## 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 dynamics,

a valuable teaching resource for course instructors, and a useful reference for selfstudy

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 concise

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 closedloop

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) ordinary

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 sinusoidal

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.

4. Derive and analyze mathematical models (ODEs) for low-order mechanical

systems, both translational and rotational systems, that are composed

of inertial elements, spring elements, and damping devices.

5. Derive and analyze mathematical models (ODEs) for low-order electrical

systems (circuits) composed of resistors, capacitors, inductors, and operational

amplifiers.

6. Derive (from ODEs) and manipulate Laplace transfer functions and

block diagrams representing output-to-input relationships of discrete elements

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 general

organization is the following: Chapters 1-10 deal primarily with the ODEs and behaviors

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, motivating

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, integral,

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 control

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 maturity

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 engineers

in practice and to engineering colleges. Furthermore, MATLAB-like software that uses commandline

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/).

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 programming.

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 emphases

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 textbook,

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.

most demanding course instructor can, while still treating the students fairly, cover this

entire book in a three-credit, one-semester course. Therefore, instructors who wish to use

this as a one-semester course textbook should decide in advance which parts of the book

are essential to the course and which parts they cannot cover in the time allotted for the

course. If, for example, it is essential to cover all of Chapters 13-17 on classical control,

an instructor might elect to skip some or all of the “new” sections in Chapters 1, 7, 8, and

10, and to skip Chapters 11 and 12 on systems with two degrees of freedom, but to cover

most everything else in the book. Sections 6-4, 6-5, and 8-11 deal primarily with computational

methods for calculating approximate time-response solutions of first- and secondorder

ODEs; the contents of these sections are nicely compatible with the chapters in

which they reside; but they are not essential to the reader’s understanding of system dynamics,

so they can be omitted from course coverage without great loss. On the other

hand, I discourage the omission of Chapter 5 on basic electrical systems, not only because

I believe the material is important to most engineers, but also because such systems

provide many examples and homework problems later in the book.

The homework problems at the ends of chapters are very important to the learning

objectives of this book. I wrote each problem statement while at the same time preparing

the solution, in order to help make the statements as clear, correct, and unambiguous as

possible. In many cases, I stated a result, such as a Laplace transform, in a chapter’s text

but left as a homework problem the proof or other development of that result. When

teaching a lesson from the course notes that grew into this book, I would often not lecture

on the material of the reading that I had assigned for the lesson. Instead, I would assume

that the students had, in fact, completed and understood the assigned reading, summarize

the main results of that reading and ask if there were questions about it, then, after responding

to any questions, spend most remaining lesson time discussing some of the related

homework problems.

A major focus of this book is computer calculation of system characteristics and

responses and display of the results graphically, with use of MATLAB commands and

programs. However, the book employs, for the most part, basic MATLAB commands

and operations (aside from array operations), such as those on hand calculators; there is

very little use of advanced MATLAB operations and functions, because these can produce

results without the user having to understand the processes of production. For this

introductory material, I think it is important that the computer and software function as a

“super calculator”, which relieves the user of the drudgery of calculations, especially

complex and/or repetitive calculations, but still requires the user to understand the process

well enough to be able to design and program the calculations and graphical displays.

Since 1967 with the publication of Dynamics of Physical Systems by Robert H.

Cannon, Jr., most textbooks on introductory system dynamics have included very few, if

any, applications specifically relevant to aerospace engineering. Therefore, I have tried

to include in this book at least some relevant examples and homework problems. These

include rolling dynamics of flight vehicles in Chapter 3, spacecraft actuators in Chapters

3, 10, and 12, aerospace motion sensors in Chapters 9 and 10, aeroelasticity in Chapters

11 and 12, attitude control of spacecraft and aircraft in Chapters 14-16, and an analysis of

aeroelastic flutter in the final homework problem of Chapter 16.

I favor illustrating and validating theory, whenever possible, with measured data.

