The magnitude of a vector is simply its length; hence, in a graphical representation as in Figure 1.5.1, the magnitude is simply the geometrical length.. Observe, for example, in Figure
Trang 2MECHANICAL SYSTEMS
Trang 4C RC PR E S S
Boca Raton London New York Washington, D.C
Harold Josephs Ronald L HustonMECHANICAL SYSTEMS
Trang 5This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials
or for the consequences of their use.
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Visit the CRC Press Web site at www.crcpress.com
© 2002 by CRC Press LLC
No claim to original U.S Government works International Standard Book Number 0-8493-0593-4 Library of Congress Card Number 2002276809 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0
Printed on acid-free paper
Library of Congress Cataloging-in-Publication Data
Josephs, Harold.
Dynamics of mechanical systems / by Harold Josephs and Ronald L Huston.
p ; cm.
Includes bibliographical references and index.
ISBN 0-8493-0593-4 (alk paper)
1 Mechanical engineering I Huston, Ronald L., 1937- II Title.
TJ145 J67 2002.
CIP
Trang 6This is a textbook intended for mid- to upper-level undergraduate students in engineeringand physics The objective of the book is to give readers a working knowledge of dynamics,enabling them to analyze mechanical systems ranging from elementary and fundamentalsystems such as planar mechanisms to more advanced systems such as robots, spacemechanisms, and human body models The emphasis of the book is upon the fundamentalprocedures underlying these dynamic analyses Readers are expected to obtain skillsranging from the ability to perform insightful hand analyses to the ability to developalgorithms for numerical/computer analyses In this latter regard, the book is alsointended to serve as an independent study text and as a reference book for beginninggraduate students and for practicing engineers
Mechanical systems are becoming increasingly sophisticated, with applications ing greater precision, improved reliability, and extended life These enhanced requirementsare spurred by a demand for advanced land, air, and space vehicles; by a correspondingdemand for advanced mechanisms, manipulators, and robotics systems; and by a need
requir-to have a better understanding of the dynamics of biosystems The book is intended requir-toenable its readers to make engineering advances in each of these areas The authors believethat the skills needed to make such advances are best obtained by illustratively studyingfundamental mechanical components such as pendulums, gears, cams, and mechanismswhile reviewing the principles of vibrations, stability, and balancing The study of thesesubjects is facilitated by a knowledge of kinematics and skill in the use of Newton’s laws,energy methods, Lagrange’s equations, and Kane’s equations The book is intended toprovide a means for mastering all of these concepts
The book is written to be readily accessible to students and readers having a background
in elementary physics, mathematics through calculus and differential equations, and mentary mechanics The book itself is divided into 20 chapters, with the first two chaptersproviding introductory remarks and a review of vector algebra The next three chaptersare devoted to kinematics, with the last of these focusing upon planar kinematics Chapter
ele-6 discusses forces and force systems, and Chapter 7 provides a comprehensive review ofinertia including inertia dyadics and procedures for obtaining the principal moments ofinertia and the corresponding principal axes of inertia
Fundamental principles of dynamics (Newton’s laws and d’Alembert’s principle) arepresented in Chapter 8, and the use of impulse–momentum and work–energy principles
is presented in the next two chapters with application to accident reconstruction Chapters
11 and 12 introduce generalized dynamics and the use of Lagrange’s equation and Kane’sequations with application to multiple rod pendulum problems The next five chaptersare devoted to applications that involve the study of vibration, stability, balancing, cams,and gears, including procedures for studying nonlinear vibrations and engine balancing.The last three chapters present an introduction to multibody dynamics with application
to robotics and biosystems
Application and illustrative examples are discussed and presented in each chapter, andexercises and problems are provided at the end of each chapter In addition, each chapterhas its own list of references for additional study Although the earlier chapters providethe basis for the latter chapters, each chapter is written to be as self-contained as possible,with excerpts from earlier chapters provided as needed
Trang 8The book is an outgrowth of notes the authors have compiled over the past three decades
in teaching various courses using the subject material These notes, in turn, are basedupon information contained in various texts used in these courses and upon the authors’independent study and research
The authors acknowledge the inspiration for a clearly defined procedural study ofdynamics by Professor T R Kane at the University of Pennsylvania, now nearly 50 yearsago The authors particularly acknowledge the administrative support and assistance ofCharlotte Better in typing and preparing the entire text through several revisions Thework of Xiaobo Liu and Doug Provine for preparation of many of the figures is alsoacknowledged
Trang 10Ronald L Huston, Ph.D., P.E., is distinguished research professor and professor ofmechanics in the Department of Mechanical, Industrial, and Nuclear Engineering at theUniversity of Cincinnati He is also a Herman Schneider chair professor Dr Huston hasbeen at the University of Cincinnati since 1962 In 1978, he served as a visiting professor
