刚体的基本运动课件

刚体的基本运动1Chapter7:BasicmotionsofarigidbodyTheshapeanddimensionsofarigidbodyshouldbeconsideredinvestigatingitsmotion.However,duetoitsunchangedshape,.§7-1Translationalmotionofarigidbody[Examples]translationandrotationBasicmotionsRectilineartranslationcurvilineartranslationRotationaboutafixedaxisKinematics2刚体的基本运动[例]

由于研究对象是刚体,所以运动中要考虑其本身形状和尺寸大小,又由于刚体是几何形状不变体,所以研究它在空间的运动学

是指刚体的平行移动和转动§7-1刚体的平行移动(平动)基本运动3OBrotatesaboutafixedaxisThemotionofCDisatranslationThemotionsofABandofthecamaretranslationsItisnotnecessarytodetermineitspositionbyfindingthepositionsofallitspoints.Instead,itspositionisdeterminedcompletelybydescribingthepositionofalineoraplaneinit.Kinematics4OB作定轴转动CD作平动AB、凸轮均作平动运动学位置就不必一个点一个点地确定,只要根据刚体的各种运动形式,确定刚体内某一个有代表性的直线或平面的位置即可。51.Definitionoftranslatorymotionofarigidbody:

Thedirectionofthelinelinkingarbitrarytwopointsintherigidbodyneverchangesduringitsmotion.NotetheequationsofmotionofthepointsAandB,i.e.

and ,wehaveThelengthanddirectionofABdonotchangeatallduringitsmotion.

Itstrajectorymaybeastraightlineacurve[Example]Kinematics6运动学一.刚体平动的定义:

刚体在运动中,其上任意两点的连线始终保持方向不变。由A,B两点的运动方程式:而[例]AB在运动中方向和大小始终不变

它的轨迹可以是直线可以是曲线7Conclusions:2.Featuresoftranslationalmotion:Alltheparticlesofarigidbodywithtranslationalmotionhavethesamemotion,i.e.theirtrajectories,velocitiesandaccelerationsareidentical.Thetranslationofarigidbodycanbesimplifiedtothemotionofaparticleinit.Kinematics8运动学得出结论:即二.刚体平动的特点:

平动刚体在任一瞬时各点的运动轨迹形状,速度,加速度都一样。

即:平动刚体的运动可以简化为一个点的运动。9§7-2Rotationofarigidbodyaboutafixedaxis1.Featureofrotationaboutafixedaxisanditssimplification

Feature:Thereisafixedlinecalledaxis,andeverypointoftherigidbodywhichisnotonthisaxismovesalongacircularpathinaplaneperpendiculartothisfixedaxis.2.Angleofrotationandequationofrotation

---angleof

rotation,whoseunitisrad.

=f(t)---equationofrotationSigndefinition:followsright-handrule.counter-clockwiseclockwiseKinematics10运动学§7-2刚体的定轴转动一.刚体定轴转动的特征及其简化特点:有一条不变的线称为转轴,其余各点都在垂直于转轴的平面上做圆周运动。二.转角和转动方程

---转角,单位弧度(rad)

=f(t)---为转动方程方向规定:从z轴正向看去,

逆时针为正顺时针为负11Unitusedinengineering:

n=rpm(roundperminute)TherelationbetweennandwisTheunitisrad/s.Iftheequationoftherotationisgivenas3.Angularvelocityandangularacceleration(1).Definitionoftheangularvelocity：Kinematics12运动学三.定轴转动的角速度和角加速度

1.角速度：工程中常用单位：

n=转/分(r/min)则n与w的关系为:单位rad/s若已知转动方程13Arigidbodyhasacceleratedrotationwhen

and

havethesamesigns,deceleratedrotationwhen

and

haveoppositesigns.

3.Uniformrotationandrotationwithuniformangularacceleration

Abodyhasuniformrotationwhen

=const.androtationwithconstantangularaccelerationwhen

=const.FrequentlyusedformulaeSimilartothemotionofaparticle.2.Angularacceleration:

Denotetheangularvelocityattas

andatt+△tas

+△,theangularaccelerationcanbedefinedasunit:rad/s2(scalar)Kinematics142.角加速度:

设当t时刻为

,t+△t时刻为

+△

与

方向一致为加速转动,

与

方向相反为减速转动

3.匀速转动和匀变速转动当

=常数,为匀速转动；当

=常数,为匀变速转动。常用公式与点的运动相类似。运动学单位:rad/s2(代数量)15

,

aredefinedforthewholerigidbody(includingallpointsinit)；

v,a

aredefinedforagivenpointinthebody(theyaredifferentfordifferentpoint).§7–3Velocityandaccelerationofapointinarigidbodyrotatingaboutafixedaxis1.Relationshipbetweenlinearvelocityandangularvelocity