I also favor using measured data to identify system dynamic characteristics based on

mathematical models (e.g., time constants and natural frequencies), and system basic

properties (e.g., mass, stiffness, and damping). Accordingly, I included quite a lot of material

in this book about identification of first- and second-order systems, especially in

Chapters 9 and 10. Photographs of instructional laboratory structures and motion data

measured from those structures are included, for examples, in Section 7-6 on distributedparameter

structures, and in homework Problems 7.10 and 12.5. Other applications of

real and simulated experimental data appear in several homework problems.

I welcome feedback about this book from anyone who reads it. Please send your

comments to my VPI & SU email address, whallaue@vt.edu. I will be grateful to learn

of any errors that readers detect and report to me. I retain all of the source wordprocessor

files, so I am able to correct errors and replace any defective file with the corrected

version. I regard the basic organization of the book as fixed, so that, except to correct

major, serious errors, I will not revise the chapters and appendices so extensively as

to disrupt the original page numbering, equation numbering, Table of Contents, and Index.

I am ready and willing, however, to add files that supplement chapter and appendix

contents, when such additions will improve the book. In particular, I would welcome

new examples and homework problems that are clearly relevant to aerospace engineering,

while still being compatible with the introductory level of the book. If you send to me

any such educational and motivational gem and if I decide that it satisfies my criteria,

then I will be most pleased to add it as a supplementary file and to acknowledge your

contribution.

## TABLE OF CONTENTS

Section Title Page

Chapter1

Introduction; examples of 1st and 2nd order systems; example

analysis and MATLAB1 graphing

1-1 Introduction 1-1

1-2 Linear, time-invariant (LTI) systems and ordinary differential equations

(ODEs)

1-1

1-3 The mass-damper system: example of 1st order, linear, time-invariant

(LTI) system and ordinary differential equation (ODE)

1-3

1-4 A short discussion of engineering models 1-5

1-5 The mass-damper system (continued): example of solving the 1st order,

LTI ODE for time-history response, given a pulse excitation and an

initial condition (IC)

1-6

1-6 The mass-damper system (continued): numerical/graphical evaluation

of time-history response using MATLAB

1-9

1-7 Some notes regarding good engineering graphical practice, with

reference to Figure 1-2

1-10

1-8 Plausibility checks of system response equations and calculations 1-11

1-9 The mass-damper-spring system: example of 2nd order LTI system and

ODE

1-12

1-10 The mass-damper-spring system: example of solving a 2nd order LTI

ODE for time response

1-13

1-11 Homework problems for Chapter 1 1-18

1 MATLAB ® is a registered trademark of The MathWorks, Inc.

TC-1

Table of Contents

Section Title Page

Chapter

2

Complex numbers and arithmetic; Laplace transforms;

partial-fraction expansion

2-1 Review of complex numbers and arithmetic 2-1

2-2 Introduction to application of Laplace transforms 2-5

2-3 More about partial-fraction expansion 2-11

2-4 Additional useful functions and Laplace transforms: step, sine, cosine,

definite integral

2-12

2-5 Homework problems for Chapter 2 2-15

Chapter

3

Mechanical units; low-order mechanical systems; simple

transient responses of 1st order systems

3-1 Common mechanical units 3-1

3-2 Calculation of mass from measured weight 3-2

3-3 Reaction wheel: a rotational 1st order system 3-3

3-4 Simple transient responses of 1st order systems, 1st order time constant

and settling time

3-4

3-5 Aileron-induced rolling of an airplane or missile 3-8

3-6 Translational spring and viscous damper (dashpot) 3-11

3-7 More examples of damped mechanical systems 3-13

3-8 Homework problems for Chapter 3 3-17

TC- 2

Table of Contents

Section Title Page

Chapter

4

Frequency response of 1st order systems; transfer function;

