INTELLIGENT POWERTRAIN DESIGN

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INTELLIGENT POWERTRAIN DESIGN
The BOND GRAPH Methodology for Modeling of
Continuous Dynamic Systems
Jimmy C. Mathews Advisors Dr. Joseph Picone
Dr. David Gao Powertrain Design Tools
Project
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Outline

• Dynamic Systems and Modeling
• Bond Graph Modeling Concepts
• Sample Applications of Bond Graph Modeling
• The Generic Modeling Environment (GME) and Bond
  Graph Modeling
• 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

Physical System
STEP 1 Develop an engineering model STEP 2
Write differential equations STEP 3
Determine a solution STEP 4 Write a
program
Schematic Model
The Big Question??
Classical Methods, Block Diagrams OR Bond Graphs
GME Matlab/Simulink
Output Data Tables Graphs
Differential Equations
Simulation and Analysis Software
Fig 1. Modeling Dynamic Systems 1
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• Bond Graphs vs. Block Diagrams 5, 8

• Block Diagrams
  
  Early attempt to deal with
  heterogeneity, closely related to the emergence
  of automatic control, nice example of information
  hiding, very successful and good environments
  like Simulink, Easy V, and VisSim available
  presently.
• Familiar and versatile graphical notation to
  represent Signal Flow.
• Downsides i. Do not provide a
  suitable notation for depicting physical system
  models because not all block diagrams represent
  physical processes. ii. Energetic
  Coupling between elements/systems - - - energy
  exchange implies interaction, i.e. a bilateral,
  two-way influence of each system on the other.
  
  Block diagrams fundamentally depict a unilateral
  influence of one system on another. Hence, to
  describe energetic interaction of two
  systems/elements in terms of signal flow, the
  output of one should be the input of another and
  vice versa.

iii. When two systems interact energetically, we
must have the block representation as in figure 2
(or its converse). In contrast, the block
diagrams shown below might represent possible
operations on signals or information, but neither
represents any possible energetic interaction.
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• Bond Graphs vs. Block Diagrams (contd..)

• Bond Graphs
  
  Close correspondence between
  the bond graph and the physical system it
  represents.
• Conserves the physical structural information as
  well as the nature of sub-systems which are often
  lost in a block diagram.
• Can be directly derived from the IPM. When the
  IPM is changed, only the corresponding part of a
  bond graph has to be changed. Advantage of making
  the model very amenable to modification for
  model development and what if? situations.
• Account for all the energy in physical systems
  and provide a common link among various
  engineering systems. 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
  e(t) and displacement F(t).
• The 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.
• Some more advantages will be discussed after
  dealing with the concept of causality.

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• Bond Graph Methodology

• Invented by Henry Paynter in the 1961, later
  elaborated by his students Dean C. Karnopp and
  Ronald C. Rosenberg
• A Bond Graph is 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 bond-graph model consists of subsystems which
  can either describe idealized elementary
  processes or non-idealized processes.
  Non-idealized processes can either be non-linear
  equation models or bond graph sub models 3.
• Subsystems can have two type of ports power
  ports and signal ports.
• Power ports specify both an effort variable and
  flow variable. Signal ports specify only one
  variable, a flow or an effort or a mathematical
  variable.

• Two types of knots in bond graphs, 0 junctions
  and 1 junctions represent domain-independent
  generalizations of Kirchoffs laws.
• Connects are called bonds, indicate power between
  various subsystems. The half arrows indicates
  positive power flow orientation. The full arrows
  indicate signal flows.
• Bond is characterized by the value of an
  instantaneous power, computed as the product of
  effort and flow variables (e.g. voltage and
  current in the electrical domain).

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• The Bond Graph Modeling Formalism

• Bond Graphs Reach?

Fig 4. Multi-Energy Systems Modeling using Bond
Graphs
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The Bond Graph Modeling Formalism (contd..)

• Two different physical domains are considered
  the Electrical and the Mechanical domains.
• Electrical Domain
• To facilitate conversion of electrical circuits
  to bond graphs, represent different elements
  (Voltage Source, Resistor, Capacitor, Inductor)
  with visible ports (figure 5).
• To these ports, we connect power bonds denoting
  energy exchange between elements.

• Mechanical Domain
• Mechanical elements like Force, Spring, Mass,
  Damper are similarly dealt with.

