Jimmy C. Mathews
Advisors: Dr. Joseph Picone
Dr. David Gao
The BOND GRAPH Methodology for Modeling of Continuous
Dynamic Systems and its Application in Powertrain Design
INTELLIGENT POWERTRAIN DESIGN
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Outline
•
Dynamic Systems and Modeling
•
Bond Graph Modeling Concepts
Introduction and basic elements of bond graphs
Causality and state space equations
•
System Models and Applications using the Bond Graph
Approach
Electrical Systems
Mechanical Systems
•
The Generic Modeling Environment (GME) and Bond Graph
Modeling
•
Some Future Concepts
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•
Dynamic Systems
Related sets of processes and reservoirs (forms in which matter or energy exists) through
which material or energy flows,
characterized by continual change
.
•
Common Dynamic Systems
electrical, mechanical, hydraulic, thermal among numerous others.
•
Real

time Examples
moving automobiles, miniature electric circuits, satellite positioning systems
•
Physical systems
Interact, store energy, transport or dissipate energy among subsystems
•
Ideal Physical Model (IPM)
The starting point of modeling a physical system is mostly the IPM.
•
To perform simulations, the IPM must first be transformed into
mathematical descriptions, either using
Block diagrams
or
Equation
descriptions
•
Downsides
–
laborious procedure
,
complete derivation of the mathematical
description has to be repeated in case of any modification to the IPM [3].
Dynamic Systems and Modeling
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Computer Aided Modeling and Design of Dynamic Systems
•
Basic Concepts
Physical
System
Schematic
Model
Classical Methods, Block
Diagrams OR Bond
Graphs
Simulation and
Analysis
Software
Output
Data Tables &
Graphs
STEP 1: Develop an ‘engineering model’
STEP 2: Write differential equations
STEP 3: Determine a solution
STEP 4: Write a program
The Big
Question??
Differential
Equations
GME +
Matlab/Simulink
Fig 1.
Modeling Dynamic Systems [1]
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Bond Graph Methodology
•
Invented by Henry Paynter in 1961, later elaborated by his students Dean C. Karnopp and
Ronald C. Rosenberg
•
An
abstract
representation of a system where a collection of components interact with each
other through energy ports and are placed in a system where energy is exchanged [2]
•
A domain

independent graphical description of dynamic
behavior of physical systems
•
Consists of subsystems which can either describe
idealized elementary processes or non

idealized
processes [3]
•
System models will be constructed using
a uniform
notations for all types of physical system
based on
energy flow
•
Powerful tool for modeling engineering systems, especially when different physical domains
are involved
•
A form of object

oriented physical system modeling
Fig 2.
Subsystems of a bond graph [3]
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•
Bond Graphs
•
Conserves the
physical structural information
as well as the
nature of sub

systems
which are
often lost in a block diagram.
•
When the IPM is changed, only the corresponding parts of a bond graphs have to be changed
.
Amenable to modification for ‘model development’ and ‘what if?’ situations.
•
Use analogous power and energy variables in all domains, but allow the special features of
the separate fields to be represented.
•
The only physical variables required to represent all energetic systems are
power variables
[effort (e) & flow (f)] and
energy variables
[momentum
p (t) and displacement q (t)].
•
Dynamics of physical systems are derived by the application of
instant

by

instant
energy
conservation. Actual inputs are exposed.
•
Linear and non

linear elements are represented with the same symbols; non

linear kinematics
equations can also be shown.
•
Provision for
active bonds
. Physical information involving information transfer, accompanied by
negligible amounts of energy transfer are modeled as
active bonds
.
The Bond Graph Modeling Formalism
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The Bond Graph Modeling Formalism (contd..)
•
A Bond Graph’s Reach
Electrical
Mechanical
Translation
Mechanical
Rotation
Hydraulic/Pneumatic
Chemical/Process
Engineering
Thermal
Magnetic
Figure 3.
Multi

Energy Systems Modeling using Bond Graphs
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The Bond Graph Modeling Formalism (contd..)
•
Introductory Examples
•
Electrical Domain
Power Variables
:
Electrical Voltage (u) & Electrical Current (i)
Power in the system:
P =
u * i
Constitutive Laws:
u
R
=
i
* R
u
C
= 1/C * (∫
i
dt)
u
L
= L * (d
i
/dt); or i = 1/L * (∫
u
L
dt)
Fig
4
.
A series RLC circuit [
4]
Fig.
5
Electric elements with power ports [
4]
Represent different elements with visible
ports
(
figure 5
)
To these ports, connect
power bonds
denoting energy exchange
The voltage over the elements are
different
The current through the elements is the
same
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The R
–
L

