Modeling&Simulation ME 6105
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Object Oriented Modeling with Dymola Systems Realization Laboratory GW Woodruff School of Mechanical Engineering Systems Realization Laboratory Product and Systems Lifecycle Management Center Georgia Institute of Technology Lecture Objectives Relate method for modeling lumpedparameter energy based systems to objectoriented modeling Explore objectoriented modeling in Dymola Explain how Dymola works Review From Last Lecture A Systems Modeling Approach 1 Identify the lumped system components 2 Describe the connections or interactions between the components 3 Describe the internal behavior of the components 4 Make sure the system of equations is square Ultimate Goal Convert this systematic method into an automated computer algorithm Node 2 3 4 Node 3 2 V V V V 5 VV IC rource Square DAE System V1 11 VzV1VW 11120 V2 12 V3 V42RI3 3I420 V3 3 CdV5 V6dt215 15160 V4 14 VizV 1620 V5 15 V2V3 2320 V6 16 V4V5 V10 12 unknowns 12 equations Principles of ObjectOriented Modeling Enablin Model Reuse Modularity Classesampinstances Generic parametric models are defined as classes ReS39Stor Class Example 3 model class forresistors vs a model instance forR5Q Encapsulation V Z R Only interface is exposed All details are hidden inside the class Example interface two electrical ports I Abstraction User can deal with model at a high level of abstraction Example no need to know equations to use a resistor model Inheritance Child class is a specialization of a parent class T Example both resistors and capacitors are children of the class of electrical components with two ports uic r of ola Resistor1 i 3 i Ground l 9 Z 3 E 5 LJouoedeo Menu Help Documentation Getting Started with Dymola gum gem Capsular v Resmom v v t t t mud J u u 2 u a a s o a 1 x mm 5 vnle smaxemumhumd mmaaamu mmmeWax J true 1 H How Does D mola Work All models are expressed in the model Capacitor quotIdeal linear electrical Modellca modeling capacitorquot language extends Interfaces0nePort paraneter SICapacitance Dropping a model on c1 quotCapacitancequot the canvas equation Instantiating a 1 Wet end Capacitor Modelica class How Does mola Work I Connecting ports I Is done automatically Instantiating connection s also expressed in equations Modelica Kirchhoff s laws model myCircuit ModelicaElectricalAnalogBasicGround Groundl Mndp1ira F1Pnfrina1 Analon Ra in RP i for Resistorl ModelicaElectricalAnalogBasicCapacitor Capacitorl Modp39lira F hat hcira39l Ana39lon Qanma 3 equation connectConstantVoltage1p Resistor1p connectResistor1n Capacitor1p 39 n n connectConstantVoltage1n Ground1p end nyCircuit Dymola Uses The Modelica Language Modeling Language Generic Not a programming language No specific discipline Declarative Targetssystemsengineering Equationbased Hybrid systems Not procedural Mixed continuous Nonmlusal discrete event systems Fma or Fma0 Comparable to Object Oriented VHDLAMS Encapsulation VerilogA Abstraction Inheritance Dymola User Interface Torn Sorted Equations Code Generator C Code C Compiler Executable Simulation An Overview of the Modelica Standard Library Lots of information about syntax and semantics of Modelica caReference Packages Waltladen jmudelca For working With signals amp controls generate waveforms transfer functions sampling logical operators lookup tables interpolation Defines constants pi e G h Motors and generators DC asynchronous induction 39 Multiphase power grids power electronics thyristors etc Only to be used when writing Modelica see Blocks otherwise Only 3D and 1D for planar motion use 3D mm 51mm Media liquids and gasses no component models w El Media 4 ii j Slums lt E g SlaeGranh i Dmemal 1 FluidHeHlFlow Z a Hea ransisr Utilities for printing and file access E a Writes 4 Use these units when defining variables Very simple flow of fluids With thermal properties Signal Connector Types I Signals k lnPort Bluetriangle arrow 7 7 OutPort White triangle blue border mpm S Vector of signals Real 7 r Integer Signals Intergerlnport Orange triangle lntegei ToReall RealTolnteger1 lntegerOutPort White triangle orange bor er Vector ofintegers gt I R I Boolean Signals BooleanlnPort