Modeling Energetic Systems with Multiple Energy Domains

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Modeling Energetic Systems with Multiple Energy
Domains

• ME4803 Motion Control
• Wayne Book
• Georgia Tech

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Energy DomainsCommon Features

• Product of effort (e) and flow (f) define power
  (P)
• Mechanical e force f velocity
• Fluid e pressure f volumetric flow
• Electrical e emf (voltage) f current
• Magnetic e magnetomotive force (Ni) f
  d(flux)/dt
• Integral of flow is displacement q
• Integral of effort is momentum p
• Integral of power is energy E

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The Tetrahedron of StateGeneral Case
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Tetrahedron for Mechanical Systems
Force, N
Displacement, m
momentum
Velocity, m/s
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Tetrahedron for Fluid Systems
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Tetrahedron for Electrical Systems
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Linkages between other vertices?
capacitance
Force displacement/compliance
resistance
Force damping x velocity
inertia
Momentum mass x velocity
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Representation of Energy Ports(Bond Graphs)
Energy port
Effort variable
e1
Sign of positive energy flow
Other components in system
R1
f1
Causal stroke
Flow variable
Component symbol (resistance)
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Resistive Elements in 4 Domains Dissipates
energy a static function between e and f
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For Capacitor, q static function(e)
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The Capacitor Stores Energy by displacement
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Capacitive Elements in 4 Domains
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The InertanceStores energy by momentum
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For Inertia, f static function(p)
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Inertance Elements in Four Domains
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What 2-ports Look Like
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What 3-ports Look Like
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Conservation principles in Bond Graphs
Single effort
Single flow
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Causality Distinguishes Inputs from
Outputs(independent from dependent, cause from
effect)
The causal stroke is by the component which
specifies the flow
The component on the other end specifies the
effort
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More Causal Possibilities and the transformer
and gyrator concepts
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Energy Ports are Demanded by Bond Graphs, but
Block Diagrams will Accomodate
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Using Bond Graphs(and other energy port
representations)

• Derive equations of motion
• Identify components
• Construct graph (connections and conservation)
• Simplify
• Assign sign conventions (half arrows)
• Determine causality (causal strokes)
• Identify causality issues (differential causality
  indicates algebraic constraints)
• Examine for state variables (system order) and
  causality conflicts
• Simulate
• Note that 1c) through 3 can be done automatically
  by computer (Dymola, Enport)

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Simple Electrical Graph Example
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More Complex Electrical Example
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Rules for Connections (T Mech)

• Establish 1 junction for each relevant, distinct
  velocity (absolute or relative)
• Velocity sources impose velocity constraints
  (including velocity 0)
• Insert springs and damping between relevant 1
  junctions by connecting through 0 junction.
• Force sources attach to 1 junctions
• Attach inertias to 1 junctions (which sum forces)
• Eliminate zero velocity 1 junctions and bonds
• Any 0 or 1 junctions with only two bonds are
  replaced by a single bond

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Simple Mechanical (Translation)
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More Complex Mechanical (Translation)
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Hydraulic Graph Construction

• Establish 0 junction for each relevant pressure
• Attach components to 1 junctions placed between
  appropriate 0 junctions
• Assign sign conventions to power
• Define reference pressure and eliminate that
  pressure 0 junction and associated bonds
• Simplify 0 or 1 junctions with only 2 bonds

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Simple Hydraulic
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More Complex Hydraulic
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Formulation of Equations from Bond Graphs

• ME4803-Motion Control
• Wayne Book

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Formulation of System Equations

• Equations for same system (say 4th order) can be
  in different forms
• one 4th order equation
• two, second order equations
• four, first order equations dx/dt f(x, u)
• First order equations are most versatile for
  simulation, analysis (linear and nonlinear)
• Choice of variables affects ease in deriving
  equations

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To remember the nomenclature

• F(1)ow is constant
• Eff(0)rt is constant
• Sign half arrow shows positive energy flow e f gt 0

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Causality Indications (review)Order of
Assignment of Causal Strokeindicating input of
effort
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Three Cases of Causality

• Basic equation formulation when causality of
  all bonds is determined by imposing
• source as input constraint
• Integral causality constraint (storage elements)
• Consequences of 0-, 1-junctions and TF, GY
• Extended formulation I -- causality of some bonds
  remains undetermined after above steps
• Extended formulation II -- derivative causality
  of some storage elements results (effort input to
  C, flow input to I)

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Example 1
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Assignment of Causality Example 1
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Example 2
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Assignment of CausalityExample 2
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Basic Formulation of Equations

• Identify
• bonds
• source variables
• energy variable derivatives for storage elements
• coenergy variables
• Write constitutive equations for storage elements
• Use structure and causality to eliminate coenergy
  variables

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Basic Formulation (cont.)

• Identify variables
• Write constitutive equations for storage elements
• Use structure and causality to eliminate coenergy
  variables
• express energy variable derivatives in junction
  variables
• replace coenergy variables until only inputs and
  energy variables remain
• The results are the state equations

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Case 2 Incomplete causality results
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Extended Formulaton ICausality Incompletely
Specified
Pick one R for effort in
Example with remaining acausal bonds
Each arbitrary selection that is necessary means
at least one algebraic equation to eliminate
intermediate variable in favor of state variables
Or pick for flow in
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Case 3 Conflict in integral causality
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Extended Formulation IIDerivative Causality of
Some Element
Assign causality for source (no propagation)
Assign causality for one C
Propagates to R
Propagates to other C
Results in differential Causality