Transcript Continuous System Modeling

M athematical M odeling of Physical S ystems
Bond Graphs II
• In this class, we shall deal with the effects of
algebraic loops and structural singularities on the
bond graphs of physical systems.
• We shall also analyze the description of mechanical
systems by means of bond graphs.
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M athematical M odeling of Physical S ystems
Table of Contents
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October 11, 2012
Algebraic loops
Structural singularities
Bond graphs of mechanical systems
Selection of state variables
Example
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M athematical M odeling of Physical S ystems
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U0.e
U0.f
R2.e
R2.f
R2.e
R1.f
U0.e
R1.f
Choice
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R2.e
R3.f
U0 .e = f(t)
U0 .f = L1 .f + R1 .f
dL1 .f /dt = U0 .e / L1
R3 .f = R1 .f – R2 .f
R2 .e = R3 · R3 .f
R2 .f = R2 .e / R2
R1 .f = R1 .e / R1
R1 .e = U0 .e – R2 .e
R1.e
R1.f
U0.e
L1.f
Algebraic Loops
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M athematical M odeling of Physical S ystems
Structural Singularities
 Structural
Singularity
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© Prof. Dr. François E. Cellier
L1.e
R1.f
L1.e
L1.f
U0 .e = f(t)
U0 .f = C1 .f + R1 .f
U0.e
U0.f
U0.e
R1.f
L1.e
R2.f
R1.e
R1.f
U0.e
C1.f
Causality conflict
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M athematical M odeling of Physical S ystems
Bond Graphs of Mechanical Systems I
• The two adjugate variables of the mechanical translational
system are the force f as well as the velocity v.
• You certainly remember the classical question posed to
students in grammar school: If one eagle flies at an altitude
of 100 m above ground, how high do two eagles fly?
Evidently, position and velocity are intensive variables and
therefore should be treated as potentials.
• However, if one eagle can carry one sheep, two eagles can
carry two sheep. Consequently, the force is an extensive
variable and therefore should be treated as a flow variable.
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M athematical M odeling of Physical S ystems
Bond Graphs of Mechanical Systems II
• Sadly, the bond graph community chose the
reverse definition.
“Velocity” gives the
impression of a movement and therefore of a flow.
• We shall show that it is always possible
mathematically to make either of the two
assumptions (duality principle).
force f = potential
• Therefore:
f
v
P=f·v
velocity v = flow
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M athematical M odeling of Physical S ystems
Passive Mechanical Elements in
Bond Graph Notation
x
fI = m · dv /dt
m
fI
I :m
v
fI
fB v1
fk x1
B
k
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v2 fB
x2 fk
fB = B · D v
D x = fk / k
 Dv = (1 / k) · dfk /dt
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fB
Dv
fk
Dv
R :B
C : 1/k
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M athematical M odeling of Physical S ystems
Selection of State Variables
• The “classical” representation of mechanical
systems makes use of the absolute motions of the
masses (position and velocity) as its state
variables.
• The multi-body system representation in Dymola
makes use of the relative motions of the joints
(position and velocity) as its state variables.
• The bond graph representation selects the
absolute velocities of masses as one type of state
variable, and the spring forces as the other.
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M athematical M odeling of Physical S ystems
An Example I
The cutting forces are
represented by springs and
friction elements that are
placed between bodies at a
0-junction.
The D’Alembert principle is
formulated in the bond graph
representation as a grouping of all
forces that attack a body around a
junction of type 1.
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M athematical M odeling of Physical S ystems
An Example II
FI3 v3
v31
v3
FBb
FBb
FBa
FBa
v3
v32
F v3
v1 FBb
The sign rule follows here
automatically, and the modeler
rarely makes any mistake
relating to it.
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v2 FBa
v1
v1
v2
Fk2
Fk1
FB2
FB2
v2
v1 FI1v1 FBd
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v21 FB2
v2 FBcv2 FI2
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M athematical M odeling of Physical S ystems
References
•
Borutzky, W. and F.E. Cellier (1996), “Tearing
Algebraic Loops in Bond Graphs,” Trans. of SCS, 13(2),
pp. 102-115.
•
Borutzky, W. and F.E. Cellier (1996), “Tearing in Bond
Graphs With Dependent Storage Elements,” Proc.
Symposium on Modelling, Analysis, and Simulation,
CESA'96, IMACS MultiConference on Computational
Engineering in Systems Applications, Lille, France, vol.
2, pp. 1113-1119.
October 11, 2012
© Prof. Dr. François E. Cellier
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