Pinktriangle BooleanOutPort White triangle BODIeanF UISE1 pink border Vector of Boolean variables 39 f ldealSWltEhl period1 Electrical Connector Types I Electrical Pins Capacitort Positive Blue squares I D Negative White square with blue border C1 Voltage and current flow I Multiphase Electrical Plug Resistor1 Positive Blue circle Negative White circle with blue GO border m3 Vector of m Pins Mechanical Connector Types Translational Flange SlidingMass1 Le Green square Right White square with green border only Position and force ow Rotational Flange lnemm Le Gray square i i Right White square with gray border Equot 3 only i Angle and torque ow J 1 Mechanical Frame Le Gray rectangle Right White rectangle with black I border a b 3D position 3D orientation 3D force ow 3D torque ow 1 W3 ActuatedPrismaticl Electrical SubLibrary m HH Ground Reslalui SignaqultagE CunsIaWnllage I ElectricalElectronics rem we Basic HEalingREsismr SnapVUiiaga RampVDllagE Ideal Lines Ga iiu Semiconductor Sensors 03quot de Capacitor SmaVullagE ExpSlnEVultagE Sources FEE lquotquotquotquotD I D D v Indumoi Translormer ExpnnznlialsV PulsEVuhagE El gt3 rep 434 Gylatur EMF SaWTnuthnll Tiapezuiqull ag I Onedimensional translation Scalar velocity and force lulerlaces SlidingMass Slup Good for simple Cartesian r m W rCEquot 390 A A mechanisms Rod Spring Damper SpringDamper Do not use forJomts in 3D mechanisms rug gt n p a S a phlrwra ElastnGap Pnsilinn Accelerate Move 77 39gt worn a Fixed Force RelaliveSlales Sensors Mechanical Rotational SubLibrary 397 3 a at LussyGEar GaaIElliciEnm inertia IdanGEal I Onedimensional rotation Q Q 51 211 39ii Scalarangularvelocity and Gear Gear ldealPIanEtary IdEBIGEBIRZT torque Goodforgeartrains i 11 W FEB Do notuseforjointsin3D me aT mechanisms Position Acneleiate 5pm Damper A av ta quotFlaw Muve leed SpringDampEr Elastacklash gt9 u Zip Torque RelalNeStates Bearinanctiun Clutch Signals Block SubLibrary El inlagialnr lelntagmlm Cluck Constant I Mathematical operations on signals continuous Derivative FllSIUldEl Step Ramp Math Sources Secondordei Pl Sine EXpSlnE I Similarto MatlabSimulink PID LimPlD Exponentials Pulse tits A B TianaterFuncli SlaleSpace SawTUuth Trapezoid Be Careful with Sign Conventions All three models will result in the same signs forthe torques and velocities The gear ratio is affected 15 versus 51 D poem395 am of roman 02 a 425 1 33 Bah l imqiinnr v 1 gun 5 sine Be Careful with Sign Conventions sinei F 2 lfan internal variable is we created an arrow in the icon defines the positive direction traqHzll l Eg Spring has an internal PM a variable f What are the signs of the forces in each of the connectors lrquzll l Exercise 1 Model a DC Motor Motor Equations di v R1 LE Km K a BackEMF Electro Motive Force Quad Ki JdJ Ba Ki Motortorque A motor consists of Electrical resistor inductor and backEMF in series Mechanical motor torque driving load rotor inertia and bearing friction Best Practice Organize Models in a Package Interfaces Connectordefinitions Abstract classes eg twopin Ideal ldealized models used in component definitions Packages I Components The system components Cumpgnems I Examples Q lnlerfaces jldeal jExamples Etta lt Examples of how the components can be configured into systems Can also include tests of component models Exercise 1 Model a DC Motor Exercise 2 Model a Train Model the train as a selfcontained model What are the inputs and outputs of the train model Include the following components phenomena A DC motor from previous exercise Agearbogtlt Wheels conversion from rotational to translational Mass of the train As an extension Air resistance Assume Train moves on a straight horizontal line No controller is included 9 use constant voltage source Exercise 2 Model a Train Exercise 2 Model a Train Which assumptions have you made in your model I In which situations would this model be useful How could you further re ne the model Become familiarwith Modelica Standard Libraries Electrical Mechanical translational and rotational Signal Learn How to use the Standard Libraries to Model multidisciplinary systems Atrain example Further Reading Peter Fritzson Principles of ObjectOriented Modeling and