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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

         • 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
             Electrical Systems
             Mechanical Systems

         • The Generic Modeling Environment (GME) and Bond Graph
         • 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
• 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
                                                                                    The Big
                          Classical Methods, Block                                 Question??
                             Diagrams OR Bond

                                                              GME +
                                     Differential         Matlab/Simulink            Output
                                     Equations                                    Data Tables &

                                                 Simulation and
     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                                     Hydraulic/Pneumatic

                                              Thermal                               Chemical/Process
         Electrical                                                                   Engineering

                                 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
                                                                           The current through the elements is the
                       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


                                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..)

      Lets compare! We see the following analogies between the mechanical and electrical

•     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
                                        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
  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
    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-
                                          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 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

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

                                               C                                                  C
                        L            Integral Causality (Preferred)                 Derivative Causality

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

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

                                U1                             U3



                                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


                                     Se : U     1          I:L


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


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


                                L1   R2

                                                          R1   R2        R3

  •     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

                                                                       Transmission Ratio

                                                                                              ωi          τL        ω
                                                                                            1       TF          0
                                                                                                           τR       ωR

                                                                                    Differential Ratio

                                                                       Bond Graph without Drive Shaft Compliance [9]

         A Drive Train Schematic [9]
                                                         Drive Shaft Compliance

                                           TF        C
                                                                  τL        ω
                                            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
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,"           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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