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```									       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

Intelligent Powertrain Design                                 Page 1 of 42
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].

Intelligent Powertrain Design                                                      Page 2 of 42
Computer Aided Modeling and Design of Dynamic Systems
• Basic Concepts

Physical                        STEP 1: Develop an ‘engineering model’
System                               STEP 2: Write differential equations
STEP 3: Determine a solution
Schematic                           STEP 4: Write a program
Model
The Big
Classical Methods, Block                                 Question??
Diagrams OR Bond
Graphs

GME +
Equations                                    Data Tables &
Graphs

Simulation and
Analysis
Software
Fig 1. Modeling Dynamic Systems [1]

Intelligent Powertrain Design                                                           Page 3 of 42
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
Fig 2. Subsystems of a bond graph [3]
energy flow

• Powerful tool for modeling engineering systems, especially when different physical domains
are involved
• A form of object-oriented physical system modeling

Intelligent Powertrain Design                                                      Page 4 of 42
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.

Intelligent Powertrain Design                                                  Page 5 of 42
The Bond Graph Modeling Formalism (contd..)
• A Bond Graph’s Reach

Mechanical
Rotation

Mechanical                                     Hydraulic/Pneumatic
Translation

Thermal                               Chemical/Process
Electrical                                                                   Engineering

Magnetic
Figure 3. Multi-Energy Systems Modeling using Bond Graphs

Intelligent Powertrain Design                                                       Page 6 of 42
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
Fig 4. A series RLC circuit [4]
Constitutive Laws:                       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
The current through the elements is the
same
Fig. 5 Electric elements with power ports [4]
Intelligent Powertrain Design                                                                        Page 7 of 42
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]

Intelligent Powertrain Design                                                              Page 8 of 42
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)
Intelligent Powertrain Design                                                                  Page 9 of 42
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.

Intelligent Powertrain Design                                                   Page 10 of 42
The Bond Graph Modeling Formalism (contd..)
Table 1. Domains with corresponding flow, effort, generalized displacement and generalized
momentum
f                 e               q = ∫f dt               p = ∫e dt
flow              effort          generalized              generalized
displacement              momentum
Electromagnetic                       i                 u               q = ∫i dt                λ = ∫u dt
current           voltage            charge                 magnetic flux
Mechanical                            v                 f               x = ∫v dt                p = ∫f dt
Translation                        velocity           force          displacement               momentum

Mechanical Rotation                  ω                  T               θ = ∫ω dt                b = ∫T dt
angular velocity      torque       angular displacement           angular
momentum
Hydraulic /                           φ                P               V = ∫φ dt                τ = ∫P dt
Pneumatic                        volume flow        pressure           volume                momentum of a
flow tube
Thermal                             T                   FS             S = ∫fS dt
temperature        entropy flow        entropy

Chemical                             μ                FN               N = ∫fN dt
chemical potential   molar flow       number of moles

Intelligent Powertrain Design                                                              Page 11 of 42
The Bond Graph Modeling Formalism (contd..)
• Foundations of Bond Graphs
Based on the assumptions that satisfy basic principles of physics;            e
a. Law of Energy Conservation is applicable
A           B
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 power-
conjugated variables
Power bond viewed as interaction
Fig. 8 Energy flow between two sub models represented by
of energy and bilateral signal flow         ports and bonds [4]

Intelligent Powertrain Design                                                             Page 12 of 42
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

Intelligent Powertrain Design                                                               Page 13 of 42
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

Intelligent Powertrain Design                                                              Page 14 of 42
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.

Intelligent Powertrain Design                                                              Page 15 of 42
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

Intelligent Powertrain Design                                                        Page 16 of 42
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]

Intelligent Powertrain Design                                                                          Page 17 of 42
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]

Intelligent Powertrain Design                                                                Page 18 of 42
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]

Intelligent Powertrain Design                                                               Page 19 of 42
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]

Intelligent Powertrain Design                                                    Page 20 of 42
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]

Intelligent Powertrain Design                                                       Page 21 of 42
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!

Intelligent Powertrain Design                                                  Page 22 of 42
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]

Intelligent Powertrain Design                                                    Page 23 of 42
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
Elements    Representation                     Interpretation
Type
e
Se
e                        Se
Se
f
f
Fixed                                                                           e
Sf
e                        Sf
Sf
f                                      f
e1                     e2
Constrained              TF
e1                e2
f1
TF                                TF
f2
n                         f1       n           f2
OR                       e1                    e2
e1                e2
f1
TF                                TF
f2
n                                  n
f1                    f2
Intelligent Powertrain Design                                                   Page 24 of 42
The Bond Graph Modeling Formalism (contd..)
Causality
Elements                              Representation
Type
GY                e1              e2                        e1                    e2
f1
GY                    OR
f1
GY
f2                                              f2
r                                         r
Constrained 0 Junction                                               OR    any other combination where
exactly one bond brings in the effort
variable
0
1 Junction
OR    any other combination where
exactly one bond has the causal
stroke away from the junction
1
C            Integral Causality (Preferred)                 Derivative Causality

C                                                  C
Preferred
L            Integral Causality (Preferred)                 Derivative Causality

L                          L
Intelligent Powertrain Design                                                             Page 25 of 42
The Bond Graph Modeling Formalism (contd..)

