Introduction to Dynamics and Control in Mechanical Engineering Systems

One of the first books to provide in-depth and systematic application of finite element methods to the field of stochastic structural dynamics The parallel developments of the Finite Element Methods in the 1950’s and the engineering applications of stochastic processes in the 1940’s provided a combined numerical analysis tool for the studies of dynamics of structures and structural systems under random loadings. In the open literature, there are books on statistical dynamics of structures and books on structural dynamics with chapters dealing with random response analysis. However, a systematic treatment of stochastic structural dynamics applying the finite element methods seems to be lacking. Aimed at advanced and specialist levels, the author presents and illustrates analytical and direct integration methods for analyzing the statistics of the response of structures to stochastic loads. The analysis methods are based on structural models represented via the Finite Element Method. In addition to linear problems the text also addresses nonlinear problems and non-stationary random excitation with systems having large spatially stochastic property variations.

Table of Contents:
Series Preface xiii Preface xv Acknowledgments xvii 1 Introduction 1 1.1 Important Difference between Static and Dynamic Responses 1 1.2 Classification of Dynamic Systems 2 1.3 Applications of Control Theory 3 1.4 Organization of Presentation 4 References 5 2 Review of Laplace Transforms 7 2.1 Definition 8 2.2 First and Second Shifting Theorems 10 2.3 Dirac Delta Function (Unit Impulse Function) 10 2.4 Laplace Transforms of Derivatives and Integrals 11 2.5 Convolution Theorem 11 2.6 Initial and Final Value Theorems 13 2.7 Laplace Transforms of Periodic Functions 13 2.8 Partial Fraction Method 15 2.9 Questions and Solutions 16 2.10 Applications of MATLAB 22 Exercise Questions 26 References 27 3 Dynamic Behaviors of Hydraulic and Pneumatic Systems 29 3.1 Basic Elements of Liquid and Gas Systems 29 3.1.1 Liquids 30 3.1.3 Remarks 31 3.2 Hydraulic Tank Systems 32 3.2.1 Non-interacting Hydraulic Tank Systems 32 3.2.2 Interacting Hydraulic Tank Systems 34 3.3 Nonlinear Hydraulic Tank and Linear Transfer Function 35 3.4 Pneumatically Actuated Valves 37 3.5 Questions and Solutions 39 Appendix 3A: Transfer Function of Two Interacting Hydraulic Tanks 49 Exercise Questions 52 4 Dynamic Behaviors of Oscillatory Systems 57 4.1 Elements of Oscillatory Systems 57 4.2 Free Vibration of Single Degree-of-Freedom Systems 59 4.3 Single Degree-of-Freedom Systems under Harmonic Forces 63 4.4 Single Degree-of-Freedom Systems under Non-Harmonic Forces 65 4.5 Vibration Analysis of Multi-Degrees-of-Freedom Systems 67 4.5.1 Formulation and Solution for Two-Degrees-of-Freedom Systems 67 4.5.2 Vibration Analysis of a System with a Dynamic Absorber 72 4.5.3 Normal Mode Analysis 73 4.6 Vibration of Continuous Systems 77 4.6.1 Vibrating Strings or Cables 78 4.6.2 Remarks 80 4.7 Questions and Solutions 81 Appendix 4A: Proof of Equation (4.19b) 97 Exercise Questions 99 References 104 5 Formulation and Dynamic Behavior of Thermal Systems 105 5.1 Elements of Thermal Systems 105 5.1.1 Thermal Resistance 105 5.1.2 Thermal Capacitance 106 5.1.3 Thermal Radiation 107 5.2 Thermal Systems 107 5.2.1 Process Control 107 5.2.2 Space Heating 108 5.2.3 Three-Capacitance Oven 109 5.3 Questions and Solutions 112 Exercise Questions 117 6 Formulation and Dynamic Behavior of Electrical Systems 121 6.1 Basic Electrical Elements 121 6.2 Fundamentals of Electrical Circuits 122 6.2.1 