Introduction to Linear, Time-Invariant, Dynamic Systems for Students of Engineering William L. Hallauer, Jr. INTRODUCTION TO LINEAR, TIME-INVARIANT, DYNAMIC SYSTEMS FOR STUDENTS OF ENGINEERING William L. Hallauer, Jr. Permission of the author is granted for reproduction in any form of this book or any part of this book, provided that there is appropriate attribution, for example, by a citation in the References section of a technical article, and provided that the reproduction is not made for the purpose of charging a fee to users (e.g., students in a class) beyond the cost of reproduction. However, plagiarism from this book is prohibited, and any use of the material in this book to produce profit or income beyond the cost of reproduction is prohibited. © 2016 by William L. Hallauer, Jr. Introduction to Linear, Time-Invariant, Dynamic Systems for Students of Engineering is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. For details of use requirements, please see https://creativecommons.org/licenses/by-nc/4.0/. Trademark Notice: Product or corporate names printed herein may be trademarks or registered trademarks. They are printed only for unique identification and for explana- tion, without intent to infringe. INTRODUCTION TO LINEAR, TIME-INVARIANT DYNAMIC SYSTEMS FOR STUDENTS OF ENGINEERING William L. Hallauer, Jr. AUTHOR’S PREFACE I taught many times the college undergraduate, junior-level, one-semester course entitled “AOE 3034, Vehicle Vibration and Control” in the Department of Aerospace and Ocean Engineering (AOE) at Virginia Polytechnic Institute and State University (VPI & SU). I was dissatisfied using commercially available textbooks for AOE 3034, so I began writing my own course notes, and those notes grew into this book. Although this project began with preparation of informal handout notes, the completed book is a formal college engineering textbook, complete with homework problems at the end of each chapter, a detailed Table of Contents, a list of References, and a detailed Index. I hope that this book will be understandable and enlightening for students of engineering system dynam- ics, a valuable teaching resource for course instructors, and a useful reference for self- study and review. The content of this book is based primarily on topics that the faculty of AOE and VPI & SU elected to include in AOE 3034 during the 1990s and early 2000s. The con- cise course description is: “Free and forced motions of first-order systems. Free and forced motions of second-order systems, both undamped and damped. Frequency and time responses. Introduction to control, transfer functions, block diagrams, and closed- loop system characteristics. Higher-order systems.” A more detailed course description is provided by the following list of primary learning objectives, which were developed to satisfy requirements of the agency that accredits engineering college degrees in the United States: At the completion of AOE 3034, the student should be able to: 1. Solve first-, second-, and higher-order, linear, time-invariant (LTI) or- dinary differential equations (ODEs) with forcing, using both time-domain and Laplace-transform methods. 2. Solve for the frequency response of an LTI system to periodic sinusoi- dal excitation and plot this response in standard form (log magnitude and phase versus frequency). 3. Explain the role of the “time constant” in the response of a first-order LTI system, and the roles of “natural frequency”, “damping ratio”, and “resonance” in the response of a second-order LTI system. AP-1 Author’s Preface 4. Derive and analyze mathematical models (ODEs) for low-order me- chanical systems, both translational and rotational systems, that are com- posed of inertial elements, spring elements, and damping devices. 5. Derive and analyze mathematical models (ODEs) for low-order electri- cal systems (circuits) composed of resistors, capacitors, inductors, and op- erational amplifiers. 6. Derive (from ODEs) and manipulate Laplace transfer functions and block diagrams representing output-to-input relationships of discrete ele- ments and of systems. 7. Define and evaluate “stability” for an LTI system. 8. Explain “proportional”, “integral”, and “derivative” types of feedback control for single-input, single-output (SISO), LTI systems. 9. Sketch the locus of characteristic values, as a control parameter varies, for a feedback-controlled SISO, LTI system. 