cover

Dynamic System Modeling

and Control

by

Hugh Jack

(Draft Version 3.1, September 3, 2009)

© Copyright 1993-2009 Hugh Jack

This work is licensed under a

Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.

For uses not covered under the license please contact Hugh Jack (jackh@gvsu.edu)

© Copyright 1993-2009 Hugh Jack

1. INTRODUCTION 1.1

1.1 BASIC TERMINOLOGY 1.1

1.2 EXAMPLE SYSTEM 1.4

1.3 SUMMARY 1.4

1.4 PRACTICE PROBLEMS 1.4

2. TRANSLATION 2.1

2.1 INTRODUCTION 2.1

2.2 MODELING 2.3

2.2.1 Free Body Diagrams 2.4

2.2.2 Mass and Inertia 2.4

2.2.3 Gravity and Other Fields 2.8

2.2.4 Springs 2.10

2.2.5 Damping and Drag 2.18

2.2.6 Cables And Pulleys 2.21

2.2.7 Friction 2.23

2.2.8 Contact Points And Joints 2.25

2.3 SYSTEM EXAMPLES 2.25

2.4 OTHER TOPICS 2.35

2.5 SUMMARY 2.36

2.6 PRACTICE PROBLEMS 2.36

2.7 PRACTICE PROBLEM SOLUTIONS 2.41

2.8 ASSIGNMENT PROBLEMS 2.45

3. ANALYSIS OF DIFFERENTIAL EQUATIONS 3.1

3.1 INTRODUCTION 3.1

3.2 EXPLICIT SOLUTIONS 3.2

3.3 RESPONSES 3.16

3.3.1 First-order 3.17

3.3.2 Second-order 3.23

3.3.3 Other Responses 3.28

3.4 RESPONSE ANALYSIS 3.31

3.5 NON-LINEAR SYSTEMS 3.33

3.5.1 Non-Linear Differential Equations 3.34

3.5.2 Non-Linear Equation Terms 3.38

3.5.3 Changing Systems 3.41

3.6 CASE STUDY 3.47

3.7 SUMMARY 3.51

3.8 PRACTICE PROBLEMS 3.51

3.9 PRACTICE PROBLEM SOLUTIONS 3.57

3.10 ASSIGNMENT PROBLEMS 3.64

4. NUMERICAL ANALYSIS 4.1

4.1 INTRODUCTION 4.1

4.2 THE GENERAL METHOD 4.1

4.2.1 State Variable Form 4.2

4.3 NUMERICAL INTEGRATION 4.10

4.3.1 Numerical Integration With Tools 4.10

4.3.2 Numerical Integration 4.15

4.3.3 Taylor Series 4.21

4.3.4 Runge-Kutta Integration 4.23

4.4 SYSTEM RESPONSE 4.29

4.4.1 Steady-State Response 4.30

4.5 DIFFERENTIATION AND INTEGRATION OF EXPERIMENTAL DATA 4.31

4.6 ADVANCED TOPICS 4.33

4.6.1 Switching Functions 4.33

4.6.2 Interpolating Tabular Data 4.36

4.6.3 Modeling Functions with Splines 4.37

4.6.4 Non-Linear Elements 4.39

4.7 PRACTICAL ELEMENTS OF COMPUTER MATH 4.39

4.7.1 Numbering Systems 4.40

4.7.2 Speed 4.41

4.7.3 Accuracy 4.42

4.8 CASE STUDY 4.43

4.9 SUMMARY 4.50

4.10 PRACTICE PROBLEMS 4.51

4.11 PRACTICE PROBLEM SOLUTIONS 4.56

4.12 ASSIGNMENT PROBLEMS 4.77

5. ROTATION 5.1

5.1 INTRODUCTION 5.1

5.2 MODELING 5.2

5.2.1 Inertia 5.3

5.2.2 Springs 5.7

5.2.3 Damping 5.12