at Stanford University, and from 1979 to 1980 he was division director of civil and ical engineering at the National Science Foundation From 1990 to 1996, Dr Huston was
mechan-a director of the Monmechan-arch Resemechan-arch Foundmechan-ation He is the mechan-author of over 140 journmechan-alarticles, 142 conference papers, 4 books, and 65 book reviews and is a technical editor of
Applied Mechanics Reviews, and book review editor of the International Journal of Industrial
reconstruction His research interests are in multibody dynamics, human factors, chanics, and ergonomics and safety Dr Huston received his B.S degree (1959), M.S degree(1961), and Ph.D (1962) from the University of Pennsylvania, Philadelphia He is aLicensed Professional Engineer and a Fellow of the American Society of MechanicalEngineers
Trang 12Chapter 1 Introduction 1
1.1 Approach to the Subject 1
1.2 Subject Matter 1
1.3 Fundamental Concepts and Assumptions 2
1.4 Basic Terminology in Mechanical Systems 3
1.5 Vector Review 5
1.6 Reference Frames and Coordinate Systems 6
1.7 Systems of Units 9
1.8 Closure 11
References 11
Problems 12
Chapter 2 Review of Vector Algebra 15
2.1 Introduction 15
2.2 Equality of Vectors, Fixed and Free Vectors 15
2.3 Vector Addition 16
2.4 Vector Components 19
2.5 Angle Between Two Vectors 23
2.6 Vector Multiplication: Scalar Product 23
2.7 Vector Multiplication: Vector Product 28
2.8 Vector Multiplication: Triple Products 33
2.9 Use of the Index Summation Convention 37
2.10 Review of Matrix Procedures 38
2.11 Reference Frames and Unit Vector Sets 41
2.12 Closure 44
References 44
Problems 45
Chapter 3 Kinematics of a Particle 57
3.1 Introduction 57
3.2 Vector Differentiation 57
3.3 Position, Velocity, and Acceleration 59
3.4 Relative Velocity and Relative Acceleration 61
3.5 Differentiation of Rotating Unit Vectors 63
3.6 Geometric Interpretation of Acceleration 66
3.7 Motion on a Circle 66
3.8 Motion in a Plane 68
3.9 Closure 71
References 71
Problems 71
Chapter 4 Kinematics of a Rigid Body 77
4.1 Introduction 77
4.2 Orientation of Rigid Bodies 77
Trang 134.3 Configuration Graphs 79
4.4 Simple Angular Velocity and Simple Angular Acceleration 83
4.5 General Angular Velocity 85
4.6 Differentiation in Different Reference Frames 87
4.7 Addition Theorem for Angular Velocity 90
4.8 Angular Acceleration 93
4.9 Relative Velocity and Relative Acceleration of Two Points on a Rigid Body 97
4.10 Points Moving on a Rigid Body 103
4.11 Rolling Bodies 106
4.12 The Rolling Disk and Rolling Wheel 107
4.13 A Conical Thrust Bearing 110
4.14 Closure 113
References 113
Problems 114
Chapter 5 Planar Motion of Rigid Bodies — Methods of Analysis 125
5.1 Introduction 125
5.2 Coordinates, Constraints, Degrees of Freedom 125
5.3 Planar Motion of a Rigid Body 128
5.3.1 Translation 129
5.3.2 Rotation 130
5.3.3 General Plane Motion 130
5.4 Instant Center, Points of Zero Velocity 133
5.5 Illustrative Example: A Four-Bar Linkage 136
5.6 Chains of Bodies 142
5.7 Instant Center, Analytical Considerations 147
5.8 Instant Center of Zero Acceleration 150
Problems 156
Chapter 6 Forces and Force Systems 163
6.1 Introduction 163
6.2 Forces and Moments 163
6.3 Systems of Forces 165
6.4 Zero Force Systems 170
6.5 Couples 170
6.6 Wrenches 173
6.7 Physical Forces: Applied (Active) Forces 177
6.7.1 Gravitational Forces 177
6.7.2 Spring Forces 178
6.7.3 Contact Forces 180
6.7.4 Action–Reaction 181
6.8 First Moments 182
6.9 Physical Forces: Inertia (Passive) Forces 184
References 187
Problems 187
Chapter 7 Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 199
7.1 Introduction 199
7.2 Second-Moment Vectors 199
Trang 147.3 Moments and Products of Inertia 200
7.4 Inertia Dyadics 203
7.5 Transformation Rules 205
7.6 Parallel Axis Theorems 206
7.7 Principal Axes, Principal Moments of Inertia: Concepts 208
7.8 Principal Axes, Principal Moments of Inertia: Example 211
7.9 Principal Axes, Principal Moments of Inertia: Discussion 215
7.10 Maximum and Minimum Moments and Products of Inertia 223
7.11 Inertia Ellipsoid 228
7.12 Application: Inertia Torques 228
References 230
Problems 230
Chapter 8 Principles of Dynamics: Newton’s Laws and d’Alembert’s Principle 241
8.1 Introduction 241
8.2 Principles of Dynamics 242
8.3 d’Alembert’s Principle 243
8.4 The Simple Pendulum 245
8.5 A Smooth Particle Moving Inside a Vertical Rotating Tube 246
8.6 Inertia Forces on a Rigid Body 249
8.7 Projectile Motion 251
8.8 A Rotating Circular Disk 253
8.9 The Rod Pendulum 255
8.10 Double-Rod Pendulum 258
8.11 The Triple-Rod and N-Rod Pendulums 260
8.12 A Rotating Pinned Rod 263
8.13 The Rolling Circular Disk 267
8.14 Closure 270
References 270
Problems 271
Chapter 9 Principles of Impulse and Momentum 279
9.1 Introduction 279
9.2 Impulse 279
9.3 Linear Momentum 280
9.4 Angular Momentum 282
9.5 Principle of Linear Impulse and Momentum 285
9.6 Principle of Angular Impulse and Momentum 288
9.7 Conservation of Momentum Principles 294
9.8 Examples 295
9.9 Additional Examples: Conservation of Momentum 301
9.10 Impact: Coefficient of Restitution 303
9.11 Oblique Impact 306
9.12 Seizure of a Spinning, Diagonally Supported, Square Plate 309
9.13 Closure 310
Problems 311
Chapter 10 Introduction to Energy Methods 321
10.1 Introduction 321
10.2 Work 321
10.3 Work Done by aCouple 326
Trang 1510.4 Power 327
10.5 Kinetic Energy 327
10.6 Work–Energy Principles 329
10.7 Elementary Example: A Falling Object 332
10.8 Elementary Example: The Simple Pendulum 333
10.9 Elementary Example — A Mass–Spring System 336
