Impulse-Bond Graphs
Bondgraphic modeling of discrete transition processes
ICBGM 2007, San Diego
Authors: Dirk Zimmer and François E. Cellier,
ETH Zürich, Institute of Computational Science, Department of Computer Science
Overview
•
Motivation
•
Definition of impulse bonds
•
Mechanical impulse-bond graphs
•
Derivation of an IBG from a regular BG
•
Limitations
•
Conclusions
ETH
Zürich
Department of Computer Science
Institute of Computational Science
© Dirk Zimmer, January 2007, Slide 2
Motivation I
ETH
Zürich
Department of Computer Science
Institute of Computational Science
•
Impulse Bond Graphs (IBGs) have been primarily developed to describe
discrete transition processes in mechanical systems.
•
Such transitions usually represent elastic or semi-elastic collisions. In these
cases, the transition model is an intermediate model that interrupts the
continuous process.
•
Discrete transitions might also represent non-elastic collisions (for instance a
transition from friction to stiction). Such transitions are typically reducing
the degrees of freedom in the overall system. Hence they represent a
transition between two different continuous modes.
© Dirk Zimmer, January 2007, Slide 3
Motivation II
ETH
Zürich
Department of Computer Science
Institute of Computational Science
•
Since normal bonds describe a continuous process, they are obviously unable
to describe a discrete transition.
•
In general, we observe that a discrete change of a bondgraphic variable
(effort, flow) is accompanied by an impulse quantity of its dual counterpart.
•
Based on this observation we developed a new type of bonds that enables us
to represent a transition model in a bondgraphic fashion. We call these
bonds: Impulse bonds.
•
Although impulse bond graphs (IBGs) are primarily intended for mechanical
system, they can be embedded into the general bondgraphic framework.
© Dirk Zimmer, January 2007, Slide 4
ETH
Impulse Bonds.
•
Zürich
Department of Computer Science
Institute of Computational Science
An impulse bond is a pseudo-bond, where the product of the adjugated
variables represents an amount of work. It is represented by a two-headed
harpoon:
p
fm
p
N
fm
•
The regular impulse bond describes an impulse of effort p that leads to a
sudden change of flow f from fpre to fpost, where fm = (fpre+fpost)/2.
•
Hence an impulse bond represents a sudden transmission of energy between
its vertex elements and not a continuous power flow.
© Dirk Zimmer, January 2007, Slide 5
ETH
Impulse Bonds.
•
•
•
Department of Computer Science
Institute of Computational Science
It is a prerequisite for any kind of
impulse modeling that the integral
curve of e is irrelevant. Hence we can
suppose e to be of rectangular shape.
We suppose, that the impulse relevant
storage and transformation elements
are all linear. Hence the flow f is
linearly changing.
The work W is the integrated power
curve and can now be transformed
into the product W = p · fm , where
– p = ∫e dt
– fm = (fpre+ fpost)/2
Zürich
e
e·fpost
e·fpre
t
W
t+ε
© Dirk Zimmer, January 2007, Slide 6
First Example
ETH
Zürich
Department of Computer Science
Institute of Computational Science
• Let us model the elastic collision
between two rigid bodies in a
mechanical system.
• The model structure before and after
the collision is not affected. The
continuous part can therefore
sufficiently be described by a single
bond-graph.
• The collision causes an impulse of
force that leads to a discrete change of
velocity. This transition is modeled by
the corresponding impulse-bond
graph.
© Dirk Zimmer, January 2007, Slide 7
ETH
1st Example: Continuous Model
Zürich
Department of Computer Science
Institute of Computational Science
gy
Se
•
The gravity affects only the
vertical domain.
-t
•

The collision affects only the
horizontal domain.