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The Bond Graph Modeling Formalism (contd..)
The R L - C circuit The power being exchanged
by a port with the rest of the system is a
product of the voltage and the current P u
i The equations for the resistor, capacitor and
inductor are u_R i R u_C 1/C (?idt) u_L
L (di/dt) or i 1/L (?u_L dt)
1
Fig 6. The RLC Circuit 4
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The Bond Graph Modeling Formalism (contd..)

• The Spring-Mass-Damper System
• Port variables on the bond graph elements are
  force on the element port and velocity of the
  element port. P F v
• The equations for the damper (damping
  coefficient, a), spring (coefficient, KS) and
  mass are F_d a v
• F_s KS (?v dt) 1/CS (? vdt)
• F_m m (dv/dt) or v 1/m (?F_m dt) Also,
  F_a force

Fig 7. The Spring Mass Damper System 4
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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??
• Various physical domains are distinguished that
  each 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.
• Also note, the bond graph modeling language is
  domain-independent.

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The Bond Graph Modeling Formalism (contd..)
Table 1. Domains with corresponding flow, effort,
generalized displacement and generalized momentum
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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.
• 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 called power-conjugated variables.

(directed bond from A to B)
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The Bond Graph Modeling Formalism (contd..)

• Two views of Interpretation of Power Bond
• 1. As an interaction of energy connected
  subsystems for a load to each other by their
  energy exchange embodies an exchange of a
  physical quantity. 2. As a bilateral
  signal flow interpreted as effort and flow
  flowing in opposite direction, thus determining
  the computational direction of the bond
  variables w.r.t. one of the sub models, effort
  is the input and flow is the output and vice
  versa for the other sub model.
• Determining the direction of Effort and Flow
• During modeling it need not be decided what the
  computational direction of the bond variables is,
  however it is necessary to derive the
  mathematical model (set of differential
  equations) from the graph. Process of
  determining the computational direction of the
  bond variables is called causal analysis
  indicated in the graph by the so-called causal
  stroke, (indicating the direction of the effort),
  called the causality of the bond (figure 9).

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The Bond Graph Modeling Formalism (contd..)

• Bond Graph Elements
• Bond graph elements are drawn as letter
  combinations (mnemonic codes) indicating the type
  of element. The bond graph elements are the
  following
• 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 1junctions, for ideal connecting
  two or more sub models

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The Bond Graph Modeling Formalism (contd..)

• Storage Elements
• Two types C elements I elements qtype
  and ptype 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

Equations for linear capacitor and linear
spring dq/dt i, u (1/C) q dx/dt v,
F k x (1/C) x
For a capacitor, C F is the capacitance and for
a spring, K N/m is the stiffness and C m/N
the compliance.
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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 f
Fig. 11 Examples of I - elements 4
Equations for linear inductor and linear
mass d?/dt u, i (1/L) ? dp/dt
F, V (1/m) p 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. 12 Examples of Resistors 4
Electrical resistance value ? given by Ohms
law u R I If the resistance value can be
controlled by an external signal, the resistor is
a modulated resistor, with mnemonic MR. E.g.
hydraulic tap the position of the tap is
controlled from the outside, and it determines
the value of the resistance parameter. In the
thermal domain, the dissipator irreversibly
produces thermal energy, the thermal port is
drawn as a kind of source of thermal energy. The
R becomes an RS.
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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. Source elements are used to give a variable
a fixed value, for example, in case of a point in
a mechanical system with a fixed position, a Sf
with value 0 is used (fixed position means
velocity zero).
Fig. 13 Examples of Sources 4
When a system part needs to be excited, often a
known signal form is needed, which can be modeled
by a modulated source driven by some signal form
(figure 14).
Fig. 14 Example of Modulated Voltage Source 4
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The Bond Graph Modeling Formalism (contd..)
Transformers (toothed wheel, electromotor,
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. Only one dimensionless parameter n is
required to describe effort transduction and flow
transduction. n is a defined as follows e1
and f1 belong to the bond pointing towards TF.
Fig. 15 Examples of Transformers 4
If n is not constant, it becomes an input signal
to the modulated transformer, MTF.
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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. R
has a physical dimension (same as R-element),
since r is the relation between effort and flow.
Fig. 16 Examples of Gyrators 4
Gyrator is defined by one bond pointing towards
and other bond pointing away. If r is not
constant, the gyrator is a modulated gyrator, a
MGY.
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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. The power
direction determines the sign of flows all
inward pointing bonds get a plus and all outward
pointing bonds get a minus. 0 junction can be
interpreted as the generalized Kirchoffs Current
Law. Additionally, equality of efforts (like
electrical voltage) at a parallel connection.
Fig. 17 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). Represe
nts 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. Again, the power
direction determines the sign of flows all
inward pointing bonds get a plus and all outward
pointing bonds get a minus. 1- junction can be
interpreted as the generalized Kirchoffs 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. 18 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. A port that has incoming power bond
consumes power. E.g. R, C. If power is negative,
it flows in the opposite direction of the
half-arrow. R, C, and I elements have an
incoming bond (half arrow towards the element) as
standard, which results in positive parameters
when modeling reallife components. For source
elements, the standard is outgoing, as sources
mostly deliver power to the rest of the
system. For TF and GYelements (transformers
and gyrators), the standard is to have one bond
incoming and one bond outgoing, to show the
natural flow of energy. These are constraints
on the model! Duality The role of effort and
flow in the storage elements (C, I) are
interchanged. They are each others dual form.
A gyrator can be used to decompose an I-element
to a GY and C element and vice versa.
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The Bond Graph Modeling Formalism (contd..)

