modeling of dynamic systems: notes on bond graphs version...

Modeling of Dynamic Systems: Notes on BondGraphs

Version 1.0Copyright 2015

Diane L. Peters, Ph.D., P.E.

Spring 2015

Contents

1 Overview of Dynamic Modeling 5

2 Bond Graph Basics 72.1 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Inertance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Source of Effort . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Source of Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Gyrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.9 Common Effort Junction . . . . . . . . . . . . . . . . . . . . . 152.10 Common Flow Junction . . . . . . . . . . . . . . . . . . . . . 162.11 Simplification of Bond Graphs . . . . . . . . . . . . . . . . . . 172.12 Assigning Causality . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Mechanical Systems 233.1 Mechanical Translation . . . . . . . . . . . . . . . . . . . . . . 233.2 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; someof these methods work within a single domain, or field, while others aremore general. In previous courses, you may have used Newton’s Laws orthe Lagrangian to derive equations for a system; for a purely mechanicalsystem, these methods will yield a dynamic model of the system. You mayhave also seen Kirchoff’s voltage and current laws used to derive equationsfor an electrical circuit; these equations are the dynamic model of the givenelectrical system. Within the hydraulic domain, you may have been exposedto Bernoulli’s equation or the Navier-Stokes equation in a fluids class; theseprinciples also allow you to derive a dynamic model for a system, if it’s inthe 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 allowmodels to be developed for multi-domain systems. They may seem ratherabstract, but this abstract nature allows them to be used to effectively de-scribe mechanical, electrical, and hydraulic components, and to unite themin a single framework. There are many different books and academic paperson 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 energyas a “common currency” between different domains, such as mechanical,electrical, fluid, thermal, acoustic, etc. For each domain, an effort and aflow are defined. Every bond, or connection between two elements in a bondgraph, is associated with an effort and a flow, and the product of these twoquantities 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 thesecoordinates are generalized momentums, p and generalized displacements,q. In the linear mechanical domain, these are simply the momentum anddisplacement; 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 pshown. This figure is known as the Tetrahedron of State.

2.1. CAUSALITY 9

Table 2.1: Key Quantities in Various DomainsDomain Effort Flow Momentum DisplacementMechanical Force Velocity Linear LinearTranslation Momentum DisplacementMechanical Moment Angular Angular AngularRotation Velocity Momentum DisplacementElectrical Electric Current Flux Linkage Charge

Potential(Voltage)

Hydraulic Pressure Volumetric Pressure VolumeFlow Momentum

Bond graphs are constructed of energy storage elements, energy dissi-pation elements, junctions, transformers and gyrators, and sources. Theseelements are described below. The various energy storage and dissipationelement in the different domains are listed in Table 2.2.

Table 2.2: Key Quantities in Various DomainsElement Type

Domain I C RMechanical Translation Mass Linear Spring DamperMechanical Rotation Mass Moment Torsional Spring Rotary DamperElectrical Inductor Capacitor ResistorHydraulic Fluid Tank Pipe Resistance

Inertia or Orifice

2.1 Causality

Bonds connected to an element in a bond graph have causal strokes to indicatewhether effort is being imposed on the element, or imposed by it. If thecausal stroke is near the element, then effort is being imposed on it, and itresponds with a flow; if the causal stroke is away from the element, then itis imposing an effort on the system, and the system responds to that effortwith a flow. Sources have a required causality, based on what type of sourcethey are, as noted below; junctions, transformers, and gyrators have rulesgoverning what possible combinations of causal strokes are valid; and other


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

inertance element is characterized by the relation f =1

Ip, where I is the

parameter characterizing the inertance. This leads, through conservation ofenergy, 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 strokeat the end of the bond nearest to the element as shown in Figure 2.3, themomentum 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 shownin Figure 2.4. In many cases, the relationship is linear, and the compliance

element is characterized by the relation e =1

Cq, where C is the parameter

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 causalstroke at the end of the bond farthest from the element as shown in Figure


2.5, the displacement associated with it will be an independent state of thesystem. Compliances store energy in the form of potential energy, or energyof 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. Thisenergy is not destroyed, since total energy is conserved, but it is convertedinto a form where it cannot be easily recovered. Resistance elements may beeither 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 theequation e = Rf . The concepts of integral and derivative causality do notapply to resistance elements, and either of the causalities shown in Figure 2.7is equally valid. The way the causal strokes are placed does have an influenceon 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 coveredin the book by Karnopp, Margolis, and Rosenberg which is listed in thebibliography 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 thesystem responds with a particular flow. Sources of effort may be forces,torques, pressures, or electric potential (voltage), as shown in Table 2.1. Bydefinition, since an effort is being imposed on the system, the causal strokefor a source of effort /bf must be located away from the element, as shownin 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 thesystem responds with an effort. Sources of flow may be linear or angular


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, asshown 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

e2 =1

me1 (2.1)

f2 = mf1 (2.2)

where m is the modulus of the transformer. There are two valid possibilitiesfor causality on a transformer, as shown in Figure 2.10. In both cases, onecausal stroke is located at the element, and the other is located away fromit. Examples of transformers in the mechanical domain are given in Table2.10. Another example of a transformer is a piston driven by a fluid, wherethe output force (an effort) is related through the area to the input pressure(an effort).

