Copyright

by

Ji-Hoon Choi

2006

The Dissertation Committee for Ji-Hoon Choi Certifies that this is the approved

version of the following dissertation:

MODEL BASED DIAGNOSTICS OF MOTOR AND PUMPS

Committee:

Michael D. Bryant, Supervisor

Benito Fernandez-Rodriguez

Mircea D. Driga

Gustavo de Veciana

Eric P. Fahrenthold

MODEL BASED DIAGNOSTICS OF MOTOR AND PUMPS

by

Ji-Hoon Choi, M.S., B.S.

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

The University of Texas at Austin

December 2006

Dedication

To my family for their love, support, and encouragement

v

Acknowledgements

I wish to express the sincere gratitude to my advisor, Dr. Michael D. Bryant, who

supplied encouragement, guidance, technical, and financial support through this thesis.

None of this work could have taken place without his engineering experience and insight.

I would also like to thank my committee members, Dr. Benito Fernandez-Rodriguez, Dr.

Eric P. Fahrenthold, Dr. Mircea D. Driga, and Dr. Gustavo de Veciana for their interests

in my work and carefully reviewing this dissertation. Additional appreciation also must

go to all of my friends who always give me help and encouragement. Finally, I am

always indebted to my family for the love and patience they show me.

vi

Model Based Diagnostics of Motor and Pumps

Publication No._____________

Ji-Hoon Choi, Ph.D.

The University of Texas at Austin, 2006

Supervisor: Michael D. Bryant

The purpose of machine fault diagnosis is to 1) detect, 2) identify, and 3) predict

components which are more likely to fail. This study describes a fault diagnosis method

that utilizes models, parameter estimation, and Shannon’s communication theory.

A centrifugal pump driven by a squirrel cage induction motor is selected as an

objective system to be monitored using the proposed diagnostic method. Separate bond

graph models for a motor and a pump are combined to emulate dynamics of a motor-

pump system. A test-bed was built and parameters in the model are “tuned” by

comparing simulations to sensor measured data, and altering parameters until simulations

agree with data. Degradations in different components are induced. As a fault progresses,

measured signals change, and for simulations to mimic measurements, parameters must

change. Inspecting deviations of parameters from their nominal values allows detection

and isolation of faults since parameters of the model have direct one to one

correspondence to components in the physical system.

An analogy was made between a machine and a communication channel, yielding

a “machine communications channel” to estimate severity of faults. Design of

vii

communications systems is aided by theorems of Shannon, which establish minimum

signal to noise ratios for acceptable transmission and reception. The transmitter activates

a communication channel with an input signal. Noise in the channel, along with

component kinematics and dynamics, alters the signal. When the channel operates

properly, the signal is “received” at output within tolerances.

Faults disrupt functionality, and change the response. Faults generate “noise”, the

difference between an actual signal and the desired signal. Unless the signal to noise ratio

is kept sufficiently high, the receiver cannot reconstruct the signal within a desired

tolerance, and the channel malfunctions. In terms of a machine channel, the machine

cannot perform its task within tolerance. Shannon’s theory applied to machinery can

establish performance limits, and develop failure criteria to assess functionality of new or

degraded machinery.

In this study, signals from healthy and faulty systems, and the difference between

them, noise, become important diagnostic tools to assess severity of faults. Fourier

transform of these signals give spectral densities, needed to estimate channel capacities

and information rates. Shannon’s theorems of communication theory assess channel

capacity, the maximum amount of information a machine (channel) can successfully

transmit and receive. If this is less than the rate of information needed to perform a given

job, Shannon’s theorems predict machine malfunction.

Faults such as a damaged stator circuit in a motor will be introduced into the test

equipment built for this study and diagnosed by the proposed method. We will explain

the motor-pump bond graph model, and then present results of fault diagnoses via

experiments, simulations, parameter estimation, and fault severity assessment.

viii

Table of Contents

List of Tables .......................................................................................................... x

List of Figures ........................................................................................................ xi

Chapter 1: Introduction ........................................................................................ 1 1.1 Motivation and Objectives....................................................................... 1 1.2 Overview of Diagnostic Methods ............................................................ 2

1.2.1 Model Free Techniques................................................................ 3 1.2.2 Model Based Techniques............................................................. 5

1.3 Approach.................................................................................................. 8 1.3.1 Construction of Detailed Model Using Bond Graphs.................. 9 1.3.2 Parameter Tuning....................................................................... 10 1.3.3 Applying Information Theory for Assessing Fault Severity...... 11

1.4 Fundamentals for Motor-Pump Modeling ............................................. 13 1.4.1 Squirrel Cage Induction Motor .................................................. 13 1.4.2 Centrifugal Pump....................................................................... 15 1.4.3 Bond Graphs .............................................................................. 19

1.5 Review of Shannon’s Communication Theorem................................... 20 1.5.1 Communication Channel ........................................................... 20 1.5.2 Shannon’s Information Theory .................................................. 21 1.5.3 Channel Capacity of an Analog Channel................................... 23

1.6 Outline.................................................................................................... 26

Chapter 2: Modeling of Systems........................................................................ 28 2.1 Squirrel Cage Induction Motor Model................................................... 28 2.2 Centrifugal Pump Model........................................................................ 32

Chapter 3: Parameter Tuning ............................................................................. 34 3.1 Experimental Setup................................................................................ 34 3.2 Fault Detection and Parameter Tuning .................................................. 34

3.2.1 Fault in a Stator Circuit.............................................................. 37

ix

3.2.2 Closing Valve at Outlet Pipe...................................................... 42 3.2.3 Bearing Contaminated with dirt................................................. 44

Chapter 4: Fault Evaluation by Information Theory.......................................... 46 4.1 Analogy of Machine to Communication Channel ................................. 46 4.2 Application of Shannon’s Theorem....................................................... 48

Chapter 5: Centrifugal Pump Model with the Interaction between Volute and Impeller ........................................................................................................ 52

5.1 Flow between Impeller and Volute Tongue.................................. 52 5.2 Interaction between Volute and Impeller...................................... 55

Chapter 6: Conclusion and Future Work ........................................................... 61 6.1 Summary and Conclusion...................................................................... 61 6.2 Suggested Future Work.......................................................................... 62

Appendices............................................................................................................ 64 A. Power Transfer between Motor and Pump ............................................. 64 B. Forces Exerted on Impeller of Centrifugal Pump ................................... 67

Glossary ................................................................................................................ 68 Chapter 1 and 4 ............................................................................................ 68 Chapter 2 , 3 and 5 ....................................................................................... 69

Bond graphs ........................................................................................ 69 Induction Motor .................................................................................. 69 Centrifugal Pump................................................................................ 70

Bibliography ......................................................................................................... 73

Vita ...................................................................................................................... 78

x

List of Tables

Table 3.1 Parameters of a motor-pump................................................................. 36

Table 3.2 Sensitivities of system responses.......................................................... 41

Table 3.3 Parameters tuning data.......................................................................... 41

Table 4.1 Sensitivity of information rate R to tolerance α ................................... 48

xi

List of Figures

Figure 1.1 Structure of a diagnosis system [Frank, 2000] ...................................... 6

Figure 1.2 Fault diagnosis system using bank of estimators [Frank, 1990] ........... 7

Figure 1.3 Proposed diagnosis system. ................................................................... 9

Figure 1.4 Output torque caused by a cracked gear tooth..................................... 11

Figure 1.5 Cross sectional view of squirrel cage induction motor. ...................... 14

Figure 1.6 Cross section of a centrifugal pump and flow path (dotted curves) .... 16

Figure 1.7 General components of a centrifugal pump......................................... 16

Figure 1.8 Equivalent bond graphs for a system................................................... 20

Figure 1.9 Shannon and Weaver model of a communication channel [Shannon, 1948].

.......................................................................................................... 21

Figure 1.10 (a) Sampling of a function of bandwidth Bω , (b) Distance of a point from

origin (c) Transmitted and received signals in 2 BTω dimensional signal

space (Figures were reproduced from [Raisbeck, 1965]) ................ 25

Figure 2.1 Structure of tested motor-pump........................................................... 31

Figure 2.2 Kim and Bryant [2000]’s bond graph of an induction motor.............. 31

Figure 2.3 Tanaka et al. [2000]’s bond graph of a centrifugal pump system....... 33

Figure 3.1 Test system setup................................................................................. 35

Figure 3.2 Currents in healthy condition and with damaged stator circuit........... 39

Figure 3.3 Magnified view of current (A) in Figure 3.2 with tuned response after

adjusting stator coil resistances........................................................ 39

Figure 3.4 Measured (dotted lines) and tuned (solid lines) rotational velocity by stator

coil resistances (upper) and by motor inductances (bottom)............ 40

Figure 3.5 Tuned pressures ................................................................................... 40

xii

Figure 3.6 Tuned pressures by hydraulic loss at outlet pipe, Rout ......................... 43

Figure 3.7 Flow volume rates by hydraulic loss at outlet pipe, Rout ..................... 43

Figure 3.8 Rotational velocity affected by contaminated bearing ........................ 44

Figure 3.9 Frequency analysis of rotational velocities in Figure 3.8.................... 45

Figure 3.10 Effect of tuning bearing resistance Rbr on simulated rotational velocity45

Figure 4.1 (a) Channel capacity vs. added resistances (2.5, 4.5, 6 Ω) in stator circuit

(b) Normalized channel capacities................................................... 50

Figure 4.2 Power spectra of current ia to calculate channel capacities (A), (B), and (C)

in Figure 4.1 ..................................................................................... 50

Figure 4.3 Current ia to calculate channel capacities (A), (B), and (C) in Figure 4.1

.......................................................................................................... 51

Figure 4.4 (a) Channel capacity vs. valve angle (b) Normalized channel capacities

.......................................................................................................... 51

Figure 5.1 Flows in a pump system [Tanaka et al., 2000].................................... 54

Figure 5.2 Flows in a pump .................................................................................. 54

Figure 5.3 Continuities of flows with an equivalent bond graph representation .. 55

Figure 5.4 Bond graph model of the pump system with clearance flow .............. 55

Figure 5.5 Pump geometry.................................................................................... 56

Figure 5.6 Plots from equations (5.13) and (5.14), (a) vs. from Iversen [1960], (b)59

Figure 5.7 Updated pump model with the interaction between volute and impeller

using rotor bending sub-model and volute pressure distribution..... 60

Figure A.1 Control volume and flow velocities of pump impeller....................... 66

Figure A.2 Modulated gyrator for Eulerian turbomachine ................................... 66

1

Chapter 1: Introduction

1.1 MOTIVATION AND OBJECTIVES

Maintenance costs of machines can accrue to purchase prices within a year of

operation; downtime losses can far exceed this in minutes [Intel, 1991]. Small reductions

in life-cycle maintenance costs can yield substantial savings. In the case of safety critical

systems such as airplanes and nuclear power plants, the failure of even a small

component can cause miseries [Kinnaert, 2003]. Critical towards avoiding failures are

effective machine diagnostic and prognostic methods.

To achieve these aims, machine degradation analysis has been developed to

predict and identify components about to fail, and to detect failures before catastrophes

[Isermann, 1997]. Most industry tools are signal-based and key on very specific features

of waveforms or spectra. However, since designs, manufacture, process dynamics, and

operating conditions vary with machines and changes over time, these features vary

markedly, making these tools unreliable [Isermann and Ballé, 1997].

The purpose of this research is to develop a fault diagnosis method that can

estimate system condition and identify components about to fail. The developed method

is applied to an induction motor and a centrifugal pump. In North America, three-phase

induction motors consume between 40~50% of all generated electrical capacity and most

of their applications involve pumps and fans in processes of industries for heating,

cooling, pumping, conveyors, etc [Durocher et. al., 2004]. Centrifugal pumps are most

popular among pumps including rotary, reciprocating, and diaphragm pumps. Largest

customers for centrifugal pumps are municipal wastewater plants, municipal water

treatment (drinking water), chemical plants, refining industries, pharmaceutical

2

industries, etc. Early faults detection in motors and pumps can increase reliability, safety,

and life-cycles of overall systems, resulting in significant reduction of energy and costs to

operate and maintain processes of industries, considering the prevalence of motors and

pumps in industry [Hellman, 2002].

1.2 OVERVIEW OF DIAGNOSTIC METHODS

The procedure of fault diagnosis can be summarized with the following steps

[Isermann and Ballé, 1997]:

• Fault detection: Determination of the occurrence of a fault in a system and the

time of detection.

• Fault isolation: Determination of the location of the fault.

• Fault identification: Determination of the type or nature of the fault.

• Fault analysis: Analysis including the size (severity) and urgency of corrective

action.

• Fault accommodation: The reconfiguration of the system using healthy

components (maintenance) or changing operating conditions (process

optimization) based on risk analysis.

According to the range of performance, fault diagnosis methods are classified as FD (for

Fault Detection) or FDI (for Fault Detection and Isolation) or FDIA (for Fault Detection,

Isolation, Identification, and Analysis) method [Frank et al., 2000].

Focusing on the reduction of lifetime costs, well-developed fault diagnosis

methods give the following benefits.