• Causal Analysis
• Causal analysis is the determination of the
  signal direction of the bonds. The energetic
  connection (bond) is now interpreted as a
  bi-directional signal flow. 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
• Four different types of constraints need to be
  discussed prior to following a systematic
  procedure for bond graph formation and causal
  analysis.
• Causal Constraints
• Fixed Causality (Se, Sf)
• Fixed causality is the case when equations allow
  only one of the two port variables to be the
  outgoing variable. An effort source (Se) has by
  definition always its effort variable as signal
  output, and has the causal stroke outwards. This
  causality is called effort-out causality or
  effort causality. A flow source (Sf) clearly has
  a flow-out causality or flow causality.
• May occur at non-linear elements, where the
  equations for that port cannot be inverted (e.g.
  division by zero).

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The Bond Graph Modeling Formalism (contd..)
Constrained Causality (TF, GY, 0-junction,
1-junction) Constrained causality is defined
when a relations exist between the causalities of
the different ports of the element. At a TF, one
of the ports has effort-out causality and the
other has flow-out causality.
OR Similarly, at a GY, both ports have either
effort-out causality or flow-out causality. At
a 0junction, where all efforts are the same,
exactly one bond must bring in the effort. This
implies that 0junctions always have exactly one
causal stroke at the side of the junction. The
causal condition at a 1junction is the dual form
of the 0-junction. All flows are equal, thus
exactly one bond will bring in the flow, implying
that exactly one bond has the causal stroke away
from the 1junction. 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.
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The Bond Graph Modeling Formalism (contd..)
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. (figures 10
11). Indifferent causality (Linear
R) Indifferent causality is used, when there are
no causal constraints! At a linear R, it does not
matter which of the port variables is the
output. There is no difference choosing the
current as incoming variable and the voltage as
outgoing variable, or the other way around.
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Examples

• Electrical Circuit 1 (R-L-C) and its Bond Graph
  model

U2
U3
U1
STEP 1 Determine which physical domains exist
in the system and identify all basic elements
like C, I, R, Se, Sf, TF, GY. Give each element a
unique name. STEP 2 Indicate a reference effort
for each domain in the Ideal Physical Model
(reference velocity with positive direction for
the mechanical domains). Note that references in
the mechanical domain have a direction. Generatio
n of the connection / junction structure. STEP
3 Identify all other efforts (mechanical
domains velocities) and give them unique
names. STEP 4 Draw these efforts (mechanical
velocities), and not the references, graphically
by 0junctions (mechanical 1junctions). Keep if
possible, the same layout as the IPM.
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• Examples (contd..)

