ANALYSIS OF LAMINATED CONICAL SHELL STRUCTURES USING HIGHER ORDER MODELS. and J. Herskovits c

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1 ANALYI OF LAMINATED CONICAL HELL TUCTUE UING HIGHE ODE MODEL I. F. Pino Corria a Criovão M. Moa oar b* Caro A. Moa oar b an. Hrovi c a ENIDH - Dparamno Máquina Maríima Paço Arco Oira Poruga. b IDMEC/IT - Iniuo Engnharia Mcânica - Iniuo uprior Técnico Av. ovico Pai 9- Liboa Poruga. c COPPE/UF Univria Fra o io aniro CP585 CEP95-97 rai. ATACT In hi papr i prn a numrica mho for h rucura anayi of amina conica h pan uing a quariara ioparamric fini mn ba on h highr orr har formaion hory. Th ipacmn xprion u for h ongiuina an circumfrnia componn of h ipacmn fi ar givn by powr ri of h ranvra coorina an h coniion of ro r in h op an boom urfac of h h i impo. Th hap funcion u for h ranvra ipacmn ar C conforming an h fini mn i a conica/cyinrica pan wih 8 no an gr of from. Th mo prn prform aic anayi wih arbirary bounary coniion an oa a w * Corrponing auhor. T.: ; Fax.: E-mai ar: cmmoar@afa.i.u.p (C.M. Moa oar) Pino Corria a.

2 ignvau probm (fr vibraion an bucing). Iuraiv xamp ar prn an icu. KEYWOD: Fini mn Conic pan Lamina h Highr orr mo aic Fr vibraion ucing. INTODUCTION On of h mo wiy u hap in h aropac aronauic an in gnra inuri i h h of rvouion ihr in curv pan or in conica fruum. Th avn of amina compoi wih hir pcific mchanica characriic incra h man for numrica oo for h anay of h rucur wih high accuracy an o achiv fficin opimi ign. inc h mi-96 numrou ica numrica ouion ba in h fini mn ha bn coninuouy vop for h anayi of uch rucur wha i cary crib in a comprhniv urvy by Yang a. []. For hin ioropic h i i w rcogni ha h caica hory of hin h ba in h Lov-Kirchhoff hypoh a crib by Krau [] coniu a goo approach. For mora hicn amina h a fir orr har formaion hory (FDT) i nough o a wih h ffc of ranvr har formaion an roary inria ahough har corrcion facor ough o b u. Highr orr har formaion (HDT) hori wih impo coniion on vanihing of h urfac har r ar n for amina anioropic h an wr fir prn by y a. []. A common approach o a wih axiymmric h wih arbirary appi oa i o u Fourir ri xpanion for h componn of h ipacmn an oa in orr of h circumfrnia variab. y hi way a wo-imnion Pino Corria a.

3 probm i ranform in a of uncoup on-imnion probm which corrpon o h harmonic of h Fourir xpanion. Combin wih a cririou choo of h appropria fini mn hi mho a o vry fficin ouion a ho prn by hinman a. [] an Moa oar a. [5]. a o axiymmric h Pino Corria a. [6] u a mi-anayic fini mn ouion ba in an HDT mo for h anayi of amina anioropic conica h uing h quivan ing ayr hory. Whn axiymmric gomry i no ongr aow by cuou circumfrnia chang in h maria propri or bounary coniion i bcom ncary o u a wo-imniona anayi ovr h h urfac. A much n approach ha i w crib in Yang a. [] coni in h u of a arg numbr of pa mn o mo h h gomry. Ahough h impiciy inhrn o pa mn hi mho ha h iavanag of ariing rronou bning momn in h inr mn bounary u o h chang of urfac orinaion. himarai a. [7] prn an ioparamric quariara har formab h mn for h anayi of gnray amina h of rvouion. Thi mn ha no h cornr no hav 8 gr of from (of) an h g no hav of o a oa of 6 of for h mn. rnipiy an Hrmiian hap funcion ar u for h in-pan an ranvra ipacmn rpcivy. Th coninuiy of h ranvra ipacmn an i rivaiv i aur an h mn i of h yp C conforming fini mn. haar a. [8] u a rfin mo wih a highr orr ipacmn fi combin wih a picwi inar C coninuo ig-ag funcion for h in-pan ipacmn for h anayi of anioropic amina h of rvouion. Th fini mn Pino Corria a.

