related inertia force

(𝐟𝑢𝑛𝑏), then EOM of the complete assembled rotor-bearing-foundation system may be

written as

[𝑀𝑠]{�̈�} + ([𝐶𝑠] + Ω[𝐺𝑠]){�̇�} + [𝐾𝑠]{𝐝} = {𝐟𝑢𝑛𝑏} (4.38)

where the subscript 𝑠 denotes the system, and 𝐝 can be written as

𝟏.𝟐𝟓𝟏

√𝟐= 𝟎.𝟖𝟖𝟒𝟔

30.406 31.5116

𝒇𝒏:𝟑𝟏.𝟏𝟑

𝒂:𝟏.𝟐𝟓𝟏


96

{𝐝} = [𝑢1, 𝑣1, 𝜃1, Ψ1, 𝑢2, 𝑣2, 𝜃2, Ψ2, … , 𝑢𝑛, 𝑣𝑛, 𝜃𝑛, Ψ𝑛]T (4.39)

where the subscript 𝑛 denotes the number of nodes in the system. The response and

unbalance force vectors may be expressed as

{𝐝(𝑡)} = {𝐃}𝑒𝑗𝜔𝑡 , and {𝐟𝑢𝑛𝑏(𝑡)} = {𝐅𝑢𝑛𝑏}𝑒𝑗𝜔𝑡 (4.40)

where {𝐃} and {𝐅𝑢𝑛𝑏} are the complex displacement and unbalance force vectors,

respectively. By substituting Equation (4.40) into Equation (4.38), we obtain

{𝐃} [−𝜔𝑖2[𝑀𝑠] + 𝑗𝜔𝑖[𝐶𝑠 + Ω𝐺𝑠] + [𝐾𝑠]] = {𝐅𝑢𝑛𝑏} (4.41)

where 𝜔 is the rotational speed of the machine in radians/second. The left hand side of

Equation (4.41) is known as the dynamic stiffness matrix (also called impedance

matrix) and usually denoted to as (𝐙). The size of each element matrix of the rotor-

bearing system is 8×8, and the size of the full-system matrices is 4(n) × 4(n). It is clear

from Figure ‎4.6 that the second node of each element is also the first node of the

adjacent element. Thus, the DOFs of the common nodes are affected by both elements.

In other words, element 1 affects only DOFs 1 to 8, element 2 affects only DOFs 5 to

12, element 3 affects only DOFs 9 to 17 and so on.

Assume a flexible rotor that includes 𝑛 number of nodes. Thus, after assembling the

elements matrices, and by considering the 𝑁𝑡ℎ measured frequency, Equation (4.41)

can be rewritten as:


97

(4.42)

After assembling the shaft matrices, the disc and foundation matrices should be added.

Assume the disc is positioned at the 4th

node, and the two massless bearings are located

at the first and last nodes. Hence Equation (4.42) becomes:

The rotor corresponding to Equation (4.43) is illustrated in Figure ‎4.6.

(4.43)

[ 24 0 0 0 −12 0 0 6𝑙0 24 0 0 0 −12 −6𝑙 00 0 8𝑙2 0 0 6𝑙 2𝑙2 00 0 0 8𝑙2 −6𝑙 0 0 2𝑙2

−12 0 0 −6𝑙 24 0 0 00 −12 6𝑙 0 0 24 0 00 −6𝑙 2𝑙2 0 0 0 8𝑙2 06𝑙 0 0 2𝑙2 0 0 0 8𝑙2]

[ 312 0 0 0 54 0 0 −13𝑙0 312 0 0 0 54 13𝑙 00 0 8𝑙2 0 0 −13𝑙 −3𝑙2 00 0 0 8𝑙2 13𝑙 0 0 −3𝑙2

54 0 0 13𝑙 312 0 0 00 54 −13𝑙 0 0 312 0 00 13𝑙 −3𝑙2 0 0 0 8𝑙2 0

−13𝑙 0 0 −3𝑙2 0 0 0 8𝑙2 ]

[M]en-1 [G]en-1

[K]en-1

[M]e1

[M]e2

[M]en-2

[G]e1

[G]e2

[G]e3

[G]en-2

[K]e1

[K]e2

[K]e3

[K]en-2

[M]e3

+ D (𝐸𝐼

𝑙3)

+D 𝑗𝜔𝑁 𝐅𝑢𝑛𝑏(𝑗𝜔𝑁) = −D𝜔𝑁2 (

𝜌𝐴𝑙

420)

(𝜌𝐴𝑙

420) (312) +𝑚𝑑 0 0 0

0 (𝜌𝐴𝑙

420) (312) +𝑚𝑑 0 0

0 0 (𝜌𝐴𝑙

420) (8𝑙2) + 𝐼𝑑 0

0 0 0 (𝜌𝐴𝑙

420) (8𝑙2) + 𝐼𝑑

𝐺1𝑣 + 𝐺𝑓1𝑣 0

0 𝐺1ℎ + 𝐺𝑓1ℎ

(𝜌𝐴𝑙

420) (156) +𝑀𝑓1𝑣 0

0 (𝜌𝐴𝑙

420) (156) +𝑀𝑓1ℎ

(𝐸𝐼

𝑙3) (12) + 𝐾𝑓𝑛𝑣 0

0 (𝐸𝐼

𝑙3) (12) +𝐾𝑓𝑛ℎ

(𝐸𝐼

𝑙3) (12) +𝐾𝑓1𝑣 0

0 (𝐸𝐼

𝑙3) (12) + 𝐾𝑓1ℎ

(𝜌𝐴𝑙

420) (156) +𝑀𝑓𝑛𝑣 0

0 (𝜌𝐴𝑙

420) (156) +𝑀𝑓𝑛ℎ

[M]en-1

[G]en-1

[K]en-1

[M]e1

[M]e2

[M]e3

[M]en-2

[G]e1

[G]e2

[G]e3

[G]en-2

[K]e1

[K]e2

[K]e3

[K]en-2

+ D

+ D𝑗𝜔𝑁 𝐅𝑢𝑛𝑏(𝑗𝜔𝑁) = −D𝜔𝑁2

𝐺𝑛𝑣 + 𝐺𝑓𝑛𝑣 0

0 𝐺𝑛ℎ + 𝐺𝑓𝑛ℎ


98

Figure ‎4.6 A simple schematic representation of the rotor system used to demonstrate the matrix assembly

Signal processing 4.3

The measured vibration signals during the transient operation conditions (i.e. run-up and

run-down) are very helpful for identifying faults in rotating machines. After storing the

time domain signal of the machine’s‎ transient operation in a computer, further

processing on the signal can be performed to identify and diagnose the different rotor

faults [152, 153].

4.3.1 3D waterfall plot

When acquiring rotor vibration during‎ machine’s‎ run-up/run-down, the time domain

signal is usually divided into short segments. Then, an instantaneous spectrum for each

of these segments is obtained (i.e. short-time Fourier transform (STFT)) [17]. The mean

speed of each time segment is obtained from the RPM vs. time plot, and each spectrum

is stamped by the corresponding mean speed. The RPM vs. time plot is obtained with

the help of tachometer signal measured on a reference shaft of the rotating machine. The

set of spectra produced is usually called ‘RPM‎map’.‎The‎RPM‎map can be plotted in

the form of 3D waterfall plot [154].

The 3D waterfall plot, also known as spectrum cascade plot, is a three-dimensional plot

with frequency on the 𝑥-axis, vibration amplitude on the 𝑦-axis and time/rotational

speed on the 𝑧-axis (i.e. vertical to 𝑥 and 𝑦 axes) [155]. The waterfall plot clearly

displays the alteration in spectral information with speed/time. Therefore, this kind of

plot can be a very useful tool for the identification of rotating machine malfunctions

including rotor mass unbalance. Also, one of the advantages of the waterfall plot is that

Front view Side view of 𝑭𝐛𝟐

𝒙

𝑭𝒃𝟐𝒚

𝑭𝒃𝟐𝒙

𝒛

𝒚 𝒚

𝑭𝐛𝟐 𝑭𝐛𝟏

𝒛 𝒙

E 1 E 2 E 3 E n-2 E n-1

No

de 2

No

de 3

No

de 5

No

de 4

No

de n

-1

No

de n

-2

E 4

No

de n

No

de 1


99

it can identify the machine’s critical speeds (i.e. resonant condition) that are being

excited during machine’s run-up/run-down. The critical speeds are shown as peaks at

fixed frequencies. Typical 3D waterfall plot is displayed in Figure ‎4.7.

Figure ‎4.7 Typical 3D waterfall plot

4.3.2 Order tracking

When malfunctions exist in rotating machines such as rotor unbalance, misalignment

and rotor rub, they produce vibrations at a frequency that is equal to machine’s‎

rotational speed or one of its harmonics/sub-harmonics [17, 156]. The harmonics/sub-

harmonics of the rotation frequency of the machine are known as orders [156].

Therefore, the rotation speed of the machine is referred to as order 1 (also known as

1×), twice the rotational speed is referred to as order 2 (i.e. 2×) and so forth. To

enhance understanding, consider a periodic function with five terms

𝑥(𝑡) = 2 sin(0.5 ∗ 2𝜋𝑓𝑡) + 2 sin(2𝜋𝑓𝑡) + 2 sin(2 ∗ 2𝜋𝑓𝑡) + 2 sin(3 ∗ 2𝜋𝑓𝑡) + 2 sin(4 ∗ 2𝜋𝑓𝑡) (4.44)

for 0 ≤ 𝑡 ≤ 0.1sec and 𝑓 = 60 Hz.

The time waveform corresponding to the signal in Equation (4.44) and its vibration

spectrum (i.e. vibration amplitude versus frequency) are shown in Figure ‎4.8(a) and

Figure ‎4.8(b) respectively. In this particular example, the vibration amplitude at 60 Hz

is referred to as order 1 (i.e. 1×), 30 Hz as order 0.5 (i.e. 0.5×), 120 Hz as order 2 (i.e.

2×), and so on as shown in Figure ‎4.8 [53].

Mag

(g

)

Frequency (Hz)


100

Figure ‎4.8 (a) Time waveform, (b) vibration spectrum of the time waveform

When the vibration related to the rotational speed occurs at the same frequency as the

machines critical speed, very high and dangerous levels of vibrations take place [156].

The harmonic as well as sub-harmonic orders of the rotation frequency of the machine

can be analysed using order tracking technique. The technique is called order tracking

as the rotation frequency and its harmonics are being tracked and used for further

analysis [157]. The order track diagram is a 2-dimensional plot that has the time (or

machine speed in case of transient operation) in the 𝑥-axis and the frequency component

of a particular order of the rotation frequency in the 𝑦-axis [158].

4.3.3 Bode plot

It has been found that each rotor fault is related to the rotation frequency or one of its

orders. For instance, the rotor mass unbalance is associated with the first order (1×) of

the rotating machine, and misalignment is mainly related to the second order (2×) [53,

159]. Furthermore, assume a gear with 22 teeth; the 22nd

order will show the gear mesh

frequency. When dealing with rotor mass unbalance, an extra plot is required (i.e.

machine speed in the 𝑥-axis and the phase angle corresponding to the frequency

component of a certain order of the rotation frequency in the 𝑦-axis). The plot of the

frequency component and its phase with machine speed is known as Bode plot [160]. A

(a)

(b) 𝟏 ×

𝟎.𝟓 ×

𝟒 ×

𝟑 × 𝟐 ×


101

typical Bode plots of the 1× and 2× frequency components for a machine that coasts-up

from 300 to 2700 RPM are shown in Figures 4.9 and 4.10 respectively.

Figure ‎4.9 Typical Bode plot of the 1× shaft displacement for machine coast-up

Figure ‎4.10 Typical Bode plot of the 2× shaft displacement for machine coast-up


102

Summary 4.4

This chapter started by explaining the steps of modelling a simple rotating machine. It

began with explaining the modelling of each part of the machine (i.e. shaft, disc and

foundation). Then, the different parts were assembled to form the complete rotating

machine. After that, the influence of the damping on the rotor dynamic model is

presented. The second part of this chapter briefly explained the signal processing used

in the current research project.

‎CHAPTER 5 SENSITIVITY ANALYSIS OF IN-SITU ROTOR BALANCING

103

CHAPTER 5

SENSITIVITY ANALYSIS OF IN-SITU

ROTOR BALANCING


104

This chapter is a reformatted version of the following papers:

1. Title: Sensitivity analysis of in-situ rotor balancing

Authors: Sami M. Ibn Shamsah, Jyoti K. Sinha and Parthasarathi

Mandal

Status: Published in IMechE 11th

VIRM Conference Proceedings

2. Title: Precaution during the field balancing of rotating machines


Mandal

Status: Published in Journal of Maintenance Engineering

Abstract: Rotor unbalance is one of the common faults in any rotating machine which

occurs regularly during their operation. This may cause an unacceptable level of

vibration leading to failure of the machine. Hence the regular field balancing is

generally carried out to keep the machine vibration within an acceptable limit. The

influence coefficient (IC) method is an acceptable field balancing method for rotating

machines. However, the current industrial practice is to apply the IC method at a single

machine rotating speed for the unbalance estimation. In the current study, the IC method

is used to estimate unbalance from vibration data acquired from a small experimental

rig at a single rotor speed as well as a wide range of speeds during machine run-up. It

was observed that the inclusion of more speeds during the run-up significantly improves

the unbalance estimation when compared to the estimation at a single speed. The current

paper presents the experimental rig, unbalance experiments conducted and the

sensitivity analysis of the unbalance estimation using speed ranging from a single speed

to multiple speeds.

Keywords: Rotating machines, rotor unbalance, influence coefficient method, order

tracking










105

Introduction 5.1

The achievement of desired reliability targets of industrial rotating machines is often

hampered by the existence of various rotor-related faults including unbalance,

misalignment, cracks, shaft rub, etc. Faults in rotating machines are unpreventable due

to a number of reasons such as manufacturing/installation imperfections as well as

wears and tears owing to day-to-day operations. These rotor faults are major sources of

unwanted high vibrations in rotating machines such as compressors, pumps and gas

turbines. Since this class of machines plays very vital roles in the achievement of most

industrial objectives, it is therefore imperative to continuously sort for approaches that

simplify as well as enhance the detection and diagnosis of these faults at their early

stages. Although there are various commonly encountered rotor-related faults,

unbalance is amongst the most prevalent in rotating machines.

Over the years, vibration-based condition monitoring (VCM) techniques are

successfully used to detect and diagnose rotor unbalance [29, 51, 88, 99]. Amongst all

vibration based rotor balancing techniques, the most predominant approaches are the

modal balancing (MB) and the influence coefficient (IC) balancing methods [53]. The

MB method was firstly applied in the 1960s by Bishop et al. [31]. In order to apply this

balancing method efficiently, a highly skilled person with sound knowledge of rotor-

dynamics is required [43]. On the contrast, the IC method does not need any prior

knowledge of the dynamics of the rotor [17]. It requires only the vibration responses of

the rotor at different trial masses. It assumes that the rotor system is linear and the

influence of the individual unbalances can be superposed to give the influence of a set

of unbalances [29]. Therefore, due to its simplicity, the IC method is accepted as the

rotor balancing method in industries and plants around the world.

Comprehensive reviews of literature pertaining to the IC balancing method were

conducted by Darlow [29], Foiles et al. [88], Parkinson [101] and Zhou and Shi [19].

Yu [70] and Lee et al. [71] used the IC method to estimate the rotor unbalance at a

constant rotor speed. Zhou et al. [72] were successfully able to balance a rotor-bearing

system using the IC method during speed varying period. Dyer and Ni [73] have

extended the IC technique to the active control and on-line estimation. In their study,

they have successfully implemented an adaptive control scheme that combines flexible


106

rotor balancing method and the on-line estimation of the IC using an active balancing

system. Recently, Xu and Fan [74] have used the IC method to balance a rigid rotor

using two balancing planes.

The current study aims to experimentally investigate the effect of rotor unbalance

estimation using the IC method to understand the influence of rotor speed ranging from

a single speed to multiple speeds on the unbalance estimation accuracy. The paper

presents the experimental rig, unbalance experiments conducted and the results of the

sensitivity analysis of the unbalance estimation using rotor speed ranging from a single

speed to multiple speeds.


Figure ‎5.1 shows the experimental apparatus used for the experiments. The rig consists

of a solid shaft of length 1000mm and diameter 20mm. It is made of steel and supported

on relatively flexible foundation through two greased lubricated ball bearings. The shaft

also carries a steel balancing disc of 130mm diameter and 20mm thickness at midspan

between the two bearings. The disc contains staggered threaded holes (M5) in two

different pitch diameters, which are 70 and 120mm. The angle between two adjacent

holes for a particular pitch diameter is 30 degree. A flexible coupling made of

Aluminium connects the shaft to a 0.75kW, 3-Phase, 3000 RPM motor. Each of the two

ball bearings is mounted on flexible steel horizontal beam (530mm × 25mm × 3mm).

Each horizontal beam is screwed to two rectangular steel blocks (107mm × 25mm ×

25mm) that are screwed to a thick base plate (580mm × 150mm × 15mm) as shown in

Figure ‎5.1. Both base plates of the two bearing pedestals are mounted on a massive steel

platform. The bearing near the motor is denoted as B1 and the other bearing is denoted

as B2.

Experiments 5.3

The rotor was run-up in a speed range of 0 to 3000 RPM. A modal test was carried out

and two critical speeds of the machine (18 and 39 Hz) were found within the run-up

speed range. Two accelerometers with a sensitivity of 100mV/g were installed on each

of the two bearing housings. The accelerometers were mounted in the vertical and

horizontal directions at each bearing housing. The vibration responses were measured in


107

both directions during eight machine run-ups (i.e. one with residual unbalance and

seven with added unbalance at a radius of 6cm). Since the rotor vibrations due to mass

unbalance are synchronous to the rotational speed, the measured vibration responses of

each run were order tracked to get the synchronous vibration components (both

amplitudes and phases) for all measurement locations. The order tracked responses (1×

responses) in the speed range from 420 to 2820 RPM (i.e. 7 to 47 Hz) with a spacing of

60 RPM (1 Hz) were then used for the unbalance estimations. The added unbalances

used for the different runs are listed in Table ‎5.1. Figure ‎5.2 demonstrates the unbalance

and phase angle with respect to the laser tacho sensor. Figure ‎5.3 shows a typically

measured vibration responses at the bearings B1 and B2 in the vertical direction for

different runs.

