- •1. INTRODUCTION
- •1.1 BASIC TERMINOLOGY
- •1.2 EXAMPLE SYSTEM
- •1.3 SUMMARY
- •1.4 PRACTICE PROBLEMS
- •2. TRANSLATION
- •2.1 INTRODUCTION
- •2.2 MODELING
- •2.2.1 Free Body Diagrams
- •2.2.2 Mass and Inertia
- •2.2.3 Gravity and Other Fields
- •2.2.4 Springs
- •2.2.5 Damping and Drag
- •2.2.6 Cables And Pulleys
- •2.2.7 Friction
- •2.2.8 Contact Points And Joints
- •2.3 SYSTEM EXAMPLES
- •2.4 OTHER TOPICS
- •2.5 SUMMARY
- •2.6 PRACTICE PROBLEMS
- •2.7 PRACTICE PROBLEM SOLUTIONS
- •2.8 ASSIGNMENT PROBLEMS
- •3. ANALYSIS OF DIFFERENTIAL EQUATIONS
- •3.1 INTRODUCTION
- •3.2 EXPLICIT SOLUTIONS
- •3.3 RESPONSES
- •3.3.1 First-order
- •3.3.2 Second-order
- •3.3.3 Other Responses
- •3.4 RESPONSE ANALYSIS
- •3.5 NON-LINEAR SYSTEMS
- •3.5.1 Non-Linear Differential Equations
- •3.5.2 Non-Linear Equation Terms
- •3.5.3 Changing Systems
- •3.6 CASE STUDY
- •3.7 SUMMARY
- •3.8 PRACTICE PROBLEMS
- •3.9 PRACTICE PROBLEM SOLUTIONS
- •3.10 ASSIGNMENT PROBLEMS
- •4. NUMERICAL ANALYSIS
- •4.1 INTRODUCTION
- •4.2 THE GENERAL METHOD
- •4.2.1 State Variable Form
- •4.3 NUMERICAL INTEGRATION
- •4.3.1 Numerical Integration With Tools
- •4.3.2 Numerical Integration
- •4.3.3 Taylor Series
- •4.3.4 Runge-Kutta Integration
- •4.4 SYSTEM RESPONSE
- •4.4.1 Steady-State Response
- •4.5 DIFFERENTIATION AND INTEGRATION OF EXPERIMENTAL DATA
- •4.6 ADVANCED TOPICS
- •4.6.1 Switching Functions
- •4.6.2 Interpolating Tabular Data
- •4.6.3 Modeling Functions with Splines
- •4.6.4 Non-Linear Elements
- •4.7 CASE STUDY
- •4.8 SUMMARY
- •4.9 PRACTICE PROBLEMS
- •4.10 PRACTICE PROBLEM SOLUTIONS
- •4.11 ASSIGNMENT PROBLEMS
- •5. ROTATION
- •5.1 INTRODUCTION
- •5.2 MODELING
- •5.2.1 Inertia
- •5.2.2 Springs
- •5.2.3 Damping
- •5.2.4 Levers
- •5.2.5 Gears and Belts
- •5.2.6 Friction
- •5.2.7 Permanent Magnet Electric Motors
- •5.3 OTHER TOPICS
- •5.4 DESIGN CASE
- •5.5 SUMMARY
- •5.6 PRACTICE PROBLEMS
- •5.7 PRACTICE PROBLEM SOLUTIONS
- •5.8 ASSIGNMENT PROBLEMS
- •6. INPUT-OUTPUT EQUATIONS
- •6.1 INTRODUCTION
- •6.2 THE DIFFERENTIAL OPERATOR
- •6.3 INPUT-OUTPUT EQUATIONS
- •6.3.1 Converting Input-Output Equations to State Equations
- •6.3.2 Integrating Input-Output Equations
- •6.4 DESIGN CASE
- •6.5 SUMMARY
- •6.6 PRACTICE PROBLEMS
- •6.7 PRACTICE PROBLEM SOLUTIONS
- •6.8 ASSGINMENT PROBLEMS
- •6.9 REFERENCES
- •7. ELECTRICAL SYSTEMS
- •7.1 INTRODUCTION
- •7.2 MODELING
- •7.2.1 Resistors
- •7.2.2 Voltage and Current Sources
- •7.2.3 Capacitors
- •7.2.4 Inductors
- •7.2.5 Op-Amps
- •7.3 IMPEDANCE
- •7.4 EXAMPLE SYSTEMS
- •7.5 ELECTROMECHANICAL SYSTEMS - MOTORS
- •7.5.1 Permanent Magnet DC Motors
- •7.5.2 Induction Motors
- •7.5.3 Brushless Servo Motors
- •7.6 FILTERS
- •7.7 OTHER TOPICS
- •7.8 SUMMARY
- •7.9 PRACTICE PROBLEMS
- •7.10 PRACTICE PROBLEM SOLUTIONS
- •7.11 ASSIGNMENT PROBLEMS
- •8. FEEDBACK CONTROL SYSTEMS
- •8.1 INTRODUCTION
- •8.2 TRANSFER FUNCTIONS
- •8.3 CONTROL SYSTEMS
- •8.3.1 PID Control Systems
- •8.3.2 Manipulating Block Diagrams
