Modeling of Dynamic Systems: Notes on Bond Graphs Version 1.0 Copyright Diane L. Peters, Ph.D., P.E.

Transcription

1 Modeling of Dynamic Systems: Notes on Bond Graphs Version 1.0 Copyright 2015 Diane L. Peters, Ph.D., P.E. Spring 2015

2 2

3 Contents 1 Overview of Dynamic Modeling 5 2 Bond Graph Basics Causality Inertance Compliance Resistance Source of Effort Source of Flow Transformer Gyrator Common Effort Junction Common Flow Junction Simplification of Bond Graphs Assigning Causality Mechanical Systems Mechanical Translation Mechanical Rotation 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 4 CONTENTS

5 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 systems. 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 describe 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 6 CHAPTER 1. OVERVIEW OF DYNAMIC MODELING

7 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 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.

9 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 dissipation 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 I C R 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 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 between 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 inertance element is characterized by the relation f = 1 p, where I is the I parameter characterizing the inertance. This leads, through conservation of energy, to the relation e = ṗ. 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.

11 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 between 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 element is characterized by the relation e = 1 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 Element 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 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

13 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 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 output effort to an input effort, and an output flow to an input flow. Transformers can join different domains, or they may operate within the same domain. The transformer is characterized by the equations e 2 = 1 m e 1 (2.1) f 2 = mf 1 (2.2) where m is the modulus of the transformer. There are two valid possibilities for causality on a transformer, as shown in Figure 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 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). Figure 2.10: Valid Causalities for Transformer

15 2.8. GYRATOR 15 Note that, in some textbooks, the transformer might be represented by TR or TF. 2.8 Gyrator A gyrator is an idealized energy conserving element that relates an output effort to an input flow, and an output flow to an input effort. Gyrators can also join different domains. The gyrator is characterized by the equations e 2 = 1 m f 1 (2.3) f 2 = me 1 (2.4) As with a transformer, there are two valid possibilities for causality. For the gyrator, either both causal strokes are located at the element, or both are located away from it, as shown in Figure A DC motor is an example of a gyrator, where the output torque (an effort) is related to the input current (a flow). Figure 2.11: Valid Causalities for Gyrator In some textbooks, the gyrator may be represented as GY. 2.9 Common Effort Junction A 0 junction, also known as a common effort junction, is an element which neither dissipates nor stores power, and for which the effort on every bond is identical. Since power is conserved at this junction, and the efforts on the various bonds are equal by definition, the sum of the flows must be zero; all flows that enter the junction must leave it. If the arrow points

16 16 CHAPTER 2. BOND GRAPH BASICS into the junction, then the sign of the flow is assumed to be positive; if the arrow points away from the junction, then the flow is assumed to be negative. When assigning causality, exactly one causal stroke is located at the junction, and all others must be located away from it, as shown in Figure The equations characterizing this junction are given below. e 1 = e 2 = e 3 =... = e n (2.5) f 1 + f 2 + f f n = 0 (2.6) Figure 2.12: Typical 0 Junction Physical examples of 0 junctions are nodes in a circuit, points where various pipes are joined in a fluid system, and the force across a massless element such as a spring or damper. These are discussed further in the sections on various types of systems Common Flow Junction A 1 junction, also known as a common flow junction, is an element which neither dissipates nor stores power, and for which the flow on every bond is identical. Since power is conserved at this junction, and the flows on the various bonds are equal by definition, the sum of the efforts must be zero. If the arrow points into the junction, then the sign of the effort is assumed to be positive; if the arrow points away from the junction, then the effort is assumed to be negative. When assigning causality, exactly one causal stroke is located away from the junction, and all others must be located at it, as shown in Figure The equations characterizing this junction are given below.

