Transcript Document

INTELLIGENT POWERTRAIN DESIGN
The BOND GRAPH Methodology for Modeling of Continuous
Dynamic Systems and its Application in Powertrain Design
Jimmy C. Mathews
Advisors: Dr. Joseph Picone
Dr. David Gao
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 and Modeling
• 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].
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Computer Aided Modeling and Design of Dynamic Systems
• Basic Concepts
STEP 1: Develop an ‘engineering model’
Physical
System
STEP 2: Write differential equations
STEP 3: Determine a solution
Schematic
Model
STEP 4: Write a program
The Big
Question??
Classical Methods, Block
Diagrams OR Bond
Graphs
Differential
Equations
Fig 1. Modeling Dynamic Systems [1]
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GME +
Matlab/Simulink
Output
Data Tables &
Graphs
Simulation and
Analysis
Software
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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]
Fig 2. Subsystems of a bond graph [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
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The Bond Graph Modeling Formalism
• 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.
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The Bond Graph Modeling Formalism (contd..)
• A Bond Graph’s Reach
Mechanical
Rotation
Hydraulic/Pneumatic
Mechanical
Translation
Thermal
Electrical
Magnetic
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Chemical/Process
Engineering
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:
Constitutive Laws:
P=u*i
Fig 4. A series RLC circuit [4]
uR = i * R
uC = 1/C * (∫i dt)
uL = L * (di/dt); or i = 1/L * (∫uL dt)
Represent different elements with visible
ports (figure 5)
To these ports, connect power bonds
denoting energy exchange
The voltage over the elements are
different
Fig. 5 Electric elements with power ports [4]
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The current through the elements is the
same
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The Bond Graph Modeling Formalism (contd..)
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
1
Fig 6. The RLC Circuit and its equivalent Bond Graph [4]
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The Bond Graph Modeling Formalism (contd..)
• 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:
Fd = α * v
Fs = KS * (∫v dt) = 1/CS * (∫ v dt)
Fm = m * (dv/dt); or v = 1/m * (∫Fm dt); Also, Fa = force
Fig 7. The Spring Mass Damper System and
its equivalent Bond Graph [4]
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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The Bond Graph Modeling Formalism (contd..)
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.
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The Bond Graph Modeling Formalism (contd..)
Table 1. Domains with corresponding flow, effort, generalized displacement and generalized
momentum
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
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
FS
entropy flow
S = ∫fS dt
entropy
Chemical
μ
chemical potential
FN
molar flow
N = ∫fN dt
number of moles
Mechanical Rotation
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The Bond Graph Modeling Formalism (contd..)
• Foundations of Bond Graphs
e
Based on the assumptions that satisfy basic principles of physics;
B
A
a. Law of Energy Conservation is applicable
f
b. Positive Entropy production
c. Power Continuity
(directed bond from A to B)
• 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).
Energy flow along the bond has
the physical dimension of power,
being the product of two variables
effort and flow called powerconjugated variables
Power bond viewed as interaction
of energy and bilateral signal flow
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Fig. 8 Energy flow between two sub models represented by
ports and bonds [4]
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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
dq/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
dp/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 1st or 3rd 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..)
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. 12 Examples of Sources [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).
Fig. 13 Example of Modulated Voltage
Source [4]
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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 domaintransformation.
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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The Bond Graph Modeling Formalism (contd..)
Causal Constraints: Four different types of constraints need to be discussed prior to
following a systematic procedure for bond graph formation and causal analysis
Causality
Type
Elements
Representation
e
Se
e
f
Se
Fixed
TF
e1
f1
f
e
f
TF
n
OR
e1
f1
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Se
e
Sf
Sf
Constrained
Interpretation
TF
n
Sf
e2
f2
e2
f
e1
e2
TF
f1
e1
n
f2
e2
TF
f2
f1
n
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f2
The Bond Graph Modeling Formalism (contd..)
