Jimmy C. Mathews

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16 Νοε 2013 (πριν από 3 χρόνια και 7 μήνες)

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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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Intelligent Powertrain Design

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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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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Future Concepts (contd..)

Fig 20.
The Simulation Generation Process [7]

Bond Graph Interpreters
in GME ??


Creating Bond Graph Interpreters

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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,"

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