•
The corresponding transformers
are modulated by the pendulum
angle.
Dq
1
t

mTFy
mTF
v
•
1
y
mTF
mTFx
x
Collision?
fy
fx
vx
1
Dq
0
C
1
Iy
I
I=m1
Ix
I
I=m1
The position sensor Dq triggers
the collision.
C=c
R
R=d
I1
I
I=m2
© Dirk Zimmer, January 2007, Slide 8
ETH
1st Example: Transition Model
•
•
This impulse bond graph represents a
linear system of equations.
•
•
Department of Computer Science
Institute of Computational Science
TFy
Iy
I
TF
I=m1
The impulse is triggered by the
impulse switch element ISw:
fm = 0 : at the time of collision.
p = 0 : otherwise.
This specific switch is neutral with
respect to energy since the product
p·fm is always zero.
In general, impulse switches can
dissipate or sometimes even generate
energy.
Zürich
TFx
1
Ix
1
TF
ISw
p=0 - > fm=0
p
fm
I
I=m1
0
I1
I
I=m2
© Dirk Zimmer, January 2007, Slide 9
ETH
1st Example: Transition Model
•
Obviously, the impulse bond graph
inherited its structure from its
continuous parent model.
Department of Computer Science
Institute of Computational Science
TFy
Iy
I
TF
I=m1
TFx
•
A small number of fixed conversion
rules enables the modeler to derive
the IBG from an existing regular BG
in a convenient way.
1
p=0 - > fm=0
This
allows
a
modeler
to
automatically transfer the knowledge
contained in the regular BG to the
corresponding IBG.
Ix
1
TF
ISw
•
Zürich
p
fm
I
I=m1
0
I1
I
I=m2
© Dirk Zimmer, January 2007, Slide 10
Derivation Rules I
•
Zürich
Department of Computer Science
Institute of Computational Science
Effort sources, capacitive and resistive elements do neither cause nor
transmit any effort impulse and can therefore be neglected if they are
connected to a 1-junction. If they are connected to a 0-junction, they
have to be replaced by a source of zero effort.
Se
C
R
•
ETH



Se
C
R
All sensor elements can be removed.
Dp
Dq


Dp
Dq
© Dirk Zimmer, January 2007, Slide 11
Derivation Rules II
•
Department of Computer Science
Institute of Computational Science


0
1
Sources of flow determine the flow variable and consequently also the
average flow variable fm. Therefore these elements remain unchanged.
Sf
•
Zürich
All junctions remain.
0
1
•
ETH

Sf
Linear transformers or gyrators also project the impulse variable and the
average by the same linear factor. Thus, also these elements remain
unchanged.
TF

TF
© Dirk Zimmer, January 2007, Slide 12
ETH
Derivation Rules III
•
Department of Computer Science
Institute of Computational Science
All modulating signals must be replaced by a constant signal for the time of
the impulse. Hence modulated transformers must become linear transformers.
mTF
•
Zürich

TF
Inductances or inductive fields are still denoted by the same symbol, but they
represent now different equations.
I
e = I · (df / dt)

