Table Of Content
Principles of
Engineering Mechanics
Volume 2
Dynamics- The Analysis ofMotion
MATHEMATICAL CONCEPTSAND METHODS
IN SCIENCEAND ENGINEERING
Series Editor: Angelo Miele
Mechanical Engineering andMathematicalSciences
Rice University
Latestvolumesinthisseries:
PRINCIPLESOFENGINEERINGMECHANICS:
VOLUME2.DYNAMICS-THEANALYSISOFMOTION· MillardF. Beatty. ir.
CONSTRAINEDOPTIMIZATIONANDIMAGESPACEANALYSIS:
VOLUME 1.SEPARATIONOFSETSANDOPTIMALITY
CONDITIONS • FrancoGiannessi
ADVANCEDDESIGNPROBLEMSINAEROSPACEENGINEERING:
VOLUME 1.ADVANCEDAEROSPACESYSTEMS• EditorsAngeloMiele
andAidaFrediani
UNIFIEDPLASTICITYFORENGINEERINGAPPLICATIONS·SolR.Bodner
THEORYANDAPPLICATIONSOFPARTIALDIFFERENTIAL
EQUATIONS• PieroBassaniniandAlanR.Elcrat
NONLINEAREFFECTSINFLUIDSANDSOLIDS• EditorsMichaelM.Carroll
andMichaelA.Hayes
Principles of
Engineering Mechanics
Volume 2
Dynamics- The Analysis of Motion
Millard F. Beatty,Jr.
ProfessorEmeritus,DepartmentofEngineeringMechanics
UniversityofNebraska-Lincoln.Lincoln.NE
~
Springer
MillardF.Beatty,Jr.,PhD
ProfessorEmeritus,Department ofEngineeringMechanics
UniversityofNebraska-Lincoln
Lincoln,NE68588-0526
E-mail:[email protected]
LibraryofCongressControlNumber:20059350I0
ISBN-lO:0-387-23704-6
ISBN-13:9780387237046
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Preface
This volume, which begins withChapter5,isa vectortreatmentofthe principles
ofmechanicswrittenprimarilyforadvancedundergraduateandfirstyeargraduate
students specializing in engineering.A substantial part of the material, however,
exclusiveofobviousadvancedtopics,hasbeentailoredtoprovideafirstcoursefor
sophomore orjuniorlevel undergraduates having two years of university mathe
maticsthrough differential equations,which maybetakenconcurrently. Thebook
also serves as a sound, self-study review for advanced students specializing in
mechanics, engineeringscience,engineeringphysics,andapplied mathematics. It
isprerequisite, of course, that the reader be familiar with vector and elementary
matrix methods, the primitive concepts of mechanics,and the basic kinematical
equations developed in Volume I,which comprises the firstfour chapters of this
two volume series.In addition, it is recommended that advanced studentsbe fa
miliar with the elements of tensor algebra and finite rigid body rotations treated
there inChapter3.
The arrangement of the subject into the separate parts kinematics and dy
namics has always seemed to me the best didactic approach, and it still does.
Unfortunately,theluxury ofaslow-paced twosemestercourse onthesesubjectsis
seldom afforded.Therefore,inaconcentrated,fast-paced single semestercourse,
especiallyforadiverse group ofundergraduates,aleveland varietyoftopics may
bechosen for presentationto address studentneeds and tosatisfy curriculum ob
jectives.Butagreatdealofthesubject matteralsocanbeassigned firsttoreading
and later discussed briefly in lectures. Assigned reading of my discourse on the
foundation principles of classical mechanics, for example, might offer an oppor
tunityforaninterestingdialogue among students andtheinstructor.Atthebottom
line,however,theclassroomusageofthismaterial mustbeassessed bytheteacher
inkeeping withtheclasslevel,thecourse objectives, thecurriculumschedule, and
thedesired emphasis.
Naturally, the presentation isinfluenced by my personal interests and back
ground in mechanical engineering, engineering science, and mechanics. Conse
quently, my approach definitely is somewhat more sophisticated and mathemat
ical than iscommonly found in traditional textbooks on engineering mechanics.
vii
viii Preface
In keeping with this approach, the prerequisite mathematics, principally that of
the seventeenth and eighteenthcenturies, is used withoutapology.Nevertheless,
aware that many readers may not have mastered these prerequisite materials, I
haveexercisedcaretoreinforcetheessentialtoolsbothdirectlywithin thetextand
indirectlyintheillustrations,exercisesand problemsselectedforstudy.Elements
of the theory of linear differential equations of the second order, for example,
arepresented;hyperbolicfunctions areintroducedandillustratedgraphically;the
analyticgeometryofconic sectionsisreviewed; and advanced studentswillfinda
thoughtful discussion on the subjectofelliptic functions and integrals applied in
several examples.Theseand othertopics areillustratedinmanyphysicalapplica
tions throughout the book; and several exercises and problems expand on related
details. Instances where material may be omitted without loss of continuity are
clearlyindicated.