Figure 2.10: Valid Causalities for Transformer

2.8. GYRATOR 15

Note that, in some textbooks, the transformer might be representedby“TR” or “TF”.

2.8 Gyrator

A gyrator is an idealized energy conserving element that relates an outputeffort to an input flow, and an output flow to an input effort. Gyrators canalso join different domains. The gyrator is characterized by the equations

e2 =1

mf1 (2.3)

f2 = me1 (2.4)

As with a transformer, there are two valid possibilities for causality. For thegyrator, either both causal strokes are located at the element, or both arelocated away from it, as shown in Figure 2.11. A DC motor is an example ofa 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 whichneither dissipates nor stores power, and for which the effort on every bondis identical. Since power is conserved at this junction, and the efforts onthe various bonds are equal by definition, the sum of the flows must bezero; all flows that enter the junction must leave it. If the arrow points


into the junction, then the sign of the flow is assumed to be positive; if thearrow 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 2.12. Theequations characterizing this junction are given below.

e1 = e2 = e3 = . . . = en (2.5)

f1 + f2 + f3 + . . .+ fn = 0 (2.6)

Figure 2.12: Typical “0” Junction

Physical examples of “0” junctions are nodes in a circuit, points wherevarious pipes are joined in a fluid system, and the force across a masslesselement such as a spring or damper. These are discussed further in thesections on various types of systems.

2.10 Common Flow Junction

A “1” junction, also known as a common flow junction, is an element whichneither dissipates nor stores power, and for which the flow on every bondis identical. Since power is conserved at this junction, and the flows on thevarious 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 assumedto be positive; if the arrow points away from the junction, then the effort isassumed to be negative. When assigning causality, exactly one causal strokeis located away from the junction, and all others must be located at it, asshown in Figure 2.13. The equations characterizing this junction are givenbelow.

2.11. SIMPLIFICATION OF BOND GRAPHS 17

f1 = f2 = f3 = . . . = fn (2.7)

e1 + e2 + e3 + . . .+ en = 0 (2.8)

Figure 2.13: Typical “1” Junction

Physical examples of “1” junctions are wires in an electrical circuit with-out any junctions, pieces of pipe in a fluid power system with no branches,and a location in a mechanical system where elements are connected andmove together. These are discussed further in the sections on various typesof systems.

2.11 Simplification of Bond Graphs

In order to have a complete, valid bond graph, you should simplify yourinitial bond graph, as appropriate, and then assign causality to all of theelements. 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 onthe system. There are several other simplifications you can make, in the casewhen transformers and gyrators are directly coupled to one another:

1. Two transformers connected directly to each other can be replaced by asingle transformer. The modulus of the new transformer is the productof the moduli for each of the individual transformers; i.e., if the firsttransformer has the equations

f2 = r1f1 (2.9)

e2 =1

r1e1 (2.10)


and the second transformer has the equations

f3 = r2f2 (2.11)

e3 =1

r2e2 (2.12)

then the equivalent transfomer will have the equations

f3 = r2f2 = r2 (r1f1) = r1r2f1 (2.13)

e3 =1

r2e2 =

1

r2

(1

r1e1

)=

1

r1r2e1 (2.14)

So, the modulus of the new transformer is given by

req = r1r2 (2.15)

Note that the order of the transformers doesn’t matter.

2. Two gyrators connected directly to each other can be replaced by asingle transformer. In this case, the order DOES matter, as you can seefrom the development of the equations for the equivalent transformermodulus.

The first gyrator is characterized by the relations

e2 = m1f1 (2.16)

f2 =1

m1

e1 (2.17)

and the second gyrator is characterized by the relations

e3 = m2f2 (2.18)

f3 =1

m2

e2 (2.19)

The resulting transformer, then, has the relations

e3 = m2f2 = m2

(1

m1

e1

)=m2

m1

e1 (2.20)

f3 =1

m2

e2 =1

m2

(m1f1) =m1

m2

f1 (2.21)

2.11. SIMPLIFICATION OF BOND GRAPHS 19

So, the equivalent transformer has a modulus

req =m1

m2

(2.22)

Note that if the gyrators’ order was reversed, then the modulus wouldbe inverted. Transformers can be combined in either order, but gyratorscannot.