17 2.11. SIMPLIFICATION OF BOND GRAPHS 17 f 1 = f 2 = f 3 =... = f n (2.7) e 1 + e 2 + e e n = 0 (2.8) Figure 2.13: Typical 1 Junction Physical examples of 1 junctions are wires in an electrical circuit without any junctions, pieces of pipe in a fluid power system with no branches, and a location in a mechanical system where elements are connected and move together. These are discussed further in the sections on various types of systems Simplification of Bond Graphs In order to have a complete, valid bond graph, you should simplify your initial bond graph, as appropriate, and then assign causality to all of the elements. In simplifying the bond graph, you can remove any 1 or 0 which has either 1 or 2 connections, since such a junction has no effect on the system. There are several other simplifications you can make, in the case when transformers and gyrators are directly coupled to one another: 1. Two transformers connected directly to each other can be replaced by a single transformer. The modulus of the new transformer is the product of the moduli for each of the individual transformers; i.e., if the first transformer has the equations f 2 = r 1 f 1 (2.9) e 2 = 1 r 1 e 1 (2.10)

18 18 CHAPTER 2. BOND GRAPH BASICS and the second transformer has the equations f 3 = r 2 f 2 (2.11) e 3 = 1 r 2 e 2 (2.12) then the equivalent transfomer will have the equations f 3 = r 2 f 2 = r 2 (r 1 f 1 ) = r 1 r 2 f 1 (2.13) e 3 = 1 e 2 = 1 ( ) 1 e 1 = 1 e 1 (2.14) r 2 r 2 r 1 r 1 r 2 So, the modulus of the new transformer is given by r eq = r 1 r 2 (2.15) Note that the order of the transformers doesn t matter. 2. Two gyrators connected directly to each other can be replaced by a single transformer. In this case, the order DOES matter, as you can see from the development of the equations for the equivalent transformer modulus. The first gyrator is characterized by the relations e 2 = m 1 f 1 (2.16) f 2 = 1 m 1 e 1 (2.17) and the second gyrator is characterized by the relations e 3 = m 2 f 2 (2.18) f 3 = 1 m 2 e 2 (2.19) The resulting transformer, then, has the relations e 3 = m 2 f 2 = m 2 ( 1 m 1 e 1 ) = m 2 m 1 e 1 (2.20) f 3 = 1 m 2 e 2 = 1 m 2 (m 1 f 1 ) = m 1 m 2 f 1 (2.21)

19 2.11. SIMPLIFICATION OF BOND GRAPHS 19 So, the equivalent transformer has a modulus r eq = m 1 m 2 (2.22) Note that if the gyrators order was reversed, then the modulus would be inverted. Transformers can be combined in either order, but gyrators cannot. 3. A transformer and gyrator connected directly to each other can be replaced by a single gyrator. The order of these items DOES matter. Consider, first, the case when the transformer is first and the gyrator is second. The transformer is characterized by the relations and the gyrator is characterized by the relations The resulting gyrator, then, has the relations f 2 = rf 1 (2.23) e 2 = 1 r e 1 (2.24) e 3 = mf 2 (2.25) f 3 = 1 m e 2 (2.26) e 3 = mf 2 = m (rf 1 ) = mrf 1 (2.27) f 3 = 1 m e 2 = 1 ( ) 1 m r e 1 = 1 mr e 1 (2.28) So, the equivalent gyrator has a modulus m eq = mr (2.29) 4. Now, consider a gyrator first, followed by a transformer. The gyrator has the equations e 2 = mf 1 (2.30) f 2 = 1 m e 1 (2.31)

20 20 CHAPTER 2. BOND GRAPH BASICS and the transformer has the equations f 3 = rf 2 (2.32) e 3 = 1 r e 2 (2.33) The equivalent gyrator then has the equations e 3 = 1 r e 2 = 1 r (mf 1) = m r f 1 (2.34) ( ) 1 f 3 = rf 2 = r m e 1 = r m e 1 (2.35) So, the equivalent gyrator has a modulus m eq = m r (2.36) There are two more simplifications, which are shown graphically in Figure 2.14.