• Reduction of costs due to production outages

• Reduction of operating costs such as energy consumption, service, and

maintenance

3

• Reduction of related acquisition costs from increased lifetime

• Increased safety for human, environment, and machine

Fault detection and diagnosis techniques have been classified into two distinctive

groups according to the use of mathematical model of a system; model free (signal based)

and model based techniques.

1.2.1 Model Free Techniques

FDI methods which do not use mathematical models of systems are classified as

model free technique. Their simplicity and relative ease of implementation have made

them popular throughout almost every industry.

Limit checking: Due to the simplicity and reliability in steady state, the methods of

tracking changes on measurable inputs and outputs signals of a dynamic system have

been popular [Isermann, 1997]. Signals from systems are compared against fixed

thresholds. A fault is declared when a signal exceeds a threshold value. While these

methods are simple and straightforward, the limits are valid only if the system operates

approximately in a steady state. Because most fixed thresholds don’t consider the system

dynamics and control action, rapid change of operating point in normal range enervates

these methods. Loose tolerance may be insensitive to a fault since the alarm waits until a

threshold is violated. On the other hand, tight tolerance may lead to many false alarms

[Isermann, 1997]. Therefore these methods only allow the detection of relatively large

deviations from normal operating conditions and not incipient changes [Kinnaert, 2003].

Consequently enough time for counteractions such as other operations, reconfiguration,

maintenance or repair may not be guaranteed [Isermann, 1997]. Also these methods can

4

trigger a confusing multitude of alarms and make isolation almost impossible because

even a single component fault may affect many plant variables [Gertler, 1998].

Physical redundancy: In this approach, multiple sensors are installed to measure the same

physical quantity, making it possible to identify sensor fault by checking any severe

inconsistency between the measurements. At least three sensors are needed to form a

voting scheme for the isolation of faulty sensors. Hence the extra cost and extra weight of

redundant hardware make it difficult to apply this method in some occasion [Gertler,

1998].

Special sensors: These sensors are installed explicitly for detection and diagnosis. They

are either limit sensors which perform limit checking in hardware from measured signals

or other special sensors which measure some fault-indicating physical quantity, such as

sound, vibration, elongation, etc [Gertler, 1998].

Spectrum analysis: In this approach, a fault is detected and identified in frequency

domain because most plant variables exhibit a distinctive frequency spectrum under

normal condition. Any deviation from normal spectrum can be a signature of

abnormality. Certain types of faults can be isolated due to their own characteristic

signature in the spectrum facilitating fault isolation [Gertler, 1998]. However this method

is sensitive to external influences, causing overlapping signatures which cannot be

identified from actual fault.

Logic reasoning: This method is based on characteristic behaviors of a system under

consideration. Comparing the current behavior against previously stored behavior of a

5

system detects abnormalities. The simplest techniques consist of trees of logical rules of

the “IF (symptom AND symptom) THEN (conclusion)” type. Each conclusion can, in-

turn, serve as a symptom in the next rule, until the final conclusion is reached. These

methods are usually complementary to the methods outlined above [Gertler, 1998].

1.2.2 Model Based Techniques

To overcome the deficiencies of model free methods, the need for model based

methods has been gathering strength. Early detection of faults by means of physical

models consists of computing the state of fault-relevant variables from a sufficient base

of acquired data according to only physical relationships [Hellman, 2002]. It has been

shown in many research efforts that the early detection can be achieved by gathering

more information by using the relationship between the quantities in the form of

mathematical models [Isermann, 1997]. The well developed model based techniques

provide such advantages [Isermann, 1997] as

• Early detection of small faults with abrupt change of incipient time behavior

• Diagnosis of faults in the actuator, process components, or sensors at the same

time

• Detection of faults in closed loops

• Supervision of process in transient states.

Figure 1.1 [Frank, 2000] shows typical diagnosis systems: residual generation and

residual evaluation. The process model, which simulates fault free state of the process,

estimates the measured process variables using the same process inputs to the physical

system, and compares the estimated one with the measured one to generate difference

between them. This difference is called residual. If there is no fault, modeling

6

uncertainty, and measurement noise, there will be no deviation from zero. The deviation

of residuals from zero may suggest the existence of possible fault(s).

After the generation of residuals, decision logic determines whether the residuals

are within a certain tolerance of the normal value, and decides which are the most

degraded components in the system [Isermann, 1984; Kinnaert, 2003]. If necessary,

residuals will be further processed before decision to distinguish faults from unknown

inputs, model uncertainty, etc.

Residual generation

Process

Processmodel

Residualprocessing

Decisionlogic

Residual evaluation

Model based fault diagnosis system

OutputInput

Residual Knowledge of faults

-

Figure 1.1 Structure of a diagnosis system [Frank, 2000]

Based on a wide variety of published literature by experts in the field on the use

of residual generation for fault diagnostics, three methods have been generally identified

for residual generation, which are state estimators, parity equations, and parameter

estimation.

State estimators: The underlying idea used in this approach involves the use of observers

or filters to estimate the outputs of the system from measurements. The weighted output

estimation error generated by observers or filters are used as residuals [Patton and Chen,

1993]. Representative schemes are Luenberger observers in a deterministic case [Duan

7

and Patton, 1998] and Kalman filters [Grewal and Andrews, 1993; Zarchan and Musoff,

2000] in a stochastic case. For fault diagnosis, bank of filters (or observers) are structured

as in Figure 1.2, where each of filters is readily tuned to a particular system condition.

The residuals from bank of estimators are further processed to isolate and identify faults.

Actual system

Estimator 2

Estimator 1

Estimator n

Decis

ion logic

Alarms

Input Output

Fault Disturbance

Figure 1.2 Fault diagnosis system using bank of estimators [Frank, 1990]

Parity equation: This approach checks the consistency between outputs and inputs of the

mathematical relationships fabricated from a system model. Equations from the model

have to be manipulated to set input-output mathematical relationships, subject to a linear

dynamic transformation. The transformed residuals serve for fault detection and isolation

[Gertler, 1998]. This approach offers design freedom in that the transformation can be

set, in accordance with the limits of causality and stability. A detailed description of the

parity equation approach is provided in [Chow and Wilsky, 1984] and [Gertler, 1998].

Parameter estimation: In this method, parameters in a model are adjusted until the model

responses closely mimic corresponding measurements. Hence this method has been

8

considered as a natural approach to the detection and identification of parametric faults.

A nominal parameter set is first obtained from fault-free situation. Then continuous

monitoring and parameter estimation follow to detect changes in these parameters.

Deviations from the nominal value of parameters serve as a source of detection and

isolation of faults. The residual in the parameter estimation approach is defined as the

parameter estimation error. Isermann has proposed several fault diagnostic schemes

based on parameter estimation techniques [Isermann, 1993].

1.3 APPROACH

Figure 1.3 represents the proposed approach according to the structure of

diagnosis system introduced in Figure 1.1. Detailed models with direct correspondence

between components in the machine and elements in the model are formulated using

bond graphs technique [Kim and Bryant, 2000]. Some of dynamic features of the

machine are collected using sensors to generate residuals, the difference between

simulation from the model and measurement from experiment. To find out the cause of

the difference, selected candidate parameters in the model are tuned to match simulation

with measurement. Other than random tuning of parameters, such numerical methods as

least square, neural networks, genetic algorithm, and others, can be applied to automate

this matching. Then, theorems of Shannon’s information theory are applied to the

residuals [Bryant, 1998], the difference between healthy and degraded signal to evaluate

the severity of the fault. If the residuals exceed given tolerances, Shannon’s theorems

declare a fault and assess the relative severity of the fault.

The approach in Figure 1.3 will be applied to a squirrel cage induction motor and

centrifugal pump to show the validity of the method. Compared to direct current motors,

squirrel cage induction motors are less expensive and generally require less maintenance

9

[Robert, 1987]. Centrifugal pumps are dominant in the pump industry, because of higher

reliability and lower maintenance cost from fewer moving parts [Volk, 1996]. Details on

each step proposed follow in the next sections.

Residual generation

Process

Bond graphsmodel

Fault identification: Parameter tuning

Fault severity: Shannon’s information theorems

Model based fault diagnosis system

OutputInput

Residual Knowledge of faults

-

Figure 1.3 Proposed diagnosis system

1.3.1 Construction of Detailed Model Using Bond Graphs

Important to model based fault diagnosis are exact and detailed models which

describe a machine’s behavior, including functional condition [Frank et al., 2000]. To test

and diagnose a system, the model should:

• Exhibit a direct one to one correspondence between elements of the model and

components/items in the physical system.

• Incorporate almost of all known effects of the device into the model, including

faults.

• Clearly define how the machine should behave, creating a reference of good

health for diagnostics.

This study will utilize bond graphs introduced in the late 1950’s by Henry M.

Paynter [1960]. This approach allows study of the structure or interconnection of a

system model, which is a direct reflection of the physics. The nature of parts of the model

and the manner in which the parts interact is represented by a graphical format. Bond

10

graphs can describe the dynamics of any physical system: mechanical, electrical, fluidic,

thermodynamic, etc. Bond graphs are maps of how and where power flows through, and

energy is stored in, a physical system. Bond graphs are similar to circuit analysis

techniques, applying Kirchoff like conservation laws to balance the physical effects

generated by sources, resistances, capacitances, inertances, transformers, and other

elements. Bond graphs are also modular: an overall system model can be created by

linking together models of individual components or sub-systems. For example, a model

for a combined system such as a pump with a motor can be simply realized by just

linking appropriate bonds in the motor model to matching bonds in the pump model,

developed separately. Refinement of the model (where is needed.) is also very easily

accomplished. State equations, mathematical description of a physical system can then be

extracted from the bond graph, and used for diagnostics.

1.3.2 Parameter Tuning

Once the model of a physical system is constructed, parameters in the model have

to be identified. Some parameters can be directly measured or calculated, others

approximated through experience, or blind assumption. Given data measured from the

real machine, parameters are adjusted (tuned) until simulations closely mimic

measurements. For a healthy system, this produces a set of nominal parameter values that

describes a fault-free situation. As a fault progresses, measured signals change, and for

simulations to mimic measurements, parameters must change. Deviations from the

nominal values of parameters can allow detection, isolation, and assessment of faults,

since parameters have a direct correspondence with specific components (and faults)

[Gertler, 1998]. Hence, degradation of a system is defined as unintended change of

parameter(s).

11

For example, if a crack appears and propagates in the root of a gear tooth (Figure

1.4), the bending compliance of that tooth must increase [Howard, 2001]. The increased

compliance perturbs the torque transmitted across gears, as shown in the upper curves by

the actual (solid) and ideal (dashed) curves. The difference between actual and ideal is

influence of the fault, shown as the bottom curve. Changing (or tuning) the compliance

parameter value of that tooth in the gearbox model will simulate this. The initial

compliance can be estimated analytically from geometry and material properties of a

tooth, or experimentally by adjusting the compliance parameter in the model, to cause

simulations to match sensor data.

In this study, defects will be intentionally introduced into physical systems to

create and test degraded cases. The origin of the artificial fault will be located and

reconfirmed by parameter tuning. By tuning a model, and then following the progressive

changes of parameters according to data from a degraded system, faults (or residuals) can

be sensed and tracked. To reduce computational cost, parameters frequently suspected as

sources of faults were pre-selected and sensitivity of the measured states to changes in

selected parameters were investigated via simulations.

Gear

Crack

Pinion actual Ideal

0

Noise/error/difference

Time

Torque

Figure 1.4 Output torque caused by a cracked gear tooth.

1.3.3 Applying Information Theory for Assessing Fault Severity

After parameter tuning, states (simulations results from a model) of a system in

health and sickness are used to evaluate the degree of system availability. In this research,

12

an analogy between a system and a communications channel [Bryant, 1998] is

constructed to diagnose severity of faults and concomitant effects on the system. In a

communication system, a transmitter sends a message with information over the

communication channel. Noise and effects of changed dynamics inherent in the channel

distort the signal, making it difficult for the receiver to reconstruct the original message.

Design of communications systems is aided by powerful theorems of Shannon, which

establish minimum signal to noise ratios for error free transmission. Unless the signal to

noise ratio is kept sufficiently high, the receiver cannot resolve the signal’s message

within acceptable tolerances, and the communication channel fails to perform its function

[Shannon, 1948]. If a machine treated as a communications channel and faults considered

as noise, then system health can be diagnosed with Shannon’s theorems to predict

impending functional failure [Bryant, 1998].

Shannon’s theorems [Shannon, 1948], appraise (1) the channel capacity C, the

maximum rate information (in bits per second) can be successfully sent over a channel

under existence of noise, and (2) the average rate of information R that must be sent to

transmit and successfully receive a given message. It can be restated that C characterizes

a machine’s condition, and R characterizes the load applied on the machine, in the

previous analogy. Both R and C are estimated using the power contained in transmitted

signals and noises [Stremler, 1982]. If R ≤ C, the information will be received intact,

otherwise not. If a communications design obeys this, it works; otherwise, not [Shannon,

1948]. Therefore it is defined in this research that if R ≤ C, the machine can perform its

task within machine designer’s intent (tolerance α), otherwise not. In analogy to

structures, C can be viewed as ‘allowable strength of a system’, R as ‘applied load to the

system’. Then the inequality, R ≤ C has the meaning of ‘applied load ≤ allowable load’.