5.2.4 Levers 5.14

5.2.5 Gears and Belts 5.15

5.2.6 Friction 5.19

5.2.7 Permanent Magnet Electric Motors 5.22

5.3 OTHER TOPICS 5.23

5.4 DESIGN CASE 5.23

5.5 SUMMARY 5.28

5.6 PRACTICE PROBLEMS 5.28

5.7 PRACTICE PROBLEM SOLUTIONS 5.36

5.8 ASSIGNMENT PROBLEMS 5.47

6. INPUT-OUTPUT EQUATIONS AND TRANSFER FUNCTIONS 6.1

6.1 INTRODUCTION 6.1

6.2 THE DIFFERENTIAL OPERATOR 6.1

6.3 INPUT-OUTPUT EQUATIONS 6.4

6.3.1 Converting Input-Output Equations to State Equations 6.6

6.4 TRANSFER FUNCTIONS 6.9

6.4.1 Integrating Input-Output Equations 6.11

6.5 DESIGN CASE 6.14

6.6 SUMMARY 6.24

6.7 PRACTICE PROBLEMS 6.24

6.8 PRACTICE PROBLEM SOLUTIONS 6.28

6.9 ASSIGNMENT PROBLEMS 6.32

6.10 REFERENCES 6.33

7. CONVOLUTION 7.1

7.1 INTRODUCTION 7.1

7.2 UNIT IMPULSE FUNCTIONS 7.1

7.3 IMPULSE RESPONSE 7.3

7.4 CONVOLUTION 7.5

7.5 NUMERICAL CONVOLUTION 7.6

7.6 SUMMARY 7.9

7.7 PRACTICE PROBLEMS 7.9

7.8 PRACTICE PROBLEM SOLUTIONS 7.10

7.9 ASSIGNMENT PROBLEMS 7.10

8. ELECTRICAL SYSTEMS 8.1

8.1 INTRODUCTION 8.1

8.2 MODELING 8.1

8.2.1 Resistors 8.2

8.2.2 Voltage and Current Sources 8.4

8.2.3 Capacitors 8.8

8.2.4 Inductors 8.10

8.2.5 Op-Amps 8.11

8.3 IMPEDANCE 8.16

8.4 EXAMPLE SYSTEMS 8.18

8.5 ELECTROMECHANICAL SYSTEMS - MOTORS 8.26

8.5.1 Permanent Magnet DC Motors 8.26

8.5.2 Induction Motors 8.28

8.5.3 Brushless Servo Motors 8.29

8.6 FILTERS 8.32

8.7 OTHER TOPICS 8.33

8.8 SUMMARY 8.33

8.9 PRACTICE PROBLEMS 8.34

8.10 PRACTICE PROBLEM SOLUTIONS 8.39

8.11 ASSIGNMENT PROBLEMS 8.44

9. FEEDBACK CONTROL SYSTEMS 9.1

9.1 INTRODUCTION 9.1

9.2 TRANSFER FUNCTIONS 9.1

9.3 CONTROL SYSTEMS 9.3

9.3.1 PID Control Systems 9.5

9.3.2 Manipulating Block Diagrams 9.7

9.3.3 A Motor Control System Example 9.12

9.3.4 System Error 9.17

9.3.5 Controller Transfer Functions 9.21

9.3.6 Feedforward Controllers 9.21

9.3.7 State Equation Based Systems 9.22

9.3.8 Cascade Controllers 9.24

9.4 EMBEDDED CONTROL 9.24

9.5 SUMMARY 9.26

9.6 PRACTICE PROBLEMS 9.26

9.7 PRACTICE PROBLEM SOLUTIONS 9.34

9.8 ASSIGNMENT PROBLEMS 9.42

10. PHASOR ANALYSIS 10.1

10.1 INTRODUCTION 10.1

10.2 PHASORS FOR STEADY-STATE ANALYSIS 10.1

10.3 VIBRATIONS 10.8

10.4 PROGRAMS 10.10

10.5 SUMMARY 10.12

10.6 PRACTICE PROBLEMS 10.13

10.7 PRACTICE PROBLEM SOLUTIONS 10.14

10.8 ASSIGNMENT PROBLEMS 10.16

11. BODE PLOTS 11.1

