Systems dynamics for mechanical engineers

Matthew A. Davies · Tony L. Schmitz

System

Dynamics for

Mechanical

Engineers

System Dynamics for Mechanical

Engineers

Matthew A. Davies • Tony L. Schmitz

System Dynamics

for Mechanical

Engineers

Matthew A. Davies

University of North Carolina at Charlotte

Charlotte, NC, USA

Tony L. Schmitz

University of North Carolina at Charlotte

Charlotte, NC, USA

ISBN 978-1-4614-9292-4 ISBN 978-1-4614-9293-1 (eBook)

DOI 10.1007/978-1-4614-9293-1

Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2014947522

# Springer Science+Business Media New York 2015

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To our Lord and Savior, Jesus Christ

Preface

In this textbook, we describe the fundamentals of system dynamics using Laplace

transform techniques and frequency domain approaches as the primary analytical

tools. It is aimed at the mechanical engineering student and, therefore, begins with a

thorough discussion of the modeling of mechanical systems to provide the backdrop

for the entire text. Once the fundamentals of mechanical system behavior are devel￾oped, the topic is broadened to include electrical, electromechanical, and thermal

systems. Wherever possible, analogies between the less familiar systems and their

mechanical counterparts are drawn upon to help clarify the subject matter. The topics

in the book are concluded with a discussion of block diagrams, feedback control

systems, and frequency response of dynamic systems including an introduction to

vibrations. Example computational techniques using MATLAB® are incorporated

throughout the text. The book is based upon undergraduate courses in system

dynamics and mechanical vibrations that the authors currently teach. It is designed

to be used in either a traditional 15-week semester or two quarters spanning 3–

4 months. It is appropriate for undergraduate engineering students who have

completed the basic courses in mathematics (through differential equations) and

physics and the introductory mechanical engineering courses including statics and

dynamics.

We organized the book into 11 chapters. The chapter topics are summarized here.

• Chapter 1—This chapter defines the concept of a dynamic system as it is

commonly used in engineering. It gives examples of such systems and, in a

broad sense, describes the importance of system dynamics in engineering. To

prepare the reader for Chap. 2, it also links the idea of a system model to the

mathematical concept of a differential equation.

• Chapter 2—This chapter describes the Laplace transform, the primary analysis

and solution technique used in this book, and supporting topics.

• Chapter 3—This chapter introduces the fundamental lumped parameter elements

used to model mechanical systems. These include translational, rotational, and

transmission elements.

• Chapter 4—This chapter introduces modeling of a mechanical system with

translation mechanical elements using the undamped and damped simple

harmonic oscillator. The models are solved for common inputs. The concepts

vii

of transfer function, characteristic equation, natural frequency, and damping

ratio are introduced.

• Chapter 5—This chapter extends the concepts in Chap. 4 to include models with

rotational degrees of freedom.

• Chapter 6—This chapter analyzes dynamic systems with transmission elements

and includes the associated geometric and power constraints.

• Chapter 7—This chapter examines electrical circuits composed of resistors,

capacitors, and inductors. The mathematical analogies between electrical and

mechanical elements are discussed.

• Chapter 8—This chapter discusses electromechanical systems including electric

motors and other electromagnetic actuators including voice coils. This discus￾sion further emphasizes the mathematical analogies between mechanical and

electrical elements.

• Chapter 9—This chapter describes bulk heat transfer showing the analogies

between mechanical, electrical, and thermal elements. It also provides an intro￾duction to proportional-integral-derivative feedback control in the context of a

temperature control system.

• Chapter 10—This chapter condenses the book concepts into the formal language

of block diagrams. Feedback and control systems are discussed in more detail.

• Chapter 11—This chapter describes the behavior of dynamic systems subjected

to sinusoidal and other periodic inputs. It is a precursor to a mechanical

vibrations course.

The text is written with the mechanical engineer in mind. This includes the

organization, selection of examples, and range of topics. It will provide the engi￾neering student not only with sound fundamentals, but also with the confidence to

address new, multidisciplinary systems that are found in practice. It will equip the

engineer with techniques to analyze the dynamics of modern systems.

We conclude by acknowledging the many contributors to this text. These

naturally include our instructors, colleagues, collaborators, and students.