- •8.3.3 A Motor Control System Example
- •8.3.4 System Error
- •8.3.5 Controller Transfer Functions
- •8.3.6 Feedforward Controllers
- •8.3.7 State Equation Based Systems
- •8.3.8 Cascade Controllers
- •8.4 SUMMARY
- •8.5 PRACTICE PROBLEMS
- •8.6 PRACTICE PROBLEM SOLUTIONS
- •8.7 ASSIGNMENT PROBLEMS
- •9. PHASOR ANALYSIS
- •9.1 INTRODUCTION
- •9.2 PHASORS FOR STEADY-STATE ANALYSIS
- •9.3 VIBRATIONS
- •9.4 SUMMARY
- •9.5 PRACTICE PROBLEMS
- •9.6 PRACTICE PROBLEM SOLUTIONS
- •9.7 ASSIGNMENT PROBLEMS
- •10. BODE PLOTS
- •10.1 INTRODUCTION
- •10.2 BODE PLOTS
- •10.3 SIGNAL SPECTRUMS
- •10.4 SUMMARY
- •10.5 PRACTICE PROBLEMS
- •10.6 PRACTICE PROBLEM SOLUTIONS
- •10.7 ASSIGNMENT PROBLEMS
- •10.8 LOG SCALE GRAPH PAPER
- •11. ROOT LOCUS ANALYSIS
- •11.1 INTRODUCTION
- •11.2 ROOT-LOCUS ANALYSIS
- •11.3 SUMMARY
- •11.4 PRACTICE PROBLEMS
- •11.5 PRACTICE PROBLEM SOLUTIONS
- •11.6 ASSIGNMENT PROBLEMS
- •12. NONLINEAR SYSTEMS
- •12.1 INTRODUCTION
- •12.2 SOURCES OF NONLINEARITY
- •12.3.1 Time Variant
- •12.3.2 Switching
- •12.3.3 Deadband
- •12.3.4 Saturation and Clipping
- •12.3.5 Hysteresis and Slip
- •12.3.6 Delays and Lags
- •12.4 SUMMARY
- •12.5 PRACTICE PROBLEMS
- •12.6 PRACTICE PROBLEM SOLUTIONS
- •12.7 ASIGNMENT PROBLEMS
- •13. ANALOG INPUTS AND OUTPUTS
- •13.1 INTRODUCTION
- •13.2 ANALOG INPUTS
- •13.3 ANALOG OUTPUTS
- •13.4 NOISE REDUCTION
- •13.4.1 Shielding
- •13.4.2 Grounding
- •13.5 CASE STUDY
- •13.6 SUMMARY
- •13.7 PRACTICE PROBLEMS
- •13.8 PRACTICE PROBLEM SOLUTIONS
- •13.9 ASSIGNMENT PROBLEMS
- •14. CONTINUOUS SENSORS
- •14.1 INTRODUCTION
- •14.2 INDUSTRIAL SENSORS
- •14.2.1 Angular Displacement
- •14.2.1.1 - Potentiometers
- •14.2.2 Encoders
- •14.2.2.1 - Tachometers
- •14.2.3 Linear Position
- •14.2.3.1 - Potentiometers
- •14.2.3.2 - Linear Variable Differential Transformers (LVDT)
- •14.2.3.3 - Moire Fringes
- •14.2.3.4 - Accelerometers
- •14.2.4 Forces and Moments
- •14.2.4.1 - Strain Gages
- •14.2.4.2 - Piezoelectric
- •14.2.5 Liquids and Gases
- •14.2.5.1 - Pressure
- •14.2.5.2 - Venturi Valves
- •14.2.5.3 - Coriolis Flow Meter
- •14.2.5.4 - Magnetic Flow Meter
- •14.2.5.5 - Ultrasonic Flow Meter
- •14.2.5.6 - Vortex Flow Meter
- •14.2.5.7 - Positive Displacement Meters
- •14.2.5.8 - Pitot Tubes
- •14.2.6 Temperature
- •14.2.6.1 - Resistive Temperature Detectors (RTDs)
- •14.2.6.2 - Thermocouples
- •14.2.6.3 - Thermistors
- •14.2.6.4 - Other Sensors
- •14.2.7 Light
- •14.2.7.1 - Light Dependant Resistors (LDR)
- •14.2.8 Chemical
- •14.2.8.2 - Conductivity
- •14.2.9 Others
- •14.3 INPUT ISSUES
- •14.4 SENSOR GLOSSARY
- •14.5 SUMMARY
- •14.6 REFERENCES
- •14.7 PRACTICE PROBLEMS
- •14.8 PRACTICE PROBLEM SOLUTIONS
- •14.9 ASSIGNMENT PROBLEMS
- •15. CONTINUOUS ACTUATORS
- •15.1 INTRODUCTION
- •15.2 ELECTRIC MOTORS
- •15.2.1 Basic Brushed DC Motors
- •15.2.2 AC Motors
- •15.2.3 Brushless DC Motors
- •15.2.4 Stepper Motors
- •15.2.5 Wound Field Motors
- •15.3 HYDRAULICS
- •15.4 OTHER SYSTEMS
