engm 541 2011 01 lecture 1 modeling equilibria 2
TRANSCRIPT
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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ENGMENGM 541, ENGM 670541, ENGM 670--X5X5
&& MECE 758MECE 758--X5X5Modeling and Simulation of Engineering SystemsModeling and Simulation of Engineering Systems
WinterWinter 20112011
Lecture 1:Lecture 1:
Introduction; Course Overview;Introduction; Course Overview;
Modeling Physical Systems,Modeling Physical Systems,
LumpedLumped--Parameter Equilibrium SystemsParameter Equilibrium Systems
M.G. LipsettM.G. LipsettDepartment of Mechanical EngineeringDepartment of Mechanical Engineering
University of AlbertaUniversity of Albertahttp://www.ualberta.ca/~mlipsett/ENGM541/ENGM541.htm
MG Lipsett, 2011 2
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
IntroductionIntroduction Engineering systems often comprise complicated assemblies of
components, which can have complex behaviours that are difficult to predict
Internet Sources: www.coolestgadgets.com; www.nasa.gov; www.microway.com.au; www.pbs.org; www.emercedesbenz.com; www.syncrude.com
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 3
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
MathematicalMathematical AnalysisAnalysis in Engineeringin Engineering
Engineering analysis: formulating governing equations that
describe the behaviour of physical and technologicalsystems, for the purpose of analysis and design
Numerical analysis: solving mathematical equations usingalgorithms
Scientific computing: development of reliable numericalmodels that can be tested in a range of cases (includingknown benchmarks)
MG Lipsett, 2011 4
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
What is Modeling?What is Modeling? A modelis a representation of knowledge
Rules, physical analogs, algebraic equations of physical laws
A systemis a bounded region comprising known elementsthat each interact in understandable ways
Applied numerical modeling has joined empiricalexperimentation and analytical methods for solvingproblems of mathematical physics
The types of systems of interest in this course include:
Models of physical systems Mechanical, electrical, thermal, structural, hydraulic, etc.
Combinations of different physical systems (mixed systems)
Models of material, energy, and information flow forengineering decisions Production systems
Economics
Scheduling
Inventory, and so on, and so on,
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 5
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
What is Simulation?What is Simulation?
Simulations are solutions of equations that are functions of time
For continuous systems, we develop (and solve) differentialequations
Examples:
Vehicle dynamics
Thermofluid interactions
Industrial processes
Biological processes
Climate change, and so on, and so on,
Often the equations can not be solved in closed-form
Sometimes simulations are based on empirical understanding oftime-varying behaviour that is not expressed as differentialequations (correlations, discrete events, etc.). These are valuablefor systems that are not characterised well by differential equations.
MG Lipsett, 2011 6
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Scientific Computing at a GlanceScientific Computing at a Glance
(Adapted from A. Quarteroni, Mathematical Models in Science and Engineering,
Notices of the American Mathematical Society Jan 2009)
Interestingproblem
Data fromthe problem
Understandingof the problem
Defining the system
Uncertainty
Sensitivity
Parameter identification
Statistical analysis
Modeling thesystem
Geometry and mesh/network
Governing equations & analysis
Numerical approximation
Algorithms for solving
Computer
simulation &post-processing
Visualisation of results
Validation /Verification Comparison to known results
Benchmark cases
Experiments
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 7
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Decision
Engineering Analysis for Design at a GlanceEngineering Analysis for Design at a Glance
Possible
solution
Design
Performance
specifications
Model of
System behaviour
Problem
definition
Assessment of
Performance
of proposed solution
Modeling to predict
how a design will perform
is key to a successfulsolution
MG Lipsett, 2011 8
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
ENGM 541 Course IntroductionENGM 541 Course Introduction Why do engineers need to learn about modeling and
simulation?