Resistors Connected in Series 122 6.2.2 Resistors Connected in Parallel 123 6.2.3 Kirchhoff’s Laws 124 6.4 Electromechanical Systems 126 6.4.1 Armature-Controlled DC Motor 127 6.4.2 Field-Controlled DC Motor 129 6.4.3 DC Generator 130 6.5 Questions and Solutions 131 Exercise Questions 134 References 135 7 Dynamic Characteristics of Transducers 137 7.1 Basic Theory of the Tachometer 137 7.2 Principles and Applications of Oscillatory Motion Transducers 138 7.2.1 Equation of Motion 139 7.2.2 Design Considerations of Two Types of Transducer 140 7.3 Principles and Applications of Microphones 141 7.3.1 Moving-Coil Microphone 141 7.3.2 Condenser Microphone 144 7.4 Principles and Applications of the Piezoelectric Hydrophone 146 7.5 Questions and Solutions 148 Appendix 7A: Proof of Approximated Current Solution 150 Exercise Questions 153 References 154 8 Fundamentals of Control Systems 155 8.1 Classification of Control Systems 156 8.2 Representation of Control Systems 156 8.3 Transfer Functions 156 8.3.1 Transfer Function of Elements in Cascade Connection 157 8.3.2 Transfer Function of Elements in Parallel Connection 157 8.3.3 Remarks 158 8.4 Closed-Loop Control Systems 158 8.4.1 Closed-Loop Transfer Functions and System Response 159 8.4.2 Summary of Steps for Determination of Closed-Loop Transfer Functions 161 8.5 Block Diagram Reduction 161 8.5.1 Moving Starting Points of Signals 161 8.5.2 Moving Summing Points 162 8.5.3 System Transfer Function by Block Diagram Reduction 162 8.6 Questions and Solutions 164 Exercise Questions 170 References 172 9 Analysis and Performance of Control Systems 173 9.1 Response in the Time Domain 173 9.2 Transient Responses as Functions of Closed-Loop Poles 175 9.3 Control System Design Based on Transient Responses 177 9.4 Control Types 180 9.4.2 Integral Control 181 9.4.3 Derivative Control 181 9.5 Steady-State Errors 182 9.5.1 Unit Step Input 182 9.5.2 Unit Ramp Input 183 9.5.3 Unit Parabolic Input 183 9.6 Performance Indices and Sensitivity Functions 184 9.6.1 Performance Indices 184 9.6.2 Sensitivity Functions 185 9.7 Questions and Solutions 185 Exercise Questions 190 10 Stability Analysis of Control Systems 195 10.1 Concept of Stability in Linear Control Systems 195 10.2 Routh–Hurwitz Stability Criterion 195 10.3 Applications of Routh–Hurwitz Stability Criterion 197 10.4 Questions and Solutions 202 Exercise Questions 208 References 210 11 Graphical Methods for Control Systems 211 11.1 Root Locus Method and Root Locus Plots 211 11.1.1 Rules for Root Locus Plots of Negative Feedback Control Systems 212 11.1.2 Construction of Root Loci 213 11.2 Polar and Bode Plots 215 11.3 Nyquist Plots and Stability Criterion 221 11.3.1 Conformal Mapping and Cauchy’s Theorem 221 11.3.2 Nyquist Method and Stability Criterion 223 11.4 Gain Margin and Phase Margin 226 11.5 Lines of Constant Magnitude: M Circles 229 11.6 Lines of Constant Phase: N Circles 233 11.7 Nichols Charts 234 11.8 Applications of MATLAB for Graphical Constructions 236 11.8.1 Root Locus Plots 236 11.8.2 Bode Plots 243 11.8.3 Nyquist Plots 249 Exercise Questions 257 References 260 12 Modern Control System Analysis 261 12.1 State Space Method 261 12.2 State Transition Matrix 262 12.3 Relationship between Laplace Transformed State Equation and Transfer Function 264 12.4 Stability Based on Eigenvalues of the Coefficient Matrix 267 12.6 Stabilizability and Detectability 277 12.7 Applications of MATLAB 277 Appendix 12A: Solution of System of First-Order Differential Equations 286 Appendix 12B: Maclaurin’s Series 291 Appendix 12C: Rank of A Matrix 294 Exercise Questions 294 References 296 Index 297