- •15.5 SUMMARY
- •15.6 PRACTICE PROBLEMS
- •15.7 PRACTICE PROBLEM SOLUTIONS
- •15.8 ASSIGNMENT PROBLEMS
- •16. MOTION CONTROL
- •16.1 INTRODUCTION
- •16.2 MOTION PROFILES
- •16.2.1 Velocity Profiles
- •16.2.2 Position Profiles
- •16.3 MULTI AXIS MOTION
- •16.3.1 Slew Motion
- •16.3.1.1 - Interpolated Motion
- •16.3.2 Motion Scheduling
- •16.4 PATH PLANNING
- •16.5 CASE STUDIES
- •16.6 SUMMARY
- •16.7 PRACTICE PROBLEMS
- •16.8 PRACTICE PROBLEM SOLUTIONS
- •16.9 ASSIGNMENT PROBLEMS
- •17. LAPLACE TRANSFORMS
- •17.1 INTRODUCTION
- •17.2 APPLYING LAPLACE TRANSFORMS
- •17.2.1 A Few Transform Tables
- •17.3 MODELING TRANSFER FUNCTIONS IN THE s-DOMAIN
- •17.4 FINDING OUTPUT EQUATIONS
- •17.5 INVERSE TRANSFORMS AND PARTIAL FRACTIONS
- •17.6 EXAMPLES
- •17.6.2 Circuits
- •17.7 ADVANCED TOPICS
- •17.7.1 Input Functions
- •17.7.2 Initial and Final Value Theorems
- •17.8 A MAP OF TECHNIQUES FOR LAPLACE ANALYSIS
- •17.9 SUMMARY
- •17.10 PRACTICE PROBLEMS
- •17.11 PRACTICE PROBLEM SOLUTIONS
- •17.12 ASSIGNMENT PROBLEMS
- •17.13 REFERENCES
- •18. CONTROL SYSTEM ANALYSIS
- •18.1 INTRODUCTION
- •18.2 CONTROL SYSTEMS
- •18.2.1 PID Control Systems
- •18.2.2 Analysis of PID Controlled Systems With Laplace Transforms
- •18.2.3 Finding The System Response To An Input
- •18.2.4 Controller Transfer Functions
- •18.3.1 Approximate Plotting Techniques
- •18.4 DESIGN OF CONTINUOUS CONTROLLERS
- •18.5 SUMMARY
- •18.6 PRACTICE PROBLEMS
- •18.7 PRACTICE PROBLEM SOLUTIONS
- •18.8 ASSIGNMENT PROBLEMS
- •19. CONVOLUTION
- •19.1 INTRODUCTION
- •19.2 UNIT IMPULSE FUNCTIONS
- •19.3 IMPULSE RESPONSE
- •19.4 CONVOLUTION
- •19.5 NUMERICAL CONVOLUTION
- •19.6 LAPLACE IMPULSE FUNCTIONS
- •19.7 SUMMARY
- •19.8 PRACTICE PROBLEMS
- •19.9 PRACTICE PROBLEM SOLUTIONS
- •19.10 ASSIGNMENT PROBLEMS
- •20. STATE SPACE ANALYSIS
- •20.1 INTRODUCTION
- •20.2 OBSERVABILITY
- •20.3 CONTROLLABILITY
- •20.4 OBSERVERS
- •20.5 SUMMARY
- •20.6 PRACTICE PROBLEMS
- •20.7 PRACTICE PROBLEM SOLUTIONS
- •20.8 ASSIGNMENT PROBLEMS
- •20.9 BIBLIOGRAPHY
- •21. STATE SPACE CONTROLLERS
- •21.1 INTRODUCTION
- •21.2 FULL STATE FEEDBACK
- •21.3 OBSERVERS
- •21.4 SUPPLEMENTAL OBSERVERS
- •21.5 REGULATED CONTROL WITH OBSERVERS
- •21.7 LINEAR QUADRATIC GAUSSIAN (LQG) COMPENSATORS
- •21.8 VERIFYING CONTROL SYSTEM STABILITY
- •21.8.1 Stability
- •21.8.2 Bounded Gain
- •21.9 ADAPTIVE CONTROLLERS
- •21.10 OTHER METHODS
- •21.10.1 Kalman Filtering
- •21.11 SUMMARY
- •21.12 PRACTICE PROBLEMS
- •21.13 PRACTICE PROBLEM SOLUTIONS
- •21.14 ASSIGNMENT PROBLEMS
- •22. SYSTEM IDENTIFICATION
- •22.1 INTRODUCTION
- •22.2 SUMMARY
- •22.3 PRACTICE PROBLEMS
- •22.4 PRACTICE PROBLEM SOLUTIONS
- •22.5 ASSIGNMENT PROBLEMS
- •23. ELECTROMECHANICAL SYSTEMS
- •23.1 INTRODUCTION
- •23.2 MATHEMATICAL PROPERTIES
- •23.2.1 Induction
- •23.3 EXAMPLE SYSTEMS
- •23.4 SUMMARY
- •23.5 PRACTICE PROBLEMS
- •23.6 PRACTICE PROBLEM SOLUTIONS
- •23.7 ASSIGNMENT PROBLEMS
- •24. FLUID SYSTEMS
- •24.1 SUMMARY