Most engineering problems are too complicated or complexto solve analytically
Engineers rely on numerical modeling and simulation toanalyse and design systems that have time-varying aspects
Engineering managers use models of technologies andbusiness processes for decision making
You may want do develop models to solve a technical orbusiness problem, by designing a solution and modelinghow you expect it to perform
You may need to interpret the results of models created byothers
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 9
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
General Course OutlineGeneral Course Outline
Understanding concepts of formulating mathematical models
based on physics (and other rules of interaction) between theelements of a system
Formulating governing equations and choosing solutionmethods for different types of analyses of physical systems
Understanding advantages and limitations of numericalsolution methods
Understanding simple models for financial decisions andtechnological systems that have event-based dynamics
Using modeling and simulation for design
Presenting and interpreting analysis and simulation results
Analysing engineering systems and processes using generalpurpose programs: MATLABand SIMULINK
MG Lipsett, 2011 10
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
ENGM 541 Course Overview (1)ENGM 541 Course Overview (1) Lecture Room:
Time Slots:
Instructor:
Office:
Office Hours:
TA:
Course Text:
E-Class &Course Web Site:
ETLE 2-001
Lectures: Wednesdays 5:00 pm 8:00 pm
Laboratories: Thursdays 5:00 pm 8:00 pm in ETLE 2-005
(required for ENGM 541 only)
MG Lipsett ([email protected])
Room 5-8J, Mechanical Engineering Building
(5th Floor West)
Wednesdays 1:003:00 pm (other times by appointment)
Masoud Mashkournia
Modeling and Analysis of Dynamic Systems,
by R. Esfandiari & B. Lu (CRC Press)
http://www.ualberta.ca/~mlipsett/ENGM541/ENGM541.htm- Lecture slides
- Assignments
- FAQ and announcements
- Worked examples and sample test questions
CHECK ECLASS & THE WEB SITE OFTEN !!
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 11
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
ENGM 541 Course Overview (2)ENGM 541 Course Overview (2)
Marks:Marks:
Assignments: 25%Will be due in class and cannot be accepted after solutions are posted
ENGM 541 Labs: 5%
ENGM 541 Project: 15% (ENGM 670 & MECE 758: 20%)Individual, criteria to be announced, due April 6 2011 (before the exam)
Midterm Examination: 20%Wednesday March 2, 2011, 5:00 pm 7:00 pm in ETLE 2-001
Final Examination: 30%Wednesday April 13, 2011, 5:00 pm 7:30 pm in ETLE 2-001
Examinations will be open book & open notes Calculators are allowed but communication features must be
turned off (no computers)
MG Lipsett, 2011 12
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
ENGM 670 & MECE 758 Course OutlineENGM 670 & MECE 758 Course Outline Lectures will be the same for ENGM 541, ENGM 758, and
ENGM 670
But there are additional requirements for grad students:
Supplementary readings
MECE 758: more on physical systems
ENGM 670 more on technological systems
More assignment problems
Additional scope for the individual project Different exam questions
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 13
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
ENGMENGM 670 & MECE 758670 & MECE 758 Course Overview (2)Course Overview (2)
Marks:Marks:
Assignments: 25%Will be due in class and cannot be accepted after solutions are posted
Lab attendance is not required; but you are responsible forbeing able to do the Matlab coding covered in the labs
Project: 20%Individual, criteria to be announced, due April 6 2011 (before the exam)
Midterm Examination: 25%Wednesday March 2, 2011, 5:00 pm 7:00 pm in ETLE 2-001
Final Examination: 30%Wednesday April 13, 2011, 5:00 pm 7:30 pm in ETLE 2-001
Examinations will be open book & open notes
Calculators are allowed but communication features must beturned off (no computers)
MG Lipsett, 2011 14
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
GeneralGeneral CourseCourse Success FactorsSuccess Factors Keys to success:
Do the homework to master model building
Try the examples in MATLAB
Check E-Class and the web site often
FAQ, worked examples, sample tests
Ask questions! (but think first)
This is a demanding course but you will gain avaluable approach to analysis and design
We have to unlearn some things to do generalsystems analysis correctly
We will also learn by doing
Ill do my best to be interesting
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 15
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
ENGM 541 Course Overview (4)ENGM 541 Course Overview (4)
University policy: suspected cheating or plagiarism will bereported and investigated
Professional ethics and integrity
Do the right thing.It will gratify some people
and astonish the rest. -Mark Twain
MG Lipsett, 2011 16
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Your Instructor: MG LipsettYour Instructor: MG Lipsett
Professional Engineer since 1986
Research
Reliability of complex systems (anomalies, machinery diagnostics)
Robotics and automation (excavation, remote embedded sensing)
More sustainable processes for oilsands bitumen production andreclamation
Industrial Experience Atomic Energy of Canada Ltd (R&D in robotic inspection, hazardous
waste site remediation, reliability)
Syncrude Canada Ltd (mining automation & space robotics
teleoperation, extraction process R&D, mine maintenance & reliability)
Seven years in leadership and management roles (Operations, R&D,Projects)
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 17
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Engineering AnalysisEngineering Analysis
Types of analysis:
Two means of modeling physical systems:
Once a model has been developed, then numerical
procedures can be used to study system behaviour usingcomputers
MG Lipsett, 2011 18
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Modeling Physical SystemsModeling Physical Systems Consider a beam:
This is an inherently continuous structure. When weanalyse this beam for deflections, natural frequencies, etc.,we can start from one of two approaches.
