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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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