Simulation With Modelica 21 VWeyIEEE Computer Society Press 2003 ISBN 047147163 Modelica Tutorial httpvwwvmodelicaorgdocumentsModelicaTutorial14pdf Should contain everything you need to know I Modelica Speci cation 0 httpvwvw mndplira quot quot 39 quot I 039 pdf Advanced reference document Example Solutions Systems Realization Laboratory DCMotor Solution name Systems Realization Laboratory Uncertainty in Design Chris Paredis GW Woodruff School of Mechanical Engineering Systems Realization Laboratory Product and Systems Lifecycle Management Center Georgia Institute of Technology Ult0ngt entif Alternatives At 012 won y 0 gtU O O UEOU Decompose and 11 l A2ltE0u 39 At 21529 232539 on UO pr mm tr r unce aIn y 0 gtUO preferences A 0m 39UOnz O gtUO k W Choose the best Choices Alternatives Outcomes Preferences alternatives behavioral simulations Setting the Context Identify the decision situation and understand ob39ectives Sensitivity analysis Further analysis no Implement de0is n Overview of Uncertainty Module Representation of Uncertainty Computation with Uncertain Information Software Support for Uncertainty in Design Sensitivity Analysis Advanced Topics amp Discussion Lecture Objectives Motivating Questions Why do we care about uncertainty in Design What are some common sources of uncertainty What mathematical formalism should one use for uncertainty Background Statistics Probability Theory Discussion Revisit criteria for selecting uncertainty representation Required Reading Clemen Chapter 7 amp 8 Some Sources of Uncertainty What are some sources of uncertainty in design General categories Specific examples Why do we care about uncertainty Failure to consider uncertainty properly can lead to Under designquot o System does not meet requirements frequently enough o Eg High likelihood of failure in structural design example Over designquot 0 System meets requirements but it costs too much to do it o Eg Designing for worst case in tolerance stackup example I Questions How do designers currently deal with uncertainty How should designers deal with uncertainty The Role of Uncertainty in Design Problems 0 Outcomes of deCIsion A 0 alternatives are 1 g 0 uncertain 021 Depending on the 02k unknown state of the 0n1 world the actual AWltEOW2 outcome that is realized 0 will be different Choice Alternatives Outcomes Do We Know Anything About The Outcomes Yes we have some knowledge about the behavior of the world around us We formulate our knowledge about the outcomes of decision alternatives in models Model Expression of our Knowledge in a Mathematical Formalism Model Requires Mathematical Formalism 3 5 35 8 A scale allows us to determine the mass of an object By applying the rules of calculus we can infer the mass of two combined objects Whatare Good Formalisms for Representing Uncertainty Whatare the Characteristics of such Formalisms Characteristics of Good Models of Uncertainty Representation Expressivity Captures all available information amp knowledge Does not imply anything else Has clear unambiguous operational definition Inference Computation Allows the decision makerto infer information relevant to decision Internally consistent Inference is computationally inexpensive Decision Making Yields coherent decisions No sure loss Ideal characteristics that are not satisfied in any formalism Possible Representations of Uncertainty Others DempsterSchaeferstructures Fuzzy numbers possibility theory Fuzzy random numbers Random fuzzy numbers Safety Factors Probability theory Probabilities of specific outcomes Distributionfunction Intervals Upperand lower bounds no knowledge of likelihood Could generalize to sets Pbox Intervalvalued probabilities hybrid representation Possible Representations of Uncertainty Safety Factors Probability theory Probabilities of specific outcomes Distributionfunction Uncertainty Formalism used in Normative Decision Theory Intervals Upperand lower bounds no knowledge of likelihood Could generalize to sets b x Intervalvalued probabilities hybrid representation Others DempsterSchaeferstructures Fuzzy numbers possibility theory Fuzzy random numbers Random fuzzy numbers Lecture Objectives Motivating Questions Why do