Figure ‎5.1 Photographic representation of the experimental rig

Accelerometer

Tachometer

𝒛

𝒚

𝒙

𝒙

𝒚

𝒚 𝒙

Shaft

Disc

Bearing

Motor

Flexible coupling

𝒛

𝒚

Bearing B2

Bearing B1

𝟑𝟎°


108

Table ‎5.1 Unbalance and phase of 8 runs

Run no. Added unbalance (𝐠𝐜𝐦@𝛉°)

𝑟𝑢𝑛0 Residual unbalance 𝑒0

𝑟𝑢𝑛1 3g × 6cm @ 60° = 18gcm @ 60° 𝑒0 + 𝑒1

𝑟𝑢𝑛2 3g × 6cm @ 120° = 18gcm @ 120° 𝑒0 + 𝑒2

𝑟𝑢𝑛3 5g × 6cm @ 30° = 30gcm @ 30° 𝑒0 + 𝑒3

𝑟𝑢𝑛4 5g × 6cm @ 60° = 30gcm @ 60° 𝑒0 + 𝑒4

𝑟𝑢𝑛5 7g × 6cm @ 30° = 42gcm @ 30° 𝑒0 + 𝑒5

𝑟𝑢𝑛6 7g × 6cm @ 60° = 42gcm @ 60° 𝑒0 + 𝑒6

𝑟𝑢𝑛7 7g × 6cm @ 120° = 42gcm @ 120° 𝑒0 + 𝑒7

Figure ‎5.2 Demonstration of the added unbalance (mass and phase angle)

Figure ‎5.3 Typical measured 1× displacement responses in vertical direction for the experimental

runs 1, 4 and 6 at (a) B1 and (b) B2

𝒙

𝒚

Tachometer Laser beam

Reflective tape

Steel shaft

Disc

Added unbalance

Phase angle

(a)

(b)


109

Unbalance estimation 5.4

The displacement vector 𝐝𝑝 of any machine at the bearing pedestals in the vertical and

horizontal directions during machine 𝑝th run-up can be written as

𝐝𝑝 = [𝐝𝑝(𝑓1) 𝐝𝑝(𝑓2) … 𝐝𝑝(𝑓𝑘) … 𝐝𝑝(𝑓𝑁)]T (5.1)

where 𝑓1, 𝑓2,‎…,‎𝑓𝑁 are the rotor run-up speeds in Hz.

At a single rotor speed 𝑓𝑘, the displacement vector 𝐝𝑝(𝑓𝑘) is written as

𝐝𝑝(𝑓𝑘) = [y1,𝑝(𝑓𝑘) x1,𝑝(𝑓𝑘) y2,𝑝(𝑓𝑘) x2,𝑝(𝑓𝑘)]T (5.2)

The variables y and x are the measured vertical and horizontal displacements at bearing

housing where the subscripts 1 and 2 represent bearings B1 and B2 respectively.

The sensitivity matrix for the IC method [17] can be written as

𝐒𝑝0 = 1

𝑒𝑝

[ 𝐝𝑝(𝑓1) − 𝐝0(𝑓1)

𝐝𝑝(𝑓2) − 𝐝0(𝑓2)

⋮𝐝𝑝(𝑓𝑁) − 𝐝0(𝑓𝑁)]

(5.3)

where 𝑒𝑝 is the added unbalance at run number 𝑝.

The residual rotor unbalance 𝑒0 can be calculated using the following equation:

𝐒𝑝0𝑒0 = 𝐝0 (5.4)

Equation (5.4) can further be modified to calculate the added unbalance directly for

each run;

𝐒𝑝0𝑒𝑞 = (𝐝𝑞 − 𝐝0) (5.5)


110

where the subscript 𝑞 in Equation (5.5) represents the estimation of the unbalance for

the 𝑞th run. Table ‎5.2 lists the different scenarios (cases) used for the estimation of the

sensitivity matrices and the added unbalances.

Table ‎5.2 Sensitivity using different runs

Case no. Sensitivity Unbalance

Case I 𝑟𝑢𝑛3 − 𝑟𝑢𝑛0 𝑟𝑢𝑛4 − 𝑟𝑢𝑛0 𝑒4

Case II 𝑟𝑢𝑛4 − 𝑟𝑢𝑛0 𝑟𝑢𝑛1 − 𝑟𝑢𝑛0 𝑒1

Case III 𝑟𝑢𝑛5 − 𝑟𝑢𝑛0 𝑟𝑢𝑛3 − 𝑟𝑢𝑛0 𝑒3

Case IV 𝑟𝑢𝑛5 − 𝑟𝑢𝑛0 𝑟𝑢𝑛4 − 𝑟𝑢𝑛0 𝑒4

Case V 𝑟𝑢𝑛6 − 𝑟𝑢𝑛0 𝑟𝑢𝑛4 − 𝑟𝑢𝑛0 𝑒4

Case VI 𝑟𝑢𝑛7 − 𝑟𝑢𝑛0 𝑟𝑢𝑛2 − 𝑟𝑢𝑛0 𝑒2

Results and observations 5.5

Initially, the unbalance was estimated for all cases in Table ‎5.2 at a single rotor speed of

7 Hz (420 RPM), and then more speeds from the machine run-up were added gradually

to observe the influence on the unbalance estimation. The inclusion of multiple speeds

in the unbalance estimation is represented as Speed range and is defined as

Speed range = 𝑓1 + 𝑛(𝑑𝑓) = 7 Hz + 𝑛(1) (5.6)

where 𝑛 = 0, 1, … , 40.

The estimated unbalances (both amplitudes and phases) at different speed ranges for all

cases listed in Table ‎5.2 are shown in Figure ‎5.4. Actual added unbalance for each case

as well as the critical speeds of the rig are also shown in Figure ‎5.4 for the easy

comparison.


111

(a)

(b)


112

(c)

(d)


113

Figure ‎5.4 The actual and estimated unbalances and phase angles for (a) case I, (b) case II, (c) case

III, (d) case IV, (e) case V and (f) case VI

(e)

(f)


114

It can be observed from Figure ‎5.4 that using a single rotor speed to compute the rotor

unbalance might not provide the best unbalance estimation, hence leads to poor

balancing. In addition, it can be noticed that the estimated rotor unbalances using the

critical speeds and speeds around them are showing larger error in the estimation.

However, the inclusion of more run-up speeds in the unbalance estimation generally

provides much better and stable unbalance estimation in terms of both unbalance

amplitude and phase. The differences between the estimated and actual rotor unbalance

and phase is relatively small with higher speed range for all cases.

Further exercise for the unbalance estimation is also carried out at different single

speeds. It is observed that the estimated amplitudes and phases are scattered around the

actual unbalance when different single speeds are used to estimate rotor unbalance. It is

typically shown in Figure ‎5.5 for Case-II. This study also suggests that the utilisation of

a single machine speed may or may not estimate accurate rotor unbalance.

Figure ‎5.5 A typical comparison between the actual and estimated rotor unbalance at different

single speeds for Case II


115

Conclusion 5.6

The current study investigates the application of field rotor balancing using the IC

method at a single speed as well as during machine run-up. Experiments were

conducted on a small laboratory rig with one balancing plane. It was shown that the

inclusion of more speeds during coast-up considerably improves the unbalance

estimation when compared to the estimation at a single speed. Therefore, it is better to

include a range of run-up or run-down speeds of the machine for effective machine

balancing.

‎CHAPTER 6 MULTI-PLANES ROTOR UNBALANCE ESTIMATION…

116

CHAPTER 6

MULTI-PLANES ROTOR UNBALANCE

ESTIMATION USING INFLUENCE

COEFFICIENT METHOD


117

This chapter is a reformatted version of the following paper:

Title: Multi-planes rotor unbalance estimation using influence coefficient

method

Authors: Sami M. Ibn Shamsah, Jyoti K. Sinha and Parthasarathi Mandal

Status: Published in the Twelfth International Conference on Vibration

Engineering and Technology of Machinery (VETOMAC XII)

Abstract: The rotor mass unbalance is probably the most common rotor fault in any

rotating machine. If the rotor mass unbalance exceeds the allowable limit, it will cause

an excessive vibration which may lead to machine failure. Therefore, the regular in-situ

rotor balancing is often required to keep machine vibration within an acceptable level.

Current practice in industries is to use a single machine speed for balancing the rotor by

the Influence Coefficient (IC) method. This paper investigates the effect on the

unbalance estimation by the application of the IC method using measured machine

vibrations at a single speed as well as multiple speeds simultaneously during machine’s

transient operation (run-up or run-down). This study is carried out on an experimental

rotating rig with two balancing planes. The current paper presents the laboratory rig,

unbalance experiments carried out and the sensitivity analysis of the unbalance

estimation using measured machine response at speed ranging from a single speed to

multiple speeds.


tracking





118

Introduction 6.1

Rotating machines form the bedrock of power plants as they are used in most of their

operations. Therefore, the reliability of this category of machines is of vital importance

for power plants. These machines are susceptible to a broad variety of faults owing to

many reasons such as machining errors and manufacturing shortcomings. The rotor is

one‎ of‎ the‎ rotating‎ machine’s‎ main‎ parts‎ as‎ well‎ as‎ the bearings and supporting

structure. The rotor mass unbalance, which occurs when the mass is unequally

distributed around the axis of rotation, is known as the most common malfunction that

causes the entire machine to vibrate [53]. If the amount of mass unbalance exceeds the

allowable limit, it might lead to detrimental effects on the entire machine. As a

consequence, machine downtime and unscheduled maintenance actions are required.

This, in turn, influences the operating cost negatively. Therefore, the regular balancing

of rotating machines is necessary to ensure safe machine operation and long service life.

Rotating machines such as steam turbine generators, pumps, motors, and so on are

balanced using either the influence coefficient (IC) method or the modal balancing

[161]. The former of these methods is widely used as it does not require any prior

knowledge of the underlying physics of the machine [17]. It requires machine runs with

trial masses that are equal to the number of rotor balancing planes in addition to

machine run with residual unbalance to obtain the ICs between the rotor and bearings

[17]. The IC balancing approach was firstly proposed in the 1930s [54].

Lee et al. [71] and Darlow [29] presented review papers concerning the IC balancing

technique. Sinha [17] explained the application of the IC balancing method on rotating

machines with single/multiple balancing planes and provided some examples. Den

Hartog [58] briefly discussed the IC technique for two balancing planes. Hopkirk [55]

formulated the two planes, two sensors, single speed balance procedure using ICs in the

manner as most modern procedures. Goodman [56] presented a least square approach to

extend the technique to multi-planes balancing using data acquired at different machine

speeds and measurement locations. This method was refined and verified in 1972 by

Tessarzik et al. [57].

In industry, the IC balancing method is generally applied using machine response

acquired at a single machine speed to balance the rotating machine. Therefore, the main


119

thrust of this paper is to study the influence of applying the IC balancing method on a

rotating machine with multiple planes using vibration measurements acquired at single

speeds as well as multiple speeds in a single band.


A laboratory scaled experimental rotating rig used in the study is shown in Figure ‎6.1.

The rig consists of a solid mild steel shaft of dimensions 1000mm (length) × 20mm

(diameter) which is supported on flexible supporting foundations through two ball

bearings. Two identical steel balancing discs of dimensions 130mm (outer diameter) ×

20mm (thickness) are attached to the shaft at distances of 240mm and 665mm from the

bearing closer to the motor. The disc that is closer to the motor is denoted as d1 and the

other disc as d2.

The rotor of the rig is connected to a motor shaft via a flexible coupling. The supporting

foundation of each bearing consists of a horizontal steel beam (530mm × 25mm ×

8mm) which is secured on the top of two rectangular steel blocks (107mm × 25mm ×

25mm) as shown in Figure ‎6.1. The complete rig is mounted on a massive steel

platform. The bearing near the motor is symbolised as B1 and the other bearing as B2.


120

Figure ‎6.1 Mechanical layout of the test rig

Machine runs and data acquisition 6.3

The machine was run-up slowly from 300 to 3000 RPM (i.e. 5 to 50 Hz). A total of 6

run-ups were carried out where each run lasted for 135s. The machine speed was

increased linearly with time. The vibration responses on both bearing pedestals in the

vertical and horizontal directions were recorded using four accelerometers (sensitivity

0.1V/g) for each run-up. The measured vibration responses were then order tracked to

𝒙, horizontal

𝒚, vertical

𝒛

𝒚

𝒙

Bearing B2

Bearing B1

Disc d2

Disc d1


121

obtain the 1× vibration component responses at bearings B1 and B2 in both vertical and

horizontal directions. Two critical speeds (i.e. around 22 and 31 Hz) were observed

within the run-up speed range (see Figure ‎6.2). The measured vibration responses in the

speed range from 15 to 40 Hz (i.e. 900 to 2400 RPM) with an interval of 0.5 Hz (30

RPM) were then used in the unbalance estimation. The different scenarios used for the

unbalance estimation are presented in Table ‎6.1.

Figure ‎6.2 Typical measured 1× displacement responses in horizontal direction for the

experimental runs 3 and 5 at bearings (a) B1 and (b) B2

1st critical speed

2nd

critical speed

1st critical speed

2nd

critical speed

(a)

(b)

122

Table ‎6.1 Mass unbalances and phase angles of 6 machine runs

Run no.

Added unbalance (𝐠𝐜𝐦@𝛉°)

Disc d1 Disc d2

𝑟𝑢𝑛0 Residual unbalance 𝑒01 Residual unbalance 𝑒02

𝑟𝑢𝑛1 3g × 6cm @ 30° = 18gcm @ 30° 𝑒01 + 𝑒11 Residual unbalance 𝑒02

𝑟𝑢𝑛2 Residual unbalance 𝑒01 3g × 6cm @ 30° = 18gcm @ 30° 𝑒02 + 𝑒22

𝑟𝑢𝑛3 3g × 6cm @ 90° = 18gcm @ 90° 𝑒01 + 𝑒31 5g × 6cm @ 30° = 30gcm @ 30° 𝑒02 + 𝑒32



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123

Application of IC method 6.4

The application of the IC balancing method on rotating machine with multiple

correction planes requires machine runs equal to the number of planes plus one.

Therefore, in the case of two correction planes, at least three machine transient

operations (i.e. run-ups/downs) are required to perform balancing using the IC method.

The displacement vector 𝐃𝑚 of any machine at the bearing pedestals in the vertical and

horizontal directions during 𝑚𝑡ℎ machine’s run-up can be written as

𝐃𝑚 = [𝐃𝑚(𝑓1) 𝐃𝑚(𝑓2) … 𝐃𝑚(𝑓𝑘) … 𝐃𝑚(𝑓𝑁)]T (6.1)

where 𝑓1, 𝑓2,‎…,‎𝑓𝑁 are the rotor run-up speeds in Hz with 𝑓𝑁 as the maximum speed.

At a single rotor speed 𝑓𝑘, the displacement vector 𝐃𝑚(𝑓𝑘) is written as per the

following equation:

𝐃𝑚(𝑓𝑘) = [y1,𝑚(𝑓𝑘) x1,𝑚(𝑓𝑘) y2,𝑚(𝑓𝑘) x2,𝑚(𝑓𝑘)]T (6.2)

where y1,𝑚 and x1,𝑚 are displacements in the vertical and horizontal directions,

respectively, at bearing B1 for run number 𝑚, similarly the displacements y2,𝑚 and x2,𝑚

at bearing B2. The displacement vectors 𝐃0, 𝐃1 and 𝐃2 are used to construct the

sensitivity matrix 𝐒 as shown in Equation (6.3).

𝐒 =

[ 𝐃1(𝑓1)−𝐃0(𝑓1)

𝑒11𝐃1(𝑓2)−𝐃0(𝑓2)

𝑒11

⋮𝐃1(𝑓𝑁)−𝐃0(𝑓𝑁)

𝑒11

𝐃2(𝑓1)−𝐃0(𝑓1)

𝑒22𝐃2(𝑓2)−𝐃0(𝑓2)

𝑒22

⋮𝐃2(𝑓𝑁)−𝐃0(𝑓𝑁)

𝑒22 ]

(6.3)

where 𝑒11 is the added unbalance to disc d1 at run1 and 𝑒22 is the added unbalance to

disc d2 at run2.


124

The sensitivity matrix together with the vibration measurements acquired at run0 are

used to obtain the residual rotor mass unbalances for discs d1 and d2 (i.e. 𝑒01 and 𝑒02)

as shown in the following equation

[𝐒] [𝑒01𝑒02] = [𝐃0] (6.4)

The added mass unbalances to both discs can be calculated for each run-up/down by

modifying Equation (6.4) as;

[𝐒] [𝑒𝑞1𝑒𝑞2] = [𝐃𝑞 − 𝐃0] (6.5)

where the subscripts 𝑞1 and 𝑞2 in Equation (6.5) represent the estimation of the mass

unbalances added to discs d1 and d2 for the 𝑞𝑡ℎ run, respectively.

Unbalance estimation and discussion 6.5

The IC method was used to estimate the rotor unbalance for all cases in Table ‎6.1. First,

the unbalance estimation was carried out using the measured vibrations at the machine

speed 15 Hz (900 RPM). Then, more speeds from the machine run-up were added

further to observe the influence on the rotor unbalance estimation. The inclusion of

multiple machine speeds in the unbalance estimation is represented as Speed range and

is defined as

Speed range = 𝑓1 + 𝑛(𝑑𝑓) = 15 Hz + 𝑛(0.5) (6.6)

where 𝑛 = 0, 1, … , 50.

The estimated amplitudes and phase angles of the added mass unbalances of discs d1

and d2 for runs 3, 4 and 5 at different speed ranges are presented in Figures 6.3, 6.4 and

6.5. For the sake of easy comparison, the actual added unbalances of both discs for each

case as well as the critical speeds are also shown in the figures.


125

Figure ‎6.3 Actual and estimated unbalances (amplitude and phase) of (𝒓𝒖𝒏𝟑 − 𝒓𝒖𝒏𝟎) at (a) disc

d1 and (b) disc d2; * : estimated unbalance, : actual added unbalance

(b)

(a)


126

Figure ‎6.4 Actual and estimated unbalances (amplitude and phase) of (𝒓𝒖𝒏𝟒 − 𝒓𝒖𝒏𝟎) at (a) disc


(a)

(b)


127

Figure ‎6.5 Actual and estimated unbalances (amplitude and phase) of (𝒓𝒖𝒏𝟓 − 𝒓𝒖𝒏𝟎) at (a) disc


It can be clearly seen from Figures 6.3, 6.4 and 6.5 that when using machine vibration

responses acquired at a range of multiple speeds in a single band, the unbalance

estimation is significantly enhanced. Moreover, it can be observed that as the speed

range increases, the error between the estimated and actual unbalances (both amplitude

and phase) reduces significantly. It can also be noticed that the accuracy of the

(a)

(b)


128

unbalance estimation using vibration measurements at both critical speeds is small

comparing to higher speed ranges.

Concluding remarks 6.6

The application of the IC balancing method using vibration data acquired from a small

test rig with two balancing planes at single and multiple speeds in a single band has

been studied in the current paper. The results indicate that the unbalance estimation at

both planes using measured vibration at a single machine speed may not be good

enough. However, the use of measured machine responses at more speeds from the

machine run-up/down increases the accuracy of the unbalance estimation. Thus, it is

better to use the full speed range of the transient operation of the rotating machine for

accurate and precise machine balancing.