C circuit
The common current becomes a “1

junction” in the bond graphs.
Note: the current through all connected bonds is the same, the voltages sum to zero
The Bond Graph Modeling Formalism (contd..)
Fig 6.
The RLC Circuit and its equivalent Bond Graph [4]
1
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The Bond Graph Modeling Formalism (contd..)
Fig 7.
The Spring Mass Damper System and
its equivalent Bond Graph [4]
•
Mechanical Domain
Mechanical elements like Force, Spring, Mass, Damper are similarly dealt with.
Power variables
:
Force (F) & Linear Velocity (v)
Power in the system: P =
F
*
v
Constitutive laws:
F
d
=
α
* v
F
s
= K
S
* (∫
v
dt) = 1/C
S
* (∫ v dt)
F
m
= m * (d
v
/dt); or
v
= 1/m * (∫
F
m
dt); Also,
F
a
= force
The common velocity becomes a “1

junction” in the bond graphs. Note: the velocity of all
connected bonds is the same, the forces sum to zero)
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Analogies
!
Lets compare! We see the following analogies between the mechanical and electrical
elements:
•
The
Damper
is analogous to the
Resistor
.
•
The
Spring
is analogous to the
Capacitor
, the mechanical
compliance
corresponds with the
electrical
capacity
.
•
The
Mass
is analogous to the
Inductor
.
•
The
Force
source is analogous to the
Voltage
source.
•
The common
Velocity
is analogous to the loop
Current
.
Notice that the bond graphs of both the RLC circuit and the Spring

mass

damper system are
identical. Still wondering how??
•
The bond graph modeling language is
domain

independent
.
•
Each of the various physical domains is characterized by a particular conserved quantity.
Table 1
illustrates these domains with corresponding flow (f), effort (e), generalized
displacement (q), and generalized momentum (p).
•
Note that
power = effort x flow
in each case.
The Bond Graph Modeling Formalism (contd..)
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f
flow
e
effort
q =
∫f dt
generalized
displacement
p =
∫e dt
generalized
momentum
Electromagnetic
i
current
u
voltage
q =
∫i dt
charge
λ
= ∫u dt
magnetic flux
linkage
Mechanical
Translation
v
velocity
f
force
x =
∫v dt
displacement
p = ∫f dt
momentum
Mechanical Rotation
ω
angular velocity
T
torque
θ
=
∫
ω
dt
angular displacement
b = ∫
T
dt
angular
momentum
Hydraulic /
Pneumatic
φ
volume flow
P
pressure
V
=
∫
φ
dt
volume
τ
=
∫P dt
momentum of a
flow tube
Thermal
T
temperature
F
S
entropy flow
S =
∫f
S
dt
entropy
Chemical
μ
chemical potential
F
N
molar flow
N =
∫f
N
dt
number of moles
Table 1.
Domains with corresponding flow, effort, generalized displacement and generalized
momentum
The Bond Graph Modeling Formalism (contd..)
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The Bond Graph Modeling Formalism (contd..)
•
Foundations of Bond Graphs
Based on the assumptions that satisfy basic principles of physics;
a. Law of Energy Conservation is applicable
b. Positive Entropy production
c. Power Continuity
•
Closer look at Bonds and Ports
Power port
or
port
: The contact point of a sub model where an ideal connection will be
connected; location in a system where energy transfer occurs
Power bond
or
bond
: The connection between two sub models; drawn by a single line (
Fig. 8
)
Bond denotes ideal energy flow between two sub models; the energy entering the bond on
one side immediately leaves the bond at the other side (
power continuity
).
A
B
e
f
(directed bond from A to B)
Fig. 8
Energy flow between two sub models represented by
ports and bonds [4]
Energy flow along the bond has
the physical dimension of power,
being the product of two variables
effort
and
flow
called power

conjugated variables
Power bond viewed as
interaction
of energy
and
bilateral signal flow
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The Bond Graph Modeling Formalism (contd..)
•
Bond Graph Elements
–
9 elements
Drawn as letter combinations (
mnemonic codes
) indicating the type of element.
C
storage element for a
q

