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

1. 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

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

3. 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.

4. 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.

5. e f 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.

6. 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:

7. Bond graphs: Example

8. Bond graphs: Example

9. Bond graphs: Example

10. 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.

11. 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.

12. Bondgraphs for mechanics 1 • Unfortunately, the BondLib doesn’t feature mechanical applications. • Various other approaches to this subject are insufficient and/or outdated.

13. 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.

14. fx vx } f3 v fy vy t  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

15. 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.

16. Multibond graphs: Example Multibond graph of a planar pendulum

17. 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

18. Multibond graphs: Example 2 Model of a free crane crab:

19. Multibond graphs: Example 2

20. Multibond graphs: Example 2

21. Multibond graphs: Example 2

22. 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.

23. 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.

24. 3D Mechanics: Components • Basic elements: • Joints:

25. 3D Mechanics: Components • Force elements: • Ideal rolling objects:

26. 3D Mechanics: Example 1 Model of an uncontrolled bicycle

27. 3D Mechanics: Example 1 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

28. Animation Window: 3D Mechanics: Example 1 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

29. 3D Mechanics: Example 1 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

30. 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.

31. 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.

32. Additional work • Modeling of mutual gravitational attraction • Alternative approach to the multibondgraphic modeling of 3D-Systems • Modeling of mutual collisions • Modeling of hard impacts…

33. 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.

34. 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.

35. 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.

36. The End