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

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CS 367 Model-Based ReasoningLecture 15
(03/12/2002)

• Gautam Biswas

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Todays Lecture

• Last Lectures
• Modeling with Bond Graphs
• Todays Lecture
• Review
• Bond Graphs and Causality
• State Space Equations from Bond Graphs
• More Complex Examples
• 20-SIM

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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

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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

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Review Junctions

• Electrical Domain 0- enforces Kirchoffs current
  law, 1- enforces Kirchoffs 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.

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Component Behaviors
Mechanics Electricity Hydraulic Thermal
Effort e(t) F, force V, voltage P, pressure T, temperature
Flow f(t) v, velocity i, current Q, volume flow rate , heat flow rate
Momentum p ?e.dt P, momentum ?, flux p ?P.dt ?P.dt Pp
Displacement q ?f.dt x, distance q, charge q ?Q.dt volume Q, heat energy
Power P(t)e(t).f(t) F(t).v(t) V(t).i(t) P(t).Q(t)
Energy E(p)?f.dp E(q)?e.dq ?v.dP (kinetic) ?F.dx (potential) ?i.d ? ?v.dq ?Q.dp ?P.dq
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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
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Electrical Circuit Example
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Electrical Circuits Example 2
Try this one
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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

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Example Mechanical Model
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Mechanical Model Example 2
Try this one
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Behavior of System State Space Equations

• Linear System
• Nonlinear System

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State Equations

• Linear
• Nonlinear

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State Space Standard form
Single nth order form
n first-order coupled equations
In general, can have any combination in between
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More complex example
g
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More complex example (2)
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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
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Causality for basic multiports
Note that a lot of the causal considerations are
based on algebraic relations
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Causality Assignment Procedure
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Causality Assignment Example
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Causality Assignment Double Oscillator
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Causality Assignment Example 3
Try this one
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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

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Equation generation procedure
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Equation generation example
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Equation Generation Example 2
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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.
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H. W. Problem 2

1. Bond graph.
2. Derive state equations in terms of energy
   variables.
3. Simulate in 20-Sim with diff. Parameter values.
   Comment on results.

Input Velocity at bottom of tire
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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

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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)