we care about uncertainty in Design What are some common sources of uncertainty What mathematical formalism should one use for uncertainty gtBackground Read Chapter 7 0f Stalls quotMaking Hard Decisionsquot Probability Theory for a quick review Discussion Revisit criteria for selecting uncertainty representation Statistics versus Probability Theory Both deal with probabilities but with different interpretations Statistics Analysis of frequencies of past events Probability Theory Predicting the likelihood of future events Fundamental Interpretations of Probabilities Many interpretations argued throughout history You probably learned the frequentist interpretation first Probability ofan event isthe limit value of long run frequency of outcomes Eg Coin toss Pheads heads tosses Frequentist interpretation breaks down for certain events Probability of rain tomorrow Probability ofGT winning against Duke on Saturday Many other examples ls probability meaningful beyond relative frequencies Subjective Probabilities I Probability expresses your willingness to bet or act Probability of an event relative amount you are willing to pay to engage in a bet that pays 1 if the eventoccurs pays 0 otherwise 9 probability betl1 important you should determine the amount for which you are willing to both buy and sell the bet the fair price Subjective but Unambiguous 9 the meaning is welldefined and consistentacross different events Operational definition 9 especially important in support of decision making Criteria for Acceptable Probability Values Internall Consistent Coherence Example GT plays against Duke I believe GT has a 50 chance of winning I believe Duke has a 40 chance of winning Are these acceptable probability values Must satisfy nosureloss criterion Dutch book Kolmogorov39s Axioms For any event E 0 S PE 1 For the space of all possible events 9 PQ 1 For disjoint events PE1 UE2 UmUEn ZIP5 11 Criteria for Acceptable Probability Values Externall Consistent What is the wrong with the following belief lam willing to pay 06 for a bet that pays 1 if afair coin flip results in heads 0 otherwise What is the relationship between a frequentist interpretation of an inherently random event and a subjective probability of that event Beliefs should be consistent with scientific factual information How much would you be willing to pay for a coin ip with a bent coin Representing Ignorance in Probabilities What is the probability of drawing a black ball Scenario 1 You have a box and you know thatthere are the same number of white and black balls Scenario 2 You have a box from which you have drawn n balls half black and the rest white Scenario 3 You have a box with white and black balls but no knowledge as to the quantities Representing Ignorance in Probabilities What is the probability of drawing a black ball Scenario 1 You have a box and you know thatthere are the same number of white and black balls Scenario 2 You have a box from which you have drawn n balls half black and the rest white Scenario 3 You have a box with white and black balls but no knowledge as to the quantities The subjective probability of the nextbal drawn being black is 05 all three cases lnformation Entropy of a Random Variable le is a discrete random variable with distribution given by PXxkpk for k12n then the entropy ofX is defined as HltXgt 2 pk logpk k1 le is a continuous random variable with probability density px then the entropy of X is defined as x HltXgt i pltxgtlogpltxgtgtdx Lgtlt Interpretation of Information Entropy Intuitive amount ofinformationthat existsin event Certain event 9 onlyl outcome with probability off 9 HO Coinflip 1 1 1 1 HX lo 10 1b1t 2 g2 2 g2 Entropy rate ofa data source Number of bits per symbol needed to encode it Eg entropy of the English text is between 06 and 13 bits per character Modeling Ignorance Choose the distribution that is consistent with the available information but implies the least amount of information beyond what is truly known 9 Maximum Entropy Distribution Since probability theory does not allow one to express ignorance explicitly we have to use the probability distribution that comes as close as possible Maximum Entropy