This presentation ofdynamics focuses steadily on illustrating the predictive
value ofthemethods andprinciplesofmechanicsindescribingandexplainingthe
motion and physicalbehaviorofparticles andrigid bodies under various kinds of
forces and torques.The emphasis throughout this book is principally analytical,
notcomputational,thoughmanynumericalexamplesandproblemsareprovidedto
exposetherelevantaspectsofanexampleortoillustratecertaintheoreticaldetails.
Exampleshavebeen selectedfortheirinterestandinstructivevalueandtohelpthe
student achieve understanding of the various concepts,principles, and analytical
methods. In some instances, experimentalresults, factual situations, and applica
tionsthatconfirmanalyticalpredictionsaredescribed.Ihopethehistorical remarks
and several relevant short stories add interestandreality tothesubject. Numerous
assignmentproblems,ranging from easy, straightforwardextensionsorreinforce
ments of the subject matterto more difficult problemsthat challengethe creative
skills ofbetterstudents, areprovidedatthe end ofeach chapter.Toassist the stu
dent'sstudy ofdynamics,answers totheodd-numberedproblemsareprovidedat
the back ofthe book.
Itisaxiomaticthatphysicalintuitionorinsightcannotbetaught. Ontheother
hand,competence in mathematical and physical reasoning may be developed so
that these special human qualitiesmay beintelligentlycultivatedthrough study of
physicalapplications that mirror the world around us and throughpractice ofthe
rational process of reasoning from firstprinciples.With these attributes in mind,
one objective of this volume is to help the engineering student develop confi
dence intransformingproblemsintoappropriatemathematicallanguagethat may
be manipulated to derive substantive and useful physical conclusions or specific
numerical results. I intend that this treatment should present a penetrating and
interesting treatise on the elements of classical mechanics and their applications
to engineering problems; therefore, this text is designed to deepen and broaden
thestudent'sunderstandingandtodevelop hisorhermastery ofthefundamentals.
However,toreapaharvestfromtheseedssownhere,itisimportantthatthestudent
workthrough manyoftheexamplesandproblemsprovidedforstudy.When teach
ingthis courseoverthe years,Ihaveusuallyassignedsixproblemsperweek, two
Preface ix
perclassperiod,roughly ninety persemester.Iftheassignmentproblemschosen
rangefromeasytodifficult,Idon'tthinkitiswisetoassignmore.Themereunder
standingthatonemayapplytheoretical conceptsandformulastosolveaparticular
problem is not equivalent to possession of the knowledge and skills required to
produce itssolution.These talents grow only from experience in dealing repeat
edly with these matters.My viewof the importanceofsolvingmany problems is
expressedfurtheratthebeginningoftheproblem setforChapter5,thefirstchapter
ofthisvolume.The attitude emphasized there isechoed throughoutthistext.It is
my hope that these books on kinematics and dynamics may provide engineering
students and others with solid mathematical and mechanical foundations for fu
tureadvanced studyoftopicsinmechanicaldesignanalysis,advanced kinematics
ofmechanismsand analyticaldynamics,mechanical vibrationsandcontrols,and
areasofcontinuummechanicsofsolidsand fluids.
The ContentsofThis Volume
Volume 2: Dynamics, concerns the analysis of motion based on classical
foundation principlesofmechanicsduetoNewtonandEuler,latercastinagener
alized energy basedformulation byLagrange.Afreshdevelopmentofaclassical
subjectlikethisisseldom seen.Ibelieve,however,thatthereader willfindwithin
these pages many fresh developments, beginning with my discussion of the fun
damental classical principlesof mechanics and concluding withthe presentation
oftheelegantanalytical formulation ofthe Lagrange equationsofmotion.And in
between, thereaderwillfindhere arefreshingly different, consistent,logical, and
gradual development and careful application of all of the classical principles of
mechanicsforaparticle,asystemofparticles,arigidbody,actually anynumberof
particlesand rigid bodies,subjected tovariouskindsofforcesand torques.Some
unusual,hopefully interesting thingstolookforaresketchedbelow.