STEP 5 Identify all effort differences
(mechanical velocity(flow) differences) needed
to connect the ports of all elements enumerated
in Step 1. Differences have a unique name. STEP
6 Construct the effort differences using a
1junction (mechanical flow differences with a
0junction) and draw them as such in the
graph. STEP 4 0 0 0 STEP 5, 6
0 1 0 1 0 STEP 7 The junction structure is
now ready and the elements can be connected.
Connect the port of all elements found at step 1
with the 0junctions of the corresponding efforts
or effort differences (mechanical 1junctions of
the corresponding flows or flow
differences). STEP 8 Simplify the resulting
graph by applying the following simplification
rules 1. A junction between two
bonds can be left out, if the bonds have a
through power direction (one bond incoming, the
other outgoing). 2. A bond between two the same
junctions can be left out, and the junctions can
join into one junction. 3. Two separately
constructed identical effort or flow differences
can join into one effort or flow difference.
U2
U3
U1
0 U23
0 U12
U1
U2
U3
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Examples (contd..)
STEP 7
STEP 8
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Examples (contd..)
The Causality Assignment Algorithm STEP 1a.
Chose a fixed causality of a source element,
assign its causality, and propagate this
assignment through the graph using the causal
constraints. Go on until all sources have their
causalities assigned. STEP 1b. Chose a not yet
causal port with fixed causality (non-invertible
equations), assign its causality, and propagate
this assignment through the graph using the
causal constraints. Go on until all ports with
fixed causality have their causalities
assigned. STEP 2 Chose a not yet causal port
with preferred causality (storage elements),
assign its causality, and propagate this
assignment through the graph using the causal
constraints. Go on until all ports with preferred
causality have their causalities assigned.
1a.
2.
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Examples (contd..)
STEP 3 Chose a not yet causal port with
indifferent causality, assign its causality, and
propagate this assignment through the graph using
the causal constraints. Go on until all ports
with indifferent causality have their causalities
assigned.
3.

• Electrical Circuit 2 and its Bond Graph model

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• Examples (contd..)

• A DC Motor and its Bond Graph model

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• Generation of Equations from Bond Graphs

• 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.

Fig. 19 Bond Graph of a series RLC circuit
For the given RLC circuit, Se e1 U e2
R f2 (de3/dt) (1/C) f3 (df4/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 (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 0 U (df4/dt
) -1/L -R/L f4 1/L
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• Generation of Equations from Bond Graphs (contd..)

Some Points to Note One of the most important
features of bond graphs is easy determination of
causality. For computer algorithms to solve
equations, representing the physics of real
systems, it is essential that proper input and
output causality be maintained. State variables
and computational problems are known completely
after assigning causality, even before the
modeler derives a single equation. Modeling in
terms of bond graphs helps one focus on modeling
the physical effects without bothering about the
computational issues such as generation of a
consistent system of equations. B.G. on one hand
relate closely to the structure of the system
being modeled, while on the other hand, they
contain enough information to derive other system
representations like state-space equations. B.G
can be drawn or a B.G. description of the system
can be created before causality is considered. In
contrast, causality has to be considered before a
block diagram can be drawn. E.g. the decision as
to whether a resistor has a voltage or current as
output has to be made before a block diagram can
be constructed. In B.G., causality can be
automatically assigned after the system has been
described.
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The Bond Graph Metamodeling Environment in GME
39
Applications in GME Metamodeling Environment

• RLC Circuit

40
Applications in GME Metamodeling Environment
(contd..)
DC Motor model
41
Future Concepts

• Defining the GME Approach for analysis of Bond
  Graphs 1

42
Future Concepts (contd..)

• Creating Bond Graph Interpreters

43
Future Concepts (contd..)

• Advanced Bond Graph Techniques
• Expansion of Bond Graphs to Block Diagrams
• Bond Graph Modeling of Switching Devices
• Bond Graphs as Object-oriented physical-systems
  modeling
• Hierarchical modeling using Bond Graphs
• Use of port-based approach for Co-simulation

44
References

• Granda J. J, Computer Aided Design of Dynamic
  Systems http//gaia.csus.edu/grandajj/
• Wong Y. K., Rad A. B., Bond Graph Simulations of
  Electrical Systems, The Hong Kong Polytechnic
  University, 1998
• http//www.ce.utwente.nl/bnk/bondgraphs/bond.htm
• Broenink J. F., "Introduction to Physical Systems
  Modeling with Bond Graphs,"University of Twente,
  Dept. EE, Netherlands.
• Granda J., Reus J., "New developments in Bond
  Graph Modeling Software Tools The Computer Aided
  Modeling Program CAMP-G and MATLAB," California
  StateUniversity, Sacramento
• http//www.bondgraphs.com/about2.html
• Vashishtha D., Modeling And Simulation of Large
  Scale Real Time Embedded Systems, M.S. Thesis,
  Vanderbilt University, May 2004
• Hogan N. "Bond Graph notation for Physical System
  models," IntegratedModeling of Physical System
  Dynamics