04/19/23 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
04/19/23 3
Review: Modeling with Bond Graphs
• Based on concept of reticulation• Properties of system lumped into processes with distinct
parameter valuesLumped Parameter Modeling
• Dynamic System Behavior: function of energy exchange between components
• State of physical system – defined by distribution of
energy at any particular timeDynamic Behavior: Current State + Energy exchange
mechanisms
04/19/23 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
04/19/23 5
Review: Junctions
Electrical Domain: 0- enforces Kirchoff’s current law, 1- enforces Kirchoff’s voltage lawMechanical 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 velocityHydraulic Domain: 0- conservation of volume flow rate, when a set of pipes join1- 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
Q
Q
Q
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Building Electrical Models
For each node in circuit with a distinct potential create a 0-junctionInsert 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 bondsIf explicit ground potential, delete corresponding 0-junction and its adjacent bondsSimplify bond graph (remove extraneous junctions)
Hydraulic, thermal systems similar, but mechanical different
04/19/23 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 directionsEliminate 0 velocity 1-junctions and their bondsSimplify bond graph
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Behavior of System: State Space Equations
Linear System
Nonlinear System
vectoroutputpyuDxCy
vectorinputmu
vectorstatenxuBxAx
1:..
1:
1:;..
vectoroutputpyuxy
vectorinputmu
vectorstatenxuxx
1:),(
1:
1:;),(
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State Equations
Linear
Nonlinear
mnmnnnnnn
mmnn
mmnn
ububxaxax
ububxaxax
ububxaxax
uBxAx
.........
.
..
.........
.........
..
1111
212121212
111111111
),( uxx
),....,,,....,(
.
..
),....,,,....,(
),....,,,....,(
11
1122
1111
mnnn
mn
mn
uuxxx
uuxxx
uuxxx
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State Space: Standard form
vx
k.xb.vvm.
vandxvariablesstateoftermsinWrite
Or
xkxbxm
xbxkxm
0...
...
Single nth order form
n first-order coupled equations
In general, can have any combination in between
04/19/23 16
More complex example
g
gmm
kxmm
kkx
dt
d
mmk
m
kx
dt
d
variablesystemasxwithformorderFourth24
)11
(..
.)()
11()(
2112
21
212
211
2
22
2
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More complex example (2):
gmmxkxmmxm
gmxkxmxm
xandxntsdisplacemetwowithformorderSecond
)(.)(.
....
214242131
2314131
43
2
213
2423122
13111
2143
4
...
...
,
vx
vvx
gmxkxkvm
gmxkvm
vandvandxandxwithformorderFirst
04/19/23 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
Bef
A
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Causality for basic multiports
Note that a lot of the causal considerations are based onalgebraic relations
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Generate equations from Bond Graphs
Step 1: Augment bond graph by adding1) Numbers to bonds
2) Reference power direction to each bond
3) A causal sense to each e,f variable of bond
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Equation generation example
65
5
2
2
6
5
2
2
6
62645
5
5.
533432
.
)(
.)()(
2
23
RC
q
I
p
R
e
I
p
R
efffq
C
qtE
efRtEeetEp
IPR
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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.
m1 m2
k1 F0(t)
k2
b1 b2
04/19/23 29
H. W. Problem 2 :
Input : Velocity at bottom of tire
(a) Bond graph.
(b) Derive state equations in terms of energy variables.
(c) Simulate in 20-Sim with diff. Parameter values. Comment on results.
V
g
K B
V
g
kV0(t)
M
m
04/19/23 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).dVFluid Port: a place where we can define an average pressure, P and a
volume flow rate, QExamples of ports: (i) end of a pipe or tube
(ii) threaded hole in a hydraulic pump