11.1 INTRODUCTION 11.1

11.2 BODE PLOTS 11.5

11.3 STRAIGHT LINE APPROXIMATIONS 11.10

11.3.1 Second Order Underdamped Terms 11.18

11.3.2 Lone Ds on the Top or Bottom 11.21

11.4 FREQUENCY RESPONSE FUNCTIONS 11.24

11.5 SIGNAL SPECTRUMS 11.25

11.6 SUMMARY 11.27

11.7 PRACTICE PROBLEMS 11.27

11.8 PRACTICE PROBLEM SOLUTIONS 11.31

11.9 ASSIGNMENT PROBLEMS 11.43

11.10 LOG SCALE GRAPH PAPER 11.45

12. ROOT LOCUS ANALYSIS 12.1

12.1 INTRODUCTION 12.1

12.2 ROOT-LOCUS ANALYSIS 12.1

12.3 SUMMARY 12.10

12.4 PRACTICE PROBLEMS 12.11

12.5 PRACTICE PROBLEM SOLUTIONS 12.13

12.6 ASSIGNMENT PROBLEMS 12.24

13. NONLINEAR SYSTEMS 13.1

13.1 INTRODUCTION 13.1

13.2 SOURCES OF NONLINEARITY 13.1

13.2.1 Non-Linear Relationships 13.1

13.3 NON-LINEAR ELEMENTS 13.2

13.3.1 Time Variant 13.3

13.3.2 Switching 13.3

13.3.3 Deadband 13.4

13.3.4 Saturation and Clipping 13.7

13.3.5 Hysteresis and Slip 13.8

13.3.6 Delays and Lags 13.9

13.4 SUMMARY 13.10

13.5 PRACTICE PROBLEMS 13.10

13.6 PRACTICE PROBLEM SOLUTIONS 13.10

13.7 ASIGNMENT PROBLEMS 13.10

14. CONTINUOUS SENSORS 14.1

14.1 INTRODUCTION 14.1

14.2 INDUSTRIAL SENSORS 14.2

14.2.1 Angular Displacement 14.3

Potentiometers 14.3

14.2.2 Encoders 14.4

Tachometers 14.8

14.2.3 Linear Position 14.8

Potentiometers 14.8

Linear Variable Differential Transformers (LVDT) 14.9

Moire Fringes 14.11

Accelerometers 14.12

14.2.4 Forces and Moments 14.15

Strain Gages 14.15

Piezoelectric 14.18

14.2.5 Liquids and Gases 14.20

Pressure 14.21

Venturi Valves 14.22

Coriolis Flow Meter 14.23

Magnetic Flow Meter 14.24

Ultrasonic Flow Meter 14.24

Vortex Flow Meter 14.24

Positive Displacement Meters 14.25

Pitot Tubes 14.25

14.2.6 Temperature 14.25

Resistive Temperature Detectors (RTDs) 14.26

Thermocouples 14.26

Thermistors 14.28

Other Sensors 14.30

14.2.7 Light 14.30

Light Dependant Resistors (LDR) 14.30

14.2.8 Chemical 14.31

pH 14.31

Conductivity 14.31

14.2.9 Others 14.32

14.3 INPUT ISSUES 14.32

14.4 SENSOR GLOSSARY 14.35

14.5 SUMMARY 14.36

14.6 REFERENCES 14.37

14.7 PRACTICE PROBLEMS 14.37

14.8 PRACTICE PROBLEM SOLUTIONS 14.38

14.9 ASSIGNMENT PROBLEMS 14.40

15. ANALOG INPUTS AND OUTPUTS 15.1

15.1 INTRODUCTION 15.1

15.2 ANALOG INPUTS 15.3

15.3 ANALOG OUTPUTS 15.10

15.4 NOISE REDUCTION 15.12

15.4.1 Shielding 15.14

15.4.2 Grounding 15.16

15.5 CASE STUDY 15.16

