CHAPTER 3: BOND GRAPH SYNTHESIS - UTRGV Faculty Web

CHAPTER 3:BOND GRAPHSYNTHESIS& EQUATIONDERIVATIONSamanthaRamirez1

What type of mathematicalequations are needed?How are these equationssystematically derived?How are the individualconstitutive relations of thecomponents connected togenerate a mathematical model?2

Objectives:OBJECTIVES&OUTCOMESTo effectively use bondgraphs to formulatemodels that facilitatederiving mathematicalrepresentations ofdynamic systems,To be able tosystematically derivemathematicalrepresentations usingbond graphs, andTo understand the flowof information within asystem dynamicsmodel and its relationto mathematicalrepresentations.Outcomes: Upon completion,you should be able tosynthesize bond graphmodels of mechanical,electrical, andhydraulic systems,annotate bond graphsto indicate appropriatepower flow andcausality, andderive mathematicalmodels in the form ofdifferential andalgebraic equationsusing bond graphrepresentations.3

Power bond labelse fefe R-Elements Dissipate Energy Direct algebraic relationship between e & ffffeeef C-Elements Store Potential Energy Derivative Causality I-Elements Store Kinetic Energy Derivative Causality4

Sources Supply energy Transformers Convert energy Power through Gyrators Convert energy Power through𝑒1 𝑛𝑒2𝑛𝑓1 𝑓2𝑒1 𝑟𝑓2𝑟𝑓1 𝑒25

1-Junction Common flow Summation of efforts 0-Junction Common effort Summation of flows6

Power goes from the systemto R-, C-, and I-elements Sources generally assumed to supplypower to the system Effort sources specify effort into thesystem Flow sources specify flow into thesystem 2-ports have a power throughconvention7

Junctions with twobonds (power in, powerout) can be simplifiedinto a single bondAdjacent junctions ofthe same type areactually the samejunction and can becollapsedWhat if power is not showing apower in, power outconvention?8

BOND GRAPHSYNTHESIS:MECHANICALTRANSLATION& ROTATION1.Identify distinct velocities (linear/angular)2.Insert the force/torque-generating 1-portsand the energy-conserving 2-ports3.Assign power directions4.Eliminate zero velocity (linear/angular)sources5.Simplify6.Assign causality9

R-ElementDamper or frictionC-ElementSpringI-ElementMassEffort SourceExternal forceFlow SourceVelocity source or shakerTransformerLever or rocker arm1-JunctionCommon velocity; Sum of forces0-JunctionCommon force; Sum of velocities10

Mechanical TranslationExample 1Figure 3.3Figure 3.4

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Mechanical TranslationExample 2Figure 3.6

R-ElementBearing or frictionC-ElementTorsion spring or shaftI-ElementRotational inertiaEffort SourceExternal torque (motor)Flow SourceAngular velocity source (motor)TransformerGear pair or chain and sprockets1-JunctionCommon angular velocity; Sum of moments (torques)0-JunctionCommon moment (torque); Angular velocity differential14

MECHANICAL ROTATION EXAMPLE 115

MECHANICAL ROTATION EXAMPLE 217

Generate a bond graphto predict the responseof the system.19

BOND GRAPHSYNTHESIS:ELECTRIC &HYDRAULICCIRCUITS1.Identify distinct voltages/pressures2.Insert 1-port circuit elements and energyconverting 2-ports3.Assign power directions4.Eliminate explicit ground/atmosphericpressure (or reference pressure)5.Simplify6.Assign causality20

rEffort SourceBattery or voltage sourceFlow SourceIdeal current sourceTransformerTransformer1-JunctionCommon current; KVL0-JunctionCommon voltage: KCL21

ELECTRIC CIRCUIT EXAMPLE 122

ELECTRIC CIRCUIT EXAMPLE 224

Electrical elementsconnected between thesame pair of voltages Equivalencies can beused to simplify circuitbranches connected inparallel Circuit elementsconnected in parallelshare a common voltagedrop across them26

Electric CircuitExample 3Figure 3.1227

R-ElementValve or surface roughnessC-ElementAccumulatorI-ElementSlug of fluidEffort SourceDisplacement pump or pressure sourceFlow SourceCentrifugal pump or ideal flow sourceTransformerN/A1-JunctionCommon flow; Sum of pressure drops around a loop0-JunctionCommon pressure; Sum of flows into a junction28