Distributions px Known upper and lower bounds Uniform distribution Known mean and variance Normal distribution Interpretation What are the implications Probability Distributions in Practice lecture developed in part by Stephanie Thompson Systems Realization Laboratory GW Woodruff School of Mechanical Engineering Systems Realization Laboratory Product and Systems Lifecycle Management Center Georgia Institute of Technology Lecture Overview Eliciting Distributions Process for eliciting probability distributions Heuristics and biases Parametric Distributions Types of distributions and common uses Maximum Likelihood Estimation Principles of MLE Goodnessoffit testing Eliciting Discrete Probabilities Discrete probabilities can be elicited by asking lottery questions How much would you willing to pay for a gamble in which you earn 1 if an event occurs and O if it does not The subjective probability of event occurring pricel1 You must be willing to both buy and sell the bet ie it must be a fair price for the bet Tip Use a bisection approach to find what the fair price is between 0 and 1 concrete questions are easier to answer Eliciting Continuous Probabilities Continuous probabilities can also be elicited by asking lottery questions but the event changes How much would you willing to pay for a gamble in which you earn 1 ifX ltgtltand OifXgtgtlt 9 the subjective probability of X lt X pricel1 Each bet reveals one point on a CDF Example What will be the price of 1 liter of Euro 95 one year from today at the gas station at CORA Example Eliciting a Continuous Probability What will be the price of 1 liter of Euro 95 one year from today at the gas station at CORA People are Not Very Good in Probability Assessment People use heuristics 9 introducing bias Representativeness Situation In a sharp turn in an Atlanta suburb A driver who was speeding significantly lost control of his car and crashed into a ditch Rank the type of car you think was involved in this accident from most to least likely a BMW m3 b Ford Mustang 0 Toyota Camry Based on work by Nobel Prize winners Tversky and Kahneman People are Not Very Good in Probability Assessment People use heuristics 9 introducing bias Representativeness Situation In a sharp turn in an Atlanta suburb A driver who was speeding significantly lost control of his car and crashed into a ditch Rank the type of car you think was involved in this accident from most to least likely a BMW m3 b Ford Mustang 0 Toyota Camry How similar a sample is to a typical representative of a class is a poor indicator of the probability of that sample being in the class Based on work by Nobel Prize winners Tversky and Kahneman Other Examples of Assessment Bias Availability The assessment of the probability that an event will occur is linked to the ease with which the DM can remember similar events Anchoring and Adjustment Consider current situation and make adjustments to assess probability of a similar future situation 9 we tend to underadjust Motivational Bias DM may not express true believe because there is an incentive to bias the decision in a particular direction Lecture Overview Eliciting Distributions Process for eliciting probability distributions Heuristics and biases EgtParametric Distributions Types of distributions and common uses Maximum Likelihood Estimation Principles ofMLE Goodnessoffit testing Modeling Uncertainty Little or No Explicit Knowledge Elicit subjective probability For continuous distributions fit a CDF to elicited points Experimental data 1 Find the best fit forthe distribution parameters 2 Check whether best fit is a good fit goodnessoffit testing Which Distribution I Parametric probability I Common Probability distribution Distributions Findthequotbestfitquotamong 39 Exponentlal common parametricdistributions 39 Urilform Samplethis parametric 39 Tr39anQUIar distributionduringsimulation 39 Normal Lognormal CDF Beta Gamma 1 Weibull Simple Distributions Uniform Distribution Exponential Distribution a Only lower and upper bounds A are known 39 Triangular Distribution Time between random events constant arrival rate 71quotx20 xlt0 a c b fx jg Quickfirstguess ofdistribution Normal Distribution I Use