‎CHAPTER 7 SENSITIVITY ANALYSIS OF THE INFLUENCE COEFFICIENT…

129

CHAPTER 7

SENSITIVITY ANALYSIS OF THE

INFLUENCE COEFFICIENT

BALANCING METHOD FOR MULTIPLE

PLANES ROTOR BALANCING WITH

REDUCED NUMBER OF SENSORS


130

This chapter is reformatted version of the following papers:

1. Title: Reliable machine balancing for efficient maintenance


Mandal

Status: Submitted to Journal of Quality in Maintenance Engineering

2. Title: Rotor unbalance estimation with reduced number of sensors

Authors: Sami M. Ibn Shamsah and Jyoti K. Sinha

Status: Published in Machines Journal

Abstract: Rotor mass unbalance inevitably presents in almost all rotating machines. If

the vibration amplitude level due to rotor mass unbalance exceeds a set allowable limit,

it may lead to catastrophic machine failure. This is generally not acceptable to any

industry as it results in machine downtime and unscheduled costly maintenance actions.

Hence, the regular field balancing is generally carried out to keep the machine vibration

within an acceptable level. The influence coefficient (IC) method is an acceptable on-

site vibration based balancing approach for rotating machines. However, the current

industrial applications of the IC method generally use vibration measurements acquired

at single machine speed using two vibration sensors at a bearing pedestal for the

unbalance estimation. In the present study, the same concept of the IC balancing method

is applied again but with using vibration data acquired from small test rig at a single

machine speed as well as multiple speeds in a single band during machine run-up. Four

scenarios are presented in the current paper. In the first scenario, the IC balancing

method is applied on a test rig with one balancing disc using vibration measurements

acquired in vertical and horizontal directions at single machine speeds. In the second

scenario, the IC balancing method is applied on a test rig with one balancing disc using

measured vibration responses in vertical and horizontal directions at multiple machine

speeds. Then, the IC balancing method is applied on a test rig with two balancing discs

using vibration measurements acquired at two directions as well as one direction at

multiple machine speeds. It is observed that the inclusion of a range of run-up/down





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http://www.mdpi.com/2075-1702/4/4/19


131

machine speeds in a single band significantly improves the unbalance estimation.

Furthermore, it is noticed that the application of the IC method using a single vibration

sensor per bearing location provides as accurate results as when applying the method

using two sensors per bearing location.


tracking

Introduction 7.1

Rotating machines are considered as the cornerstone of most energy/power plants,

petrochemical factories, etc., as they are used in most of their operations. For that

reason, the reliability and maintainability of this sort of machines are of paramount

importance for these plants. Typical rotating machines are made up of several integrated

components including rotating system, bearings, supporting structure, electric motor,

etc. There is a long list of malfunctions that often hinder the industrial rotating machines

from achieving their anticipated reliability targets [17, 78, 82]. Since this class of

machines plays a crucial role in the achievement of most industrial objectives, it is

therefore vitally important to continuously seek after new methods that simplify as well

as enhance the identification and diagnosis of rotating machine malfunctions at their

early stages [17, 53].

Rotor mass unbalance is one of the most common malfunctions in rotating machines

which occurs repeatedly throughout their operations [10]. This stimulus develops when

the mass is asymmetrically distributed around the axis of rotation [162]. If the machine

vibration due to rotor mass unbalance exceeds the maximum permissible levels, it may

lead to machine failure. As a consequence, machine downtime and unscheduled

maintenance actions will be required, which in turn influence the operating cost

negatively. Moreover, high levels of rotor mass unbalance may result in a significant

cutback of the machine fatigue life [12]. Therefore, the regular field balancing is

essential to keep the machine vibration within a permissible level. Owing to the fact that

machine vibration signals carry plentiful and valuable information concerning dynamics

and health of the machine parts, vibration-based rotor fault detection methods have been

immensely used for detecting rotor related faults, including rotor mass unbalance [53].


132

Many vibration based balancing techniques have been proposed in the literature [14-17,

31, 43, 48, 51, 56, 70, 71]. Generally, rotating machines balancing approaches can be

divided into two main groups, namely the influence coefficient (IC) balancing and the

modal balancing [54, 55, 72, 161, 163, 164]. In order to apply the latter method

efficiently, a highly skilled engineer with thorough knowledge of rotor-dynamics is

needed [43]. Darlow [50] presented some shortcomings of the modal balancing

approach. The IC balancing method, however, does not require any prior knowledge of

the underlying dynamics of the machine [17, 163, 164]. It only requires machine runs

with trial weights [17]. Therefore, due to its simplicity and high efficiency, the IC

method has become the traditional in-situ balancing approach in most industries around

the world.

Literature review papers pertaining to the IC balancing method were presented by

Darlow [29], Fang et al. [165], Zhou and Shi [19], Dyer and Ni [73] and Parkinson

[101]. Lund and Tonnesen [16] and Tessarzik et al. [57] have used the IC approach for

the rotor balancing. In his book, Sinha [17] has explained comprehensively the

application of the IC balancing approach for rotating machine with a single as well as

multiple balancing planes. He also presented graphical as well as mathematical

approaches, and provided some examples to enhance understanding. Yu [70] as well as

Lee et al. [71] used the IC method to estimate the rotor unbalance at a constant rotor

speed. Ibn Shamsah et al. [163, 164] have used this method and highlighted that the use

of more speeds simultaneously during the machine’s transient operation (i.e. run-up or

run-down) gives much better rotor unbalance estimates than at a single machine speed.

The current industrial application of the IC balancing method is to use vibration

response acquired at a single machine speed in multiple mutually perpendicular

directions at each bearing pedestal for the estimation of the rotor mass unbalance. It is

believed that the utilisation of the machine vibration measurements at multiple

orthogonal directions provides better machine dynamics, and hence the estimated

unbalance is likely to be more accurate. Therefore, the main thrust of this paper is to

investigate the application of the IC balancing method using only one sensor at a

bearing pedestal. The second objective of the current paper is to study the influence of

applying the IC balancing method using vibration measurements acquired at single

speeds and multiple speeds in a single band.


133

Theory of IC balancing method 7.2

Assume that the displacement vectors in the vertical and horizontal directions at all

bearing pedestals in a machine are 𝐝0, 𝐝1, 𝐝2,‎…,‎𝐝𝑛, where the subscripts denote the

1𝑠𝑡, 2𝑛𝑑, 3𝑟𝑑,‎…,‎(𝑛 + 1)𝑡ℎ machine run-ups respectively. The displacement vectors for

the different machine run-up speeds can be written as

𝐝0 = [𝐝0(𝑓1) 𝐝0(𝑓2) … 𝐝0(𝑓𝑘) … 𝐝0(𝑓𝑁)]T

𝐝1 = [𝐝1(𝑓1) 𝐝1(𝑓2) … 𝐝1(𝑓𝑘) … 𝐝1(𝑓𝑁)]T

𝐝2 = [𝐝2(𝑓1) 𝐝2(𝑓2) … 𝐝2(𝑓𝑘) … 𝐝2(𝑓𝑁)]T

⋮ 𝐝𝑛 = [𝐝𝑛(𝑓1) 𝐝𝑛(𝑓2) … 𝐝𝑛(𝑓𝑘) … 𝐝𝑛(𝑓𝑁)]

T

(7.1)

where 𝑓1, 𝑓2,‎…,‎𝑓𝑁 are the machine run-up speeds in Hz, with 𝑓𝑁 as the maximum

speed. Note that the displacement vector 𝐝0 is always for the machine 1𝑠𝑡 run with

residual rotor unbalance (i.e. without any trial masses on the balancing planes) that

needs to be balanced. Assume that the rotating machine has ℎ number of bearing

pedestals. Hence, the displacement vectors at a single rotor speed 𝑓𝑘 can be written as

𝐝1(𝑓𝑘) = [y1,1(𝑓𝑘) x1,1(𝑓𝑘) … yℎ,1(𝑓𝑘) xℎ,1(𝑓𝑘)]T

𝐝2(𝑓𝑘) = [y1,2(𝑓𝑘) x1,2(𝑓𝑘) … yℎ,2(𝑓𝑘) xℎ,2(𝑓𝑘)]T

⋮

𝐝𝑛(𝑓𝑘) = [y1,𝑛(𝑓𝑘) x1,𝑛(𝑓𝑘) … yℎ,𝑛(𝑓𝑘) xℎ,𝑛(𝑓𝑘)]T

(7.2)

where yℎ,𝑛 and xℎ,𝑛 are the displacements at the ℎ𝑡ℎ bearing pedestal in the vertical and

horizontal directions respectively for the (𝑛 + 1)𝑡ℎ machine run. In case of 𝑛 balancing

planes, the sensitivity matrix 𝐒 is constructed using the measured vibration responses as

written in Equation (7.3).

𝐒 =

[ 𝐝1(𝑓1)−𝐝0(𝑓1)

𝑒1,1

𝐝1(𝑓2)−𝐝0(𝑓2)

𝑒1,1

⋮𝐝1(𝑓𝑁)−𝐝0(𝑓𝑁)

𝑒1,1

𝐝2(𝑓1)−𝐝0(𝑓1)

𝑒2,2

𝐝2(𝑓2)−𝐝0(𝑓2)

𝑒2,2

⋮𝐝2(𝑓𝑁)−𝐝0(𝑓𝑁)

𝑒2,2

…

𝐝𝑛(𝑓1)−𝐝0(𝑓1)

𝑒𝑛,𝑛

𝐝𝑛(𝑓2)−𝐝0(𝑓2)

𝑒𝑛,𝑛

⋮𝐝𝑛(𝑓𝑁)−𝐝0(𝑓𝑁)

𝑒𝑛,𝑛 ]

(7.3)


134

where 𝑒1,1 is the added unbalance to the 1𝑠𝑡 balancing plane at the 2𝑛𝑑 machine run-up,

𝑒2,2 is the added unbalance to the 2𝑛𝑑 balancing plane at the 3𝑟𝑑 machine run-up and

𝑒𝑛,𝑛 is the added unbalance to the 𝑛𝑡ℎ balancing plane at machine run-up number 𝑛 + 1.

The residual rotor mass unbalance 𝐞0 for each balancing plane can then be calculated as

shown in Equation (7.4), below:

[𝐒]2ℎ𝑁×𝑛 [

𝑒1,0𝑒2,0⋮𝑒𝑛,0

]

𝑛×1

= [𝐝0]2ℎ𝑁×1 (7.4)

The rotor mass unbalance 𝐞 is a complex value and can further be written as

[ 1 𝑗 0 0 0 0 … 0 00 0 1 𝑗 0 ⋮ … 0 00 0 0 0 1 𝑗 ⋮ 0 00 0 0 0 0 0 ⋱ ⋮ ⋮⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 𝑗 0 00 0 0 0 0 … 0 1 𝑗]

𝑛×2𝑛

[ 𝑟𝑒𝑎𝑙(𝑒1,0)

𝑖𝑚𝑎𝑔(𝑒1,0)

𝑟𝑒𝑎𝑙(𝑒2,0)

𝑚𝑎𝑔(𝑒2,0)

⋮𝑟𝑒𝑎𝑙(𝑒𝑛,0)

𝑖𝑚𝑎𝑔(𝑒𝑛,0)]

2𝑛×1

= 𝐓𝐞𝟎 (7.5)

where 𝐓 is the transformation matrix. Substituting Equation (7.5) into Equation (7.4)

gives

[𝐒]2ℎ𝑁×𝑛𝐓𝑛×2𝑛[𝐞𝟎]2𝑛×1 = [𝐝0]2ℎ𝑁×1 (7.6)

Equation (7.6) can be rewritten as

[𝐒𝑇]2ℎ𝑁×2𝑛[𝐞𝟎]2𝑛×1 = [𝐝0]2ℎ𝑁×1 (7.7)

where 𝐒𝑇 = 𝐒𝐓. The real and imaginary parts in Equation (7.7) are separated as

[𝑟𝑒𝑎𝑙 (𝐒𝑇)2ℎ𝑁×2𝑛𝑖𝑚𝑎𝑔 (𝐒𝑇)2ℎ𝑁×2𝑛

]4ℎ𝑁×2𝑛

[𝐞𝟎]2𝑛×1 = [𝑟𝑒𝑎𝑙(𝐝0)2ℎ𝑁×1𝑖𝑚𝑎𝑔(𝐝0)2ℎ𝑁×1

]4ℎ𝑁×1

(7.8)

Hence,


135

[𝐒𝑠]4ℎ𝑁×2𝑛[𝐞𝟎]2𝑛×1 = [𝐝0𝑠]4ℎ𝑁×1 (7.9)

Rearranging Equation (7.9) gives

𝐞𝟎 = [𝐒𝑠]+𝐝0𝑠 (7.10)

where [𝐒𝑠]+ is the Moore-Penrose pseudo-inverse of [𝐒𝑠] (i.e. [𝐒𝑠]

+ = ( 𝐒𝑠T𝐒𝑠)

−1𝐒𝑠T).

Equation (7.4) can be further modified to calculate the added unbalances at the different

planes simultaneously directly for each machine’s‎transient‎operation;

[𝐒]2ℎ𝑁×𝑛 [

𝑒1,𝑞𝑒2,𝑞⋮𝑒𝑛,𝑞

]

𝑛×1

= [𝐝𝑞 − 𝐝0]2ℎ𝑁×1 (7.11)

where the subscript 𝑞 in Equation (7.11) represents the (𝑞 + 1)𝑡ℎ machine run-up. Thus,

Equation (7.11) can be rewritten as

𝐞q = [𝐒𝑠]+𝐝q,0𝑠

(7.12)

In case of measurements at one direction per bearing pedestal, the displacement vectors

are written as

𝐝0(𝑓𝑘) = [r1,0(𝑓𝑘) r2,0(𝑓𝑘) … rh,0(𝑓𝑘)]T

𝐝1(𝑓𝑘) = [r1,1(𝑓𝑘) r2,1(𝑓𝑘) … rh,1(𝑓𝑘)]T

𝐝2(𝑓𝑘) = [r1,2(𝑓𝑘) r2,2(𝑓𝑘) … rh,2(𝑓𝑘)]T

⋮𝐝𝑛(𝑓𝑘) = [r1,𝑛(𝑓𝑘) r2,𝑛(𝑓𝑘) … rh,𝑛(𝑓𝑘)]

T

(7.13)

where r1,𝑛(𝑓𝑘), r2,𝑛(𝑓𝑘) and rh,𝑛(𝑓𝑘) are the radial displacements at the 1𝑠𝑡, 2𝑛𝑑 and h𝑡ℎ

bearings pedestals respectively at the rotor speed 𝑓𝑘 for the (𝑛 + 1)𝑡ℎ machine’s



136

Example-1: rig with one balancing plane 7.3

Example 1 is related to the application of the IC balancing method on a small rotating

machine with one balancing plane, and measurements are taken at vertical and

horizontal directions. First, the unbalance is estimated using measured vibrations at

single machine speeds. Then, the unbalance is estimated using measured vibrations at

multiple rotor speeds together in a single band. A total of 6 machine run-ups are carried

out, and 4 unbalance estimation scenarios are used.

7.3.1 Experimental setup

A photograph of the test rig used for the experiments is provided in Figure ‎7.1. The

rotor consists of a solid steel shaft with a length of 1000mm and a diameter of 20mm

and is supported on somewhat flexible foundation through two greased lubricated ball

bearings. The shaft carries a steel balancing disc of 130mm diameter and 20mm

thickness at midspan between the two bearings. Two flanges with a diameter of 40mm

and thickness of 20mm each are added to the disc to ensure that the disc will not slip

during the rotor rotation. The disc contains staggered M5 tapped holes in two different

pitch diameters, which are 70mm and 120mm. The angle between two neighbouring

holes for a particular pitch diameter is 30°.

The rotor-bearing system is connected to an electrical motor via flexible coupling made

of Aluminium. Each of the two ball bearings is mounted on a thin steel horizontal beam

(530mm × 25mm × 3mm). Each horizontal beam is secured atop two rectangular steel

blocks (107mm × 25mm × 25mm) that are screwed to a thick base plate (580mm ×

150mm × 15mm). The supporting structure is mounted on a massive steel platform

using 4 screws. An anti-vibration (TICO) pad which has a thickness of 12mm is

attached to the bottom side of the steel platform to mitigate noise and vibration. The

bearing near the motor is denoted as B1 and the other bearing as B2.


137

Figure ‎7.1 Photographs of the rig (a) assembled rig, (b) balancing disc, (c) flexible coupling

7.3.2 Instrumentation

The dynamic response of the system is measured by Integrated Circuit Piezoelectric

(ICP) accelerometers which have a sensitivity of 100 mV/g. The sensors are mounted in

the vertical and horizontal directions of each bearing pedestal. The rotational speed of

the shaft was measured using a laser tachometer (Figure ‎7.2). The measured analogue

signals from the accelerometers and tachometer are conditioned and converted into

𝒛

𝒚

𝒙

Shaft

Balancing disc

Motor

Bearing B2

Bearing B1

Safety guard Control panel

Horizontal

beam

(a)

(b) (c)


138

digital signals using DAQ device with 4-input channels. The DAQ system is connected

to a personal computer (PC) to record the data and store it for further analysis. The

speed of the motor was controlled by software in the PC.

Figure ‎7.2 Laser tachometer

7.3.3 Modal tests

Modal testing has been conducted on the test rig at zero rotor speed. The impact

response technique [17] was used for the modal testing. The rig was excited using a 1.1

mV/g instrumented hammer. Then, the response of the machine was measured by seven

100 mV/g ICP accelerometers. The modal test was done first in the vertical direction.

Then, the accelerometers were repositioned in the horizontal direction, and the modal

test was conducted again. The first four natural frequencies by appearance were

observed at 17.09, 29.91, 31.13 and 58.59 Hz. Figure ‎7.3 shows a typical measured

Frequency Response Function (FRF) computed from the modal analysis.

Laser

tachometer

Reflective

tape


139

Figure ‎7.3 Typical measured FRF plots of the rotor at distance of 75cm from bearing B1, (a) vertical direction,

(b) horizontal direction

(a)

(b)


140

7.3.4 Experiments carried out

The vibration responses were measured in both bearing housings during 6 machine’s

transient operations (one run-up with residual unbalance and five run-ups with different

added mass unbalances at a radius of 6cm). For each machine run, the rotor was run-up

linearly in a speed range of 0 to 50 Hz. The measurements at each bearing are taken in

vertical and horizontal directions (see Figure ‎7.4). Since the rotor vibrations due to mass

unbalance are synchronous to the rotational speed, the measured vibration responses of

each run were order tracked to get the 1× (i.e. one multiplied by rotating frequency)

vibration component at the four measurement locations. The order tracked 1× vibration

components in the speed range from 7 to 47 Hz with a spacing of 1 Hz were then used

for the unbalance estimation. Two critical speeds were found in the speed range (i.e. at

around 18 and 29 Hz). The added unbalances used for the different runs are listed in

Table ‎7.1. Table ‎7.2 lists the different scenarios used for the estimation of the added

unbalances.