10.10 Skidding Vehicle Speeds: Accident Reconstruction Analysis 338
10.11 A Wheel Rolling Over a Step 341
10.12 The Spinning Diagonally Supported Square Plate 342
10.13 Closure 344
References (Accident Reconstruction) 344
Problems 344
Chapter 11 Generalized Dynamics: Kinematics and Kinetics 353
11.1 Introduction 353
11.2 Coordinates, Constraints, and Degrees of Freedom 353
11.3 Holonomic and Nonholonomic Constraints 357
11.4 Vector Functions, Partial Velocity, and Partial Angular Velocity 359
11.5 Generalized Forces: Applied (Active) Forces 363
11.6 Generalized Forces: Gravity and Spring Forces 367
11.7 Example: Spring-Supported Particles in a Rotating Tube 369
11.8 Forces That Do Not Contribute to the Generalized Forces 375
11.9 Generalized Forces: Inertia (Passive) Forces 377
11.10 Examples 379
11.11 Potential Energy 389
11.12 Use of Kinetic Energy toObtain Generalized Inertia Forces 394
11.13 Closure 401
References 401
Problems 402
Chapter 12 Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 415
12.1 Introduction 415
12.2 Kane’s Equations 415
12.3 Lagrange’s Equations 423
12.4 The Triple-Rod Pendulum 429
12.5 The N-Rod Pendulum 433
12.6 Closure 435
References 436
Problems 436
Chapter 13 Introduction to Vibrations 439
13.1 Introduction 439
13.2 Solutions of Second-Order Differential Equations 439
13.3 The Undamped Linear Oscillator 444
13.4 Forced Vibration of an Undamped Oscillator 446
13.5 Damped Linear Oscillator 447
13.6 Forced Vibration of a Damped Linear Oscillator 449
13.7 Systems with Several Degrees of Freedom 450
13.8 Analysis and Discussion of Three-Particle Movement: Modes of Vibration 455
Trang 1613.9 Nonlinear Vibrations 458
13.10 The Method of Krylov and Bogoliuboff 463
13.11 Closure 466
References 466
Problems 467
Chapter 14 Stability 479
14.1 Introduction 479
14.2 Infinitesimal Stability 479
14.3 A Particle Moving in a Vertical Rotating Tube 482
14.4 A Freely Rotating Body 485
14.5 The Rolling/Pivoting Circular Disk 488
14.6 Pivoting Disk with a Concentrated Mass on the Rim 493
14.6.1 Rim Mass in the Uppermost Position 498
14.6.2 Rim Mass in the Lowermost Position 502
14.7 Discussion: Routh–Hurwitz Criteria 505
14.8 Closure 509
References 509
Problems 510
Chapter 15 Balancing 513
15.1 Introduction 513
15.2 Static Balancing 513
15.3 Dynamic Balancing: A Rotating Shaft 514
15.4 Dynamic Balancing: The General Case 516
15.5 Application: Balancing of Reciprocating Machines 520
15.6 Lanchester Balancing Mechanism 525
15.7 Balancing of Multicylinder Engines 526
15.8 Four-Stroke Cycle Engines 528
15.9 Balancing of Four-Cylinder Engines 529
15.10 Eight-Cylinder Engines: The Straight-Eight and the V-8 532
15.11 Closure 534
References 534
Problems 534
Chapter 16 Mechanical Components: Cams 539
16.1 Introduction 539
16.2 A Survey of Cam Pair Types 540
16.3 Nomenclature and Terminology for Typical Rotating Radial Cams with Translating Followers 541
16.4 Graphical Constructions: The Follower Rise Function 543
16.5 Graphical Constructions: Cam Profiles 544
16.6 Graphical Construction: Effects of Cam–Follower Design 545
16.7 Comments on Graphical Construction of Cam Profiles 549
16.8 Analytical Construction of Cam Profiles 550
16.9 Dwell and Linear Rise of the Follower 551
16.10 Use of Singularity Functions 553
16.11 Parabolic Rise Function 557
16.12 Sinusoidal Rise Function 560
16.13 Cycloidal Rise Function 563
16.14 Summary: Listing of Follower Rise Functions 566
Trang 1716.15 Closure 568
References 568
Problems 569
Chapter 17 Mechanical Components: Gears 573
17.1 Introduction 573
17.2 Preliminary and Fundamental Concepts: Rolling Wheels 573
17.3 Preliminary and Fundamental Concepts: Conjugate Action 575
17.4 Preliminary and Fundamental Concepts: Involute Curve Geometry 578
17.5 Spur Gear Nomenclature 581
17.6 Kinematics of Meshing Involute Spur Gear Teeth 584
17.7 Kinetics of Meshing Involute Spur Gear Teeth 588
17.8 Sliding and Rubbing between Contacting Involute Spur Gear Teeth 589
17.9 Involute Rack 591
17.10 Gear Drives and Gear Trains 592
17.11 Helical, Bevel, Spiral Bevel, and Worm Gears 595
17.12 Helical Gears 595
17.13 Bevel Gears 596
17.14 Hypoid and Worm Gears 597
17.15 Closure 599
17.16 Glossary of Gearing Terms 599
References 601
Problems 602
Chapter 18 Introduction to Multibody Dynamics 605
18.1 Introduction 605
18.2 Connection Configuration: Lower Body Arrays 605
18.3 A Pair of Typical Adjoining Bodies: Transformation Matrices 609
18.4 Transformation Matrix Derivatives 612
18.5 Euler Parameters 613
18.6 Rotation Dyadics 617
18.7 Transformation Matrices, Angular Velocity Components, and Euler Parameters 623
18.8 Degrees of Freedom, Coordinates, and Generalized Speeds 628
18.9 Transformations between Absolute and Relative Coordinates 632
18.10 Angular Velocity 635
18.11 Angular Acceleration 640
18.12 Joint and Mass Center Positions 643
18.13 Mass Center Velocities 645
18.14 Mass Center Accelerations 647
18.15 Kinetics: Applied (Active) Forces 647
18.16 Kinetics: Inertia (Passive) Forces 648
18.17 Multibody Dynamics 650
18.18 Closure 651
References 651
Problems 652
Chapter 19 Introduction to Robot Dynamics 661
19.1 Introduction 661
19.2 Geometry, Configuration, and Degrees of Freedom 661
19.3 Transformation Matrices and Configuration Graphs 663
Trang 1819.4 Angular Velocity of Robot Links 665
19.5 Partial Angular Velocities 667
19.6 Transformation Matrix Derivatives 668
19.7 Angular Acceleration of the Robot Links 668
19.8 Joint and Mass Center Position 669
19.9 Mass Center Velocities 671
19.10 Mass Center Partial Velocities 673
19.11 Mass Center Accelerations 673
19.12 End Effector Kinematics 674
19.13 Kinetics: Applied (Active) Forces 677