10. Use MATLAB1 as a tool to study the time and frequency responses of LTI systems. Rather that summarizing the contents of this book chapter by chapter, I invite the reader of this preface to peruse the detailed Table of Contents. However, the book’s gen- eral organization is the following: Chapters 1-10 deal primarily with the ODEs and be- haviors of first-order and second-order dynamic systems; Chapters 11 and 12 touch on the ODEs and behaviors of mechanical systems having two degrees of freedom, i.e., fourth-order systems; Chapters 13 and 14 introduce classical feedback control, motivat- ing the concept with what I believe is a unique approach based on the standard ODE of a second-order dynamic system; Chapter 15 presents the basic features of proportional, in- tegral, and derivative types of classical control; and Chapters 16 and 17 discuss methods for analyzing the stability of classical control systems. The principal parts of Chapters 1- 16 are focused on the ten primary learning objectives listed above. I added Chapter 17 on frequency-response stability analysis because I feel that an introduction to classical con- trol theory and design is incomplete without that subject, even though it was not included in AOE 3034. The general minimum prerequisite for studying this book is the intellectual matur- ity of a junior-level (third-year) college student in an accredited four-year engineering curriculum. More specifically, a reader of this book should already have passed standard first courses in engineering dynamics and ODEs. It will be helpful if, but probably is not 1 MATLAB ® is a registered trademark of The MathWorks, Inc. MATLAB is widely available to engi- neers in practice and to engineering colleges. Furthermore, MATLAB-like software that uses command- line language similar to MATLAB’s, and functions similarly to MATLAB in many respects, is available for download from the Internet, for example, GNU Octave (http://www.gnu.org/software/octave/). AP-2 Author’s Preface mandatory that, the reader has studied basic electrical circuits, perhaps in an introductory college physics course. It is necessary that the reader has studied basic computer pro- gramming. MATLAB computer programs and commands appear throughout this book, so the reader should be able to understand MATLAB commands. However, MATLAB commands are generally clearly expressed in standard English and standard arithmetic notation, so a person who has done any computer programming, even if that was not with MATLAB, probably can follow the computer commands and command sequences in this book. Familiarity with matrix notation and matrix arithmetic operations also will be helpful, especially for Chapters 11 and 12. My students who took at the same time AOE 3034 and a mathematics course on operational methods (primarily Laplace transforms) often found that the combination of those courses was unusually complementary and beneficial to their comprehension of the material. A mathematical second-order system is represented in this book primarily by a single second-order ODE, not in the state-space form by a pair of coupled first-order ODEs. Similarly, a two-degrees-of-freedom (fourth-order) system is represented in Chapters 11 and 12 by a pair of coupled second-order ODEs, not in the state-space form by four coupled first-order ODEs. A reader who can understand the mathematics and dynamics of relatively simple systems expressed here in classical second-order form probably will have little trouble making the transition in more advanced literature to the general state-space representation of higher order systems. This book deals mostly with specific idealized models of basic physical systems, such as mass-damper-spring mechanisms and single-loop electrical circuits. The empha- ses are on fundamental ODEs and fundamental system response characteristics. I have chosen, therefore, not to burden the reader with bond graph modeling, the general and powerful, but complicated, modern tool for analysis of dynamic systems. However, the material in this book is an appropriate preparation for the bond graph approach presented in, for example, System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems, 5th edition, by Dean C. Karnopp, Donald L. Margolis, and Ronald C. Rosenberg, published by John Wiley & Sons, 2012. I intended originally that Chapters 1-16 of the course notes (before they grew into a book) could be covered in a normally-paced course of three fifty-minute lessons per week in a standard college semester of fourteen weeks duration. Even so, instructors of AOE 3034, including myself, had difficulty squeezing all of that content into forty-two lessons. Furthermore, in the process of converting the course notes into a complete text- book, I added material that is relevant and interesting (to me, at least) in many complete “new” sections to the ends of Chapters 12, 7, 8, 10, 14, and 16. And, as mentioned above, I also added a complete “new” Chapter 17. Consequently, I doubt that even the 2 I added Section 1-10, which deals with mass-spring systems, after working with several graduate students whose research subjects were design, analysis, and testing of aerodynamic sensors that include mechanical components. These graduate students had not recently reviewed elementary system dynamics, and so were unfamiliar with fundamental concepts such as natural frequency and resonance. I decided, therefore, to make Chapter 1 a succinct summary of basic mechanical-system dynamics (excluding feedback control), suitable for quick review by graduate students or any engineers who specialize in other areas but need to understand at least the most basic of this book’s lessons.