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

William Hallauer, Virginia Tech

Copyright Year: 2016

Publisher: A.T. Still University

Language: English

## Formats Available

## Conditions of Use

Attribution-NonCommercial

CC BY-NC

## Reviews

This on-line textbook is a challenging combination of system dynamics and responses, mechanical vibrations, mechanical and electrical systems, rigid body dynamics, and feedback control. Covered are free and forced, undamped and damped responses,... read more

This on-line textbook is a challenging combination of system dynamics and responses, mechanical vibrations, mechanical and electrical systems, rigid body dynamics, and feedback control. Covered are free and forced, undamped and damped responses, in both the frequency and time domain.

The textbook focuses on linear time-invariant (LTI) systems, with time- and Laplace-solutions of the governing ordinary differential equations (ODEs). First-, second-, and fourth-order systems are included and considered. The time constant is presented for first-order LTI systems, and natural frequency, damping ratio, and resonance are presented for second-order LTI systems. Translational and rotational systems mechanical systems are included, with inertial, spring, and damping elements. Electrical circuit LTI systems with resistors, capacitors, inductors, and operational amplifiers are also presented. MATLAB software is applied as a tool and for examples throughout the book. Some examples and applications specific to Aerospace Engineering are presented throughout. System identification from experimental data and homework problems with real and simulated data are featured.

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This on-line textbook is a challenging combination of system dynamics and responses, mechanical vibrations, mechanical and electrical systems, rigid body dynamics, and feedback control. Covered are free and forced, undamped and damped responses, in both the frequency and time domain.

The textbook focuses on linear time-invariant (LTI) systems, with time- and Laplace-solutions of the governing ordinary differential equations (ODEs). First-, second-, and fourth-order systems are included and considered. The time constant is presented for first-order LTI systems, and natural frequency, damping ratio, and resonance are presented for second-order LTI systems. Translational and rotational systems mechanical systems are included, with inertial, spring, and damping elements. Electrical circuit LTI systems with resistors, capacitors, inductors, and operational amplifiers are also presented. MATLAB software is applied as a tool and for examples throughout the book. Some examples and applications specific to Aerospace Engineering are presented throughout. System identification from experimental data and homework problems with real and simulated data are featured.

The stated prerequisites are: 1. The student should be, at minimum, a junior in an accredited four-year engineering curriculum; and 2. The student should have already completed standard first courses in engineering dynamics and differential equations. Other desired, but not strictly necessary, stated prerequisites are a course in basic electrical circuits, a course in basic computer programming, familiarity with MATLAB commands, knowledge of matrix notation and matrix arithmetic operations (basic course in linear algebra), and knowledge of Laplace transforms.

Here are some limitations of the book, as identified by the author. The bond graph approach is not presented, but author says this book prepares the student to tackle that subject. No state-space representation nor solutions nor control are presented, even for the fourth-order systems. The author stated that his textbook as grown into too much material for a three-hour semester course – so an instructor must pick and choose topics rather than try to present it all.

On my shelf I have at least four separate textbooks each covering one of these topics in depth (system dynamics and responses, mechanical vibrations, mechanical and electrical systems, rigid body dynamics, and feedback control). At 439 pages, the current book is long, but short in terms of complete coverage for each of these related but distinct subjects. In my opinion, the math level is beyond standard engineering juniors, and seems to be written more for electrical, mechanical, and aerospace engineering graduate students.

The sections that I read in detail are comprehensive, accurate, error-free and unbiased, clear to read, consistent in notation, modular in the sense that portions can be used easily from the whole, the organization is logical, and it is free of grammatical errors. Overall, I believe this on-line textbook is very strong and should be seriously considered for adoption by instructors of courses in linear systems dynamics and related topics. This textbook is a serious contender, with long-range potential in longevity and topics coverage, that could support at least four related, required courses in linear systems dynamics at my university.

## 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

## Ancillary Material

## About the Book

This is a complete college textbook, including a detailed Table of Contents, seventeen Chapters (each with a set of relevant homework problems), a list of References, two Appendices, and a detailed Index. The book is intended to enable students to:

- Solve first-, second-, and higher-order, linear, time-invariant (LTI) ordinary 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 sinusoidal excitation and plot this response in standard form;
- 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; - Derive and analyze mathematical models (ODEs) of low-order mechanical systems, both translational and rotational, that are composed of inertial elements, spring elements, and damping devices;
- Derive and analyze mathematical models (ODEs) of low-order electrical circuits composed of resistors, capacitors, inductors, and operational amplifiers;
- Derive (from ODEs) and manipulate Laplace transfer functions and block diagrams representing output-to-input relationships of discrete elements and of systems;
- Define and evaluate
*stability*for an LTI system; - Explain
*proportional*,*integral*, and*derivative*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 general 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,
*i.e.*, fourth-order systems; - Chapters 13 and 14 introduce classical feedback control;
- Chapter 15 presents the basic features of proportional, integral, 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 maturity 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, __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 by two coupled second-order ODEs, not in the state-space form by four coupled first-order ODEs. The book does __not__ use *bond graph* 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 of __basic__ (not __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. **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;
- 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.