Impulse-Bond Graphs
Authors: Dirk Zimmer and François E. Cellier,
ETH Zürich, Institute of Computational Science, Department of Computer Science
Bondgraphic modeling of discrete transition processes
ICBGM 2007, San Diego
© Dirk Zimmer, January 2007, Slide 2
Department of Computer ScienceInstitute of Computational Science
ETH Zürich
• Motivation
• Definition of impulse bonds
• Mechanical impulse-bond graphs
• Derivation of an IBG from a regular BG
• Limitations
• Conclusions
Overview
© Dirk Zimmer, January 2007, Slide 3
Department of Computer ScienceInstitute of Computational Science
ETH Zürich
• 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.
Motivation I
© Dirk Zimmer, January 2007, Slide 4
Department of Computer ScienceInstitute of Computational Science
ETH ZürichMotivation II
• 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 5
Department of Computer ScienceInstitute of Computational Science
ETH ZürichImpulse Bonds.
• 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:
• 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.
f p m
N p
f m
© Dirk Zimmer, January 2007, Slide 6
Department of Computer ScienceInstitute of Computational Science
ETH ZürichImpulse Bonds.
• 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
t t+ ε
We f· p re
e f p os t·e
© Dirk Zimmer, January 2007, Slide 7
Department of Computer ScienceInstitute of Computational Science
ETH ZürichFirst Example
• 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 8
Department of Computer ScienceInstitute of Computational Science
ETH Zürich1st Example: Continuous Model
Collision?
Dq
1 I Ix
I = m 1 mTF
mTFx 1
I Iy
I = m 1
I I 1
I = m 2
mTF mTFy
1
D q x
S e g y
0
1 C C = c R = d R
f
v
y
y
f
v
x
x
-t
t
• The gravity affects only the vertical domain.
• The collision affects only the horizontal domain.
• The corresponding transformers are modulated by the pendulum angle.
• The position sensor Dq triggers the collision.
© Dirk Zimmer, January 2007, Slide 9
Department of Computer ScienceInstitute of Computational Science
ETH Zürich1st Example: Transition Model
ISw p = 0 - > fm=0
1 I Ix
I = m 1 1
I Iy
I = m 1
I I 1
I = m 2
0
TF TFy
TF TFx
f p m
• This impulse bond graph represents a linear system of equations.
• 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.
© Dirk Zimmer, January 2007, Slide 10
Department of Computer ScienceInstitute of Computational Science
ETH Zürich1st Example: Transition Model
ISw p = 0 - > fm=0
1 I Ix
I = m 1 1
I Iy
I = m 1
I I 1
I = m 2
0
TF TFy
TF TFx
f p m
• Obviously, the impulse bond graph inherited its structure from its continuous parent model.
• A small number of fixed conversion rules enables the modeler to derive the IBG from an existing regular BG in a convenient way.
• This allows a modeler to automatically transfer the knowledge contained in the regular BG to the corresponding IBG.
© Dirk Zimmer, January 2007, Slide 11
Department of Computer ScienceInstitute of Computational Science
ETH Zürich
• 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.
• All sensor elements can be removed.
Derivation Rules I
Se Se
C C
R R
Dp Dp
Dq Dq
© Dirk Zimmer, January 2007, Slide 12
Department of Computer ScienceInstitute of Computational Science
ETH ZürichDerivation Rules II
• All junctions remain.
• Sources of flow determine the flow variable and consequently also the average flow variable fm. Therefore these elements remain unchanged.
• Linear transformers or gyrators also project the impulse variable and the average by the same linear factor. Thus, also these elements remain unchanged.
0 0
1 1
Sf Sf
TF TF
© Dirk Zimmer, January 2007, Slide 13
Department of Computer ScienceInstitute of Computational Science
ETH Zürich
• All modulating signals must be replaced by a constant signal for the time of the impulse. Hence modulated transformers must become linear transformers.
• Inductances or inductive fields are still denoted by the same symbol, but they represent now different equations.
• Finally, one needs to include the ISw Element.
• The resulting IBG can than be simplified.
Derivation Rules III
mTF TF
Ie = I · (df / dt)
Ip = 2·I·(fm - fpre)
© Dirk Zimmer, January 2007, Slide 14
Department of Computer ScienceInstitute of Computational Science
ETH Zürich2nd Example
• 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 15
Department of Computer ScienceInstitute of Computational Science
ETH Zürich2nd Example
• 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 16
Department of Computer ScienceInstitute of Computational Science
ETH Zürich
• 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.
2nd Example
© Dirk Zimmer, January 2007, Slide 17
Department of Computer ScienceInstitute of Computational Science
ETH Zürich2nd Example
a ba b
a ba b
© Dirk Zimmer, January 2007, Slide 18
Department of Computer ScienceInstitute of Computational Science
ETH Zürich2nd Example: BG
© Dirk Zimmer, January 2007, Slide 19
Department of Computer ScienceInstitute of Computational Science
ETH Zürich2nd Example: IBG
© Dirk Zimmer, January 2007, Slide 20
Department of Computer ScienceInstitute of Computational Science
ETH Zürich2nd Example: Results
• 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 systemto a size of 10. The corres-ponding simulation result is shown on the right. The plot displays the angular velocity
0 1 2 3 4 5 6 7
6
7
8
9
10
11
12
13
14
[ra
d/s]
Revolute1.w
© Dirk Zimmer, January 2007, Slide 21
Department of Computer ScienceInstitute of Computational Science
ETH ZürichLinearity
• 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 22
Department of Computer ScienceInstitute of Computational Science
ETH ZürichNon-linearities
• 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 23
Department of Computer ScienceInstitute of Computational Science
ETH ZürichOther domains
f p m
e q m
C=c
2
C2
C=c
1
C1
ground
sw itch
step
0
• 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:
• The flow impulse bond can be used for instance in electric circuits to represent an impulse of current, i. e. a transmission of charge.
© Dirk Zimmer, January 2007, Slide 24
Department of Computer ScienceInstitute of Computational Science
ETH ZürichConclusions I
• 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 25
Department of Computer ScienceInstitute of Computational Science
ETH ZürichConclusions II
• 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 impulse-bond graphs remain purely a modeling tool so far.
• The restriction to linear elements impairs the generality of IBGs in non-mechanical domains.
The End
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