general method for derivation of frequency response

4-1 Definition of frequency response 4-1

4-2 Response of a 1st order system to a suddenly applied cosine, cosω t 4-1

4-3 Frequency response of the 1st order damper-spring system 4-4

4-4 Period, frequency, and phase of periodic signals 4-8

4-5 Easy derivation of the complex frequency-response function for

standard stable 1st order systems

4-10

4-6 Transfer function, general definition 4-11

4-7 Frequency-response function from transfer function, general derivation 4-12

4-8 Homework problems for Chapter 4 4-15

Chapter 5 Basic electrical components and circuits

5-1 Introduction 5-1

5-2 Passive components: resistor, capacitor, and inductor 5-1

5-3 Operational amplifier (op-amp) and op-amp circuits 5-9

5-4 RC band-pass filter 5-12

5-5 Homework problems for Chapter 5 5-13

Chapter

6

General time response of 1st order systems by application of

the convolution integral

6-1 The convolution transform and its inverse, the convolution integral 6-1

TC- 3

Table of Contents

Section Title Page

6-2 General solution of the standard stable 1st order ODE + IC by

application of the convolution integral

6-2

6-3 Examples of 1st order system response 6-3

6-4 General solution of the standard 1st order problem: an alternate

derivation

6-6

6-5 Numerical algorithm for the general solution of the standard 1st order

problem

6-7

6-6 Homework problems for Chapter 6 6-11

Chapter

7

Undamped 2nd order systems: general time response;

