齿轮减速器系统可变固有特性动力学研究

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:1999209228;:2000202225::http:ööö:100026893(2001)0120064205,,(,710072)DYNAMICANALYSISOFVARIABLENATURALCHARACTERISTICFORGEARDECELERATORSHAORen2ping,SHENYun2wen,SUNJin2cai(DepartmentofMechanicalEngineering,NorthwesternPolytechnicalUniversity,Xian710072,China):,,,;(),,,:;;;;;:TH11311:AAbstract:Ageardeceleratorisacomplexelasticsystemwithmultipletransmission.Itsdynamiccharacteristicappearsasnonlinearvaryingphenomenabecauseofthenonlinearinfluenceofgearmeshingstiffnessandtransmissionerroraswellasthenonlinearactionofbearingsupportingstiffness.Thesystemstiffnesswasanalyzedalwaysbyafixed2constantmethod;infact,itdidnotaccordwiththepracticeofengineering.Inthispaper,thefluctuationofgearmeshingstiffness,theinfluenceoftransmissionerror,andtheactionofbearingsupportingstiffnessareconsideredalltogether.Andthetheoreticmodelinganddynamicresponseanalysisaredoneforthetransmissionsystemwithbi2gradetapercylindergears.ThenonlineartermsofthetransmissionerroraredealtwiththemethodofFourierseriesexpansion.Furthermore,thetheoreticanalysisiscomparedwiththeexperimentalresults.Itisshownthatthemeshingstiffnessofgeartransmissionvariesevidentlybetweenthesingle2toothedmeshingzoneanddouble2toothedmeshingzone,butitvariesslowlyinthesamemeshingzone.Thedynamiccharacteristicsofthedecelerator,i.e.thenaturalfrequency,themodeshapeandsoon,varywithmeshingperiod.Itappearstobeavariablenaturalcharacteristic.Therefore,themultiplegeartransmissionsystemshallbeanalyzedaccordingtothemeanmeshingstiffnessofthesingle2toothedmeshingzoneanddouble2toothedmeshingzonerespectively,andindoingso,itmaymeettherequirementsofengineering.Keywords:geardecelerator;meshingstiffness;transmissionerror;bearingstiffness;dynamicnaturalcharacteristic;dynamicanalysis,,,,[1],,,Fourier,,,1,1,T1,T2Hi(i=1,,5),z1,yi(i=1,,6),xi(i=3,,6),ei1ei2,rbi22120011ACTAAERONAUTICAETASTRONAUTICASINICAVol.22No.1Jan.20011Fig.1Dynamicmodelofdecelerator(i=2,,5)Kt1,Kt2()ei1,ei2(),,Mxb+Cxa+K(t)x=F(t)(1):MM=diag[I1,I2,I3,I4,I5,m2,m2,m3,m3,m4,m4,m5,m5]x=[H1H2H3H4H5z1y1y3x3y4x4y6x6]TF(t)=[T1,rb26ni=1Ki1ei1,-rb36ni=1Ki1ei1,rb46Ni=1Ki2ei2,-rb56Ni=1Ki2ei2,-T2,6ni=1Ki1ei1,0,-6ni=1Ki1ei1,0,6Ni=1Ki2ei2,0,-6Ni=1Ki2ei2,0]TKi1Ki2(i=1,2)12,i=1,i=2;nN(n=2,N=2);C,13;Kt[2],,Kv$KKt1=Kv1+$K1(t),Kt2=K{2+$K2(t),K(t)=K+$K(t)K(t)=K11K12K13K14K21K22K23K24K31K32K33K34K41K42K43K441313:K11=-KH1-KH10-KH1KH1+Kt1r2b2-Kt1rb3rb20-Kt1rb2rb3KH2+Kt1r2b3K12=00000Kt1rb2-KH20-Kt1rb3K13=0000-Kt1rb200Kt1rb30;K14=[0]34K21=00-KH20000Kt1rb2-Kt1rb3K22=KH2+Kt2r2b4-Kt2rb5rb40-Kt2rb5rb4KH3+Kt2r2b5000Kz1+Kt1K23=0000000-Kt10K24=Kt2rb40-Kt2rb40-Kt2rb50Kt2rb50000034K31=0000-Kt1rb2Kt1rb3000K32=00000-Kt1000K33=Ky1000Ky3+Kt1000Kx3;K34=[0]34K41=[0]43K42=Kt2rb4-Kt2rb50000-Kt2rb4-Kt2rb50000;K43=[0]43K44=Ky4+Kt20-Kt200Kx400-Kt20Ky6+Kt20000Kx644:x=xq+$x,F(t)=Fv+$F(t),xa=$xa,xb=$xb,Fv=Kvxq,:K(t)x=Kv$x+Kvxq+$K(t)xq+$K(t)$x$K(t)$x,(1)M$xb+C$xa+Kv$x=$F(t)-$K(t)Kv-1Fv(2):Fv=[T1000-T200000000]T;$F(t)=[0rb26ni=1Ki1ei1-rb36ni=1Ki1ei1561:rb46Ni=1Ki2ei2-rb56Ni=1Ki2ei26ni=1Ki1ei10-6ni=1Ki1ei106Ni=1Ki2ei20-6Ni=1Ki2ei20]T:$F(t)$K(t),$K(t)Kt1Kt2$K(t)=5K(t)5Kt1$Kt1+5K(t)5Kt2$Kt2(3)Tz$Tz,$Tz$Kt1$Kt2,$Tz$K(t),(2):ei1$P1ei2$P2,(3)$F(t)-$K(t)Kv-1Fv=$P1+$P2=$P(4):$P1=$F1(t)-5K(t)5Kt1$Kt1Kv-1Fv;$P2=$F2(t)-5K(t)5Kt2$Kt2Kv-1Fv;Ki1,Ki2,ei1,ei2,,,Xz1Xz2[3]$P1=6Lj=1[Ajcos(jXz1t)+Bjsin(jXz1t)]$P2=6Mj=1[Cjcos(jXz2t)+Djsin(jXz2t)](L=1,2,3;M=1,2,3,)(5):Aj=2Tz1Tz10$P1cos(jXz1t)dt;Bj=2Tz1Tz10$P1sin(jXz1t)dt;Cj=2Tz2Tz20$P2cos(jXz2t)dt;Dj=2Tz2Tz20$P2sin(jXz2t)dt.$P=$P1+$P2=6Lj=1[Ajcos(jXz1t)+Bjsin(jXz1t)]+6Mj=1[Cjcos(jXz2t)+Djsin(jXz2t)](6)Xj=jX,$x1=Ejcos(Xj1t)+Fjsin(Xj1t)$x2=Gjcos(Xj2t)+Hjsin(Xj2t){$x}j={$x1}+{$x2}(7):Xj1=jXz1;Xj2=jXz2$x1$x2(2)M$xb+C$xa+Kv$x=$PKv-X2j1MXj1C-Xj1CKv-X2j1MEjFj=AjBj(8)Kv-X2j2MXj2C-Xj2CKv-X2j2MGjHj=CjDj(9)(8)(9),Ej,Fj,GjHj,()$x=6$xj(10)2:(kgm2)I1=0116931,I2=516776610-4,I3=1108632410-3,I4=4184601510-3,I5=1156454210-3;(kg)m2=2160874,m3=1174783,m4=3143439,m5=4127211;(Nmörad)KH1=213487168104,KH2=10169318104,KH3=9126741187104;[4],,(:Nöm):Kz1=5110107,Ky1=3146107,Ky3=4195107,Kx3=3163107,Ky6=5124107,Kx6=4168107,,P=715kW,780römin,T1=911827Nm:1z1=28,z2=35,2z3=60,z4=40:1m=215,2m=210[5],Weber,,,12,Tz1Tz223,,,,,5:(1);(2);(3)12;(4)126622;(5),1TziTzi0Ktidt1()45,511,,1213,,,21Fig.2Firstgrademeshingstiffnesswithmeshingperiod31Fig.3Secondgrademeshingstiffnesswithmeshingperiod1Table1Naturalfrequencyofsystem(Theory):Hz1234567137.43627353.14894415.45941493.95776508.07681516.46127583.09326237.63849353.25376415.89941494.78517508.08542516.65381583.68754337.51087353.02531415.89303494.19928508.07824516.49254583.34271437.56461353.18245415.46752494.55196508.08113516.52464583.45717537.49113353.17162415.83041494.22485508.07962516.54124583.5672389101112131712.20257877.01113948.127811503.044145486.627727823.957512712.47088879.34319951.394651505.825607498.5156210662.59803712.34875878.26147951.384331503.544737496.339117825.870844712.36439878.21831948.142671505.331305486.9635510662.550055712.35497878.11254950.692431504.592817047.943189939.204694(12)Fig.4Naturalfrequency(122th

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