19.14 Kinetics: Passive (Inertia) Forces 680
19.15 Dynamics: Equations of Motion 681
19.16 Redundant Robots 682
19.17 Constraint Equations and Constraint Forces 684
19.18 Governing Equation Reduction and Solution: Use of Orthogonal Complement Arrays 687
19.19 Discussion, Concluding Remarks, and Closure 689
References 691
Problems 691
Chapter 20 Application with Biosystems, Human Body Dynamics 701
20.1 Introduction 701
20.2 Human Body Modeling 702
20.3 A Whole-Body Model: Preliminary Considerations 703
20.4 Kinematics: Coordinates 706
20.5 Kinematics: Velocities and Acceleration 709
20.6 Kinetics: Active Forces 715
20.7 Kinetics: Muscle and Joint Forces 716
20.8 Kinetics: Inertia Forces 719
20.9 Dynamics: Equations of Motion 721
20.10 Constrained Motion 722
20.11 Solutions of the Governing Equations 724
20.12 Discussion: Application and Future Development 727
References 730
Problems 731
Appendix I Centroid and Mass Center Location for Commonly Shaped Bodies with Uniform Mass Distribution 735
Appendix II Inertia Properties (Moments and Products of Inertia) for Commonly Shaped Bodies with Uniform Mass Distribution 743
Index 753
Trang 201
Introduction
This book presents an introduction to the dynamics of mechanical systems; it is basedupon the principles of elementary mechanics Although the book is intended to be self-contained, with minimal prerequisites, readers are assumed to have a working knowledge
of fundamental mechanics’ principles and a familiarity with vector and matrix methods.The readers are also assumed to have knowledge of elementary physics and calculus Inthis introductory chapter, we will review some basic assumptions and axioms and otherpreliminary considerations We will also begin a review of vector methods, which we willcontinue and expand in Chapter 2
Our procedure throughout the book will be to develop a general methodology which
we will then simplify and specialize to topics of interest We will attempt to illustratethe concepts through examples and exercise problems The reader is encouraged to solve
as many problems as possible Indeed, it is our belief that a basic understanding of theconcepts and an intuitive grasp of the subject are best obtained through solving theexercise problems
Dynamics is a subject in the general field of mechanics, which in turn is a discipline ofclassical physics Mechanics can be divided into two divisions: solid mechanics and fluidmechanics Solid mechanics may be further divided into flexible mechanics and rigidmechanics Flexible mechanics includes such subjects as strength of materials, elasticity,viscoelasticity, plasticity, and continuum mechanics Alternatively, aside from statics,dynamics is the essence of rigid mechanics Figure 1.2.1 contains a chart showing thesesubjects and their relations to one another
is concerned primarily with the analysis of forces and force systems and the determination
of equilibrium configurations In contrast, dynamics is a study of the behavior of movingrigid body systems As seen in Figure 1.2.1, dynamics may be subdivided into three sub-subjects: kinematics, inertia, and kinetics
Trang 212 Dynamics of Mechanical Systems
regard to the cause of the motion matics includes an analysis of the posi-
Kine-t i o n s , d i s p l a c e m e n Kine-t s , Kine-t r a j e c Kine-t o r i e s ,velocities, and accelerations of the mem-bers of the system. Inertia is a study of themass properties of the bodies of a systemand of the system as a whole in variousconfigurations. Kinetics is a study of forces
Forces are generally divided into twoclasses: applied (or “active”) forces and iner- tia (or “passive”) forces Applied forcesarise from contact between bodies andfrom gravity; inertia forces occur due to themotion of the system
The study of dynamics is based upon several fundamental concepts and basic assumptionsthat are intuitive and based upon common experience: time, space, force, and mass Time
is a measure of change or a measure of a process of events; in dynamics, time is assumed
to be a continually increasing, non-negative quantity Space is a geometric region whereevents occur; in the study of dynamics, space is usually defined by reference frames orcoordinate systems Force is intuitively described as a push or a pull The effect of a forcedepends upon the magnitude, direction, and point of application of the push or pull; aforce is thus ideally suited for representation by a vector. Mass is a measure of inertiarepresenting a resistance to change in motion; mass is the source of gravitational attractionand thus also the source of weight forces
In our study we will assume the existence of an inertial reference frame, which is simply
a reference frame where Newton’s laws are valid More specifically, we will assume theEarth to be an inertial reference frame for the range of systems and problems considered
in this book
Newton’s laws may be briefly stated as follows:
1 In the absence of applied forces, a particle at rest remains at rest and a particle
in motion remains in motion, moving at a constant speed along a straight line
2 A particle subjected to an applied force will accelerate in the direction of theforce, and the acceleration will be proportional to the magnitude of the forceand inversely proportional to the mass of the particle Analytically, this may beexpressed as
(1.3.1)where F is the force (a vector), m is the particle mass, and a is the resultingacceleration (also a vector)
3 Within a mechanical system, interactive forces occur in pairs with equal tudes but opposite directions (the law of action and reaction)
magni-F= ma
FIGURE 1.2.1
Subdivisions of mechanics.