21 2.12. ASSIGNING CAUSALITY 21 Figure 2.14: Bond Graph Simplification 2.12 Assigning Causality To assign causality, follow this procedure: 1. Begin with the sources. Start with one of the sources, assign its causality, and then assign any causalities that are not optional; for example, if a source of flow is connected to a 1, assigning the appropriate

22 22 CHAPTER 2. BOND GRAPH BASICS causality to that source will dictate what the causality MUST be on the other bonds on that 1. Similarly, if a source of effort is connected to a 0, assigning causality to that source will determine the causality on every bond on the 0. Go as far as you can for each source before going on to the next source. 2. Once the sources are assigned, you ll assign the energy storage elements. Start with any energy storage element that isn t yet assigned, and assign its preferred (integral) causality, and then assign any causalities that are not optional. Note that, in some cases, this may require that other energy storage elements take on the non-preferred, or derivative, causality. Go as far as you can for each energy storage element, then go on to the next unassigned element and repeat the procedure. 3. At this point, you may find that all of the elements have been assigned; however, there may be cases where you have some unassigned R elements. If so, then you can assign an arbitrary causality to one of these elements. Next, assign any causalities that are no longer optional, going as far as you can, and then repeat the procedure until all elements are assigned. In some cases, you will have one or more energy storage elements in derivative causality. This means that these energy storage elements are NOT independent, and they will not be associated with a state variable. This will be explained more fully when the derivation of equations is covered.

23 Chapter 3 Mechanical Systems In mechanical systems, the energy storage elements are inertia (mass or mass moment) and compliances, or springs, which may be either linear or torsional. The dashpot or damper, either linear or rotary, is the resistive element. Transformers may take several forms, with typical exclusively mechanical transformers listed in Table 3.1. Table 3.1: Examples of Mechanical Transformers Element Domains Modulus Gears Mechanical Rotation/Mechanical Rotation r 1 /r 2 Gear & Rack Mechanical Rotation/Mechanical Translation r 1 Lever Mechanical Rotation/Mechanical Rotation l 1 /l Mechanical Translation In mechanical translation, the inertance, compliance, and resistive elements are the mass, spring, and damper. The parameter I is associated with the mass, C is associated with the spring constant, and R is associated with the damping constant. Note that while I = m and R = b, C = 1 k. Sources of effort are forces, and sources of flow are imposed velocities. Each 1 junction will be associated with a particular velocity (such as the velocity of a moving mass), and a 0 will be associated with a particular force (such as the force across an ideal spring or ideal damper). In constructing mechanical bond graphs, you can follow this procedure: 23

24 24 CHAPTER 3. MECHANICAL SYSTEMS 1. Identify the 1 junctions by finding the unique velocities in the system. 2. Attach the inertia elements (I) to the relevant 1 junctions. 3. Identify the 0 junctions by finding the forces in the system. These will typically be the forces across springs and dampers. 4. Attach the compliance and resistance elements (C and R) to the relevant 0 junctions. 5. Connect the 1 and 0 junctions to form a single bond graph. 6. Simplify by eliminating any unnecessary elements, such as 1 and 0 junctions with only 1 or 2 connections. 7. Assign causality to the bond graph. Examples of mechanical translation, and the appropriate bond graphs, are given below. An explanation of how these bond graphs are constructed will be given through videos posted on the class Blackboard site. EXAMPLE 1.1: Given the system shown, construct a bond graph and assign appropriate causality.

25 3.1. MECHANICAL TRANSLATION 25 EXAMPLE 1.2: Given the system shown, construct a bond graph and assign appropriate causality.

26 26 CHAPTER 3. MECHANICAL SYSTEMS 3.2 Mechanical Rotation In mechanical rotation, the inertance, compliance, and resistive elements are the mass moment, torsional spring, and rotational damper. As in the linear case, while I = J and R = b, C = 1, so you need to be alert to this when developing your equations. Sources of effort are torques (AKA moments), and k sources of flow are imposed angular velocities. The process for construction is identical to that for mechanical translation, with the rotational instead of linear quantities. Note that, in many rotational problems that involve gears or wheels that roll without slipping, you will see derivative causality. This will become important when deriving system equations. An example of mechanical rotation, and the appropriate bond graph, is given below. EXAMPLE 1.3: In the given system, three rollers are pinned at their centers, and roll without slipping on each other. A torque, T, is applied to the first roller; the third roller has a rotational damping associated with it, with damping constant b. Construct a bond graph and assign appropriate causality.

