Engineering Basic Mechanics Ⅱ Dynamics 工程基礎(chǔ)力學(xué) Ⅱ 動力學(xué) 課件 Chapter 2 Fundamental Kinematics of a Rigid Body

Chapter2FundamentalKinematicsofaRigidBody§2.1Translationalmotionofarigidbody§2.2Rotationofarigidbodyaboutafixedaxis§2.3Velocityandaccelerationofapointin

arigidbodyrotatingaboutafixedaxis

Maincontents1.

DefinitionThedirectionofthelinelinkingarbitrarytwopointsintherigidbodyneverchangesduringitsmotion.2.1Translationalmotionofarigidbody2.

FeaturesDifferentiate:Allparticlesinarigidbodywithtranslationalmotionhavethesametrajectories.Arbitrarytwoparticlesinarigidbodywithtranslationalmotionhavethesamevelocitiesandaccelerationsinthesameinstant.Themotionregularitiesofalltheparticlesinarigidbodywithtranslationalmotionarecompletelysame,sothetranslationofarigidbodycanbesimplifiedtothemotionofaparticleinit.2.1TranslationalmotionofarigidbodyExamplesoftranslationalmotion

2.1Translationalmotionofarigidbody1.

DefinitionTherearetwofixedpointsduringthemovementofarigidbody,itiscalledtherotationaboutafixedaxis.Theaxisisafixedlinethroughthetwofixedpoints.2.2Rotationofarigidbodyaboutafixedaxis2.FeaturesThedistancesbetweeneverypointintherigidbodyandthefixedaxisremainconstants.Everypointintherigidbodywhichisnotonthisaxismovesalongacircularpathinaplaneperpendiculartothisfixedaxis.3.Equationofrotationφ

isanalgebraicquantity.Signdefinitionofφ:followsright-handrule.φ

isthemonotropiccontinuousfunctionoftimet,whentherigidbodyrotates.

2.2Rotationofarigidbodyaboutafixedaxis4.Angularvelocityandangularaccelerationω

isanalgebraicquantitySigndefinitionofω:followsright-handrule.Unit：(1)Angularvelocity:

Inordertodescribethespeedanddirectionoftherotationofarigidbody,theangularvelocityisdefinedas,2.2Rotationofarigidbodyaboutafixedaxis4.Angularvelocityandangularacceleration(2)Angularacceleration:

Inordertodescribethespeedofangularvelocitychangedwithtimeoftherotationofarigidbody，theangularaccelerationisdefinedas,α

isanalgebraicquantitySigndefinitionofα:followsright-handrule.Unit：Arigidbodyhasacceleratedrotationwhenαand

havethesamesigns,deceleratedrotationwhenα

and

haveoppositesigns.

=const,uniformrotation.

α

=const,rotationwithconstantangularacceleration.2.2Rotationofarigidbodyaboutafixedaxis2.3Velocityandaccelerationofapointinarigidbodyrotatingaboutafixedaxis1.Themotionequationdefinedbycurvilinearcoordinate

2.Velocityofapoint

Differentiatetheexpressionabovewithrespecttotimet,gives

∵,

,∴itcanbeobtainedas,Directionofvelocity:alongthetangentline,pointtotherotationdirection.3.AccelerationofapointNormalacceleration:Tangentialacceleration:TheaccelerationofpointM:Thedirectionofacceleration:2.3Velocityandaccelerationofapointinarigidbodyrotatingaboutafixedaxis4.Thedistributionregularitiesofthevelocityandaccelerationintherotatedrigidbody(1)Ateverytime,thevelocityandaccelerationofapointareproportionaltoR.(2)Ateverytime,thedirectionsofthevelocityandaccelerationofapointareperpendiculartoR.Theangle

betweentheaccelerati