Trang 22Introduction 3
Particular terminology is associated with dynamics, and specifically with mechanicalsystem dynamics, which we will use in the text We will attempt to define the terms as
we need them, but it might also be helpful to mention some of them here:
purposes, dynamic events will occur
locating the points of a space Typical reference frames employ Cartesian axessystems
in the description of its motion and of its response to forces applied to it “Small”
is, of course, a relative term A body considered as a particle may be small insome contexts but not in others (for example, an Earth satellite or an automobile).Particles are generally identified with points in space, and they generally havefinite masses
or constant, such as a sandstone The number of particles in a body is usuallyquite large A reference frame may be regarded, for kinematic purposes, as arigid body whose particles have zero masses
move The number of degrees of freedom possessed by a particle, body, or system
is defined as the number of geometric parameters (for example, coordinates,distances, or angles) needed to uniquely describe the location, orientation, and/
or configuration of the particle, body, or system
Con-straints can be either geometric (holonomic) or kinematic (nonholonomic)
trans-mitting, and/or changing forces and motion
The three general categories of machines are:
transmit motion between rotating shafts
objec-tives are to transmit motion between a rotating member and a nonrotatingmember The term “cam” is sometimes also used to describe a gear tooth
motion of a rigid body or the motion of a point of a body along a curve
• A link is a connective member of a machine or a mechanism A link maintains
a constant distance between two points of a mechanism, although links may beone way, such as cables
• A driver is an “input” link that stimulates a motion
Trang 234 Dynamics of Mechanical Systems
elements of a mechanism Two elements brought together by a joint are times called kinematic pairs Figure 1.4.1 shows a number of commonly used joints(or kinematic pairs)
some-• A kinematic chain is a series of links that are either joined together or are in contactwith one another A kinematic chain may contain one or more loops A loop is
a chain whose ends are connected An open chain (or “open tree”) contains noloops, a simple chain contains one loop, and a complex chain involves more thanone loop or one loop with open branches
Joint Name
Schematic Representation
Trang 24Introduction 5
Because vectors are used extensively in the text,
it is helpful to review a few of their fundamental
concepts We will expand this review in Chapter
2 Mathematically, a vector may be defined as an
element of a vector space (see, for example,
Ref-erences 1.1 to 1.3) For our purposes, we may think
of a vector simply as a directed line segment
Figure 1.5.1 shows some examples of vectors
as directed line segments In this context, vectors
are seen to have several characteristics:
magni-tude, orientation, and sense The magnitude of a
vector is simply its length; hence, in a graphical
representation as in Figure 1.5.1, the magnitude
is simply the geometrical length (Observe, for example, that vector 2B has a length andmagnitude twice that of vector B.) The orientation of a vector refers to its inclination inspace; this inclination is usually measured relative to some fixed coordinate system The
repre-sentation Observe, for example, in Figure 1.5.1 that vectors A and –A have opposite sense.The combined characteristics of orientation and sense are sometimes called the direction
of a vector
In this book, we will use vectors to represent forces, velocities, and accelerations Wewill also use them to locate points and to indicate directions The units of a vector arethose of its magnitude In turn, the units of the magnitude depend upon the quantity thevector represents For example, if a vector represents a force, its magnitude is measured
in force units such as Newtons (N) or pounds (lb) Alternatively, if a vector representsvelocity, its units might be meters per second (m/sec) or feet per second (ft/sec) Hence,vectors representing different quantities will have graphical representations with differentlength scales (A review of specific systems of units is presented in Section 1.7.)
Because vectors have the characteristics of magnitude and direction they are distinctfrom scalars, which are simply elements of a real or complex number system For example,the magnitude of a vector is a scalar; the direction of a vector is not a scalar To distinguishvectors from scalars, vectors are printed in bold-face type, such as V Also, because themagnitude of a vector is never negative (length is
never negative), absolute-value signs are used to
designate the magnitude, such as V.
In the next chapter, we will review algebraic
oper-ations of vectors, such as the addition and
multipli-cation of vectors In preparation for this, it is helpful
to review the concept of multiplication of vectors by
scalars Specifically, if a vector V is multiplied by a
scalar s, the product, written as sV, is a vector whose
magnitude is sV, where s is the absolute
value of the scalar s The direction of sV is the same
as that of V if s is positive and opposite that of V if
s is negative Figure 1.5.2 shows some examples of
products of scalars and vectors
-(3/2)V
Trang 256 Dynamics of Mechanical Systems
Two kinds of vectors occur so frequently that they deserve special attention: zero vectors
and unit vectors A zero vector is simply a vector with magnitude zero A unit vector is a
vector with magnitude one; unit vectors have no dimensions or units
Zero vectors are useful in equations involving vectors Unit vectors are useful for
separating the characteristics of vectors That is, every vector V may be expressed as the
product of a scalar and a unit vector In such a product, the scalar represents the magnitude
of the vector and the unit vector represents the direction Specifically, if V is written as:
(1.5.1)where s is a scalar and n is a unit vector, then s and n are:
(1.5.2)
This means that given any non-zero vector V we can always find a unit vector n with the
same direction as V; thus, n represents the direction of V
We can represent a reference frame by identifying it with a coordinate–axes system such
as a Cartesian coordinate system Specifically, we have three mutually perpendicular lines,
called axes, which intersect at a point O called the origin, as in Figure 1.6.1 The space is
then filled with “points” that are located relative to O by distances from O to P measured
along lines parallel to the axes These distances form sets of three numbers, called the
coordinates of the points Each point is then associated with its coordinates
The points in space may also be located relative to O by introducing additional lines
conveniently associated with the points together with the angles these lines make with
the mutually perpendicular axes The coordinates of the points may then involve these
angles
To illustrate these concepts, consider first the Cartesian coordinate system shown in
Figure 1.6.2, where the axes are called X, Y, and Z Let P be a typical point in space Then
the coordinates of P are the distances x, y, and z from P to the planes Y–Z, Z–X, and X–Y,
Trang 26Consider the task of locating P relative to the origin O by moving from O to P along lines parallel to X, Y, and Z, as shown in Figure 1.6.3 The coordinates may then be interpreted as the distances along these lines The distance d from O to P is then given by
the Pythagorean relation:
(1.6.1)
Finally, the point P is identified by either the name P or the set of three numbers (x, y, z).
