© 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

© Dirk Zimmer, February 2006, Slide 2 Overview Motivation Introduction to bond graphs Presentation of multibond graphs 3D-mechanical models Conclusions

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 5 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. efef

© Dirk Zimmer, February 2006, Slide 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: SourcesSeSf DissipativeRG StorageCI Junctions01 domain effortflow electricui mechanicfv thermalTdS/dt

© Dirk Zimmer, February 2006, Slide 7 Bond graphs: Example

© Dirk Zimmer, February 2006, Slide 8 Bond graphs: Example

© Dirk Zimmer, February 2006, Slide 9 Bond graphs: Example

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 12 Bondgraphs for mechanics 1 Unfortunately, the BondLib doesn’t feature mechanical applications. Various other approaches to this subject are insufficient and/or outdated.

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 14 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 } f3vf3v tt fyvyfyvy fxvxfxvx Composition of a multibond for planar mechanics

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 16 Multibond graphs: Example Multibond graph of a planar pendulum

© Dirk Zimmer, February 2006, Slide 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

© Dirk Zimmer, February 2006, Slide 18 Multibond graphs: Example 2 Model of a free crane crab:

© Dirk Zimmer, February 2006, Slide 19 Multibond graphs: Example 2

© Dirk Zimmer, February 2006, Slide 20 Multibond graphs: Example 2

© Dirk Zimmer, February 2006, Slide 21 Multibond graphs: Example 2

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 24 3D Mechanics: Components Basic elements: Joints:

© Dirk Zimmer, February 2006, Slide 25 3D Mechanics: Components Force elements: Ideal rolling objects:

© Dirk Zimmer, February 2006, Slide 26 3D Mechanics: Example 1 Model of an uncontrolled bicycle

© Dirk Zimmer, February 2006, Slide 27 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 Animation Window:

© Dirk Zimmer, February 2006, Slide 28 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 Animation Window:

© Dirk Zimmer, February 2006, Slide 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

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 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…

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 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.

© Dirk Zimmer, February 2006, Slide 36 The End