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1、机械系统动力学机械系统动力学机电工程学院机电工程学院 机械设计系机械设计系ChenZhaobo(陈照波)Tel:86412057E-mail:机械楼一楼1020室2021/9/231参考书1.闻邦椿等闻邦椿等机械振动理论及应用机械振动理论及应用 高等教育出版社高等教育出版社2.胡海岩胡海岩 机械振动基础机械振动基础 北京航空航天大学出版社北京航空航天大学出版社 3.师汉民师汉民 机械振动系统机械振动系统 华中科技大学出版社华中科技大学出版社4.W.T.Thomson 振动理论及应用振动理论及应用 清华大学出版社清华大学出版社2021/9/232考核办法累加式1.大作业大作业1 10%2.大作业
2、大作业2 10%3.平时表现平时表现 10%4.期终考试期终考试 70%2021/9/233Preface2021/9/234Whatisdynamics？Dynamicsfocusesonunderstandingwhyobjectsmovethewaytheydo.Dynamics=Kinematics+KineticsKinematicsisthestudyofthemotionofpointmassesorrigidbodies.Kinetics isthestudyoftheforceswhichcauseandaffectmotion.2021/9/235Whatismechani
3、calsystem?lStructurelMechanismlMachine2021/9/236Developmentstagesofdynamicsofmechanicalsystems:lStaticanalysislKineto-staticanalysislDynamicanalysislElasto-dynamicanalysis2021/9/237Whytostudydynamicsofmechanicalsystem?lHigherspeedlHigherprecisionlMoreflexiblelMorecomplicated2021/9/238ResonanceWhenaf
4、orcingfrequencyisequaltoanaturalfrequency2021/9/239Prerequisites:Themostimportantprerequisiteisordinarydifferentialequations.Youshouldbepreparedtoreviewundergraduatedifferentialequationsifnecessary.COURSE GOALS:1.Tobecomeproficientatmodelingvibratingmechanicalsystems.2.Toperformdynamicanalysissuchas
5、freeandforcedresponseofSDOFandMDOFsystems.3.Tounderstandconceptsofmodalanalysis.4.Tounderstandconceptsinpassiveandactivevibrationcontrolsystems.2021/9/2310Chapter1Singledegreeoffreedomsystems2021/9/2311ObjectivesRecognizeaSDOFsystemBeabletosolvethefreevibrationequationofaSDOFsystemwithandwithoutdamp
6、ingUnderstandtheeffectofdampingonthesystemvibrationApplynumericaltoolstoobtainthetimeresponseofSDOFsystem2021/9/2312SingledegreeoffreedomsystemsWhenone variablecandescribethemotionofastructureorasystemofbodies,thenwemaycallthesystema1-D system orasingledegreeoffreedom(SDOF)system.2021/9/2313LkmxmstU
7、nloadedSpringBody inequilibrium(at rest)Body in motionAt equilibrium,kst=mgUndampedSpring-MassSystem2021/9/2314k(st+x)x mg=mx.FreeBodyDiagram2021/9/2315EquationsofmotionSumforces:Rearrangetoyieldthefamiliarequationofmotion:2021/9/2316PhysicalmodelMathematicalmodellRestoringforce2021/9/2317lDampingfo
8、rce2021/9/2318UndampedFreeVibration2021/9/2319DifferentialequationSolvingODElProposedsolution:lIntoODEyougetthecharacteristicequation:2021/9/2320lGiving:lTheproposedsolutionbecomes:lForsimplicity,letsdefine:lGiving:2021/9/2321Letsmanipulatethesolution2021/9/2322Recall2021/9/2323Manipulatingthesoluti
9、onlSolutionwehave:lRewriting:lGiving:2021/9/2324FurthermanipulationlSolutionwehave:lLet:lGiving:2021/9/2325Differentformsofthesolution2021/9/2326Note!2021/9/2327NaturalfrequencylInthepreviouslyobtainedsolution:lThefrequencyofvibrationislItdependsonlyonthecharacteristicsofthevibrationsystem.Thatiswhy
10、itiscalledthenatural frequencyofvibration.2021/9/2328Naturalfrequencylnaturalfrequencyfromstaticdeflection.lnaturalfrequencyfromenergymethod.2021/9/2329Recall:InitialConditions2021/9/2330Amplitude&phasefromICs2021/9/2331Undampedfreeresponse2021/9/2332AddingDamping2021/9/2333DampinglDampingissomeform