few 1 Errors of various types 2 f x e Sum ofa large number of 27502 other quantities central limit 20 theorem rrx Maximum entropy distribution when only mean and 04 variance are known NOTE when estimating 3 mean and variance use Student distribution tdist Parameters rt 0 Lognormal Distribution I fame02 I Use fxe 2 72 Timeto perform a task 35427502 Product ofa large number of other quantities Quantities that are always positive and the distribution is skewed towards zero x LNu039Z ltgt1nx Nu0392 Parameters y 039 7 US 10 is 20 25 an 35 41 u I Beta Distribution I Use xuq711 xaze1 When lower and upper x 2 W 0 lt x lt1 bounds exist 1 2 Rough initial model but more refined then triangular Commonly used in Bayesian probability theory easy to compute posterior distributions Parameters a1 a2 Weibull Distribution fx a x 1e x x gt O Use Timeto complete a task x Timeto failure for a piece of Fx 1 8 equipment If the failure rate decreases overtime then alpha lt 1 If the failure rate is constant overtime then alpha 1 If the failure rate increases overtime then alpha gt 1 Parameters 1 Example Fitting a Normal Distribution Normal distribution 1 J s 32 x 2 427502 Step 1 determinethe best t e 20 eXnlei quot11 eWMkgim mz 11 Maximum Likelihood Estimation Choose the parameters of the distribution such that the likelihood L6 fx1x2xn 6 that the sample data is drawn from the distribution is maximized fx1xzx l lt9isthe conditional probability density function of the distribution evaluated at points x Example for normal distribution Mn knacewam mo 1 n 329702 Example Maximum Likelihood Estimation I Maximize Lp02 to obtain maximum likelihood estimators of the mean and variance V 70341 He zo2 11 maxLu72 where Lu72 2 W 27m argmaX Lu0392 arg max lnLu 01 70 LE lnLuo zln 1 22xr z 27m2 39 Example Maximum Likelihood Estimation I Take the derivative wrt p and solve for maximum InLLO39ZIH 1 iHLZEW lzj 272702 B1 2039 1mm o lt 2gtjixL ugt 3 2039Z 1 A 1 2 xi I Repeat for 02 to get MLE forthe variance Decision Making in Design Systems ealization Laboraton C h S P a 8 SW Woodruff School of Mechanical Engineering Systems Realization Laboratory Product and Systems Lifecycle Management Center Georgia Institute of Technology Lecture Objectives Motivate this course s focus on decision making in design Describe the elements of design decisions Identify why decisions are difficult Introduce a modeling language to decompose and structure decision problems READING Clemen Chapters 1 and 2 What is Design Designing is the process of converting information about needs and requirements for a productsystem into a complete specification of that productsystem Our perspective Design is a type of problemsolving Design involves deciding on the most preferred solution alternative Generate Evaluate Select Alternatives Alternatives Alternative Formulation Analysis Interpretation Georgiaiir i is m39r39rech J Product Development Phases Characterized by product development phase Decisions HHHHHHHHHHHH 4 Product LifeCycle gt I Pahl and Beitz classification of design activities Clarification of task Conceptual design Embodimentdesign Detail design i6 titlePegs Product Development Scope Requirements Design activities characterized by scope and level of abstraction Transmission Requirements Engine Requirements Frame Requirements Frame Transmission E Engine I Decision Making is a Key Aspect of Designing Decisions provide a common structure to most design activities Same structure across different design phases different problem domains different problem scopes etc Decisions are the milestones at which the design process moves forward eg phasegate process Feasibility amp Co nCept Detailed Design Pilo t Production g Prodction Design Vi Reviews Wars Jm litiEIN mm C V t G Program Design Phase Pilot Phase ReV39eWS Approval Approval Approval Geo iatlu39 Um swab J What is a Decision A Decision is A present action to achieve a future outcome An irrevocable allocation of resources in the sense that it would take additional resources perhaps prohibitive in amount to change the allocation Matheson Howard A Decision Maker is An authority with power to allocate an organization s resources Decision