mmps 19 engl.ppt [recovered]
TRANSCRIPT
Mathematical Modeling of Physical Systems
Multi-bond Graphs
• We shall today look at vectors of bonds, calledl i b dmulti-bonds.
• Especially when dealing with 2D and 3Dmechanics, the d’Alembert principle must beapplied to each degree of freedom separately.
• Each equation looks structurally the same.• This leads naturally to a demand for multi-bondThis leads naturally to a demand for multi bond
graphs.
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
Table of Contents
• Planar pendulumPlanar pendulum• Multi-bonds
M lti b d h lib• Multi-bond graph library• Multi-bond graph basics• Multi-port transformers
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
A Planar Pendulum• Let us model the following planar pendulum:
Holonomic Constraint
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Mathematical Modeling of Physical Systems
dvx
dtm = -F sin(φ)
dvy
dtm = -F cos(φ) + mg
x = ℓ sin(φ)
v = ℓ cos(φ) φ.
y = ℓ cos(φ)
v = ℓ sin(φ) φ.dt dtvx = ℓ cos(φ) φ vy = -ℓ sin(φ) φ
Se:mg
F i ( )
Se:mg
vy mg
F i ( ) F ( )
1 vx
F sin(φ)
1vy
F cos(φ)0
F sin(φ)
ℓ cos(φ) φ. 0
F cos(φ)
ℓ sin(φ) φ.
vx
0Dqx
TF
Causality Conflict
I:m
vx Fx
I:m
vy Fyφ
I:m I:m
φ = asin( x / ℓ )
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
AnalysisIt has been possible to describe the motion of the planarpendulum by a bond graph enhanced by activated bonds forthe description of the holonomic constraint. Unfortunately,the bond graph doesn’t tell us much that we didn’t knowalready
We shouldn’t have to derive the equations first in order to be able
already.
to derive the bond graph from them.
The resulting bond graph didn’t preserve the topological properties The resulting bond graph didn t preserve the topological properties of the system in any recognizable form.
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
Multi-bonds• Multi-bond graphs are a vector extension of the regular
bond graphs.
• A multi-bond contains a freely selectable number ofregular bonds of identical or similar domains.
} ff
fxvx Composition of a
l i b d f} f3vτ
fyvy
multi-bond for planar mechanics
• All bond graph component models are adjusted in asuitable fashion
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
suitable fashion.
Mathematical Modeling of Physical Systems
Multi-bond Graph Library• A Dymola library for modeling systems by means
of multi-bond graphs has been developed.• The library has been designed with an interface
that looks as much as possible like that of theoriginal BondLib library.
• Just like the original library, also the new multi-bond graph library contains sub-librariessupporting modelers in modeling systems from
ti l li ti d i i ll fparticular application domains, especially frommechanics.
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
Planar Pendulum III
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Mathematical Modeling of Physical Systems
Planar Pendulum IV
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Mathematical Modeling of Physical Systems
Planar Pendulum V
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Mathematical Modeling of Physical Systems
Planar Pendulum VI
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Mathematical Modeling of Physical Systems
Planar Pendulum VII
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Mathematical Modeling of Physical Systems
Planar Pendulum VIII
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Mathematical Modeling of Physical Systems
Planar Pendulum IX
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Mathematical Modeling of Physical Systems
Planar Pendulum X
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Mathematical Modeling of Physical Systems
Planar Pendulum XIa-causal signal vector
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Mathematical Modeling of Physical Systems
Planar Pendulum XII
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Mathematical Modeling of Physical Systems
Multi-bond Graph Basics• The basic multi-bond graph models contain little that is
surprising They represent essentially natural extensions ofsurprising. They represent essentially natural extensions ofthe regular bond graph models.
• A few points are worth mentioning though. First, there isp g g ,the defaults model that must be included in each multi-bond graph model. It contains only a single parameter, thedimensional parameter, n, that specifies, how many bondseach multi-bond contains by default.Th d f lt d l t b f d i h lti b d• The defaults model must be referenced in each multi-bondgraph model as an outer model.
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
Multi-bond Graph Basics II
• If the multi-bond graph model inherits one of the partialmodels, this has already been taken care of.
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
Multi-bond Graph Basics III• A second difference concerns the use of junctions.
Whereas the general bond graph library provides separateWhereas the general bond graph library provides separatejunction models for 2..6 bond connections, the multi-bondgraph library offers only junctions with either 4 or 8connectors. Yet, individual connectors may be leftunconnected as needed.
• A third difference is in the use of transformers andgyrators. The multi-bond graph library offers a muchlarger variety of different transformer and gyrator modelslarger variety of different transformer and gyrator modelswhen compared to the regular bond graph library.
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
Multi-port Transformers
e1 e2TF Transformation: e1 = M · e2 (1)
f 1 f 2
TFM Energy Conservation: e1
T · f1 = e2T · f2 (2)
(3) (M ·e2 )T · f1 = e2T · f2
(4)f2 = MT · f1
The transformer may either be described by means ofequations (1) and (2) or using equations (1) and (4).
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
Multi-port Transformers II• The transformer that looks most similar to the TF element of the
regular bond graph library is the flow multi-port transformer. Thecardinality of the bonds on the two sides doesn’t have to be identical.
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
Multi-port Transformers III• Yet, since M doesn’t usually have an inverse, an effort
transformer model must also be providedtransformer model must also be provided.
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
Multi-port Transformers IV• Also offered are modulated versions of multi-port
transformers and gyratorstransformers and gyrators.• Yet, this is still insufficient. Special transformers for
particular purposes ought to be provided as well, since theyp p p g p , yare being used frequently in mechanics.
• We already met the translational transformer.y f• Also provided is a prismatic transformer.• The special transformers are contained in the 2Dp
mechanics sub-library, since they are only useful in thatcontext.
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Mathematical Modeling of Physical Systems
Multi-port Transformers V
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Mathematical Modeling of Physical Systems
Multi-bond Graph Basics IV
• Finally, although the library offers causal multi-bonds, these are much less useful than the causalregular bonds, because many multi-bonds havemixed computational causality. Hence causalmulti-bonds are rarely used in practice.
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
Planar Pendulum XIII
mixed causality
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Mathematical Modeling of Physical Systems
References I
• Zimmer D (2006) A Modelica Library forZimmer, D. (2006), A Modelica Library forMultiBond Graphs and its Application in 3D-Mechanics, MS Thesis, Dept. of ComputerMechanics, MS Thesis, Dept. of ComputerScience, ETH Zurich.
i d C lli (2006) “ h• Zimmer, D. and F.E. Cellier (2006), “TheModelica Multi-bond Graph Library,” Proc. 5th
Intl Modelica Conference Vienna Austria Vol 2Intl. Modelica Conference, Vienna, Austria, Vol.2,pp. 559-568.
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012
Mathematical Modeling of Physical Systems
References II
• Cellier, F.E. and D. Zimmer (2006), “WrappingM l i b d G h A S d A hMulti-bond Graphs: A Structured Approach toModeling Complex Multi-body Dynamics,”Proc 20th European Conference on ModelingProc. 20th European Conference on Modelingand Simulation, Bonn, Germany, pp. 7-13.
Start Presentation© Prof. Dr. François E. CellierNovember 8, 2012