type variable
,
e.g. capacitor (stores charge), spring (stores displacement)
L
storage element for a
p

type variable
,
e.g. inductor (stores flux linkage), mass (stores momentum)
R
resistor dissipating free energy,
e.g. electric resistor, mechanical friction
Se, Sf
sources,
e.g. electric mains (voltage source), gravity (force source),
pump (flow source)
TF
transformer,
e.g. an electric transformer, toothed wheels, lever
GY
gyrator,
e.g. electromotor, centrifugal pump
0, 1
0 and 1 junctions, for ideal connection of two or more sub

models
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The Bond Graph Modeling Formalism (contd..)
•
Storage Elements
Two types; C
–
elements & I
–
elements;
q
–
type
and
p
–
type
variables are conserved
quantities and are the result of an accumulation (or
integration
) process; they are the
state
variables
of the system.
C
–
element
(capacitor, spring, etc.)
q
is the conserved quantity, stored by accumulating the net flow,
f
to the storage element
Resulting balance equation
d
q
/dt =
f
Fig. 9
Examples of C

elements [4]
An element relates
effort
to the
generalized displacement
1

port element that stores and gives up energy without loss
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The Bond Graph Modeling Formalism (contd..)
I
–
element
(inductor, mass, etc.)
p
is the conserved quantity, stored by accumulating the net effort,
e
to the storage element.
Resulting balance equation
d
p
/dt =
e
Fig. 10
Examples of I

elements [4]
For an inductor, L [H] is the inductance and for a mass, m [kg] is the mass. For all other
domains, an I
–
element can be defined.
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The Bond Graph Modeling Formalism (contd..)
R
–
element
(electric resistors, dampers, frictions, etc.)
R
–
elements dissipate free energy and energy flow towards the resistor is always positive.
Algebraic relation between effort and flow, lies principally in 1
st
or 3
rd
quadrant.
e
= r * (
f
)
Fig. 11
Examples of Resistors [4]
If the resistance value can be controlled by an external signal, the resistor is a modulated
resistor, with mnemonic
MR
. E.g. hydraulic tap
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The Bond Graph Modeling Formalism (contd..)
Fig. 12
Examples of Sources [4]
Sources
(voltage sources, current sources, external forces, ideal motors, etc.)
Sources represent the system

interaction with its environment. Depending on the type of the
imposed variable, these elements are drawn as
Se
or
Sf
.
Fig. 13
Example of Modulated Voltage
Source [4]
When a system part needs to be excited by a known signal form, the source can be modeled
by a modulated source driven by some signal form (
figure 13
).
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The Bond Graph Modeling Formalism (contd..)
Transformers
(toothed wheel, electric transformer, etc.)
An ideal transformer is represented by
TF
and is power continuous (i.e. no power is stored or
dissipated). The transformations can be within the same domain (toothed wheel, lever) or
between different domains (electromotor, winch).
e1
= n *
e2
&
f2
= n *
f1
Efforts are transduced to efforts and flows to flows;
n
is the
transformer ratio
.
Fig. 14
Examples of Transformers [4]
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The Bond Graph Modeling Formalism (contd..)
Gyrators
(electromotor, pump, turbine)
An ideal gyrator is represented by
GY
and is power continuous (i.e. no power is stored or
dissipated). Real

life realizations of gyrators are mostly transducers representing a domain

transformation.
e1
= r *
f2
&
e2
= r *
f1
r is the gyrator ratio and is the only parameter required to describe both equations.
Fig. 15
Examples of Gyrators [4]
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The Bond Graph Modeling Formalism (contd..)
Junctions
Junctions couple two or more elements in a power continuous way; there is no storage or
dissipation at a junction.
0
–
junction
Represents a
node at which all efforts of the connecting bonds are equal
. E.g. a parallel
connection in an electrical circuit.
The sum of flows of the connecting bonds is zero, considering the sign.
0
–
junction can be interpreted as the generalized Kirchoff’s Current Law.
Equality of efforts (like electrical voltage) at a parallel connection.
Fig. 16
Example of a 0

