Dynamic Systems Modeling and Analysis Hung V. Vu California State University, Long Beach Ramin S. Esfandiari California State University, Long Beach THE McGRAW-HILL COMPANIES, INC. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto
CONTENTS Preface xv Complex Analysis, Differential Equations, and the Laplace Transform 1 1.1 Complex Numbers in Rectangular Form 1 Addition and Multiplication of Complex Numbers: Rectangular Form / Complex Conjugation: Rectangular Form / Division of Complex Numbers: Rectangular Form Problem Set 1.1 3 1.2 Representation of Complex Numbers in Polar Form 3 Multiplication and Division of Complex Numbers: Polar Form / Complex Conjugation: Polar Form / Powers of a Complex Number / Roots of a Complex Number Problem Set 1.2 8 1.3 Complex Variables and Functions 8 Problem Set 1.3 11 1.4 Linear Differential Equations with Constant Coefficients 11 Standard Forms / Solution of Linear ODEs with Constant Coefficients / Homogeneous, Linear ODEs with Constant Coefficients / Free Response of First-Order Systems / Free Response of Second-Order Systems / Nonhomogeneous, Linear ODEs with Constant Coefficients Problem Set 1.4 21 1.5 Laplace Transformation 22 Special Functions / Linearity of Laplace and Inverse Laplace Transforms Problem Set 1.5 29 1.6 Laplace Transform of Derivatives and Integrals 30 Solving Initial-Value Problems Problem Set 1.6 34 1.7 Inverse Laplace Transformation 34 Partial Fraction Method / Partial Fraction Method: An Alternative Approach / Convolution Method
vi Contents Problem Set 1.7 43 1.8 Shift on r-axis 44 Periodic Functions Problem Set 1.8 49 1.9 Applications: System Response 50 First-Order Systems / Second-Order Systems Problem Set 1.9 61 1.10 Initial- and Final-Value Theorems 61 Problem Set 1.10 64 Summary 64 Problems 69 2 Matrix Analysis 76 2.1 Vectors and Matrices 76 Matrix Operations / Matrix Transpose Problem Set 2.1 81 2.2 Determinant, Inverse, and Rank of a Matrix 82 Properties of Determinants / Determination of Inverse Adjoint Matrix / Linear Systems of Algebraic Equations Cramer's Rule Problem Set 2.2 88 2.3 Eigenvalues and Eigenvectors 89 Solving the Eigenvalue Problem / Special Matrices Problem Set 2.3 95 2.4 Matrix Transformation and Diagonalization 95 Exponential of a Matrix / Exponential of a Diagonal Matrix / Exponential of a Diagonalizable Matrix Problem Set 2.4 101 Summary 101 Problems 105 3 System Model Representation 108 3.1 Configuration Form 108 Second-Order Matrix Form Problem Set 3.1 112 3.2 State-Space Representation 113 Methodology / General Formulation / Nonuniqueness of State Variables / Decoupling Problem Set 3.2 124
Contents vii 3.3 Input-Output (I/O) Equation 125 Strategy Problem Set 3.3 128 3.4 Transfer Function 128 Relation Between State-Space Form and Transfer Function / Single Input-Single Output (SISO) Systems / Multiple Input-Multiple Output (MIMO) Systems Problem Set 3.4 137 3.5 State-Space Representation from the Input-Output Equation 138 Problem Set 3.5 143 3.6 Linearization 144 Graphical Interpretation / Taylor Series Expansion / Determination of the Operating Point / Numerical Solution of a Nonlinear Model / Fourth-Order Runge-Kutta Method Problem Set 3.6 163 Summary 164 Problems 167 Mechanical Systems 172 4.1 Introduction 172 4.2 Mechanical Elements 173 Mass Element / The Right-Hand Rule (RHR) / Spring Element / Damper Element 4.3 Equivalence 179 4.4 Degrees of Freedom 181 Rolling Constraints / Holonomic Constraints and Nonholonomic Constraints Problem set 4.1-4.4 186 4.5 Translational systems 190 Newton's Second Law / Free-Body Diagram and Sign Convention / Modeling of a System with Viscous Damping / Modeling of a System with Coulomb Damping / D'Alembert's Principle / Gravity and Differential Equation of Spring-Mass Systems / Systems with Displacement Input / Transfer Functions and State-Space Representation for SDOF Systems / Two-Degree-of-Freedom (TDOF) Systems / Systems with Massless Junctions / Skeleton Approach Problem Set 4.5 210
