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