15.6 SUMMARY 15.17

15.7 PRACTICE PROBLEMS 15.17

15.8 PRACTICE PROBLEM SOLUTIONS 15.17

15.9 ASSIGNMENT PROBLEMS 15.18

16. CONTINUOUS ACTUATORS 16.1

16.1 INTRODUCTION 16.1

16.2 ELECTRIC MOTORS 16.1

16.2.1 Basic Brushed DC Motors 16.3

16.2.2 AC Motors 16.7

16.2.3 Brushless DC Motors 16.15

16.2.4 Stepper Motors 16.17

16.2.5 Wound Field Motors 16.19

16.3 HYDRAULICS 16.23

16.4 OTHER SYSTEMS 16.24

16.5 SUMMARY 16.25

16.6 PRACTICE PROBLEMS 16.25

16.7 PRACTICE PROBLEM SOLUTIONS 16.26

16.8 ASSIGNMENT PROBLEMS 16.27

17. MOTION CONTROL 17.1

17.1 INTRODUCTION 17.1

17.2 MOTION PROFILES 17.2

17.2.1 Velocity Profiles 17.2

17.2.2 Position Profiles 17.12

17.3 MULTI AXIS MOTION 17.15

17.3.1 Slew Motion 17.16

Interpolated Motion 17.17

17.3.2 Motion Scheduling 17.18

17.4 PATH PLANNING 17.20

17.5 CASE STUDIES 17.22

17.6 SUMMARY 17.24

17.7 PRACTICE PROBLEMS 17.24

17.8 PRACTICE PROBLEM SOLUTIONS 17.25

17.9 ASSIGNMENT PROBLEMS 17.26

18. LAPLACE TRANSFORMS 18.1

18.1 INTRODUCTION 18.1

18.2 APPLYING LAPLACE TRANSFORMS 18.3

18.2.1 A Few Transform Tables 18.4

18.3 MODELING TRANSFER FUNCTIONS IN THE s-DOMAIN 18.9

18.4 FINDING OUTPUT EQUATIONS 18.12

18.5 INVERSE TRANSFORMS AND PARTIAL FRACTIONS 18.15

18.6 EXAMPLES 18.23

18.6.1 Mass-Spring-Damper Vibration 18.23

18.6.2 Circuits 18.25

18.7 ADVANCED TOPICS 18.27

18.7.1 Input Functions 18.27

18.7.2 Initial and Final Value Theorems 18.28

18.8 LAPLACE IMPULSE FUNCTIONS 18.29

18.9 A MAP OF TECHNIQUES FOR LAPLACE ANALYSIS 18.30

18.10 SUMMARY 18.31

18.11 PRACTICE PROBLEMS 18.32

18.12 PRACTICE PROBLEM SOLUTIONS 18.37

18.13 ASSIGNMENT PROBLEMS 18.42

18.14 REFERENCES 18.52

19. CONTROL SYSTEM ANALYSIS 19.1

19.1 INTRODUCTION 19.1

19.2 CONTROL SYSTEMS 19.1

19.2.1 PID Control Systems 19.3

19.2.2 Analysis of PID Controlled Systems With Laplace Transforms 19.5

19.2.3 Finding The System Response To An Input 19.8

19.2.4 Controller Transfer Functions 19.13

19.3 ROOT-LOCUS PLOTS 19.13

19.3.1 Approximate Plotting Techniques 19.17

19.4 DESIGN OF CONTINUOUS CONTROLLERS 19.21

19.5 SUMMARY 19.21

19.6 PRACTICE PROBLEMS 19.22

19.7 PRACTICE PROBLEM SOLUTIONS 19.27

19.8 ASSIGNMENT PROBLEMS 19.27

20. STATE SPACE ANALYSIS 20.1

20.1 INTRODUCTION 20.1

20.2 OBSERVABILITY 20.13

20.3 CONTROLLABILITY 20.15

20.4 OBSERVERS 20.18

20.5 SUMMARY 20.18

