A Concise Introduction to Mechanics of Rigid Bodies: Multidisciplinary Engineering

197 Pages·2017·3.988 MB·English

by  L. Huang (auth.)

L. Huang
A Concise
Introduction
to Mechanics
of Rigid Bodies
Multidisciplinary Engineering
Second Edition
A Concise Introduction to Mechanics of
Rigid Bodies
L. Huang
A Concise Introduction to
Mechanics of Rigid Bodies
Multidisciplinary Engineering
Second Edition
123
L.Huang
SchoolofEngineering,
ComputerandMathematicalSciences
AucklandUniversityofTechnology
Auckland,NewZealand
ISBN978-3-319-45040-7 ISBN978-3-319-45041-4 (eBook)
DOI10.1007/978-3-319-45041-4
LibraryofCongressControlNumber:2016951927
©SpringerInternationalPublishingSwitzerland2012,2017
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ForXin,Zhongjia,and Zhongjin
Preface to the Second Edition
Inthisneweditionofthebook,mainlythepartsonrotationalmotionsareenhanced.
Several methods of rotation representation including Euler angles, exponential
coordinates, and quaternions are added, and the Rodrigues formula is introduced
to describe equivalent/effectiveaxis–angle representation. Accordingly,kinematic
analysis based on those representations has been added. Errors and omissions
identifiedinthefirsteditionhavebeenrectifiedinthisnewedition.
Thoughthebooknowcontainsmorepages,itretainsthecomprehensivenessin
coverageandcompactnessinsizewelcomedbyreadersfromavarietyofareaswho
want to study rigid body mechanics but have usually found the relevant material
scatteredamongmultiplevolumesintraditionaltextbooks.
Auckland,NewZealand L.Huang
vii
Preface to the First Edition
Mechanics is a mature engineering subject. Why do we need another mechanics
book?
This book presents rigid-body mechanics in a compact form for a multidis-
ciplinary engineering program. Typically, such programs include mechatronics,
which covers mechanical engineering, electrical and electronic engineering, and
computer engineering, and biomechanics, which is related to human physiology,
sports, and engineering mechanics. In terms of course structure, they have the
followingfeatures:
• Awiderangeofsubjectsinvariousareasistaught.
• Limitedteachingtimeisallocatedtoeachsubject.
• Thereisasharptransitionfromfundamentaltospecializedsubjects.
Mechanics is one of the most important subjects in traditional mechanical
engineeringprograms.Itusuallytakestwosemesterstocoveralltherelevanttopics,
arranged in the following order: mechanics of particle motion (one dimension),
planar motion (two dimensions), and rigid body motion in space. In comparison,
thetimeallottedtotheteachingofmechanicsismuchshorterinamultidisciplinary
engineering program. It is hard to find a concise presentation of the material that
coverstheessentialprinciplesofmechanicsandthatcanbetaughtinalimitedtime.
Many existing textbooksfocusalmost exclusivelyon the mechanicsof particle
and planar motions, which are easily visualized and can be readily analyzed
with geometrical and graphicalmethods. On the other hand, multibody dynamics
in three-dimensional space is the cornerstone of many specialized courses in
multidisciplinaryengineeringprograms(e.g.,robotics).Thereisahugegapbetween
what is covered in traditional textbooks on mechanics and the requirements for
specialized courses (e.g. [7] and [8]) on the motions of rigid bodies and complex
systemsinthree-dimensionalspace.Thusabookisneededthatnarrowsthegapby
focusingonthemechanicsofrigidbodies,withparticlemotionandplanarmotion
asspecialcases.
Themainmotivationforwritingthisbookwastoproduceaworkthatiscompact
butcomprehensiveinitscoverageoftheessentialprinciplesofrigid-bodydynamics.
Toachievethis,thefollowingapproacheswereadoptedinthebook:
ix
x PrefacetotheFirstEdition
• Thethree-dimensionaldynamicsofarigidbodyisdealtwithfromthebeginning
of the book to the end. Mechanics in relation to particle motion and planar
motionaretreatedasspecialcases.Thisisincontrasttothetraditional,sequential
coverageoftopics:particlemotion!motioninaline!planarmotion!three-
dimensionalmotion.
• Matrix and vector manipulationsare used extensivelyto make the presentation
concise and clear. This is in contrast to the traditional approach, which is
dominatedbyscalarandvectormanipulations.Thisisfineforlow-dimensional
motionbuttooclumsyfordescribingmotionsandmasspropertieswithrespect
to different reference frames. Matrix and vector notations can greatly simplify
mathematicalexpressionsandrevealtheessentialprinciplesgoverningmotions
in high dimensions; this is also in line with the general approach taken in
advanced specialized courses like roboticsand mechanisms. In this regard,the
