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1、TheoreticalQuestion2APiezoelectricCrystalResonatorunderanAlternatingVoltageConsiderauniformrodofunstressedlengthℓandcross-sectionalareaA(Figure2a).ItslengthchangesbyDℓwhenequalandoppositeforcesofmagnitudeFareappliedtoitsendsfacesnormally.ThestressTonthe
2、endfacesisdefinedtobeF/A.Thefractionalchangeinitslength,i.e.,Dℓ/ℓ,iscalledthestrainSoftherod.Intermsofstressandstrain,Hooke’slawmaybeexpressedasor(1)whereYiscalledtheYoung’smodulusoftherodmaterial.NotethatacompressivestressTcorrespondstoF<0andadecreasei
3、nlength(i.e.,Dℓ<0).SuchastressisthusnegativeinvalueandisrelatedtothepressurepbyT=–p.Forauniformrodofdensityr,thespeedofpropagationoflongitudinalwaves(i.e.,soundspeed)alongtherodisgivenbyyxℓFzFFigure2aDℓA(2)Theeffectofdampinganddissipationcanbeignoredina
4、nsweringthefollowingquestions.PartA:mechanicalpropertiesxFigure2cxpvunstressedwavemotionFigure2b¥x=0ppvcompressedunstressedAuniformrodofsemi-infinitelength,extendingfromx=0to¥(seeFigure2b),hasadensityr.Itisinitiallystationaryandunstressed.Apistonthenste
5、adilyexertsasmallpressureponitsleftfaceatx=0foraveryshorttimeDt,causingapressurewavetopropagatewithspeedutotheright.(a)Ifthepistoncausestherod’sleftfacetomoveataconstantvelocityv(Figure2b),whatarethestrainSandpressurepattheleftfaceduringthetimeDt?Answer
6、smustbegivenintermsofr,u,andvonly.[1.6points](b)Consideralongitudinalwavetravelingalongthexdirectionintherod.Foracrosssectionatxwhentherodisunstressed(Figure2c),letx(x,t)beitsdisplacementattimetandassume(3)wherex0andkareconstants.Determinethecorrespondi
7、ngvelocityv(x,t),strainS(x,t),andpressurep(x,t)asafunctionofxandt.[2.4points]PartB:electromechanicalproperties(includingpiezoelectriceffect)Consideraquartzcrystalslaboflengthb,thicknessh,andwidthw(Figure2d).Itslengthandthicknessarealongthex-axisandz-axi
8、s.Electrodesareformedbythinmetalliccoatingsatitstopandbottomsurfaces.Electricalleadsthatalsoserveasmountingsupport(Figure2e)aresolderedtotheelectrode’scenters,whichmaybeassumedtobestationaryforlongitudinaloscillationsalongthexdir