- •24.2 MATHEMATICAL PROPERTIES
- •24.2.1 Resistance
- •24.2.2 Capacitance
- •24.2.3 Power Sources
- •24.3 EXAMPLE SYSTEMS
- •24.4 SUMMARY
- •24.5 PRACTICE PROBLEMS
- •24.6 PRACTICE PROBLEMS SOLUTIONS
- •24.7 ASSIGNMENT PROBLEMS
- •25. THERMAL SYSTEMS
- •25.1 INTRODUCTION
- •25.2 MATHEMATICAL PROPERTIES
- •25.2.1 Resistance
- •25.2.2 Capacitance
- •25.2.3 Sources
- •25.3 EXAMPLE SYSTEMS
- •25.4 SUMMARY
- •25.5 PRACTICE PROBLEMS
- •25.6 PRACTICE PROBLEM SOLUTIONS
- •25.7 ASSIGNMENT PROBLEMS
- •26. OPTIMIZATION
- •26.1 INTRODUCTION
- •26.2 OBJECTIVES AND CONSTRAINTS
- •26.3 SEARCHING FOR THE OPTIMUM
- •26.4 OPTIMIZATION ALGORITHMS
- •26.4.1 Random Walk
- •26.4.2 Gradient Decent
- •26.4.3 Simplex
- •26.5 SUMMARY
- •26.6 PRACTICE PROBLEMS
- •26.7 PRACTICE PROBLEM SOLUTIONS
- •26.8 ASSIGNMENT PROBLEMS
- •27. FINITE ELEMENT ANALYSIS (FEA)
- •27.1 INTRODUCTION
- •27.2 FINITE ELEMENT MODELS
- •27.3 FINITE ELEMENT MODELS
- •27.4 SUMMARY
- •27.5 PRACTICE PROBLEMS
- •27.6 PRACTICE PROBLEM SOLUTIONS
- •27.7 ASSIGNMENT PROBLEMS
- •27.8 BIBLIOGRAPHY
- •28. FUZZY LOGIC
- •28.1 INTRODUCTION
- •28.2 COMMERCIAL CONTROLLERS
- •28.3 REFERENCES
- •28.4 SUMMARY
- •28.5 PRACTICE PROBLEMS
- •28.6 PRACTICE PROBLEM SOLUTIONS
- •28.7 ASSIGNMENT PROBLEMS
- •29. NEURAL NETWORKS
- •29.1 SUMMARY
- •29.2 PRACTICE PROBLEMS
- •29.3 PRACTICE PROBLEM SOLUTIONS
- •29.4 ASSIGNMENT PROBLEMS
- •29.5 REFERENCES
- •30. EMBEDDED CONTROL SYSTEM
- •30.1 INTRODUCTION
- •30.2 CASE STUDY
- •30.3 SUMMARY
- •30.4 PRACTICE PROBLEMS
- •30.5 PRACTICE PROBLEM SOLUTIONS
- •30.6 ASSIGNMENT PROBLEMS
- •31. WRITING
- •31.1 FORGET WHAT YOU WERE TAUGHT BEFORE
- •31.2 WHY WRITE REPORTS?
- •31.3 THE TECHNICAL DEPTH OF THE REPORT
- •31.4 TYPES OF REPORTS
- •31.5 LABORATORY REPORTS
- •31.5.0.1 - An Example First Draft of a Report
- •31.5.0.2 - An Example Final Draft of a Report
- •31.6 RESEARCH
- •31.7 DRAFT REPORTS
- •31.8 PROJECT REPORT
- •31.9 OTHER REPORT TYPES
- •31.9.1 Executive
- •31.9.2 Consulting
- •31.9.3 Memo(randum)
- •31.9.4 Interim
- •31.9.5 Poster
- •31.9.6 Progress Report
- •31.9.7 Oral
- •31.9.8 Patent
- •31.10 LAB BOOKS
- •31.11 REPORT ELEMENTS
- •31.11.1 Figures
- •31.11.2 Graphs
- •31.11.3 Tables
- •31.11.4 Equations
- •31.11.5 Experimental Data
- •31.11.6 Result Summary
- •31.11.7 References
- •31.11.8 Acknowledgments
- •31.11.9 Abstracts
- •31.11.10 Appendices
- •31.11.11 Page Numbering
- •31.11.12 Numbers and Units
- •31.11.13 Engineering Drawings
- •31.11.14 Discussions
- •31.11.15 Conclusions
- •31.11.16 Recomendations
- •31.11.17 Appendices
- •31.11.18 Units
- •31.12 GENERAL WRITING ISSUES
- •31.13 WRITERS BLOCK
- •31.14 TECHNICAL ENGLISH
- •31.15 EVALUATION FORMS
- •31.16 PATENTS
- •32. PROJECTS
- •32.2 OVERVIEW
- •32.2.1 The Objectives and Constraints
- •32.3 MANAGEMENT
- •32.3.1 Timeline - Tentative
- •32.3.2 Teams
- •32.4 DELIVERABLES
- •32.4.1 Conceptual Design
- •32.4.2 EGR 345/101 Contract
- •32.4.3 Progress Reports
- •32.4.4 Design Proposal