11、offriction!lInsolids,frictionbetweenmoleculesresultindampinglInfluids,viscosityistheformofdampingthatismostobservedlInthiscourse,wewillusetheviscousdampingmodel;i.e.dampingproportionaltovelocity2021/9/2334Spring-mass-dampersystemsFromNewtonsLaw:2021/9/2335Solution(datesto1743byEuler)Dividetheequatio
12、nofmotionbymWherethedampingRatioisgivenby:(dimensionless)2021/9/2336Whichisnowanalgebraicequationin&substituteintoequationofmotionHerethediscriminant,determinesthenatureoftheroots.2021/9/23371)Rootsarerepeated&equal.Threepossibilities:Calledcriticallydamped2021/9/2338Nooscillationoccurs.CriticalDamp
13、ing2021/9/23392021/9/2340Over-Damping2021/9/2341MostInterestingCase!2021/9/2342Under-dampingDampednaturalfrequency2021/9/2343Under-Dampedfreeresponse2021/9/2344Under-Dampedfreeresponse2021/9/2345LogarithmicdecrementLogarithmicdecrementisnaturallogarithmofratioofanytwosuccesiveamplitudesandisequaltof
14、ollowingform2021/9/2346Dampingestimates2021/9/2347Homework#11.Theamplitudeofvibrationofanundampedsystemismeasuredtobe1mm.thephaseshiftismeasuredtobe2radandthefrequency5rad/sec.Calculatetheinitialconditions.2.Usingtheequation:evaluatetheconstantA1andA2intermsoftheinitialconditions.3.Anautomobileismod
15、eledas1000kgmasssupportedbyastiffnessk=400000N/m.Whenitoscillates,themaximumdeflectionis10cm.whenloadedwiththepassengers,themassbecomes1300kg.calculatethechangeinthefrequency,velocityamplitude,andaccelerationifthemaximumdeflectionremain10cm2021/9/2348Homework#14.Usethegivendatatoplottheresponseofthe
16、SDOFsystem5.SolvetheequationAndplottheresponse.2021/9/2349单自由度线性系统的强迫振动单自由度线性系统的强迫振动简谐激励简谐激励 激励力随时间的变化规律为正弦或余弦函数。周期激励周期激励 非周期激励非周期激励2021/9/2350简谐激励下的强迫振动简谐激励下的强迫振动2021/9/2351简谐激励下的强迫振动简谐激励下的强迫振动2021/9/2352简谐激励下的强迫振动简谐激励下的强迫振动2021/9/2353简谐激励下的强迫振动简谐激励下的强迫振动2021/9/2354简谐激励下的强迫振动简谐激励下的强迫振动则有则有令令2021/9/
17、2355简谐激励下的强迫振动简谐激励下的强迫振动00.511.52-20-10010203040rX(dB)z=0.01z=0.1z=0.3z=0.5z=1幅频特性幅频特性 r=1附近发生共振附近发生共振lz z=0时时,r=1 共振共振 z z 增大，共振点增大，共振点 r 12021/9/2356简谐激励下的强迫振动简谐激励下的强迫振动相频特性相频特性00.511.5200.511.522.533.5rPhase(rad)z=0.01z=0.1z=0.3z=0.5z=12021/9/2357简谐激励下的强迫振动简谐激励下的强迫振动2021/9/2358简谐激励下的强迫振动简谐激励下的强迫振
18、动00.20.40.60.8051015202530z00.20.40.60.800.20.40.60.81zrpeak上图显示共振峰值随阻尼的上图显示共振峰值随阻尼的减小而增大减小而增大下图显示共振频率随阻尼的减下图显示共振频率随阻尼的减小而增大小而增大注意共振峰值只有当注意共振峰值只有当 z zti)F(ti)脉冲激励脉冲激励j jthth 脉冲激励作用下的脉冲激励作用下的t t时刻响应时刻响应ti xi t2021/9/2381非周期激励下的强迫振动非周期激励下的强迫振动 脉冲响应函数法脉冲响应函数法 卷积积分的性质卷积积分的性质2021/9/2382脉冲响应函数法应用示例脉冲响应函数法
19、应用示例无阻尼系统无阻尼系统 无阻尼系统单位脉冲响应函数无阻尼系统单位脉冲响应函数mkx(t)t1t2F00初始条件引起的响应初始条件引起的响应外力作用期间的响应外力作用期间的响应2021/9/2383脉冲响应函数法应用示例脉冲响应函数法应用示例无阻尼系统无阻尼系统 2021/9/2384脉冲响应函数法应用示例脉冲响应函数法应用示例无阻尼系统无阻尼系统 002021/9/2385脉冲响应函数法应用示例脉冲响应函数法应用示例无阻尼系统无阻尼系统 总总的的响响应应0246810-0.100.10.20.3Time(s)Displacement x(t)2021/9/2386是静态变形的2倍2021/9/2387作作 业业 3 计算单自由度系统的响应：计算单自由度系统的响应：1 一个有阻尼的弹簧质量系统，已知一个有阻尼的弹簧质量系统，已知m=196kg,k=19600N/m,c=2940Ns/m,作用在质量块上的激振力为作用在质量块上的激振力为F(t)=160sin（19t）N，试求忽略阻尼及考虑阻尼的两种情况中，试求忽略阻尼及考虑阻尼的两种情况中，系统的振幅放大因子及位移。系统的振幅放大因子及位移。2 计算单自由度无阻尼系统对如图计算单自由度无阻尼系统对如图所示矩形激励力的响应：所示矩形激励力的响应：2021/9/2388