More details about Shannon’s theorems are introduced in Section 1.5.

13

1.4 FUNDAMENTALS FOR MOTOR-PUMP MODELING

This section provides background information on selected systems (squirrel cage

induction motor and centrifugal pump) and applied modeling method (bond graphs) in

this study

1.4.1 Squirrel Cage Induction Motor

Electric motors convert electric energy into rotary motion. Among various electric

motors, three-phase induction motors are workhorses of industry for widespread use:

heating, cooling, refrigeration, pumping, conveyors, etc [Devaney and Eren, 2004].

Simple and rugged construction, high reliability and efficiency, easy maintenance, and

low costs made them popular. More than 90% of all motors in industry worldwide are AC

induction motors [Peltola, 2002].

An induction motor has two major sub systems: a rotating rotor and a static stator.

Induction machines can have a wound rotor, or a squirrel cage rotor. Wound rotor

induction machines insert resistance between slip rings, making possible high starting

torques with moderate starting currents, smooth accelerations under heavy load, less

heating during starting, and adjustable speed [Pansini, 1989]. Disadvantages include

higher purchase and maintenance costs, and less ruggedness than squirrel cage types.

Widely used squirrel cage induction motors are more simple, rugged, and

inexpensive. The squirrel cage rotor is a structure of steel core laminations mounted on a

shaft, solid bars of conducting material in the rotor slots, end rings, and usually a fan. In

large machines, the rotor bars may be of copper alloy, driven into the slots and brazed to

the end rings. Rotors of up to 50cm diameter usually have die-cast aluminum bars. The

14

core laminations for such rotors are stacked in a mold, which is then filled with molten

aluminum. In this single economical process, the rotor bars, end rings and cooling fan

blades are cast at the same time [McPherson, 1981]. Figure 1.5 is a schematic of a

squirrel cage induction motor.

When energized by an AC supply voltage, the stator coils form a rotating

magnetic field that cuts through the rotor, inducing a voltage in the rotor bars. Resulting

currents in these bars induce a secondary magnetic field in the rotor, which attempts to

align with the stator magnetic field. However, because the stator magnetic field rotates,

the rotor field, and consequently, the rotor, chase the stator field, following slightly

behind. This is motor action [Lawrie, 1987]. The induction motor speed depends on the

speed of the rotating stator field.

Motor failures can be classified into four areas [Durocher and Feldmeier, 2004]:

• Bearing failure: 41%

• Stator turn faults: 37%

• Rotor bar failure: 10%

• Other: 12%

shaft

stator

rotor bar

bearing

end ring fan blade

bearing

rotor

Figure 1.5 Cross sectional view of squirrel cage induction motor.

15

1.4.2 Centrifugal Pump

Centrifugal pumps convert energy of a prime mover (an electric motor or turbine)

into kinetic (or velocity) energy and then into pressure energy of a pumped fluid. Energy

changes in a pump occur by virtue of the impeller and the volute (or diffuser). Figure 1.6

which depicts a side cross section of a centrifugal pump, indicates movement of the fluid.

The rotating impeller converts rotational energy into kinetic energy of fluid. The

stationary volute converts the kinetic energy into pressure. The process fluid (a liquid)

enters the suction nozzle, and proceeds into the center of the impeller. When the impeller

rotates, centrifugal forces push fluid sitting in the flow paths between the vanes outward.

As fluid leaves the eye (or center) of the impeller, a low-pressure area is created,

causing more liquid to flow toward the inlet from the suction nozzle. Because the

impeller blades are curved, the fluid is pushed in both tangential and radial directions by

the centrifugal forces.

Centrifugal force imparts kinetic energy to the fluid. The amount of energy is

proportional to the translational velocity at the edge or vane tip of the rotating impeller.

An impeller with greater rotating speed or blade size will impart greater energy to the

fluid.

The kinetic energy of fluid flowing from an impeller is harnessed by the volute

(casing). The volute, a curved funnel with increasing area to the discharge port, catches

and slows liquid from the impeller. As cross-section area increases along the volute, fluid

speed reduces and pressure increases. In the discharge nozzle, the fluid further

decelerates, and pressure increases.

The head (pressure in terms of height of liquid) developed can be related to the

kinetic energy at the periphery of the impeller. A pump should induce flow, and not

create pressure. Pressure indicates the amount of resistance to flow.

16

As a result of the dynamic principle of energy transfer and of internal interactions

(especially between diffuser and impeller), the laws governing the energy transformation

in centrifugal pumps are more complex than for positive displacement pumps [Bachus

and Custodio, 2003; Tuzson, 2000].

DischargeImpeller

Volute

Volutecasing

Suctioneye

Impellerrotation

Vane

Flow path(dotted curves)

Figure 1.6 Cross section of a centrifugal pump and flow path (dotted curves)

Discharge

nozzle

Suctionnozzle

Impeller Seal

Volute

Shaft

Bearing

Coupling

Wearring

Figure 1.7 General components of a centrifugal pump

17

Other than impeller and volute, which are crucial for centrifugal pumps, general

components in Figure 1.7 are summarized:

Volute casing: A volute casing helps balance the hydraulic pressure on the shaft of the

pump. This balance occurs best at the manufacturer's recommended capacity. Running

volute-style pumps at a capacity lower than the manufacturer’s recommendation can put

lateral stress on the shaft of the pump, increasing wear-and-tear on seals, bearings, and

shaft. Double-volute casings are used when the radial thrusts become significant at off-

design capacities.

Seal chamber and stuffing box: Seal chamber and Stuffing box both refer to a chamber,

either integral or separate from the pump case housing, that forms the region between the

shaft and casing where sealing media are installed. When sealing is achieved by a

mechanical seal, the chamber is commonly referred to as a seal chamber. When sealing is

achieved by packing, the chamber is referred to as a stuffing box. Both seal chamber and

stuffing box protect against leakage, where the shaft passes through the pump pressure

casing. Sub-ambient pressure at the bottom of the chamber prevents air leakage into the

pump. When pressure is above atmospheric, the chambers prevent liquid leakage out of

the pump. Cooling or heating arrangement of the seal chambers and stuffing boxes

provide proper temperature control. A major problem with centrifugal pumps delivering

ultra-pure, highly toxic, sterile or delicate fluids is the shaft seal, which seals the rotating

drive shaft against the casing. According to Exxon, 80% of pumps in the chemical

industry are withdrawn from service because of mechanical seal failure, with the

remaining 20% withdrawn for failure of bearings, couplings and other associated items

18

[Bernard, 1991]. Maintenance costs are approximately twice a pump’s value in the first

five years of life [Schöb, 2002].

Bearing housing: The bearing housing encloses the shaft bearings, which keep the shaft

or rotor in correct alignment with stationary parts, despite action of radial and transverse

loads. The bearing housing also includes an oil reservoir for lubrication, and jacket for

cooling, by circulating water.

Wear rings: A wear ring provides an easily and economically renewable leakage joint

between the impeller and the casing. If the clearance between impeller and casing

becomes too large, pump efficiency reduces, causing heat and vibration problems. Most

manufacturers require disassembly of the pump, to check the wear ring clearance, and

replacement of the rings when clearance doubles.

Shaft: A centrifugal pump shaft transmits torques encountered during starting and steady

operation, while supporting the impeller and other rotating parts. Shaft deflections must

not exceed the minimum clearance between the rotating and stationary parts.

Coupling: Couplings transmit torque between two shafts, and can compensate for

mismatch of mating shaft sizes and alignments. Shaft couplings can be broadly classified

as rigid or flexible. Rigid couplings are used in applications without possibility for any

misalignment. Flexible shaft couplings compensate selection, installation and

maintenance errors.

19

Auxiliary piping systems: Auxiliary piping systems include tubing, piping, isolating

valves, control valves, relief valves, temperature gauges, thermocouples, pressure gauges,

sight flow indicators, orifices, fluid reservoirs, and all related vents and drains.

1.4.3 Bond Graphs

In this study, induction motor and centrifugal pump physics are integrated into a

composite bond graph model. Usually, a mechanical system consists of several machine

components; and each machine component consists of primitive components or elements.

For degradation analyses, we require a one-to-one correspondence between

machine components and fundamental sites of faults to locate and pinpoint anomalies in

system behavior. As shown in Figure 1.8, several equivalent bond graphs can be

presented for the same healthy machine model. In Figure 1.8-(a), the parallel springs are

modeled via two different bond graphs. Each give identical dynamic behavior, but the

first maintains a one-to-one correspondence between machine components and bond

graph elements. Figure 1.8-(b) also shows similar principles for a gear train modeled as

simple transformers.

With a one-to-one correspondence, degradations in real systems can be instilled in

a bond graph model by varying bond graph parameters, adding noise (effort or flow)

sources, or changing the power pathways. From the model, state equations can be derived

[Karnopp et al., 1990].

20

k

2k

:1/k

:1/2k

or

:1/3k0

1 1

0

C

C

1

C

10

(a)

or. . . .2 4/3 . .8/3

1 : 2

3 : 4

TF TF TF

(b)

Figure 1.8 Equivalent bond graphs for a system.

1.5 REVIEW OF SHANNON’S COMMUNICATION THEOREM

1.5.1 Communication Channel

Figure 1.9 is known as the Shannon-Weaver Model which consists of transmitter

(signal source), channel media, and receiver. Message X from an information source,

encoded in signal x(t) are injected into the channel by the transmitter. The receiver

accepts a signal from the channel that contains the transmitted signal y(t) altered by the

dynamics of the channel, and corrupted by noise n(t) added by channel. If the signal to

noise ratio S/N of y(t), the ratio of the average power

[ ]2

0

1( ) lim ( )T

TS P y t y t dt

T→∞= = ∫ (1.1)

of the signals y(t) to the average power of the noise,

[ ]2

0

1( ) lim ( )T

TN P n t n t dt

T→∞= = ∫ (1.2)

21

is maintained sufficiently high, message X can be transferred to a destination with little

or almost no obfuscation of information. Otherwise, received message X’ will be a poor

reproduction of message X. Here t is time, and T is some long time period in practice.

Noise, n(t)

Received signaly(t)x(t)

Message, X

TransmitterInformationsource Receiver Destination

Noise source

Σ

Receivedmessage, X’

Figure 1.9 Shannon and Weaver model of a communication channel [Shannon, 1948].

1.5.2 Shannon’s Information Theory

In the late 1940s Claude Shannon, a research mathematician at Bell Telephone

Laboratories, formulated a mathematical theory of communication that gave the first

systematic framework in which to optimally design telephone systems. The main

questions motivating this was how to design telephone systems to carry the maximum

amount of information, and how to correct for distortions on the lines [Cover and

Thomas, 1991].

Shannon’s theorems established absolute bounds on the performance of

communication systems, estimating how much information could be transmitted over a

noisy channel. To complete his analysis of the communication channel, Shannon

introduced the entropy rate R, a quantity that measured a source's information production

rate and also a measure of the information carrying capacity, called the communication

channel capacity C defined as the maximum rate of reliable information transmission

through the channel.

RC max= (1.3)

22

Shannon [1954] defined the channel capacity and the entropy rate in a time continuous

channel with additive white Gaussian noise:

2log 1 iB

SCN

ω = +

(1.4)

where Si is the average power of the desired signal, N is the power of the noise and Bω

is the bandwidth of the channel in hertz.

Shannon [1954] integrated equation (1.4) over infinitesimally small bands to

estimate the capacity of channels with non-flat bands as

20

( )log( )

B S fC dfN f

ω =

∫ (1.5)

where S(f) and N(f) represent power spectral densities of each corresponding signal over

bandwidth Bω with respect to frequency f.

2log ii

i

SRN

ω

=

(1.6)

where ( ) ( )i iN P y t y t= − is the maximum allowed root mean square (RMS) error

between received y(t) and desired yi(t) signals and iω is the signal bandwidth. RMS

pertains to the square root of the power measure defined in equations (1.1) and (1.2). Rate

R in equation (1.6) depends on details of a desired transmission: Si is a 2-norm estimate

of the amount of information contained in a desired message, bandwidth ωi gauges the

desired speed of transmission, and fidelity measure Ni sets a tolerance on received errors.

With the quantities defined in equations (1.4) and (1.6), Shannon estimated

channel’s effectiveness with inequality

CR ≤ (1.7)

which compares the desired rate of information transmission R to the maximum possible

transmission rate C. If the entropy rate R is equal to or less than the channel capacity C,

then there exists a coding technique which enables transmission over the channel with an

23

arbitrarily small frequency of errors. This restriction holds even in the presence of noise

in the channel. A converse to this theorem states that it is not possible to transmit

messages without errors if CR > .

1.5.3 Channel Capacity of an Analog Channel*

This section explains the meaning of channel capacity of an analog channel by

graphical representation and helps to understand how the capacity of the channel can be

estimated using the equation (1.4).

An analog signal x(t) in Figure 1.10-(a) is transmitted through an analog channel

with a finite bandwidth Bω , free of distortion, but with uniform Gaussian noise of known

power. If the signal contains no frequencies higher than Bω cycles per second,

Shannon’s sampling theorem guarantees that a sample set containing a series of points spaced ( )1/ 2 Bω seconds apart uniquely determines the signal. If we collect samples

during a finite time T, then the number of sampled points will be 2 BTω .

