# INTRODUCTION TO LINEAR TIME INVARIANT DYNAMIC SYSTEMS FOR STUDENTS OF ENGINEERING William L. Hallauer

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## 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
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
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
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,
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.
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
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.

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
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
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
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
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
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
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
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
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
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