Modeling of Dynamic Systems: Notes on Bond Graphs Version 1.0 Copyright 2015 Diane L. Peters, Ph.D., P.E. Spring 2015 2 Contents 1 Overview of Dynamic Modeling 5 2 Bond Graph Basics 7 2.1 Causality . 9 2.2 Inertance . 10 2.3 Compliance . 11 2.4 Resistance . 12 2.5 Source of Effort . 13 2.6 Source of Flow . 13 2.7 Transformer . 14 2.8 Gyrator . 15 2.9 Common Effort Junction . 15 2.10 Common Flow Junction . 16 2.11 Simplification of Bond Graphs . 17 2.12 Assigning Causality . 21 3 Mechanical Systems 23 3.1 Mechanical Translation . 23 3.2 Mechanical Rotation . 26 4 Electrical Systems 27 5 Hydraulic Systems 29 6 Multi-Domain Systems 31 7 Deriving State-Space Equations from Bond Graphs 33 8 Practice Problems 37 3 4 CONTENTS Chapter 1 Overview of Dynamic Modeling There are a variety of different methods for modeling dynamic systems; some of these methods work within a single domain, or field, while others are more general. In previous courses, you may have used Newton's Laws or the Lagrangian to derive equations for a system; for a purely mechanical system, these methods will yield a dynamic model of the system. You may have also seen Kirchoff's voltage and current laws used to derive equations for an electrical circuit; these equations are the dynamic model of the given electrical system. Within the hydraulic domain, you may have been exposed to Bernoulli's equation or the Navier-Stokes equation in a fluids class; these principles also allow you to derive a dynamic model for a system, if it's in the fluid domain. We use a method called bond graphs to develop dynamic models of sys- tems. Construction of a bond graph is one of several methods which allow models to be developed for multi-domain systems. They may seem rather abstract, but this abstract nature allows them to be used to effectively de- scribe mechanical, electrical, and hydraulic components, and to unite them in a single framework. There are many different books and academic papers on bond graphs, so this is just a brief overview of the basics. 5 6 CHAPTER 1. OVERVIEW OF DYNAMIC MODELING Chapter 2 Bond Graph Basics The bond graph technique for dynamic systems modeling is based on energy as a \common currency" between different domains, such as mechanical, electrical, fluid, thermal, acoustic, etc. For each domain, an effort and a flow are defined. Every bond, or connection between two elements in a bond graph, is associated with an effort and a flow, and the product of these two quantities is the power transmitted on that bond. 7 8 CHAPTER 2. BOND GRAPH BASICS Figure 2.1: Tetrahedron of State The state of a system is described by generalized coordinates, where these coordinates are generalized momentums, p and generalized displacements, q. In the linear mechanical domain, these are simply the momentum and displacement; in other domains, they are different, as detailed in Table 2.1. These are shown in Figure 2.1, with the relationship between e, f, q, and p shown. This figure is known as the Tetrahedron of State. 2.1. CAUSALITY 9 Table 2.1: Key Quantities in Various Domains Domain Effort Flow Momentum Displacement Mechanical Force Velocity Linear Linear Translation Momentum Displacement Mechanical Moment Angular Angular Angular Rotation Velocity Momentum Displacement Electrical Electric Current Flux Linkage Charge Potential (Voltage) Hydraulic Pressure Volumetric Pressure Volume Flow Momentum Bond graphs are constructed of energy storage elements, energy dissi- pation elements, junctions, transformers and gyrators, and sources. These elements are described below. The various energy storage and dissipation element in the different domains are listed in Table 2.2. Table 2.2: Key Quantities in Various Domains Element Type Domain ICR Mechanical Translation Mass Linear Spring Damper Mechanical Rotation Mass Moment Torsional Spring Rotary Damper Electrical Inductor Capacitor Resistor Hydraulic Fluid Tank Pipe Resistance Inertia or Orifice 2.1 Causality Bonds connected to an element in a bond graph have causal strokes to indicate whether effort is being imposed on the element, or imposed by it. If the causal stroke is near the element, then effort is being imposed on it, and it responds with a flow; if the causal stroke is away from the element, then it is imposing an effort on the system, and the system responds to that effort with a flow. Sources have a required causality, based on what type of source they are, as noted below; junctions, transformers, and gyrators have rules governing what possible combinations of causal strokes are valid; and other 10 CHAPTER 2. BOND GRAPH BASICS elements have a preferred causality. Details on the rules for each element, and the preferred causal strokes, are given in the sections below. 