4 formuaion compri an igh-no quariara h mn an ony C coninuiy funcion ar u in h gnrai ipacmn fi. Kan a. [9] prn a mo uing a highr orr ipacmn fi voping a nin-non ioparamric quariara bi-quaraic (Lagragan famiy) fini mn for h anayi of inar an gomricay non-inar probm in amina h. Tigh a. [] u a HDT mo wih impo coniion of ro ranvr har r a h uppr an boom fac of h h appi o cyinrica h pan ma of amina compoi. From Hamion Princip h quaion of moion an h aocia bounary coniion ar riv which ar hn ov uing h Garin chniqu. Paaoo a. [] vop an HDT mo wih impo coniion of vanihing ranvr har r a h h urfac for cyinrica pan amina h. Th quariara fini mn wih 8 no an 6 of u a hap funcion Lagrang quaraic funcion for h in-pan ipacmn an roaion an Hrmiian poynomia for h ranvr ipacmn. Th mn i C nonconforming bcau h coninuiy of h norma op of h ranvr ipacmn i no aur bwn aacn mn. In h prn wor a quariara ioparamric mn i vop ba in h appropria rain ipacmn raion. A HDT ipacmn fi i u wih h impo coniion of ro ranvra har r a h urfac of h h. For comparaiv purpo a FDT mo wih har corrcion facor i ao vop an vaua. Th prn HDT mn ha 8 no an of an h hap funcion u ar Lagrang quaraic funcion an Hrmiian poynomia rpcivy for h in-pan ipacmn an roaion an for h ranvr ipacmn. In hi wor Pino Corria a.

5 h mn i C conforming bcau h coninuiy of boh h norma an h angnia op of h ranvr ipacmn ar fufi bwn aacn mn. An quivan ing ayr aumpion i foow o a wih h amina rucur of h h. DEVELOPMENT OF THE MODEL A highr orr har formaion hory mo i u ba on a gnric ipacmn fi givn by Paaoo []: u u u * * ( θ ) ( ) β β v ( ) ( ) v v * θ β * θ β () θ θ w ( θ) w whr: u v w ar ipacmn in h mi pan of h amina rfrr o h oca ax β β θ ar h roaion of h norma o h mi pan abou h an θ { } ax an i h im variab. Th ipacmn fi i fin a: u v w. Th funcion u * v * β * β * θ ar highr orr rm in h Tayor ri xpanion ao fin in h mi pan. A h funcion pn ony of θ an an h vau θ ar h principa raiu of curvaur of h urfac in h θ an ircion. Accoring o h Lagrangian formuaion h rain ipacmn raion ar obain by pciaiing h gnra hr-imniona rain ipacmn raion from h Grn rain nor xpr in arbirary orhogona curviinar coorina a in aaa []. In orr o fin h gomric ca facor an h curvaur raiu i i ncary o u om concp from urfac hory. Th rain-ipacmn Pino Corria a. 5

6 xprion ar hn uc for an arbirary conica h. For h HDT mo an from h coniion ha h ranvr har r vanih on h h op an boom urfac h h ( ) ( ) γ θ ± γ θ ± () θ i i poib o xpr funcion u * v * β * β * θ in rm of u v β an β θ. Du o h compxiy of h h rain ipacmn raion wih a highr orr ipacmn fi i wa ncary o u a ymboic manipuaor []. Th ipacmn fi a hn h foowing form: u( θ) u v( θ) v w( θ) w w β ϕ inφ β h inφ inφ w ( v βθ) ϕ βθ θ h w βθ θ () which in a mor compac form i givn by: u v ϕ ( ) w () Th yp of h ipacmn fi u i fin by h vau ϕ an ϕ for HDT an FDT rpcivy. In h ca of a fir orr har formab hory (FDT) ipacmn fi i i ncary o u har corrcion facor in h har r quaion. Th inar rain-ipacmn raion for a gnric conica gomry ar rprn by a powr ri of h ranvra coorina : Pino Corria a. 6