20.6 PRACTICE PROBLEMS 20.19

20.7 PRACTICE PROBLEM SOLUTIONS 20.19

20.8 ASSIGNMENT PROBLEMS 20.19

20.9 BIBLIOGRAPHY 20.19

21. STATE SPACE CONTROLLERS 21.1

21.1 INTRODUCTION 21.1

21.2 FULL STATE FEEDBACK 21.2

21.3 OBSERVERS 21.5

21.4 SUPPLEMENTAL OBSERVERS 21.11

21.5 REGULATED CONTROL WITH OBSERVERS 21.11

21.6 LQR 21.22

21.7 LINEAR QUADRATIC GAUSSIAN (LQG) COMPENSATORS 21.24

21.8 VERIFYING CONTROL SYSTEM STABILITY 21.24

21.8.1 Stability 21.25

21.8.2 Bounded Gain 21.26

21.9 ADAPTIVE CONTROLLERS 21.28

21.10 OTHER METHODS 21.31

21.10.1 Kalman Filtering 21.32

21.11 SUMMARY 21.32

21.12 PRACTICE PROBLEMS 21.33

21.13 PRACTICE PROBLEM SOLUTIONS 21.33

21.14 ASSIGNMENT PROBLEMS 21.33

22. SYSTEM IDENTIFICATION 22.1

22.1 INTRODUCTION 22.1

22.2 SUMMARY 22.10

22.3 PRACTICE PROBLEMS 22.10

22.4 PRACTICE PROBLEM SOLUTIONS 22.10

22.5 ASSIGNMENT PROBLEMS 22.10

23. ELECTROMECHANICAL SYSTEMS 23.1

23.1 INTRODUCTION 23.1

23.2 MATHEMATICAL PROPERTIES 23.1

23.2.1 Induction 23.1

23.3 EXAMPLE SYSTEMS 23.9

23.4 SUMMARY 23.16

23.5 PRACTICE PROBLEMS 23.16

23.6 PRACTICE PROBLEM SOLUTIONS 23.16

23.7 ASSIGNMENT PROBLEMS 23.16

24. FLUID SYSTEMS 24.1

24.1 SUMMARY 24.1

24.2 MATHEMATICAL PROPERTIES 24.1

24.2.1 Resistance 24.2

24.2.2 Capacitance 24.4

24.2.3 Power Sources 24.6

24.3 EXAMPLE SYSTEMS 24.8

24.4 SUMMARY 24.10

24.5 PRACTICE PROBLEMS 24.10

24.6 PRACTICE PROBLEMS SOLUTIONS 24.10

24.7 ASSIGNMENT PROBLEMS 24.10

25. THERMAL SYSTEMS 25.1

25.1 INTRODUCTION 25.1

25.2 MATHEMATICAL PROPERTIES 25.1

25.2.1 Resistance 25.1

25.2.2 Capacitance 25.3

25.2.3 Sources 25.4

25.3 EXAMPLE SYSTEMS 25.4

25.4 SUMMARY 25.7

25.5 PRACTICE PROBLEMS 25.7

25.6 PRACTICE PROBLEM SOLUTIONS 25.7

25.7 ASSIGNMENT PROBLEMS 25.7

26. OPTIMIZATION 26.1

26.1 INTRODUCTION 26.1

26.2 OBJECTIVES AND CONSTRAINTS 26.2

26.3 SEARCHING FOR THE OPTIMUM 26.6

26.4 OPTIMIZATION ALGORITHMS 26.9

26.4.1 Random Walk 26.9

26.4.2 Gradient Decent 26.10

26.4.3 Simplex 26.10

26.5 SUMMARY 26.10

26.6 PRACTICE PROBLEMS 26.10

26.7 PRACTICE PROBLEM SOLUTIONS 26.10

26.8 ASSIGNMENT PROBLEMS 26.10

27. FINITE ELEMENT ANALYSIS (FEA) 27.1

27.1 INTRODUCTION 27.1

27.2 FINITE ELEMENT MODELS 27.2

27.3 FINITE ELEMENT MODELS 27.4

27.4 SUMMARY 27.12

27.5 PRACTICE PROBLEMS 27.13

27.6 PRACTICE PROBLEM SOLUTIONS 27.13

27.7 ASSIGNMENT PROBLEMS 27.13