Figure 1.10-(b) shows a point identified by three numbers x1, x2, and x3 in three-

dimensional space. Similarly, the signal x(t) can be considered as a point

1 2 2, ,...,BTx x x ω which is identified by 2 BTω coordinates, in a space of 2 BTω

dimensions with 2 BTω mutually perpendicular axes. The Power P of the signal x(t) can

be calculated from the discrete sample data as 2

2

1

12

BT

nnB

P xT

ω

ω =

= ∑ (1.8)

where xn is the nth sample of x(t). The distance, d, from the origin to a point in the 2 BTω -

dimensional space is 2

2 2

1

2BT

n Bn

d x TPω

ω=

= =∑ or 2 Bd TPω= , (1.9)

* This section summarizes a part of chapter 3 in Raisbeck [1964].

24

where equation (1.8) was applied to the last equality of the first of equations (1.9). In

other words, the distance between two points in the space is proportional to the square

root of the difference of the power of the two signals.

In Figure 1.10-(c), a signal x(t) with 2 BTω coordinates plots as a point in the

space. Since noise of power N also transmits through the channel, and the noise has

components that are random and uncertain, the noise power N adds a cloud of uncertainty

about the point that represents the signal. In the space, this generates a geometric sphere

with radius 2 BTNω , via equation (1.9). An output signal corrupted by noise of power

N will approximately lie within a sphere of radius 2 BTNω centered around the point

1 2 2, ,...,BTx x x ω representing the input signal x(t), of which the position is assumed to be

known before transmission. The spheres of uncertainty due to the noise are represented as

gray circles in Figure 1.10-(c). Approximating the power of any output signal from this

channel as P+N, all possible outputs of any transmitted signal of power P with noise power N should be contained within a sphere of radius ( )2 BT P Nω + , see Figure 1.10-

(c). The illustration in Figure 1.11-(c) shows the geometry for 2 2BTω = , i.e., 2D.

Channel capacity is obtained geometrically from Figure 1.10-(c). The gray

spheres represent all possible receptions of specific signals of power P and noise N. The

total volume in the hyperspace of M small gray spheres must be less than or equal to the

volume of the boundary outer sphere. This gives

( )( ) ( )2 22 2

B BT T

B BK T P N MK TNω ω

ω ω+ ≥ (1.10)

where K is a scaling constant for the space whose numerical value is not important. The

exponents 2 BTω arise from calculating the volume in the hyper-sphere of 2 BTω

dimensions. Arranging equation (1.10) gives 1 log log 1B

PMT N

ω ≤ +

. (1.11)

25

The ratio P/N is the well-known signal-to-noise ratio. log /M T in equation (1.11)

represents the average rate of information transfer R, see equation (1.6). Equation (1.11)

provides an upper bound on the channel capacity of this channel. After a complicated

mathematical development and with simplifying assumption (refer to [Shannon, 1954;

Raisbeck, 1965]), the lower bound on the channel capacity is 1lub log log 1B

PMT N

ω = +

(1.12)

where lub signifies least upper bound. The bound in equation (1.12), defined as the

channel capacity, can guarantee with confidence that there exist codes which permit

transmission at a rate as close as desired to the channel capacity

log 1BPCN

ω = +

. (1.13)

Equation (1.13) is equivalent to equation (1.4), with Si replaced by P.

Radius

Volume

2 BTNω

( )22

BT

BK TNω

ω

( )2 BT P Nω +

( )( )22

BT

BK T P Nω

ω +

Space of 2ωΒT dimensions

Radius

Volume

Transmittedsignal

Locus ofreceived

signal

(a)

(c)

12 Bω

T2ωΒT samples in T

t

x(t)

x1x2

x3

d3

2 2 23 1 2 3d x x x= + +

(b)(shown in 2 dimensions)

Figure 1.10 (a) Sampling of a function of bandwidth Bω , (b) Distance of a point from origin (c) Transmitted and received signals in 2 BTω dimensional signal space (Figures were reproduced from [Raisbeck, 1965])

26

1.6 OUTLINE

In this chapter, a fault diagnosis method that utilizes models, parameters

estimation, and Shannon’s communication theory were proposed along with the

description of trend of fault diagnosis methods. Fundamentals on squirrel cage induction

motor, centrifugal pump, and bond graphs were noted. Brief introduction about

communication channel and some quantities from Shannon’s information theorems were

also provided.

Chapter 2 will explain the model for motor-pump system. The motor model

updated by Kim and Bryant [2000] and the pump model developed by Tanaka et al.

[2000] will be combined to describe the behavior of the test apparatus built for this study.

Each part of the bond graph model will be matched to a component in the physical

system, which will promote understanding of the behavior of the model in healthy or

faulty condition.

In Chapter 3, the structure of test apparatus including motor, pump, piping,

sensors, and data measurement devices will be introduced. Intentional faults will be

injected in the test setup and identified by tuning parameters in the model. Result of

simulations and experiments will demonstrate the validity of proposed method.

In Chapter 4, an analogy between machine and communication channel will be

introduced to use Shannon’s information theorems in machine diagnostics. Channel

capacity C and information rate R will be obtained using measurements from Chapter 3 to

assess motor-pump’s functionality.

In Chapter 5, the centrifugal pump model introduced in Chapter 2 will be

modified by examining the effect of pressure distribution enclosing impeller. Applied

forces to impeller will be obtained by integrating the pressure distribution with respect to

27

circumferential area of the impeller. This chapter was included as a suggestion for further

study on fault diagnostics of motor-pump systems.

Chapter 6 summarizes all the study implemented, gives conclusions, and

recommends future work.

28

Chapter 2: Modeling of Systems

The motor-pump model utilized in this study is briefly introduced. A squirrel cage

induction motor model, Figure 2.2, and a centrifugal pump model, Figure 2.3, were

linked, to study the dynamics of a motor-pump system, Figure 2.1.

2.1 SQUIRREL CAGE INDUCTION MOTOR MODEL

Many 3-phase AC induction motor models employ state space two-reaction

theory [Park, 1929], including Ghosh and Bhadra’s bond graph [Ghosh and Bhadra,

1993] of an induction machine, which employs a mutually perpendicular a-b model in a

stationary reference frame, linked with the three phase current source inverter. Kim and

Bryant [2000] altered this to partition the electrical, magnetic and mechanical energy

domains, Figure 2.2. Here, MSe:Va , MSe:Vb and MSe:Vc indicate the 3-phase alternating

stator voltages, R:Rs,Rsm model resistive losses in the stator windings, and GY:ns models

the stator coil, transition from electric to magnetic domain, where modulus ns is the

number of turns. The battery of transformers TF:mk convert the 3-phase (a-b-c) into a

rotating phasor vector (α-β) with 1 3 2m = , 632 −== mm , 24 =m , and

25 −=m ; these transformers derived from relations of Hancock [1974],

−−−

=

=

c

b

a

c

b

a

iii

iii

ii

21210616132

34sin32sin0sin34cos32cos0cos

32

ππππ

β

α (2.1)

The two-port C fields represent stator and rotor field interaction [Karnopp, 2003].

Constitutive laws si i i si

ri i i ri

M a bM b c

ϕϕ

=

, (2.2)

relate magneto motive force M to flux ϕ, via the reluctance matrix with elements

29

2

2si ri

imisi ri

n LaL L L

=−

, 2si r mi

imisi ri

n n LbL L L

−=

−, and

2

2r si

imisi ri

n LcL L L

=−

(2.3)

that depend on rotor and stator turns rn and sin , self inductances riL and siL for

rotor and stator, and mutual inductance miL between rotor and stator. The subscript

,i α β= denotes the motor phases. Voltage induced by time varying flux cutting the

metal rotor bar circuits is represented as the battery of gyrators, which have moduli nr

related to the number of turns. The modulated transformers MTF:mrαk,mrβk relate angular

position of the rotor relative to the flux field. To incorporate individual rotor bars into the

bond graph, a and b phase currents and voltages of the rotor should be split into separate

bar currents and voltages. The axes of a-b-c and a-b phases are stationary with respect to

the stator, but because the rotor rotates relative to these axes, bar currents must depend on

the rotation angle q of the rotor. Using Hancock [1974], ( ) ( )

2 2( 1) 2 2( 1)cos sin

rk r k r k

r r

i i mr i mr

k ki in n n n

α α β β

α βπ πθ θ

= +

− − = + + +

(2.4)

In equation (2.4), irk represents the current in the kth rotor bar (k = 1, 2, … n), iar and ibr

are rotor currents in a and b phases, and n (= 34 in this study) is the number of rotor bars.

Equation (2.4) is modeled via the 0 junction in the shaded area at the top right corner of

Figure 2.2. Elements R:Rrk represent resistive losses in the rotor circuits, and MGY:rk

convert rotor bar currents into electro-magnetic torque Te for a Pp-pole machine as

1 12

n np

e k k rkk k

PT T r i

= =

= =∑ ∑ (2.5)

where the moduli of the modulated gyrators are

( ) ( )k r r k r r kr n mr n mrβ α α βϕ ϕ= − (2.6)

30

The final transformer TF has modulus 2m pm P= in equation (2.5). Bearing friction is

R:Rbr. Power, p1 drives the rest of the system. The state equations, extracted from the

bond graph are

2 21 1

2 22 2 4

2 23 3 5

1

1

1

sa ss a sa a s

s s sa

sb s sb sb b s s

s s sb

sc s sc sc c s s

s s sc

R nV R V Mm n n R m

R n nV R V M Mm n n R m m

R n nV R V M Mm n n R m m

α α

α β

α β

ϕ = − + ⋅ + + − + + ⋅ + + − + + ⋅ +

2 24 2 4

2 25 3 5

1

1

sb s ss b sb b s s

s s sb

sc s sc sc c s s

s s sc

R n nV R V M Mm n n R m m

R n nV R V M Mm n n R m m

β α β

α β

ϕ = − + + ⋅ + + − + + ⋅ +

( ) ( ) ( )2( )2 2

1 1 1

1p p p

p

P P Prr

r k rk k k P rk k kk k kr r r m

MM hmr R mr mr R r mrn n n m J

βααϕ +

= = =

= − ⋅ − ⋅ ⋅ − ⋅⋅∑ ∑ ∑

( ) ( )

( )

2 22

( ) ( ) ( )2 21 1

2

( )1

1

p p

p p p

p p

p

p

p

P Pr r

r k r k P k k P r k Pk P k Pr r

P

k P kk Pr m

M Mmr R mr mr Rn n

h r mrn m J

β αβϕ − − −

= + = +

−= +

= − ⋅ − ⋅ ⋅

− ⋅⋅

∑ ∑

∑

( ) ( )( )1 1

p p

p

P Prr

k k k k P brk km r m r

MM hh r mr r mr Rm n m n J

βα+

= =

= ⋅ + ⋅ −⋅ ⋅∑ ∑ , 1

m

hm J

θ =

(2.7)

where each symbol is represented in Figure 2.2. In equation (2.7), subscripts r, s, a, b, c,

α, and β stand for rotor, stator, a-b-c phase, and α-β phase, respectively.

31

o-ring

washer

sleeveshim

seal

impeller

washer

screw

casingkey

shaftstator

rotor bar

bearing

end ring fan blade

bearing

inductionmotor

Figure 2.1 Structure of tested motor-pump

p1

1

1

1

1

1

1

0

0

0

0

0

0

1

1

GY

GY

GY

TF

C

C

GY

GY

GY

GY

MTF

MTF

MTF

MTFMGY

MGY TF

R

R

R

R

R

R

R

R

I

GY MTF

GY MTF0 1 MGY

R

1

1R

TF

TF

TF

TF

Rs:

Rs:

Rs:

Rsm:

Rsm:

Rsm:

ns. .

. .

. .

ns

ns

. .m1

. .m2

: m3m4. .

. .m5

nr. .nr. .

nr. .nr. .

nr. .

nr. .mrβn. .

mrαn. .

mrα1. .mrβ1. .

mrαk. .mrβk. .

: Rr1

: Rrk

: Rrn

A rotor bar

MechanicalMagnetic Mathematical ElectricalMath.Electrical Magnetic

n Rotor bars

r1. .

rk. .

rn. .

Stator coils

Rbr: J

MSe

MSe

MSe

. .Va

. .Vc

. .Vbmm. .

Msα

Mrα

Msβ

Mrβ

ω

sαϕ

rβϕsβϕ

h

rαϕ

. .

Figure 2.2 Kim and Bryant [2000]’s bond graph of an induction motor

32

2.2 CENTRIFUGAL PUMP MODEL

A centrifugal pump, a simple but crucial part of many industrial plants, can have

considerable impact on overall cost and performance of the plant. Early detection of

faults in pumps can save considerable downtime and replacement costs. In this study, a

physical model using bond graphs method is employed, to detect causes and assess

consequences of faults. The pump and induction motor models introduced earlier will be

unified, forming a whole system for fault diagnosis.