- •32.4.5 The Final Report
- •32.5 REPORT ELEMENTS
- •32.5.1 Gantt Charts
- •32.5.2 Drawings
- •32.5.3 Budgets and Bills of Material
- •32.5.4 Calculations
- •32.6 APPENDICES
- •32.6.1 Appendix A - Sample System
- •32.6.2 Appendix B - EGR 345/101 Contract
- •32.6.3 Appendix C - Forms
- •33. ENGINEERING PROBLEM SOLVING
- •33.1 BASIC RULES OF STYLE
- •33.2 EXPECTED ELEMENTS
- •33.3 SEPCIAL ELEMENTS
- •33.3.1 Graphs
- •33.3.2 EGR 345 Specific
- •33.4 SCILAB
- •33.5 TERMINOLOGY
- •34. MATHEMATICAL TOOLS
- •34.1 INTRODUCTION
- •34.1.1 Constants and Other Stuff
- •34.1.2 Basic Operations
- •34.1.2.1 - Factorial
- •34.1.3 Exponents and Logarithms
- •34.1.4 Polynomial Expansions
- •34.1.5 Practice Problems
- •34.2 FUNCTIONS
- •34.2.1 Discrete and Continuous Probability Distributions
- •34.2.2 Basic Polynomials
- •34.2.3 Partial Fractions
- •34.2.4 Summation and Series
- •34.2.5 Practice Problems
- •34.3 SPATIAL RELATIONSHIPS
- •34.3.1 Trigonometry
- •34.3.2 Hyperbolic Functions
- •34.3.2.1 - Practice Problems
- •34.3.3 Geometry
- •34.3.4 Planes, Lines, etc.
- •34.3.5 Practice Problems
- •34.4 COORDINATE SYSTEMS
- •34.4.1 Complex Numbers
- •34.4.2 Cylindrical Coordinates
- •34.4.3 Spherical Coordinates
- •34.4.4 Practice Problems
- •34.5 MATRICES AND VECTORS
- •34.5.1 Vectors
- •34.5.2 Dot (Scalar) Product
- •34.5.3 Cross Product
- •34.5.4 Triple Product
- •34.5.5 Matrices
- •34.5.6 Solving Linear Equations with Matrices
- •34.5.7 Practice Problems
- •34.6 CALCULUS
- •34.6.1 Single Variable Functions
- •34.6.1.1 - Differentiation
- •34.6.1.2 - Integration
- •34.6.2 Vector Calculus
- •34.6.3 Differential Equations
- •34.6.3.1.1 - Guessing
- •34.6.3.1.2 - Separable Equations
- •34.6.3.1.3 - Homogeneous Equations and Substitution
- •34.6.3.2.1 - Linear Homogeneous
- •34.6.3.2.2 - Nonhomogeneous Linear Equations
- •34.6.3.3 - Higher Order Differential Equations
- •34.6.3.4 - Partial Differential Equations
- •34.6.4 Other Calculus Stuff
- •34.6.5 Practice Problems
- •34.7 NUMERICAL METHODS
- •34.7.1 Approximation of Integrals and Derivatives from Sampled Data
- •34.7.3 Taylor Series Integration
- •34.8 LAPLACE TRANSFORMS
- •34.8.1 Laplace Transform Tables
- •34.9 z-TRANSFORMS
- •34.10 FOURIER SERIES
- •34.11 TOPICS NOT COVERED (YET)
- •34.12 REFERENCES/BIBLIOGRAPHY
- •35. A BASIC INTRODUCTION TO ‘C’
- •35.2 BACKGROUND
- •35.3 PROGRAM PARTS
- •35.4 HOW A ‘C’ COMPILER WORKS
- •35.5 STRUCTURED ‘C’ CODE
- •35.7 CREATING TOP DOWN PROGRAMS
- •35.8 HOW THE BEAMCAD PROGRAM WAS DESIGNED
- •35.8.1 Objectives:
- •35.8.2 Problem Definition:
- •35.8.3 User Interface:
- •35.8.3.1 - Screen Layout (also see figure):
- •35.8.3.2 - Input:
- •35.8.3.3 - Output:
- •35.8.3.4 - Help:
- •35.8.3.5 - Error Checking:
- •35.8.3.6 - Miscellaneous:
- •35.8.4 Flow Program:
- •35.8.5 Expand Program:
- •35.8.6 Testing and Debugging:
- •35.8.7 Documentation
- •35.8.7.1 - Users Manual:
- •35.8.7.2 - Programmers Manual:
- •35.8.8 Listing of BeamCAD Program.
