2005-2006 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Chan, CY

Li, JS

Liu, GR

Lou, SM

Zhao, DF

PolyU Institutional Repository

?????????????????????? ???　???　???　???(浙江科技学院机电系 ,杭州 , 310000)C. Y. Chan(香港理工大学 ,香港 )?　?　活动标形法的论述首推苏步青译 ,佐佐木重夫著《微分几何学》 ,还有的著作从外微分形式引入活动标形 .本文论证取曲面的正交曲率网为参数网时 ,曲面论的基本公式就是活动标形的微分形式 ,并用其分析了点啮合齿轮传动误差 .???　活动标形　接触分析　曲率网Moving Frame Aand Its ApplicationIn The Error AnalysisOf Point Contact GearingLiu Huran( Dept. o f Mechanica l Engineering , Zhejiang Univ er sity o f Science and Technolog y, 310000)Abstract　 This paper demonst rates that the ba sic fo rmulas o f surface theo ry a re just the differential forma-tion of the mov ing frame ,if the ver tical curva ture w eb is cho sen as the param eter w eb. And applied the theor yinto the ana ly sis of the t ransmission er ro r o f the po int contac t g earing.Keywords　 mov ing frame　 conta ct ana ly sis　 curv at ur e w eb　　　　 1　 The moving frame and the fundamental theory of　　　　 the differential geologySuppose that the tw o parameter curv es u and v have been cho sen as the curv ature w eb,then w e wi ll have the follow ing equation F= 0,M= 0. On the surfaces there are 3 vectorsw hich a re perpendicular wi th each other,e1= ru / E ,e2= rv G ,e3= e1×e2The dif ferentiate e 'can be w ri t ten in the fo rm of linear combina tion of ei(ei )u=∑ aijej , (ei )v=∑ bkjej? 25?? 3?2005? 9? 　　　　???????M AT HEM AT ICAL THEORY AN D APPLICAT IONS　　　　　　 　　Vol. 25 No. 3　　 Sep. 2005 ???????　????: 2005? 1? 2?ai j and bk j formed a dissymmetric ma trix respectiv ely, that i s ak j+ aji= 0,bij+ bji= 0　　　　　 e1u= ruu E+ ( 1 / E )uru ,e1v= ruv / E+ ( 1 / E ) v ru　　　　　 e2u= ruv G+ ( 1 / E )urv ,e2v= rvv / G+ ( 1 / G )v rvSince that F= ru rv= 0,M= nruv= 0　　　　　a12= - a21= e2ue1= ruv ru / EF= - EV EF　　　　　a13= e1ue3= ruun / E = L / E　　　　　a23= e2ue3= ruvn / G= 0by the same reason ,b12= Gu / GE.b13= 0,b23= N / G　　　　　de= (e1 )u du+ (e1 )vdv=∑ dki jej　　　　　dki j= aijdu+ bi jdv= - dkji　　　　　dk12= ( - Ev du+ gudv ) /2 eg= kgqdk1+ kg2dk2In the equa tion , kg 1 and kg 2 a re the geodetic disto rt curv atures of the curv e u and v re-spectiv ely.Since that ,dk1= E du dk2= Gdv　　　　　dk23= N / G= k2dk2 , dk13= L / E = kdk1In the equation , k1 and k2 a re the main curv atures o f the curv e u and v respectively. Inthe mov ing coordinate system tha t is fixed on the surface, the mov ing f rame has the fol lowingdi fferential form:　　　　　de1= dk12e2+ dk13e3　　　　　de2= - dk12e1+ dk23e3　　　　　de3= - dk13e1- dk23e2　　　　　dr= dk1e1+ dk2e2 ( 1)What should be st ressed is, the curv ature w eb as parameter w eb exists in theo ry, but i tprobably can 't be found in general si tuation. Wha t 's mo re, i t is di fficult to express in the formo f appa rent function. However, w e needn 't use such parameters in practical calculation in theexample below .2　 The analysis of the movement of the pointcontact meshing by the Moving FrameSuppose the∑ ( 1) and∑ ( 2) are tw o surfaces, establish the moving frames on the twosurfaces respectiv ely, acco rding to equ. ( 1)　　　　　de1 (i )= dk12(i )e2(i )+ dk13( i)e3( i)　　　　　de2 (i )= - dk12 (i)e1 (i)+ dk2 3(i )e3(i )　　　　　de3 (i )= - dk13 (i)e1 (i)+ dk2 3(i )e2(i )　　　　　dr( i)= dk1 (i) e1 (i )+ dk2(i )e2(i )　　 i= 1, 2, ( 2)　　　　　dk1( i) = E (I ) du( i)65　? 3?　　　　　　　　　　　??????????????????????　　　　　