- •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
bode plots - 10.1
10. BODE PLOTS
Topics:
• Bode plots
Objectives:
•To be able to describe the response of a system using Bode plots
10.1INTRODUCTION
When a phasor transform is applied to a transfer function the result can be expressed as a magnitude and angle that are functions of frequency. The magnitude is the gain, and the angle is the phase shift. In the previous chapter these values were calculated for a single frequency and then multiplied by the input values to get an output value. At different frequencies the transfer function value will change. The transfer function gain and phase angle can be plotted as a function of frequency to give an overall picture of system response.
bode plots - 10.2
Aside: Consider a ’graphic equalizer’ commonly found on home stereo equipment. The spectrum can be adjusted so that high or low tones are emphasized or muted. The position of the sliders adjusts the envelope that the audio signal is filtered through.
The sliders trace out a Bode gain plot. In theoretical terms the equalizer can be described with a transfer function. As the slides are moved the transfer function is changed, and the bode plot shifts. In the example below the slides are positioned to pass more of the lower frequencies. The high frequencies would not be passed clearly, and might sound somewhat muffled.
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gain |
f(Hz) |
Figure 10.1 Commonly seen Bode plot
The mass-spring-damper transfer function from the previous chapter is expanded in Figure 10.2. In this example the transfer function is multiplied by the complex conjugate to eliminate the complex number in the denominator. The magnitude of the resulting transfer function is the gain, and the phase shift is the angle. Note that to correct for the quadrant of the phase shift angles pi radians is subtracted for certain frequency values.
bode plots - 10.3
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x( ω |
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------------F( ω |
) |
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x( ω |
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j( 3000ω ) + ( 2000 – 1000ω |
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( –j) ( 3000ω |
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2) |
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F------------( ω |
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---------------------------------------------------------------------- |
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------------------------------------------------------------------------------- |
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j( 3000ω ) + ( 2000 – 1000ω |
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x( ω |
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F------------( ω ) |
------------------------------------------------------------------------- |
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( 2000 – 1000ω |
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x( ω |
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------------ |
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------------------------------------------------------------------------------ |
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2) 2 |
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------------------------------------------------------------------------------ |
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F( ω |
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+ ( 3000ω |
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= |
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atan |
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----------------------------------------- |
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= atan |
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------------ |
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for ( ω |
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≤ |
2) |
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– 1000ω |
2 + 2000 |
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2–ω |
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for ( ω |
> |
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Figure 10.2 A phasor transform example
The results in Figure 10.2 are normally left in variable form so that they may be analyzed for a range of frequencies. An example of this type of analysis is done in Figure 10.3. A set of frequencies is used for calculations. These need to be converted from Hz to rad/s before use. For each one of these the gain and phase angle is calculated. The gain gives a ratio between the input sine wave and output sine wave of the system. The magnitude of the output wave can be calculated by multiplying the input wave magnitude by the gain. (Note: recall this example was used in the previous chapter) The phase angle can be added to the input wave to get the phase of the output wave. Gain is normally converted to ’dB’ so that it may cover a larger range of values while still remaining similar numerically. Also note that the frequencies are changed in multiples of tens, or magnitudes.
bode plots - 10.4
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0.001 |
0.006283 |
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0.010.06283
0.10.6283
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1 |
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Note: the gain values will cover many magnitudes of values. To help keep the graphs rational we will
"compress" the values by converting them to dB (decibels) using the following formula.
gaindb = 20log ( gain)
Note: negative phase angles mean that the mass motion lags the force.
Note: The frequencies chosen should be chosen to cover the points with the greatest amount of change.
Figure 10.3 A phasor transform example (continued)
In this example gain is defined as x/F. Therefore F is the input to the system, and x is the resulting output. The gain means that for each unit of F input to the system, there will be gain*F=x output. If the input and output are sinusoidal, there is a difference in phase between the input and output wave of θ (the phase angle). This is shown in Figure 10.4, where an input waveform is supplied with three sinusoidal components. For each of the frequencies a gain and phase shift are calculated. These are then used to calculate the resulting output wave, relative to the input wave. The resulting output represents the steady-state response to the sinusoidal output.