Figure ‎7.4 Typical arrangement of accelerometers on bearing pedestal

𝒙, horizontal

𝒚, vertical


141

Table ‎7.1 List of 6 machine runs with different added unbalances (mass and phase angles)


𝑟𝑢𝑛0 Residual unbalance 𝑒0

𝑟𝑢𝑛1 7g × 6cm @ 30° = 42gcm @ 30° 𝑒0 + 𝑒1

𝑟𝑢𝑛2 3g × 6cm @ 120° = 18gcm @ 120° 𝑒0 + 𝑒2

𝑟𝑢𝑛3 5g × 6cm @ 30° = 30gcm @ 30° 𝑒0 + 𝑒3

𝑟𝑢𝑛4 5g × 6cm @ 60° = 30gcm @ 60° 𝑒0 + 𝑒4

𝑟𝑢𝑛5 3g × 6cm @ 60° = 18gcm @ 60° 𝑒0 + 𝑒5

Table ‎7.2 Different scenarios used for the estimation of the added unbalance

Case no. Sensitivity Unbalance % error

Case I

𝑟𝑢𝑛1 − 𝑟𝑢𝑛0

𝑟𝑢𝑛2 − 𝑟𝑢𝑛0 𝑒2 ±4%,+4°

Case II 𝑟𝑢𝑛3 − 𝑟𝑢𝑛0 𝑒3 ±14%,+1°

Case III 𝑟𝑢𝑛4 − 𝑟𝑢𝑛0 𝑒4 ±5%,+2°

Case IV 𝑟𝑢𝑛5 − 𝑟𝑢𝑛0 𝑒5 ±17%, 0°

Sensitivity analysis of unbalance estimation 7.4

The effect of estimating the rotor unbalance using measured machine responses of the

vertical and horizontal directions at single and multiple machine speeds is investigated

in the next two sub-sections.

7.4.1 Using single speed

The measured vibration data of machine run numbers 0 and 1 in Table ‎7.1 were

substituted in Equation (7.3) to construct the sensitivity matrix. Then, the displacement

vector of each rotor speed was used individually in Equation (7.11) and the rotor

unbalance was calculated. Typical estimated rotor mass unbalances using the individual

measured data are presented in Figure ‎7.5. Actual added unbalances as well as the

critical speeds of the rig are also shown in Figure ‎7.5 for the purpose of easy

comparison.


142

(a)

(b)


143

Figure ‎7.5 Comparison between the actual and estimated rotor unbalance for (a,b) case III, (c,d)

case IV; ○: estimated unbalance using different single speeds, ☆: error, : actual added

unbalance

(c)

(d)


144

7.4.2 Using speed range

Initially, the unbalance was estimated using measured vibration data from vertical and

horizontal directions at a single rotor speed of 7 Hz. Then, more speeds from the

machine run-up were added gradually, to observe the influence of including vibration

measurements at multiple rotor speeds together in a single band on the unbalance

estimation. The inclusion of multiple speeds in the unbalance estimation is represented

as Speed range and is defined as

Speed range = 𝑓1 + 𝑘(𝑑𝑓) = 7 Hz + 𝑘(1) (7.14)

where 𝑘 = 0, 1, … , 40.

Typical estimated rotor mass unbalances (both amplitudes and phases) at different speed

ranges are shown in Figure ‎7.6.


145

(a)

(b)


146

Figure ‎7.6 Comparison between the actual and estimated rotor unbalance for (a,b) case III, (c,d)

case IV; ○: estimated unbalance using speed range, ☆: error, : actual added unbalance

It can be observed from Figure ‎7.6 that using vibration measurements from a single as

well as few machine speed ranges to compute the rotor unbalance might not provide

accurate unbalance estimation, hence probably leads to poor balancing. Furthermore, it

(c)

(d)


147

can be noticed that the estimated rotor unbalances using the critical speeds and speeds

around them show larger error. However, the inclusion of vibration measurements from

more run-up speeds generally provides relatively accurate and stable unbalance

estimation in terms of both unbalance amplitude and phase. It can also be observed that

the unbalance estimation becomes stable from speed range 39 Hz onwards. Hence, the

maximum error of both amplitudes and phases in speed ranges from 39 to 47 Hz are

listed in Table ‎7.2.

Example-2: rig with two balancing planes 7.5

The same test apparatus used in the previous example is used in this case, but with two

modifications. The first modification is that a second disc that is identical to the first

one is added to the rig. The first and second discs are positioned at distances of 240mm

and 665mm from bearing B1, respectively. The second modification is using thicker

steel horizontal beams (530mm × 25mm × 8mm). The modified test rig is shown in

Figure ‎7.7. The disc closer to bearing B1 is symbolised as D1 and the other disc as D2.

A similar modal test procedure as in the previous rig has been conducted here. The first

four obtained natural frequencies by appearance were identified at 24.41 Hz, 31.13 Hz,

53.71 Hz and 84.23 Hz. Typical FRF is shown in Figure ‎7.8.

Figure ‎7.7 Photograph of the test rig with two balancing discs

𝒛

𝒚

𝒙

Bearing B2

Bearing B1

Disc D2

Disc D1


148

Figure ‎7.8 Typical measured FRF plots of the rotor at distance of 75cm from bearing B1 in (a)

vertical and (b) horizontal directions

(a)

(b)


149

7.5.1 Experiments carried out

The vibration responses were measured in both bearing housings during 13 machine

run-ups. The first run-up was with residual unbalance and the rest 12 run-ups were with

different added mass unbalances at a fixed radius of 6cm. For each machine run, the

rotor was run-up linearly in a speed range of 5 to 50 Hz. The measurements at each

bearing were taken in the vertical, horizontal and 45-degree directions (see Figure ‎7.9).

The order tracked 1× vibration components in the speed range from 15 to 40 Hz with a

spacing of 0.5 Hz were used for the unbalance estimation.

The added mass unbalances utilised for the different runs are listed in Table ‎7.3. The

different scenarios used for the calculation of the sensitivity and estimation of the added

unbalances are given in Table ‎7.4. Two critical speeds were found in the speed range

(i.e. at around 24 and 31 Hz).

Figure ‎7.9 A typical accelerometer installation at a bearing in 3 directions

𝒙, horizontal

𝒚, vertical 𝒓, radial(𝟒𝟓°)

150

Table ‎7.3 List of 13 machine run-ups with different added unbalances (mass and phase angles)


Disc D1 Disc D2

𝑟𝑢𝑛0 Residual unbalance 𝑒1,0 Residual unbalance 𝑒2,0

𝑟𝑢𝑛1 3g × 6cm @ 30° = 18gcm @ 30° 𝑒1,0 + 𝑒1,1 Residual unbalance 𝑒2,0

𝑟𝑢𝑛2 Residual unbalance 𝑒1,0 3g × 6cm @ 30° = 18gcm @ 30° 𝑒2,0 + 𝑒2,2

𝑟𝑢𝑛3 7g × 6cm @ 60° = 42gcm @ 60° 𝑒1,0 + 𝑒1,3 3g × 6cm @ 150° = 18gcm @ 150° 𝑒2,0 + 𝑒2,3




𝑟𝑢𝑛7 5g × 6cm @ 330° = 30gcm @330° 𝑒1,0 + 𝑒1,7 3g × 6cm @ 210° = 18gcm @ 210° 𝑒2,0 + 𝑒2,7






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Table ‎7.4 Different scenarios used for the added unbalance estimation

Case no. Sensitivity Unbalance Disc D1 Disc D2 % error, D1 % error, D2

Case I



𝑟𝑢𝑛3 − 𝑟𝑢𝑛0 𝑒1,3 𝑒2,3 ±24% , ±5° ±49% , ±11°

Case II 𝑟𝑢𝑛4 − 𝑟𝑢𝑛0 𝑒1,4 𝑒2,4 ±13% , ±5° +30% , ±12°

Case III 𝑟𝑢𝑛5 − 𝑟𝑢𝑛0 𝑒1,5 𝑒2,5 ±20% , +7° +34% , ±8°

Case IV 𝑟𝑢𝑛6 − 𝑟𝑢𝑛0 𝑒1,6 𝑒2,6 +23% , ±7° ±22% , +13°

Case V 𝑟𝑢𝑛7 − 𝑟𝑢𝑛0 𝑒1,7 𝑒2,7 +40% , ±5° ±27% , +25°

Case VI 𝑟𝑢𝑛8 − 𝑟𝑢𝑛0 𝑒1,8 𝑒2,8 +27% , ±4° +27% , ±22°

Case VII 𝑟𝑢𝑛9 − 𝑟𝑢𝑛0 𝑒1,9 𝑒2,9 ±19% , +6° +30% , ±12°

Case VIII 𝑟𝑢𝑛10 − 𝑟𝑢𝑛0 𝑒1,10 𝑒2,10 ±13% , ±13° +30% , ±3°

Case IX 𝑟𝑢𝑛11 − 𝑟𝑢𝑛0 𝑒1,11 𝑒2,11 ±4% , +11° +25% , ±5°

Case X 𝑟𝑢𝑛12 − 𝑟𝑢𝑛0 𝑒1,12 𝑒2,12 +20% , ±13° +10% , ±9°

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CHAPTER 7 SENSITIVITY ANALYSIS OF THE INFLUENCE COEFFICIENT…

152

Sensitivity analysis of unbalance estimation 7.6

This section presents a comparison between the application of the IC unbalance

estimation approach using measured response at two orthogonal directions and one

direction.

7.6.1 Using vertical and horizontal responses

The measured vibration responses at vertical and horizontal directions were used

together to estimate the rotor unbalance of all cases in Table ‎7.4. First, the unbalance

estimation was carried out using measured vibrations at the first machine speed 15 Hz.

Then, more speeds from the machine coast-up were added steadily. Same definition of

Speed range as in Equation (7.14) is used here but with changing 𝑓1 to 15 Hz, 𝑑𝑓 to 0.5

Hz and 𝑘 to 0, 1, … , 50. Typical estimated added unbalances of both discs at different

machine speed ranges are presented in Figures 7.10 to 7.13.


153

(a)

(b)


154

Figure ‎7.10 Case I (a) disc D1 (𝒆𝟏,𝟑), (b) error of unbalance in disc D1, (c) disc D2(𝒆𝟐,𝟑), (d) error

of unbalance in disc D2; ○: estimated unbalance, ☆: error, : actual added unbalance

(c)

(d)


155

(a)

(b)


156

Figure ‎7.11 Case II (a) disc D1 (𝒆𝟏,𝟒), (b) error of unbalance in disc D1, (c) disc D2(𝒆𝟐,𝟒), (d) error

of unbalance in disc D2; ○: estimated unbalance, ☆: error, : actual added unbalance

(c)

(d)


157

(a)

(b)


158

Figure ‎7.12 Case IV (a) disc D1 (𝒆𝟏,𝟔), (b) error of unbalance in disc D1, (c) disc D2(𝒆𝟐,𝟔), (d) error of

unbalance in disc D2; ○: estimated unbalance, ☆: error, : actual added unbalance

(d)

(c)


159

(a)

(b)


160

Figure ‎7.13 Case IX (a) disc D1 (𝒆𝟏,𝟏𝟏), (b) error of unbalance in disc D1, (c) disc D2(𝒆𝟐,𝟏𝟏), (d) error of

unbalance in disc D2; ○: estimated unbalance, ☆: error, : actual added unbalance

It can be clearly seen from Figures 7.10 to 7.13 that when using vibration measurements

acquired from vertical and horizontal directions together at a range of multiple machine

speeds in a single band, the unbalance estimation is significantly enhanced for both

(c)

(d)


161

balancing planes. Moreover, it can be observed that as the speed range increases, the

error between the estimated and actual unbalances becomes fairly small. It has been

found that the unbalance estimation becomes stable from speed range 35 Hz onwards.

Hence, the maximum error of both amplitudes and phases in speed ranges 35 to 40 Hz

are also listed in Table ‎7.4. It can also be noticed that the accuracy of the unbalance

estimation using vibration measurements at both critical speeds is low comparing to

higher speed ranges.

7.6.2 Using radial responses only

Instead of using two vibration sensors at a bearing pedestal in both horizontal and

vertical directions, just a single sensor is used at each pedestal, hence reducing the

number of sensors by 50%. Although the vibration measurement at any one direction is

possible to use in the proposed balancing approach, it is preferred to use measurements

at 45 degree to the vertical/horizontal directions (i.e. 𝒓 in Figure ‎7.9) in the current

study. The reason is that amongst all other directions, the measurements at this direction

are likely to have the most significant content of vibration behaviour from both

orthogonal directions. Same unbalance estimation scenarios and speed range definition

as in sub-section 7.6.1 (Tables 7.3 and 7.4) are repeated here. Table ‎7.5 presents the

different unbalance estimation cases. Typical estimated added unbalances of both discs

at different speed ranges are provided in Figures 7.14 to 7.17.

162

Table ‎7.5 List of 10 scenarios used for the added unbalance estimation

Case no. Sensitivity Unbalance Disc D1 Disc D2 % error, D1 % error, D2

Case I



𝑟𝑢𝑛3 − 𝑟𝑢𝑛0 𝑒1,3 𝑒2,3 +26% , ±1° ±5% , ±10°

Case II 𝑟𝑢𝑛4 − 𝑟𝑢𝑛0 𝑒1,4 𝑒2,4 +16% , ±10° +56% , ±4°

Case III 𝑟𝑢𝑛5 − 𝑟𝑢𝑛0 𝑒1,5 𝑒2,5 +10% , ±14° +28% , ±4°

Case IV 𝑟𝑢𝑛6 − 𝑟𝑢𝑛0 𝑒1,6 𝑒2,6 +30% , +2° ±6% , ±4°

Case V 𝑟𝑢𝑛7 − 𝑟𝑢𝑛0 𝑒1,7 𝑒2,7 +37% , +5° ±20% , +5°

Case VI 𝑟𝑢𝑛8 − 𝑟𝑢𝑛0 𝑒1,8 𝑒2,8 +40% , ±5° ±27% , +25°

Case VII 𝑟𝑢𝑛9 − 𝑟𝑢𝑛0 𝑒1,9 𝑒2,9 +4% , ±8° +21% , ±9°

Case VIII 𝑟𝑢𝑛10 − 𝑟𝑢𝑛0 𝑒1,10 𝑒2,10 +15% , +6° +21% , ±13°

Case IX 𝑟𝑢𝑛11 − 𝑟𝑢𝑛0 𝑒1,11 𝑒2,11 +26% , +8° +20% , ±12°

Case X 𝑟𝑢𝑛12 − 𝑟𝑢𝑛0 𝑒1,12 𝑒2,12 +33% , ±7° +17% , ±5°

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(a)

(b)


164

Figure ‎7.14 Case I (a) disc D1 (𝒆𝟏,𝟑), (b) error of unbalance in disc D1, (c) disc D2(𝒆𝟐,𝟑),

(d) error of unbalance in disc D2; ○: estimated unbalance, ☆: error, : actual added

unbalance

(c)

(d)


165

(a)

(b)


166

Figure ‎7.15 Case II (a) disc D1 (𝒆𝟏,𝟒), (b) error of unbalance in disc D1, (c) disc D2 (𝒆𝟐,𝟒),


unbalance

(c)

(d

)


167

(a)

(b)


168

Figure ‎7.16 Case IV (a) disc D1 (𝒆𝟏,𝟔), (b) error of unbalance in disc D1, (c) disc D2 (𝒆𝟐,𝟔),


unbalance

(c)

(d)


169

(a)

(b)


170

Figure ‎7.17 Case IX (a) disc D1 (𝒆𝟏,𝟏𝟏), (b) error of unbalance in disc D1, (c) disc D2

(𝒆𝟐,𝟏𝟏), (d) error of unbalance in disc D2; ○: estimated unbalance, ☆: error, : actual

added unbalance

It can be observed from Figures 7.14 to 7.17 that when using machine vibration

responses measured at only one direction (i.e. 45-degree direction) at a range of

(c)

(d)


171

multiple speeds in a single band, the unbalance estimation is reasonably accurate.

Moreover, it can be observed that the unbalance estimation becomes stable and fairly

accurate after passing 35 Hz. Therefore, the maximum amplitude and phase errors of

both balancing planes in the speed range of 35 to 40 Hz for all cases are listed in

Table ‎7.5.

Comparison of results 7.7

Figure ‎7.18 shows a grouped bar chart that compares the results of the estimated rotor

mass unbalance using orthogonally mounted pair of sensors at a bearing pedestal and a

single sensor per bearing pedestal (i.e. mounted at 45 degree to vertical/horizontal

directions). The results of the estimated rotor unbalance using measured vibrations at

vertical direction only as well as horizontal direction only are also shown in

Figure ‎7.18. It can be noticed from Figure ‎7.18 that the estimated added unbalances

using vibration data from two sensors as well as single sensor per bearing pedestal are

of good accuracy.


172

Figure ‎7.18 Grouped bar chart of the comparison between actual and estimated unbalances

for (a) disc D1, and (b) disc D2

Overall observations 7.8

It can be observed from Section 7.4 that as the speed range increases, the rotor mass

unbalance estimation (both amplitude and phase) greatly enhanced. Therefore, it is

better to use vibration measurements at multiple rotor speeds together in a single band

for effective machine balancing. Furthermore, it can be noticed that the accuracy of the

unbalance estimation using vibration measurements at critical speeds is low comparing

to higher speed ranges.

Comparing Figures 7.10 to 7.13 with Figures 7.14 to 7.17 respectively, it is easy to see

that the unbalance estimation using measured machine vibration at one direction is as

accurate and precise as when using measured vibration at two orthogonal directions.

Therefore, the current industrial application of the IC method can be improved by using

(a)

(b)


173

a single sensor per bearing pedestal (at 45-degree direction) instead of two sensors. The

reduction of vibration sensors by half is of great benefit to any industry that has rotating

machines because it will reduce the time and computational effort in the signal

processing significantly, hence estimating rotor unbalance faster and more effective.


This paper studied the application of the IC balancing method using vibration data

acquired from a small test rig with single/multiple balancing planes at single speed as

well as various speed ranges. Moreover, the influence of estimating the rotor unbalance

using single vibration sensor per bearing location instead of two has been investigated.

First, the IC method has been applied on a small size laboratory rig with one balancing

plane using vibration measurements acquired from vertical and horizontal directions.

Then, the experimental apparatus was modified by adding a second balance disc, and

the same estimation procedure was repeated. After that, the same concept of the IC

unbalance estimation method was used again but with reduced number of sensors.

Instead of using two vibration sensors at a bearing pedestal in both vertical and

horizontal directions, only one sensor at a bearing pedestal is used; hence reducing the

number of sensors by half.