- •35.9 PRACTICE PROBLEMS
- •36. UNITS AND CONVERSIONS
- •36.1 HOW TO USE UNITS
- •36.2 HOW TO USE SI UNITS
- •36.3 THE TABLE
- •36.4 ASCII, HEX, BINARY CONVERSION
- •36.5 G-CODES
- •37. ATOMIC MATERIAL DATA
- •37. MECHANICAL MATERIAL PROPERTIES
- •37.1 FORMULA SHEET
- •38. BIBLIOGRAPHY
- •38.1 TEXTBOOKS
- •38.1.1 Slotine and Li
- •38.1.2 VandeVegte
- •39. TOPICS IN DEVELOPMENT
- •39.1 UPDATED DC MOTOR MODEL
- •39.2 ANOTHER DC MOTOR MODEL
- •39.3 BLOCK DIAGRAMS AND UNITS
- •39.4 SIGNAL FLOW GRAPHS
- •39.5 ZERO ORDER HOLD
- •39.6 TORSIONAL DAMPERS
- •39.7 MISC
- •39.8 Nyquist Plot
- •39.9 NICHOLS CHART
- •39.10 BESSEL POLYNOMIALS
- •39.11 ITAE
- •39.12 ROOT LOCUS
- •39.13 LYAPUNOV’S LINEARIZATION METHOD
- •39.14 XXXXX
- •39.15 XXXXX
- •39.16 XXXXX
- •39.17 XXXXX
- •39.18 XXXXX
- •39.19 XXXXX
- •39.20 XXXXX
- •39.21 SUMMARY
- •39.22 PRACTICE PROBLEMS
- •39.23 PRACTICE PROBLEM SOLUTIONS
- •39.24 ASSGINMENT PROBLEMS
- •39.25 REFERENCES
- •39.26 BIBLIOGRAPHY
state space analysis - 27.1
27. FINITE ELEMENT ANALYSIS (FEA)
Topics:
Objectives:
27.1 INTRODUCTION
state space analysis - 27.2
27.2FINITE ELEMENT MODELS
•Consider a central node ’i’ connected to neighboring nodes with springs.
nb
y
Ks
x
na |
Ks |
ni |
Ks |
nc
Fi |
Ks |
nd
•Equations can be written to relate the position of node i to the surrounding nodes
state space analysis - 27.3
and the applied force.
Ks( yb – yi)
Ks( xi – xa)
Ks( xc – xi)
Fi = ( Fix, Fiy)
Ks( yi – yd)
+
∑Fx = Fix – Ks( xi – xa) + Ks( xc – xi) = 0 xi( 2Ks) + xa( –Ks) + xc( –Ks) = Fix
+ 
∑Fx = Fiy – Ks( yi – yd) + Ks( yb – yi) = 0 yi( 2Ks) + yb( –Ks) + yd( –Ks) = Fiy
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state space analysis - 27.4
27.3FINITE ELEMENT MODELS
•Consider a central node ’i’ connected to neighboring nodes with springs.
na |
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nb |
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nd |
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Fd |
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• Equations can be written to relate the position of node i to the surrounding nodes and the applied force. The ’x’ and ’y’ values are deflections from the unloaded state. The ’Ks’ value is based on the material stiffness and the geometry of the elements.
Ks( xb – xa)
( Fax, Fay)
Ks ( ya – yd)
Ks( xb – xa)
( Fbx, Fby)
Ks( yb – yc)
+
∑Fx = Fax + Ks ( xb – xa) = 0
Fax = xa( Ks) + xb( –Ks)
+ 
∑Fx = Fay – Ks( ya – yd) = 0
Fay = ya( Ks) + yd( –Ks)
+
∑Fx = Fbx – Ks( xb – xa) = 0
Fbx = xa( –Ks) + xb( Ks)
+ 
∑Fx = Fby – Ks( yb – yc) = 0
Fby = yb( Ks) + yc ( –Ks)
state space analysis - 27.5
Ks( yb – yc)
Ks( xc – xd)
( Fcx, Fcy)
Ks( ya – yd)
Ks( xc – xd)
( Fdx, Fdy)
+ |
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∑Fx = Fcx – Ks( xc – xd) = 0 |
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Fcx |
= xc( Ks) + xd( –Ks) |
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∑Fx = Fcy + Ks( yb – yc) = 0 |
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Fcy |
= yb( –Ks) + yc( Ks) |
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∑Fx = Fdx + Ks( xc – xd) = 0 |
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Fdx |
= xc( –Ks) + xd( Ks) |
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+ |
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∑Fx = Fdy + Ks( ya – yd) = 0 |
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Fdy |
= ya( –Ks) + yd( Ks) |
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Fax |
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–K |
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K |
s |
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x |
a |
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Fay |
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K |
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–K |
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y |
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Fb |
x |
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–K |
s |
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K |
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Fby |
= |
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Ks |
0 |
–Ks |
0 |
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yb |
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Fcx |
0 |
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Ks |
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–Ks |
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Fcy |
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–Ks |
0 |
Ks |
0 |
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yc |
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F |
dx |
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0 |
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–Ks |
0 |
Ks |
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xd |
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Ks |
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yd |
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Fdy |
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local stiffness matrix |
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• This can be combined into a more
state space analysis - 27.6
• A four element mesh
n1 |
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n2 |
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n3 |
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Ks |
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Ks |
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Ks |
#1 |
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#2 |
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Ks |
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n4 |
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Ks |
n5 |
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n6 |
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#3 |
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#4 |
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Ks |
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n7 |
Ks |
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n9 |
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n8 |
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state space analysis - 27.7
For element #1 |
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F1x |
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–K |
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0 |
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Local stiffness matrix |
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K |
s |
s |
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x |
1 |
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F1y |
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s |
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–K |
s |
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y |
1 |
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F2 |
x |
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–K |
s |
0 |
K |
s |
0 |
0 |
0 |
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x |
2 |
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F2y |
= |
0 |
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0 |
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Ks |
0 |
–Ks |
0 |
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y2 |
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F4x |
0 |
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0 |
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Ks |
0 |
–Ks |
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x4 |
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F4y |
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0 |
Ks |
0 |
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y4 |
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F |
5x |
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0 |
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0 |
0 |
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–Ks |
0 |
Ks |
0 |
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x5 |
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0 |
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–Ks |
0 |
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0 |
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0 |
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Ks |
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y5 |
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F5y |
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Global stiffness matrix |
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F1x |
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F1y |
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Ks |
0 |
–Ks |
0 |
0 |
0 |
0 |
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F2x |
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0 |
Ks |
0 |
0 |
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0 |
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F2y |
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0 |
Ks |
0 |
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0 |
0 |
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F3x |
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Ks |
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F3y |
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F4x |
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F4y |
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Ks |
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F5x |
