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Master Thesis: A Modelica Library for Multibond Graphs and its Application in 3D-Mechanics

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Master Thesis:A Modelica Library for Multibond Graphsand its Application in 3D-Mechanics

Author:

Dirk Zimmer

Adviser:

Prof. François E. Cellier

Responsible:

Prof. Walter Gander

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

- 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.

- 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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- 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.

- 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:

- 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.

- 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.

- Unfortunately, the BondLib doesn’t feature mechanical applications.
- Various other approaches to this subject are insufficient and/or outdated.

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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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

- 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 graph of a planar pendulum

- 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

Model of a free crane crab:

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.

- 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.

- Basic elements:
- Joints:

- Force elements:
- Ideal rolling objects:

Model of an uncontrolled bicycle

Animation Window:

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 equations

1 non-linear equation

Simulation

20 sec, 2500 output points

213 integration steps.

0.7s CPU-Time

Animation Window:

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 equations

1 non-linear equation

Simulation

20 sec, 2500 output points

213 integration steps.

0.7s CPU-Time

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 equations

1 non-linear equation

Simulation

20seconds, 2500 output points

213 integration steps.

0.7s CPU-Time

Plot Window: Lean Angle

- 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.

- 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.

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

- 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.

- 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.