The results indicate that the unbalance estimation using measured vibrations acquired at

a single machine speed may not be good enough. Furthermore, it was observed that the

certainty of reliably achieving an accurate and precise rotor mass unbalance estimation

using vibration response acquired at low-speed ranges is low. On the other hand, the

application of the IC method using vibration measurements acquired at higher speed

ranges in a single band considerably improves the accuracy of the unbalance estimation.

Thus, it is better to use higher speed ranges of the run-up/run-down of the rotating

machine for accurate and precise machine balancing.

Also, it can be noticed for all cases that when using vibration measurements acquired at

critical speeds, the unbalance estimation becomes inaccurate and provides misleading

results. The unbalance estimation in the case of reduced number of sensors provided as

accurate results as when using orthogonally mounted pair of sensors at a bearing

housing as presented in Figure ‎7.18. The proposed application of the IC balancing

method with reduced number of sensors could be beneficial to industries as it helps to


174

analyse the measured data in a timely manner and hence detect unbalance and take

balancing decision faster. Furthermore, the significant reduction in the number of

sensors is expected to reduce the likelihood of tripping the rotating machine as a result

of false signal from a faulty vibration sensor.

‎CHAPTER 8 MATHEMATICAL MODEL-BASED ROTOR UNBALANCE…

175

CHAPTER 8

MATHEMATICAL MODEL-BASED

ROTOR UNBALANCE ESTIMATION

USING A SINGLE MACHINE RUNDOWN

WITH REDUCED NUMBER OF SENSORS


176


Title: Rotor unbalance estimation using a single machine rundown with

reduced number of sensors

Authors: Sami M. Ibn Shamsah and Jyoti K. Sinha

Status: Published in the Proceedings of the International Conference on

Engineering Vibration (ICoEV 2015)

Abstract: Earlier proposed methods in the literature on the rotor unbalance and

foundation model estimation using a single machine run-down data have used vibration

measurements at all bearing pedestals in both lateral and vertical orthogonal directions.

It is generally believed that the measurement in both directions provides the bigger

picture of machine dynamical behaviour. However, in the present study, the concept of

the earlier method is used again but with reduced number of sensors. Instead of using 2

vibration sensors at a bearing pedestal in both lateral and vertical directions, just a

single sensor is used at each pedestal, hence reducing the number of sensors by 50%.

The sensor is mounted in the radial direction (45 degree to both lateral and vertical

directions) so that the measured vibration data will have a significant content of

vibration behaviour from both directions. The concept is applied to a simple simulated

rig of a rotor having a balancing disc and supported on either side by a ball bearing on a

flexible foundation. The paper presents the modelling details and comparison of several

unbalance estimations in the simulated rig example.

Keywords: Rotor unbalance, foundation model, vibration monitoring, fault diagnosis,

rotating machinery

Introduction 8.1

The reliability of rotating machines is essential to industries and plants around the world

as most of their activities rely on this class of machines. Over the years, with the

development of engineering and material science, industries have tended to use high-

speed flexible rotating machines which are supported by lightweight fabricated steel

structures and run for extended periods of time. As these flexible foundations usually




177

have natural frequencies below the operating speed of the machine, their effect should

be taken into consideration in the routine condition monitoring and associated

maintenance strategy. Neto et al. [114] have highlighted the effects of the flexible

foundation on the rotating machine by presenting some real case studies.

Many researchers proposed methods for the identification of rotating machines

foundation model [117]. Overview of the different experimental modal analysis

approaches was provided by Irretier [166]. Stephenson et al. [118] applied modal

analysis methods to determine the foundation’s dynamic characteristics from a test in

terms of FRFs. Lees et al. [102] gave an overview of the research on the model-based

identification of foundation model as well as presenting a brief theory of this approach.

It has been found that using measured vibration response at the bearing locations

together with an acceptable numerical model for the rotor and a reasonably accurate

model for the bearings seems to be the most promising approach to identify the

foundation model and rotor faults simultaneously [26, 102, 124]. Some researchers

recently proposed vibration based condition monitoring (VCM) techniques that use few

sensors, without necessarily compromising the valuable information required for the

diagnosis [78].

The existing rotor unbalance and foundation model identification approaches [25, 26]

used two orthogonal vibration sensors per bearing location. These methods are not very

practical for large and complex rotating machines as they could be computationally

involved and time-consuming as well as significantly relying on experience and

engineering judgment during data analysis. Therefore, the current study aims to improve

the existing approaches by presenting a simplified and computationally efficient method

for identifying and quantifying the state of rotor unbalance and foundation model in

rotating machines using response from a single vibration sensor per bearing location

and vibration measurements from a single machine’s‎run-up.

Earlier method 8.2

Earlier rotor fault identification method [25] used vibration response measurements

from vertical and lateral directions (i.e. 𝑣𝑘 and 𝑢𝑘 in Figure ‎8.1 (b) respectively) to

estimate the state of rotor unbalance (both amplitude and phase) and foundation model


178

from a single machine coast-down (Figure ‎8.1). In their model, each node consists of 4

degrees of freedom (2 translational and 2 rotational), i.e. [𝑢, 𝑣, 𝜃,Ψ]T, as shown in

Figure ‎8.2.

Figure ‎8.1 (a) Schematic representation of the rig, (b) measurements are taken in the radial direction

𝒛

𝒙

𝒚


𝒙

𝑭𝒚

𝑭𝒙

𝒛

(b)

Bearing front view Bearing side view

𝒖𝒌

𝒗𝒌 𝒓𝒌𝟏 𝒓𝒌𝟐

𝟒𝟓°

𝟒𝟓°

(a)

𝒚 𝒚

𝑭𝟐

A

A

𝑭𝟏

𝒛 𝒙


179


combination of horizontal and vertical

Proposed method 8.3

The current approach aims to investigate the feasibility of estimating the foundation

model and state of rotor unbalance using a reduced number of vibration measurements

from a single machine’s‎ run-up. Same formulation as the earlier method [25] is used,

but instead of using two vibration sensors per bearing location, i.e. one measures

vertical response and the other measures lateral response, only one sensor is used. In

order to have significant content of vibration behaviour from vertical and horizontal

directions, the response is taken from the radial direction, i.e. 𝑟𝑘1 or 𝑟𝑘2 in Figure ‎8.1

(b). Schematic representation of the rig used in the numerical simulation is shown in

Figure ‎8.3.

Figure ‎8.3 Schematic representation of the rotor

𝒛

𝒚

(a) (b)

𝚿𝒌 𝛉𝒌

𝒖𝒌 𝒖𝒍

𝚿𝒍


𝒛

𝒙

(c) O

𝒛

𝒚

𝒖𝒍

𝒙 𝒖𝒌

𝚿𝒌 𝒗𝒌

𝛉𝒌 𝒗𝒍

𝛉𝒍

𝚿𝒍

Node 24 Node 1

Nod

e 3

Nod

e 21

Nod

e 22

Nod

e 23

Nod

e 2

E.1 . . . . E.2 E.21 E.3 . E.22 E.23


𝒛

𝒚 𝒚

𝒛 𝒙

𝑭𝟐𝒓 𝑭𝟏𝒓 𝑭𝒓

𝒙

Nod

e 15

45∘ from 𝒙 and 𝒚 axes


180

8.3.1 Theory

In the construction of typical rotating machinery, the rotor, bearings and machine

supporting structure (foundation) are often considered as the major components, where

the rotor is connected to the flexible machine supporting structure through the bearings.

The equation of motion of such structure can be written as [109]

[

𝐙𝑅,𝑖𝑖 𝐙𝑅,𝑖𝑓 0

𝐙𝑅,𝑓𝑖 (𝐙𝑅,𝑓𝑓 + 𝐙𝐹,𝑓𝑓) −𝐙𝐹,𝑓𝑖0 −𝐙𝐹,𝑖𝑓 𝐙𝐹,𝑖𝑖

] {

𝐝𝑅,𝑖𝐝𝑅,𝑓𝐝𝐹,𝑖

} = {𝐟𝑢00

} (8.1)

where Z, d and fu are the dynamic stiffness matrix, displacement and unbalance forces

respectively, and the subscripts 𝐹, 𝑅, 𝑖 and 𝑓 denote the foundation, rotor, internal

degrees of freedom and connection degrees of freedom respectively. In this method, the

dynamic stiffness matrix of each element is reduced from 8 by 8 to 4 by 4 related to

degrees of freedom [𝑢𝑘, 𝑣𝑘 , 𝜃𝑘 , Ψ𝑘, 𝑢𝑙 , 𝑣𝑙 , 𝜃𝑙 , Ψ𝑙]T to [𝑟𝑘, Ψ𝑟𝑘, 𝑟𝑙, Ψ𝑟𝑙]

T respectively.

Therefore, the total size of the dynamic stiffness matrix of the system is reduced by half.

As a result, the time and computational effort in the signal processing will reduce

significantly. The unbalance forces are assumed to be applied only at the rotor internal

degrees of freedom. It is also assumed that the bearings are rigid connections between

the rotor and foundation. The foundation at bearings locations can be modelled by using

the degrees of freedom of the rotor only; hence the foundation internal degrees of

freedom are eliminated. Therefore, Equation (8.1) is reduced to

[𝐙𝑅,𝑖𝑖 𝐙𝑅,𝑖𝑓

𝐙𝑅,𝑓𝑖 (𝐙𝑅,𝑓𝑓 + 𝐙𝐹,𝑓𝑓)] {𝐝𝑅,𝑖𝐝𝑅,𝑓

} = {𝐟𝑢0} (8.2)

Equation (8.2) may be expressed as:

[𝐙𝑅,𝑖𝑖]{𝐝𝑅,𝑖} + [𝐙𝑅,𝑖𝑓]{𝐝𝑅,𝑓} = {𝐟𝑢} (8.3)

and

[𝐙𝑅,𝑓𝑖]{𝐝𝑅,𝑖} + [(𝐙𝑅,𝑓𝑓 + 𝐙𝐹,𝑓𝑓)]{𝐝𝑅,𝑓} = {0} (8.4)


181

Equation (8.3) can be rearranged as:

{𝐝𝑅,𝑖} = [𝐙𝑅,𝑖𝑖]−1[{𝐟𝑢} − [𝐙𝑅,𝑖𝑓]{𝐝𝑅,𝑓}] (8.5)

The unknown response can be eliminated by substituting Equation (8.5) into Equation

(8.4) as follows

[𝐙𝑅,𝑓𝑖][𝐙𝑅,𝑖𝑖]−1[{𝐟𝑢} − [𝐙𝑅,𝑖𝑓]{𝐝𝑅,𝑓}] + [(𝐙𝑅,𝑓𝑓 + 𝐙𝐹,𝑓𝑓)]{𝐝𝑅,𝑓} = {0} (8.6)

It is assumed that a reasonably accurate model for the rotor is available and the response

is measured. Thus, the foundation model and unbalance forces are the only unknowns in

Equation (8.6).

By inspecting Figure ‎8.4, it can be figured out that (𝑥, 𝑦) coordinates of a point P can be

written in terms of (𝑥′, 𝑦′) as follows

𝑥 = 𝑥′𝑐𝑜𝑠(𝛽) − 𝑦′𝑠𝑖𝑛(𝛽)

𝑦 = 𝑥′𝑠𝑖𝑛(𝛽) + 𝑦′𝑐𝑜𝑠(𝛽)

In matrix form,

[𝑥𝑦] = [

cos(𝛽) −sin(𝛽)

sin (𝛽) cos (𝛽)] [𝑥′

𝑦′]

Hence,

[𝑥′

𝑦′] = [

cos(𝛽) sin(𝛽)

−sin (𝛽) cos (𝛽)] [𝑥𝑦]

Figure ‎8.4 Coordinates of a point in two systems

P

𝑥′𝑠𝑖𝑛𝛽 𝑥

𝑥′

𝑦 𝑦′

𝛽

𝛽

𝑥′𝑐𝑜𝑠𝛽

𝑦′𝑠𝑖𝑛𝛽

𝑦′𝑐𝑜𝑠𝛽


182

Therefore, to get the radial response, the vertical and horizontal responses at each

bearing are multiplied by the following transformation matrix

𝐓𝑟 = [cos(45) sin(45)−sin (45) cos (45)

] (8.7)

8.3.2 Parameter estimation

The unbalance planes are assumed to be positioned at nodes 𝑛1, 𝑛2, … , 𝑛𝑝 where 𝑝 is the

number of planes. The amplitude of unbalance may be defined as

𝑢𝑛𝑖 = 𝑚𝑢𝑖𝑒𝑖 (8.8)

where 𝑚𝑢𝑖 is the unbalance mass at plane 𝑖, and 𝑒 is the eccentric radius of the

unbalance mass. In terms of matrices, the amplitude of unbalance and the phase angles

associated to the unbalance planes can be written as [𝑢𝑛1 , 𝑢𝑛2 , 𝑢𝑛3 , … , 𝑢𝑛𝑝]T

and

[α𝑛1 , α𝑛2 , α𝑛3 , … , α𝑛𝑝]T

respectively. Therefore, the complex quantity of the rotor

unbalance may be expressed as


where 𝑎𝑛𝑖 and 𝑏𝑛𝑖 can be defined as

𝑎𝑛𝑖 = 𝑚𝑖 𝑒𝑖 cos(α) and 𝑏𝑛𝑖 = 𝑚𝑖 𝑒𝑖 cos (α +𝜋

2) = 𝑚𝑖 𝑒𝑖 sin(α) (8.10)

Hence, the unbalance forces can be written as

𝐟𝑢 = 𝜔2

{

0⋮0

𝑚1 𝑒1(𝑐𝑜𝑠(α1) + 𝑗 𝑠𝑖𝑛(α1))0⋮0

𝑚𝑝 𝑒𝑝(𝑐𝑜𝑠(α𝑝) + 𝑗 𝑠𝑖𝑛(α𝑝))

0⋮0 }

= 𝜔2

{

0⋮0

𝑎𝑛1 + 𝑗𝑏𝑛10⋮0

𝑎𝑛𝑝 + 𝑗𝑏𝑛𝑝0⋮0 }

(8.11)


183

where the locations in the unbalance force vector depend on the nodal locations of the

unbalance planes. Equation (8.11) may be further simplified as

𝐟𝑢 = 𝜔2𝐓𝐠 (8.12)


and 𝐓 is the transformation matrix that is

defined such that Equations (8.11) and (8.12) are equivalent. Substituting Equation

(8.12) into Equation (8.6) gives:

[𝐙𝑅,𝑓𝑖𝐙𝑅,𝑖𝑖−1𝐙𝑅,𝑖𝑓 − 𝐙𝑅,𝑓𝑓]{𝐝𝑅,𝑓} − [𝐙𝐹,𝑓𝑓]{𝐝𝑅,𝑓} = 𝜔

2[𝐙𝑅,𝑓𝑖𝐙𝑅,𝑖𝑖−1𝐓]𝐠 (8.13)

In order to identify the rotor unbalance in a least-square sense, the foundation

parameters are grouped into vectors 𝐯𝒐, 𝐯𝟏 and 𝐯𝟐. Assume that the foundation dynamic

stiffness matrix (𝐙𝐹,𝑓𝑓) is written in terms of mass, damping and stiffness matrices. If

there are n measured degrees of freedom at the foundation-bearing interface,

then 𝐯𝒐, 𝐯𝟏 and 𝐯𝟐 may be written as

𝐯𝒐 = [𝑘𝐹,11 𝑘𝐹,12 … 𝑘𝐹,𝑛𝑛], 𝐯𝟏 = [𝑐𝐹,11 𝑐𝐹,12 … 𝑐𝐹,𝑛𝑛], 𝐯𝟐 = [𝑚𝐹,11 𝑚𝐹,12 … 𝑚𝐹,𝑛𝑛] (8.14)

𝐯𝒐, 𝐯𝟏 and 𝐯𝟐 can be collected in one matrix as follows

𝐯 = [𝐯𝒐 𝐯𝟏 𝐯𝟐]T (8.15)

With this definition of 𝐯, there is a linear transformation such that

[𝐙𝐹,𝑓𝑓]{𝐝𝑅,𝑓} = [W]{𝐯} (8.16)

where W contains the measured response terms at each frequency. For the 𝑁𝑡ℎ measured

frequency

W(𝜔𝑁) = [W𝑜(𝜔𝑁) W1(𝜔𝑁) W2(𝜔𝑁)]T (8.17)

If all elements of the foundation mass, damping and stiffness matrices are identified,

then


184

W𝑘(𝜔𝑁) = (𝑗𝜔𝑁)𝑘

[ d𝑅𝑓,1(𝜔𝑁) ⋯ d𝑅𝑓,𝑛(𝜔𝑁) 0 ⋯ 0 ⋯ 0 ⋯ 0

0 ⋯ 0 d𝑅𝑓,1(𝜔𝑁) ⋯ d𝑅𝑓,𝑛(𝜔𝑁) 0 ⋮ ⋯ ⋮

⋮ ⋮ ⋱ 0 00 0 0 0 ⋯ 0 0 d𝑅𝑓,1(𝜔𝑁) ⋯ d𝑅𝑓,𝑛(𝜔𝑁)]

(8.18)

where 𝑘 = 0, 1, 2. Equation (8.13), then, becomes

[𝐙𝑅,𝑓𝑖𝐙𝑅,𝑖𝑖−1𝐙𝑅,𝑖𝑓 − 𝐙𝑅,𝑓𝑓]{𝐝𝑅,𝑓} − [W]{v} = 𝜔

2[𝐙𝑅,𝑓𝑖𝐙𝑅,𝑖𝑖−1𝐓]𝐠 (8.19)

Let

R(𝜔𝑁) = 𝜔𝑁2 [𝐙𝑅,𝑓𝑖𝐙𝑅,𝑖𝑖

−1 𝐓] (8.20)

and

H(𝜔𝑁) = [𝐙𝑅,𝑓𝑖𝐙𝑅,𝑖𝑖−1𝐙𝑅,𝑖𝑓 − 𝐙𝑅,𝑓𝑓]{𝐝𝑅,𝑓} (8.21)

hence, Equation (8.19) becomes

H(𝜔𝑁) − [W(𝜔𝑁)]{v} = [R(𝜔𝑁)]{𝐠} 𝑦𝑖𝑒𝑙𝑑𝑠→ [W(𝜔𝑁)]{v} + [R(𝜔𝑁)]{𝐠} = H(𝜔𝑁) (8.22)

Equation (8.22) may be rewritten as

[W(𝜔𝑁) R(𝜔𝑁)] {v𝐠} = [H(𝜔𝑁)] (8.23)

Suppose that the response is measured for frequencies starting from 1 to 𝑞 Hz. All

measurements are used at ones, and one estimation of the rotor unbalance is produced.

