The geometry of surfaces contact

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1 Applid ad ompuaioal Mchaics ( h gomry of surfacs coac J. Sigl a * J. Švíglr a a Faculy of Applid Scics UWB i Pils Uivrzií 0 00 Pils zch public civd 0 Spmbr 007; rcivd i rvisd form 0 Ocobr 007 Absrac his coribuio dals wih a gomrical xac dscripio of coac bw wo giv surfacs which ar dfid by h vcor fucios. hs surfacs ar subsiud a a coac poi by approxima surfacs of h scod ordr i accordac wih h aylor sris ad cosquly hr is drivd a diffrial surfac of hs scod ordr surfacs. Kowldg of pricipal ormal curvaurs hir dircios ad h sor (upi idicarix of his diffrial surfac ar cssary for dscripio of coac of hs surfacs. For dscripio of surfac gomry h firs ad h scod surfac fudamal sor ad a furhr mhods of h diffrial gomry ar usd. A gomrical visualisaio of obaid rsuls of his aalysis is mad. Mhod ad rsuls of his sudy will b applid o coac aalysis of ooh scrw surfacs of scrw machis. 007 Uivrsiy of Ws Bohmia. All righs rsrvd. Kywords: coac mchaics diffrial gomry firs ad scod fudamal sor Gaussia ad ma curvaur pricipal curvaurs ad hir dircios scrw machi aylor sris sor idicarix. Iroducio h aim of his papr which cras h firs par of coac aalysis of wo bodis i accordac wih h Hrz hory [] [] is h drmiaio of diffrial surfac ad is curvaurs a his coac poi. I his sudy h simplifid surfacs which rprs h complicad chical surfac ar cosidrd. h surfacs ar giv by vcor fucios. Boh surfacs ar rplacd i h coac poi by approxima surfacs of scod ordr i accordac wih h aylor sris [4] pp. 05. A h coac poi o his diffrial surfac h pricipal ormal curvaurs ad hir pricipal dircios which dfi a coac bas ar drmid. h pricipl ormal curvaurs a h coac poi mus b kow i ordr o dscrib h coac of surfacs i h mar of h Hrz hory. hrfor h drivaivs of his diffrial surfac ar cssary up o h scod ordr mad. All dscripios ar show for h surfac oly. For h surfac is valid h sam procdur. Obaid mhod ad rsuls of h soluio will b applid for drmiaio of coac of ooh surfacs of scrw comprssor roors or scrw machi wih as cosquc of forc ad ha dformaio of machi housig skw axs. Afr ha i is possibl o dal wih h displacms fild ad srss fild i a ighbourhood of h coac poi dpdig o gomry of ooh surfacs.. Problm dscripio ad ipu paramrs roaio axs ar o ad o. h iiial global coordia sysm giv by { } ; i wo scrw surfacs ad fig. cra a gral kimaic coupl i h spac. hir O is placd o h o 0 axis fig.. I is cosidrd wih rspc o a chag of is posiio ha h axis * orrspodig auhor. l.: mail: jsigl@km.zcu.cz. 647