7 γ γ γ θθ θ θ γ γ γ θθ θ θ N κ κ θθ κ θ (5) κ θ κ whr if a HDT ipacmn fi i u N an in h ca of a FDT ipacmn fi N. Furhr ai in h ucion of xprion (5) can b n in Pino Corria a. [6]. CONTITUTIVE ELATION Coniring a h ma of ayr of orhoropic maria. Gnray h fibr of h h iniviua ayr or py ar orin a an ang α in rfrnc o h h coorina ym. Thrfor h r an rain ar vaua wih rpc o h h coorina ym (θ) accoring o h ru of a con orr nor ranformaion y []. Th coniuiv raion for ha ayr ar givn by σ τ { σ σ τ } Q { τ τ } ψ Q γ θ θθ θ b (6) whr h rain vcor ar: { } θθ γ θ an γ { γ θ γ }. Th maric: b Q Q i i ( 6 ) an Q Q ( m n ) 5 ar ymmric array of ranform mn iffn for h h ayr wih rpc o h h coorina ym (θ) foowing a fourh orr nor ranformaion y []. Th cofficin ψ θ ψ ψ ha appar in h ranvr har r xprion ar h har corrcion facor givn Pino Corria a. 7

8 by Vinon [5]. Th cofficin ar ony inrouc if h FDT ipacmn fi i u. Hnc coniring h coniuiv raionhip (6) an h rain ipacmn raion (5) h fina form for h r in h h amina for h inpan an ranvr har r ar: σ τ Q Q b ψ γ N κ N κ (7) FINITE ELEMENT FOMULATION Th rucura anayi i carri ou uing an ioparamric fini mn wih conica pan gomry a hown in Figur. Th mn ha igh no h cornr no hav igh gr of from u w w w v w β βθ θ θ an h mi-i no hav wo gr of from { u } v ach for a oa of fory gr of from for h mn. Th hap funcion chon ar of C coninuiy conforming which man ha h inr mn coninuiy of h ranvra ipacmn w an of i rivaiv w w θ w θ ar fufi h ranvra ipacmn wihin h mn i givn by: w w w w η ( ) H w H H H (8) 57 θ θ who xprion ar uc a: Pino Corria a. 8

9 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 6 L L H L H L H 6 H η η η η η η η η η η η η η η η η η η η η (9) whr: an η ar h mn oca coorina an η h no coorina in h mn oca rfrnia (Fig. ) L an L η ar h mn ngh in h an η ircion an 5 or 7 inica h mn cornr no. For h in-pan ipacmn fi ampiu u an v quaraic rnipiy hap funcion Q i ar u [] an for h roaion β an β θ inar Lagrang funcion L i [] ar chon. Th hap funcion marix wih imnion (5 ) i hn fin a: q L L H H H H Q Q Q Q N q i i i i i i i i i i () whr: ( ) 7 5 i v u w w w w v u q i i β β θ θ θ () i h vcor wih h gr of from of h mn. Th rain ipacmn raion can b arrang in a marix form in rm of h mn gr of from a for h in-pan rain: () q q N L L N N Pino Corria a. 9

10 an ao for h har rain γ L N L N q N q () From h powr of arrangmn of xprion () an () ubania impificaion ari in h finiion of h iffn marix. Th fir orr (inar) an con orr (non inar) rain nrgi h inic nrgy an h xrna oa wor of an mn ar givn by: U V σ U n V σ T V ρ & & Ω p A () V V V A whr p rprn h vcor of h appi urfac oa n an σ ar h non inar rain an iniia r vcor rpcivy. Th oa ponia nrgy of h mn i givn by: ( U U ) T Ω Π (5) Th govrning quaion of h mn ar hn obain by uing h Hamion Princip in h oa ponia nrgy quaion (5) yiing h foowing xprion for h iffn ma an gomric iffn maric which for an HDT ipacmn fi ar givn by: Pino Corria a.

11 ( ) η I H G F I H G F E H G F E D G F E D F E D A I H G F I H G F E H G F E D G F E D F E D A K (6) whr: ( ) ( ) Q I H FG E D A am NL b (7) ( ) ( ) Q I H G F E D A am NL (8) Th acobian i givn by: am θ (9) an am i h man raiu of h ayr. Whn h FDT ipacmn fi i u h ψ har corrcion facor [5] mu b inrouc in h appropria cofficin of xprion (8). Th conin ma marix for an HDT ipacmn fi i givn by: Pino Corria a.