To illustrate the use of additional lines and angles, consider the cylindrical coordinate
system shown in Figure 1.6.4 In this system, a typical point P is located relative to the origin O by the coordinates (r, θ, z) measuring: (1) the distance r along the newly introduced
radial line, (2) the inclination angle θ between the radial line and the X-axis, and (3) the distance z along the line parallel to the Z-axis, as shown in Figure 1.6.4.
By comparing Figures 1.6.3 and 1.6.4 we can readily obtain expressions relating Cartesianand cylindrical coordinates Specifically, we obtain the relations:
(1.6.2)
and
(1.6.3)
As a third illustration, consider the coordinate system shown in Figure 1.6.5 In this
case, a typical point P is located relative to the origin O by the coordinates (ρ, φ, θ)measuring the distance and angles as shown in Figure 1.6.5 Such a system is called a
spherical coordinate system.
θθ
θ
Trang 27By comparing Figures 1.6.3 and 1.6.5, we obtain the following relations between theCartesian and spherical coordinates:
(1.6.4)
and
(1.6.5)
The uses of vectors and coordinate systems are closely related To illustrate this, consider
again the Cartesian coordinate system shown in Figure 1.6.6 Let the unit vectors nx, ny,
and nz be parallel to the X-, Y-, and Z-axes, as shown Let p be a position vector locating
P relative to O (that is, p is OP) Then it is readily seen that p may be expressed as the
vector sum (see details in the next chapter):
ρφθ
Trang 281.7 Systems of Units
In this book, we will use both the English and the International unit systems On occasion,
we will want to make conversions between them Table 1.7.1 presents a listing of unitconversion factors for commonly occurring quantities in mechanical system dynamics
m/sec 2 ft/sec 2 3.280 840 × 10 0 m/sec 2 in./sec 2 3.937 008 × 10 1 m/sec 2 cm/sec 2 1.000 000 * × 10 2
in./sec 2 ft/sec 2 8.333 333 × 10 –2 in./sec 2 m/sec 2 2.540 000 * × 10 –2 in./sec 2 cm/sec 2 2.540 000 * × 10 0
cm/sec 2 ft/sec 2 3.280 840 × 10 –2 cm/sec 2 in./sec 2 3.937 008 × 10 –1 cm/sec 2 m/sec 2 1.000 000 * × 10 –2
Trang 29ft lb/sec Nm/sec (or W) 1.355 818 × 10 0
lb/in 2 (or psi) lb/ft 2 1.440 000 * × 10 2 lb/ft 2 N/m 2 (or Pa) 4.788 026 × 10 1 lb/ft 2 lb/in 2 (or psi) 6.944 444 × 10 –3 N/m 2 (or Pa) lb/in 2 (or psi) 1.450 377 × 10 –4 N/m 2 (or Pa) lb/ft 2 2.088 543 × 10 –2
ft/sec cm/sec 3.048 000 * × 10 1 ft/sec km/hr 1.097 280 * × 10 0 ft/sec in./sec 1.200 000 * × 10 1 ft/sec mi/hr (or mph) 6.818 182 × 10 –1 in./sec m/sec 2.540 000 * × 10 –2 in./sec cm/sec 2.540 000 * × 10 0 in./sec km/hr 9.144 000 * × 10 –2
Trang 30by providing exercises (or problems) for the reader These problems, appearing at the ends
of the chapters, are not intended to be burdensome but instead to serve as a learning aidfor the reader In addition, references will be provided for parallel study and for more in-depth study
TABLE 1.7.1 (CONTINUED)
Conversion Factors between English and International Unit Systems
To Convert Multiply by
in./sec mi/hr (or mph) 5.681 818 × 10 –2 mi/hr (or mph) m/sec 4.470 400 * × 10 –1 mi/hr (or mph) km/hr 1.609 344 × 10 0 mi/hr (or mph) ft/sec 1.466 667 × 10 0 mi/hr (or mph) in./sec 1.760 000 * × 10 1
Note: cm, centimeters; deg, degrees; ft, feet; g, grams; g, gravity acceleration (taken as 32.2 ft/sec2 ); HP, horsepower; in., inches; J, Joules; kg, kilograms; lb, pounds; m, meters; mi, miles; mph, miles per hour; N, Newtons; Pa, Pascals; psi, pounds per square inch; rad, radius; rpm, revolutions per minute; sec, seconds; W, watts.
* Exact by definition.
Trang 311.5 Hsu, H P., Vector Analysis, Simon & Schuster Technical Outlines, New York, 1969.
1.6 Brand, L., Vector and Tensor Analysis, Wiley, New York, 1964.
1.7 Kane, T R., Analytical Elements of Mechanics, Vol 2, Academic Press, New York, 1961 1.8 Kane, T R., and Levinson, D A., Dynamics: Theory and Applications, McGraw-Hill, New York,
1985, pp 361–371.
1.9 Likins, P W., Elements of Engineering Mechanics, McGraw-Hill, New York, 1973.
1.10 Beer, F P., and Johnston, E R., Jr., Vector Mechanics for Engineers, 6th ed., McGraw-Hill, New
York, 1996.
1.11 Yeh, H., and Abrams, J I., Principles of Mechanics of Solids and Fluids, Vol 1, Particle and Body Mechanics, McGraw-Hill, New York, 1960.
Rigid-1.12 Haug, E J., Intermediate Dynamics, Prentice Hall, Englewood Cliffs, NJ, 1992.
1.13 Meriam, J L., and Kraige, L G., Engineering Mechanics, Vol 2, Dynamics, 3rd ed., John Wiley
& Sons, New York, 1992.
1.14 Hibbler, R C., Engineering Mechanics: Statics and Dynamics, Macmillan, New York, 1974 1.15 Shelley, J P., Vector Mechanics for Engineers, Vol II, Dynamics, Schaum’s Solved Problems Series,
McGraw-Hill, New York, 1991.