Thus, Equation (8.23) is repeated 𝑞 times to give

[ W0(𝜔1) W1(𝜔1) W2(𝜔1) R(𝜔1)

W0(𝜔2) W1(𝜔2) W2(𝜔2) R(𝜔2)⋮ ⋮ ⋮ ⋮

W0(𝜔𝑞) W1(𝜔𝑞) W2(𝜔𝑞) R(𝜔𝑞)] {𝐯𝐠} =

[ H(𝜔1)

H(𝜔2)⋮

H(𝜔𝑞)] (8.24)

Equation (8.23) is a least square problem where the number of equations is more than

the number of unknowns. Therefore, to obtain the rotor unbalance, Equation (8.23) may

be rewritten as


185

{v𝐠} = [W(𝜔𝑞) R(𝜔𝑞)]

+H(𝜔𝑞) (8.25)

where [W(𝜔𝑞) R(𝜔𝑞)]+

is the Moore-Penrose pseudo-inverse of [W(𝜔𝑞) R(𝜔𝑞)], and

can be written as

[W(𝜔𝑞) R(𝜔𝑞)]+= [[W(𝜔𝑞) R(𝜔𝑞)]

T[W(𝜔𝑞) R(𝜔𝑞)]]

−1

[W(𝜔𝑞) R(𝜔𝑞)]T (8.26)

Although Equation (8.23) may be solved in a least-squares sense directly as shown in

Equation (8.25), it is not computationally efficient to invert the normal equations

matrix. Therefore, the solution via the singular-value decomposition (SVD) or QR-

decomposition approaches is more numerically stable.

The condition number of the inverse problem (i.e. Equation (8.23)) is relatively large,

which implies that any small perturbation in the input may result in significant errors in

the output data. Therefore, in order to solve this ill-conditioned problem, column scaling

regularisation is applied.

Simulated example 8.4

The proposed method was applied to a numerically simulated machine, where a flexible

rotor was mounted on two bearings fitted on a flexible foundation. The bearings were

assumed to be rigid joints between the rotor and foundation. The rotor in this example

consists of one steel shaft of 600mm long, with a nominal diameter of 20mm. Balancing

disc with a diameter of 125mm and thickness of 10mm was located at 100mm away

from the centre.‎The‎Young’s‎modulus‎ and‎density‎of‎ the‎ rotor‎ and‎disc‎material‎ are‎

200 GPa and 7850 kg/m3 respectively. The rotor was created with 23 two-noded

Timoshenko beam elements, as shown in Figure ‎8.3, where each node has 2 degrees of

freedom. The torsional and axial vibrations were assumed to be negligible.

Results and discussion 8.5

The estimation process was done through two steps. First, Equation (8.2) was used to

calculate response at the normal directions of the two bearings using assumed

foundation model and a given rotor unbalance. The response vector was ordered as


186

𝐝𝑅,𝑓 = [𝑢𝑏1 𝑣𝑏1 𝑢𝑏2 𝑣𝑏2]T (8.27)

where the subscript b denotes the bearings. Then, the responses in 𝑟𝑘1 and 𝑟𝑘2 directions

were calculated by multiplying the responses in the normal directions by Equation (8.7).

The foundation mass and damping matrices were considered as

M𝐹 = diag [5 5 5 5] kg , C𝐹 = diag [150 150 150 150] N 𝑠𝑚⁄ (8.28)

At each unbalance configuration, three different stiffness combinations were used. First,

the horizontal and vertical stiffnesses were assumed to be identical. Then, the horizontal

stiffness was reduced to 80% and 50% of the vertical stiffness. The different unbalance

and foundation stiffness configurations are listed in Table ‎8.1. The numerically

modelled machine was run-up from 1 to 400 Hz with measurements taken at a spacing

of 0.5 Hz. The parameter estimation process was done four times where every time the

response of one of the four directions (i.e. vertical, horizontal, radial vertical and radial

horizontal) was assumed as the only measured response. The results of the different

directions are presented in the next three subsections.

Table ‎8.1 The unbalance and foundation stiffness configurations for the simulated examples

Configuration

no.

Run

no.

Foundation stiffness Unbalance

(Kg.m)

Phase (deg.)

w.r.t horizontal

direction V (MN/m) H (MN/m)

1

1

10.00

10.00

3.20e-04 30.00 2 8.00

3 5.00

2

4

10.00

10.00

7.20e-04 105.00 5 8.00

6 5.00

3

7

10.00

10.00

9.00e-04 220.00 8 8.00

9 5.00

4

10

10.00

10.00

1.80e-03 320.00 11 8.00

12 5.00

8.5.1 Vertical response only

Only the vertical vibration response of the two bearings was assumed as the measured

response and used to estimate the foundation model and the amplitude and phase of the


187

rotor unbalance. The estimated rotor unbalances for the different runs are listed in

Table ‎8.2. It can be seen that the maximum error in the identified unbalances is less than

1%, which indicates that the estimated unbalances are reasonably accurate. As can be

seen in Table ‎8.2, when the measurements from the vertical direction were used in the

unbalance estimation, the estimated phase angles were found to be 90 degrees more than

the actual ones. This is due to the fact that the actual phase angles have been measured

with respect to the horizontal axis (Figure ‎8.5).

Figure ‎8.5 Phase angles with respect to vertical and horizontal axes in cases of (a) response taken at

normal directions and (b) response taken at radial directions

8.5.2 Horizontal response only

The horizontal response was assumed as the measured response, and the estimation

process was done again. When the horizontal and vertical stiffnesses are identical, the

unbalance that was estimated using the response from the horizontal direction is same as

the vertical direction. However, when the horizontal and vertical stiffnesses are

different, the estimated rotor unbalance is not same. The identified rotor unbalances for

the 12 runs are presented in Table ‎8.2.

𝒙

𝒚

𝒖

𝒗

𝛂+ 𝟒𝟓°

𝛂 + 𝟏𝟑𝟓°

unbalance 𝒙

𝒚

𝒖

𝒗

𝛂

𝛂+ 𝟗𝟎°

unbalance (a) (b)


188

Table ‎8.2 The estimated unbalance for the simulated examples using vertical and horizontal

directions separately

Ru

n n

o.

Vertical direction only Horizontal direction only

Un

ba

lan

ce

(kg

.m)

Un

ba

lan

ce

error (%

)

Ph

ase

(deg

.)

Angle

w.r.t

horizont

al

(phase-

90)

Ph

ase

error

(%)

Un

ba

lan

ce

(kg

.m)

Un

ba

lan

ce

error (%

)

Ph

ase

(deg

.)

Ph

ase

error

(%)

1 3.1781e-04 0.6833 120 30.00 0 3.1781e-04 0.6833 30 0

2 3.1781e-04 0.6833 120 30.00 0 3.1941e-04 0.1834 30 0

3 3.1781e-04 0.6833 120 30.00 0 3.1952e-04 0.1499 30 0

4 7.1509e-04 0.6824 195 105.00 0 7.1509e-04 0.6824 105 0

5 7.1509e-04 0.6824 195 105.00 0 7.1866e-04 0.1859 105 0

6 7.1509e-04 0.6824 195 105.00 0 7.1889e-04 0.1544 105 0

7 8.9387e-04 0.6807 310 220.00 0 8.9387e-04 0.6807 220 0

8 8.9387e-04 0.6807 310 220.00 0 8.9831e-04 0.1881 220 0

9 8.9387e-04 0.6807 310 220.00 0 8.9861e-04 0.1546 220 0

10 0.0017382 0.6767 50 320.00 0 0.0017 0.6767 320 0

11 0.0017382 0.6767 50 320.00 0 0.001747 0.1958 320 0

12 0.0017382 0.6767 50 320.00 0 0.0017471 0.1632 320 0

8.5.3 Radial responses only

The rotor unbalance and foundation model were, first, estimated using the measured

vibration response from the first radial direction 𝑟𝑘1 only. Then, the estimation process

was repeated using responses from the second radial direction 𝑟𝑘2 only. The results of

the different runs for both cases are shown Table ‎8.3. The rotor response may be

estimated using the identified parameters from either 𝑟𝑘1 or 𝑟𝑘2 directions. This is

because the estimated parameters which were calculated using responses from either 𝑟𝑘1

or 𝑟𝑘2 directions include a significant content of vibration behaviour from the vertical

and horizontal directions. A comparison between the actual and estimated responses is

shown in Figures 8.6 and 8.7. By looking at these two figures, it can be said that the

estimated foundation model is of a good quality.

189

‎CH

AP

TE

R 8

MA

TH

EM

AT

ICA

L M

OD

EL

-BA

SE

D R

OT

OR

UN

BA

LA

NC

E…

Table ‎8.3 The estimated unbalance for the simulated examples using 𝒓𝒌𝟏 and 𝒓𝒌𝟐 directions separately

Ru

n n

o.

𝒓𝒌𝟏 direction only 𝒓𝒌𝟐 direction only

Un

bala

nce

(Kg.m

)

Un

bala

nce

error (%

)

Ph

ase

(deg

.)

Angle w.r.t

horizontal

(phase-135)

Ph

ase

error

(%)

Un

bala

nce

(Kg.m

)

Un

bala

nce

error (%

)

Ph

ase

(deg

.)

Angle w.r.t

horizontal

(phase-45)

Ph

ase

error

(%)

1 3.1785e-04 0.6724 75 30 0 3.1785e-04 0.6724 165 30 0

2 3.1888e-04 0.3492 75 30 0 3.1769e-04 0.7220 165 30 0

3 3.1869e-04 0.4081 75 30 0 3.1733e-04 0.8331 165 30 0

4 7.1515e-04 0.6733 150 105 0 7.1515e-04 0.6733 240 105 0

5 7.1755e-04 0.3406 150 105 0 7.1475e-04 0.7294 240 105 0

6 7.1712e-04 0.4007 150 105 0 7.1393e-04 0.8426 240 105 0

7 8.9392e-04 0.6751 265 220 0 8.9392e-04 0.6751 355 220 0

8 8.9683e-04 0.3518 265 220 0 8.9355e-04 0.7167 355 220 0

9 8.9632e-04 0.4094 265 220 0 8.9255e-04 0.8274 355 220 0

10 0.001738 0.6790 5 320 0 0.001738 0.6790 95 320 0

11 0.0017438 0.3503 5 320 0 0.0017375 0.7141 95 320 0

12 0.001743 0.4069 5 320 0 0.001736 0.8253 95 320 0


190

Figure ‎8.6 Comparison between the actual and estimated responses at (a) bearing 1 𝒓𝒌𝟏 and (b)

bearing 2 𝒓𝒌𝟏, for run number 5: , actual ; , estimated

(b)

(a)


191

Figure ‎8.7 Comparison between the actual and estimated responses at (a) bearing 1 𝒓𝒌𝟐 and (b)

bearing 2 𝒓𝒌𝟐, for run number 5: , actual; , estimated

Conclusion 8.6

This paper has presented a simplified and computationally efficient method for

identifying and quantifying the state of rotor unbalance and foundation model in

rotating machines using measured vibration response at a single machine transient

operation with a single vibration sensor per bearing location. The vibration sensors have

been located in the radial direction (i.e. 45 degree to both lateral and vertical directions)

in order to include the effect of the vertical and horizontal directions. Numerical

simulation has been done on a rotor-bearing-foundation system with different unbalance

and stiffness configurations. Vibration measurements have been collected during a

(b)

(a)


192

single coast-up and used in the parameter estimation process. The estimated rotor

unbalance and foundation model of the different configurations were close to the actual

ones, which indicates the potentials of the proposed technique for practical applications.

CHAPTER 9 MULTI-PLANES ROTOR UNBALANCE IDENTIFICATION…

193

CHAPTER 9

MULTI-PLANES ROTOR UNBALANCE

IDENTIFICATION USING DATA FROM A

SINGLE MACHINE RUN-UP WITH

REDUCED NUMBER OF SENSORS


194


Title: Estimating rotor unbalance from a single run-up and using reduced sensors

Authors: Sami M. Ibn Shamsah, Jyoti K. Sinha and Parthasarathi Mandal

Status: Submitted to Journal of Sound and Vibration

Abstract: The earlier model-based rotor mass unbalance estimation methods have used two

orthogonal sensors per bearing pedestal. It is generally believed that the vibration

measurement at two orthogonal directions provides the bigger picture of machine dynamical

behaviour. However, in the present study, the concept of the earlier method is applied again

but with using only one sensor at a bearing pedestal, rather than the usual pair of sensors

arranged in orthogonal directions. The significant reduction in the number of vibration

sensors without necessarily compromising the valuable information required for the diagnosis

could be of great benefit to industries that have huge rotating machines with numerous

bearings. The reason is that the computational effort in the complex signal processing will

reduce considerably and hence the downtime of the rotating machine will reduce

significantly. The concept is applied to experimental rigs with a single as well as multiple

balancing planes and supported on either side through a stiff ball bearing on a somewhat

flexible foundation. The paper presents the experimental apparatus, unbalance estimation and

comparison of several unbalance estimations of various scenarios. The results indicate the

potentials of the proposed technique for practical applications.

Keywords: Rotor unbalance, vibration monitoring, fault diagnosis, rotating machinery





195

Introduction 9.1

Rotating machines, such as turbogenerator set, are essential components in power generation

and industrial applications. Therefore, their reliability and availability are in high demand.

The typical industrial rotating machine consists of three main elements: the rotor, the

bearings and the foundation. There are several faults that often occur to the rotors, and hence

impede the rotating machines from achieving their anticipated reliability targets [17, 78, 167,

168]. Since this class of machines plays a crucial role in the achievement of most industrial

objectives, it is therefore necessary to continuously seek for approaches that simplify as well

as improve the detection and diagnosis of rotating machine malfunctions at their early stages.

The rotor mass unbalance, which occurs when the centre of mass of the rotor is not aligned

with the centre of rotation, is one of the most common malfunctions that cause the whole

machine to vibrate [169]. If the amount of rotor mass unbalance exceeds a pre-set limit, it

may cause unexpected machine failure. Hence, costly machine repairs and unplanned plant

downtime are required. Therefore, the regular balancing of rotating machines is necessary to

guarantee safe and smooth running as well as long service life of the machines.

Many approaches for the identification of unbalance rely on vibration signals as machine

vibrations change in different ways under different fault conditions. A number of vibration-

based rotor balancing methods have been proposed in the last few decades [51, 95, 99, 106,

170]. Recently, some researchers have successfully identified and diagnosed rotor mass

unbalance using methods that rely on an accurate numerical model of the machine as well as

measured vibration response from a single machine run-down [26, 127]. The finite element

(FE) method has been found to be the most appropriate tool for the numerical modelling in

structural engineering today [27]. Often, an accurate mathematical model of the rotor and

approximate model of the bearings can be constructed using the FE method [21].

Lees et al. [102] gave an overview of the research on the model-based rotor unbalance

identification approach. Lees and Friswell [123] have presented a technique for identifying

the amplitude of rotor unbalance using machine vibration measurements from a single

machine’s transient operation (i.e. run-down) of a simulated rotating machine. Their method

requires an accurate numerical model of the rotor and approximate model of the bearings. No


196

attempts for the estimation of the phase have been made as it relies strongly on the quality of

the mathematical model of the bearings.

Later, Edwards et al. [124] as well as Lees et al. [125] have gone one step forward by

identifying both the machine supporting structure parameters as well as the phase and

amplitude of rotor mass unbalance. The results have been verified experimentally on a test rig

by using the vibration data from a single machine run-down. Sinha et al. [25] have proposed

a method that estimates both rotor unbalance and foundation model of a rotor-bearing-

foundation system. The method is similar in concept to Edwards et al. [124]. This approach

has provided a very good estimation of both state of rotor unbalance as well as foundation

model.

In the present study, the concept of the earlier model-based balancing method is enhanced by

reducing number of sensors (accelerometers). Instead of using a pair of sensors arranged in

orthogonal directions, only one sensor is used at a bearing pedestal. The significant reduction

in the number of sensors is of great benefit when it comes to the huge industrial rotating

machines as the computational effort in the complex signal processing will reduce

considerably; and hence the machine downtime will reduce significantly. Also, the significant

reduction in the number of sensors will considerably reduce the possibility of tripping the

machine because of false signal from faulty vibration sensor. The concept is applied to

experimental rigs with single/multiple balancing planes and supported on either side through

a stiff ball bearing on a somewhat flexible foundation. The paper presents the experimental

apparatus, unbalance estimation and comparison of several unbalance estimations of various

scenarios.

Earlier method 9.2

In the construction of typical rotating machinery, the rotor, journal bearings and foundation

are often considered as the principal components, where the rotor is connected to the

foundation through the bearings. The motion equation of such structure can be written [25] as

[

𝐙𝑅,𝑖𝑖 𝐙𝑅,𝑖𝑏 0

𝐙𝑅,𝑏𝑖 𝐙𝑅,𝑏𝑏 + 𝐙𝐵 −𝐙𝐵0 −𝐙𝐵 𝐙𝐵 + �̅�𝐹

] {

𝐝𝑅,𝑖𝐝𝑅,𝑏𝐝𝐹,𝑏

} = {𝐟𝑢00

} (9.1)


197

where 𝐙 is the dynamic stiffness matrix. The frequency-dependent dynamic stiffness matrix

may be written as 𝐙(ω) = −ω2𝐌+ jω𝐂 + 𝐊, where ω is the rotational speed in rad/sec, and

𝐌, 𝐂 and 𝐊 are the mass, damping and stiffness matrices of the structural system. In case of

modelling rotor with large shaft diameter, gyroscopic effects should be added to the dynamic

stiffness matrix. However, gyroscopic effects are generally small and can be ignored when

modelling shaft with small diameter [21]. 𝐙𝐵, 𝐝 and 𝐟𝑢 in Equation (9.1) are the dynamic

stiffness matrix of the bearings, displacement vector and unbalance forces respectively. The

unbalance forces are assumed to be applied only at rotor internal degrees of freedom. The

subscripts 𝐹, 𝑅, 𝐵, 𝑖 and 𝑏 in Equation (9.1) denote the foundation, the rotor, the bearings,

the internal degrees of freedom and the bearing (connection) degrees of freedom,

respectively. Due to the fact that the vibration response can be measured only at the bearing-

foundation interface and not in the internal degrees of freedom of the foundation, the internal

foundation degrees of freedom have been eliminated. Therefore, only a reduced order model

of the foundation, �̅�𝐹, can be estimated [109, 171]. In the model of the earlier method, each

element consists of 8 degrees of freedom (4 translational and 4 rotational), i.e.