12 η η η η η η η η η η η η η η η η I I I I I I I M () whr: I am NL m m ρ () In h xprion of h n orr rain nrgy h non inar rain vcor U n can b arrang in a powr ri of h coorina: () θ θ θθ θ θ θθ θ θ θθ γ γ γ γ γ γ γ γ γ M M n n n n n n n n n n n n n n n n n n Whn a HDT ipacmn fi i u M7 an in h ca of a FDT ipacmn fi M. Th n orr rain nrgy xprion () can hn b facori a in Moia a. [6] an Pino Corria a. [6]: σ σ σ V M * n * n V M n n V n V V V U () Pino Corria a.

13 Thi facoriaion i carri ou in a uch way ha in h xprion ony inar rm appar in h ipacmn fi funcion an ra rivaiv in orr o coorina an θ. Th iniia r vcor σ i now rprn by maric which ar fin by h raion: * n M... σ σ () σ σ σ M p p M p * n p * n M p p n q G N G N q whr marix G wih imnion (8 5) i givn by: L L H Q Q G (5) Th ub-maric: Q ( ) H (6 ) an L ( ) ar fin a: θ Q θ θ θ H an θ L (6) Expring h non inar rain ipacmn raion in rm of h noa gr of from h n orr rain nrgy for an mn i hn givn by: U q K q U σ (7) Pino Corria a.

14 whr K σ i h gomric iffn marix aocia o an iniia r a givn by: K σ M G Ap p N G σ p GN η (8) wih: A G p NL p am (9) In h mn iffn ma an gomric iffn maric h ingraion ovr h ircion i prform anayicay an h ingraion in h η ircion i carri ou uing numrica Gauian quaraur [7] wih a ru of poin. Foowing ha an aing in accoun h conribuion of a mn in h omain an inroucing h bounary coniion in h appropria noa vau h quiibrium quaion of h ym ar obain afr prforming h uua fini mn ambing chnic. ( K K ) δ P M & δ G () whr: M K an K G ar h ym ma iffn an gomric iffn maric P i h ym appi oa vcor an δ ar h ym gr of from. NUMEICAL APPLICATION Ioropic Pinch Cyinr In hi xamp prn by Paaoo a. [] an ioropic cyinr wih (raiu).95 in L (ngh).5 in an h (hicn).9 in i ac by wo f quiibraing poin oa of magniu P b. Th maria propri of h cyinr ar E.5 6 pi an ν.5. cau of ymmry ony on ocan of Pino Corria a.

15 h cyinr i icri a inica in Figur h ymmry bounary coniion ar givn by: x L / θ π / : : u v w w θ β β θ A fini mn mo wih n pan mn of qua i i u whr n i h numbr of acua iviion in h circumfrnia ircion. Tab how h raia ipacmn of h poin of appicaion of h forc P obain wih HDT an FDT ipacmn fi mo for variou mh. A goo agrmn i foun bwn h prn mo an h C nonconforming mn HDT ba mo prn by Paaoo a. []. For h mh maximum icrpanci of.8 % an.9% ar foun for h FDT an HDT mo rpcivy. Camp Conica Pan wih Exrna Prur A camp conica pan ma of a amina wih a py ayou [9º/º/9º] a crib by haar a. [8] i ubc o an xrna conan prur. Th maria propri of h ayr ar a foow: E L. Pa E L /E T an G G.5 E T G. E T an ν ν ν.5. A mh i u o icri on-haf of h pan. Th cnra fcion ar cacua for vra vau of h raio E L /E T uing h vop mo wih HDT an FDT ipacmn fi an ru ar prn in Tab. For comparaiv purpo ru from haar a. [8] ar ao hown. A vau ar normai a inica in h prviou wor by ETh w. A can b n in Tab h ru from h prn wor ar in pl w fair agrmn wih ho prn by haar a. [8] wih a ayrwi ba mo. Pino Corria a. 5

