Master Thesis: A Modelica Library for Multibond Graphs and its Application in 3DMechanics

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Master ThesisA Modelica Library for Multibond
Graphsand its Application in 3D-Mechanics

• Author
• Dirk Zimmer

Adviser Prof. François E. Cellier
Responsible Prof. Walter Gander
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Overview

• Motivation
• Introduction to bond graphs
• Presentation of multibond graphs
• 3D-mechanical models
• Conclusions

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Motivation

• First objectiveImplementation of a general
  modeling tool for multidimensional physical
  processes multibond graphs.
• Second objectiveThe modeling of mechanical
  systems in terms of multibond graphs.

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Introduction to bond graphs 1

• Elements of a physical system have a certain
  behavior with respect to power and energy.
• A battery is a source of energy.
• A thermal capacitance stores energy.
• A mechanical damper dissipates energy.
• Power is distributed along a junction.
• This offers a general modeling approach for
  physical systems bond graphs.

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Introduction to bond graphs 2

• Bond graphs are a modeling tool for continuous
  physical systems.
• The edges of the graph are the bonds themselves.
• A bond carries an effort and a flow variable. The
  product of them is power.

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Introduction to bond graphs 3

• The choice of effort and flow determines the
  modeling domain
• The vertex elements are denoted by a mnemonic
  code corresponding to their behavior with respect
  to energy and power

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Bond graphs Example
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Bond graphs Example
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Bond graphs Example
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Advantages of bond graphs

• Bond graphs offer a general modeling approach to
  a wide range of physical systems. They find the
  right balance between specificity and generality.
• The concept of energy and power creates a
  semantic level for each bond graph.
• Relations can more naturally be expressed in
  2D-drawings than in 1D-code.

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The Modelica/Dymola BondLib

• Bond graphs can be composed on screen by drag and
  drop.
• The resulting model can directly be simulated.
• The library features domain specific solutions,
  e.g., a library for electric systems.

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Bondgraphs for mechanics 1

• Unfortunately, the BondLib doesnt feature
  mechanical applications.
• Various other approaches to this subject are
  insufficient and/or outdated.

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Bondgraphs for mechanics 2

• Problems of mechanical bond graphs
• Mechanical processes are multidimensional
• Usage of MultiBond Graphs.
• Holonomic constraints are non-physical
• Need for extra modeling via signals.
• Mechanical bond graphs become very large
• Wrapping of the bondgraphic models.

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MultiBond Graphs

• Multibonds are a vectorial extension of bond
  graphs.
• A multibond covers an arbitrary number of single
  bonds of the same domain.
• All vertex elements are extended accordingly.

Composition of a multibond for planar mechanics
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The MultiBondLib

• A Modelica/Dymola Library for modeling Multibond
  graphs has been developed.
• It is an adaptation of the BondLib.
• Further possible applications of multibond graphs
  are
• multidimensional heat distribution
• chemical reaction dynamics
• general relativity.

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Multibond graphs Example

• Multibond graph of a planar pendulum

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Multibond graphs Sensors

• Sensor elements serve for different purposes.
  They can be used to...
• ...measure bondgraphic variables.
• ...convert bondgraphic variables to
  non-bondgraphic signals.
• ...establish algebraic relationships between
  bondgraphic elements.

Application of a bondgraphic sensor element
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Multibond graphs Example 2

• Model of a free crane crab

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Multibond graphs Example 2
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Multibond graphs Example 2
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Multibond graphs Example 2
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Wrapping

• Wrapping combines the best of two worlds
• An easy-to-use model is provided at the top
  level.
• A look inside the model reveals a familiar
  bondgraphic model.

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3D Mechanics

• A Modelica library for the object-oriented
  modeling of 3D-mechanical systems has been
  developed.Partial reimplementation of the
  MultiBody library.
• All models consist of wrapped bondgraphic models.
• 3D-specific problems had to be solved.
• Handling of different coordinate systems.
• Description of the orientation.

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3D Mechanics Components

• Basic elements
• Joints

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3D Mechanics Components

• Force elements
• Ideal rolling objects

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3D Mechanics Example 1
Model of an uncontrolled bicycle
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3D Mechanics Example 1

• Translation
• FrontRevolute.phi
• RearWheel.phi1
• RearWheel.phi2
• RearWheel.phi3
• RearWheel.phi_d1
• RearWheel.phi_d2
• RearWheel.phi_d3
• RearWheel.xA
• RearWheel.xB
• Steering.phi
• Systems of 3 and 17 linear equations
• 1 non-linear equation
• Simulation
• 20 sec, 2500 output points
• 213 integration steps.
• 0.7s CPU-Time

Animation Window
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3D Mechanics Example 1

• Translation
• FrontRevolute.phi
• RearWheel.phi1
• RearWheel.phi2
• RearWheel.phi3
• RearWheel.phi_d1
• RearWheel.phi_d2
• RearWheel.phi_d3
• RearWheel.xA
• RearWheel.xB
• Steering.phi
• Systems of 3 and 17 linear equations
• 1 non-linear equation
• Simulation
• 20 sec, 2500 output points
• 213 integration steps.
• 0.7s CPU-Time

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3D Mechanics Example 1

• Translation
• FrontRevolute.phi
• RearWheel.phi1
• RearWheel.phi2
• RearWheel.phi3
• RearWheel.phi_d1
• RearWheel.phi_d2
• RearWheel.phi_d3
• RearWheel.xA
• RearWheel.xB
• Steering.phi
• Systems of 3 and 17 linear equations
• 1 non-linear equation
• Simulation
• 20seconds, 2500 output points
• 213 integration steps.
• 0.7s CPU-Time

Plot Window Lean Angle
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3D Mechanics Kinematic Loops

• Redundant statements appear in kinematic loops
  and lead to a singularity of the model.
• Automatic removal of the redundant statements.
• Systems of non-linear equations have to be
  solved.

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Efficiency of the simulation

• Same efficiency as the MultiBody library. The
  efficiency is not impaired by the bondgraphic
  methodology
• The state selection is of major importance for
  the efficiency. Relative positions and motions of
  the joints do usually form a good set of state
  variables.
• The automatic state selection is mostly
  meaningful
• and can be improved manually if necessary.
• Kinematic loops could be closed more efficiently
  by special cut joints, that contain analytic
  solutions.

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Additional work

• Modeling of mutual gravitational attraction
• Alternative approach to the multibondgraphic
  modeling of 3D-Systems
• Modeling of mutual collisions
• Modeling of hard impacts

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Additional work Impacts

• Extension of the continuous models to hybrid
  models that allow a discrete change of motion.
• The impulse equations were derived out of the
  continuous bondgraphic models.
• Several impact models (elasticity, friction,
  shape).
• Impacts can act on kinematic loops.
• Solution is fine for small scale models.

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Conclusions

• A general solution for multibondgraphic modeling
  is provided.
• Object-oriented modeling of 2D- and 3D-mechanical
  systems is supported.
• Hybrid mechanical systems can be simulated.
• The modeling is convenient and the simulation is
  done efficiently.

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Outlook on future tasks

• Modeling of structural changes
• Modeling of friction and the transition to
  adhesion.
• Modeling of constrained joints.
• Improvement of the hybrid models.
• Bondgraphic modeling of deformable objects.

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The End