动态法测定金属的杨氏模量

MeasuretheYoung’sModulusviathedynamicmethod2013301020145陈锋1.ThematerialsEnergyconversiondevice,Low-frequencygenerator,OscilloscopeandShieldedcables2.TheprincipleForauniformfinerodwithlengthLmuchbiggerthanitsdiameterd,bothofitsendsdoesaslightlyvibration,themotionofitwillsatisfytheequation:𝜕4𝑦𝜕𝑥4+𝜌𝑆𝐸𝐽𝜕2𝑦𝜕𝑡2=0Where,𝜌isthedensityoftherod,Sisthecrosssectionoftherod,Jisthemomentofinertia,EistheYoung’sModulusandyisthevibrationdisplacementoftherod.Solvingthisequation,wecangetthat:1𝑋d4𝑋d𝑥4=𝜌𝑆𝐸𝐽1𝑇d2𝑇d𝑡2And:𝑦(𝑥,𝑡)=(𝐵1ch𝐾𝑥+𝐵2𝑠ℎ𝐾𝑥+𝐵3𝑐𝑜𝑠𝐾𝑥+𝐵4𝑠𝑖𝑛𝐾𝑥)𝐴𝑐𝑜𝑠(𝜔𝑡+𝜑)Where:𝜔=(𝐾4𝐸𝐽𝜌𝑆)1/2Thisisthefrequencyequation.Ifthesuspendinglinesuspendsthenodeofthesample,bothendsoftherodareinthefreesituation.Atthismoment,theboundaryconditionsisthebroadwiseforceandthebendingmomentiszero.𝐹=−𝜕𝑀𝜕𝑥=−𝐸𝐽𝜕3𝑦𝜕𝑥3=0𝑀=𝐸𝐽𝜕2𝑦𝜕𝑥2=0So:𝑑3𝑋𝑑𝑥3|𝑥=0=0,𝑑3𝑋𝑑𝑥3|𝑥=𝑙=0,𝑑2𝑋𝑑𝑥2|𝑥=0=0,𝑑2𝑋𝑑𝑥2|𝑥=𝑙=0Eliminatingtheequationsandusingthenumericalmethod,wecangettheKandlshouldsatisfyKl=0,4.730,7.853,10.996,14.137…Generally,wedefinethefrequencyasbasisfrequencywhenKl=4.730Whenthesamplevibrateatthebasisfrequency,therearetwonodeswhichisatadistance0.224land0.776ltotheendrespectively.Fedthe𝐾1=4.730𝑙intothefrequencyequation,wegetthenaturalvibrationangularfrequencyis:𝜔=((4.730)4𝐸𝐽𝜌𝑆)1/2AndgettheYoung’sModulus:𝐸=1.9978×10−3𝜌𝑙4𝑆𝐽𝜔2=7.8870×10−2𝑙3𝑚𝐽𝑓2Wherethemomentofinertia𝐽=∫𝑦2𝑑𝑠=𝜋𝑑464So:𝐸=1.6067𝑙3𝑚𝑑4𝑓2Where,listhelengthoftherod,disthediameter,misthemass,fistheresonancefrequencyofthesample.Actually,Ealsorelatetotherateofdiameterandlengthd/l,sotheequationshouldmultiplyamodifyingfactorR.𝐸=1.6067𝑅𝑙3𝑚𝑑4𝑓2When𝑙≫𝑑,R≈1.When𝑙isn’tmuchbiggerthand,weknowthat:d/l0.010.020.030.040.050.06R1.0011.0021.0051.0081.0141.019Whenthefrequencyoftheforceattachtotheresonancefrequency,theothersuspendinglinewillgetthemaximumamplitude.Therelationshipbetweenthenaturalfrequencyandtheresonancefrequencyis:𝑓𝑟=𝜔𝑟2𝜋=12𝜋√𝜔2−2𝛽2𝛽isthedampingcoefficient.Towardsthegeneralmetalmaterials,themaximumdampingcoefficientisthe1%ofthe𝜔,sowecanuse𝑓𝑟tocomputeinsteadoff.3.Theprocedure(1)TheYoung’sModulusofcopperandstainlesssteelisabout1.2×1011𝑁∙𝑚−2and2×1011𝑁∙𝑚−2.Wecanfollowtheequationstoestimatetheresonancefrequency.(2)Installthesamplerodandkeepithorizontal.(3)Connecttheapparatusandadjustthemtothenormalsituation.(4)Movethegeneratorandreceiverfromtheendsoftherodtothecenterandmeasurearesonancefrequencyeach5mm.Recordthedistancexbetweenthesupportingpointandtheendresonancefrequencyf.Ineachmeasurement,weshouldadjustthefrequencyofthegeneratortomaketheamplitudedisplayedontheoscilloscopeisthemaximum.Atthistimethefrequencyofthegeneratoristheresonancefrequency.(5)Therealresonancepeak’sresonancewidthissoshortthatifwechangethefrequencyslightly,theamplitudewillchangesharply.Weshoulddistinguishtherealresonancepeakfromthefakeone.4.Theresult[temperature:26℃]Copper:Number123456Avg.Length/cm17.9517.9417.9217.9117.9317.9217.93Diameter/cm0.8000.7940.7960.7920.7960.7960.796Mass/g75.29775.29975.30175.29775.29575.29775.298Suspendingpointx/cm510152025303540Resonancefrequencyf/Hz747.9745.6743.3742.0740.4740.0740.0Thebasisfrequencyatthenode(0.224l)isabout740.0HzTheuncertaintyUl=0.01000235cmTheuncertaintyUd=0.00200001cmTheuncertaintyU=0.00010009gTheuncertaintyU=0.1Hz73974074174274374474574674774874901020304050So,theuncertaintyUE=E̅∙Ur=0.9718×1010N∙−2E=(0.9531±0.09718)×1011N∙−2StainlesssteelNumber123456Avg.Length/cm17.9217.9217.9517.9317.9217.9217.93Diameter/cm0.6020.6000.6080.6020.6020.6040.603Mass/g40.41440.41540.41740.41240.41640.41440.415Suspendingpointx/cm510152025303540Resonancefrequencyf/Hz839.3835.6834.7833.5832.5831.8831.6831.6Thebasisfrequencyatthenode(0.224l)isabout831.6HzTheuncertaintyUl=0.01000108cmTheuncertaintyUd=0.00200001cmTheuncertaintyU=0.00010005gTheuncertaintyU=0.1HzSo,theuncertaintyUE=E̅∙Ur=0.2624×1011N∙−2E=(1.962±0.2624)×1011N∙−26.DiscussionErroranalysis(1)Forthecopperrod,wecouldn’tmeasuretheresonancefrequencywhenx=35mm.Ithinkthisisthenodesotheimageontheoscilloscopeisalwaysahorizontalstraightline.(2)Ontheotherhand,thescalarontherodisn’t5mmeachexactlyandthelengthoftherodistoolonger.So,wecouldn’tguaranteethatthemovingdistanceofthetwosidesisthesameeachtime.83183283383483583683783883984001020304050