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= |
0 |
0 |
0 |
–Ks 0 0 |
0 |
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F5y |
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0 |
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–Ks |
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F6x |
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–Ks |
0 |
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F6y |
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F7x |
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F7y |
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F8x |
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0 |
0 |
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0 |
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F8y |
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0 |
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0 |
0 |
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0 |
0 |
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0 |
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0 |
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F9x |
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F9y |
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x1 |
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y1 |
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0 |
0 |
0 |
0 |
0 |
0 |
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x2 |
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0 |
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–Ks |
0 0 0 0 0 0 |
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y2 |
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0 |
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0 |
0 |
0 |
0 |
0 |
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x3 |
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–Ks |
0 |
0 |
0 0 0 0 0 0 |
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y3 |
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0 |
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0 |
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x4 |
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0 |
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0 |
0 |
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y4 |
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0 |
–Ks |
0 |
0 0 0 0 0 0 |
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x5 |
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Ks |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
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y5 |
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0 |
Ks |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
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x6 |
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0 |
0 |
Ks |
0 |
0 |
0 |
0 |
0 |
0 |
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y6 |
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0 |
0 |
0 |
0 0 0 0 0 0 |
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0 |
0 |
0 |
0 0 0 0 0 0 |
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x7 |
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0 |
0 |
0 |
0 0 0 0 0 0 |
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y7 |
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0 |
0 |
0 |
0 0 0 0 0 0 |
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x8 |
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0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
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y8 |
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0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
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x9 |
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y9 |
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state space analysis - 27.8
For element #2 |
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F2x |
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–K |
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K |
s |
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2 |
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local stiffness matrix |
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K |
s |
0 |
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0 |
0 |
0 |
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–K |
s |
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y |
2 |
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F3 |
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s |
0 |
K |
s |
0 |
0 |
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x |
3 |
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F3y |
= |
0 |
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0 |
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Ks |
0 |
–Ks |
0 |
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y3 |
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F5x |
0 |
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0 |
0 |
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Ks |
0 |
–Ks |
0 |
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x5 |
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F5y |
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0 |
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–Ks |
0 |
Ks |
0 |
0 |
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y5 |
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F |
6x |
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0 |
0 |
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0 |
–Ks |
0 |
Ks |
0 |
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x6 |
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0 |
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–Ks |
0 |
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0 |
0 |
0 |
0 |
Ks |
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y6 |
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F6y |
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added to global matrix |
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F F F F
F F
F F F F F
F F
F F F F F
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1x |
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x1 |
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Ks 0 –Ks |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
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y1 |
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1y |
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0 0 0 0 |
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2x |
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0 |
Ks |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
–Ks 0 0 0 0 0 0 |
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y2 |
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2y |
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–Ks 0 2Ks |
0 |
–Ks 0 0 0 0 0 0 0 0 0 0 0 |
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x3 |
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3x |
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0 |
0 |
0 |
2Ks |
0 |
0 |
0 |
–Ks |
0 |
0 |
0 |
0 |
0 0 0 0 |
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3y |
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0 |
0 |
–Ks |
0 |
Ks |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 0 0 0 |
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y3 |
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x4 |
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4x |
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0 |
0 |
0 |
0 |
0 |
Ks |
0 |
0 |
0 |
0 |
0 |
0 |
0 0 0 0 |
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y4 |
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4y |
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0 |
0 |
0 |
0 |
0 |
0 |
Ks |
0 |
–Ks |
0 |
0 |
0 |
0 0 0 0 |
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5x |
= |
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0 0 0 –Ks |
0 |
0 |
0 |
Ks |
0 |
0 |
0 |
0 |
0 0 0 0 |
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x5 |
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5y |
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0 0 |
0 |
0 |
0 |
0 |
–Ks |
0 |
2Ks |
0 |
0 |
0 |
0 0 0 0 |
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y5 |
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x6 |
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6x |
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0 |
–Ks |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2Ks 0 |
0 |
0 0 0 0 |
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y6 |
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6y |
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0 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
Ks |
0 |
0 0 0 0 |
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7x |
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0 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
Ks |
0 0 0 0 |
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x7 |
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7y |
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
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y7 |
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8x |
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
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x8 |
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8y |
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
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y8 |
