© dirk zimmer, february 2006, slide 1 master thesis: a modelica library for multibond graphs and...

© Dirk Zimmer, February 2006, Slide 1

Master Thesis:

A Modelica Library for Multibond Graphs

and its Application in 3D-Mechanics

Author:

Dirk Zimmer

Adviser:

Prof. François E. Cellier

Responsible:

Prof. Walter Gander


Overview

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


Motivation

• First objective:Implementation of a general modeling tool for multidimensional physical processes: multibond graphs.

• Second objective:The modeling of mechanical systems in terms of multibond graphs.


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.



• 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.

e

f



• 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:

Sources Se Sf

Dissipative R G

Storage C I

Junctions 0 1

domain effort flow

electric u i

mechanic f v

thermal T dS/dt


Bond graphs: Example



S eU 0

0

I L

1

RR 1

0

CC

RR 2

i1

i1 i1

iL uL uCiC

u0

i0

u1

u2

i2

v2v1


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.


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.


Bondgraphs for mechanics 1

• Unfortunately, the BondLib doesn’t feature mechanical applications.

• Various other approaches to this subject are insufficient and/or outdated.


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.


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.

MultiBond Graphs

} f3v

t

fy

vy

fx

vx Composition of a multibond for planar mechanics


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.


Multibond graphs: Example

Multibond graph of a planar pendulum


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


Multibond graphs: Example 2

Model of a free crane crab:

yprismatic joint

revolutejoint

mass 1

mass 2

rod

x


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.


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.


3D Mechanics: Components

• Basic elements:

• Joints:


3D Mechanics: Components

• Force elements:

• Ideal rolling objects:


3D Mechanics: Example 1

Model of an uncontrolled bicycle



Translation:• FrontRevolute.phi• RearWheel.phi[1]• RearWheel.phi[2]• RearWheel.phi[3]• RearWheel.phi_d[1]• RearWheel.phi_d[2]• RearWheel.phi_d[3]• RearWheel.xA• RearWheel.xB• Steering.phi

Systems of 3 and 17 linear equations1 non-linear equation

Simulation20 sec, 2500 output points 213 integration steps. 0.7s CPU-Time

Animation Window:





Simulation20 sec, 2500 output points 213 integration steps. 0.7s CPU-Time

Animation Window:





Simulation20seconds, 2500 output points 213 integration steps. 0.7s CPU-Time

0 10 20

-0.2

-0.1

0.0

0.1

0.2

0.3

[rad

]

RearWheel.phi[2]

Plot Window: Lean Angle


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.


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.


Additional work

• Modeling of mutual gravitational attraction

• Alternative approach to the multibondgraphic modeling of 3D-Systems

• Modeling of mutual collisions

• Modeling of hard impacts…


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.


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.


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.


The End

© dirk zimmer, february 2006, slide 1 master thesis: a modelica library for multibond graphs and...

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