27 Chapter 4 Electrical Systems In electrical systems, the inertance, compliance, and resistive elements are the inductor, capacitor, and resistor. Note that I = L, R = R, and C = C. Sources of effort are voltage sources such as batteries, and sources of flow are current sources. Construction of bond graphs for electrical systems is fairly straightforward; you need to recognize how the elements of the circuit correspond to the various bond graph elements. A node in a circuit, where wires are joined together, is a 0 junction. A wire with no branches is a 1 junction. To construct the bond graph, follow this process: 1. Assign a power convention. To do this, mark the direction in which you assume current will be flowing, and then mark the appropriate voltage drops for this direction. For example, if current is flowing from left to right through a resistor, then the left end of the resistor will be marked as + and the right end as Label the nodes in the circuit, and establish a 0 junction for each one. 3. Establish a 1 junction for each wire, and join them to the 0 junctions based on the way the circuit is connected. 4. Add the sources and elements to the bond graph, connected to the 1 junctions representing the appropriate wires. 5. Remove all bonds with zero power. This means that any bonds connected to ground can be removed. If ground is not marked, then you 27

28 28 CHAPTER 4. ELECTRICAL SYSTEMS can assume a particular location as ground. Typically, this will eliminate one 0 junction and every bond connected to it. 6. Perform other appropriate simplifications, such as removing 0 and 1 junctions with either one or two bonds connected to them. 7. Assign appropriate causality. An example is given here, along with the appropriate bond graph. An explanation of the construction process is given in a video on the class s Blackboard site. EXAMPLE 1.4: Given the circuit shown, construct a bond graph and assign appropriate causality.

29 Chapter 5 Hydraulic Systems In hydraulic systems, the inertance, compliance, and resistive elements are the fluid inertia, tanks, and fluid resistance. Fluid resistance can come from an orifice, or from the friction as fluid moves through a long pipe. Sources of effort are pressures, and sources of flow are fluid flows. The fluid capacitance is given by the relation C = A, where A is the ρg cross-sectional area of the tank, ρ is the fluid density, and g is the gravitational constant. Fluid inertia is given by the relation I = ρl, where l is the A length of the pipe. Construction of bond graphs for hydraulic systems is very similar to the construction process for electrical systems, with nodes in a pipe represented by 0 junctions, and straight pipes with a fluid flowing through them represented by 1 junctions. To construct the bond graph, follow this process: 1. Label the nodes for each pressure of interest, and establish a 0 junction for each one. Remember that you may have a node corresponding to atmospheric pressure in some cases (this is often removed later, but may be needed as part of the initial construction) 2. Establish a 1 junction for each pipe, and join them to the 0 junctions based on the way the system is connected. 3. Add the sources and elements to the bond graph, connected to the 1 and 0 junctions as appropriate. 29

30 30 CHAPTER 5. HYDRAULIC SYSTEMS 4. Perform appropriate simplifications, such as removing 0 and 1 junctions with either one or two bonds connected to them. 5. Assign appropriate causality.

31 Chapter 6 Multi-Domain Systems In multi-domain systems, at least two different domains are represented. Each portion of the bond graph - mechanical, electrical, or hydraulic - can be constructed separately, using the appropriate steps, and then the various pieces are joined together with either transformers or gyrators. The choice of whether to use a transformer or a gyrator depends on the physics of the connection between the domains. If the efforts in the domains relate to each other, then a transformer is used; if effort in one domain is related to flow in the other domain, then a gyrator is used. Once a bond graph is constructed, it doesn t matter what the original domain was; the mechanics of assigning causality and deriving equations is independent of the domains. 31

