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ENGM ENGM 541, ENGM 670541, ENGM 670--X5X5
& & MECE 758MECE 758--X5X5Modeling and Simulation of Engineering SystemsModeling and Simulation of Engineering Systems
Winter Winter 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. Lipsett
Department 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 Systems
Lecture 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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© MG Lipsett, 2011 3
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 – Modeling and Simulation of Engineering Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Mathematical Mathematical Analysis Analysis in Engineeringin Engineering
• Engineering analysis: formulating governing equations that
describe the behaviour of physical and technological
systems, for the purpose of analysis and design
• Numerical analysis: solving mathematical equations using
algorithms
• Scientific computing: development of reliable numerical
models that can be tested in a range of cases (including
known benchmarks)
© MG Lipsett, 2011 4
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 – Modeling and Simulation of Engineering Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
What is Modeling?What is Modeling?
• A model is a representation of knowledge – Rules, physical analogs, algebraic equations of physical laws
• A system is a bounded region comprising known elements that each interact in understandable ways
• Applied numerical modeling has joined empirical experimentation and analytical methods for solving problems 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 for engineering decisions– Production systems
– Economics
– Scheduling
– Inventory, and so on, and so on,…
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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 Systems
Lecture 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) differential equations
• 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 of time-varying behaviour that is not expressed as differential equations (correlations, discrete events, etc.). These are valuable for 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 Systems
Lecture 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)
Interesting problem
Data from the problem
Understanding of the problem
•Defining the system
•Uncertainty
•Sensitivity
•Parameter identification
•Statistical analysis
Modeling the system
•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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© MG Lipsett, 2011 7
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 – Modeling and Simulation of Engineering Systems
Lecture 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 successful
solution
© MG Lipsett, 2011 8
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 – Modeling and Simulation of Engineering Systems
Lecture 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 complex
to solve analytically
• Engineers rely on numerical modeling and simulation to analyse and design systems that have time-varying aspects
• Engineering managers use models of technologies and business processes for decision making
• You may want do develop models to solve a technical or
business problem, by designing a solution and modeling how you expect it to perform
• You may need to interpret the results of models created by
others
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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 Systems
Lecture 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 the
elements of a system
• Formulating governing equations and choosing solution
methods for different types of analyses of physical systems
• Understanding advantages and limitations of numerical
solution methods
• Understanding simple models for financial decisions and
technological 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 general purpose programs: MATLAB® and SIMULINK®
© MG Lipsett, 2011 10
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 – Modeling and Simulation of Engineering Systems
Lecture 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 1:00–3:00 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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© MG Lipsett, 2011 11
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 – Modeling and Simulation of Engineering Systems
Lecture 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 Systems
Lecture 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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© MG Lipsett, 2011 13
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 – Modeling and Simulation of Engineering Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
ENGM ENGM 670 & MECE 758 670 & 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 for being 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 be turned 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 Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
General General Course Course 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 a valuable approach to analysis and design
• We have to “unlearn” some things to do general systems analysis correctly
• We will also learn by doing
• I’ll do my best to be interesting
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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 Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
ENGM 541 Course Overview (4)ENGM 541 Course Overview (4)
• University policy: suspected cheating or plagiarism will be
reported 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 Systems
Lecture 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 and
reclamation
• 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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© MG Lipsett, 2011 17
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 – Modeling and Simulation of Engineering Systems
Lecture 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 using computers
© MG Lipsett, 2011 18
Department of Mechanical EngineeringEngineering Management Group
ENGM 541, ENGM 670-X5, MECE 758-X5 – Modeling and Simulation of Engineering Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Modeling Physical SystemsModeling Physical Systems
• Consider a beam:
• This is an inherently continuous structure. When we
analyse this beam for deflections, natural frequencies, etc., we can start from one of two approaches.
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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 Systems
Lecture 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 can determine the deflections
• The price we pay is one of approximating the physical system 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 Systems
Lecture 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 they
be approximated by algebraic equations (e.g. finite difference expressions), for almost all non-trivial cases
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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 Systems
Lecture 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 of solution:
• We want to have a consistent way to set up problems – and to 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 Systems
Lecture 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 lumped parameters.
• We are concerned with systems of interconnected elementselements. The elements within the problem have properties that we must know before we can proceed.
• Elements are connected at nodes. nodes. Here is an example of a system network:
• Loops are paths that start at a particular node, pass through a number of elements, and return to the original node.
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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 Systems
Lecture 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 a
node.
Examples:
• Loop and node variables:
• The loop and node variables are related by the constitutive relationships 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 Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Formulating Constitutive RelationshipsFormulating Constitutive Relationships
1. State the variables
2. Describe the element
3. Sketch the constitutive relationship.
4. Use an analytic expression for the relationship
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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 Systems
Lecture 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 Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Constitutive Relationship Example #2Constitutive Relationship Example #2
1. 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 Systems
Lecture 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 are
governed by loop admissibility laws that require the value of
the 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 Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
GeneralisingGeneralising Kirchoff’sKirchoff’s LawLaw
• We use a general approach for system networks using the principles of Kirchoff’s Laws.
• Kirchoff’s Laws for electrical circuits use the physical laws
of conservation of charge (node law) and conservation of energy added or taken by a potential field (around loops, mesh law), including dissipation. Gain or loss around an
entire loop has to be zero (because there is no net change
in the location with respect to the field).
• For other types of physical systems, we construct our variable assignments so that we can exploit similar physical
laws:
– Conservation of momentum law (D’Alembert’s law for forces)
– Conservation of mass law for flows, etc., etc.
• For non-physical systems, we need similar loop & node
laws
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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 Systems
Lecture 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 Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Some Equilibrium Element TypesSome Equilibrium Element Types
Type 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 Systems
Lecture 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
expressed
2. Choose variables so as to satisfy the pertinent admissibility
requirement
3. Choose other variable type & write as many equations as
necessary to check that admissibility is satisfied.
4. Relate the loop and node variables using the constitutive
relationships.
5. Eliminate all but the chosen variables (all of one type) from the 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 Systems
Lecture 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 Systems
Lecture 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 Systems
Lecture 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 Systems
Lecture 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 δinode 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 Systems
Lecture 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 Systems
Lecture 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 Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
Case 2: Find the Displacements in the NodesCase 2: Find the Displacements in the Nodes
1) 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 Systems
Lecture 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 Systems
Lecture 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 Systems
Lecture 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 for y1, y2, go back to the definition of the non-
dimensional variables to solve for the displacement (the
loop variables); then, from their solution, we can find forces
using 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 Systems
Lecture 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 includes energy methods.
• We make up a scalar function from the constitutive relationships of all the elements in the system, and search for an extreme value of the function (e.g. minimum
potential energy).
• We go back to our original definition of a constitutive
relationship 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 Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
EnergyEnergy
• Area under the curve is the energy U in the element:
• We write p (which is a node variable) as a function of q(loop variable) and U becomes 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 Systems
Lecture 1: Course Introduction; Lumped-Parameter Equilibrium Systems
CoCo--EnergyEnergy
• Similarly to energy, with co-energy U* as a function of p only
• For all sets of state variables satisfying node (loop) admissibility, those also satisfying loop (node) admissibility will 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 Systems
Lecture 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 substituting the appropriate words for 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