Kinematics16

,

对整个刚体而言(各点都一样)；

v,a

对刚体中某个点而言(各点不一样)。运动学(即角量与线量的关系)§7-3转动刚体内各点的速度和加速度一.线速度V和角速度

之间的关系172.Relationbetweenangularacceleration

andan

,a

Kinematics18运动学二.角加速度

与an

,a

的关系19Conclusions:①visproportionaltoRanditsdirectionisperpendiculartoR.Itisdefinedaspositivewhenitsdirectionisthesameasthatof

.②Theangle

betweentheresultantaccelerationofanypointandtheradiusthroughitisidenticalandlessthan90o.

Thedistributiondiagramsofvelocitiesandaccelerationsofarotatingbodyareshownbelow:DistributionofthevelocitiesofpointsDistributionoftheaccelerationsofpointsKinematics20运动学结论:①v方向与

相同时为正,

R,与R成正比。

②各点的全加速度方向与各点转动半径夹角

都一致,且小于90o,

在同一瞬间的速度和加速度的分布图为:各点速度分布图各点加速度分布图21

Inengineeringpractice,severalgearsmeshedwitheachotherareusuallyusedtochangevelocitiesofrotationandtransmitpower,whichcanbeexplainedasfollows.§7–4TransmissionbetweentworotatingrigidbodiesI.Transmissionofgearwheels

Itisassumedthatthegearrollswithoutslipping,hence,ThetransmissionratioofgearEtogearFcanbedefinedas1.InnermeshingKinematics22运动学

我们常见到在工程中,用一系列互相啮合的齿轮来实现变速,它们变速的基本原理是什么呢?§7-4绕定轴转动刚体的传动问题一.齿轮传动因为是做纯滚动(即没有相对滑动)定义齿轮传动比1.内啮合232.ExternalmeshingNumberofteeth:Kinematics24运动学2.外啮合25SinceObviously,when,:transmissionforincreasingtherotatingvelocityofthedrivengear；

when,:transmissionfordecreasingtherotatingvelocityofthedrivengear.

Kinematics26运动学由于转速n与w有如下关系：显然当:时,,为升速转动；时,,为降速转动。27Pulleysystem:ForasystemconsistingofpulleysA,B,C,D,E,F,G,H,thetransmissionrationofthepulleysAtoHisgivenby：

,wheremrepresentsthenumberofthepulleys,andthenegativesignindicatesthattherotationdirectionofthelastpulleyisoppositetothatofthefirstone.II.Transmissionsystemconsistingofbeltandpulley(butnotduetothedifferencebetweentheirdirections)Transmissionratio:Kinematics28运动学三.链轮系:设有：A,B,C,D,E,F,G,H

轮系,则总传动比为：

其中m代表外啮合的个数；负号表示最后一个轮转向与第一个轮转向相反。二.皮带轮系传动(而不是方向不同)皮带传动29§7–5VectorrepresentationofangularvelocityandangularaccelerationKinematics1.Vectorrepresentationofangularvelocityandangularacceleration:Thedirectionsofandaredeterminedaccordingtotheright-handrule.Magnitude:30运动学§7-5角速度和角加速度的矢量表示点的速度和加速度的矢量表示一.角速度和角加速度的矢量表示按右手定则规定，的方向。312.VectorialrepresentationofthelinearvelocityandaccelerationofanypointinarigidbodyKinematics32二刚体内任一点的线速度和线加速度的矢积表示运动学331.Basicconcepts,featuresofmotionsandequations1)Basicconcepts：rectilinearmotion,curvilinearmotion(ofaparticle);translation,rotationaboutafixedaxis(ofarigidbody).2)Basicfeaturesofmotionsandbasicequations:Chapter6:KinematicsofaparticleChapter7:BasicmotionsofarigidbodyLessonforproblemsolving

Kinematics34一.基本概念和基本运动规律及基本公式1.基本概念：直线运动,曲线运动(点);

平动,定轴转动(刚体)。2.基本运动规律与公式:第六章点的运动学,刚体的基本运动习题课运动学35KinematicsMotionofaparticleaccelerationRectili-nearmotionUniformvelocityUniformacceler-ationGeneralcaseUniformvelocityUniformacceler-ationGeneralcaseCurvili-nearmotion36运动学37RotationofabodyaboutafixedaxisEquationofrotation:Angularvelocity:Angularacceleration:Uniformrotation:Rotationwithuniformlyvaryingvelocity:Kinematics38