32 32 CHAPTER 6. MULTI-DOMAIN SYSTEMS

33 Chapter 7 Deriving State-Space Equations from Bond Graphs Once a system has been represented by a bond graph, you can use the bond graph to derive the state-space equations describing the system s dynamics. The procedure for developing the equations is independent of the domain which is represented by the bond graph. The procedure is conceptually simple, although the algebra can become very complex at times, particularly in large systems. The basic steps are: 1. Select the input and energy state variables. Each source will supply an input, which will become part of the u vector in the equation ẋ = Ax + Bu. Each independent energy storage variable (those I and C elements in integral, NOT derivative, causality) will supply a state variable. If the independent energy storage element is an I element, its state variable will be a generalized momentum (p). If the independent energy storage element is a C element, its state variable will be a generalized displacement (q). 2. Write the initial set of system equations. Each 0, 1, transformer, and gyrator will generate a set of equations relating efforts and flows, and each I, C, and R element will supply necessary relations. The equations from the junctions can be thought of as the backbone of the system equations, into which the equations from the storage elements and resistances can be plugged in. 3. Reduce the initial set of equations to the proper number of equations, 33

34 34CHAPTER 7. DERIVING STATE-SPACE EQUATIONS FROM BOND GRAPHS in state-space form. This requires you to eliminate everything except the states, their derivatives, inputs, and the parameters characterizing the system. This is the step that can be algebraically tedious, although with practice you ll develop a better understanding of where to start and how to flow through the system. In order to demonstrate the procedure, some of the examples from previous sections will be used here. The answers are given here; the process for getting those answers can be found in the videos provided as an additional class resource. EXAMPLE 1.6: Given the following system and its bond graph, derive the state-space equations for the system.

35 35 [ ṗ2 q 3 ] b = m 1 m k 0 [ p2 q 3 ] + [ 1 0 ] F (t) (7.1)

36 36CHAPTER 7. DERIVING STATE-SPACE EQUATIONS FROM BOND GRAPHS

37 Chapter 8 Practice Problems 1. Given the system shown below, construct a bond graph, simplify, apply appropriate causal strokes, and derive the state equations for the system in matrix form. 2. Given the system shown below, construct a bond graph, simplify, apply appropriate causal strokes, and derive the state equations for the system in matrix form. 37

38 38 CHAPTER 8. PRACTICE PROBLEMS 3. Given the quarter-car model shown, construct a bond graph, simplify, apply appropriate causal strokes, and derive the state equations for the system in matrix form. Assuming that the outputs are the acceleration and velocity of the sprung mass, m s, write the output equation in matrix form. 4. Given the system shown below, construct a bond graph, simplify, apply appropriate causal strokes, and derive the state equations for the system in matrix form.

39 5. Given the system shown in the previous problem, change the flow source, ω, to an effort source, T, and re-solve the problem. How does this change affect the system? 6. Given the system shown below, construct a bond graph, simplify, apply appropriate causal strokes, and derive the state equations for the system in matrix form Given the system shown below, construct a bond graph, simplify, apply appropriate causal strokes, and derive the state equations for the system in matrix form.

40 40 CHAPTER 8. PRACTICE PROBLEMS 8. Given the system shown below, construct a bond graph, simplify, apply appropriate causal strokes, and derive the state equations for the system in matrix form. 9. Given the system shown below, construct a bond graph, simplify, apply appropriate causal strokes, and derive the state equations for the system in matrix form.

41 10. Given the system shown below, construct a bond graph, simplify, apply appropriate causal strokes, and derive the state equations for the system in matrix form. If the system outputs are the flow out of the second tank and the volume of water in the first tank, write the output equation for the system. 41

42 42 CHAPTER 8. PRACTICE PROBLEMS

43 Bibliography [1] Wolfgang Borutzsky, ed. Bond Graph Methodology: Development and Analysis of Multi-disciplinary Dynamic System Models, London: Springer-Verlag (2010) [2] Wolfgang Borutzsky, ed. Bond Graph Modelling of Engineering Systems: Theory, Applications and Software Support, London: Springer-Verlag (2011) [3] Dean C. Karnopp, Donald L. Margolis, & Ronald C. Rosenberg System Dynamics: Modeling and Simulation of Mechatronics Systems, Fourth Edition, Hoboken: John Wiley & Sons (2006) [4] Arun K. Samantaray & Belkacem Ould Bouamama Model-based Process Supervision: A Bond Graph Approach, London: Springer-Verlag (2008) 43