Junction [4]
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The Bond Graph Modeling Formalism (contd..)
1
–
junction
Is the dual form of the 0

junction (roles of effort and flow are exchanged).
Represents
a node at which all flows of the connecting bonds are equal
. E.g. a series
connection in an electrical circuit.
The efforts of the connecting bonds sum to zero.
1

junction can be interpreted as the generalized Kirchoff’s Voltage Law.
In the mechanical domain, 1

junction represents a
force

balance
, and is a generalization of
Newton’ third law.
Additionally, equality of flows (like electrical current) through a series connection.
Fig. 17
Example of a 1

Junction [4]
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The Bond Graph Modeling Formalism (contd..)
Some Miscellaneous Stuff!
Power Direction:
The power is positive in the direction of the power bond. If power is
negative, it flows in the opposite direction of the half

arrow.
Typical Power flow directions
R, C, and I elements have
an incoming bond
(half arrow towards the element)
Se, Sf:
outgoing bond
TF
–
and GY
–
elements (transformers and gyrators): one bond incoming and one bond
outgoing, to show the ‘natural’ flow of energy.
These are
constraints
on the model!
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The Bond Graph Modeling Formalism (contd..)
•
Causal Analysis
Causal analysis
is the determination of the signal direction of the bonds
Establishes the cause and effect relationships between the bonds
Indicated in the bond graph by a
causal
stroke;
the
causal stroke
indicates the direction of the
effort signal
.
The result is a
causal bond graph
, which can be seen as a compact block diagram.
Causal analysis covered by modeling and simulation software packages that support bond
graphs; Enport, MS1, CAMP

G, 20 SIM
Fig. 18
Causality Assignment [4]
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Causality
Type
Elements
Representation
Interpretation
Fixed
Se
Sf
Constrained
TF
OR
Se
e
f
e
Se
f
Sf
e
f
e
Sf
f
TF
e
1
e
2
f
2
f
1
n
e
1
e
2
f
2
f
1
TF
n
e
1
e
2
f
2
f
1
TF
n
TF
e
1
e
2
f
2
f
1
n
Causal Constraints:
Four different types of constraints need to be discussed prior to
following a systematic procedure for bond graph formation and causal analysis
The Bond Graph Modeling Formalism (contd..)
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Causality
Type
Elements
Representation
Constrained
GY
OR
0 Junction
1 Junction
Preferred
C
Integral Causality (Preferred) Derivative Causality
L
Integral Causality (Preferred) Derivative Causality
e
1
e
2
f
2
f
1
GY
r
e
1
e
2
f
2
f
1
GY
r
The Bond Graph Modeling Formalism (contd..)
0
1
OR
any other combination where
exactly one bond brings in the effort
variable
OR
any other combination where
exactly one bond has the causal
stroke away from the junction
C
C
L
L
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Causality
Type
Elements
Representation
Indifferent
R
OR
R
R
The Bond Graph Modeling Formalism (contd..)
Some notes on
Preferred Causality
(C, I)
Causality determines whether an integration or differentiation w.r.t time is adopted in storage
elements.
Integration has a preference over differentiation
because:
1. At integrating form, initial condition must be specified.
2. Integration w.r.t. time can be realized physically; Numerical differentiation is not physically
realizable, since information at future time points is needed.
3. Another drawback of differentiation: When the input contains a step function, the output will
then become infinite.
Therefore, integrating causality is the preferred causality
. C

element will have effort

out
causality and I

element will have flow

out causality
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•
Electrical Circuit # 1 (R

L

C) and its Bond Graph model
0
0
0
0
1
0
1
0
+

U0
U1
U2
U3
Examples
U1
U2
U3
0:
U12
0:
U23
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Intelligent Powertrain Design
0:
U12
0
1
0
1
0
Se : U
C : C
0:
U23
R : R
I : L
U1
U3
U2
Examples (contd..)
1
Se : U
C : C
R : R
I : L
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Examples (contd..)
The Causality Assignment Algorithm:
1
Se : U
C : C
R : R
I : L
1.
2.
1
Se : U
C : C
R : R
I : L
1
Se : U
C : C
R : R
I : L
3.
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Examples (contd..)
•
Electrical Circuit # 2 and its Bond Graph model
R1
R2
R3
C1
C2
L1
R1
R2
R3
C2
C1
L1
•
A DC Motor and its Bond Graph model
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Examples (contd..)
•
A Drive Train Schematic and its Bond Graph model
Bond Graph without Drive Shaft Compliance [9]
Bond Graph with Drive Shaft Compliance [9]
S
E
TF
1
TF
0
τ
L
ω
L
τ
R
ω
R
ω
i
Differential Ratio
Transmission Ratio
A Drive Train Schematic [9]
S
F
TF
1
TF
0
τ
L
ω
L
τ
R
ω
R
ω
i
Drive Shaft Compliance
0
C
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Examples (contd..)
•
Schematic for Tire and Suspension and their Bond Graph model
Suspension model for one
wheel and anti