1.16 Jong, I C., and Rogers, B G., Engineering Mechanics, Statics and Dynamics, Saunders College
Publishing, Holt, Rinehart & Winston, Philadelphia, PA, 1991.
1.17 Huston, R L., Multibody Dynamics, Butterworth–Heinemann, Stoneham, MA, 1990.
1.18 Higdon, A., and Stiles, W B., Engineering Mechanics, Vol II, Dynamics, 3rd ed., Prentice Hall,
Englewood Cliffs, NJ, 1968.
1.19 Meirovitch, L., Methods of Analytical Dynamics, McGraw-Hill, New York, 1970.
1.20 Paul, B., Kinematics and Dynamics of Planar Machinery, Prentice Hall, Englewood Cliffs, NJ, 1979 1.21 Sneck, H J., Machine Dynamics, Prentice Hall, Englewood Cliffs, NJ, 1991.
1.22 Mabie, H H., and Reinholtz, C F., Mechanisms and Dynamics of Machinery, 4th ed., Wiley, New
York, 1987.
1.23 Ginsberg, J H., Advanced Engineering Dynamics, Harper & Row, New York, 1988.
1.24 Shames, I H., Engineering Mechanics, Prentice Hall, Englewood Cliffs, NJ, 1980.
1.25 Liu, C Q., and Huston, R L., Formulas for Dynamic Analyses, Marcel Dekker, New York, 1999.
Problems
Section 1.5 Vector Review
P1.5.1: Suppose a velocity vector V is expressed in the form:
where i, j, and k are mutually perpendicular unit vectors.
a Determine the magnitude of V.
b Find a unit vector n parallel to V and having the same sense as V.
Section 1.6 Reference Frames and Coordinate Systems
P1.6.1: Suppose the Cartesian coordinates (x, y, z) of a point P are (3, 1, 4).
a Find the cylindrical coordinates (r, θ, z) of P.
b Find the spherical coordinates (ρ, φ, θ) of P
V= + +3i 4j 12 ft seck
Trang 32P1.6.2: Suppose the Cartesian coordinates (x, y, z) of a point P are (–1, 2, –5).
a Find the cylindrical coordinates (r, θ, z) of P.
b Find the spherical coordinates (ρ, φ, θ) of P.
P1.6.3: Suppose the cylindrical coordinates (r, θ, z) of a point P are (4, π/6, 2).
a Find the Cartesian coordinates (x, y, z) of P.
b Find the Spherical coordinates (ρ, φ, θ) of P.
P1.6.4: Suppose the spherical coordinates (ρ, φ, θ) of a point P are (7, π/4, π/3).
a Find the Cartesian coordinates (x, y, z) of P.
b Find the cylindrical coordinates (r, θ, z) of P.
P1.6.5: Consider the cylindrical coordinate system (r, θ, z) with z identically zero This
system then reduces to the “polar coordinate” system as shown in Figure P1.6.1
a Express the coordinates (x, y) in terms of (r, θ)
b Express the coordinates (r, θ) in terms of (x, y).
P1.6.6: See Problem 1.6.5 Let n x and n y be unit vectors parallel to the X- and Y-axes, as shown Let n r and nθ be unit vectors parallel and perpendicular to the radial line as shown
a Express n r and nθ in terms of n x and n y
b Express n x and n y in terms of n r and nθ
Section 1.7 Systems of Units
P1.7.1: An automobile A is traveling at 60 mph.
a Express the speed of A in ft/sec.
b Express the speed of A in km/sec.
Trang 33P1.7.2: A person weighs 150 lb.
a What is the person’s weight in N?
b What is the person’s mass in slug?
c What is the person’s mass in kg?
P1.7.3: An automobile A accelerates from a stop at 3 mph/sec.
a Express the acceleration in ft/sec2
b Express the acceleration in m/sec2
c Express the acceleration in g.
Trang 34Recall that the characteristics of a vector are its magnitude, its orientation, and its sense.Indeed, we could say that a vector is defined by its characteristics The concept of vectorequality follows from this definition: Specifically, two vectors are equal if (and only if)they have the same characteristics Vector equality is fundamental to the development ofvector algebra For example, if vectors are equal, they may be interchanged in vectorequations, which enables us to simplify expressions It should be noted, however, thatvector equality does not necessarily denote physical equality, particularly when the vec-tors model physical quantities This occurs, for example, with forces We will explore thisconcept later.