Tanaka et al. [2000] modeled a centrifugal pump and pipe system using the

modulated gyrator in Figure 2.3. In Figure 2.3, motor power p1 to the pump shaft

generates hydraulic power through the impeller. Some power is stored as kinetic energy

in I:J in Figure 2.2 and the liquid inertia (Iimp, Iout), consumed by mechanical (Rdisk) or

fluid losses (Rimp, Rvolute), or lost by leakage loss (Rleak) due to gaps inside pumps.

Conservation of angular momentum of the fluid in the impeller links results in [White,

1994; Tuzson, 2000] ( )g impT R Q= ⋅ and ( )i gP R ω∆ = ⋅ (2.8)

where the modulus

( ) ( )2 2 2 12 1

2 1

cot cot= ,2

impg g imp i i

i i

QR R Q r r

B Bβ βω ρ ω

π = − − −

(2.9)

and ri, Bi, and β denote radius, axial thickness, and blade angle. Subscripts 1 and 2 denote

the center and perimeter side of the impeller blade, see ‘Appendix A’ for detailed

derivation of Equations (2.8) and (2.9). Equation (2.8) models power conversion between

fluidic and mechanical with MGY: Rg in Figure 2.3 [Tanaka et al., 2000]. Fluidic power

circulates clockwise along the volute, outlet pipe, water tank, and suction pipe in

sequence. In between are minor losses (Rout, Rin: friction loss, expansion loss, contraction

loss, valve loss, etc.) general for any pipe system. In Figure 2.3, integral causalities exist

33

on inertance elements Iimp and Iout. Compliance of fluid in the tank Ctank is ignored. The

fluid volume in the pipe is much smaller than the fluid volume in the tank, and transients

are very fast, so the fluid level in the tank can be regarded as constant. Pressure

equilibrium of each inertance element in Figure 2.3 results in

( )22imp imp g imp out leak imp outI Q R R Q R Q Qω= − − − (2.10)

( ) ( )2 2out out leak imp out out in outI Q R Q Q R R Q= − − + . (2.11)

Reservoir

Suctionpipe

Outlet pipe

Qleak

Qin

ω

Qimp

Qout

Impeller

Fluidic

Qleak

Mechanical

p1

: Iimp

1

R : Rimp

R : Rleak

R : Rvolute

1

10

0

MGY1

R : Rdisk I

Rg. .C : Ctank

I

R

R

1

1

0

: Rout

: Rin

: Iout

Qimp

Qin

Qout

ω

Impeller Volute Outlet pipe

Tank

Suction pipe

T ∆Pi

Figure 2.3 Tanaka et al. [2000]’s bond graph of a centrifugal pump system

34

Chapter 3: Parameter Tuning

This chapter describes the experimental setup built for this study. Measurements

from the apparatus and simulations using the model introduced in Chapter 2 are

compared to detect and identify faults. Parameters in the model are adjusted until

simulation matches measurement.

3.1 EXPERIMENTAL SETUP

In Figure 3.1, the squirrel cage induction motor (1) (3-phase, 2 hp, 3600 rpm)

drives the centrifugal pump (2) (19 m max. head), see Figure 2.1. Measured are 3 phases

of input voltages (10) and currents (11), rotational speed of the motor (3), flow rate at the

outlet pipe (6), and pressures at the inlet (5) and the outlet (4) of the pump. Voltage

divider circuits (10) step the input 230 volt supply voltages down to within ±10 volts, for

data acquisition. Three Hall effect linear current sensors (11) measure currents from the

three power supply lines to the motor. Pressure transducers measure pressures at the inlet

(5) and the outlet (4) of the pump. The encoder (3) generates pulses at a rate proportional

to the rotational velocity. These pulses pass through a frequency-to-voltage converter

(12), which produces voltage proportional to rotational velocity. Similarly, the flow rate

through the pipe is obtained by processing the pulse signal from the paddlewheel type

flow meter, with a frequency-to-voltage converter (13).

3.2 FAULT DETECTION AND PARAMETER TUNING

To assess the model based fault detection and identification, three artificial faults

were introduced. In the motor, a fault caused by cracked component or loose connection

35

in stator circuit was emulated by connecting a resistance in series to the coil [Geiger,

1982; Isermann and Freyermuth 1991; Thomsen and Kallesoe 2006], and a motor bearing

was degraded by injecting dirt and sands into the bearing case [Isermann and Freyermuth

1991]. In the pumping system, a butterfly valve in the middle of the outlet pipe was

closed, step by step, to mimic increasing resistance in the pipe [Isermann and Freyermuth

1991]. Tests for healthy machines without faults and impaired machines with faults were

conducted. For each test, the apparatus in Figure 3.1 was switched on and system

responses were measured versus time. Using models in Figures 2.2 and 2.3, simulations

were performed for a “healthy machine”, an exemplar of desired behavior, and for a

degraded machine. Simulations were compared to the appropriate measurements to tune

the model’s parameters. Inputs to the model were the same 3 phase voltages measured via

voltage dividers (10) in Figure 3.1. Table 3.1 presents values of parameters tuned by

comparing model’s simulation to data measured from a healthy machine under normal

operation. In the sections that follow, faults will be introduced, data will be measured,

parameters will be tuned, and faults will be detected.

13

10 12

11

9

14

15

16

7

4

5

2

3

1

8

17

6

1. Induction motor2. Centrifugal pump3. Encoder4. Pressure transducer5. Pressure transducer6. Flowsensor7. Discharge valve8. Suction valve9. Tank (250 gallon water)

10. Voltage dividers11. Hall effect current sensors12. F-V converter13. F-V converter14. 3-phase input voltages15. Data acquisition board16. Inlet pipe (Length: 3m, Dia.: 2")17. Outlet pipe (Length: 5m, Dia.: 1.5")

Figure 3.1 Test system setup

36

Table 3.1 Parameters of a motor-pump

1.0281

Healthyvalue

366.70.86630.10330.13770.11620.0034

3.6e117.0e9

1.6e152.3e111.0e10

Rs

Parameters

Rsm

Rr1,...,Rr34

Ls

Lr

Lm

Rbr

Rimp

Rvolute

Rleak

Rout

Rin

Stator coil resistances (Ω)

Description

Stator magnetic losses (1/Ω)Rotor bar resistance (Ω)Stator inductances (H)Rotor inductances (H)Mutual inductances (H)Mechanical friction (N-s/m)

Loss in impeller (kg/m7)Loss in volute (kg/m7)Leakage loss (kg/m7)Loss in outlet pipe (kg/m7)Loss in inlet pipe (kg/m7)

1.1e-5Rdisk Mechanical friction (N-s/m)

0.0038028.6e72.5e6111

10.0250.05

0.011530

JIimp

Iout

ns

nr

ri1

ri2

Bi2

β1

β2

Moment of inertia (N-m2)Liquid inertia in impeller (kg/m5)Liquid inertia in outlet pipe (kg/m5)Number stator coil turnsNumber rotor coil turnsImpeller inner radius (m)Impeller outer radius (m)

Axial width at impeller outlet (m)Blade angle at impeller inlet (°)Blade angle at impeller outlet (°)

0.01Bi1 Axial width at impeller inlet (m)

37

3.2.1 Fault in a Stator Circuit

For the fault in a stator circuit, Figure 3.2 shows the change of measured 3 phase

currents (a, b, c), from healthy to degraded conditions. Currents in the second and third

columns in Figure 3.2 were measured after connecting 2.5 Ω and 4.5 Ω resistors

progressively in series to the a phase stator coil. As the resistance (fault) increases, the

time to steady state increases, and magnitudes of ia in each column of Figure 3.2 reduce.

Higher resistance simultaneously affects measured current, rotational velocity, and

pressure. Figures 3.4 and 3.5 show rotational velocities and pressures measured

simultaneous to the currents in Figure 3.2.

Table 3.2 assesses sensitivity of the measured states to changes in selected

parameters. After each parameter in table 3.2 was individually perturbed (1% of nominal

value), a simulation was performed to observe changes in system responses. The number

of ‘+’ symbols in any row in table 3.2 indicates the influence of each parameter’s change

on the system response. Table 3.2 suggests that measured currents, rotational velocities,

and pressures are sensitive to changes in stator coil resistances (Rsa, Rsb, Rsc) or motor

inductances (Ls, Lr, Lm), even though the origin of the fault is the stator resistance Rsa.

First, the motor-pump model was tuned by adjusting stator coil resistances only, and

tuned a second time by adjusting motor inductances only. The cost function for tuning

was a 2-norm defined as the sum of the square of difference between measured and

simulated rotational velocity. Currents and pressures were not considered in the cost

function. Using cost function and curve fitting tools in 20-sim to tune parameters (refer

to http://www.20sim.com/product/timedomain.html), simulations of healthy (Table 3.1),

and degraded machines (Table 3.3) are presented in Figures 3.3, 3.4, and 3.5 respectively.

Simulations nearly overlay experiments. Although Figures 3.3 and 3.5 tuned parameters

38

with rotational velocity measurements only, current and pressure simulations also overlay

current and pressure measurements.

Simulations with parameters tuned by stator coil resistances and by motor

inductances gave similar rotational velocities (Figure 3.4) and pressures (Figure 3.5).

However, the magnified details shown in the bubbles in Figure 3.4 of rotational velocities

at steady state suggests that simulations from tuning by stator coil resistances more

closely fits measurements, than tuning by motor inductances, for the resistance fault.

Since the induction motor model in Figure 2.2 represents a symmetrical electric machine,

each of Rsa, Rsb, and Rsc with the tuned values can in turn produce the rotational velocities

in Figure 3.4. Considering the measured currents in Figure 3.2, Rsa has to be largest

among the tuned resistances. Figure 3.3 compares simulated to measured current ia

(Figure 3.2), after assigning the largest value of tuned stator coil resistance to Rsa.

39

-500

50

-500

50

Cur

rent

(A)

0 0.5

-500

50

0 0.5Time (s)

0 0.5

Healthy machine

ia

ib

ic

Degradation

ia ia

ib ib

ic ic

(A)

Figure 3.2 Currents in healthy condition and with damaged stator circuit

0 0.01 0.02-100

-50

0

50

100

Time (s)

Cur

rent

(A)

0.5 0.51 0.52-20

-10

0

10

20ExperimentSimulation

Figure 3.3 Magnified view of current (A) in Figure 3.2 with tuned response after adjusting stator coil resistances

40

0

200

400

Rot

atio

nal v

eloc

ity (r

ad/s

)

0 0.2 0.4 0.6 0.8 10

200

400

Time (s)

...

.

.

........... ... .....

.

.

........... ... .....

..........................................

......

...........

.

.

Healthy machine

Healthy machine

Degradation

Degradation

Figure 3.4 Measured (dotted lines) and tuned (solid lines) rotational velocity by stator coil resistances (upper) and by motor inductances (bottom)

Pre

ssur

e (k

Pa)

Time (s)

Healthy machine

0 0.5 10 0.5 10

40

80

0 0.5 1

Experiment Tuned by resistors Tuned by inductances120

Degradation

Figure 3.5 Tuned pressures

41

Table 3.2 Sensitivities of system responses

Rotationalspeed Currents Pressure

(Flow rate)

++

.

.

.

++

+

+

.

.

. ++

++

.

.

++

+

Rs

Parameters

Rr1,...,Rr34

Ls, Lr, Lm Rbr, Rdisk

Rimp

Rout

Rin, Rvolute, Rleak

Sensitivities

+++

+++ +++ +++

.

Table 3.3 Parameters tuning data

Subscripts a, b, c, α , and β denote magnetic axes.

1.0281

Healthyvalue

0.1033

0.1377

0.1162

2.5 (Ω) 4.5 (Ω)Rsa (Ω)

Parameters

Rsb (Ω)Rsc (Ω)Lsα (H)Lsβ (H)Lrα (H)Lrβ (H)Lmα (H)Lmβ (H)

Connected resistor

Tuning byresistances

Tuning byinductances

2.05251.09590.52960.1037

5.06681.3719

0.10310.13820.13790.1152

1.39310.10410.10370.13870.13820.11430.11480.1154

42

3.2.2 Closing Valve at Outlet Pipe

Fluid loss, Rout in the centrifugal pump model of Figure 2.3 models pipe line

losses such as friction loss, expansion loss, contraction loss, valve loss, etc. The butterfly

valve (7) in the middle of the outlet pipe was closed in 10° increments to mimic

increasing resistance. The valve can be adjusted from fully open 0° to fully closed

90° .

Closing the valve from 0° to 40° had little effect on measured currents and

rotational velocity, but pressure signals increased significantly. From table 3.2, Rout was

selected as the parameter for tuning, since it increases outlet pressure significantly, with

little effect on currents and rotational velocity. Rimp was deselected, since increasing Rimp

decreases outlet pressure.

Figure 3.6 shows the measured pressure as valve angle changed from 0° to 40°,

and the simulated pressure obtained by adjusting Rout from 2.3e11, to 2.4e11, 2.7e11,

3.1e11, and 3.3e11 (kg/m7). Accordingly increased pressure obstructs the flow through

the pipe as in Figure 3.7. Flow volume rates in Figure 3.7 were measured from the

flowsensor (6) in Figure 3.1. Changing Rout had negligible effect on current and rotational

velocity, as implied by table 3.2.