2.2 Inertance The energy storage element known as inertance exhibits a relationship be- tween flow and generalized momentum. This relation may be non-linear, as shown in Figure 2.2. In many cases, the relationship is linear, and the 1 inertance element is characterized by the relation f = p, where I is the I parameter characterizing the inertance. This leads, through conservation of energy, to the relation e =p _. Figure 2.2: Relation Between Flow and Momentum for Inertance Element When the inertance element is in integral causality, with the causal stroke at the end of the bond nearest to the element as shown in Figure 2.3, the momentum associated with it will be an independent state of the system. Inertances store energy in the form of kinetic energy, or energy of motion. 2.3. COMPLIANCE 11 Figure 2.3: Integral and Derivative Causality for Inertance Element 2.3 Compliance The energy storage element known as compliance exhibits a relationship be- tween effort and displacement. This relation may be non-linear, as shown in Figure 2.4. In many cases, the relationship is linear, and the compliance 1 element is characterized by the relation e = q, where C is the parameter C characterizing the compliance. This element also exhibits the relation f =q _. Figure 2.4: Relation Between Effort and Displacement for Compliance Ele- ment When the compliance element is in integral causality, with the causal stroke at the end of the bond farthest from the element as shown in Figure 12 CHAPTER 2. BOND GRAPH BASICS 2.5, the displacement associated with it will be an independent state of the system. Compliances store energy in the form of potential energy, or energy of position. Figure 2.5: Integral and Derivative Causality for Compliance Element 2.4 Resistance The element known as resistance does not store energy; it dissipates it. This energy is not destroyed, since total energy is conserved, but it is converted into a form where it cannot be easily recovered. Resistance elements may be either non-linear or linear, as shown in Figure 2.6. Figure 2.6: Relation Between Effort and Flow for Resistance Element 2.5. SOURCE OF EFFORT 13 For a linear resistance element, the effort and flow are related by the equation e = Rf. The concepts of integral and derivative causality do not apply to resistance elements, and either of the causalities shown in Figure 2.7 is equally valid. The way the causal strokes are placed does have an influence on the structure of the equations - there is a concept called an algebraic loop - but this is beyond the scope of this course. If you're interested, it is covered in the book by Karnopp, Margolis, and Rosenberg which is listed in the bibliography for these notes. Figure 2.7: Causality Options for Resistance Element 2.5 Source of Effort A source of effort is a source which imposes an effort on a system, and the system responds with a particular flow. Sources of effort may be forces, torques, pressures, or electric potential (voltage), as shown in Table 2.1. By definition, since an effort is being imposed on the system, the causal stroke for a source of effort /bf must be located away from the element, as shown in Figure 2.8. Figure 2.8: Causality Required for Effort Source 2.6 Source of Flow A source of flow is a source which imposes a flow on a system, and the system responds with an effort. Sources of flow may be linear or angular 14 CHAPTER 2. BOND GRAPH BASICS velocities, volumetric flow of fluid, or electric current, as shown in Table 2.1. By definition, since an effort is being imposed on the source by the system, the causal stroke for a source of flow must be located at the element, as shown in Figure 2.9. Figure 2.9: Causality Required for Flow Source 2.7 Transformer A transformer is an idealized energy conserving element that relates an out- put effort to an input effort, and an output flow to an input flow. Transform- ers can join different domains, or they may operate within the same domain. The transformer is characterized by the equations 1 e = e (2.1) 2 m 1 f2 = mf1 (2.2) where m is the modulus of the transformer. There are two valid possibilities for causality on a transformer, as shown in Figure 2.10. In both cases, one causal stroke is located at the element, and the other is located away from it. Examples of transformers in the mechanical domain are given in Table 2.10. Another example of a transformer is a piston driven by a fluid, where the output force (an effort) is related through the area to the input pressure (an effort).