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 19
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
LumpedLumped--Parameter ModelParameter Model
The properties of the continuous system are visualised as
being separate from one another
The beam is modeled as a linkage mechanism
We find a set of algebraic equations from which we candetermine the deflections
The price we pay is one of approximating the physicalsystem at the modeling level.
MG Lipsett, 2011 20
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Continuous ModelContinuous Model Alternatively, the beam is modeled by deriving differential
equations that represent the continuous system
The solution to the differential equations requires that theybe approximated by algebraic equations (e.g. finitedifference expressions), for almost all non-trivial cases
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 21
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Solving Algebraic Equations of the ModelSolving Algebraic Equations of the Model
In either case, we are solving algebraic equations.
After the modeling is complete, we choose the type ofsolution:
We want to have a consistent way to set up problems andto solve them.
MG Lipsett, 2011 22
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Equilibrium Problems for LumpedEquilibrium Problems for Lumped--Parameter SystemsParameter Systems We are looking for steady-state solutions to problems where
the continuous system has been modeled using lumpedparameters.
We are concerned with systems of interconnectedelementselements. The elements within the problem have propertiesthat we must know before we can proceed.
Elements are connected at nodes.nodes. Here is an example of asystem network:
Loops are paths that start at a particular node, pass througha number of elements, and return to the original node.
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 23
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Loop and Node VariablesLoop and Node Variables
A system will have both loop and node variables.
Loop variables describe the path around the loop.Examples:
Node variables describe variables that come together at anode.
Examples:
Loop and node variables:
The loop and node variables are related by the constitutiverelationships of the elements.
MG Lipsett, 2011 24
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Formulating Constitutive RelationshipsFormulating Constitutive Relationships1. State the variables
2. Describe the element
3. Sketch the constitutive relationship.
4. Use an analytic expression for the relationship
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7/31/2019 ENGM 541 2011 01 Lecture 1 Modeling Equilibria 2
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MG Lipsett, 2011 25
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Constitutive Relationship Example #1Constitutive Relationship Example #1
1. State variables:
2. Describe element:
3. Sketch:
4. Write analytical relationship:
MG Lipsett, 2011 26
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Constitutive Relationship Example #2Constitutive Relationship Example #21. State variables:
2. Describe element:
3. Sketch:
4. Write analytical relationship:
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MG Lipsett, 2011 27
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Admissibility LawsAdmissibility Laws
The node laws satisfy the admissibility requirement that the
node variable is conserved at a node
The loop laws are similar (but different). Loop variables aregoverned by loop admissibility laws that require the value ofthe loop variable at a node to have only one value
MG Lipsett, 2011 28
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
GeneralisingGeneralising KirchoffsKirchoffs LawLaw We use a general approach for system networks using the
principles of Kirchoffs Laws.
Kirchoffs Laws for electrical circuits use the physical lawsof conservation of charge (node law) and conservation ofenergy added or taken by a potential field (around loops,mesh law), including dissipation. Gain or loss around anentire loop has to be zero (because there is no net changein the location with respect to the field).
For other types of physical systems, we construct ourvariable assignments so that we can exploit similar physicallaws:
Conservation of momentum law (DAlemberts law for forces)
Conservation of mass law for flows, etc., etc.
For non-physical systems, we need similar loop & nodelaws
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MG Lipsett, 2011 29
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Example: LumpedExample: Lumped--Parameter Electrical NetworkParameter Electrical Network
C
L
R1
R2
MG Lipsett, 2011 30
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Some Equilibrium Element TypesSome Equilibrium Element TypesType Node Variable Loop Variable
Mechanical
Electrical
Fluid Flow
Heat Transfer
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MG Lipsett, 2011 31
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
General Procedure for Setting Up A ProblemGeneral Procedure for Setting Up A Problem
1. Choose the variable in which you want your final equations
expressed2. Choose variables so as to satisfy the pertinent admissibility
requirement
3. Choose other variable type & write as many equations asnecessary to check that admissibility is satisfied.
4. Relate the loop and node variables using the constitutiverelationships.
5. Eliminate all but the chosen variables (all of one type) fromthe equations. Substitute in the equations, and group terms.
