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Bond Graphs II

In this class, we shall deal with the effects of algebraic loops and structural singularities on the bond graphs of physical systems. We shall also analyze the description of mechanical systems by means of bond graphs. Bond Graphs II. Algebraic loops Structural singularities

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Bond Graphs II

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  1. In this class, we shall deal with the effects of algebraic loops and structural singularities on the bond graphs of physical systems. We shall also analyze the description of mechanical systems by means of bond graphs. Bond Graphs II

  2. Algebraic loops Structural singularities Bond graphs of mechanical systems Selection of state variables Example Table of Contents

  3. U0.e R1.e R2.e R2.e U0.e U0 .e = f(t) U0 .f = L1 .f + R1 .f dL1 .f /dt = U0 .e / L1 R2.e Choice U0.f R1.f R2.f R3.f R1.f R1.f L1.f U0.e R3 .f = R1 .f –R2 .f R2 .f= R2 .e / R2 R1 .f= R1 .e / R1 R1 .e = U0 .e –R2 .e Algebraic Loops R2 .e= R3 · R3 .f

  4. Causality conflict  Structural Singularity U0.e R1.e L1.e L1.e U0.e L1.e C1.f U0.f R1.f R1.f R1.f R2.f L1.f U0.e Structural Singularities U0 .e = f(t) U0 .f = C1 .f + R1 .f

  5. The two adjugate variables of the mechanical translational system are the force f as well as the velocity v. You certainly remember the classical question posed to students in grammar school: If one eagle flies at an altitude of 100 m above ground, how high do two eagles fly? Evidently, position and velocity are intensive variables and therefore should be treated as potentials. However, if one eagle can carry one sheep, two eagles can carry two sheep. Consequently, the force is an extensive variable and therefore should be treated as a flow variable. Bond Graphs of Mechanical Systems I

  6. Sadly, the bond graph community chose the reverse definition. “Velocity” gives the impression of a movement and therefore of a flow. We shall show that it is always possible mathematically to make either of the two assumptions (duality principle). Therefore: f P = f · v v force f = potential velocity v = flow Bond Graphs of Mechanical Systems II

  7. x m I : m v fB fk fk fB f B I R : B fB fI fk Dv v x x v Dx = fk / k 2 1 2 1 C : 1/k  Dv = (1 / k) · dfk /dt Dv Passive Mechanical Elements in Bond Graph Notation fI = m · dv /dt fB = B ·Dv k

  8. The “classical” representation of mechanical systems makes use of the absolute motions of the masses (position and velocity) as its state variables. The multi-body system representation in Dymolamakes use of the relative motions of the joints (position and velocity) as its state variables. The bond graph representation selects the absolute velocities of masses as one type of state variable, and the spring forces as the other. Selection of State Variables

  9. An Example I The cutting forces are represented by springs and friction elements that are placed between bodies at a 0-junction. The D’Alembert principle is formulated in the bond graph representation as a grouping of all forces that attack a body around a junction of type 1.

  10. v1 v2 v3 v3 v3 v3 v1 v31 v21 v2 v32 v1 v1 v1 v2 v2 v2 FI3 FBa FBa FBb FBb F FBb FBa Fk2 Fk1 FB2 FB2 FBd FB2 FBc FI2 FI1 An Example II The sign rule follows here automatically, and the modeler rarely makes any mistake relating to it.

  11. Borutzky, W. and F.E. Cellier (1996), “Tearing Algebraic Loops in Bond Graphs,” Trans. of SCS, 13(2), pp. 102-115. Borutzky, W. and F.E. Cellier (1996), “Tearing in Bond Graphs With Dependent Storage Elements,” Proc. Symposium on Modelling, Analysis, and Simulation, CESA'96, IMACS MultiConference on Computational Engineering in Systems Applications, Lille, France, vol. 2, pp. 1113-1119. References

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