2 J. J. Sigl. al. / Applid ad ompuaioal Mchaics X ( o 0 is displacd io h posiio o. his axis displacm is drmid by radius vcors ra r 0 B ad displacm vcors u 0 A u B. h radius vcors ar xprssd by a homogous coordias. L hs surfacs coac islf a h poi which ar giv by h followig radius vcors r θ r θ ( whr i { } is h surfac i coordia sysm of h Euclida affi spac E ad i θ ar curviliar coordias of his poi i o h surfac i. h drmiaio of h i i coac poi of surfacs ad is o subjc of his soluio. his problm will b solvd sparaly wih h craio of idividual surfacs. For h soluio hs followig paramrs ar slcd. h iiial posiio of h axis o which is markd as o 0 is coicid wih h hird bas vcor fig.. h posiio of h axis o drmi hs followig paramrs r [ 0 0 ] [ 0 0 ] u [ ] A [ ] A 0 r B 0 u. h surfac B is dfid wih hs paramrs ; r 05; z ; H ; 75 ad similary h surfac has hs paramrs ; H 4; 75. h xplaaio of hs paramrs is i h x chapr. For drmiaio of h axis o posiio ar usd followig wo paramrs. h firs o is roaio 80 [ ] ad mas a roaio arroud h ormal li a h coac poi h scod o disac has a fucio for a visual dmosraio oly which dfis h disac bw coac pois o h coac ormal li fig.. A urig of h surfac owards h fixd coordia sysm f fig. is giv by h coordia φ 0 [ ]. h coac poi o h surfac is drmid by [ / 0] θ π Ω ad o h surfac by [ ] Ω. θ 75 π / 0. Gomry dfiiio of problm h fixd coordia sysm f was iroducd. his sysm is drmid by h vcors r r u u fig.. Bcaus h surfac has o dgr of frdom hr is crad a A0 B0 A B acual coordia sysm which coordia is φ. h quaio for rasformaio of h coordia sysm io is whr ( r r ( f f a A cos si 0 0 si cos 0 0 f f f f r O f f ( b is h rasformaio marix of h vcor coordias i h coordia sysm a b io a h coordia sysm a r is h radius vcor of h poi A o h surfac a i h coordia sysm a ad a ib ( j is h j-h coordia of h i-h bas vcor of h coordia sysm b xprssd i h coordia sysm a. h displacm of h origi O 0 is 648

3 J. J. Sigl. al. / Applid ad ompuaioal Mchaics X ( r r + u + r. (4 O f A0 A A0 f Boh surfacs ad ar dfid i h acual coordia sysms ad by vcor fucios as follows roaio o O 0 O f 0 B u B o disac r A0 f A f f o 0 u A r B0 B 0 A 0 r A0 O 0 Fig.. Visualisaio of dfii surfacs ad coordia sysms. H r ( r θ cosθ + r cos cos θ siθ + r cos si θ z si + θ [ θ ] 0 π Ω Ω 0 π whr r z ar radiuss of a auloid H is a high pr o roa i h dircio of h hird coordia is a rvoluio muliplicaor ad h Ω is h rag o which h vcor fucio r is giv is ipu. h spcial cass of h surfac dscribd by h (5 ar for xampl llipsoid auloid circl surfac orkscrw surfac hlicoid scrw auloid c. h surfac is dfid by H cos cos cos si si r r θ θ θ + θ π [ θ ] Ω Ω 0 π 0 π whr is a radius of his scrw surfac H is a high pr o roa i h dircio of h hird coordia is a rvoluio muliplicaor. his surfac is somims calld as h orkscrw surfac. h θ φ ar curviliar coordias o h surfac i. h ag vcors filds drmiig a bas of curviliar coordias i vry poi of surfac ar π (5 (6 649

4 J. J. Sigl. al. / Applid ad ompuaioal Mchaics X ( H θ siθ r cos si θ cosθ + r cos cos θ 0 r( θ θ π (7 ( θ ( θ r [ si cos θ si si θ cos 0]. (8 r r z h ormal ad h ui ormal vcor is ( θ ( θ. (9 0 O h surfac is slcd arbirary poi is radius vcor i h is r θ sysm { O ; i } r which drmis h coac poi. A his poi is sablishd a coordia h rasformaio marix of his sysm io is r r. (0 A similar coordia sysm { O ; i } is crad for h surfac i which origi will b o li h coac poi of h surfac. his sysm is dfid wih h rasformaio marix ( disac roaio π fig.. O h surfac is drmid arbirary poi is radius vcor i h is ( θ r r which drmis h coac poi o h surfac. A his poi is drmid a gral coordia sysm o h surfac ; } which h firs bas is coliar wih h ad h hird { 0 bas is coliar wih h 0 fig.. h rasformaio marix of h coordia sysm io h coordia sysm has hus h form r r ( h ivrs marix. h coac pois ad ar idical by h paramr disac 0 ad hrfor hy cra h coac poi. 4. h approximaio of surfacs wih scod ordr surfacs a h coac pois h surfac i { } i is subsiud a h poi i i θ i i by h aylor sris of h vcor fucio dfiig h surfac i up o h scod ordr fig.. As a illusraio h surfac a h coac poi is hus subsiud wih followig approxima surfac ( θ r( θ r r ( θ r θ + θ θ + + θ ( r( θ r θ r θ + θ θ + θ θ + (. θ θ θ 650