Two fundamental ideas useful in relating mathematical and physical quantities are theconcepts of fixed and free vectors A fixed vector has its location restricted to a line fixed
in space To illustrate this, consider the fixed line L as shown in Figure 2.2.1 Let v be avector whose location is restricted to L, and let the location of v along L be arbitrary Then
v is a fixed vector Because the location of v along L is arbitrary, v might even be called a
sliding vector.Alternatively, a free vector is a vector that may be placed anywhere in space if itscharacteristics are maintained Unit vectors such as nx, ny, and nz shown in Figure 2.2.1are examples of free vectors Most vectors in our analyses will be free vectors Indeed, wewill assume that vectors are free vectors unless otherwise stated
Trang 3516 Dynamics of Mechanical Systems
Vectors obey the parallelogram law of addition This is a simple geometric algorithm forunderstanding and exhibiting the powerful analytical utility of vectors To see this, con-sider two vectors A and B as in Figure 2.3.1 Let A and B be free vectors To add A and
B, let them be connected “head-to-tail” (without changing their characteristics) as in Figure2.3.2 That is, relocate B so that its tail is at the head of A Then, the sum of A and B, calledthe resultant, is the vector R connecting the tail of A to the head of B, as in Figure 2.3.3.That is,
(2.3.1)The vectors A and B are called the components of R
The reason for the name “parallelogram law” is that the same result is obtained if thehead of B is connected to the tail of A, as in Figure 2.3.4 The two ways of adding A and
B produce a parallelogram, as shown The order of the addition — that is, which vector
is taken first and which is taken second — is therefore unimportant; hence, vector addition
is commutative That is,
Trang 36Review of Vector Algebra 17
Vector subtraction may be defined from vector addition Specifically, the difference of
two vectors A and B, written as A – B, is simply the sum of A with the negative of B That is,
(2.3.3)
An item of interest in vector addition is the magnitude of the resultant, which may be
determined using the geometry of the parallelogram and the law of cosines For example,
in Figure 2.3.5, let θ be the angle between A and B, as shown Then, the magnitude of the
resultant R is given by:
(2.3.4)
Example 2.3.1: Resultant Magnitude
To illustrate the use of Eq (2.3.4), suppose the magnitude of A is 15 N, the magnitude of
B is 12 N, and the angle θ between A and B is 60˚ Then, the magnitude of the resultant R is:
(2.3.5)
Observe from Eq (2.3.4) that if we double the magnitude of both A and B, the magnitude
of the resultant R is also doubled Indeed, if we multiply A and B by any scalar s, R will
also be multiplied by s That is,
(2.3.6)
This means that vector addition is distributive with respect to scalar multiplication
Next, suppose we have three vectors A, B, and C, and suppose we wish to find their
resultant Suppose further that the vectors are not parallel to the same plane, as, for
example, in Figure 2.3.6 The resultant R is obtained in the same manner as before That
is, the vectors are connected head to tail, as depicted in Figure 2.3.7 Then, the resultant
R is obtained by connecting the tail of the first vector A to the head of the third vector C
as in Figure 2.3.7 That is,
Trang 3718 Dynamics of Mechanical Systems
Example 2.3.2: Resultant Magnitude in Three Dimensions
As an illustration, suppose C is perpendicular to the plane of vectors A and B of Example
2.3.1 and suppose the magnitude of C is 10 N Then, from the results of Eq (2.3.5), the
magnitude of R is:
(2.3.8)
This procedure has several remarkable features First, as before, the order of the
compo-nents in Eq (2.3.7) is unimportant That is,
(2.3.9)
Second, to obtain the resultant R we may first add any two of the vectors, say A and B,
and then add the resultant of this sum to C This means the summation in Eq (2.3.7) is
associative That is,
(2.3.10)
This feature may be used to obtain the magnitude of the resultant by repeated use of the
law of cosines as before
Third, observe that configurations exist where the magnitude of the resultant is less
than the magnitude of the individual components Indeed, the magnitude of the resultant
could be zero
Finally, with three or more components, the procedure of finding the magnitude of the
resultant by repeated use of the law of cosines is cumbersome and tedious
An attractive feature of vector addition is that the resultant magnitude may be obtained
by strictly analytical means — that is, without regard to triangle geometry This is the
original reason for using vectors in analysis We will explore this further in the next section
FIGURE 2.3.6
Vectors A, B, and C to be added.
FIGURE 2.3.7 Resultant of vectors A, B, and C of Figure 2.3.6.
Trang 382.4 Vector Components
Consider again Eq (2.3.7) where we have the vector sum:
(2.4.1)
Instead of thinking of this expression as a sum of components, consider it as a
represen-tation of the vector R Suppose further that the components A, B, and C happen to be
mutually perpendicular and parallel to coordinate axes, as shown in Figure 2.4.1 Then,
by the Pythagoras theorem, the magnitude of R is simply:
(2.4.2)
To develop these ideas still further, suppose that nx, ny, and nz are unit vectors parallel
to X, Y, and Z, as in Figure 2.4.2 Then, from our discussion in Chapter 1, we see that A,
B , and C can be expressed in the forms:
Trang 39Then, the magnitude of R is:
(2.4.5)
A question that arises is what if A, B, and C are not mutually perpendicular? How then
can we find the magnitude of the resultant? A powerful feature of the vector method isthat the same general procedure can be used regardless of the directions of the components
All that is required is to express the components as sums of vectors parallel to the X-, Y-, and Z-axes (or other convenient mutually perpendicular directions) For example, suppose
a vector A is inclined relative to the X-, Y-, and Z-axes, as in Figure 2.4.3 Let θx, θy, and
θz be the angles that A makes with the axes Next, let us express A in the desired form:
(2.4.6)
where Ax, Ay, and Az are vector components of A parallel to X, Y, and Z A x, Ay, and Az
may be considered as “projections” of A along the X-, Y-, and Z-axes Their magnitudes
are proportional to the magnitude of A and the cosines of the angles θx, θy, and θz That is,
(2.4.7)
where a x , a y , and a z are defined as given in the equations As before, let nx, ny, and nz be
unit vectors parallel to X, Y, and Z Then, A x, Ay, and Az can be expressed as:
θθθ
θθθ
A=a xnx+a yny+a znz=Acosθxnx+Acosθyny+Acosθznz
Trang 40Then, the magnitude of A is simply:
Example 2.4.1: Vector Addition Using Components
To illustrate the use of these ideas, consider again the vectors of Examples 2.3.1 and 2.3.2
(see Figures 2.3.6 and 2.3.7) Specifically, let A be parallel to the Y-axis with a magnitude
of 15 N Let B be parallel to the Y–Z plane and inclined at 60˚ relative to the Y-axis Let the magnitude of B be 12 N Let C be parallel to the X-axis with a magnitude of 10 N
(Figure 2.4.4) As before let nx, ny, and nz be unit vectors parallel to the X-, Y-, and Z-axes.
Then, A, B, and C may be expressed as:
Example 2.4.2: Direction Cosines
A particle P is observed to move on a curve C in a Cartesian reference frame R, as shown
in Figure 2.4.5 The coordinates of P are functions of time t Suppose that at an instant of interest x, y, and z have the values 8 m, 12 m, and 7 m, respectively Determine the orientation angles of the line of sight of an observer of P if the observer is at the origin
O Specifically, determine the angles θ, θ , and θ of OP with the X-, Y-, and Z-axes.
A=(a x2+ +a y2 a z2)1 2/
n=cosθxnx+cosθyny+cosθznz
1=cos2θx+cos2θy+cos2θz