Analysis Descriptive theory How do people solve design problems I Prescriptive theory How should people solve design problems Normative theory How would people solve design problems under idealized circumstances The decision makeris o Fully informed o Fully rational o Able to process knowledge and information accurately at zero cost Elements of a Decision gtU011 gtU012 gtU01k 1 1 1 39U021 gtU0 gtU02k Select Al QQQ GOO gtU0n1 gtU0n2 1 2 k 21 22 2k r11 r12 OHk gt U0nk 0 0 A1 6 A2ltz An E Choices Alternatives Outcomes Preferences Why is Decision Making Hard How to Formulate a Design Decision understand ob39ectives 0 39Ult0gt A n 02 Identify Alternatives 01k U 01k P 0 mo Decomposeand 1 1 model the problem Decision A2 022 gtU022 Select Al roblemstructure 0 Um Encertain I 0 Um reterenctes Anlt0nz gtUlt0nzgt p 0 gtU0 k k 39 Choosethe best Choices Alternatives Outcomes Prelerenoes alternatives Sensitivity analysis Furtheranal sis b no Implement dec n Decomposing Decision Problems Influence Diagrams Decompose and organize decisions into decision elements Represent decision structure as a graph Product eg consequences orvalues Influence Diagrams Meaning of the Arrows Arrows can mean either relevance or temporal sequence Relevance Relevance Sequence 0 The probability of C F is computed G an H are known depends on A and B from D and E before deciding l Influence Diagram Examples 391 Tomoml Damping Payload Max x Maximum Upward Accelemuon of VAVA Damping 9 90quot Coef uem Q 2 391 5 Amphtude ofTermn Legend Daemon 0 chance Event In uence Arc I d Warm An influence diagram is not a flowchart There cannot be any cycles in an influence diagram Influence diagrams can be developed incrementally Decisions are key milestones in the design process Decision analysis is a prescriptive decision theory Key elements are alternatives outcomes preferences I What makes decisions hard Identifying the scope of the decision Identifying the decision alternatives Identifying the sources of uncertainty that influence the outcome of the decision Identifying the decision maker39s preferences The structure of a decision problem can be represented in an influence diagram Georgiatimg 119 d mh t i7 Modeling of Energybased Systems Systems Realization Laboratory GW Woodruff School of Mechanical Engineering Systems Realization Laboratory Product and Systems Lifecycle Management Center Georgia Institute of Technology Lecture Objectives Identify how energybased systems modeling fits in the context of the course Characterize systems in terms of energybased lumpedparameter models Identify when lumpedparameter modeling is appropriate Develop a systematic method for modeling lumped parameter systems A Generate Evaluate Select Alternatives Alternatives Alternative Formulation W Interpretation Modeling the structure of design problems Formulation Modeling deSIgn objectives Modeling energybased systems Modeling uncertainty Analysis IVIOdeIIng discreteevent systems Modeling preferences Interpretation Information technologies for simulationbased design What are the abstractions that have been made In Which scenarios does it make sense to make such abstractions Mathematical Formalism for LumpedParameter Models Ordinary Differential Equations ODE x 111 y gx Examples Spring I Damper m kx bx Differential Algebraic Equations DAE ODE with algebraic constraints x fxZtau nxzru 0 Examples y gx o o 9 9 Fourbar mechanism Why a Systematic Modeling Approach Consider a double pendulum Equation of motion using a Lagrangian formulation l K I I w 39W 4 l K Doable but how about a 3dimensional spherical double pendulum Why a Systematic Modeling Approach world y Objectoriented modeling Define the behavior of individual system components Define their interactions 3D Double Spherical Pendulum leoiieuds in 039LJJ Japuiirto poq An algorithm automatically converts component equations connection equations into system 1 equahons g g 9 Henyauds Modeling Approach A Systems Perspective 1 Identify the lumped system components 2 Describe the connections or interactions between the components 3 Describe the internal behavior of the components 4 Make sure the system of equations is square Ultimate Goal Convert