43

0 0.2 0.4 0.6 0.8 10

40

80

120

160

Time (s)

Pre

ssur

e (k

Pa)

ExperimentSimulation

fully open(valve angle: 0°)

10°20°30°40°

Valve angle Degradation

Figure 3.6 Tuned pressures by hydraulic loss at outlet pipe, Rout

0 0.5 1 1.5-1

0

1

2

3

4

5

6

7

8

x 10-4

Time (s)

Flow

rate

(m3 /s

)

9

0°

40°

Degradation

ExperimentSimulation

Valve angle

Steady state flow rate (m3/s)

6.99x10-4

6.97x10-4

6.50x10-4

6.25x10-4

5.29x10-4

010203040

Valve angle (°)

Figure 3.7 Flow volume rates by hydraulic loss at outlet pipe, Rout

44

3.2.3 Bearing Contaminated with dirt

To vary the resistance of the ball bearing (40 mm diameter) in the induction

motor, dirt and sand were injected into the available space inside the bearing casing.

Added substances didn’t affect measurements much but generated a small vibration in the

rotational velocity in Figure 3.8. Sidebands and increased magnitude of peaks can be seen

in the frequency response of rotational velocity, for the contaminated bearing, see Figure

3.9-(b). This reflects the noisy measurement by contamination. The magnified view in

Figure 3.8 couldn’t be reproduced in simulations by simply adjusting bearing resistance

Rbr in the model. Increasing the bearing resistance Rbr in the model reduces the rise time

and decreases the steady state value of the rotational velocity, see the simulation in

Figure 3.10. The detailed physics of ball bearings, with presence of external particles,

needs to be included in the model, to detect bearing contamination by impurities. This is

beyond the scope of this dissertation.

0.2 0.4 0.6 0.8Time (s)

Bearing with dirtHealthy Bearing

00

100

200

300

400

Rot

atio

nal v

eloc

ity (r

ad/s

)

1

Figure 3.8 Rotational velocity affected by contaminated bearing

45

Frequency (Hz)

Rot

atio

nal v

eloc

ity (d

B)

10-2

100

102

(a) Healthy bearing

0 100 200 300 400 50010

-2

100

102

(b) Bearing with dirt

Figure 3.9 Frequency analysis of rotational velocities in Figure 3.8

0 0.2 0.4 0.6 0.8 10

100

200

300

400

Time (s)

Rot

atio

nal v

eloc

ity (r

ad/s

)

10×Rbr

Rbr

2×Rbr 3×Rbr

Degradation

Figure 3.10 Effect of tuning bearing resistance Rbr on simulated rotational velocity

46

Chapter 4: Fault Evaluation by Information Theory

4.1 ANALOGY OF MACHINE TO COMMUNICATION CHANNEL

Bryant [Bryant, 1998; Choi and Bryant, 2002-(1); Lee et al., 2006] showed the

possibility that Shannon’s communication theory can be applied to the fault diagnosis of

machine systems by analogy. A machine component (or system) accepts a signal from an

upstream component, by its function alters that signal, and then passes the signal on to

the next downstream component. In the analogy a machine is a communications channel.

When operating properly, the signal is received. Faults in the machine which disrupt

operation alter the flow of signal. Faults will be viewed as agents that alter system

parameter or contaminate the signal with noise. Unless the signal to noise ratio is kept

sufficiently high, downstream components cannot resolve the signal message error free,

and the machine malfunctions.

In Figure 1.9, input signal x(t) containing information is “transmitted” and

“received” as output y(t) over a “machine channel”. Faults in the machine disrupt the

flow of signal and add “noise”, “…any unwanted component in a received signal [Fish,

1994]”, tantamount to the difference y(t)-yi(t) between actual received signal y(t) and the

signal yi(t) received within tolerances α if the machine had no faults. Output y(t) is x(t)

altered by channel dynamics, but with noise n(t) added.

When applied to machinery, the inequality in equation (1.7) predicts a machine’s

ability to perform a task, given the machine’s available resource described by C, and the

demands of the task, described by R. If R ≤ C the system functions within specification,

otherwise not. Thus, C and R are analogous to a strength of, and load onto, the machine

channel. In this study, y(t) and yi(t) were defined as sets of data measured with and

without any intentional fault. The difference in y(t) and yi(t) was considered as fault (or

47

noise) n(t). Power spectra S(f) and N(f) obtained from Fourier transforms of y(t) and n(t),

were integrated according to equation (1.5), to estimate the machine channel capacity C.

From the inequality in equation (1.7), R consolidates a minimally required channel

capacity to achieve a given task within a tolerance. The amount of a tolerance can be

adjusted by setting the largest acceptable deviation, or noise Ni in equation (1.6). If we

have a task which has to be implemented within a maximum error range of some percent ±α of the error-free execution yi, then ( ) ( )in t y tα≤ , and from equation (1.6), (1.1),

and (1.2) the information rate R is 2

2 2 22

1 ( ) 1log log 2 log1 ( )

ii

i i ii

i

y t dtS TRN y t dt

T

ω ω ωαα

= = =

×

∫

∫. (4.1)

In machine systems, signal transmission R generally should be constant, because a

machine or component operation is often repetitive. The user of a machine can arbitrarily

decide for what value of tolerance, the machine “works” satisfactorily. Industrial

machinery can tolerate large errors, but “malfunctions” when errors in its output

variable(s) exceed some percentage α of the desired value. If α = 0.1 (10 % tolerance),

and 1000iω = Hz, then

( ) 32 2

11000 log 6.64 100.1

R = = ×

(bits per second, bits/s)

Here the integration bandwidth iω for the ideal signal yi(t) was set in accordance with

Shannon’s sampling theorem. Shannon’s sampling theorem suggests that the bandwidth

should be )(2

121

tsi ∆== ωω , where sω is the sampling bandwidth, and

0.0005t∆ = second is the data sampling time step employed in this study [Stremler,

1982]. Table 4.1 shows various information rates for selected tolerances.

48

Table 4.1 Sensitivity of information rate R to tolerance α

α R (kbits/s) with ω = 1 kHz0.10.20.30.4

6.64054.6415

0.50.60.70.8

3.47222.64252.00001.47321.02860.64350.30390.9

4.2 APPLICATION OF SHANNON’S THEOREM

The channel capacity was calculated for fault cases in Sections ‘3.2.1 Fault in a

Stator Circuit’ and ‘3.2.2 Closing Valve at Outlet Pipe’, using equation (1.5).

Considering the sampling frequency 2000 Hz, bandwidth frequency Bω in equation

(1.5) was set as 1000 Hz according to Shannon’s sampling theorem [Shannon and

Weaver 1948]. 1000 samples, which lie in steady state, were selected to calculate channel

capacities, assuming that all data was measured during operation.

Variation of channel capacity C, for various measurements of states for a ‘fault in

a stator circuit’ case in Section 3.2.1, is shown in Figure 4.1-(a). Supplementary

experiment for a-phase coil with 6 Ω resistor was implemented to generate the last

channel capacity values for each signal in Figure 4.1-(a). The values corresponding to 2.5

Ω for each channel capacity was used for normalization. The normalized channel

capacity in Figure 4.1-(b) shows the relative variance of channel capacity of each signal.

As the resistance of the stator coil (y axis in Figure 4.1) increases, the channel capacity of

current ia tends to decrease consistently as (A)→(B)→(C) as in Figure 4.1-(b). Figures

4.2 and 4.3 show measured current ia in frequency and time domains, which were utilized

49

to calculate channel capacities (A), (B), and (C) in Figure 4.1. The bandwidth ω of 1000

Hz was applied in equation (4.16) to obtain channel capacity, but only 500 Hz ranges are

shown in Figure 4.2, for readability. Measured sinusoids for healthy and degraded

currents were synchronized, given the same phase as in Figure 4.3, to avoid phase error

when estimating noise. For the synchronization, the healthy and degraded current data

sets, which minimize the sum of squares of the difference in each 1000 consecutive

sample set, were selected. Note that current ia deviates from the healthy signal as the

degradation worsens. Accordingly, channel capacity decreases as (A)→(B)→(C) as in

Figure 4.1-(b). Synchronization had a minor effect on the calculation of channel

capacities for pressure and rotational velocity signals, since for these signals steady state

DC values are relatively large. The dashed lines in Figure 4.1-(a) represent information

rates R for 10 %, 20 %, and 70% error tolerances, respectively. Each value (6.64 kbits/s,

4.64 kbits/s, and 1.03 kbit/s) is obtained from equation (4.1) with ω = 1000 (Hz) and α =

0.1, 0.2, and 0.7. If the channel capacity of a signal drops below one of these lines, the

system may fail to perform its function within the corresponding tolerance. Appropriate

tolerances for each signal for normal operation of the machine could be decided by

accumulated performance data, expert’s experience, etc.

Similar to the ‘fault in a stator circuit’ example, the channel capacity C was

calculated and presented for the ‘Closing Valve at Outlet Pipe’ case in Figure 4.4. For the

final channel capacity value of each signal in Figure 4.4, additional measurement was

implemented with closing the valve at 50° . As was expected in the sensitivity analysis in

table 3.2, the channel capacity for the pressure signal decreases a large amount. An

identical bandwidth (ω = 1000 Hz) case was applied. For an error tolerance of 0.2, the

system works acceptably in regions (A) and (B), where the condition C R≥ in equation

(1.7) is satisfied.

50

2 70

2

4

6

8

10

12

14

Cha

nnel

cap

acity

(bits

/s)

2 70.6

0.7

0.8

0.9

1.0

1.1

Nor

mal

ized

cha

nnel

cap

acity

x 1031.2Degradation

α = 0.1

α = 0.2

Degradation

ia

P

iaibic

ω

(A)

(B)

(C)α = 0.7

(a) (b)Added resistance in stator circuit (Ω)

Figure 4.1 (a) Channel capacity vs. added resistances (2.5, 4.5, 6 Ω) in stator circuit (b) Normalized channel capacities (P, ω, ia, ib, and ic denote calculated channel capacities from pressure, rotational velocity, and 3-phase currents measurements, respectively)

0 100 200 300 400 500Frequency (Hz)

-100

0

150

-100

0

150

-100

0

150

Pow

er S

pect

rum

Mag

nitu

de (d

B)

Deg

rada

tion

(A), 2.5 ΩHealthy

(C), 6.0 ΩHealthy

(B), 4.5 ΩHealthy

(a)

(b)

(c)

Figure 4.2 Power spectra of current ia to calculate channel capacities (A), (B), and (C) in Figure 4.1

51

Cur

rent

(A)

-10

0

10

-10

0

10

-10

0

10

0.25 sec. at steady state

Healthy(A), 2.5 ΩFault (Noise)

Healthy(B), 4.5 ΩFault (Noise)

Healthy(C), 6.0 ΩFault (Noise)

Deg

rada

tion(b)

(a)

(c)

Figure 4.3 Current ia to calculate channel capacities (A), (B), and (C) in Figure 4.1

10 500

0.2

0.4

0.6

0.8

1.0

1.2

Nor

mal

ized

cha

nnel

cap

acity

Cha

nnel

cap

acity

(bits

/s)

0

2

4

6

8

10

12

14 x 103

5010

Degradation Degradation

P

iaibic

ω

P

(A)

α = 0.2

α = 0.1

α = 0.7

(B)

(C)

Valve angle (degree)(a) (b)

Figure 4.4 (a) Channel capacity vs. valve angle (b) Normalized channel capacities (P, ω, ia, ib, and ic denote channel capacity calculation using measured pressure, rotational velocity, and 3-phase currents, respectively)

52

Chapter 5: Centrifugal Pump Model with the Interaction between Volute and Impeller

The centrifugal pump model in Figure 2.3 was modified to include rotordynamics

of a motor-pump which is critical to rotating turbomachinery, when vibration of the

system is an issue. The focus was on obtaining radial forces acting on an impeller

induced by the fluid flowing through the impeller to the volute enclosing the impeller

(Refer to Section 1.4.2 for details on the inside structure of centrifugal pumps). The

impeller is connected to a shaft, the shaft is supported by bearings, and the bearings are

fixed in the housing of the pump (ex, the motor-pump in Figure 2.1). Because of these

connections, the radial forces to the impeller will disturb every part of the system. Faults

varying the radial forces applied to the impeller could be sensed and identified by

observing the rotordynamic behavior of the shaft of a motor-pump, or by analyzing

vibrations from any part of the system. For the model to provide this information, the

radial forces obtained by applying elementary fluid mechanics to the interaction between

the impeller and the volute, were utilized as a link which connects the bond graph model

of a centrifugal pump in Figure 2.3 with the finite element bond graph model of a rotor in

[Choi and Bryant, 2002-(2)]. Briefly this chapter will suggest a way of injecting the

rotordynamics of a motor-pump into a bond graph model.

5.1 Flow between Impeller and Volute Tongue

Figure 5.2 shows flow continuity in the centrifugal pump introduced in Figure 1.6

with Qcl, the flow between impeller and volute tongue. Qimp and Qout represent the flow

from the impeller inlet and the flow toward the outlet pipe, respectively. Leakage flow

Qleak flows from the inside of the pump back to the pump inlet through the space between

the impeller (or shroud) and the pump housing, see Qleak in Figure 5.1.