I
p = 2·I·(fm - fpre)
•
Finally, one needs to include the ISw Element.
•
The resulting IBG can than be simplified.
© Dirk Zimmer, January 2007, Slide 13
2nd
ETH
Example
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Department of Computer Science
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•
Let us create a simple, academic model of a
piston engine.
•
This is a planar mechanical model that includes a
kinematic loop: There are 4 joints that each
define one degree of freedom, but the final model
owns only one degree of freedom.
•
The ignition is triggered when the piston’s
position reaches a certain threshold.
•
The ignition is regarded as a discrete event that
causes a force impulse so that each ignition will
add a constant amount of energy into the system.
© Dirk Zimmer, January 2007, Slide 14
2nd
Example
ETH
Zürich
Department of Computer Science
Institute of Computational Science
•
The model below represents the continuous part, and has been created with
components that contain wrapped planar mechanical multi-bond graphs:
•
The components feature icons that make the model intuitively understandable.
© Dirk Zimmer, January 2007, Slide 15
2nd
Example
ETH
Zürich
Department of Computer Science
Institute of Computational Science
•
Unwrapping the model leads to a multi-bond graph. The unwrapping is not
necessary for simulation, it is only done here to reveal the underlying
bondgraphic model.
•
The multi-bond graph uses planar mechanical multi-bonds, where the first two
components belong to the translational domain, and the third component
describes the rotational domain. All variables are resolved with respect to the
inertial system.
•
Whereas the bond graph cares about the dynamics, the signals care about the
positional state of the system.
© Dirk Zimmer, January 2007, Slide 16
2nd
ETH
Example
r=
{0
,0
}
fixed
Department of Computer Science
Institute of Computational Science
a
b
a
b
a spring
c=
0
b
a revolute
b
Zürich
a
b
a
b
r=
{0
,0
}
fixed
an
prismatic
=
{1
}
b ,0
© Dirk Zimmer, January 2007, Slide 17
2nd
Example: BG
ETH
Zürich
Department of Computer Science
Institute of Computational Science
© Dirk Zimmer, January 2007, Slide 18
2nd
Example: IBG
ETH
Zürich
Department of Computer Science
Institute of Computational Science
© Dirk Zimmer, January 2007, Slide 19
•
•
ETH
Example: Results
Zürich
Department of Computer Science
Institute of Computational Science
The ISw elements contains a non-linear equation:
– p ·| fm| = Eexplosion : at the time of ignition.
– p=0
: otherwise.
Hence, this IBG describes a non-linear system of equation.
Dymola reduces the system
to a size of 10. The corresponding simulation result is
shown on the right. The plot
displays the angular velocity
Revolute1.w
14
13
12
11
[rad/s]
2nd
10
9
8
7
6
0
1
2
3
4
5
6
7
© Dirk Zimmer, January 2007, Slide 20
Linearity
ETH
Zürich
Department of Computer Science
Institute of Computational Science
•
An IBG must consist of linear elements to be valid. The only exception is
the ISw element.
•
Otherwise the product of the adjugated variables would not represent the
correct amount of work anymore.
•
Fortunately, all mechanical IBGs are linear, because all potential non-linear
elements of the continuous domain vanish.
– Non-linear capacitances and resistances disappear
– Non-linear modulation by position becomes constant.
– The inductance are always linear (Newton’s law)
© Dirk Zimmer, January 2007, Slide 21
Non-linearities
ETH
Zürich
Department of Computer Science
Institute of Computational Science
•
Impulse modeling on non-linear storage elements is principally possible,
but the usability of IBGs is drastically impaired.
•
The product of the adjugated variables becomes meaningless
•
Junctions cannot be considered to be energy neutral anymore.
•
Transformers elements must be linear to enable impulse modeling in
general.
•
Non-linear storage elements must be integrable into the form:
fpost = h(p,fpre), where h is a non-linear function.
© Dirk Zimmer, January 2007, Slide 22
ETH
Other domains
Zürich
Department of Computer Science
Institute of Computational Science
•
One can define impulse bonds also for other domains. This generates the
need for dual type of impulse bonds.
•
Hence, one distinguishes between the effort impulse bond and the flow
impulse bond:
em
q
p
fm
step
The flow impulse bond
can be used for instance in electric circuits to
represent an impulse of0current, i. e. a transmission of charge.
C=c2
C2
C=c1
sw itch
C1
•
ground
© Dirk Zimmer, January 2007, Slide 23
Conclusions I
ETH
Zürich
Department of Computer Science
Institute of Computational Science
•
Impulse-bond graphs have been applied for the development of the
MultiBondLib. The MultiBondLib is a free Modelica Library for general
multi-bond graphs.
•
The library additionally contains also mechanical components based upon
wrapped MBGs. Especially an extensive set of hybrid mechanical
components is provided.
•
The corresponding impulse-equations of these hybrid components have
been derived by the methodology of impulse-bond graphs.
•
Originally it was intended to wrap the graphical models of the BG and the
IBG together, but this caused practical difficulties, since the two graphical
models obstructed each other.
© Dirk Zimmer, January 2007, Slide 24
Conclusions II
ETH
Zürich
Department of Computer Science
Institute of Computational Science
•
IBGs represent a convenient way to describe discrete transition processes in
a bondgraphic fashion. They are especially suited for mechanics.
•
We think that IBG are valuable for the understanding and teaching of
discrete transition processes in physical systems.
•
The derivation rules enable a convenient transfer of knowledge.
•
Currently we do not provide an implementation for IBGs that is able to
conveniently interact with its continuous parent model. Hence impulsebond graphs remain purely a modeling tool so far.
•
The restriction to linear elements impairs the generality of IBGs in nonmechanical domains.
© Dirk Zimmer, January 2007, Slide 25
The End