Constraint Models: Bond Graphs

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Constraint ModelsBond Graphs

• COMP155 / EMGT155
• Nov 10 2008

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Constraint Models

• Constraint models are about balance
• Causality is included only for explanation
• Not part of the model itself
• Many laws of nature are symmetric and specified
  in terms of balances between various things

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Constraint Examples

• Biology
• change in population growth - death reaction
• Thermodynamics
• heat of system heat entering heat leaving
  heat produced
• Mechanics
• sum of forces acting on an object 0
• Electricity
• sum of currents at a node 0
• voltages around a closed circuit 0

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Bond Graphs - Overview

• Bond Graphs are a graphical tool for
  capturing the energy structure of systems
• originally defined by H. M. Paynter (MIT) in 1959
• A bond is flow path for energy and information
  between system components.
• Effort in one direction results in flow in
  the other direction

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Origin of Bond Graphs
electricalsubsystem
hydraulic subsystem
mechanicalsubsystem
from An Epistemic Prehistory of Bond Graphs by
H. M. Paynter
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Origin of Bond Graphs
electricalsubsystem
hydraulic subsystem
mechanicalsubsystem
from An Epistemic Prehistory of Bond Graphs by
H. M. Paynter
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Effort and Flow
www.bondgraphs.com
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Example Domain Mechanical
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Example Domain Electrical
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A Mechanical System
damper(friction)
spring
mass
gravity
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Equivalent Bond Graph
Junction connecting components represents a
system constraint.Flows (velocity) are equal and
effort (force) sums to zero.
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An Electrical System

• Kirchoffs Current Law sum of currents
  through a node is zero i1 i2 i3

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Equivalent Bond Graph
Straightforward topological mapping from circuit
diagram.
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Simplified Bond Graph
Previous model was simplified using 20-sim
tool. Junction connecting components represents a
system constraint.Flows (current) are equal and
effort (voltage) sums to zero.
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Effort and Flow Sources

• Effort source
• Flow source

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R-Elements

• R-elements are energy dissipaters
• electrical resistors, mechanical dampers or
  dashpots, porous plugs in fluid lines
• symbol
• meaning
• e R f
• power e f R f2

effort resistanceflow
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C-Elements

• C-elements store and return energy without loss
• electrical capacitors, mechanical springs and
  torsion bars, gravity tanks, accumulators
• symbol
• meaning

flow (velocity, current)is the causedeformation
or charge (q)is the effect
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I-Elements

• I-elements are inertial elements
• electrical inductors, mechanical and fluid mass
• symbol
• meaning

effort (force, voltage)is the causeflow
(velocity, current)is the effect
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Junctions

• Junctions are connections between bonds
• junctions conserve power
• junctions are reversible
• 0 junctions
• equality of effort
• flow sums to zero
• 1 junctions
• equality of flow
• effort sums to zero

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1 Junction Example (1)
power conservation e1 f1 e2 f2 e3 f3 e4f4
0 equality of flow f1 f2 f3 f4 yields,
effort sums to zero e1 e2 e3 e4 0
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1 Junction Example (2)
power conservation e1 f1 - e2 f2 e3 f3 - e4f4
0 equality of flow f1 f2 f3 f4 yields,
effort sums to zero e1 - e2 e3 - e4 0
1
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0 Junction Example (1)
power conservation e1 f1 e2 f2 e3 f3 e4f4
0 equality of effort e1 e2 e3
e4 yields, flow sums to zero f1 f2 f3 f4
0
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0 Junction Example (2)
power conservation e1 f1 - e2 f2 e3 f3 - e4f4
0 equality of effort e1 e2 e3
e4 yields, flow sums to zero f1 - f2 f3 - f4
0
0
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A Mechanical System
spring(capacitor)
damper friction(resistor)
mass(inductor)
gravity force(effort source)
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1 Junction
power conservation eS fS - eC fC - eR fR - eIfI
0 equality of flow fS fC fR fI effort
sums to zero eS - eC - eR - eI 0
eC
fC
eR
eS
fR
fS
eI
fI
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Effort Source Gravity
parameters real effort -9.8 variables
real flow equations p.e effort flow
p.f
constant parameter
constant effort may result in variable flow
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R-Element Damper
parameters real r 1.0 equations
p.e r p.f
resistance constant parameter
effort resistance flowforce friction
velocity
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C-Element Spring
parameters real c 1.0 equations state
int(p.f) p.e state / c
spring constant constant parameter
state ? flow displacement ? velocity
effort state / c force displacement / c
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I-Element Mass
parameters real i 1.0 equations state
int(p.e) p.f state / i
mass (inertia) constant parameter
state ? effort momentum ? force
flow state / i velocity momentum / mass
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Equations Computed by 20-sim
removed equations OneJunction\p1.e
spring\p.e OneJunction\p3.f spring\p.f OneJunc
tion\p4.e gravity\p.e OneJunction\p1.f
spring\p.f OneJunction\p4.f spring\p.f OneJunc
tion\flow spring\p.f mass\p.f
spring\p.f gravity\p.f spring\p.f OneJunction\
p2.f spring\p.f gravity\flow
spring\p.f damper\p.f spring\p.f OneJunction\p
2.e damper\p.e OneJunction\p3.e mass\p.e
static equations gravity\p.e
gravity\effort dynamic equations spring\p.e
spring\state / spring\c spring\p.f mass\state
/ mass\i damper\p.e damper\r
spring\p.f mass\p.e (gravity\p.e - damper\p.e)
- spring\p.e system equations spring\state
int (spring\p.f, spring\state_initial) mass\state
int (mass\p.e, mass\state_initial)
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Static Equations

• gravity\p.e gravity\effort
• Effort supplied by gravity is a constant
• Set in 20-sim as value of parameter gravity/effort

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Dynamic Equations

• spring\p.e spring\state / spring\c
• effort of the spring is the spring state
  (displacement) divided by the spring constant
• spring constant is set as a parameter
• spring\p.f mass\state / mass\i
• springs flow (velocity) is the mass state
  (momentum) divided by mass inertia
• since all velocitys are the same, masss
  velocity is substituted
• mass inertia is set as a parameter

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Dynamic Equations

• damper\p.e damper\r spring\p.f
• effort of damper is resistance of damper times
  velocity
• since all velocitys are the same, springs
  velocity is substituted
• damper resistance is set as a parameter
• mass\p.e (gravity\p.e - damper\p.e) -
  spring\p.e
• mass effort is what is left once damper and
  spring effort have been subtracted form gravity
  effort

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System Equations

• spring\state int (spring\p.f,
  spring\state_initial)
• spring state (deformation) is integral of spring
  flow (velocity)
• mass\state int (mass\p.e, mass\state_initial)
• mass state (momentum) is integral of mass effort

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Simulation Results
spring state displacement mass state momentum