viii Contents 4.6 Rotational systems 213 The Moment Equation / Angular Momentum and Moments of Inertia / Modeling of Rigid Bodies in Planar Motion 4.7 Mixed Systems: Translational and Rotational 226 Problem Set 4.6^.7 236 4.8 Gear-Train Systems 238 Problem set 4.8 243 4.9 Lagrange's Equations 244 Kinetic Energy / Potential Energy / Nonconservative Forces Problem Set 4.9 258 Summary 261 Problems 263 Electrical, Electronic, and Electromechanical Systems 265 5.1 Introduction 265 5.2 Electrical Systems 265 Passive Electrical Elements / Kirchhoff's Laws 5.3 Electronic Systems: Operational Amplifiers 275 Problem Set 5.1-5.3 277 5.4 Electromechanical Systems 279 Elemental Relations of Electromechanical Systems / Armature-Controlled DC Motors / Field-Controlled DC Motors Problem Set 5.4 288 5.5 Impedance Methods ' 290 Electrical Systems / Equivalence / Voltage Divider and Transfer Function / Mechanical Systems Problem Set 5.5 295 Summary 295 Problems 296 Fluid and Thermal Systems 304 6.1 Introduction 304 6.2 Thermodynamics 305 Chemistry / Ideal Gases / Gaseous Mixtures / Intensive and Extensive Properties / Conservation of Energy / Specific Heats
Contents ix 6.3 Fluid Mechanics 311 Laminar and Turbulent Flows / Fluid Resistance 6.4 Pneumatic Systems 315 Fundamentals / Mathematical Modeling 6.5 Liquid-Level Systems 320 Fundamentals / Mathematical Modeling 6.6 Thermal Systems: Heat Transfer 323 Fundamentals / Mathematical Modeling 6.7 Analogous Systems 327 Summary 328 Problems 331 System Response 334 7.1 Types of Response 334 7.2 Transient Response of First-Order Systems 335 Free Response / Forced Response / Step Response / Ramp Response 7.3 Transient Response of Second-Order Systems 339 Free Response / Forced Response / Response to a Unit Impulse / Unit-Impulse Response / Response to a Unit Step / Unit-Step Response Problem Set 7.1-7.3 355 7.4 Transient Response of Higher-Order Systems 356 Problem Set 7.4 361 7.5 Steady-State Response to Sinusoidal Input 362 Bode Plot / First-Order System / Second-Order System 7.6 Response to an Arbitrary Input Convolution Integral 370 Problem Set 7.5-7.6 374 7.7 Solving the State Equation 375 Homogeneous State Equation / Laplace Transform Approach / State-Transition Matrix / Nonhomogeneous State Equation / Laplace Transform Approach Problem Set 7.7 381 7.8 Modal Decomposition 381 7.9 Fourier Analysis 385 Fourier Series / Fourier Cosine and Sine Series / Convergence of Fourier Series / Interval Extension / System Response via Fourier Series
x Contents Problem Set 7.8-7.9 Summary Problems Introduction to Vibrations 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 Introduction Natural Frequencies and Mode Shapes Unsymmetrical Systems / Symmetrical Systems / Automobile Suspension Problem Set 8.1-8.2 Modal Matrix Orthogonality / Orthonormality / Damping Problem Set 8.3 Logarithmic Decrement Problem Set 8.4 Beat Phenomenon Problem Set 8.5 Frequency Response of Vibration Systems Simple Mechanical System / Vibration Isolation / System with Support Motion / Rotating Unbalanced Mass Problem Set 8.6 Damped Vibration Absorber Frequency Response / Transient Response Problem Set 8.7 Modal Analysis: System Response Eigenvalue Problem / Eigensolutions: Natural Frequencies and Mode Shapes / Orthonormality / System Response via Modal Analysis Vibration Testing Summary Problems 394 394 396 401 401 403 426 427 436 437 440 441 447 448 459 461 469 471 479 479 482 Block Diagram Representation 486 9.1 Block Diagrams 486 Block Diagram Operations / Closed-Loop Systems 9.2 Block Diagram from Governing Equations 493 Block Diagram from State-Space Model / Block Diagram from Input-Output Equation