53

The volute tongue in Figure 5.2 divides pressurized flow Qpress through the

impeller, into Qnet and Qcl. Qcl passes through the clearance (tongue clearance) between

the volute tongue and the impeller, and returns to the volute. After circulating along the

volute, Qcl discharges toward the pump outlet with Qnet, forming Qout. A large tongue

clearance increases losses by allowing too much flow to return to the volute; a small

tongue clearance causes strong pressure fluctuations at the blade passing frequency

[Tuzson, 2000, Brennen, 1994].

To model the clearance flow Qcl into model, flow continuities of the pump system

in Figures 5.1 and 5.2 are schematized in Figure 5.3-(a) with equivalent bond graphs in

Figure 5.3-(b). Figure 5.4 shows the modified Tanaka’s model in Figure 2.3 to describe

the clearance flow.

In the centrifugal pump and pipe system sections in Figure 5.4, three integral

(independent) causalities exist on inertance energy storage elements Iimp, Ipipe, and Icl. The

compliance of the fluid in the tank C is ignored since the flow in the pipe is much smaller

than the fluid in the tank, and transients are very fast, so that the fluid level in the tank

can be regarded as constant. Considering pressure equilibrium of each inertance element

in Figure 5.4 results in the following state equations from bond graph.

( )

( )( )

2 2

2 2 2

2 2

imp imp i i leak out leak imp imp

pipe out leak imp leak net out in out net cl

cl cl net out net cl cl

I Q g R Q R R Q

I Q R Q R R R R Q R Q

I Q R Q R R Q

ω= + − +

= − + + + +

= − +

(5.1)

54

Reservoir

Suction pipe

Outlet pipe

QinQimp

Qleak

Qout

Impeller

Volutecasing

Torque

iω

Figure 5.1 Flows in a pump system [Tanaka et al., 2000]

Qcl

Qimp

Impeller

Volute

iω Qcl

Qout

QpressQout =Qnet +Qcl

Qnet =Qpress - Qcl

Qnet

Qpress=Qout

Volutetongue

Volutethroat

Figure 5.2 Flows in a pump

55

11

0

0

1

1

Qimp

1

Qin

Qleak

QoutQpress

Qcl

0 1Qnet

0

1Qcl

QleakQimp

Qin

Qpress

Qnet

Qout

(a) (b)

Figure 5.3 Continuities of flows with an equivalent bond graph representation

1 RleakMGYi 11

0

0

Irot

Rm

Iimp 1

0 C

1

Rout

Rin

iω Qimp

1

Ipipe

Pump Pipe system

Qin

Qleakig

Qout

Se

Rimp

Rnet

Qpress

1Qcl

0 Qnet0

1Liquid inertiain impeller

Rotationalinertia

Mechanicalloss

Lossin impeller

Leakageloss

Lossin volute

Loss in suctionpipe system

Tank

Loss in outletpipe system

Liquid inertiain pipeRclIcl

Loss in volute

Motor

Liquid inertiain clearance flow

Figure 5.4 Bond graph model of the pump system with clearance flow

5.2 Interaction between Volute and Impeller

The volute collects flow from the impeller. The collected flow generates a

pressure distribution around the impeller, which results in radial forces toward the

impeller. The effect of radial forces exerted on the impeller by the pressure enclosing the

periphery of the impeller is not small, at off-design flow rates [Agostinelli et al., 1960;

Iversen et al., 1960]. In this study, the pressure distribution obtained by Iversen et al.

[Iversen et al., 1960] was utilized to include the effect of the radial forces. Considering

56

the volute section in Figure 5.5, the moment balance about the central axis in the

direction of θ is

( )( ) ( ) ( )

( )( ) 2

sin cos2i i s i s i

i i t i

dPAPr A dA P dP r P dA r dA r

Q dQ V dV r QVr C dQr

α τ α

ρ ρ ρ

− + + + + −

= + + − − (5.2)

Neglecting second-order terms in differentials, equation (5.2) reduces to

( ) ( ) 2sin sins s tAdP PdA P dA dA VdQ QdV C dQα τ α ρ ρ ρ− − + − = + − . (5.3)

For reasonable volute area variations [Iversen et al., 1960], ( )sin sdA dAα = ,

cos 1.0α ≈ . From continuity, QVA

= ; 2dQ QdV dAA A

= − (5.4)

The friction force generated by the shear stress can be represented as

( )2 2

2

/2 2

v is v i

Q A fw rQdA f w rd dA

ρ ρτ θ θ

= =

(5.5)

Substituting equation (5.4) and (5.5) into (5.3) gives 2 2

22 3 32

2t v iQ Q C fw rQdP dQ dA dQ d

A A A Aρρ θ

− = − − +

(5.6)

Impeller

Volute

ri

rv

ct

x

y iωG0

wv

AG0

V

ri

QAP

V+dV

Q+dQ

(P+dP)(A+dA)

dθ

dQ

C2t

sdAτ2 s

dPP dA +

α

Figure 5.5 Pump geometry

57

In order to integrate equation (5.6), variation of the volute cross-section area A

and flow rate Q were assumed as linearly increasing quantities with respect to q [Iversen

et al., 1960]. It was also assumed that flow rate Q discharged evenly from all

circumferential areas of the impeller outlet. Then the flow becomes

( )2

θ θπ

= = + impcl

QQ Q Q (5.7)

The leakage flow leakQ , assumed small (usually 1~2 % of the total flow [Tuzson,

2000]), was ignored in this derivation, but may be injected as a function of q or just as a

constant according to situations or needs. The cross-section area of the volute A was

assumed rectangular in Figure 5.5, thus the area

( )0v v t vA w G w c K θ= = + (5.8)

where the cross sectional gap of the volute G0 was also assumed to increase linearly with

the ratio Kv (in meter) from the volute tongue clearance ct, according to the volute design.

Rearranging equation (5.6) 2 2

22 2 2 3 2 3

0 0 0 02 2ρ θ

π π

= − + + −

imp t impv i

v v v v

QQ C QK Q frQdP dw G w G w G w G

(5.9)

Integrating equation (5.9) with respect to q gives the pressure distribution,

( )

( ) ( )( )

( )( )

( )( ) ( )

2

2

2 3 2

22

2 2

4 2.

16

2 2 4 ln

π

θ

πρθπ θ

π θ

− + − − +

− + = + + +

+ − + + ⋅ +

t imp v cl v i

t v

i imp t imp v cl

v v t v

v i imp v v t imp t v

c Q K Q K fr

c K

frQ c Q K QP const

K w c K

K fr Q K w C Q c K

(5.10)

Radial forces exerted on the impeller can be obtained by integrating the pressure

distribution, equation (5.10), with respect to the impeller outlet area and angle q from 0 to

2p. This gives

58

( )2

20

cosπ

θ θ= ∫x i iF P B rd (5.11)

( )2

20

sinπ

θ θ= ∫y i iF P B rd (5.12)

where Bi2 is the thickness of the impeller exit (See Figure A.1. in ‘Appendix A’).

Arranging results from equations (5.11) and (5.12) gives

( )

21 2 3 2

1 2 3 2 2( , , )

= + +

= + + =

x x imp x imp cl x t imp

x imp x cl x t imp x imp cl t imp

F c Q c Q Q c C Q

c Q c Q c C Q g Q Q C Q (5.13)

( )

21 2 3 2

1 2 3 2 2( , , )

= + +

= + + =

y y imp y imp cl y t imp

y imp y cl y t imp y imp cl t imp

F c Q c Q Q c C Q

c Q c Q c C Q g Q Q C Q (5.14)

Here Qcl2 terms in equations (5.13) and (5.14) are ignored under the assumption that Qcl is

much smaller than Qimp. Explicit description of coefficients 1( 1)x yc , 2( 2)x yc , and 3( 3)x yc

in equations (5.13) and (5.14) are shown in ‘Appendix B’. Figure 5.6 compares plots

from equations (5.13) and (5.14) with plots presented in [Iversen, 1960], using parameter

values given in the same literature. ‘Force’ and ‘Angle from tongue’ in Figure 5.6 are obtained by 2 2

x yF F+ and ( )arctan y xF F respectively.

The powers delivered to the rotor via the impeller without any loss are,

( )2( , , )⋅ = = ∆ ⋅x x x imp cl t imp x x impF e g Q Q C Q e P Q (5.15)

( )2( , , )⋅ = = ∆ ⋅y y y imp cl t imp y y impF e g Q Q C Q e P Q (5.16)

where xP∆ and yP∆ are pressure changes by impeller motion, xe and ye in Figure

5.5. From equations (5.15) and (5.16),

( ) ( ) 2

( ) ( ) 2 ( )

( , , )

( , , )

= ⋅

∆ = ⋅

x y x y imp cl t imp

x y x y imp cl t x y

F g Q Q C Q

P g Q Q C e (5.17)

Equation (5.17) suggests that the interaction between the volute and the impeller can be

modeled as a modulated gyrator (MGY) connecting efforts with flows ( ( )x yF with impQ

and ( )x yP∆ with ( )x ye ) via modulus (gx(y)). The pump model in Figure 5.7 includes the

59

bond graph representation of equation (5.17) as MGYx(y). Forces xF and yF can be

applied to the rotor bending sub-model (shown as ellipses) represented in [Choi and

Bryant, 2002-(2)].

0 50 100 150 200 250 3000

10

20

30

40

50

Forc

e (P

ound

s)

0 50 100 150 200 250 300-100

-80-60-40-20

020406080

100

Capacity (GPM)

Ang

le fr

om to

ngue

(°)

(a) (b)

Figure 5.6 Plots from equations (5.13) and (5.14), (a) vs. from Iversen [1960], (b)

60

1 RleakMGYi 11

0

0

Irot

Rm

Iimp

1

0 C

1

Rout

Rin

iω Qimp

1

I

Pump Pipe system

Qin

Qleak( , )ωi i impg Q

Qout

Se

Rimp

Rnet

Qpress

1Qcl

0 Qnet0

1

Liquid inertiain impeller

Rotationalinertia

Mechanicalloss Loss

in impeller

Leakageloss

Lossin volute

Loss in suctionpipe system

Tank

Loss in outletpipe system

Liquid inertiain pipeRclIcl

Lossin volute

Motor

Liquid inertiain clearance flowxe yexF yF

MGYyMGYx

Rotorbending

x-dir.

Rotorbending

y-dir.

2( , , ) :x imp cl tg Q Q C

FEM bending elements

xP∆ yP∆

2: ( , , )y imp cl tg Q Q C

Figure 5.7 Updated pump model with the interaction between volute and impeller using rotor bending sub-model and volute pressure distribution

61

Chapter 6: Conclusion and Future Work

6.1 SUMMARY AND CONCLUSION

Detailed models with direct correspondence between components in the machine

and elements in the model were formulated, and information from signals was instilled

into models by tuning parameters to estimate the condition of components in the

machine. An apparatus and model of a motor-pump were utilized to demonstrate

detection and identification of faults. Parameters in the model were tuned to make

simulations mimic measurements. From tests and simulations, we observed:

• Measurements for healthy and degraded conditions can be closely simulated via

tuning parameters in the model.

• Proper selection of parameters for tuning allows more efficient tuning, and

pinpointing origin of faults.

• The amount of change in parameters correlated to severity of faults.

• Tuning parameters from data permitted detection and identification of faults in a

motor-pump system.

• Utilizing the modular nature of bond graphs to generate causal models of physical

systems will permit extending the method to more complicated systems.

For fault severity assessment using Shannon’s information theory, a machine was

viewed as a communications channel which must transmit and receive critical

information (given task) despite noise generated by system faults. With measurements,

the channel capacities and information rates for machines in healthy and various

degraded states were calculated. We observed:

62

• Channel capacities calculated from signals corrupted by faults decrease with

increasing fault severity.

• Monitoring changes of channel capacities calculated from different signals can

help identify faults, since each channel capacity varies differently according to a

specific fault.

• The amount of degradation can be relatively assessed using Shannon’s

information theory theorems.

6.2 SUGGESTED FUTURE WORK

Important to model based fault diagnosis is availability of appropriate and precise

models. Even though our models include and describe many physical phenomena, there

still exist unmodeled dynamics and incomplete sub models. Several suggestions can be

made to improve the current model:

• Detailed dynamics of bearings needs to be added to account for vibrations from

bearing faults.

• Effects of non-uniform pressure distribution around the pump impeller due to

fluid interaction between impeller and volute needs to be considered (A simple

example of deriving forces due to static pressure around impeller in Chapter 5

could be one of ways to relate the model in this study to the non-uniform pressure

effects. Further development and research are required.).

• Rotor dynamics considering all known effects from bending, torsion, bearings,

impeller, magnetic field, vibration generated by flow through tongue clearance of

centrifugal pump (or blade passing frequency), etc needs to be considered to

emulate vibrations dominant in rotating machinery.