3. A transformer and gyrator connected directly to each other can bereplaced by a single gyrator. The order of these items DOES matter.Consider, first, the case when the transformer is first and the gyratoris second. The transformer is characterized by the relations

f2 = rf1 (2.23)

e2 =1

re1 (2.24)

and the gyrator is characterized by the relations

e3 = mf2 (2.25)

f3 =1

me2 (2.26)

The resulting gyrator, then, has the relations

e3 = mf2 = m (rf1) = mrf1 (2.27)

f3 =1

me2 =

1

m

(1

re1

)=

1

mre1 (2.28)

So, the equivalent gyrator has a modulus

meq = mr (2.29)

4. Now, consider a gyrator first, followed by a transformer. The gyratorhas the equations

e2 = mf1 (2.30)

f2 =1

me1 (2.31)


and the transformer has the equations

f3 = rf2 (2.32)

e3 =1

re2 (2.33)

The equivalent gyrator then has the equations

e3 =1

re2 =

1

r(mf1) =

m

rf1 (2.34)

f3 = rf2 = r

(1

me1

)=

r

me1 (2.35)

So, the equivalent gyrator has a modulus

meq =m

r(2.36)

There are two more simplifications, which are shown graphically in Figure2.14.

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 causal-ity, and then assign any causalities that are not optional; for example,if a source of flow is connected to a “1”, assigning the appropriate


causality to that source will dictate what the causality MUST be onthe other bonds on that “1”. Similarly, if a source of effort is connectedto a “0”, assigning causality to that source will determine the causalityon every bond on the “0”. Go as far as you can for each source beforegoing 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 as-sign its preferred (integral) causality, and then assign any causalitiesthat are not optional. Note that, in some cases, this may require thatother energy storage elements take on the non-preferred, or derivative,causality. Go as far as you can for each energy storage element, thengo 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 ofthese elements. Next, assign any causalities that are no longer optional,going as far as you can, and then repeat the procedure until all elementsare assigned.

In some cases, you will have one or more energy storage elements inderivative causality. This means that these energy storage elements are NOTindependent, and they will not be associated with a state variable. This willbe explained more fully when the derivation of equations is covered.

Chapter 3

Mechanical Systems

In mechanical systems, the energy storage elements are inertia (mass or massmoment) 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 mechanicaltransformers listed in Table 3.1.

Table 3.1: Examples of Mechanical TransformersElement Domains ModulusGears Mechanical Rotation/Mechanical Rotation r1/r2Gear & Rack Mechanical Rotation/Mechanical Translation r1Lever Mechanical Rotation/Mechanical Rotation l1/l2

3.1 Mechanical Translation

In mechanical translation, the inertance, compliance, and resistive elementsare the mass, spring, and damper. The parameter “I” is associated with themass, “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 ofa moving mass), and a “0” will be associated with a particular force (such asthe force across an ideal spring or ideal damper). In constructing mechanicalbond graphs, you can follow this procedure:

23

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. Thesewill typically be the forces across springs and dampers.

4. Attach the compliance and resistance elements (C and R) to the rele-vant “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, aregiven below. An explanation of how these bond graphs are constructed willbe given through videos posted on the class Blackboard site.

EXAMPLE 1.1: Given the system shown, construct a bond graph and assignappropriate causality.

3.1. MECHANICAL TRANSLATION 25

EXAMPLE 1.2: Given the system shown, construct a bond graph and assignappropriate causality.

26 CHAPTER 3. MECHANICAL SYSTEMS

3.2 Mechanical Rotation

In mechanical rotation, the inertance, compliance, and resistive elements arethe mass moment, torsional spring, and rotational damper. As in the linear

case, while I = J and R = b, C =1

k, so you need to be alert to this when de-

veloping your equations. Sources of effort are torques (AKA moments), andsources of flow are imposed angular velocities. The process for constructionis identical to that for mechanical translation, with the rotational instead oflinear quantities.

Note that, in many rotational problems that involve gears or wheels thatroll without slipping, you will see derivative causality. This will become im-portant 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 cen-ters, and roll without slipping on each other. A torque, T , is applied tothe first roller; the third roller has a rotational damping associated with it,with damping constant b. Construct a bond graph and assign appropriatecausality.

Chapter 4

Electrical Systems

In electrical systems, the inertance, compliance, and resistive elements arethe 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 arecurrent sources.

Construction of bond graphs for electrical systems is fairly straightfor-ward; you need to recognize how the elements of the circuit correspond tothe various bond graph elements. A node in a circuit, where wires are joinedtogether, is a “0” junction. A wire with no branches is a “1” junction. Toconstruct the bond graph, follow this process:

1. Assign a power convention. To do this, mark the direction in which youassume current will be flowing, and then mark the appropriate voltagedrops for this direction. For example, if current is flowing from left toright through a resistor, then the left end of the resistor will be markedas “+” and the right end as “-”.