刚体定轴转动转动方程:角速度:角加速度:匀速转动:匀变速运动:运动学392.Stepsforsolvingproblemsandthingstoconsidercarefully1)Stepsforsolvingproblems:①Analyzetheproblemcarefullyandmakesurewhattheknownandwhattheunknownsare.②Selectapropercoordinatesystem,namelyarectangularcoordinatesystemoranaturalcoordinatesystemwhichcandescribetheequationsofmotionmostconveniently.③Performthedifferentialorintegralcalculationaccordingtotheknownvariables.

Finallydeterminetherelatedconstantsusinginitialconditions.2)Payspecialattentionto：①Geometricrelationsanddirectionsofmotion,②Eliminatingtheparameter“t”whileobtainingtheequationforthetrajectories,Selectionoftheappropriatecoordinatesystems.

Kinematics40二.解题步骤及注意问题1.解题步骤:①弄清题意,明确已知条件和所求的问题。②选好坐标系：直角坐标法，自然法。根据已知条件进行微分，或积分运算。用初始条件定积分常数。对常见的特殊运动，可直接应用公式计算。运动学２．注意问题：①几何关系和运动方向。②求轨迹方程时要消去参数“t”。坐标系（参考系）的选择。

413.Examples[Example1]AlocomotivemovesalongacurvedtrackofradiusR=300mwithuniformlyvaryingvelocity.Thelengthofthecurvedtrackisl=200m.Thevelocityofthelocomotivewhenitentersthetrackisv0=30km/h，andthatwhenitleavesitisv1＝48km/h.Finditsaccelerationswhenitentersandleavesthecurvedtrack,respectively.Kinematics42三.例题[例1]列车在R=300m的曲线上匀变速行驶。轨道上曲线部分长l=200m，当列车开始走上曲线时的速度v0=30km/h，而将要离开曲线轨道时的速度是v1＝48km/h。求列车走上曲线与将要离开曲线时的加速度？运动学43Solution：Sinceitmovewithuniformlyvaryingvelocity,

const.Employing

andnotingwehave

Whenitentersthecurve，ResultantaccelerationWhenitleavesthecurve，ResultantaccelerationKinematics44解：由于是匀变速运动，则常量。由公式而由已知列车走上曲线时，全加速度列车将要离开曲线时，全加速度运动学45〔Example2〕FindtheinitialvelocityandaccelerationofaprojectileshowninthefigurewhichhasitshitpointatA.Analysis:OnlywhentheprojectilereachesitshighestpointA, thenvy＝0.Solution：thedisplacementcanbeexpressedas

AtpointAvy=0.ThetimeittakesfromtheorigintopointAis:Kinematics46〔例2〕已知如图，求时正好射到A点且用力最小。分析：只有在A点，vy＝0且为最大高度时，用力才最小。解：由

由于在A点时，vy=0，所以上升到最大高度A点时所用时间为：运动学47Substitutingtheaboveequationinto①and②，yields

Substitutinginto③wehaveKinematics48将上式代入①和②，得：

运动学将代入③，得49①Thenumberofrevolutionofthepulleyin3seconds；②ThedisplacementofweightBin3seconds；③ThevelocityofweightBat

ｔ＝3s；④TheaccelerationofpointContherimofthepulley

attheinitialtime；⑤TheaccelerationofpointCatt＝3s.

.Find：〔Example3〕

TheaccelerationofweightAinthefigureisanditsinitialvelocityis（constant）Kinematics50〔例3〕已知：重物A的（常数）初瞬时速度方向如图示。求：①滑轮3s内的转数；②重物B在3s内的行程；③重物B在ｔ＝3s时的速度；④滑轮边上C点在初瞬时的加速度；⑤滑轮边上C点在ｔ＝3s时的加速度。

运动学51）constant（（）Solution:①Iftheropeisnotextensiblewehave

②），③（Kinematics(revolutions)52）常数（（）解：①

因为绳子不可以伸长，所以有

运动学②），③（53④

Att=0，⑤Att=3s，Kinematics54④

t=0时，

⑤t=3s时，运动学55[Example4]

Awheelrotatesstartingatrestwithconstantangularacceleration.OM＝0.4m.Atacertaintime,theaccelerationofpointMismeasuredtobe .Find：Theequationofrotation;Thevelocityandthecentr