Impulse-Bond Graphs

1
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

2
Overview

• Motivation
• Definition of impulse bonds
• Mechanical impulse-bond graphs
• Derivation of an IBG from a regular BG
• Limitations
• Conclusions

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Motivation I

• 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.

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Motivation 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.

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Impulse 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 (fprefpost)/2.
• Hence an impulse bond represents a sudden
  transmission of energy between its vertex
  elements and not a continuous power flow.

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Impulse 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

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First 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.

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1st Example Continuous Model

• 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.

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1st Example Transition Model

• 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 pfm is always zero.
• In general, impulse switches can dissipate or
  sometimes even generate energy.

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1st Example Transition Model

• 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.

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Derivation Rules I

• 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.

Se ? Se
C ? C
R ? R
Dp ? Dp
Dq ? Dq
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Derivation 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
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Derivation Rules III

• 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.

mTF ? TF
I e I (df / dt) ? I p 2I(fm - fpre)
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2nd 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 pistons
  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.

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2nd 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.

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2nd Example

• 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.

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2nd Example
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2nd Example BG
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2nd Example IBG
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2nd 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

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Linearity

• 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 (Newtons law)

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Non-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.

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Other domains

• 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.

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Conclusions 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.

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Conclusions 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.

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The End