16 Th ru from h FDT ba approach away how owr vau which inica a iffnr mo. Lamina Orhoropic Pinch Cyinr A impy uppor cro py cyinrica h wih py ayou [9 / /9 ] an gomric raion L/ (ngh/raiu) a prn by haar a. [8] i anay. Th maria propri of h ayr ar a foow: E /E 5 G /E.5 G /E. an ν ν.5. Two iamricay oppoi pach urfac forc wih pach i L p θ p whr L p L/5 an θ p π/5 ar appi a h mi of h cyinr. In h fini mn mo ony on quar of h cyinr i icri an h bounary coniion conir ar a foow: x : u v w β β θ x L : v w β θ θ π / : v w θ β θ Thr mh (5 5 an ) ar u for h pan mn. In Tab i i hown h normai raia fcion ru for /h (raiu/hicn) 5 an. Th maximum raia fcion ar normai wih rpc o an aiciy ouion an hn compar wih ru from wo mo: a high orr ayrwi hory an a Lov-Kirchhoff ba hory boh prn by haar a. [8]. For comparaiv purpo ru obain by Pino Corria a. [6] wih a mianayic axiymmric fini mn mo ar ao hown. In ha ca h mo wa icri in a mh of ring mn of qua ngh. A goo agrmn i foun bwn boh fini mn mo (mi-anayic axiymmric ring an h pan) Pino Corria a. 6

17 an h arnaiv ayrwi an Kirchhoff mo (for /h 5). A maximum icrpancy of % an % ( mh) i foun rpcivy. Naura Frqunci of a Lamina Cyinrica Pan Th naura frqunci of a quar cyinrica pan h a crib by Tigh a. [] ar cacua. Th amina pan ha a py ayou givn by [-5 /5 ]. Th pan gomric aa i fin by /h5 an h vau of h raio b/h vari bwn an. Th maria u for h cyinrica compoi h pan ui in hi ca i a graphi/poxy (A 5) wih h foowing propri: E.79 GPa E 9.65 GPa G G. GPa G.8 G ν. an ρ.5 g.m -. For h bounary coniion h foowing impy uppor impoiion ar ma: x b : v w βθ θ b : u w β Monoonicay convrgnc i obrv a can b n in Tab whr for h ca of /h5 an b/h h vau of h fir frqunci ar how o convrg vry quicy wih h rfining mh: an. In Figur i i hown h naura frquncy for a impy uppor h pan wih raio /h5 for icr vau of b/h. Th vau cacua ar in goo agrmn wih ho prn by Tigh a. [] uing an anayica mho ov wih h Garin chniqu. Criica ucing Exrna Prur of a Lamina Conic Pan Th variaion of h criica bucing xrna prur for a amina conica h pan wih h py ang i anay. Th conic pan i camp a h g an h amina ha four ayr wih py ayou [θ/-θ]. Th h i a conic pan wih Pino Corria a. 7

18 imnion: L(ngh). m T (op raiu). m (boom raiu).5 m θ (anguar ampiu)5º an h(hicn). m. Th maria u for h amina pan ui in hi ca i graphi/poxy wih h foowing propri: E8. GPa E. GPa G G.8 GPa G.5 GPa ν.5 an ρ.5 g.m -. In h fini mn mo a mh i u an convrgnc wa aiy obain in a ui ca. In Figur i i hown h vauaion of h criica bucing xrna prur for a camp amina conica h pan vru h py ayou ang. A maximum vau for h criica bucing xrna prur i obrv for a py ayou of [75º/-75º]. CONCLUION A cyinrica/conica pan hap fini mn mo ba in a highr orr ipacmn fi ha bn vop. Th quariara h mn ha 8 no wih a oa of gr of from an u C conforming hap funcion for h ranvra ipacmn. inc i ha bn impo h vanihing of h ranvr har r on h h op an boom urfac hr i no n o u any har corrcion facor in h mn iffn marix. Th mo i uiab for conica hap pan ma of hic or hin amina h wih arbirary bounary coniion an oa. Th convrgnc of h ouion obain ha bn prov in h ru prn om of hn in comparion wih ohr arnaiv ouion. Pino Corria a. 8