27.8 BIBLIOGRAPHY 27.13

28. FUZZY LOGIC 28.1

28.1 INTRODUCTION 28.1

28.2 COMMERCIAL CONTROLLERS 28.7

28.3 REFERENCES 28.8

28.4 SUMMARY 28.8

28.5 PRACTICE PROBLEMS 28.8

28.6 PRACTICE PROBLEM SOLUTIONS 28.8

28.7 ASSIGNMENT PROBLEMS 28.8

29. NEURAL NETWORKS 29.1

29.1 SUMMARY 29.9

29.2 PRACTICE PROBLEMS 29.10

29.3 PRACTICE PROBLEM SOLUTIONS 29.10

29.4 ASSIGNMENT PROBLEMS 29.10

29.5 REFERENCES 29.10

30. EMBEDDED CONTROL SYSTEM 30.1

30.1 INTRODUCTION 30.1

30.2 CASE STUDY 30.3

30.3 SUMMARY 30.3

30.4 PRACTICE PROBLEMS 30.3

30.5 PRACTICE PROBLEM SOLUTIONS 30.3

30.6 ASSIGNMENT PROBLEMS 30.3

31. STATE DIAGRAMS FOR SEQUENTIAL CONTROL 31.1

31.1 INTRODUCTION 31.1

31.1.1 State Diagram Example 31.4

31.1.2 Conversion to C Code 31.7

31.2 EXAMPLE SYSTEM 31.12

31.3 SUMMARY 31.12

31.4 PRACTICE PROBLEMS 31.12

32. WRITING 32.1

32.1 FORGET WHAT YOU WERE TAUGHT BEFORE 32.1

32.2 8. Proofread your report.WHY WRITE REPORTS? 32.2

32.3 THE TECHNICAL DEPTH OF THE REPORT 32.3

32.4 TYPES OF REPORTS 32.3

32.5 LABORATORY REPORTS 32.4

An Example First Draft of a Report 32.5

An Example Final Draft of a Report 32.12

32.6 RESEARCH 32.12

32.7 DRAFT REPORTS 32.12

32.8 PROJECT REPORT 32.12

32.9 OTHER REPORT TYPES 32.14

32.9.1 Executive 32.14

32.9.2 Consulting 32.15

32.9.3 Memo(randum) 32.15

32.9.4 Interim 32.15

32.9.5 Poster 32.15

32.9.6 Progress Report 32.16

32.9.7 Oral 32.17

32.9.8 Patent 32.17

32.10 LAB BOOKS 32.18

32.11 REPORT ELEMENTS 32.19

32.11.1 Figures 32.20

32.11.2 Graphs 32.21

32.11.3 Tables 32.21

32.11.4 Equations 32.22

32.11.5 Experimental Data 32.23

32.11.6 Result Summary 32.24

32.11.7 References 32.24

32.11.8 Acknowledgments 32.25

32.11.9 Abstracts 32.25

32.11.10 Appendices 32.25

32.11.11 Page Numbering 32.26

32.11.12 Numbers and Units 32.26

32.11.13 Engineering Drawings 32.26

32.11.14 Discussions 32.27

32.11.15 Conclusions 32.27

32.11.16 Recommendations 32.28

32.11.17 Appendices 32.28

32.11.18 Units 32.28

32.12 GENERAL WRITING ISSUES 32.28

32.13 WRITERS BLOCK 32.29

32.14 TECHNICAL ENGLISH 32.30

32.15 EVALUATION FORMS 32.31

32.16 PATENTS 32.33

33. PROJECTS 33.1

33.1 33.1

33.2 OVERVIEW 33.1

33.2.1 The Objectives and Constraints 33.2

33.3 MANAGEMENT 33.3

33.3.1 Timeline - Tentative 33.3

33.3.2 Teams 33.4

33.4 DELIVERABLES 33.5