5 J. J. Sigl. al. / Applid ad ompuaioal Mchaics X ( disac N roaio 0 0 O B o f N A o 0 o O 0 Fig.. Approxima surfacs wih rmshd approxima surfacs. his approxima surfac is xprssd i h coordia sysm wih h quaio r r r. ( h approxima surfac is xprssd i h coordia sysm likwis r r r. (4 For drmiaio of h diffrial surfac a h coac poi h firs ad h scod coordia of hs approxima surfacs ad hav o b slcd o a orhoormal plai grid i h coordia sysm fig.. his rasformaio is giv wih h sysm of hr oliar quaios { } N r r r (5 whr coordias N N r r M M M is a boudary of h discr irval N idx idicas h w surfac. h (5 is rwrid io h form F( x N 0 so r r r 0 (6 whr h ukows vcor x N θ N N r. For h soluio of his quaios sys- 65

6 J. J. Sigl. al. / Applid ad ompuaioal Mchaics X ( i m h Nwo s mhod is usd. his w approxima surfac N i i { } fig. 4 is acually giv i h form ( i N r u u f u u [ ] Ω N whr Ω N is h wo- u u i dimsioal discr rgio ad f i is a fucio of wo variabls o h is hus idical wih h i. o N Ω N i B i. h surfac roaio o N i O u B O f o 0 A u A f O Fig.. Viw of approxima surfacs wih rmshd approxima surfacs alog h ormal li. 5. iffrial surfac ad ir gomry a h coac poi h diffrial surfac is dscribd i h coordia sysm wih h quaio r r r r r. (7 N N N u u i i i h diffriaios hav o b prformd umrically hus diffriaio ( r u u wih rspc o u i i { } givs wo ag vcor filds drmiig h bas of h local curviliar coordias ad four vcor filds giv wih h scod drivaivs r u u r u u ( u u r ( u u i j { }. (8 i i ij i j h ormal ad h ui ormal vcor is u u u ( u u ( u u 0. (9 h covaria coordias of h firs fudamal sor g ij [] pp. 86 o h surfac a h coac poi u u [ 0 0] Ω ar dfid by h do produc of h ag vcors i 65

7 J. J. Sigl. al. / Applid ad ompuaioal Mchaics X ( G g i j { }. (0 ij i j h covaria coordias of h scod fudamal sor h ij [] pp. 99 o h surfac a h coac poi ar dfid by H h r i j { }. ( ij ij 0 O N o 0 0 roaio disac A 0 O N u B B o o Fig. 4. h rmshd approxima surfacs. h Gaussia curvaur K ad h ma curvaur H of his surfac a h coac poi is giv [] pp. 5 by d ( H h h K d ( G g g ( h ( g H g h g h + g h gg ( g. ( h pricipal ormal curvaurs κ ar drmid from h followig quaios sysm K κκ H ( κ + κ κ H ± H K. ( rmiaio of h pricipal curvaurs ad h dircios of hir ormal plas lads up o h gralizd problm of ig valus which is dscribd [6] pp. 88 wih h quaio Hx λgx. (4 h soluio of his quaio givs h ig valus λ i ha ar pricipal ormal curvaurs or xrm curvaurs acually ad ig vcors v i. hs ig valus ad vcors ar wrid 65

8 dow as follows J. J. Sigl. al. / Applid ad ompuaioal Mchaics X ( κ 0 [ ] 0 κ V v v. (5 h local coordia sysm a a poi o a surfac is grally a affi coordia sysm. I his cas a h coac poi hr is h local coordia sysm g { ; orhogoal. h ig vcors xprssd i h coordia sysm ar 0} V v v v v (6 g g g g g ad h agl bw ig vcors v i is a ach poi [u u ] of ( u u vv v v v v v v acos π. (7 curv of ormal curvaurs κ o max v E κ ( δ E c roaio Normal curvaur κ [-] κ max u B v E δ sor idicarix r I ( δ u A κ mi o c Fig. 5. Viw alog h ormal li o sor idicarix curv of ormal curvaurs ad hir dircios of diffrial surfac a coac poi. hs ui ig vcors dfi h w orhoormal coordia sysm E (xrm curvau- ; ha i is calld h coac bas as wll. h hi- rs a h coac poi giv by { } rd bas vcor is dfid wih i E v v. (8 E For h x sp i is cssary a h poi o cra a auxiliary orhoormal coordia ;. h rasformaio marix of his sysm io is sysm { } c o 0 ic 654