reader is expected to be familiar with concepts and manipulations of vectors
andmatrices,whicharecoveredinundergraduatecourseslikelinearalgebraor
engineering mathematics. Chapter 1 of this book is designed for readers who
mightneedtoreviewthebasicsintherelevantareas.
• Theconceptsofobservationframeanddescriptionframeareintroducedandare
reflected by a set of new vector notations to define kinematic variables more
clearly.
The main parts of the book concern the description and analysis of position,
velocity and acceleration, inertial properties, and the establishment of equations
of motion throughthree main methods:Newton–Eulerformulation,D’Alembert’s
principle, and Lagrange equations. Some examples are taken from existing books
onengineeringmechanicsandbiomechanics.Theyareaddressedwiththemethod-
ologiesandapproachespresentedinthebook,andeachsteptoreachthesolutionis
painstakinglyandrigorouslyexplained.Therearenoquestionsetsforpracticeafter
each chapter since rich sourcesof questionscan be foundin manyexisting books
onmechanics(e.g.[3]and[5]).
Thoughsome importanttopicslike the impulse-momentummethod,collisions,
vibrations,andforwarddynamics(derivationofarigidbody’smotionfromforces
ortorques)arenotcoveredduetospaceconstraints,whatispresentedinthebook
should providea sufficientfoundationfor dealing with those topics. Statics is not
coveredhereeithersinceitcanbetreatedasaspecialcaseofdynamics.
Asanoteofclarification,dynamicsinthisbookmeansthestudyoftherelation
between changesin motion and the causes of those changes (forces and torques).
Insome books,thisis called kinetics,and dynamicsincludesboth kinematicsand
kinetics.
The book comprises the following chapters. Chapter 1 covers preliminary
mathematicalknowledgeneededtostudythebook.Itfocusesonvectorandmatrix
operations which has been discussed in [1] and [2]. Chapter 2 is on kinematics,
Chap.3isondynamics,andChap.4presentscasestudies.
PrefacetotheFirstEdition xi
Thebookissuitableforundergraduatesandpostgraduatestudentsinmultidisci-
plinaryengineeringprogramslikemechatronicsandbiomechanics.Itisalsosuitable
forstudentsinmechanicalengineeringandengineersandresearchersinterestedin
rigid-bodydynamics.
Auckland,NewZealand L.Huang
Contents
1 PreliminariesonVectors,Matrices,ComplexNumbers
andQuaternions ............................................................. 1
1.1 Vectors................................................................... 1
1.1.1 Definition........................................................ 1
1.1.2 Operations....................................................... 3
1.2 Matrices ................................................................. 9
1.2.1 Definition........................................................ 9
1.2.2 Operations....................................................... 11
1.3 ComplexNumbers...................................................... 19
1.3.1 Definition........................................................ 19
1.3.2 Operations....................................................... 20
1.4 Quaternions.............................................................. 22
1.4.1 Definition........................................................ 22
1.4.2 Operations....................................................... 22
1.4.3 SomeDefinitionsandRelations................................ 23
1.5 SpecialVectors,Matrices,andTerms.................................. 25
2 OrientationandPositionRepresentation.................................. 27
2.1 CoordinateFrames...................................................... 27
2.2 ObservationFrame,DescriptionFrame,andVectorNotation........ 30
2.3 Orientation............................................................... 33
2.3.1 RotationMatrix[2,8]........................................... 33
2.3.2 Equivalent/EffectiveAxisandAngle.......................... 35
2.3.3 ExponentialCoordinates ....................................... 39
2.3.4 Active/PassiveInterpretationsofRotationMatrix
andOrientationfromSuccessiveRotations ................... 40
2.3.5 EulerAngles .................................................... 42
2.3.6 RPYAngles..................................................... 43
2.3.7 QuaternionofRotation......................................... 44
2.3.8 Cayley–KleinMatrix ........................................... 46
2.4 Position.................................................................. 48
2.4.1 PositionofaRigidBodyandPositionofaPoint ............. 48
2.4.2 PassiveandActiveRepresentationofPosition................ 49
2.5 Examples................................................................ 50
xiii