this systematic method into an automated computer algorithm Systems Perspective VRI Node 2 3 4 Node 3 2 V V V V 5 V Vvource I C dt Modeling Energy Flow I Port aka connector Location where energy is exchanged I 2 associated variables Across variable Quantity that is measured across two ports Through variable Quantity that is measured through a port across PortA Port B L Component I through gt Across and Through Variables Example Electrical Resistor across POITA E A A Port B Power through gt g voltage V currentl gt39 WW 39 Across and Throu Variables Across Through JACFOSS JThFOUQh I Voltage Current Flux Charge Electrical M V L A CD Vs q C or As I I f I Velocity Force Displacement Momentum ransaiona V InS F N x m p NS Angular Vel Torque Angle Twist Rotational a rads 7 Nm 9 rad T Nms H d I Pressure Volume Flow Press Moment Volume V mm 17 Nmz q m3s r Nsmz Vm3 Ch I ChemPot MolarFloW NA ofMoles emlca M Jmole v moles n mole Thermo Temperature Entropy Flow NA Entropy Dynamic T K dSdt WK S JK Connecting Components I Node equations a1 a2 a I mlnnlI q5m520 I Kirchhoff s voltage and current laws Node I Stick to yoursign conventions I Component II eg positive current into the pin or positive force from left to right I Component II Internal Component Dynamics voltage V Differential algebraic relationship between across and through variables How many equations current I gt equations unknowns Squareness unknowns 2 ports assuming 1 through and 1 across variable per port equations p0rts constitutive equations gt constitutive equations pOIts Node I II Node Equations Equations Constitutive Equations I llama Systems Perspective Unknowns and Equations V1 1 V2V1Vsuwce 1220 V2 2 V3V4R3 3420 V3 3 CdV5 V6dtI5 5620 V4 4 V12V6 1620 V5 15 V22V3 2320 V6 16 V4ZV5 4520 12 unknowns 12 equations uare NOT SQUARE Current equations are not independent Remove one 1 12 2 0 Replace with V1 2 o Grounding an across variable Implicit Differential Algebraic Equation System V1 11 V10 V2 12 V3 V4 RI3 13140 V3 13 CdV5 V6dt215 15160 V4 4 Vle 1620 V5 15 V22V3 2320 V6 16 V42V5 4520 12 unknowns 12 equations General Form of Implicit DAE FCC xju l 0 How to Solve this Implicit DAE System Desirable form Explicit ODE xzfxtu 39 Systematic process 1 Eliminate alias variables x1 x2 or xl x3 2 Revisit the unknowns Derivatives are unknown dxl dt gt dxl dxl Corresponding integrated variables are known xl J d dt t 3 Sort the equations and assign causality from knownsto unknowns Use graphbased sorting algorithm by Tarjan 1 Eliminate Alias Variables VZVl KW V V4 2 R13 9 substitutions 3 equations in CdVV6dt 5 3 unknowns V1 0 V1V6 V V V6 i0 V V V2V3 V32V2 392 your ce l05 V5V VZVlRI1 1 I I 13140 I quot 1 CdV4dt I1 I 0 39 I 0 2 Revisit the Unknowns 3 equations in 3 unknowns V K011 V Kource V2 V4 R11 V2 V RI Coll394 dt 11 CW 11 unknowns V2 dV4 I1 3 Sort the Equations and Assign Causality V Isource IZ Isource VViRI1 111V2V4R Coll4 I1 ell4 IlC unknowns V2 ell1 1 state V4 dV4 1 One State Equation some dl RC V4 Original Model Number of components 79 Variables 844 Constants 2 2 scalars Parameters 232 484 scalars Unknowns 610 1765 scalars Differentiated variables 26 scalars Equations 533 Nontrivial 358 Generated using the Dymola software Results for Double Spherical Pendulum Translated Model Constants 1359 scalars Free parameters 70 scalars Parameter depending 326 scalars Continuous time states 12 scalars Timevarying variables 134 scalars Alias variables 371 scalars Sizes of linear systems of equations 38 45 Sizes after manipulation of the linear systems 17 8 Sizes of nonlinear systems of equations 29 Sizes after manipulation of the nonlinear systems 8 Introduced energybased lumpedparameter models Important in systemlevel considerations where dynamics play a role Developed a systematic method for creating energy based lumpedparameter models Identify the lumped system components and their ports Describe the connections or interactions between the components Describe the internal behavior ofthe components Make sure the system of equations is square Convert implicit DAE into egtltplicit DAE or ODE I We will come back to this in a later lecture
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