2. Label the nodes in the circuit, and establish a “0” junction for eachone.

3. Establish a “1” junction for each wire, and join them to the “0” junc-tions 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 con-nected to ground can be removed. If ground is not marked, then you

27

28 CHAPTER 4. ELECTRICAL SYSTEMS

can assume a particular location as ground. Typically, this will elimi-nate 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 expla-nation of the construction process is given in a video on the class’s Blackboardsite.

EXAMPLE 1.4: Given the circuit shown, construct a bond graph and assignappropriate causality.

Chapter 5

Hydraulic Systems

In hydraulic systems, the inertance, compliance, and resistive elements arethe fluid inertia, tanks, and fluid resistance. Fluid resistance can come froman orifice, or from the friction as fluid moves through a long pipe. Sourcesof effort are pressures, and sources of flow are fluid flows.

The fluid capacitance is given by the relation C =A

ρg, where A is the

cross-sectional area of the tank, ρ is the fluid density, and g is the gravita-

tional constant. Fluid inertia is given by the relation I =ρl

A, where l is the

length of the pipe.

Construction of bond graphs for hydraulic systems is very similar to theconstruction process for electrical systems, with nodes in a pipe representedby “0” junctions, and straight pipes with a fluid flowing through them rep-resented by “1” junctions. To construct the bond graph, follow this process:

1. Label the nodes for each pressure of interest, and establish a “0” junc-tion for each one. Remember that you may have a node correspondingto atmospheric pressure in some cases (this is often removed later, butmay be needed as part of the initial construction)

2. Establish a “1” junction for each pipe, and join them to the “0” junc-tions 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 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.

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 - canbe constructed separately, using the appropriate steps, and then the variouspieces are joined together with either transformers or gyrators. The choiceof whether to use a transformer or a gyrator depends on the physics of theconnection between the domains. If the efforts in the domains relate to eachother, then a transformer is used; if effort in one domain is related to flow inthe 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 assigningcausality and deriving equations is independent of the domains.

31

32 CHAPTER 6. MULTI-DOMAIN SYSTEMS

Chapter 7

Deriving State-Space Equationsfrom Bond Graphs

Once a system has been represented by a bond graph, you can use the bondgraph to derive the state-space equations describing the system’s dynamics.The procedure for developing the equations is independent of the domainwhich is represented by the bond graph. The procedure is conceptuallysimple, although the algebra can become very complex at times, particularlyin large systems. The basic steps are:

1. Select the input and energy state variables. Each source will supplyan input, which will become part of the “u” vector in the equationx = Ax + Bu. Each independent energy storage variable (those Iand C elements in integral, NOT derivative, causality) will supply astate variable. If the independent energy storage element is an “I”element, its state variable will be a generalized momentum (p). If theindependent energy storage element is a “C” element, its state variablewill 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 storageelements and resistances can be plugged in.

3. Reduce the initial set of equations to the proper number of equations,

33

34CHAPTER 7. DERIVING STATE-SPACE EQUATIONS FROMBONDGRAPHS

in state-space form. This requires you to eliminate everything exceptthe states, their derivatives, inputs, and the parameters characterizingthe system. This is the step that can be algebraically tedious, althoughwith practice you’ll develop a better understanding of where to startand how to flow through the system.

In order to demonstrate the procedure, some of the examples from pre-vious sections will be used here. The answers are given here; the process forgetting those answers can be found in the videos provided as an additionalclass resource.

EXAMPLE 1.6: Given the following system and its bond graph, derive thestate-space equations for the system.

35

[p2q3

]=

− b

m−k

1

m0

[ p2q3

]+

[10

]F (t) (7.1)

36CHAPTER 7. DERIVING STATE-SPACE EQUATIONS FROMBONDGRAPHS

Chapter 8

Practice Problems

1. Given the system shown below, construct a bond graph, simplify, ap-ply appropriate causal strokes, and derive the state equations for thesystem in matrix form.


37

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 thesystem in matrix form. Assuming that the outputs are the accelerationand velocity of the sprung mass, ms, write the output equation inmatrix form.


39

5. Given the system shown in the previous problem, change the flowsource, ω, to an effort source, T , and re-solve the problem. How doesthis change affect the system?



41

10. Given the system shown below, construct a bond graph, simplify, ap-ply appropriate causal strokes, and derive the state equations for thesystem in matrix form. If the system outputs are the flow out of thesecond tank and the volume of water in the first tank, write the outputequation for the system.

Bibliography

[1] Wolfgang Borutzsky, ed. Bond Graph Methodology: Developmentand 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 SystemDynamics: Modeling and Simulation of Mechatronics Systems, FourthEdition, Hoboken: John Wiley & Sons (2006)

[4] Arun K. Samantaray & Belkacem Ould Bouamama Model-based ProcessSupervision: A Bond Graph Approach, London: Springer-Verlag (2008)

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