Charlotte, NC, USA Matthew A. Davies

Charlotte, NC, USA Tony L. Schmitz

viii Preface

Contents

1 Introduction .......................................... 1

1.1 What Is a System? ................................. 1

1.2 System Boundaries ................................. 2

1.3 Modeling and Analysis Tools . . . ...................... 5

1.4 Continuous Time Motions Versus Dynamic

“Snapshots” . ..................................... 7

1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Laplace Transform Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Definition of the Laplace Transform . . . . . . . . . . . . . . . . . . . . 14

2.3 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Laplace Transforms of Common Functions . . . . . . . . . . . . . . . 22

2.6 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . . . . 29

2.6.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6.2 Laplace Transform of a Time-Delayed Function . . . . . . 29

2.6.3 Laplace Transform of a Time Derivative . . . . . . . . . . . . 31

2.6.4 Initial and Final Value Theorems . . . . . . . . . . . . . . . . . 33

2.7 Inverting Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7.1 Distinct Real Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7.2 Complex Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.7.3 Repeated Real Poles . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.7.4 Special Case That Often Occurs with Step Inputs

to Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.8 Using MATLAB® to Find Laplace and Inverse

Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.9 Solving Differential Equations Using Laplace Transforms . . . . 46

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Elements of Lumped Parameter Models . . . . . . . . . . . . . . . . . . . . . 53

3.1 Introduction . . .................................... 53

3.2 Inertial Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

ix

3.3 Linear Spring Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Linear Damping Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 Combinations of Springs and Dampers . . . . . . . . . . . . . . . . . . . 62

3.6 Transmission Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.6.1 Levers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6.2 Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.6.3 Rack and Pinion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Transient Rectilinear Motion of Mechanical Systems . . . . . . . . . . . 77

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2.1 Initial Velocity Only (x(0) = 0, x_(0) = v0, F(t) = 0) . . . . 79

4.2.2 Impulse Input (x(0) = 0, x_(0) = 0, F(t) = P0
δ(t)) . . . . . 80

4.2.3 Step Input (x(0) = 0, x_(0) = 0, F(t) = F0
u(t)) . . . . . . . . 81

4.3 Damped Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4 The Effect of Gravitational Loads . . . . . . . . . . . . . . . . . . . . . . 92

4.5 Transfer Functions and the Characteristic Equation . . . . . . . . . . 97

4.6 Multiple Degree of Freedom Systems . . . . . . . . . . . . . . . . . . . 108

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5 Transient Rotational Motion of Mechanical Systems . . . . . . . . . . . 123

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2 The Simple Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2.1 Simple Undamped Pendulum with Initial Velocity

Only (θ(0) ¼ 0, _

θ(0) ¼ ω0, M(t) ¼ 0) . . . . . . . . . . . . . . . 126

5.2.2 Simple Damped Pendulum with Initial Velocity

(θ(0) ¼ 0, _

θ(0) ¼ ω0, M(t) ¼ 0) . . . . . . . . . . . . . . . . . . . 132

5.2.3 Simple Damped Pendulum with Step Input

(θ(0) ¼ 0, _

θ (0) ¼ 0, M(t) ¼ MO
u(t)) . . . . . . . . . . . . . . 137

5.3 Pendulum-Like Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.4 Rotational Drive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.4.1 No Input Angle (θin ¼ 0) . . . . . . . . . . . . . . . . . . . . . . . 147

5.4.2 Nonzero Input Angle (θin 6¼ 0) . . . . . . . . . . . . . . . . . . . 150

5.5 Multiple Degree of Freedom Rotational Systems . . . . . . . . . . . 155

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6 Combined Rectilinear and Rotational Motions:

Transmission Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.2 System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.3 Systems with Transmission Elements . . . . . . . . . . . . . . . . . . . . 167

x Contents

6.4 Levers in Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.5 Gears in Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.6 Other Transmission Elements . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.7 Higher Degree of Freedom Systems

and Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

7 Electric Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.2 Electrical Element Input/Output Relationships . . . . . . . . . . . . . 206

7.3 Impedances in Series and Parallel . . . . . . . . . . . . . . . . . . . . . . 210

7.4 Kirchhoff’s Laws for Circuit Analysis . . . . . . . . . . . . . . . . . . . 211

7.5 Differential Equation Methods . . . . . . . . . . . . . . . . . . . . . . . . . 212

7.6 Impedance Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7.7 Operational Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