Causality
Elements                         Representation
Type
R                                          OR
Indifferent                                        R                              R

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

Intelligent Powertrain Design                                                       Page 26 of 42
Examples
•     Electrical Circuit # 1 (R-L-C) and its Bond Graph model

U2
U1                             U3

+
-

U0

0                  0                0
U1                 U2               U3

0: U12             0:   U23

0     1            0     1          0

Intelligent Powertrain Design                                            Page 27 of 42
Examples (contd..)

R:R             I:L

0: U12         0: U23

1         0        C:C
Se : U       0       1      0
U1             U2                U3

R:R

Se : U     1          I:L

C:C

Intelligent Powertrain Design                                         Page 28 of 42
Examples (contd..)
The Causality Assignment Algorithm:

1.                 R:R                      2.            R:R

Se : U                  1   I:L                   Se : U    1               I:L

C:C                                    C:C

3.   R:R

Se : U     1         I:L

C:C

Intelligent Powertrain Design                                            Page 29 of 42
Examples (contd..)
•     Electrical Circuit # 2 and its Bond Graph model
R1                                      L1
C1             C2

C2

L1   R2
R3

R1   R2        R3
C1

•     A DC Motor and its Bond Graph model

Intelligent Powertrain Design                                  Page 30 of 42
Examples (contd..)
•     A Drive Train Schematic and its Bond Graph model

SE
Transmission Ratio

TF
ωi          τL        ω
L
1       TF          0
τR       ωR

Differential Ratio

Bond Graph without Drive Shaft Compliance [9]

A Drive Train Schematic [9]
SF
Drive Shaft Compliance

TF        C
τL        ω
ωi
L
1        0      TF          0
τR        ωR

Bond Graph with Drive Shaft Compliance [9]

Intelligent Powertrain Design                                                                            Page 31 of 42
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]

Intelligent Powertrain Design                                     Page 32 of 42
Generation of Equations from Bond Graphs

R:R                    •   A causal bond graph contains all information to derive the
set of state equations.
2              •   Either a set of Ordinary first-order Differential Equations
(ODE) or a set of Differential and Algebraic Equations
1                               (DAE).
Se : U                  1            I:L
4          •   Write the set of mixed differential and algebraic equations.
3                      •   For a bond graph with n bonds, 2n equations can be
formed, n equations each to compute effort and flow or
C:C                        their derivatives.
Fig. 19 Bond Graph of a series RLC    •   Then, the algebraic equations are eliminated, to get final
circuit                                   equations in state-variable form.

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)

Intelligent Powertrain Design                                                             Page 33 of 42
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

Intelligent Powertrain Design                                                                Page 34 of 42
The Bond Graph Metamodeling Environment in GME

Intelligent Powertrain Design                     Page 35 of 42
Applications in GME Metamodeling Environment
•     RLC Circuit

Intelligent Powertrain Design                   Page 36 of 42
Applications in GME Metamodeling Environment (contd..)
•     DC Motor

Intelligent Powertrain Design                             Page 37 of 42
Applications in GME Metamodeling Environment (contd..)

DC Motor model

Intelligent Powertrain Design                             Page 38 of 42
Future Concepts
•     Defining the GME Approach for analysis of Bond Graphs [1]

Conventional Approach          Probable GME / Matlab Approach
1. Determination of Physical System       1. Identify the physical system elements
and specifications from the               and represent a word Bond Graph.
requirements.                          2. Represent a bond graph model in
2. Draw a functional Block Diagram.          GME.
3. Transform the physical system into a   3. GME interpreters generate equations
schematic.                                in a suitable form (e.g. state-space
4. Use Schematic and obtain a                variable matrix form) suitable for
mathematical model, a block diagram       analysis in Matlab.
or a state representation.             4. Use Matlab, to analyze, design and
5. Reduce the block diagram to a close       test.
loop system.
6. Analyze, design and test.

Intelligent Powertrain Design                                              Page 39 of 42
Future Concepts (contd..)

•     Creating Bond Graph Interpreters

Bond Graph Interpreters
in GME ??

Fig 20. The Simulation Generation Process [7]

Intelligent Powertrain Design                                                   Page 40 of 42
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

Intelligent Powertrain Design                          Page 41 of 42
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

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,"           Integrated
Modeling of Physical System Dynamics

9.      Karnopp D., “System Dynamics: Modeling and simulation of mechatronic systems”

Intelligent Powertrain Design                                                                   Page 42 of 42

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