# Introduction to Linear, Time-Invariant, Dynamic Systems for Students of Engineering

**Author:**William Hallauer, Virginia Tech

**Category:**Engineering, Mathemeatics

**Pages:**439

**Tags:**linear | Download

**Description:**

- Solve first-, second-, and higher-order, linear, time-invariant (LTI) orÂdinary differential equations (ODEs) with initial conditions and excitation, using both time-domain and Laplace-transform methods;
- Solve for the frequency response of an LTI system to periodic sinusoiÂdal excitation and plot this response in standard form;
- Explain the role of theÂ
*time constant*Â in the response of a first-order LTI system, and the roles ofÂ*natural frequency*,Â*damping ratio*, andÂ*resonance*Â in the response of a second-order LTI system; - Derive and analyze mathematical models (ODEs) of low-order meÂchanical systems, both translational and rotational, that are comÂposed of inertial elements, spring elements, and damping devices;
- Derive and analyze mathematical models (ODEs) of low-order electriÂcal circuits composed of resistors, capacitors, inductors, and opÂerational amplifiers;
- Derive (from ODEs) and manipulate Laplace transfer functions and block diagrams representing output-to-input relationships of discrete eleÂments and of systems;
- Define and evaluateÂ
*stability*Â for an LTI system; - ExplainÂ
*proportional*,Â*integral*, andÂ*derivative*Â types of feedback control for single-input, single-output (SISO), LTI systems; - Sketch the locus of characteristic values, as a control parameter varies, for a feedback-controlled SISO, LTI system;
- Use MATLAB as a tool to study the time and frequency responses of LTI systems.

The book’s genÂeral organization is:

- Chapters 1-10 deal primarily with the ODEs and behaviors of first-order and second-order dynamic systems;
- Chapters 11 and 12 discuss the ODEs and behaviors of mechanical systems having two degrees of freedom,Â
*i.e.*, fourth-order systems; - Chapters 13 and 14 introduce classical feedback control;
- Chapter 15 presents the basic features of proportional, inÂtegral, and derivative types of classical control;
- Chapters 16 and 17 discuss methods for analyzing the stability of classical control systems.

The general minimum prerequisite for understanding this book is the intellectual maturÂity of a junior-level (third-year) college student in an accredited four-year engineering curriculum. A mathematical second-order system is represented in this book primarily by a single second-order ODE,Â __not__Â in theÂ *state-space*Â form by a pair of coupled first-order ODEs. Similarly, a two-degrees-of-freedom (fourth-order) system is represented by two coupled second-order ODEs, not in the state-space form by four coupled first-order ODEs. The book doesÂ __not__Â useÂ *bond graph*Â modeling, the general and powerful, but complicated, modern tool for analysis of complex, multidisciplinary dynamic systems. The homework problems at the ends of chapters are very important to the learning objectives, so the author attempted to compose problems of practical interest and to make the problem statements as clear, correct, and unambiguous as possible. A major focus of the book is computer calculation of system characteristics and responses and graphical display of results, with use ofÂ __basic__Â (notÂ __advanced__) MATLAB commands and programs. The book includes many examples and homework problems relevant to aerospace engineering, among which are rolling dynamics of flight vehicles, spacecraft actuators, aerospace motion sensors, and aeroelasticity. There are also several examples and homework problems illustrating and validating theory by using measured data to identify first- and second-order system dynamic characteristics based on mathematical models (*e.g.*, time constants and natural frequencies), and system basic properties (*e.g.*, mass, stiffness, and damping). Applications of real and simulated experimental data appear in many homework problems. The book contains somewhat more material than can be covered during a single standard college semester, so an instructor who wishes to use this as a one-semester course textbook should not attempt to cover the entire book, but instead should cover only those parts that are most relevant to the course objectives.

## About the Contributors

### Author

**William L. Hallauer, Jr.Â **is an Adjunct Professor in the Department of Aerospace and Ocean Engineering at Virginia Tech.

Education:

- B.S. in Mechanical Engineering, Stanford University, 1961-65;
- S.M. in Aeronautics and Astronautics, Massachusetts Institute of Technology, 1965-66;
- Ph.D. in Aeronautics and Astronautics, Stanford University, 1969-74.

Employment in Higher Education:

- Virginia Polytechnic Institute and State University (Aerospace and Ocean Engineering, Mechanical Engineering), 1974-87, 1989-91, 2000-05;
- The United States Air Force Academy (Engineering Mechanics), 1987-89, 1994-99.

Employment in Industry:

- Boeing Company (Commercial Airplane Group), 1966-69;
- Lockheed Missiles and Space Company, 1973-74;
- Dynacs Engineering Company, Inc. (contractor for the U.S. Air Force), 1992-94.

Primary Technical Areas of Learning, Teaching, and Research:

- Structures, structural dynamics, and fluid-structure interaction (theory and computation);
- Experimental analysis of structural dynamics, including electrical and electromechanical systems used in experiments;
- Active control of vibration in highly flexible structures;
- Composition of research articles and instructional material.

## Table of Contents

- Chapter 1 Introduction; examples of 1st and 2nd order systems; example analysis and MATLAB graphing
- Chapter 2 Complex numbers and arithmetic; Laplace transforms; partial-fraction expansion
- Chapter 3 Mechanical units; low-order mechanical systems; simple transient responses of 1st order systems
- Chapter 4 Frequency response of 1st order systems; transfer function; general method for derivation of frequency response
- Chapter 5 Basic electrical components and circuits
- Chapter 6 General time response of 1st order systems by application of the convolution integral
- Chapter 7 Undamped 2nd order systems: general time response; undamped vibration
- Chapter 8 Pulse inputs; Dirac delta function; impulse response; initialvalue theorem; convolution sum
- Chapter 9 Damped 2nd order systems: general time response
- Chapter 10 2nd order systems: frequency response; beating response to suddenly applied sinusoidal (SAS) excitation
- Chapter 11 Mechanical systems with rigid-body plane translation and rotation
- Chapter 12 Vibration modes of undamped mechanical systems with two degrees of freedom
- Chapter 13 Laplace block diagrams, and additional background material for the study of feedback-control systems
- Chapter 14 Introduction to feedback control: output operations for control of rotational position
- Chapter 15 Input-error operations: proportional, integral, and derivative types of control
- Chapter 16 Introduction to system stability: time-response criteria
- Chapter 17 Introduction to system stability: frequency-response criteria
- Appendix A: Table and derivations of Laplace transform pairs
- Appendix B: Notes on work, energy, and power in mechanical systems and electrical circuits
- Index for all Chapters and Appendices