33.4.1 Conceptual Design 33.5

33.4.2 EGR 345/101 Contract 33.5

33.4.3 Progress Reports 33.6

33.4.4 Design Proposal 33.6

33.4.5 The Final Report 33.7

33.5 REPORT ELEMENTS 33.8

33.5.1 Gantt Charts 33.8

33.5.2 Drawings 33.9

33.5.3 Budgets and Bills of Material 33.9

33.5.4 Calculations 33.10

33.6 APPENDICES 33.10

33.6.1 Appendix A - Sample System 33.10

33.6.2 Appendix B - EGR 345/101 Contract 33.18

33.6.3 Appendix C - Forms 33.19

34. ENGINEERING PROBLEM SOLVING 34.1

34.1 BASIC RULES OF STYLE 34.1

34.2 EXPECTED ELEMENTS 34.1

34.3 SEPCIAL ELEMENTS 34.2

34.3.1 Graphs 34.2

34.3.2 EGR 345 Specific 34.2

34.4 SCILAB 34.2

34.5 TERMINOLOGY 34.3

35. MATHEMATICAL TOOLS 35.1

35.1 INTRODUCTION 35.1

35.1.1 Constants and Other Stuff 35.2

35.1.2 Basic Operations 35.3

Factorial 35.4

35.1.3 Exponents and Logarithms 35.4

35.1.4 Polynomial Expansions 35.5

35.1.5 Practice Problems 35.6

35.2 FUNCTIONS 35.9

35.2.1 Discrete and Continuous Probability Distributions 35.9

35.2.2 Basic Polynomials 35.9

35.2.3 Partial Fractions 35.11

35.2.4 Summation and Series 35.14

35.2.5 Practice Problems 35.16

35.3 SPATIAL RELATIONSHIPS 35.17

35.3.1 Trigonometry 35.17

35.3.2 Hyperbolic Functions 35.22

Practice Problems 35.23

35.3.3 Geometry 35.24

35.3.4 Planes, Lines, etc. 35.41

35.3.5 Practice Problems 35.43

35.4 COORDINATE SYSTEMS 35.45

35.4.1 Complex Numbers 35.45

35.4.2 Cylindrical Coordinates 35.48

35.4.3 Spherical Coordinates 35.49

35.4.4 Practice Problems 35.50

35.5 MATRICES AND VECTORS 35.51

35.5.1 Vectors 35.51

35.5.2 Dot (Scalar) Product 35.52

35.5.3 Cross Product 35.57

35.5.4 Triple Product 35.59

35.5.5 Matrices 35.59

35.5.6 Solving Linear Equations with Matrices 35.64

35.5.7 Practice Problems 35.65

35.6 CALCULUS 35.70

35.6.1 Single Variable Functions 35.70

Differentiation 35.70

Integration 35.73

35.6.2 Vector Calculus 35.77

35.6.3 Differential Equations 35.79

First-order Differential Equations 35.80

Guessing 35.81

Separable Equations 35.81

Homogeneous Equations and Substitution 35.82

Second-order Differential Equations 35.83

Linear Homogeneous 35.83

Nonhomogeneous Linear Equations 35.84

Higher Order Differential Equations 35.86

Partial Differential Equations 35.86

35.6.4 Other Calculus Stuff 35.87

35.6.5 Practice Problems 35.87

35.7 NUMERICAL METHODS 35.93