9 J. J. Sigl. al. / Applid ad ompuaioal Mchaics X ( c c c c o 0 0 o ( c S v E o curv of ormal curvaurs κ max S κ ( δ E Normal curvaur κ [-] κ max v E f c disac sor idicarix r I ( δ roaio c κ mi ag pla a h poi o Fig. 6. sor idicarix curv of ormal curvaurs ad oscula circls i h pricipl dircios h poi S i is h circl cr. whr o [0 0 0]. h affi rasformaio of a orhoormal coordia sysm c io a affi coordia sysm g is giv by (0 0 / a ( x c r g c g r c 0 0 / si ( x c whr φ is h agl bw vcors ad x i is a vcor coordia. Bcaus h bas i vcors ar orhogoal i h solvd cas h agl φ π/. h rlaio for h ormal i curvaur ( u u κ δ [] pp. 07 i h ormal pla giv by vcors 0 ad (δ a h coac poi is whr ( u u κ κ δ ( H g ( g ( g ( g i j hij i j gij G ( 655

10 J. J. Sigl. al. / Applid ad ompuaioal Mchaics X ( cosδ ( δ δ δ δ 0 π { } δ g c g c i j ( c si (δ is a ui vcor i h agial pla. h quaio of h sor idicarix i.. upi idicarix [] pp. 09 is ( g ( g i j h r r r H r ( ij whr r i is a poi coordia o h idicarix curv fig. 5 ad 6. I hs picurs hr is h curv of h sor idicarix illusrad o a scal ad h curv of ormal curvaurs o a scal /κ max. h paramric xprssio of h idicarix curv a h poi ca b i r r δ r + δ δ 0 π i { } κ ( δ (4 I I g i whr i (δ is a coordia of h ui vcor (δ. 6. oclusio his work which occupis by h gomry of surfacs ad hir diffrial surfac a h coac poi is h prlimiary par of h coac aalysis of wo surfacs basd o h Hrz hory. h aim of his prsd aalysis is h drmiaio of h diffrial surfac of boh surfacs ad is curvaurs a h coac poi. osquly h coac bas ha cras coordia sysm is drmid. his horical sudy of h coac gomry will b implmd o h coac of ooh surfacs of scrw machis i opraio mod wh h axs of ooh surfacs ar skw. I his cas h origial coac curv bw ooh surfacs chags io h poi coac which causs a icras of h valu of ormal forc a his poi of mor h ighy ims [5] wih rspc o ormal forc a gral poi of coac curv i cas of o-dformd paralll posiio of roors. his ffc ca b h caus of a damag of ooh surfacs. h diffrial surfac which dscribs h rlaiv disac of approxima surfacs ad i h ighbourhood of h coac poi is fudamal o h soluio of h coac aalysis. h x sp of his work will b h drmiaio of h displacm fild ad srss fild i h ighbourhood of h coac poi of ooh scrw surfacs of scrw machis. Ackowldgm h work has b suppord by h projc MSM of h Miisry of Educaio of h zch public. frcs [] B. Budiský B. Kpr Základy difrciálí gomri s chickými aplikacmi SNL 970. [] H. Hrz Übr di Brührug fsr lasisch Körpr Joural für ri ud agwad Mahmaik 9 88 pp [] H. Hrz Übr di Vrilug dr ruckkräf i im lasisch Kriscylidr. Schlömilch s Zischrif für Mahmaic ud Physic Volum 8 88 pp [4] J. Jagr Nw Soluios i oac Mchaics WI Prss Gra Briai 005. [5] J. Švíglr V. Machulda J. Sigl formaio of Scrw Machi Housig udr Forc Fild ad is osqucs Schraubmaschi 006 VI-Brich 9 VI Vrlag GmbH üssldorf 006 pp [6] E. Viásk chický průvodc 67 Numrické mody SNL Praha