- •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
differential equations - 3.1
3. ANALYSIS OF DIFFERENTIAL EQUATIONS
Topics:
•First and second-order homogeneous differential equations
•Non-homogeneous differential equations
•First and second-order responses
•Non-linear system elements
•Design case
Objectives:
•To develop explicit equations that describe a system response.
•To recognize first and second-order equation forms.
3.1INTRODUCTION
In the previous chapter we derived differential equations of motion for translating systems. These equations can be used to analyze the behavior of the system and make design decisions. The most basic method is to select a standard input type (a forcing function) and initial conditions, and then solve the differential equation. It is also possible to estimate the system response without solving the differential equation as will be discussed later.
Figure 3.1 shows an abstract description of a system. The basic concept is that the system changes the inputs to outputs. Say, for example, that the system to be analyzed is an elevator. Inputs to the system would be the mass of human riders and desired elevator height. The output response of the system would be the actual height of the elevator. For analysis, the system model could be developed using differential equations for the motor, elastic lift cable, mass of the car, etc. A basic test would involve assuming that the elevator starts at the ground floor and must travel to the top floor. Using assumed initial values and input functions the differential equation could be solved to get an explicit equation for elevator height. This output response can then be used as a guide to modify design choices (parameters). In practice, many of the assumptions and tests are mandated by law or by groups such as Underwriters Laboratories (UL), Canadian Standards Association (CSA) and the European Commission (CE).
differential equations - 3.2
inputs
forcing function
Figure 3.1
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system |
outputs |
Note: By convention inputs |
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are on the left, and outputs |
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response |
are on the right. |
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differential |
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function |
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equations |
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A system with and input and output response |
There are several standard input types used to test a system. These are listed below in order of relative popularity with brief explanations.
•step - a sudden change of input, such as very rapidly changing a desired speed from 0Hz to 50Hz.
•ramp - a continuously increasing input, such as a motor speed that increases constantly at 10Hz per minute.
•sinusoidal - a cyclic input that varies continuously, such as wave height that is continually oscillating at 1Hz.
•parabolic - an exponentially increasing input, such as a motor speed that is 2Hz at 1 second, 4rad/s at 2 seconds, 8rad/s at 3 seconds, etc.
After the system has been modeled, an input type has been chosen, and the initial conditions have been selected, the system can be analyzed to determine its behavior. The most fundamental technique is to integrate the differential equation(s) for the system.
3.2 EXPLICIT SOLUTIONS
Solving a differential equation results in an explicit solution. This equation provides the general response as a function of time, but it can also be used to find frequencies and other characteristics of interest. This section will review techniques used to integrate first and second-order homogenous differential equations. These equations correspond to systems without inputs, also called unforced systems. Non-homogeneous differential equations will also be reviewed.
The basic types of differential equations are shown in Figure 3.2. Each of these equations is linear. On the left hand side is the integration variable ’x’. If the right hand side is zero, then the equation is homogeneous. Each of these equations is linear because each of the terms on the left hand side is simply multiplied by a linear coefficient.
differential equations - 3.3
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second-order homogeneous |
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second-order non-homogeneous |
Ax + Bx + Cx = Df( t) |
Figure 3.2 Standard equation forms
A general solution for a first-order homogeneous differential equation is given in Figure 3.3. The solution begins with the solution of the homogeneous equation where a general form is ’guessed’. Substitution leads to finding the value of the coefficient ’Y’. Following this, the initial conditions for the equation are used to find the value of the coefficient ’X’. Notice that the final equation will begin at the initial displacement, but approach zero as time goes to infinity. The e-to-the-x behavior is characteristic for a firstorder response.
differential equations - 3.4
Given the general form of a first-order homogeneous equation,
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A( –Y) + B = 0
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Y = --
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Therefore the general form is,
B
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A
xh = Xe
Note: The general form below is useful for finding almost all homogeneous equations
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Next, use the initial conditions to find the remaining unknowns.
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xh = Xe B
x0 = Xe
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x0 = X
Therefore the final equation is,
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x( t) = x0e A
Figure 3.3 General solution of a first-order homogeneous equation
differential equations - 3.5
Solve the following differential equation given the initial condition.
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x + 2x = 0 |
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ans. x( t) = 3e–2t
Figure 3.4 Drill Problem: First order homogeneous differential equation
The general solution to a second-order homogeneous equation is shown in Figure 3.5. The solution begins with a guess of the homogeneous solution, and the solution of a quadratic equation. There are three possible cases that result from the solution of the quadratic equation: different but real roots; two identical real roots; or two complex roots. The three cases result in three different forms of solutions, as shown. The complex result is the most notable because it results in sinusoidal oscillations. It is not shown, but after the homogeneous solution has been found, the initial conditions need to be used to find the remaining coefficient values.
differential equations - 3.6
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Note: There are three possible outcomes of finding the roots of the equations: two different real roots, two identical real roots, or two complex roots. Therefore there are three fundamentally different results.