6. Non-dimensionalise the variables.
MG Lipsett, 2011 32
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Example: LumpedExample: Lumped--Parameter Mechanical SystemParameter Mechanical System
To model this system, we have two possible approaches:1) Find the forces in the springs (node variables)
2) Find the displacements of the carts (loop variables)
K/6
K/6
K/3 K/2
2P P
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MG Lipsett, 2011 33
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Case 1: Find the Forces in the SpringsCase 1: Find the Forces in the Springs
1) Choose a set of node variables (forces at nodes).
2) Satisfy node admissibility.
MG Lipsett, 2011 34
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Case 1: Forces in Springs (2)Case 1: Forces in Springs (2)3) Choose loop variables (displacements across elements)
and ensure they satisfy loop admissibility.
K/6
K/6
K/3 K/2
2P P
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MG Lipsett, 2011 35
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Case 1: Forces in Springs (3)Case 1: Forces in Springs (3)
4) Apply constitutive relationships. For linear spring element,
this will be: fi = ki inode variable loop variable
MG Lipsett, 2011 36
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Case 1: Forces in Springs (4)Case 1: Forces in Springs (4)5) Substitute into the loop equations.
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MG Lipsett, 2011 37
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Case 1: Forces in Springs (5)Case 1: Forces in Springs (5)
6) Try to express in non-dimensional form.
MG Lipsett, 2011 38
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Case 2: Find the Displacements in the NodesCase 2: Find the Displacements in the Nodes1) Choose a set of loop variables.
2) Satisfy loop admissibility.
K/6
K/6
K/3 K/2
2P P
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MG Lipsett, 2011 39
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Case 2: Displacement of Nodes (2)Case 2: Displacement of Nodes (2)
3) Choose node variables (forces at nodes) and ensure they
satisfy node admissibility.
4) Apply constitutive relationships.
MG Lipsett, 2011 40
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Case 2: Displacement of Nodes (3)Case 2: Displacement of Nodes (3)5) Substitute into node equations.
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MG Lipsett, 2011 41
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Case 2: Displacement of Nodes (4)Case 2: Displacement of Nodes (4)
Are we done yet? Well, not quite.
From the solution fory1,y2, go back to the definition of the non-dimensional variables to solve for the displacement (theloop variables); then, from their solution, we can find forcesusing the constitutive relationships.
These two methods are called Direct ApproachesDirect Approaches.
MG Lipsett, 2011 42
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Extremum FunctionsExtremum Functions The other way of formulating the equations governing
systems is to use extremum functions. This includesenergy methods.
We make up a scalar function from the constitutiverelationships of all the elements in the system, and searchfor an extreme value of the function (e.g. minimumpotential energy).
We go back to our original definition of a constitutiverelationship to define two quantities:
1. Content U(energy)
2. Co-Content U* (co-energy)
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MG Lipsett, 2011 43
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
EnergyEnergy
Area under the curve is the energy Uin the element:
We writep (which is a node variable) as a function of q(loop variable) and Ubecomes a function of q only.
MG Lipsett, 2011 44
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
CoCo--EnergyEnergy
Similarly to energy, with co-energy U* as a function ofp only
For all sets of state variables satisfying node (loop)admissibility, those also satisfying loop (node) admissibilitywill render the co-energy (energy) an extreme value.
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MG Lipsett, 2011 45
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 Modeling and Simulation of Engineering SystemsLecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Break Time: Flexibility of Thinking ProblemsBreak Time: Flexibility of Thinking Problems
8D 24H = 1W
3P = 6
HH & MH @ 12 = N or M
4J+4Q+4K = All the FC
S&M&T&W&T&F&S are D of W
23Y 3Y = 2D
E 8 = Z
Y + 2D = T
C + 6D = NYE
Y S S A = W
NN = GN
N + P + SM = S of C
1 + 6Z = 1M
R = R = R
1B in the H = 2 in the B
Each problem is an equation, which can be solved by substitut ing the appropriate wordsfor the letters. Examples:
3F = 1Y (3 Feet = 1 Yard)4LC = GL (4 Leaf Clover = Good Luck)
Source: A Whack on the Side of the Head, R.von Oech