[𝑢𝑘, 𝑣𝑘, 𝜃𝑘 , Ψ𝑘 , 𝑢𝑙 , 𝑣𝑙 , 𝜃𝑙 , Ψ𝑙]T, as shown in Figure ‎9.1.


combination of horizontal, vertical and radial planes

𝒛

𝒚

(a) (b)

𝚿𝒌 𝛉𝒌

𝒖𝒌 𝒖𝒍

𝚿𝒍


𝒛

𝒙

(c) O

𝒛

𝒚

𝒖𝒍

𝒙

𝟒𝟓∘

𝒖𝒌

𝚿𝒌 𝒗𝒌 𝒓𝒌

𝛉𝒌 𝚿𝒓𝒌 𝒗𝒍

𝛉𝒍

𝒓𝒍 𝚿𝒍

𝚿𝒓𝒍


198

The first two rows of Equation (9.1) are used to eliminate the non-measured degrees of

freedom of the rotor (i.e. 𝐝𝑅,𝑖 and 𝐝𝑅,𝑏) as shown in Equation (9.2)

�̅�𝐹𝐝𝐹,𝑏 = 𝐙𝐵(𝑃−1𝐙𝐵 − 𝐼)𝐝𝐹,𝑏 − 𝐙𝐵𝑃

−1𝐙𝑅,𝑏𝑖𝐙𝑅,𝑖𝑖−1 𝐟𝑢 (9.2)

where 𝑃 = 𝐙𝑅,𝑏𝑏 + 𝐙𝐵 − 𝐙𝑅,𝑏𝑖𝐙𝑅,𝑖𝑖−1 𝐙𝑅,𝑖𝑏. It is assumed that reasonably accurate numerical

models of the rotor 𝐙𝑅 and bearings 𝐙𝐵 are available, and the vibration response at bearing

pedestals 𝐝𝐹,𝑏 is measured. Hence, the only unknowns in Equation (9.2) are the unbalance

forces 𝐟𝑢 and the reduced foundation model �̅�𝐹. If the bearing pedestals are assumed to be

positioned at nodes 𝑛1 to 𝑛b, then the measured displacement at the bearing-foundation

interface at the N𝑡ℎ rotor speed can be written as

𝐝𝐹,𝑏(ω𝑁) = [𝒗𝑛1(ω𝑁) 𝒖𝑛1(ω𝑁) … 𝒗𝑛b(ω𝑁) 𝒖𝑛b(ω𝑁)]T (9.3)

where 𝒗𝑛b and 𝒖𝑛b are the displacements at the bth bearing pedestal in the vertical and

horizontal directions respectively (Figure ‎9.2).


199

Figure ‎9.2 (a) A simple schematic representation of the rig, (b) vibration measurement directions of

bearing pedestal at node 𝒌

As a matter of fact, the unbalance in rotating machines is in continuous form. This is similar

to a discrete form of unbalance, on condition that the number of balancing planes is same as

the number of active modes [171]. It is a difficult task to find an exact unbalance distribution

using the existing approaches [162]. Instead, the unbalance planes are assumed to be

positioned at nodes 𝑢𝑛1 , 𝑢𝑛2 , 𝑢𝑛3 , … , 𝑢𝑛𝑝, where the subscript 𝑝 denotes the number of

balancing planes. This is demonstrated in Figure ‎9.3.

𝒛

𝒙

𝒚

Front view Side view of 𝑭𝐛𝟐

𝒙

𝑭𝒃𝟐𝒚

𝑭𝒃𝟐𝒙

𝒛

(b)

Bearing front view Bearing side view

𝒖𝒌

𝒗𝒌 𝒓𝒌

𝟒𝟓°

(a)

𝑭𝒃𝟐𝒓

𝒚 𝒚

𝑭𝐛𝟐

A

A 𝑭𝐛𝟏

𝒛 𝒙

𝟒𝟓∘


200

Figure ‎9.3 Typical rotor mass unbalance distribution along the rotor length; (a) continuous form of rotor

mass unbalance, (b) discretized form of rotor mass unbalance

The amplitude of unbalance, 𝑢, for the 𝑖th balancing plane may be defined as

𝑢𝑛𝑖 = 𝑚𝑢𝑖𝑒𝑖 (9.4)

where 𝑚𝑢𝑖 is the unbalance mass at plane 𝑖, and 𝑒 is the distance between the unbalance mass

and the geometric centre of the rotor. In terms of matrices, the amplitude of unbalance and

the phase angles associated to the unbalance planes can be written as [𝑢𝑛1 , 𝑢𝑛2 , 𝑢𝑛3 , … , 𝑢𝑛𝑝]T

and [α𝑛1 , α𝑛2 , α𝑛3 , … , α𝑛𝑝]T

respectively. Therefore, the complex quantity of the rotor

unbalance may be expressed as


where j = √−1 , and 𝑎𝑛𝑖 and 𝑏𝑛𝑖 can be defined as

𝑎𝑛𝑖 = 𝑚𝑖 𝑒𝑖 cos(α) and 𝑏𝑛𝑖 = 𝑚𝑖 𝑒𝑖 cos (α +𝜋

2) = 𝑚𝑖 𝑒𝑖 sin(α) (9.6)

To get the unbalance in the horizontal direction, Equation (9.5) should be multiplied by −𝑗 as

follows

−𝑗(𝑢𝑛𝑖 exp(jα𝑛𝑖)) = −𝑗(𝑎𝑛𝑖 + j𝑏𝑛𝑖) = −𝑗𝑎𝑛𝑖 + 𝑏𝑛𝑖 (9.7)

𝑢(𝑥)

𝒚

𝒙

𝒛

𝑢𝑛1

𝒚

𝒙

𝒛

𝑢𝑛2 𝑢𝑛3

𝑢𝑛(𝑝−1) 𝑢𝑛𝑝

(a) (b)


201

Hence, when using two measurements per bearing pedestal, the unbalance forces in the

vertical and horizontal directions can be written in a matrix form as shown in Equation (9.8),

below:

𝐟𝑢 = ω2

{

0⋮0

𝑚1 𝑒1(𝑐𝑜𝑠(α1) + 𝑗 𝑠𝑖𝑛(α1))

𝑚1 𝑒1(−𝑗cos(α1) + sin(α1))0⋮0


𝑚𝑝 𝑒𝑝(−𝑗cos(α𝑝) + sin(α𝑝))

0⋮0 }

= ω2

{

0⋮0

𝑎𝑛1 + 𝑗𝑏𝑛1−𝑗𝑎𝑛1 + 𝑏𝑛1

0⋮0

𝑎𝑛𝑝 + 𝑗𝑏𝑛𝑝−𝑗𝑎𝑛𝑝 + 𝑏𝑛𝑝

0⋮0 }

(9.8)

Equation (9.8) may be further simplified as

𝐟𝑢 = ω2𝐓𝐠 (9.9)


and 𝐓 is the transformation matrix which is

defined such that Equations (9.8) and (9.9) are equivalent. Substituting Equation (9.9) into

Equation (9.2) gives

�̅�𝐹𝐝𝐹,𝑏 = 𝐙𝐵(𝑃−1𝐙𝐵 − 𝐼)𝐝𝐹,𝑏 −ω

2(𝐙𝐵𝑃−1𝐙𝑅,𝑏𝑖𝐙𝑅,𝑖𝑖

−1 𝐓)𝐠 (9.10)

To identify the rotor unbalance in a least-square sense, the foundation parameters are grouped

into vectors v0, v1 and v2. Assume that the dynamic stiffness matrix of the reduced model of

the foundation is written in terms of mass, damping and stiffness matrices. If there is a total

of 𝑡 measured degrees of freedom at the foundation-bearing interface, then v0, v1 and v2 can

be written as:

v0 = [�̅�𝐹,11 �̅�𝐹,12 … �̅�𝐹,𝑡𝑡], v1 = [𝑐�̅�,11 𝑐�̅�,12 … 𝑐�̅�,𝑡𝑡], v2 = [�̅�𝐹,11 �̅�𝐹,12 … �̅�𝐹,𝑡𝑡] (9.11)

v0, v1 and v2 can be collected in one matrix as shown in Equation (9.12)


202

𝐯 = [v0 v1 v2]T (9.12)

With this definition of 𝐯, there is a linear transformation such that

[�̅�𝐹]{𝐝𝐹,𝑏} = [𝐖]{𝐯} (9.13)

where 𝐖 contains the measured response terms at each frequency. For the N𝑡ℎ measured

frequency

𝐖(ω𝑁) = [W0(ω𝑁) W1(ω𝑁) W2(ω𝑁)]T (9.14)

If all elements of the foundation’s mass, damping and stiffness matrices are identified, then

W𝑘(ω𝑁) = (𝑗ω𝑁)𝑘

[ 𝐝𝐹,𝑏T (ω𝑁) 0 … 0

0 𝐝𝐹,𝑏T (ω𝑁) 0

⋮ ⋮ ⋱ ⋮0 0 … 𝐝𝐹,𝑏

T (ω𝑁)]

(9.15)

where 𝑘 = 0, 1, 2. Hence, Equation (9.10) becomes

𝐙𝐵(𝑃−1𝐙𝐵 − 𝐼)𝐝𝐹,𝑏 = ω𝑁

2 (𝐙𝐵𝑃−1𝐙𝑅,𝑏𝑖𝐙𝑅,𝑖𝑖

−1 𝐓)𝐠 + [𝐖]{𝐯} (9.16)

Let

R(ω𝑁) = ω𝑁2 𝐙𝐵𝑃

−1(ω𝑁)𝐙𝑅,𝑏𝑖(ω𝑁)𝐙𝑅,𝑖𝑖−1 (ω𝑁)𝐓 (9.17)

and

H(ω𝑁) = 𝐙𝐵(ω𝑁) (𝑃−1(ω𝑁)𝐙𝐵(ω𝑁) − 𝐼)𝐝𝐹,𝑏(ω𝑁) (9.18)

Substitution of Equations (9.17) and (9.18) into Equation (9.16) gives:

[𝐖(ω𝑁)]{𝐯} + [R(ω𝑁)]{𝐠} = [H(ω𝑁)] (9.19)


203

Equation (9.19) can be rearranged as shown in Equation (9.20)

[𝐖(ω𝑁) R(ω𝑁)] {𝐯𝐠} = [H(ω𝑁)] (9.20)

Suppose that the response is measured for frequencies starting from 1 to 𝑞 Hz. All

measurements are used at once, and one estimation of the rotor unbalance is produced. Thus,

Equation (9.20) is repeated 𝑞 times to give

[ W0(ω1) W1(ω1) W2(ω1) R(ω1)

W0(ω2) W1(ω2) W2(ω2) R(ω2)⋮ ⋮ ⋮ ⋮

W0(ω𝑞) W1(ω𝑞) W2(ω𝑞) R(ω𝑞)] {𝐯𝐠} =

[ H(ω1)

H(ω2)⋮

H(ω𝑞)] (9.21)

The system of equations in Equation (9.20) is overdetermined as the number of equations is

more than the number of unknowns. Therefore, a solution can be obtained by applying the

least-squares technique as shown in Equation (9.22)

{𝐯𝐠} = [𝐖 R]

+H (9.22)

where [𝐖 R]+ is the Moore-Penrose pseudo-inverse of [𝐖 R] [172] , and can be written as

[𝐖 R]+ = [[𝐖 R]T[𝐖 R]]−1[𝐖 R]T (9.23)

Although Equation (9.20) may be solved in a least-squares sense directly as shown in

Equation (9.22), it is not adequate when the condition number is large. Thus, more accurate

solution can be achieved by applying more numerically stable approaches such as the

singular-value decomposition (SVD) and QR factorization [173]. The large condition number

of the inverse problem (Equation (9.20)) implies that any small perturbation in the input may

result in significant errors in the output data. To solve the ill-conditioned problem, extra

conditions on the solution are needed to be imposed. Therefore, column scaling regularisation

is applied to make the problem well-conditioned [173].


204

Proposed method 9.3

The current approach attempts to enhance the earlier model-based unbalance estimation

methods by investigating the feasibility of estimating the rotor unbalance using only one

vibration sensor at a bearing pedestal. Now, the measured responses at two orthogonal

directions are replaced by the measured response at one direction. Hence, Equation (9.3) can

be replaced by the following equation

𝐝𝐹,𝑏(ω𝑁) = [𝑟𝑛1(ω𝑁)… 𝑟𝑛b(ω𝑁)]

T (9.24)

In addition, Equation (9.8) is modified as shown in Equation (9.25), below:

𝐟𝑢 = ω2

{

0⋮0

𝑚1 𝑒1(𝑐𝑜𝑠(α1) + 𝑗 𝑠𝑖𝑛(α1))0⋮0


0⋮0 }

= ω2

{

0⋮0

𝑎𝑛1 + 𝑗𝑏𝑛10⋮0

𝑎𝑛𝑝 + 𝑗𝑏𝑛𝑝0⋮0 }

(9.25)

where the unbalance forces applied in Equation (9.25) are similar to the ones used in

Equation (9.8). The rest of the equations are applied again but with making the appropriate

modifications. Although the proposed approach can be applied using any unidirectional

measurements, it is preferred to use measurements at 45 degrees to the vertical/horizontal

directions (i.e. 𝑟𝑘 in Figure 9.2(b)) in the current study. The reason is that amongst all other

directions, the measurements at this direction are likely to have the most significant content

of vibration behaviour from both orthogonal directions. The bearings used in the current

study are stiff ball bearings (i.e. anti-friction bearings), hence their acceleration is regarded

equal to rotor acceleration (vibration). Therefore, they are assumed as direct connection

between the rotor and foundation in the FE model. The cross-coupling required for the

simulation of the gyroscopic effect is ignored, because the FE model of the rotor is

constructed with sensors mounted only in single plane. However, this may not impact the

rotor unbalance estimation for small and medium size rotating machines [21].


205

Experimental rig with one balancing plane 9.4

The test apparatus displayed in Figure ‎9.4 is composed of a mild solid steel shaft (1000mm

length and 20mm diameter) flexibly coupled to a three-phase, 3000 RPM, 0.75kW electric

motor. A single balancing disc of 130mm diameter and 20mm thickness is assembled to the

shaft and positioned at the midspan between two bearings. The balancing disc includes

staggered M5 tapped holes in two different pitch diameters (i.e. 70 and 120mm). The angle

between two adjacent holes for a particular pitch diameter is 30°. The entire rotor assembly is

supported on relatively flexible supporting structure through two greased lubricated stiff ball

bearings. Each ball bearing is bolted to steel horizontal beam (530mm × 25mm × 3mm)

using two bolts and nuts. Each horizontal beam is secured atop two rectangular steel blocks

(107mm × 25mm × 25mm) that are fixed on a thick base plate (580mm × 150mm × 15mm).

A 12mm thick anti-vibration pad is placed beneath the base plate to mitigate noise and

vibration. Bearing near the motor is denoted as B1 and the other bearing as B2.


206

Figure ‎9.4 Photographs of the test rig with one balancing disc

Modal tests 9.5

A modal test has been conducted on the test rig at zero RPM. Impulse-response method [17]

was used for the modal test. The test apparatus is artificially excited by using an instrumented

impact hammer that has a sensitivity of 1.1 mV/g. The response acceleration is measured by

7 Integrated Circuit Piezoelectric (ICP) accelerometers (sensitivity of 100 mV/g) which are

distributed along the rotor. Table ‎9.1 and Figure ‎9.5 show the experimentally identified

natural frequencies and a typical measured FRF plots respectively. The corresponding mode

shapes of the identified natural frequencies are presented in Figure ‎9.6.

Flexible coupling

𝒛

𝒚

𝟑𝟎°

Tachometer

𝒛

𝒚

𝒙

Shaft

Balancing disc

Motor

Bearing B2

Bearing B1

Safety guard Control panel

Horizontal

beam


207

Table ‎9.1 Experimentally identified natural frequencies of test rig with one balancing disc at zero RPM

Mode Frequency (Hz)

1 17.09

2 29.91

3 31.13

4 58.59


vertical, (b) horizontal directions

(a)

(b)


208

Figure ‎9.6 Measured mode shapes of the rig, (a) mode 1, (b) mode 2, (c) mode 3 and (d) mode 4

(a)

(c)

(d)

(b)


209

Experiments conducted 9.6

The test apparatus is coasted-up linearly from 0 to 3000 RPM (i.e. 0 to 50 Hz) and the

acceleration responses are measured by means of 3 accelerometers that are installed at the

vertical, horizontal and 45-degree to the vertical/horizontal directions of each bearing

pedestal. The arrangement of the accelerometers on the bearing housing is shown in

Figure ‎9.7. Figure ‎9.8 illustrates the unbalance and phase angle with respect to the laser tacho

sensor. A total number of 7 machine coast-ups are conducted (i.e. one with residual

unbalance and six with different added mass unbalances at a radius of 6cm). Table ‎9.2 lists

the added unbalances used in the different runs.

Figure ‎9.7 A typical accelerometer installation at a bearing in 3 directions

𝒙, horizontal

𝒚, vertical 𝒓, radial(𝟒𝟓°)


210

Figure ‎9.8 Demonstration of the added unbalance (mass and phase angle)



run0 Residual unbalance 𝑒0

run1 3g × 6cm @ 30° = 18gcm @ 30° 𝑒0 + 𝑒1

run2 7g × 6cm @ 210° = 42gcm @ 210° 𝑒0 + 𝑒2

run3 5g × 6cm @ 270° = 30gcm @ 270° 𝑒0 + 𝑒3

run4 5g × 6cm @ 60° = 30gcm @ 60° 𝑒0 + 𝑒4

run5 3g × 6cm @ 60° = 18gcm @ 60° 𝑒0 + 𝑒5

run6 3g × 6cm @ 90° = 18gcm @ 90° 𝑒0 + 𝑒6

Owing to the fact that the rotor vibrations due to mass unbalance are synchronous to the

rotational speed, the measured vibration responses of each machine’s‎ run-up were order

tracked to obtain the 1× (i.e. one multiplied by rotating frequency) vibration component

(both amplitudes and phases). Figure ‎9.9 shows the obtained critical speeds of the machine

(i.e. around 18 and 29 Hz). The order tracked 1× vibration components in the speed range

from 420 to 2820 RPM (i.e. 7 to 45 Hz) with a spacing of 60 RPM (1 Hz) were then used for

the unbalance estimation.

𝒙

𝒚

Tachometer Laser beam

Reflective tape

Steel shaft

Disc

Added unbalance

Phase angle


211

Figure ‎9.9 Typical measured 1× displacement responses in vertical direction for the machine runs 4

and 5 at bearings (a) B1 and (b) B2

Unbalance estimation 9.7

In this section, the earlier and proposed model-based unbalance estimation approaches are

applied on the test rig described in Section 9.4. Then, a comparison between the results from

both methods is presented.

9.7.1 Part 1: application of the earlier method

For the purpose of estimating the actual added unbalance, the response of run0 is subtracted

from the response of each of the other runs as shown in Table ‎9.3. The measured acceleration

responses at the vertical and horizontal directions of each bearing pedestal for the entire run-

up speed range are used in Equation (9.21) to obtain the estimated unbalance. Table ‎9.4

shows a comparison between the estimated unbalance and the actual added unbalance for

each case.