Hydraulic Circuit Example 1Figure 3.13

Hydraulic Circuit Example 2Figure 3.14

Multiple energy domainsthat are coupled throughtransducers Procedure Decompose into singleenergy domainsubsystems at thetransducers Apply energy specificguidelines to eachsubsystem Recouple usingtransducers31

A Mixed System ExampleFigure 3.16

Synthesize the bond graph for the given system.33

Synthesize the bond graph for the given system.34

Synthesize the bond graph for the given system.35

Synthesize the bond graph for the given system.36

R-ElementC-ElementI-ElementDerivative CausalityIntegral Causality37

𝐼𝑓C - ElementsI - ElementsDisplacementLinear MomentumAngleAngular MomentumChargeFlux LinkageVolumeHydraulic Momentum38

1Gyratore1f1e1f1e2f2e2f2GYGYGYGYe2f2e2f2e1f1e1f139

Synthesize simplified system bond graph Assign causality Sources firstSTATE EQUATIONDERIVATION Then energy-storing elements If unspecified bond remains, select an R-element,assign causality, and propagate Label efforts and flows on energy storingelements Apply primary conditions Apply secondary condition40

Mass-Spring-Damper ExampleFigure 3.1841

Σ𝐹 𝑚𝑥1ሷ 𝑘 𝑥2 𝑥1 𝑘𝑥1 𝑏𝑣1 𝑥 Σ𝐹 𝑚𝑥2ሷ 𝐹 𝑡 𝑘 𝑥2 𝑥1 𝑥42

Mechanical RotationExampleFigure 3.2043

Electric Circuit ExampleFigure 3.2144

Hydraulic Circuit ExampleFigure 3.2245

Mixed System ExampleFigure 3.2346

Algebraic Loops The mass-spring-dampersystem shown is a modelof two railcars beingpushed up against asnubber. What if the firstrailcar was a fully loadedcoal car and the secondan empty flatbed railcar?Figure 3.2447

Energy-storing elements in derivative causality are not dynamically independent, butrather dependent.48

Synthesize a bond graph and derive the state equations of the following system.49

Summary As illustrated in Figure 3.1 (a), generally, it is assumed that power flows from the system to energy-storing ordissipating elements. Usually, it is assumed that power flows from the source to the system. Moreover, effort sources supply effortas an input and flow sources supply flow inputs (refer to Figure 3.1 (b)). Transformers and gyrators have power through convention. As depicted in Figure 3.1 (c), the power goes inone port and out the other. Adjacent 0- or 1-junctions can be collapsed into a single junction. Common junction types adjacent to oneanother are in actuality the same junction and the attached bonds share a common effort or flow (Figure3.2). When synthesizing bond graphs for mechanical systems, we first identify distinct velocities and establish 1-junctions. For each 1-junction we identify elements that are directly associated. For example, inertias arecommonly associated with distinct velocities. Then we insert effort-generating 1-ports off of 0-junctions or 2ports between appropriate pairs of 1-junctions. Next, we eliminate zero-velocity sources and simplify.51

Summary Continued For circuits (both electric and hydraulic) we first identify distinct potentials (voltages or pressures) andestablish 0-junctions. If there are any elements directly associated with these distinct efforts, we place themdirectly off the associated junction using a bond. We then insert the 1- and 2-ports between pairs of 0-junctions.The 1-ports are placed off of 1-junctions that are inserted between pairs of 0-junctions. Next, we eliminate theground or reference pressure and simplify. Mixed systems can be dissected into subsystems, each of which is of a single energy domain. Each subsystemcan be analyzed using the associated guidelines. The subsystems interface at energy-converting transducerswhich are modeled as either transformers or gyrators. Some examples were provided in Figure 3.15. When deriving differential equations from a bond graph one must first assign causality beginning with thesources, then the energy-storying elements, and last, if necessary, the R-elements. At each stage we as- sign thecausality to an element and propagate if the causality affects adjacent junctions and/or elements. The processproceeds until all the bonds have an assigned causality. The differential equations result from applying theprimary and secondary conditions at the junctions. Algebraic loops and derivative causality require extra analysis to derive the differential equations.52

synthesize bond graph models of mechanical, electrical, and hydraulic systems, annotate bond graphs to indicate appropriate power flow and causality, and derive mathematical models in the form of differential and algebraic equations using bond graph representations. Objectives: To effect