Chapter 5: The Foundation Principles of Classical Mechanics, presents a
detailed studyandfreshdiscussion of Newton's lawsof motion, Newton'stheory
of gravitation,the role of inertial reference frames,and Newton's second law of
motion relativeto the moving Earth frame.Further, two fascinating applications
ofCoulomb'slawsoffriction having engineeringsignificance are presented. The
chapterconcludes with anadvanced topic borrowed from continuum mechanics,
hereappliedtodeducethegenerallawofmutualinteractionbetweentwoparticles.
This interesting topic hasnotpreviously appearedinanytextbook.
The kinematical equations characterizing the velocity and acceleration of a
particle intermsofrectangular,intrinsic,cylindrical, andsphericalcoordinatesare
applied inChapter6: Dynamicsof aParticle,inthe formulation ofthe equations
of motion ina varietyofspecialapplications.Besidesgravitational and frictional
forces, additional electromagnetic, elastic spring, and drag force laws are intro
duced and illustrated in some atypical examples that include the mechanics of
x Preface
an inkjetprinterand Millikan's famous oil drop experiment. Ofcourse, familiar
elementary problems, sometimes having a novel twist, appear here as well.The
elements of projectile motion with examples that include drag force effects and
the fanciful exploits of Percy Panther and Arnold Aardvark surely will attract at
tention. Several additional distinctive examplesinclude anonlinearoscillator, the
finite nonlinear radial oscillation of a particle of an incompressible rubber tube,
and problems of motion relative to the moving Earth frame that include Reich's
experimentonthedeflection offalling pellets andFoucault'sexperimentdemon
stratingtherotation oftheEarth.Theprojectiledeflection problem issolved;and,
with this in mind, the historic tragic battle oftheFalkland Islands isdescribed.
Thefirstintegralprinciples ofmechanicsforaparticle-impulse-momentum,
torque-impulse, work-energy,and the conservation laws of momentum, moment
of momentum,andenergy-aredeveloped in Chapter7: Momentum, Work, and
Energy.Alongwithseveraltypicalapplications,someadvancedtopicstreatedthere
include an introduction to elliptic functions and integrals in the problem of the
finiteamplitudeoscillationsofapendulum.The isochronouscycloidal pendulum
and Huygens's clock, and orbital motion and Kepler's laws, are other interesting
examples.
The classical principles of mechanics and first integrals for a particle are
extendedtostudyinChapter8:Dynamics ofaSystemofParticles.Theimportance
of the centerof mass ofthe system, firstmentionedinChapter5,isnowevident.
The two body problem correcting Kepler's third law is a distinctive illustration.
Inpreparationforlater workonrigid bodies, themomentofmomentum principle
with respect toa moving reference point,firstderived inChapter6for aparticle,
and thereduction toitssimplestform, iscarefullydescribed.Severalexamples of
simple systems ofparticles aredemonstrated.
Theelementsoftensoralgebra,includingtheCartesian tensortransformation
law,introducedin Volume1,Chapter3,are exploited inChapter9:The Moment
ofInertia Tensor.Heretheinertia tensor and itsgeneral propertiesarestudied and
applied tocomplexstructured(composite)bodies.Thecenterofmassconceptfor
asimple rigid body,firststudied inChapter5,isextended tocompositebodies as
well.Thetensorialformoftheparallelaxistheoremisproved,anditsinterpretation
isillustratedinexamples. The principal (extremal) valuesoftheinertiatensor and
their directionsarederivedbythe method ofLagrange multipliers; and theireasy
geometrical interpretation interms of Cauchy's momental ellipsoid isdescribed.
These topics playa key role in the general theory of rigid body motion. Some
advancedanalyticaltopicsontheprincipalvaluesofgeneralsymmetrictensorsand
themultiplicityofprincipalvaluesandcorrespondingprincipal axesareexplored
attheend.
Chapter 10: Dynamics of a Rigid Body, the most difficult of the subjects
treated in this book, focuses on Euler's generalization of Newton's principles of
mechanics, nowextendedtocharacterizetheequations ofmotionforallbodies
solids, fluids,gasses-buthereinrestrictedprincipally tothemotionsofrigidsolid
bodies. Euler's laws are set down separately and discussed in detail, somewhat