7.7.1 Inverting Op-Amp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

7.7.2 Noninverting Op-Amp Configuration . . . . . . . . . . . . . . 240

7.7.3 Other Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . 244

7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

8 Electromechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

8.2 Permanent Magnet Direct Current Motors . . . . . . . . . . . . . . . . 253

8.2.1 Motor Transfer Function . . . . . . . . . . . . . . . . . . . . . . . 256

8.2.2 Dynamic Motor Response . . . . . . . . . . . . . . . . . . . . . . 258

8.2.3 Motor Time Constants and Approximate

First-Order Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 262

8.2.4 Steady State Motor Behavior in Response

to External Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

8.3 Electric Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

8.4 Other Electromechanical Devices: Acoustic

Speaker/Voice Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

9 Thermal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

9.2 Basic Lumped Parameter Thermal Elements . . . . . . . . . . . . . . 290

9.2.1 Thermal Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

9.2.2 Thermal Resistance: Heat Exchange

with the Environment . . . . . . . . . . . . . . . . . . . . . . . . . . 291

9.3 Thermal Mass Subject to a Constant Heat Input . . . . . . . . . . . . 292

Contents xi

9.4 Thermal Mass Subjected to Heat Exchange

with the Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

9.5 Thermal Mass Subjected to Heat Exchange

with the Environment and Power Input . . . . . . . . . . . . . . . . . . 301

9.6 A Proportional Integral Derivative (PID) Thermal

Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

9.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

10 Block Diagrams and Introduction to Control Systems . . . . . . . . . . 315

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

10.2 Block Diagram Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

10.3 Feedback Loops with Proportional Gain . . . . . . . . . . . . . . . . . . 316

10.4 Feedback Loops with Proportional, Integral,

and Derivative Gains (PID Control) . . . . . . . . . . . . . . . . . . . . . 327

10.5 Block Diagram Representation of a Permanent

Magnet DC Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

10.6 Application of Block Diagrams to Servomotor Control . . . . . . . 337

10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

11 Frequency Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

11.2 Response of Spring-Mass System to a Periodic Input . . . . . . . . 348

11.3 Frequency Response Functions . . . . . . . . . . . . . . . . . . . . . . . . 353

11.4 Properties of the Frequency Response Function . . . . . . . . . . . . 365

11.5 Multiple Degree-of-Freedom Systems . . . . . . . . . . . . . . . . . . . 369

11.6 Tuned-Mass Absorber Example . . . . . . . . . . . . . . . . . . . . . . . . 373

11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

xii Contents

Introduction 1

1.1 What Is a System?

The word system has a broad modern definition. The Merriam-Webster dictionary

defines a system as “a regularly interacting or interdependent group of items

forming a unified whole.” For the engineer, a system consists of a combination of

elements which, acting together, perform a specific task. An input to a system

causes the system to exhibit a response which is observed as changes in the system

output. All of the systems we discuss in this book are causal: the input, or cause,

results in the output, or effect. Additionally, causality requires that the output

depends only on current and previous input values. Future inputs do not affect the

current output.

Systems are comprised of collections of elements that affect each other. Each

element in a system has its own input/output relationship. For example, when an

input force is applied to a linear spring, an output deflection that is proportional to

the force is obtained. Similarly, if an input voltage is applied across a resistor, an

output current flows through the resistor. System elements such as springs and

resistors defined in this way are static because the output depends on the input only

at the current time. A dynamic element produces an output that depends not only on

current inputs, but also on the previous inputs (i.e., the input history). For example,

consider a mass with a force input and a position output. The position of the mass

depends not only on the current force value, but also on previous force values. In

our analyses, this information is incorporated into the initial position and velocity of

the mass. The input/output relationship for an entire system is developed by

analyzing the interactions between all of the system elements. The output of a

static system depends only on the inputs at the current time. The output of a dynamic

system depends on the inputs and their history.

# Springer Science+Business Media New York 2015

M.A. Davies, T.L. Schmitz, System Dynamics for Mechanical Engineers,

DOI 10.1007/978-1-4614-9293-1_1

1

1.2 System Boundaries

An example of a complex system is shown in Fig. 1.1a. This is a multi-axis ultra￾precision machine tool used for manufacturing optics (lenses) and other components

with an accuracy of better than 1 μm (one-millionth of a meter1

). The machine adjusts

the position of rotating tool using five angular and translational degrees of freedom

as shown in Fig. 1.1b, and the rotating tool removes material (Fig. 1.1c) to produce

an optical component with the commanded shape. The tool is positioned relative to

the work material using five machine axes labeled X, Y, Z, B, and C. The X, Y, and Z

axes produce linear motions, while the B and C axes produce rotary motions. With

these five degrees of freedom, the machine can position the tool at an arbitrary

location and an arbitrary angle within the limitation of its work volume.

The machine itself (Fig. 1.1a) is a system shown schematically in Fig. 1.2. At

this high level, the input to the machine is a computer program that is uploaded into

the machine through its computer-based user interface. This “part program”

contains instructions that define the position of the cutting tool at each time during

Fig. 1.1 Ultra-precision machine tool used for manufacturing optical components

1 To provide a sense of scale, the diameter of a human hair is approximately 100 μm.

2 1 Introduction