If the values for Y are both real, but different, the general form is,
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If the values for Y are both real, and identical, the general form is,
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The initial conditions are then used to find the values for X1 and X2.
If the values for Y are complex, the general form is,
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Figure 3.5 Solution of a second-order homogeneous equation
As mentioned above, a complex solution when solving the homogeneous equation results in a sinusoidal oscillation, as proven in Figure 3.6. The most notable part of the solution is that there is both a frequency of oscillation and a phase shift. This form is very useful for analyzing the frequency response of a system, as will be seen in a later chapter.
differential equations - 3.7
Consider the situation where the results of a homogeneous solution are the complex
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sin ( Ct) ) |
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sin ( Ct) ) |
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x = eRt |
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X1 |
+ 2X1X2 + X2 |
– |
X1 |
+ –2X1X2 + X2 ( ( X |
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+ X ) cos ( Ct) + j( X |
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+ 2X1X2 + X2 |
– |
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x = eRt |
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4X1X2( ( X |
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+ X ) |
cos ( Ct) + j( X |
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– X ) |
sin ( Ct) ) |
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+ X ) |
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cos ( Ct) |
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+ j |
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sin ( Ct) |
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Rt |
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( X1 |
– X2) |
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X4 = atan |
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Figure 3.6 Phase shift solution for a second-order homogeneous differential equation
differential equations - 3.8
Note: Occasionally a problem solution might consist of both a sine and cosine term with the same frequency. These should normally be combined to a single term with a phase shift as shown below.
Recall the double angle formula,
sin ( ω t + θ ) = sin ω t cos θ + sin θ cos ω t
This can be written in a more common form,
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A( sin ω |
t cos θ |
+ sin θ |
cos ω |
t) |
= |
A sin ( ω |
t + θ ) |
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A cos θ |
sin ω |
t + A sin θ cos ω |
t |
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A sin ( ω |
t + θ ) |
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B sin ω |
t + C cos ω t = |
A sin ( ω |
t + θ ) |
where, |
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A = B2 + C2
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3 sin 5t + 4 cos 5t = 3 |
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B = A cos θ
C = A sin θ
= 5 sin ( 5t + 0.927)
Figure 3.7 Phase shift solution form
differential equations - 3.9
Solve the following differential equation given the initial condition.
·· |
· |
x( 0) = 1 |
· |
x( 0) = 2 |
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ans. x( t) = e–t + 3te–t
Figure 3.8 Drill Problem: Second order homogeneous differential equation
The methods for solving non-homogeneous differential equations builds upon the methods used for the solution of homogeneous equations. This process adds a step to find the particular solution of the equation. An example of the solution of a first-order nonhomogeneous equation is shown in Figure 3.9. To find the homogeneous solution the nonhomogeneous part of the equation is set to zero. To find the particular solution the final
differential equations - 3.10
form must be guessed. This is then substituted into the equation, and the values of the coefficients are found. Finally the homogeneous and particular solutions are added to get the final equation. The overall response of the system can be obtained by adding the homogeneous and particular parts. This is acceptable because the equations are linear, and the principle of superposition applies. The homogeneous equation deals with the response to initial conditions, and the particular solution deals with the response to forced inputs.
Generally, |
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· |
x( 0) = x0 |
Ax + Bx = Cf( t) |
First, find the homogeneous solution as before, in Figure 3.3.
B
–--t
A
xh = x0e
Next, guess the particular solution by looking at the form of ’f(t)’. This step is highly subjective, and if an incorrect guess is made, it will be unsolvable. When this happens, just make another guess and repeat the process. An example is given below. In the case below the guess should be similar to the exponential forcing function.
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Substitute these into the differential equation and solve for A. |
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6 4C1e |
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24C |
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Combine the particular and homogeneous solutions. |
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x = |
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Figure 3.9 Solution of a first-order non-homogeneous equation
The method for finding a particular solution for a second-order non-homogeneous differential equation is shown in Figure 3.10. In this example the forcing function is sinusoidal, so the particular result should also be sinusoidal. The final result is converted into a phase shift form.
differential equations - 3.11
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Ax + Bx + Cx = Df( t) |
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1. Find the homogeneous solution as before. |
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2. Guess the particular solution by looking at the form of ’f(t)’. This step is highly sub- jective, and if an incorrect guess is made it will be unsolvable. When this happens, just make another guess and repeat the process. For the purpose of illustration an example is given below. In the case below it should be similar to the sine function.