Causality
Type
Elements
GY
Representation
e1
f1
GY
r
e2
f2
Constrained 0 Junction
e1
OR
f1
GY
r
e2
f2
any other combination where
exactly one bond brings in the effort
variable
OR
0
1 Junction
any other combination where
exactly one bond has the causal
stroke away from the junction
OR
1
C
Integral Causality (Preferred)
Derivative Causality
C
C
Preferred
L
Integral Causality (Preferred)
L
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Derivative Causality
L
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The Bond Graph Modeling Formalism (contd..)
Causality
Type
Indifferent
Elements
Representation
R
Some notes on Preferred Causality
R
OR
R
(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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Examples
•
Electrical Circuit # 1 (R-L-C) and its Bond Graph model
U2
U1
U3
+
-
U0
0
0
0: U12
0
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0
U2
U1
1
U3
0:
0
1
U23
0
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Examples (contd..)
Se : U
0
R:R
I:L
0: U12
0: U23
1
0
1
0
U3
U2
U1
C:C
R:R
Se : U
1
I:L
C:C
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Examples (contd..)
The Causality Assignment Algorithm:
1.
1
Se : U
R:R
2.
R:R
Se : U
I:L
1
I:L
C:C
C:C
3.
Se : U
R:R
1
I:L
C:C
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Examples (contd..)
•
Electrical Circuit # 2 and its Bond Graph model
R1
C1
L1
C2
C2
L1
R2
R3
C1
•
R1
R2
R3
A DC Motor and its Bond Graph model
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Examples (contd..)
•
A Drive Train Schematic and its Bond Graph model
Transmission Ratio
SE
TF
ωi
1
τL
ω
0
TF
τR
L
ωR
Differential Ratio
Bond Graph without Drive Shaft Compliance [9]
A Drive Train Schematic [9]
SF
TF
Drive Shaft Compliance
C
τL
ωi
1
0
ω
0
TF
τR
L
ωR
Bond Graph with Drive Shaft Compliance [9]
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Examples (contd..)
•
Schematic for Tire and Suspension and their Bond Graph model
Schematic of a tire and
suspension [9]
Suspension model for one
wheel and anti-roll bar
Bond Graph of a wheel-tire
system – Vertical Dynamics [9]
Bond Graph of a wheel-tire system –
Longitudinal Dynamics [9]
Bond Graph of a wheel-tire system –
Transverse Dynamics [9]
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Generation of Equations from Bond Graphs
Se : U
R:R
•
2
•
1
1
4
I:L
3
•
•
C:C
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, 2n 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;
Hence,
e2 = R * f2;
(de3/dt) = (1/C) * f3;
(df4/dt) = (1/L) * e4;
f1 = f4; f2 = f4; f3 = f4;
e4 = e1 - e2 - e3
e1 - e2 - e3 = U – (R * f2) – e3 = U – (R * f4) – e3
(df4/dt) = (1/L) * (U – (R * f4) – e3)
- - - - - - - (i)
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Generation of Equations from Bond Graphs (contd..)
Also,
(de3/dt) = (1/C) * f3 = (1/C) * f4
- - - - - - - - (ii)
In matrix form, (dx/dt) = Ax + Bu
(de3/dt)
0
1/C
e3
=
(df4/dt)
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0
+
-1/L
-R/L
f4
U
1/L
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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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Applications in GME Metamodeling Environment (contd..)
•
DC Motor
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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..)
•
Creating Bond Graph Interpreters
Bond Graph Interpreters
in GME ??
Fig 20. The Simulation Generation Process [7]
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Future Concepts (contd..)
•
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
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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, Netherlands.
5.
Granda J. J., Reus J., "New developments in Bond Graph Modeling Software Tools: The
Computer
Aided
Modeling
Program
CAMP-G
and
MATLAB,"
California
State
University, Sacramento
6.
http://www.bondgraphs.com/about2.html
7.
Vashishtha D., “Modeling And Simulation of Large Scale Real Time Embedded Systems,” M.S.
Thesis, Vanderbilt University, May 2004
8.
Hogan
N.
"Bond
Graph
notation
for
Physical
System
models,"
Modeling of Physical System Dynamics
9.
Karnopp D., “System Dynamics: Modeling and simulation of mechatronic systems”
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Integrated