undamped vibration

7-1 Standard form for undamped 2nd order systems; natural frequency ω

n 7-1

7-2 General solution for output x(t) of undamped 2nd order systems 7-3

7-3 Simple IC response and step response of undamped 2nd order systems 7-4

7-4 Discussion of the physical applicability of step-response solutions 7-6

7-5 Dynamic motion of a mechanical system relative to a non-trivial static

equilibrium position; dynamic free-body diagram

7-7

7-6 Introduction to vibrations of distributed-parameter systems 7-9

7-7 Homework problems for Chapter 7 7-17

Chapter

8

Pulse inputs; Dirac delta function; impulse response; initialvalue

theorem; convolution sum

8-1 Flat pulse 8-1

TC- 4

Table of Contents

Section Title Page

8-2 Impulse-momentum theorem for a mass particle translating in one

direction

8-2

8-3 Flat impulse 8-3

8-4 Dirac delta function, ideal impulse 8-3

8-5 Ideal impulse response of a standard stable 1st order system 8-5

8-6 Derivation of the initial-value theorem 8-7

8-7 Ideal impulse response of an undamped 2nd order system 8-9

8-8 Ideal impulse response vs. real response of systems 8-10

8-9 Unit-step-response function and unit-impulse-response function (IRF) 8-12

8-10 The convolution integral as a superposition of ideal impulse responses 8-14

8-11 Approximate numerical solutions for 1st and 2nd order LTI systems

based on the convolution sum

8-15

8-12 Homework problems for Chapter 8 8-28

Chapter 9 Damped 2nd order systems: general time response

9-1 Homogeneous solutions for damped 2nd order systems; viscous

damping ratio ζ

9-1

9-2 Standard form of ODE for damped 2nd order systems 9-3

9-3 General solution for output x(t) of underdamped 2nd order systems 9-6

9-4 Initial-condition transient response of underdamped 2nd order systems 9-8

9-5 Calculation of viscous damping ratio ζ from free-vibration response 9-9

9-6 Step response of underdamped 2nd order systems 9-12

9-7 Ideal impulse response of underdamped 2nd order systems 9-13

TC- 5

Table of Contents

Section Title Page

9-8 Step-response specifications for underdamped systems 9-14

9-9 Identification of a mass-damper-spring system from measured response

to a short force pulse

9-18

9-10 Deriving response equations for overdamped 2nd order systems 9-20

9-11 Homework problems for Chapter 9 9-23

Chapter

10

2nd order systems: frequency response; beating response to

suddenly applied sinusoidal (SAS) excitation

10-1 Frequency response of undamped 2nd order systems; resonance 10-1

10-2 Frequency response of damped 2nd order systems 10-2

10-3 Frequency response of mass-damper-spring systems, and system

identification by sinusoidal vibration testing

10-7

10-4 Frequency-response function of an RC band-pass filter 10-10

10-5 Common frequency-response functions for electrical and mechanicalstructural

systems

10-11

10-6 Beating response of 2nd order systems to suddenly applied sinusoidal

excitation

10-14

10-7 Homework problems for Chapter 10 10-19

Chapter

11

Mechanical systems with rigid-body plane translation and

rotation

11-1 Equations of motion for a rigid body in general plane motion 11-1

11-2 Equation of motion for a rigid body in pure plane rotation 11-3

11-3 Examples of equations of motion for rigid bodies in plane motion 11-5

TC- 6

Table of Contents

Section Title Page

11-4 Homework problems for Chapter 11 11-13

Chapter

12

Vibration modes of undamped mechanical systems with two

degrees of freedom

12-1 Introduction: undamped mass-spring system 12-1

12-2 Undamped two-mass-two-spring system 12-2

12-3 Vibration modes of an undamped 2-DOF typical-section model of a

wing

12-9

12-4 Homework problems for Chapter 12 12-13

Chapter

13

Laplace block diagrams, and additional background material

for the study of feedback-control systems

13-1 Laplace block diagrams for an RC band-pass filter 13-1

13-2 Laplace block diagram with feedback branches for an m-c-k system

with base excitation

13-2

13-3 Forced response of an m-c-k system with base excitation 13-4

13-4 Homework problems for Chapter 13 13-7

Chapter

14

Introduction to feedback control: output operations for

control of rotational position

14-1 Initial definitions and terminology 14-1

14-2 More definitions, and examples of open-loop control systems 14-2

14-3 Closed-loop control of rotor position using position feedback 14-6

TC- 7

Table of Contents

Section Title Page

14-4 Transfer function of a single closed loop 14-9

14-5 Closed-loop control of rotor position using position feedback plus rate

feedback

14-11

14-6 Comments regarding classical control theory and modern control theory 14-17

14-7 Homework problems for Chapter 14 14-18

Chapter

15

Input-error operations: proportional, integral, and derivative

types of control

15-1 Initial definitions; proportional-integral-derivative (PID) control 15-1

15-2 Examples of proportional (P) and proportional-integral (PI) control 15-2

15-3 Derivation of the final-value theorem 15-12

15-4 Example of proportional-derivative (PD) control 15-13

15-5 Homework problems for Chapter 15 15-16

Chapter

16

Introduction to system stability: time-response criteria

16-1 General time-response stability criterion for linear, time-invariant

systems

16-1

16-2 Stable and unstable PD-controlled-rotor systems 16-6

16-3 Routh’s stability criteria 16-12

16-4 Loci of roots for 2nd order systems 16-18

16-5 Loci of roots for 3rd order systems 16-22

16-6 Open-loop transfer functions and loci of roots 16-31

TC- 8

Table of Contents

Section Title Page

16-7 Homework problems for Chapter 16 16-34

Chapter

17

Introduction to system stability: frequency-response criteria

17-1 Gain margins, phase margins, and Bode diagrams 17-1

17-2 Nyquist plots 17-9

17-3 The practical effects of an open-loop transfer-function pole at s = 0 +

j0

17-12

17-4 The Nyquist stability criterion 17-15

17-5 Homework problems for Chapter 17 17-23

References Refs-1

Appendix A: Table and derivations of Laplace transform pairs

A-1 Table of Laplace transform pairs used in this book A-1

A-2 Laplace transform of a ratio of two polynomials, with only simple

poles

A-4

A-3 Derivation of the Laplace transform of a definite integral A-5

A-4 Applications of the Laplace transform of a definite integral A-6

A-5 Derivation of the Laplace transform of the convolution integral A-7

TC- 9

Table of Contents

TC1- 0

Section Title Page

Appendix B: Notes on work, energy, and power in mechanical

systems and electrical circuits

B-1 Definitions of work and power B-1

B-2 Mechanical work, energy, and power (complementary to Chapter 3) B-2

B-3 Work, energy, and power in electrical circuits (complementary to

Chapter 5)

B-4

B-4 Analogies between an m-c-k mechanical system and an LRC electrical

circuit

B-8

B-5 Hysteresis and dissipation of mechanical energy by damping B-9

Index for all Chapters and Appendices Index-1