Contents xi 9.3 Block Diagram Reduction 499 Moving a Branch Point / Moving a Summing Junction / Negative and Positive Feedback / Mason's Rule Special Case / Mason's Rule General Form Summary 510 Problems 510 10 Introduction to Control Systems 516 10.1 Introduction 516 10.2 Definitions 516 10.3 Essential Components of Control Systems 518 10.4 Historical Perspective 518 10.5 General Block Diagram for Control Systems 519 10.6 Further Examples of Control Systems 520 Heating Control Systems: An Engineering Application / Heating Control Systems: Homeostasis, a Biological Condition 10.7 Poles and Zeros 522 First-Order Systems: Time Constant / Second-Order Systems: Natural Frequency and Damping Ratio / Higher-Order Systems / Damping Ratio and -Line 10.8 Transient Response Specifications 531 First-Order Systems / Second-Order Systems 10.9 Dominant Pole Concept 538 First-Order and Second-Order Systems / Higher- Order Systems 10.10 Routh-Hurwitz Stability Criterion 540 General Systems / Fourth-Order Systems / Third- Order Systems / Second-Order Systems Problem Set 10.1-10.10 543 10.11 Controller Types and Actions 545 Proportional Control (P Control) / Derivative Control (D Control) / Integral Control (I Control) / Proportional-plus-Derivative Control (PD Control) / Proportional-plus-Integral Control (PI Control) / Proportional-plus-Integral-plus-Derivative Control (PID Control) 10.12 Steady-State Error 548 Problem Set 10.11-10.12 553 10.13 Methods of Ziegler-Nichols Tuning 554 Optimizing PID Parameters in Practice
xii Contents 10.14 Speed Control System 558 Problem Set 10.13-10.14 561 Summary 562 Problems 564 Appendix A Tables 567 Appendix B Computer Simulation 571 Useful Notes for MATRIX X 571 Appendix C Useful Formulas 582 Euler's Identities 582 Taylor's Series Expansion 582 Trigonometry 582 Pythagorean Identities / Expansion Formulas / Product Formulas / Double-Angle/Half-Angle Formulas / Law of Cosines / Miscellaneous Hyperbolic Functions and Relations 584 Logarithm 584 Common Logarithm / Natural Logarithm / Decibel (db) / Change of Base / Logarithm of a Complex Number Binomial Series 585 Appendix D Answers to Odd-Numbered Problems 586 Chapter 1 586 Problem Set 1.1 / Problem Set 1.2 / Problem Set 1.3 / Problem Set 1.4 / Problem Set 1.5 / Problem Set 1.6 / Problem Set 1.7 / Problem Set 1.8 / Problem Set 1.9 / Problem Set 1.10 Chapter 2 589 Problem Set 2.1 / Problem Set 2.2 / Problem Set 2.3 / Problem Set 2.4 Chapter 3 591 Problem Set 3.1 / Problem Set 3.2 / Problem Set 3.3 / Problem Set 3.4 / Problem Set 3.5 / Problem Set 3.6 / Problems Chapter 4 594 Problem Set 4.1-4.4 / Problem Set 4.5 / Problem Set 4.6-4.7 / Problem Set 4.8 / Problem Set 4.9 / Problems
Contents xiii Chapter 5 598 Problem Set 5.1-5.3 / Problem Set 5.4 / Problem Set 5.5 / Problems Chapter 6 601 Chapter 7 602 Problem Set 7.1-7.3 / Problem Set 7.4 / Problem Set 7.5-7.6 / Problem Set 7.7 / Problem Set 7.8-7.9 Chapter 8 604 Problem Set 8.1-8.2 / Problem Set 8.3 / Problem Set 8.4 / Problem Set 8.6 / Problem Set 8.7 / Problems Chapter 9 608 Chapter 10 610 Problem Set 10.1-10.10 / Problem Set 10.11-10.12 / Problem Set 10.13-10.14 / Problems Bibliography 612 Index 615