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The Theoretical Underpinnings of the Bond Graph Methodology

In this lecture, we shall look more closely at the theoretical underpinnings of the bond graph methodology: the four base variables, the properties of capacitive and inductive storage elements, and the duality principle.

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The Theoretical Underpinnings of the Bond Graph Methodology

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  1. In this lecture, we shall look more closely at the theoretical underpinnings of the bond graph methodology: the four base variables, the properties of capacitive and inductive storage elements, and the duality principle. We shall also introduce the two types of energy transducers: the transformers and the gyrators, and we shall look at hydraulic bond graphs. The Theoretical Underpinnings of the Bond Graph Methodology

  2. The four base variables of the bond graph methodology Properties of storage elements Hydraulic bond graphs Energy transducers Electromechanical systems The duality principle The diamond rule Table of Contents

  3. Beside from the two adjugate variables e and f, there are two additional physical quantities that play an important role in the bond graph methodology: p = e · dt The Four Base Variables of the Bond Graph Methodology Generalized Momentum: q = f · dt Generalized Position:

  4. C I e = R( f ) q = C( e ) p = I( f ) Resistor: Capacity: Inductivity:   e f q p  R Arbitrarily non-linear functions in 1st and 3rd quadrants  There cannot exist other storage elements besides C and I. Relations Between the Base Variables

  5. f = C · de “Normal” capacitive equation, as hitherto commonly encountered. dt Linear Storage Elements q = C( e ) General capacitive equation: q = C · e Linear capacitive equation: Linear capacitive equation differentiated:

  6. In hydrology, the two adjugate variables are the pressure p and the volume flow q. Here, the pressure is considered the potential variable, whereas the volume flow plays the role of the flow variable. The capacitive storage describes the compressibility of the fluid as a function of the pressure, whereas the inductive storage models the inertia of the fluid in motion. [W] = [N/ m2] · [m3 / s] Phydr = p · q = kg · m -1 · s-2] · [m3 · s-1] = [kg · m2 · s-3] Hydraulic Bond Graphs I

  7. qin Compression: p C : 1/c p = c · ( qin – qout ) dp qout Dq dt Laminar Flow: q Dp Dp G : k R : 1/k p2 p1 q= k · Dp q q = k · ( p1 – p2) Turbulent Flow: q Hydro q= k · sign(Dp) · |Dp| p1 p2 Hydraulic Bond Graphs II

  8. Beside from the elements that have been considered so far to describe the storage of energy ( C and I ) as well as its dissipation(conversion to heat) ( R ), two additional elements are needed, which describe the general energy conversion, namely the Transformer and the Gyrator. Whereas resistors describe the irreversible conversion of free energy into heat, transformers and gyrators are used to model reversible energy conversion phenomena between identical or different forms of energy. Energy Conversion

  9. TF m (3) (4)  (m ·e2 ) · f1= e2 · f2 f e f e 2 1 2 1  f2 = m · f1  The transformer may either be described by means of equations (1) and (2) or using equations (1) and (4). Transformers e1 = m · e2 (1) Transformation: (2) e1· f1= e2 · f2 Energy Conservation:

  10. e2 = e1/ m e1 = m · e2 TF TF f2 = m · f1 f1 = f2 / m m m e e f f e f e f 1 1 1 2 1 2 2 2  As we have exactly one equation for the effort and another for the flow, it is mandatory that the transformer compute one effort variable and one flow variable. Hence there is one causality stroke at the TF element. The Causality of the Transformer

  11. m = 1/M m = r1 /r2 m = A Electrical Transformer Mechanical Gear Hydraulic Shock Absorber (in AC mode) Examples of Transformers

  12. GY r (3) (4)  (r ·f2 ) · f1= e2 · f2 f e f e 2 1 2 1  e2 = r · f1  The gyrator may either be described by means of equations (1) and (2) or using equations (1) and (4). Gyrators e1 = r · f2 (1) Transformation: (2) e1· f1= e2 · f2 Energy Conservation:

  13. e1 = r · f2 f2 = e1/ r GY GY f1 = e2 / r e2 = r · f1 r r f e e f f f e e 1 2 2 2 1 1 2 1  As we must compute one equation to the left, the other to the right of the gyrator, the equations may either be solved for the two effort variables or for the two flow variables. The Causality of the Gyrator

  14. Examples of Gyrators r =  The DC motor generates a torque tm proportional to the armature current ia , whereas the resulting induced Voltage ui is proportional to the angular velocity m.

  15. Causality conflict (caused by the mechanical gear) τB3 τB1 τk1 uRa FB2 ω12 Fk2 τB1 τB1 τG FG ui ua ω2 ω1 ω2 ω2 ω1 ω1 ω1 ω2 τ ia ia ia ia v v v v τJ2 uLa τJ1 v Fm -m·g Example of an Electromechanical System

  16. It is always possible to “dualize” a bond graph by switching the definitions of the effort and flow variables. In the process of dualization, effort sources become flow sources, capacities turn into inductors, resistors are converted to conductors, and vice-versa. Transformers and gyrators remain the same, but their transformation values are inverted in the process. The two junctions exchange their type. The causality strokes move to the other end of each bond. The Duality Principle

  17. i2 iL i1 i1 i1 i0 iC u0 u0 u1 u0 uC uC uC u1 i2 uC u0 iL i1 i1 i1 u0 uC iC uC u0 i0 1st Example The two bond graphs produce identical simulation results.

  18. τB3 τB1 τk1 uRa FB2 Fk2 τB1 τB1 τG FG ui ua τ -m·g uLa τJ1 Fm -m·g ω12 ω1 ω2 ia v ω12 ω1 ω2 ω1 ω2 ω1 ω1 ω2 ω2 ω1 ω2 ia ω1 ω2 ia v v v ia v v ia ia v ia ui ua τ τJ2 τB1 Fk2 uRa uLa τB1 τB1 FB2 Fm τk1 τJ2 τG τB3 FG τJ1 v ω2 ω1 ia v v 2nd Example

  19. It is always possible to dualize bond graphs only in parts. Partial Dualization It is particularly easy to partially dualize a bond graph at the transformers and gyrators. The two conversion elements thereby simply exchange their types. For example, it may make sense to only dualize the mechanical side of an electromechanical bond graph, whereas the electrical side is left unchanged. However, it is also possible to dualize the bond graph at any bond. Thereby, the “twisted” bond is turned into a gyrator with a gyration of r=1. Such a gyrator is often referred to as symplectic gyrator in the bond graph literature.

  20. Any physical system with concentrated parameters can be described by a bond graph. However, the bond graph representation is not unique, i.e., several different-looking bond graphs may represent identical equation systems. One type of ambiguity has already been introduced: thedualization. However, there exist other classes of ambiguities that cannot be explained by dualization. Manipulation of Bond Graphs

  21. More efficient Diamond k12 m2 F C:1/k12 B2 R:B2 R:B1 B12 Fk12 v12 R:B2 R:B1 v2 FB2 v1 FB1 0 v2 v2 v1 v2 FB2 v1 v1 FB1 1 v2 SE:F 0 1 F Fk12 v2 Fk12 m1 1 SE:F 1 v12 v2 Fm2 v1 Fm1 F v2 B1 v1 v2 Fm2 v1 Fm1 I:m2 I:m1 1 FB12 0 FB12 v12 FB12 I:m2 I:m1 v12 Fk12 Fk12 Fk12 +FB12 +FB12 +FB12 FB12 v12 Fk12 C:1/k12 R:B12 Different variables R:B12 The Diamond Rule 

  22. Cellier, F.E. (1991), Continuous System Modeling, Springer-Verlag, New York, Chapter 7. References

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