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For example, if we are given |
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2x + 6x + 2x = 2 sin ( 3t + 4) |
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A reasonable guess is, |
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xp = A sin ( 3t) + B cos ( 3t) |
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= 3A cos ( 3t) |
– 3B sin ( 3t) |
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xp |
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= –9A sin ( 3t) |
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– 9B cos |
( 3t) |
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Substitute these into the differential equation ans solve for A and B. |
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2( –9A sin ( 3t) – 9B cos ( 3t) ) + 6( 3A cos ( 3t) |
– 3B sin ( 3t) ) |
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2( A sin ( 3t) |
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+ B cos ( 3t) ) = 2 sin ( 3t + 4) |
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( –18A – 18B + 2A) sin ( 3t) |
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= 2 sin ( 3t + 4) |
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( – 16A – 18B) sin ( 3t) + ( 18A – 16B) |
cos ( 3t) = 2( sin 3t cos 4 + cos 3t sin 4) |
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( – 16A – 18B) sin ( 3t) + ( 18A – 16B) cos ( 3t) |
= ( 2 cos 4) sin ( 3t) |
+ ( 2 sin 4) cos ( 3t) |
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– 16A – 18B = 2 cos 4 |
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18A – 16B = 2 sin 4 |
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–1 |
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–16 –18 |
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–16 |
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2 sin 4 |
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Next, rearrange the equation to phase shift form. |
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xp = –0.0109 sin ( 3t) + 0.0823 cos ( 3t) |
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sin |
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3t + atan |
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------------------- + |
-- |
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–0.0109 |
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3. Use the initial conditions to determine the coefficients in the homogeneous solution.
Figure 3.10 Solution of a second-order non-homogeneous equation
differential equations - 3.12
When guessing particular solutions, the forms in Figure 3.11 can be helpful.
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Forcing Function |
Guess |
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A |
C |
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Ax + B |
Cx + D |
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eAx |
CeAx or |
CxeAx |
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B sin ( Ax) or B cos ( Ax) |
C sin ( Ax) |
+ D cos ( Ax) |
or Cx sin ( Ax) + xD cos ( Ax)
Figure 3.11 General forms for particular solutions
differential equations - 3.13
Solve the following differential equation given the initial condition.
·· |
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x( 0) = 0 |
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x( 0) = 0 |
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+ 2x + x = 1 |
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ans. x( t) = – e–t – te–t + 1
Figure 3.12 Drill Problem: Second order non-homogeneous differential equation
An example of a second-order system is shown in Figure 3.13. As expected, it begins with a FBD and summation of forces. This is followed with the general solution of
differential equations - 3.14
the homogeneous equation. Real roots are assumed thus allowing the problem solution to continue in Figure 3.14.
Assume the system illustrated to the right starts from rest at a height ’h’. At time ’t=0’ the system is released and allowed to move.
Mg
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My
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∑Fy = – Mg + Ksy + Kdy = –My |
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Find the homogeneous solution. |
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yh = e |
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My + Kdy + Ksy = 0 |
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MA2 + KdA + Ks = 0 |
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Let us assume that the values of M, Kd and Ks lead to the case of two different |
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positive roots. This would occur if the damper value was much larger than |
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the spring and mass values. Thus, |
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A = R1, R2 |
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yh = C1e |
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Figure 3.13 Second-order system example
The solution continues by assuming a particular solution and calculating values for the coefficients using the initial conditions in Figure 3.14. The final result is a secondorder system that is overdamped, with no oscillation.
differential equations - 3.15
Next, find the particular solution.
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yp = C |
· |
= 0 |
·· |
= 0 |
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yh |
yh |
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M( 0) + Kd( 0) + Ks( C) = Mg
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= |
yp + yh |
= ------- |
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y( 0) = h |
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y'( 0) = 0 |
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h = |
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C1 + C2 = |
h – |
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y'( t) = R1C1eR1t + R2C2eR2 t |
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0 = R1C1e0 + R2C2e0 |
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0 |
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R1C1 + R2C2 |
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C1 = |
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– |
-----C2 + C2 = |
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h – |
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Ksh – Mg |
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Ksh – Mg |
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C |
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----------------------- |
----------------- |
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C |
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-------- |
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---------------------- |
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----------------- |
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2 |
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K |
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– R |
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1 |
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2 |
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Now, combine the solutions and solve for the unknowns using the initial conditions.
y( t)
y( t)
= |
Mg |
+ |
Ksh – Mg |
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–R1 |
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R1 t |
||||||||||||||||
-------K |
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----------------------- |
K |
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----------------- |
e |
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s |
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s |
R |
1 |
– R |
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2 |
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= |
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+ |
Ksh – Mg |
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–R1 |
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R1 t |
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-------K |
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----------------------- |
K |
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----------------- |
e |
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s |
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s |
R |
1 |
– R |
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2 |
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–R2 |
Ksh – Mg |
–R1 |
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R2 t |
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+ |
-------- |
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---------------------- |
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----------------- |
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e |
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R |
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– R |
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1 |
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s |
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Ksh – Mg |
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+ |
---------------------- ----------------- e |
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K |
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Figure 3.14 Second-order system example (continued)