dk2( i) = G (i) dv( i)　　　　　dk( 12) ( i) = kg1 (i )dk1 (i )+ kg2( i) dk2(i )　　　　　dk( 23) ( i) = k2 (i )dk2 (i)　　　　　dk13( i) = k1 (i )dk1 (i)In the equation,kg1( i) and kg 2 (i) are the geodetic disto rt curv atures of the curve u (i ) and v( i)respectiv ely and k1 (i ) and k2 (i) a re the main curv atures of the curve u( i) and v (i ) respectiv ely.When the tw o bodies turn about thei r ow n axes, the dif ferential fo rm of the two movingf rame in the fix ed coo rdina te systems are respectiv ely ( that i s, the rela tionship betw een rela-tiv e dif ferentia tion and abso lute di fferentia tion) .　　　　　dE j (i )= dej (i )+ a (i )×edH(i )　　 dR( i)= dr( i)+ a( i)×R(i )dH( i) ( 3)The condi tion for the tw o surfaces to contact is　　　　　dR( 1)= dR( 2)　　 e3 ( 1)= e3( 2) ( 4)Put the equa tion ( 3) into the equa tion ( 4) , bo th sides is multiplied by e3 , consideringdr(i )e3= 0, the instantaneous ra tio is:　　　　　　 dH( 1) /dH( 2) = (a ( 1) ,R( 1) ,e3 ) /(a ( 2) ,R( 3) ,e3 ) ( 5)Di fferentia te once mo re, considering equation ( 4) , w e have:d2H( 1) /(dH( 2) ) 2=(a (1) ,dR( 1) /dH( 1) ,e3 )+ ( a( 1) , R(1) , de3 /dH( 1) ) - [( a( 2) , dR( 2) /dH( 1) ,e3 )+ (a (2) ,R( 2) , de3 /dH(1) ]dH(1) /dH(2) )(a (2) ,R( 2) , e3 )( 6)Put equation ( 2) into equation ( 3) , we have:　　　　　dR(i ) /dH( 1) = dk1(i ) /dH( 1)e1 (i )+ dk2( i) /dH( 1)e2(i )+ a(i )×R( i) ( 7)　　　　　dE 3 /dH( 1)= - k1 ( 1)dk1 (i ) /dH( 1) e1 (i) - dk2(i ) /dH( 1)e2 (i )+ a (i )×e3 ( 8)The equa tion ( 4) can be w ri t ten as:dk1( 1)e1( 1)+ dk2 ( 1)e2( 1)+ a( 1)×R( 1)dH( 1) = dk1 ( 2)e1( 2)+ dk2 ( 2)e2 ( 2)+ a( 2)×R( 2)dH( 2) ( 9)The ang le bw tw een the main directions o f the tw o surfaces isψ, then:　　　　　 e1 ( 2)= cosje1( 1)+ sinje2( 1)　　　　　 e2 ( 2)= - sinje1( 1)+ cosje2 ( 1) ( 10)Put equation ( 10) into equation ( 9) , w e have:　　　　　dk1( 2) /dH( 1)+ (a ( 1) ,R( 1) ,e1 ( 2) )dH( 2) /dH( 1)　　　　　= dk1(i ) /dH( 1) co sj+ dk2(i ) /dH( 1) sinj+ (a ( 1) ,R( 1) ,e1 ( 2) ) ( 11)　　　　　dk2( 2) /dH( 1)+ (a ( 2) ,R( 2) ,e1 ( 2) )dH( 2) /dH( 1)　　　　　= - dk1( i) /dH( 1) sinj+ dk2( i) /dH( 1) sinj+ (a( 1) , R( 1) ,e2 ( 2) ) ( 12)Di fferentia te the second equation of ( 4)　　　　　 - dk13( 1)e1( 1) - dk( 23) ( 1)e2 ( 1)+ a( 1)× e3dH( 1)　　　　　= - dk13( 2) e1 ( 2) - dk13( 2)e2( 2)+ a( 2)× e3dH( 2)Finally put the equation ( 10) into i t:66 ???????　　　　　　　　　　　　　　　　　? 25?　　　　　　 - k1 ( 2) dk1 ( 2) /dH( 1)+ a( 2)e2( 2) dH( 2) /dH( 1 )　　　　　= - k1( 1) dk1( 1) /dH( 1) cosj- k2 ( 1)dk2 (i) /dH( 1) sinj+ a( 1)e2( 2) ( 13)　　　　　 - k2 ( 2) dk2 ( 2) /dH( 1)+ a( 2)e1( 2) dH( 2) /dH( 1 )　　　　　= k1 ( 1) dk1 ( 1) /dH( 1) sinj- k2 ( 1)dk2 (i ) /dH( 1) co sj- a ( 1)e1 ( 2) ( 14)Solv e the equatoins ( 1) , ( 12) , ( 13) and ( 14) instantaneously , w e can determind　　　　　dk1( 1) /dH( 1) ,　　　　　dk2( 2) /dH( 1) ,　　　　　dk1( 2) /dH( 1) ,　　　　　dk2( 2) /dH( 1) .Then take them into equa tion ( 7)、 ( 8) , and ( 6) , we can fine the acceleration of the of drivengea r.????[ 1]　???? . ?????? . ????? , 1980.[ 2]　??? . ????????? . ????? , 1989.[ 3]　 L. Bax ter. Jr ,: Second- o rder Sur face Generation. Industria l Ma thematics, 1973, 23( 2): 1- 10.[ 4]　 M. L. Bax ter. Jr. Basic Geome try and Tooth contac t of Hypoid Gea rs. Industrial Math. 11( 2) , 1961, 1-8.67　? 3?　　　　　　　　　　　??????????????????????

Moving frame aand its application in the error analysis of point contact gearing

Abstract

Similar works

Full text

Available Versions

PolyU Institutional Repository