63

With the improvement of the current model, additional work to do can be summarized

as:

• One-to-one correspondence between tuned parameters set and a specific fault has

to be confirmed since there can be many combinations of parameters which will

generate same tuning result. Observability of the motor-pump model extended by

including tuning parameters (P) as additional states, with the form of equation

dP/dt = 0, has to be investigated to identify appropriate parameters set among

many candidate sets. Considering that the motor-pump model is nonlinear,

observability could be discussed by theoretical methods appeared in [Anguelova,

2004; Diop and Fliess, 1991; Hermann and Krener, 1977; Isidori, 1995] where the

observability is determined by rank test on the space spanned by gradients of the

Lie-derivatives of the output functions is calculated.

• More tests and simulations for common faults in motor and pump such as shaft

misalignment, bearings wear, stator turn-to-turn fault, rotor bar crack, should be

implemented.

• The accuracy and repeatability of each test need to be specified by multiple tests

and sensors with high performance.

• Parameter tunings and channel capacity characteristics for various cases and

multiple, simultaneous faults should be investigated.

• Effective and automated tuning method needs to be developed.

• Potential of information theory in fault diagnosis needs to be examined.

• Comparative study on the suggested method with other diagnostic method needs

to be implemented.

64

Appendices

A. POWER TRANSFER BETWEEN MOTOR AND PUMP

Power transfer in the rotating impeller can be found by applying conservation of

angular momentum to the control volume of annular shape in Figure A.1 [White, 1994;

Fox and McDonald, 1985; Tuzson, 2000], in the axial direction: ( ) ( ) ( ) ( )( ) ( )

2 1

2 1

i i i i

i t m i i t m i

T r C C n dA r C C n dA

rC v dA rC v dA

ρ ρ

ρ ρ

= × ⋅ − × ⋅

= −

∫ ∫

∫ ∫ (A.1)

In equation (A.1), T is the torque applied to the fluid by the impeller, ir is an impeller

radius, ρ is fluid density, n represents the unit vector which is normal to impeller

inlet or outlet surface, tC is the tangential component of the absolute velocity C of

impeller flow, and mv ( C n= ⋅ ) is the radial component of the absolute velocity C .

Considering impeller inlet/outlet area in Figure A.1 and the flow through impeller impQ ,

equation (A.1) is reduced to ( ) ( )

( ) ( ) ( ) ( )2 2 2 2 2 1 1 1 1 1

2 2 2 2 2 2 1 1 1 1 1 1

2 2 2 1 1 1

2 2

cos 2 cos 2cos cos

i t m i i i t m i i

i m i i i m i i

i imp i imp

T r C v r B r C v r B

r C v r B r C v r Br C Q r C Q

ρ π ρ π

ρ α π ρ α πρ α ρ α

= −

= −

= −

(A.2)

In equation (A.2), 1iB ( 2iB ) is the inlet (outlet) flow passage width of impeller and 1α

( 2α ) is the angle between the absolute velocity of fluid 1C ( 2C ) and the rotational

velocity of impeller 1U ( 2U ) in Figure A.1. The flow rate at impeller inlet ( 1 1 12m i iv r Bπ⋅ )

is equivalent with the flow rate at outlet ( 2 2 22m i iv r Bπ⋅ ) by continuity of the flow through

impeller, which is represented as impQ in equation (A.2). The power delivered to the

fluid by external source without any losses is thus,

65

( )( )

2 2 2 1 1 1

2 2 2 1 1 1

cos cos

cos cosi i imp i imp i

imp

i imp

T r C Q r C Q

U C U C Q

P Q

ω ρ α ρ α ω

ρ α α

⋅ = −

= −

= ∆ ⋅

(A.3)

where iω is the angular velocity of impeller, 2 2i iU r ω= , 1 1i iU r ω= , and iP∆ is the

increased pressure in fluid by applied power iT ω⋅ . Equation (A.3) is also called Euler

turbomachinery equation [White, 1994].

Using velocities in Figure A.1, equation (A.3) can be further derived that

2 2 2 2 2 2 2 22 2

2 2 22 22 2 2 2 2 2

2 2 2

cos cot cot2

cotcos cot2 2

impt m

i i

imp impi i i

i i i

QC C U v U

r B

Q QUU C U rr B B

α β βπ

βα β ω ωπ π

= = − = −

= − = −

(A.4)

where 2β is the flow angle at the impeller outlet. Similarly,

2 2 21 11 1 1 1 1 1

1 1 1

cotcos cot2 2

imp impi i i

i i i

Q QUU C U rr B B

βα β ω ωπ π

= − = −

(A.5)

Substituting equations (A.4) and (A.5) into corresponding terms in equation (A.3),

( )

2 2 2 22 12 1

2 1

2 2 2 12 1

2 1

cot cot2 2

cot cot2

imp impi i i i i i i imp

i i

impi i i i imp

i i

i imp

Q QT r r Q

B B

Qr r Q

B B

P Q

β βω ρ ω ω ω ωπ π

β βρ ω ωπ

⋅ = − − −

= − − −

= ∆ ⋅

(A.6)

From equation (A.6), we can obtain the following relationship [Tanaka, 2000]. ( )

( )i imp

i i i

T g Q

P g ω

= ⋅

∆ = ⋅ (A.7)

where gi is

( ) ( )2 2 2 12 1

2 1

cot cot= ,2

impi i i imp i i i

i i

Qg g Q r r

B Bβ βω ρ ω

π = − − −

(A.8)

The relationships in equation (A.7) suggest that centrifugal pump can be modeled as

modulated gyrator (MGY) connecting efforts with flows (T with impQ and iP∆ with

66

iω ) via the modulus (gi) [Tanaka, 2000; Paynter, 1972]. Figure A.2 describes bond graph

representation of equation (A.7).

W1

C1

U1

1α

1β1mv

W2

C2

U22α

2β

2mvCt2

iω

2β

1β ri1

ri22ir

1ir

2iB

1iB

Blade

3D control volume

Ct1

Impellerinlet

Impelleroutlet

Figure A.1 Control volume and flow velocities of pump impeller

MGY( , )i i impg Qω

TQimp

iP∆iω

Figure A.2 Modulated gyrator for Eulerian turbomachine

67

B. FORCES EXERTED ON IMPELLER OF CENTRIFUGAL PUMP

x(y) directional force, ( )x yF

( ) ( ) ( )

21( 1) 2( 2) 3( 3)

( ) 1( 1) 2( 2)2 3 2

23( 3) 2

2 4 4 2

4 2 816

8

ρπ π

ππ

− − + + + = + − + +

t v i x y t i x y v i x y imp

impx y t v v i x y i v x y cl

v v

v v x y t

c K fr I c fr I K fr I QQ

F c K K fr I frK I QK w

K w I C

where

( ) ( )2

1 2 20

1 1 2cos cos sin2

π πθ θπθ

= ⋅ = ∆ ⋅ − ∆ ⋅ + ++ ∫ t t

xv v v t t vt v

c cI d si ciK K K c c Kc K

( )2

20

1 1cos cos sinπ

θ θθ

= ⋅ = − ∆ ⋅ + ∆ ⋅ + ∫ t t

xt v v v v

c cI d ci sic K K K K

( )2

30

ln cos cos sinπ

θ θ θ

= + ⋅ = ∆ ⋅ − ∆ ⋅

∫ t tx t v

v v

c cI c K d si ciK K

( )

2

1 2 20

1 1cos sin cosπ

θ θθ

= ⋅ = ∆ ⋅ − ∆ ⋅ +

∫ t ty

v v vt v

c cI d si ciK K Kc K

( )2

20

1 1cos sin cosπ

θ θθ

= ⋅ = ∆ ⋅ − ∆ ⋅ + ∫ t t

yt v v v v

c cI d ci sic K K K K

( )2

30

ln sin sin cos ln2

π

θ θ θπ

= + ⋅ = −∆ ⋅ − ∆ ⋅ + +

∫ t t ty t v

v v t v

c c cI c K d si ciK K c K

( ) ( )2t t

v v

c cK Kci Ci Ci π∆ = − +

( ) ( )2t t

v v

c cK Ksi Si Si π∆ = − +

( ) ( )cos

x

uCi x du

u

∞

= ∫

( ) ( )sin

x

uSi x du

u

∞

= ∫

68

Glossary

CHAPTER 1 AND 4

C Channel capacity

M Number of transmitted signal

N Tolerance on error signal, average power of a noise

R Entropy rate

S Average power of a signal and noise

Si, P Average power of a signal

T Sampling period

X Sent message from information source

X’ Received message at destination

d Distance between a point and the origin in multi dimensions

f Frequency

( )f t An analog signal with respect to time

( )n t Noise in time domain

α Tolerance of noise

Bω Bandwidth of a channel

sω Sampling frequency

iω Bandwidth of a signal

( )x t Encoded input signal to a channel

( )y t , ( )iy t Output signal of a degraded and healthy machine in time domain

69

CHAPTER 2 , 3 AND 5

Bond graphs

C Bond graph element for compliance

GY Bond graph element for gyrator

I Bond graph element for inertia

MGY Bond graph element for modulated gyrator

MSe Bond graph element for modulated effort source

MTF Bond graph element for modulated transformer

R Resistive bond graph element

TF Bond graph element for transformer

Induction Motor

J Moment of inertia

sL , mL , rL Stator self inductance, mutual inductance and rotor self inductance

Mα , M β Magneto motive forces in α and β axes

sR , rR Stator and rotor resistances

pP Number of pole pairs

eT Electro-magnetic torque

aV , bV , cV Sinusoidal input voltages

h Angular momentum of shaft

iα , iβ Transformed currents from a, b, c phases to α and β axes

ai , bi , ci 3-phase currents in stator

1m ~ 5m Moduli of transformers for 3-phase to 2-phase transformation

rn , sn Number of coil turns of rotor and stator

70

p1 Transferred power from motor to pump

kr , kmr Modulated gyrator moduli of rotor (k = 1 ~ 34)

αϕ , βϕ Magnetic flux in α and β axes

ω Angular velocity of rotor

Centrifugal Pump

A Volute cross-section area

iA Impeller discharge area

sA Volute surface area for friction

1iB , 2iB Inlet and outlet flow passage width of impeller

1C , 2C Absolute fluid velocity at inlet and outlet of impeller

1tC , 2tC Tangential component of 1C and 2C

Ctank Compliance of reservoir (water tank)

xF , yF Force exerted on impeller in x and y direction

0G Volute gap between volute outer surface and impeller outlet

Iimp, Iimp Equivalent inertia of fluid in impeller

Iout, Ipipe Equivalent inertia of fluid throughout piping

Icl Equivalent inertia of fluid through tongue clearance

vK Volute gap variation coefficient

iP∆ Pressure difference through impeller

xP Force exerted on impeller in x-direction

yP Force exerted on impeller in y-direction

Q Flow rate at a volute section

clQ Flow rate at tongue clearance

impQ Impeller flow rate

71

leakQ Flow rate through the clearance space between the shroud and housing

outQ Flow rate at pump outlet

netQ Flow rate at volute exit ( net imp clQ Q Q= − )

Rcl Fluid loss in tongue clearance

diskR , Rm Mechanical loss due to fluid friction

gR , Nonlinear modulus of MGY connecting motor to pump

impR , Rimp Fluid loss in impeller

inR , Rin Fluid loss at inlet pipe

leakR , Rleak Fluid leakage loss due to gaps inside pumps

outR , Rout Fluid loss at outlet pipe

voluteR , Rnet Fluid loss at pump volute

T Torque transferred to fluid via impeller

1U , 2U Blade tip velocity at inlet and outlet of impeller

1W , 2W Relative fluid velocity at inlet and outlet of impeller

tc Clearance between impeller and volute tongue

xe , ye Eccentric displacement of impeller in x and y direction

f Friction factor

gi Nonlinear modulus of MGY connecting motor to pump

gx, gy Nonlinear modulus of MGY connecting impeller forces to a rotor

1ir , 2ir Inlet and outlet radius of impeller

1mv , 2mv Radial component of 1C and 2C

vw Volute width

1α , 2α Angle between 1C ( 2C ) and 1U ( 2U )

1β , 2β Angle between 1C ( 2C ) and 1W ( 2W )

θ Angle from volute tongue in the direction of impeller rotation

72

ρ Fluid density

τ Shear stress ω , ( iω ) Angular velocity of impeller

73

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Vita

Ji-Hoon Choi was born in Taegu, Korea on July 30, 1971, the first son of two

children of Yong Choi and Myung-Soon Kim. He received B.S. degree from Hong-Ik

University, Seoul, Korea and M.S. degree from the University of Texas at Austin, USA

in 1995 and 2001, respectively, both in mechanical engineering. Between 1995 and 1997,

he served Korean Army as an ROTC officer in a unit for military weapons maintenance.

He studied mechanical behavior of materials such as fatigue and fracture in the Graduate

School of Hong-Ik University for one year from March 1998. He then transferred to the

University of Texas at Austin in August 1999 for M.S. degree. After completing M.S.

degree, he joined the doctoral program at the same University in August 2001. He was

awarded a research assistantship from his academic advisor that supported his study. He

also worked as a teaching assistant for such courses as System Dynamics/Control and

Mechatronics.

Permanent address: 563-403 MIDOPA APT

870 Hwa-Jung

Koyang, 417-270, Korea

This dissertation was typed by Ji-Hoon Choi.