35.7.1 Approximation of Integrals and Derivatives from Sampled Data 35.93

35.7.2 Euler First-order Integration 35.94

35.7.3 Taylor Series Integration 35.94

35.7.4 Runge-Kutta Integration 35.95

35.7.5 Newton-Raphson to Find Roots 35.95

35.8 LAPLACE TRANSFORMS 35.96

35.8.1 Laplace Transform Tables 35.96

35.9 z-TRANSFORMS 35.99

35.10 FOURIER SERIES 35.102

35.11 TOPICS NOT COVERED (YET) 35.102

35.12 REFERENCES/BIBLIOGRAPHY 35.103

36. A BASIC INTRODUCTION TO `C' 36.1

36.1 WHY USE `C'? 36.1

36.2 BACKGROUND 36.2

36.3 PROGRAM PARTS 36.2

36.4 HOW A `C' COMPILER WORKS 36.11

36.5 STRUCTURED `C' CODE 36.13

36.6 ARCHITECTURE OF `C' PROGRAMS (TOP-DOWN) 36.14

36.6.1 How? 36.14

36.6.2 Why? 36.15

36.7 CREATING TOP DOWN PROGRAMS 36.16

36.8 HOW THE BEAMCAD PROGRAM WAS DESIGNED 36.17

36.8.1 Objectives: 36.18

36.8.2 Problem Definition: 36.18

36.8.3 User Interface: 36.18

Screen Layout (also see figure): 36.18

Input: 36.19

Output: 36.20

Help: 36.20

Error Checking: 36.20

Miscellaneous: 36.21

36.8.4 Flow Program: 36.22

36.8.5 Expand Program: 36.22

36.8.6 Testing and Debugging: 36.24

36.8.7 Documentation 36.25

Users Manual: 36.25

Programmers Manual: 36.26

36.8.8 Listing of BeamCAD Program. 36.26

36.9 PRACTICE PROBLEMS 36.26

37. UNITS AND CONVERSIONS 37.1

37.1 HOW TO USE UNITS 37.1

37.2 HOW TO USE SI UNITS 37.2

37.3 THE TABLE 37.2

37.4 ASCII, HEX, BINARY CONVERSION 37.6

37.5 G-CODES 37.8

38. ATOMIC MATERIAL DATA 38.1

38. MECHANICAL MATERIAL PROPERTIES 38.1

38.1 FORMULA SHEET 38.4

39. BIBLIOGRAPHY 39.1

39.1 TEXTBOOKS 39.1

39.1.1 Slotine and Li 39.1

39.1.2 VandeVegte 39.1

39.1.3 Others 39.2

40. TOPICS IN DEVELOPMENT 40.1

40.1 UPDATED DC MOTOR MODEL 40.1

40.2 ANOTHER DC MOTOR MODEL 40.4

40.3 BLOCK DIAGRAMS AND UNITS 40.8

40.4 SIGNAL FLOW GRAPHS 40.9

40.5 ZERO ORDER HOLD 40.9

40.6 TORSIONAL DAMPERS 40.9

40.7 MISC 40.10

40.8 Nyquist Plot 40.10

40.9 NICHOLS CHART 40.12

40.10 BESSEL POLYNOMIALS 40.14

40.11 ITAE 40.15

40.12 ROOT LOCUS 40.16

40.13 LYAPUNOV'S LINEARIZATION METHOD 40.16

40.14 XXXXX 40.17

40.15 XXXXX 40.18

40.16 XXXXX 40.18

40.17 XXXXX 40.18

40.18 XXXXX 40.18

40.19 XXXXX 40.18

40.20 XXXXX 40.18

40.21 SUMMARY 40.18

40.22 PRACTICE PROBLEMS 40.18

40.23 PRACTICE PROBLEM SOLUTIONS 40.19

40.24 ASSGINMENT PROBLEMS 40.19

40.25 REFERENCES 40.19

40.26 BIBLIOGRAPHY 40.19