roll bar
Bond Graph of a wheel

tire system
–
Longitudinal Dynamics [9]
Bond Graph of a wheel

tire system
–
Transverse Dynamics [9]
Schematic of a tire and
suspension [9]
Bond Graph of a wheel

tire
system
–
Vertical Dynamics [9]
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Generation of Equations from Bond Graphs
1
Se : U
C : C
R : R
I : L
1
2
4
3
Fig. 19
Bond Graph of a series RLC
circuit
•
A causal bond graph contains all information to derive the
set of state equations.
•
Either a set of
Ordinary first

order Differential Equations
(ODE) or a set of
Differential and Algebraic Equations
(DAE).
•
Write the set of mixed differential and algebraic equations.
•
For a bond graph with
n
bonds, 2
n
equations can be
formed,
n
equations each to compute effort and flow or
their derivatives.
•
Then, the algebraic equations are eliminated, to get final
equations in state

variable form.
For the given RLC circuit,
Se = e1= U;
e2 = R * f2;
(d
e3
/dt) = (1/C) * f3;
(d
f4
/dt) = (1/L) * e4;
f1 = f4; f2 = f4; f3 = f4;
e4 = e1

e2

e3
Hence,
e1

e2

e3 = U
–
(R * f2)
–
e3 = U
–
(R * f4)
–
e3
(d
f4
/dt) = (1/L) * (U
–
(R * f4)
–
e3)







(i)
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Also,
(d
e3
/dt) = (1/C) * f3 = (1/C) * f4








(ii)
In matrix form,
(d
x
/dt) = Ax + Bu
(d
e3/
dt)
0
1/C
e3
0
=
+
U
(d
f4
/dt)

1/L

R/L
f4
1/L
Generation of Equations from Bond Graphs (contd..)
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The Bond Graph Metamodeling Environment in GME
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Applications in GME Metamodeling Environment
•
RLC Circuit
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•
DC Motor
Applications in GME Metamodeling Environment (contd..)
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Applications in GME Metamodeling Environment (contd..)
DC Motor model
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Future Concepts
•
Defining the
GME
Approach for analysis of Bond Graphs
[1]
Conventional Approach
Probable
GME
/
Matlab
Approach
1.
Determination of Physical System
and specifications from the
requirements.
2.
Draw a functional Block Diagram.
3.
Transform the physical system into a
schematic.
4.
Use Schematic and obtain a
mathematical model, a block diagram
or a state representation.
5.
Reduce the block diagram to a close
loop system.
6.
Analyze, design and test.
1.
Identify the physical system elements
and represent a word Bond Graph.
2.
Represent a bond graph model in
GME
.
3.
GME
interpreters generate equations
in a suitable form (e.g. state

space
variable matrix form) suitable for
analysis in
Matlab
.
4.
Use
Matlab
, to analyze, design and
test.
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Intelligent Powertrain Design
Future Concepts (contd..)
Fig 20.
The Simulation Generation Process [7]
Bond Graph Interpreters
in GME ??
•
Creating Bond Graph Interpreters
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Intelligent Powertrain Design
•
Advanced Bond Graph Techniques
Expansion of Bond Graphs to Block Diagrams
Bond Graph Modeling of Switching Devices
Hierarchical modeling using Bond Graphs
Use of port

based approach for Co

simulation
Future Concepts (contd..)
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References
1.
Granda
J
.
J,
“Computer
Aided
Design
of
Dynamic
Systems”
http
:
//gaia
.
csus
.
edu/~grandajj/
2.
Wong
Y
.
K
.
,
Rad
A
.
B
.
,
“Bond
Graph
Simulations
of
Electrical
Systems,”
The
Hong
Kong
Polytechnic
University,
1998
3.
http
:
//www
.
ce
.
utwente
.
nl/bnk/bondgraphs/bond
.
htm
4.
Broenink
J
.
F
.
,
"Introduction
to
Physical
Systems
Modeling
with
Bond
Graphs,"
University
of
Twente,
Dept
.
EE,
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