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.