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,,(,300072):,,,,:;;;:TQ05:A:1001-4837(2003)05-0029-05NumericalSolutionfortheNaturalFrequenciesofHangingTowerNIEQing-de,DUANZhen-ya,TANWei(ChemicalEngineeringSchoolofTianjinUniversity,Tianjin300072,China)Abstract:Amathematicalexpressiontodeterminethenaturalperiodofvibrationsofhangingtowerwasob2tainedbysolvingtheequationofthetransversevibrationusingtheanalyticalmodelofelasticcontinuousbodyandthemethodofnumericalsolution.Thismethodcanbeappliedtoconstantcross-sectionandvariablecross-sectionaswell.Bycomparison,calculationresultswerefoundingoodagreementwiththatoffiniteelementanalysis.Inaddition,theinfluenceofsupportpositiononthenaturalfrequenciesofhangingtowerhadbeendiscussed.Keywords:hangingtower;methodofnumericalsolution;naturalfrequencyofvibration;naturalperiodofvibra2tion1,,,1,,,[1],2,,,,N,N(2)H1,H2,HNi[1,2]:92d4yidx4-k2iyi=0(1)ki=mi2/EiJi(2)mii,kg/m;1Eii,Pa;Jii,m42,ix:yi(x)=AiS[ki(x-ai)]+BiT[ki(x-ai)]+CiU[ki(x-ai)]+DiV[ki(x-ai)](3)yi(x)=ki{AiV[ki(x-ai)]+BiS[ki(x-ai)]+CiT[ki(x-ai)]+DiU[ki(x-ai)]}(4)yi(x)=k2i{AiU[ki(x-ai)]+BiV[ki(x-ai)]+CiS[ki(x-ai)]+DiT[ki(x-ai)]}(5)yÊi(x)=k3i{AiT[ki(x-ai)]+BiU[ki(x-ai)]+CiV[ki(x-ai)]+DiS[ki(x-ai)]}(6):Ai,Bi,Ci,Dii,STUV,(7)S[ki(x-ai)]=12[chki(x-ai)+coski(x-ai)]T[ki(x-ai)]=12[shki(x-ai)+sinki(x-ai)]U[ki(x-ai)]=12[chki(x-ai)-coski(x-ai)]V[ki(x-ai)]=12[shki(x-ai)-sinki(x-ai)](7)i,2x=ai,x-ai=0,(7):S(0)=1T(0)=U(0)=V(0)=0(8)(3)(6),:yi(ai)=Aiyi(ai)=kiBiyi(ai)=k2iCiyÊi(ai)=k3iDi(9)i,2x=ai+1,x-ai+1=Hi,(3)(6):yi(ai+1)=AiSi+BiTi+CiUi+DiViyi(ai+1)=ki(AiVi+BiSi+CiTi+DiUi)yi(ai+1)=k2i(AiUi+BiVi+CiSi+DiTi)yÊi(ai+1)=k3i(AiTi+BiUi+CiVi+DiSi)(10)(10)Si=12(chkiHi+coskiHi)Ti=12(shkiHi+sinkiHi)Ui=12(chkiHi-coskiHi)Vi=12(shkiHi-sinkiHi)(11),i+1,x-ai+1=0,(9):yi+1(ai+1)=Ai+1yi+1(ai+1)=ki+1Bi+1yi+1(ai+1)=k2i+1Ci+1yÊi+1(ai+1)=k3i+1Di+1(12)ii+1,,:03CPVTVol20.No52003yi(ai+1)=yi+1(ai+1)yi(ai+1)=yi+1(ai+1)EJiyi(ai+1)=EJi+1yi+1(ai+1)EJiyÊi(ai+1)=EJi+1yÊi+1(ai+1)(13):Ai+1=AiSi+BiTi+CiUi+DiViBi+1=kiki+1(AiVi+BiSi+CiTi+DiUi)Ci+1=Jik2iJi+1k2i+1(AiUi+BiVi+CiSi+DiTi)Di+1=Jik3iJi+1k3i+1(AiTi+BiUi+CiVi+DiSi)(14),y1(0)=0yÊ1(0)=0(15)(15)(9):C1=0,D1=0,x=ai+1iAiBiCiDi,(14)A1B1,:Ai=ai1A1+ai2B1Bi=bi1A1+bi2B1Ci=ci1A1+ci2B1Di=di1A1+di2B1(16)abcd,,x=ai+1,(16)(3)yi(x)=A1{ai1S[ki(x-ai)]+bi1T[ki(x-ai)]+ci1U[ki(x-ai)]+di1V[ki(x-ai)]}+B1{ai2S[ki(x-ai)]+bi2T[ki(x-ai)]+ci2U[ki(x-ai)]+di2V[ki(x-ai)]}(17)x=ai+1,(17):Bi=-ai1Si+bi1Ti+ci1Ui+di1Viai2Si+bi2Ti+ci2Ui+di2VA1(18)(18)(16),AiBiCiDiA1x=ai+1,(13),(14):Ai+1=0Bi+1=kiki+1[AiVi+BiSi+CiTi+DiUi]Ci+1=Jik2iJi+1k2i+1[AiUi+BiVi+CiSi+DiTi]Di+1=Di+1(19)(19)Di+1i,A1,i+1A1Di+1,:yN(H)=k2N[ANUN+BNVN+CNSN+DNTN]=0yÊN(H)=k3N[ANTN+BNUN+CNVN+DNSN]=0(20)A1Di+1,(20):F1A1+F2Di+1=0F3A1+F4Di+1=0(21)F1F2F3F4k,,(21),,:F=F1F4-F2F30(22),,F=0(F)F(10-6),(23)f=/2(23)T=1/s(24),,132051263,12Hi(1),fc1fc2,ANSYS2,,016%,11207%1%,2165%1(m)016112210(m)016161122421012(m)211077261732(kg)47001460033000(Pa)2105810112105810112105810112Hi/H00110120122501301401501601701775018019110fc1414585146861407614715181241524106741525181261471614075146841458fc2141448171404151101131768121538141913171837141913121538131768151101171404141448fc141453514586139161454518014151541064415155180161454613915148541453fc21414171319151035131716121514185217174214185212151317161510351713191414(%)c10111011801250126011901110107011101190126012501180111c201330149014401380130141015101410130138014401490133fc16111571502817898187771974612515796127197481877817897150261115fc21918212318762017161819881712201458241472014581712181988201716231876191821fc1610967146681734818271933611825156661182719338182817347146661096fc2191646231578201484181709171064201244241136201244171064181709201484231578191646(%)c1013120148201630164601517012910123401291015170164601630148201312c201891112641113311491017971105711384110570179711491111331126401891fc14166651724617066173361044173141257417316108461773617065172441666fc21511241812181518071414121311241516118167115161131124141412151807181218151124fc141636516716162661756610234170341237417036102361756616265167141636fc2141837171786151471141385121925151298181189151298121925141385151471171786141786(%)c101647019351120701252110130159501472015951101301252112070193501647c2117292142921172011881154021039216521039115401188211722142911729:cn=(fcn-fcn)/fcn3Hi/Hfc1/(fc1)0fc2/(fc2)0fc1/(fc1)0fc2/(fc2)0fc1/(fc1)0fc2/(fc2)0011001100110001100011000110000111122611203112251120011223111960121143511044114331104311429110410122511449019531144701952114570196801311303018681130101869112990186901411014110311101411030110141102901501913112320191311229019141122301611014110311101411030110141102901711303018681130101869112990186901775114490195311447019521145701968018114351104411433110431142911041019112261120311225112001122311196110110011001100011000110001100023CPVTVol20.No520034(fc1)0,(fc2)0(Hi/H)1,2(fc1/(fc1)0)(fc2/(fc2)0),333fcnn,Hz;(fcn)0Hi/H=110n,Hz5(1),;(2),016%,11207%1%,2165%;(3),,015,,;(4)0177501225,(),,,;(5),,[][1],.[J].,1982,(5):36-41.[2]W.Weaver,Jr.,S.P.Timoshenko.D.H.Young.VibrationProblemsinEngineering[M].1990.:2003-03-13:2003-04-22:(1931-),,,:(28)1512345(Hz)11129511