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9x |
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
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x9 |
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9y |
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y9 |
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state space analysis - 27.9
For element #3 |
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F4x |
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0 |
–K |
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0 |
0 |
0 |
0 |
0 |
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K |
s |
s |
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x |
4 |
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F4y |
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0 |
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K |
s |
0 |
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0 |
0 |
0 |
0 |
–K |
s |
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y |
4 |
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F5 |
x |
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–K |
s |
0 |
K |
s |
0 |
0 |
0 |
0 |
0 |
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x |
5 |
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F5y |
= |
0 |
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0 |
0 |
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Ks |
0 |
–Ks |
0 |
0 |
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y5 |
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F7x |
0 |
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0 |
0 |
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0 |
Ks |
0 |
–Ks |
0 |
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x7 |
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F7y |
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0 |
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0 |
0 |
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–Ks |
0 |
Ks |
0 |
0 |
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y7 |
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F |
8x |
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0 |
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0 |
0 |
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0 |
–Ks |
0 |
Ks |
0 |
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x8 |
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0 |
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–Ks |
0 |
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0 |
0 |
0 |
0 |
Ks |
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y8 |
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state space analysis - 27.10
For element #4 |
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F5x |
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0 |
–K |
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0 |
0 |
0 |
0 |
0 |
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K |
s |
s |
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x |
5 |
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F5y |
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0 |
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K |
s |
0 |
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0 |
0 |
0 |
0 |
–K |
s |
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y |
5 |
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F6 |
x |
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–K |
s |
0 |
K |
s |
0 |
0 |
0 |
0 |
0 |
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x |
6 |
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F6y |
= |
0 |
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0 |
0 |
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Ks |
0 |
–Ks |
0 |
0 |
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y6 |
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F8x |
0 |
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0 |
0 |
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0 |
Ks |
0 |
–Ks |
0 |
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x8 |
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F8y |
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0 |
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0 |
0 |
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–Ks |
0 |
Ks |
0 |
0 |
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y8 |
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F |
9x |
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0 |
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0 |
0 |
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0 |
–Ks |
0 |
Ks |
0 |
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x9 |
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0 |
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–Ks |
0 |
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0 |
0 |
0 |
0 |
Ks |
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y9 |
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F9y |
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• If the input forces are known, then the resulting displacements of the nodes can be calculated by inverting the matrix. Consider the matrix for a single node.
state space analysis - 27.11
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–1 |
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0 |
–K |
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0 |
0 |
0 |
0 |
0 |
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F |
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x |
a |
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K |
s |
s |
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ax |
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Fay |
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y |
a |
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0 |
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K |
s |
0 |
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0 |
0 |
0 |
0 |
–K |
s |
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x |
b |
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–K |
s |
0 |
K |
s |
0 |
0 |
0 |
0 |
0 |
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Fb |
x |
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yb |
= |
0 |
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0 |
0 |
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Ks |
0 |
–Ks |
0 |
0 |
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Fby |
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xc |
0 |
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0 |
0 |
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0 |
Ks |
0 |
–Ks |
0 |
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Fcx |
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yc |
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0 |
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0 |
0 |
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–Ks |
0 |
Ks |
0 |
0 |
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Fcy |
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xd |
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0 |
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0 |
0 |
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0 |
–Ks |
0 |
Ks |
0 |
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F |
dx |
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yd |
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0 |
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–Ks |
0 |
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0 |
0 |
0 |
0 |
Ks |
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Fdy |
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• If we assume that node ’c’ is fixed in the ’x’ and ’y’ directions, the matrix can reflect this by setting the appropriate matrix rows to zero.
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–1 |
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0 |
–K |
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0 |
0 |
0 |
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Fax |
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x |
a |
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K |
s |
s |
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y |
a |
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0 |
K |
s |
0 |
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0 |
0 |
–K |
s |
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Fa |
y |
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xb |
= |
–Ks |
0 |
Ks |
0 |
0 |
0 |
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Fbx |
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yb |
0 |
0 |
0 |
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Ks 0 |
0 |
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Fby |
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xd |
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0 |
0 |
0 |
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0 |
Ks |
0 |
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Fdx |
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yd |
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0 |
–Ks |
0 |
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0 |
0 |
Ks |
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F |
dy |
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• The displacements can then be found by selecting values for the coefficents and solving the matrix. We can select a value of 1000 for the stiffness, and a force of 10 will be