1st critical speed

2nd

critical speed

(a)

(b)


212


Case no. Unbalance

Case I run1 − run0 𝑒1

Case II run2 − run0 𝑒2

Case III run3 − run0 𝑒3

Case IV run4 − run0 𝑒4

Case V run5 − run0 𝑒5

Case VI run6 − run0 𝑒6

Table ‎9.4 Estimated unbalance for the different scenarios using pair of orthogonal sensors (at vertical and

horizontal directions) at a bearing pedestal

Case

no.

Actual added

unbalance

Estimated

unbalance

Mass

% error

Phase

difference

Case I 3g @ 30° 3.29g @ 30.0750° 8.73 0.08°

Case II 7g @ 210° 6.08g @ 200.5° 15.19 9.5°

Case III 5g @ 270° 4.8g @ 282.77° 4.2 12.77°

Case IV 5g @ 60° 5.56g @ 50.26° 10.1 9.74°

Case V 3g @ 60° 3.17g @ 47.35° 5.24 12.65°

Case VI 3g @ 90° 2.65g @ 71.7° 13.34 18.3°

9.7.2 Part 2: application of the proposed method

The measured acceleration responses at only one direction (i.e. 45-degree direction) at a

bearing pedestal for the entire run-up speed range are used together in a single band in

Equation (9.21) to obtain the estimated unbalance. Same machine run-ups and unbalance

estimation scenarios used for earlier method (i.e. Table ‎9.3) are used for the proposed

method. Table ‎9.5 shows a comparison between the estimated unbalance and the actual added

unbalance for each case.


213

Table ‎9.5 Estimated unbalance using only one sensor at a bearing pedestal (at radial direction)

Case

no.

Actual added

unbalance

Estimated

unbalance

Mass

% error

Phase

difference

Case I 3g @ 30° 3.15g @ 36.9° 4.7 6.9°

Case II 7g @ 210° 7.81g @ 203.23° 10.42 6.77°

Case III 5g @ 270° 6.32g @ 284.46° 20.9 14.46°

Case IV 5g @ 60° 5.49g @ 57.11° 8.89 2.89°

Case V 3g @ 60° 3.14g @ 63.2° 4.46 3.2°

Case VI 3g @ 90° 3.07g @ 91.09° 2.27 1.09°

9.7.3 Comparison between the earlier and proposed methods

It can be noticed from Tables 9.4 and 9.5 that the estimated unbalances (i.e. mass and phase

angle) for both earlier and proposed methods are close to the actual added unbalances. For the

sake of easy comparison, the results in Tables 9.4 and 9.5 are also presented in the form of

grouped bar chart in Figure ‎9.10. The estimation method was also applied to the radial

vertical and radial horizontal directions individually, and the results are shown in Figure ‎9.10.

As the unbalance estimation proposed in this paper mainly relies on the accuracy of the

numerical model of the rotor, the results of the estimated unbalance can be enhanced by

improving the accuracy of the mathematical model.


214

Figure ‎9.10 Grouped bar chart of the comparison between actual and estimated unbalances by both the

earlier and proposed methods, (a) mass and (b) phase angle

Modified test rig with two balancing planes 9.8

Two modifications have been made to the experimental apparatus. The first modification is

that a second balancing disc which has similar dimensions as the first one is attached to the

shaft. The disc closer to bearing B1 is symbolised as D1 and the other disc as D2. Discs D1

and D2 are positioned at distances of 240mm and 665mm from bearing B1, respectively. The

(a)

(b)


215

second modification is replacing the horizontal steel beams with thicker ones (530mm ×

25mm × 8mm). Figure ‎9.11 displays the modified test rig.

Figure ‎9.11 Photograph of the test rig with two balancing discs

A similar modal testing procedure as in the previous test rig has been conducted for the

modified rig. The experimentally identified natural frequencies and typical measured FRF

plots are provided in Table ‎9.6 and Figure ‎9.12 respectively. The corresponding mode shapes

of the identified natural frequencies are given in Figure ‎9.13.

Table ‎9.6 Experimentally identified natural frequencies of test rig with two balancing discs at zero RPM

Mode Frequency (Hz)

1 24.41

2 31.13

3 53.1

4 84.23

𝒛

𝒚

𝒙

Bearing B2

Bearing B1

Disc D2

Disc D1

DAQ hardware


216


vertical and (b) horizontal directions

(a)

(b)


217

Figure ‎9.13 Measured mode shapes of the rig, (a) mode 1, (b) mode 2, (c) mode 3 and (d) mode 4

(a)

(b)

(c)

(d)


218

9.8.1 Experiments and unbalance estimation

The modified experimental rig is coasted-up linearly from 0 to 3000 RPM (i.e. 0 to 50 Hz),

and the vibration responses are measured using only one accelerometer at a bearing pedestal

(i.e. at 45 degree direction). The obtained critical speeds of the machine (i.e. around 23 and

32 Hz) are presented in Figure ‎9.14. A total number of 8 machine coast-ups are conducted

(i.e. one with residual unbalance and seven with different added mass unbalances at a radius

of 6cm) and seven unbalance estimation scenarios are used. The order tracked 1× vibration

components in the speed range from 420 to 2820 RPM (i.e. 7 to 45 Hz) are used together in a

single band for the unbalance estimation. Tables 9.7 and 9.8 list the added unbalances utilised

for the different runs and the cases used for the unbalance estimation, respectively. Table ‎9.9

shows a comparison between the estimated unbalance and the actual added unbalance for

each case.

Figure ‎9.14 Typical measured 1× displacement responses in horizontal direction for the machine

runs 3 and 5 at bearings (a) B1 and (b) B2

1st critical speed

2nd

critical speed

(a)

(b)

219

CH

AP

TE

R 9

MU

LT

I-PL

AN

ES

RO

TO

R U

NB

AL

AN

CE

IDE

NT

IFIC

AT

ION…


Run no. Added unbalance (gcm@

°)

Disc D1 Disc D2

run0 Residual unbalance 𝑒1,0 Residual unbalance 𝑒2,0

run1 3g × 6cm @ 330° = 18gcm @ 330° 𝑒1,0 + 𝑒1,1 5g × 6cm @ 0° = 30gcm @ 0° 𝑒2,0 + 𝑒2,1








Case no. Unbalance Disc D1 Disc D2

Case I run1 − run0 𝑒1,1 𝑒2,1

Case II run2 − run0 𝑒1,2 𝑒2,2

Case III run3 − run0 𝑒1,3 𝑒2,3

Case IV run4 − run0 𝑒1,4 𝑒2,4

Case V run5 − run0 𝑒1,5 𝑒2,5

Case VI run6 − run0 𝑒1,6 𝑒2,6

Case VII run7 − run0 𝑒1,7 𝑒2,7

220

CH

AP

TE

R 9

MU

LT

I-PL

AN

ES

RO

TO

R U

NB

AL

AN

CE

IDE

NT

IFIC

AT

ION…

Table ‎9.9 Estimated unbalance for the different scenarios using 1 sensor (at radial direction) per bearing pedestal

Case no.

Disc D1 Disc D2

Actual added

unbalance(𝐠@°)

Estimated

unbalance(𝐠@°)

Mass

%error

Phase

difference(°)

Actual added

unbalance(𝐠@°)

Estimated

unbalance(𝐠@°)

Mass

%error

Phase

difference (°)

Case I 3g @ 330° 2.83g @ 333.6° 6.04 3.6° 5g @ 0° 4.51g @ 1.8° 10.85 1.8°

Case II 7g @ 60° 6.38g @ 55.9° 9.7 4.1° 3g @ 30° 2.82g @ 27.7° 6.37 2.3°

Case III 5g @ 90° 4.98g @ 81.2° 0.38 8.8° 7g @ 330° 7.29g @ 342.7° 3.92 12.7°

Case IV 3g @ 60° 3.7g @ 57.5° 19.4 2.5° 5g @ 270° 4.07g @ 280.8° 22.97 10.8°

Case V 3g @ 330° 3.18g @ 338.4° 5.57 8.4° 7g @ 0° 6.01g @ 353.5° 16.39 6.5°

Case VI 7g @ 30° 7.29g @ 19.2° 3.99 10.8° 10g @ 0° 11.82g @ 342.4° 15.39 17.6°

Case VII 3g @ 30° 3.21g @ 21.69° 6.57 8.31° 7g @ 330° 7.57g @ 339.5° 7.52 9.5°


221

It can be clearly seen in Table ‎9.9 that the estimated rotor unbalances (i.e. mass

unbalance and phase angle) using a single vibration sensor per bearing location are of

good accuracy. Therefore, based on the results in Figure ‎9.10 and Tables 9.4, 9.5 and

9.9, it can be said that the earlier mathematical model-based unbalance estimation

approaches can be simplified and enhanced by using a single vibration sensor at a

bearing pedestal instead of two orthogonal sensors. The foundation model in the current

study was obtained at one direction (i.e. 45-degree direction).


This paper has presented a simplified and computationally efficient method for

identifying and quantifying the state of rotor unbalance in rotating machines using

single vibration sensor per bearing pedestal and single machine’s transient operation.

The proposed mathematical model-based unbalance estimation method requires a

somewhat accurate numerical model of the rotor. Both the earlier and proposed model-

based unbalance estimation approaches were applied on a test rig with one balancing

disc. A comparison between the two methods showed that the unbalance estimation

using a single accelerometer per bearing pedestal provides as accurate results as when

using two orthogonal accelerometers per bearing pedestal. The proposed method was

then applied on a test rig with two balancing discs to investigate the robustness of the

method. Based on the present experimental study, it can be concluded that the use of

one vibration sensor per bearing pedestal is a viable option for the rotor unbalance

estimation. This way, the number of vibration sensors per bearing pedestal can be

reduced by half.

The significant reduction of the number of sensors could be useful to power plants that

have large rotating machines with several bearings as it will reduce the time needed for

the complex signal processing significantly with maintaining the same accuracy. Also,

the reduction of the number of sensors by half will certainly save the cost of the

vibration instrumentation and their maintenance. Moreover, the reduction of the number

of sensors helps in reducing the probability of tripping the rotating machine due to false

signal from faulty sensor.


222

Despite that the proposed unbalance estimation method has the potential for future

industrial applications, it was applied only to experimental rig with rolling-element

bearings where the bearing acceleration was regarded as equal to rotor acceleration.

Therefore, the application of the proposed method on test rigs with fluid film bearings

that have clearance is required to fully explore the potential and validate the usefulness

of the method.

‎CHAPTER 10 CONCLUSIONS AND FUTURE WORK

223

CHAPTER 10

CONCLUSIONS AND FUTURE WORK


224

Summary of research context 10.1

Most industries use rotating machines such as power generating turbines for most of

their operations. Thus, the reliability of this category of machines is of high importance

to these industries. The dynamic conditions under which rotating machines work make

them susceptible to several anomalies of which rotor related faults are very rife.

Amongst all rotating machine malfunctions, rotor mass unbalance is known as the most

common contributor to the side effect vibration. If the rotating machine’s vibration due

to rotor mass unbalance exceeds the alarm limits, it may lead to machine failure and

possibly cause catastrophic damages. As a consequence, machine downtime and

unscheduled maintenance actions are required, which in turn have significant operating

cost implications. This emphasises the importance of keeping the vibration due to mass

unbalance in safe margins by regularly balancing rotating machines.

The influence coefficient (IC) balancing approach is commonly used in industries for

on-site balancing of rotating machines. The method is generally applied using vibration

measurements acquired at a single rotor speed. The measured vibrations at a single rotor

speed possibly do not entirely reflect the dynamics of the rotating machine and may

include a high level of noise. Therefore, applying the IC method using vibration

measurements at a single rotor speed may not provide an accurate estimation of the

rotor mass unbalance and hence results in a bad balancing. Therefore, it was proposed in

this study that the current IC balancing method can be improved by using vibration

measurements acquired at multiple rotor speeds together in a single band instead of

single speed. The application of the IC balancing method using vibration response

acquired at multiple speeds in a single band was achieved in the current research

project. The results showed that the unbalance estimation was highly enhanced when

vibration at various speeds in a single band was used. Therefore, this enhancement of

the IC balancing method could be helpful for industries as it will increase the

confidence level of estimating the rotor unbalance accurately and precisely.

Earlier published papers on the IC as well as mathematical model-based unbalance

estimation approaches have used pair of sensors arranged in orthogonal directions at a

bearing pedestal. When applying these methods on rotating machines with multiple

bearings such as gas turbine, several vibration measurements are acquired at each

bearing pedestal. This results in enormous volumes of data sets that are needed to be

processed and interpreted. Consequently, the balancing process mandates a highly


225

skilled engineer, who despite the training may still yield an erroneous subjective

diagnosis. Inherently, because of such difficulties, the balancing can be lengthy and

costly. Moreover, the use of multiple sensors at a bearing pedestal could increase the

possibility of shutting the plant down due to false signal from a faulty sensor. Therefore,

an opportunity exists to simplify the IC and model-based balancing approaches by using

only one vibration sensor at a bearing pedestal rather than the usual pair of sensors

arranged in orthogonal directions. Thus, both the IC and model-based unbalance

identification methods were applied in the current research project using single

vibration sensor per bearing pedestal. The results showed that the estimated unbalance

using a single sensor per bearing pedestal, mounted at 45-degree to vertical/horizontal

directions, provides as accurate unbalance estimation as when using two sensors per

bearing location.

Main achievements 10.2

The major accomplishments of this research work are presented below with relating

them to the aims and objectives of this research project presented at the beginning of the

thesis (i.e. Section 1.3).

Objective 1

To experimentally investigate the effectiveness of the application of the influence

coefficient balancing method using vibration measurements acquired at multiple

speeds in a single band.

Achievement and contribution 1

The application of field rotor balancing using the influence coefficient method with

vibration measurements acquired at a single rotor speed as well as multiple rotor speeds

together in a single band was investigated in the current research project. First, the

proposed method was applied on a laboratory rig with a single balancing plane (Chapter

5). Then, the same concept was utilised on laboratory apparatus with multiple balancing

planes (Chapter 6). It was shown that the inclusion of vibration measurements at

multiple speeds during rotating‎machine’s‎run-up considerably improves the rotor mass

unbalance estimation when compared to the estimation at a single speed. Therefore, it is


226

concluded that the use of vibration measurements acquired at multiple rotor speeds in a

single band will significantly increase the effectiveness of machine balancing.

Objective 2

To propose and experimentally examine the effectiveness of the application of the

influence coefficient balancing method using vibration measurements at multiple

machine speeds from only one vibration sensor per bearing pedestal.


Comparison between the application of the influence coefficient balancing method

using two orthogonal vibration sensors per bearing location and only one vibration

sensor per bearing location was presented in Chapter 7. The method has been applied on

test rigs with single balancing plane as well as multiple balancing planes. To show the

robustness of the method, different unbalance estimation scenarios have been presented.

It has been found that the application of the influence coefficient method using only one

vibration sensor at a bearing pedestal provides as accurate unbalance estimation as when

using pair of vibration sensors per bearing pedestal. Therefore, the current application of

the influence coefficient balancing method can be enhanced by using only one vibration

sensor at a bearing pedestal, rather than the usual pair of sensors arranged in orthogonal

directions.

Objective 3

To develop a model-based method for identifying rotor mass unbalance using

single vibration sensor per bearing pedestal and single machine’s transient

operation (run-up/run-down).


In the present study, the concept of the earlier model-based rotor balancing method was

applied, but instead of using two vibration sensors per bearing pedestal, a single sensor

was used on each pedestal. First, the proposed method has been implemented to a

numerical model that was constructed using FE method. The unbalance estimation

method has been applied using several unbalance configurations. The unbalance has

been identified accurately for all configurations as presented in Chapter 8. Then, the


227

method has been validated experimentally on a test rig with two different configurations

(i.e. one with a single balancing plane and the other with multiple balancing planes) as

presented in Chapter 9. Several unbalance estimation scenarios have been used to

investigate the robustness of the method. The results show that the estimated unbalances

using single vibration sensor per bearing pedestal are relatively close to the estimated

unbalances using two vibration sensors per bearing pedestal. This indicates the

potentials of the proposed method for practical applications.

Overall conclusion 10.3

The application of two different balancing approaches of rotating machines (i.e.

influence coefficient and mathematical model-based balancing methods) with reduced

number of sensors was proposed in the current study. For both methods, the actual

added mass unbalances of different scenarios were estimated accurately using single

vibration sensor per bearing pedestal. Moreover, the influence coefficient balancing

method was applied using measured vibration response at multiple speeds in a single

band instead of single speed. The results showed that the application of the influence

coefficient balancing method using vibration data acquired at multiple speeds in a single

band provides more reliable unbalance estimations.

Future work 10.4

In order to utilise the full potential of the proposed balancing approaches as applicable

methods, the following future work is required

1. Performance testing of the proposed balancing methods on rotating machines with

fluid filled journal bearings.

2. Foundation should be flexible in both vertical and horizontal directions, so more

critical speeds present in the transient operation of the rotating machine.

3. Applying the proposed balancing methods on test rigs with multiple bearings as well

as multiple balancing planes.

4. Trailing the proposed methods with data from real machines in industries.

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http://www.sciencedirect.com/science/article/pii/S0094114X1400007X



APPENDICES

239

APPENDICES

APPENDICES

240

Appendix A.

A MATLAB program is written to plot the time waveform corresponding to the signal

in Equation (4.44) and its vibration spectrum

%Equation (4.44) clc clear all close all N=2^14; fs=5000; dt = 1/fs; TT= N*dt; T = 0: dt :TT ; f= 60; w=2*pi*f; x = 2*sin(0.5*w*T)+2*sin(w*T)+2*sin(2*w*T)+2*sin(3*w*T)+2*sin(4*w*T); fontsize = 14; hFig = figure(1); clf plot(T,x) set(hFig, 'Position', [200 200 1000 400]) % set(hFig, 'Position', [how far from centre to right how far from centre to right WIDTH HEIGHT]) set(gca,'FontSize',fontsize,'Fontname','times new roman'); xlabel('Time (sec)','FontSize',fontsize,'Fontname','times new roman'); ylabel('Displacement (mm)','FontSize',fontsize,'Fontname','times new roman'); set(gca,'xtick',0:0.01:0.100) set(gca,'ytick',-8:2:8) axis([0, 0.100,-8,8]) %%% uu = fft(x); u=abs(uu(1:(N)/2))/(N/2); df = 1/TT; fnyquist = fs/2; f=0:df:fnyquist-(fs/(N)); hFig = figure(2); clf plot(f,u) set(hFig, 'Position', [200 200 1000 400]) set(gca,'FontSize',fontsize,'Fontname','times new roman'); xlabel(' Frequency (Hz)','FontSize',fontsize,'Fontname','times new roman'); ylabel('Displacement (mm)','FontSize',fontsize,'Fontname','times new roman'); set(gca,'xtick',0:1800/60:18000/60) set(gca,'ytick',0:0.2:2) axis([0, 18000/60,0,2])

Enhancement of Field Balancing Methods in Rotating Machines

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