CS 367: Model-Based Reasoning Lecture 15 (03/12/2002)

1. CS 367: Model-Based ReasoningLecture 15 (03/12/2002) Gautam Biswas

2. Today’s Lecture • Last Lectures: • Modeling with Bond Graphs • Today’s Lecture: • Review • Bond Graphs and Causality • State Space Equations from Bond Graphs • More Complex Examples • 20-SIM

3. Review: Modeling with Bond Graphs • Based on concept of reticulation • Properties of system lumped into processes with distinct • parameter values • Lumped Parameter Modeling • Dynamic System Behavior: function of energy • exchange between components • State of physical system – defined by distribution of • energy at any particular time • Dynamic Behavior: Current State + Energy exchange • mechanisms

4. Review: Modeling with Bond Graphs • Exchange of energy in system through ports • 1 ports: C, I: energy storage elements; R: dissipator • 2 ports: TF, GY • Exchange with environment: through sources and sinks: Se & Sf • Behavior Generation: two primary principles • Continuity of power • Conservation of energy enforced at junctions: 3 ports 0- (parallel) junction 1- (series) junction

5. Review: Junctions • Electrical Domain: 0- enforces Kirchoff’s current law, 1- enforces Kirchoff’s voltage law • Mechanical Domain: 0- enforces geometric compatibility of single force + set of velocities that must sum to 0; 1- enforces dynamic equilibrium of forces associated with a single velocity • Hydraulic Domain: 0- conservation of volume flow rate, when a set of pipes join 1- sum of pressure drops across a circuit (loop) involving a single flow must sum to 0. Sometimes junction structures are not obvious.

6. Component Behaviors

7. Building Electrical Models • For each node in circuit with a distinct potential create a 0-junction • Insert each 1 port circuit element by adjoining it to a 1-junction and inserting the 1-junction between the appropriate of 0-junctions. • Assign power directions to bonds • If explicit ground potential, delete corresponding 0-junction and its adjacent bonds • Simplify bond graph (remove extraneous junctions) Hydraulic, thermal systems similar, but mechanical different

8. Electrical Circuit: Example

9. Electrical Circuits: Example 2 Try this one:

10. Building Mechanical Models • For each distinct velocity, establish a 1-junction (consider both absolute and relative velocities) • Insert the 1-port force-generating elements between appropriate pairs of 1-junctions; using 0-junctions; also add inertias to respective 1-junctions (be sure they are properly defined wrt inertial frame) • Assign power directions • Eliminate 0 velocity 1-junctions and their bonds • Simplify bond graph

11. Example: Mechanical Model

12. Mechanical Model: Example 2 Try this one:

13. Behavior of System: State Space Equations Linear System Nonlinear System

14. State Equations • Linear • Nonlinear

15. State Space: Standard form Single nth order form n first-order coupled equations In general, can have any combination in between

16. More complex example g

17. More complex example (2):

18. Causality in Bond Graphs • To aid equation generation, use causality relations among variables • Bond graph looks upon system variables as interacting variable pairs • Cause effect relation: effort pushes, response is a flow • Indicated by causal stroke on a bond e f B A

19. Causality for basic multiports Note that a lot of the causal considerations are based on algebraic relations

20. Causality Assignment Procedure

21. Causality Assignment: Example

22. Causality Assignment: Double Oscillator

23. Causality Assignment: Example 3 Try this one:

24. Generate equations from Bond Graphs • Step 1: Augment bond graph by adding • Numbers to bonds • Reference power direction to each bond • A causal sense to each e,f variable of bond

25. Equation generation procedure

26. Equation generation example

27. Equation Generation: Example 2

28. k1 k2 F0(t) m1 m2 b1 b2 H. W. Problem 1 : • Two springs, masses, & damper friction all linear. • F0(t) = f1 = constant. • Build bond graph; state equations. • Simulate for various parameter values.

29. V g M B K V m g k V0(t) H. W. Problem 2 : • Bond graph. • Derive state equations in terms of energy variables. • Simulate in 20-Sim with diff. Parameter values. Comment on results. Input : Velocity at bottom of tire

30. Extending Modeling to other domains • Fluid Systems • e(t) – Pressure, P(t) • f(t) – Volume flow rate, Q(t) • Momentum, p = e.dt = Pp, integral of pressure • Displacement, q = Q.dt = V, volume of flow • Power, P(t).Q(t) • Energy (kinetic): Q(t).dPp • Energy (potential): P(t).dV Fluid Port: a place where we can define an average pressure, P and a volume flow rate, Q Examples of ports: (i) end of a pipe or tube (ii) threaded hole in a hydraulic pump

31. Fluid Ports • Flow through ports transfers energy • P – force/unit area • Q – volume flow rate • P.Q = power = force . displacement / time • Moving fluid also has kinetic energy • But it can be ignored if Next time: fluid capacitors (tanks), resistances (pipes), and sources (pumps)