19 ACKNOWLEDGEMENT Th auhor wih o han h financia uppor of POCTI/FEDE/FCT an Proc: POCTI/P/EME/8/998-Pha II POCTI/P/EME/7559/ FCT/GICE an CNPq (rai). EFEENCE. Yang HTY aiga Mau A Kapania K. A urvy of rcn h fini mn. In.. Numr. Mh. Engng Krau H. Thin Eaic h. ohn Wiy & on Nw Yor y N Liu CF. A Highr Orr har Dformaion Thory of Lamina Eaic h. Inrnaiona ourna of Enginring cinc 985;():9-.. hinman I Winam. Couping wn ymmric an Aniymmric Mo in h of vouion. ourn. Comp. Ma. 978; : Moa oar CM Coriro NM arboa I. A Dicr Mo for h Dign niiviy Anayi of Muiayr Compoi h of vouion. Compoi Enginring 995;5(5): Pino Corria IF arboa I Moa oar CM Moa oar CA. A Fini Emn mi-anayica Mo for Lamina Axiymmric h: aic Dynamic an ucing. Compur an rucur ;76(-): himarai A Carr A Mo P. A har Dformab Fini Emn for h Anayi of Gnra h of vouion. Compur an rucur 989;(): haar K Varaan TK. A Highr-Orr Thory for ning Anayi of Lamina h of vouion. Compur an rucur 99;: Pino Corria a. 9

20 9. Kan T Komminni. Gomricay non-inar anayi of anwich h wih a highr-orr hory an C fini mn. Compoi rucur 99;7:-8.. Tigh KV Paaoo AN. Highr Orr Cyinrica Pan aionhip Coniring Gnra Py Layup. Compoi rucur 99;7:5-.. Paaoo AN Dnni T. Noninar Anayi of h rucur. Wahingon DC: AIAA Eucaion ri 99.. aaa A. Eaiciy - Thory an Appicaion. Maabar: Krigr Pubihing Company 99.. Char W G KO Gonn GH Lony L Monagan M Wa M. Fir Lav: A Tuoria Inroucion o MAPLE V. Nw Yor: pringr-vrag y N. Mchanic of Lamina Compoi Pa Thory an Anayi. oca aon: C..C. Pr Vinon iraowi L. Th haviour of rucur Compo of Compoi Maria. Dorrch: Marinu Nihoff Pubihr Moia Moa oar CM Moa oar CA. ucing haviour of Lamina Compoi rucur Uing a Dicr Highr-Orr Dipacmn Mo. Compoi rucur. 996;5(): ah K. Fini Emn Procur in Enginring Anayi. Lonon: Prnic- Ha haar K Varaan TK. nchmar Eaiciy ouion for Locay Loa Lamina Orhoropic Cyinrica h. AIAA ourna 99;():67-6. Pino Corria a.

21 FIGUE Figur Conica pan hap mn wih igh no howing h noa ipacmn componn an h oca coorina rfrnia ax. Pino Corria a.

22 Figur Pinch cyinr howing h icri pan. Pino Corria a.

23 8 6 FDT HDT Tigh a. [] 8 6 Figur Naura frqunci in ra. - for a amina cyinrica pan wih py ayou [-5/5] impy uppor bounary coniion an /h5 for icr vau of b/h. Pino Corria a.

24 º 5º º 5º 6º 75º 9º Figur Variaion of h criica bucing xrna prur in KPa vru py ang for a [θ/-θ] amina camp conica pan. Pino Corria a.

25 TALE Tab - aia ipacmn (in) of h poin of appicaion of h forc P for an ioropic pinch cyinr. Mh Paaoo a. [] Prn wor FDT HDT Tab Normai cnra fcion w of a camp [9º/º/9º] amina conica h pan haar a. [8] E h w pl T w. Prn wor E L /E T N-HDT FDT HDT FDT Pino Corria a. 5

26 Tab - Normai raia fcion of a h poin of appicaion of h forc for a amina orhoropic pinch cyinr (L/). Layrwi Lov- Axiymmric Prn wor /h ouion Kirchhoff mi-anayic FDT HDT [8] ouion [8] mn [6] Tab Convrgnc of h funamna frqunci wih mh rfining for a amina cyinrica pan wih /h5 b/h an py ayou [-5 /5 ]. Frquncy Mh (ra. - ) f f f Pino Corria a. 6

27 LIT OF FIGUE Figur - Conica pan hap mn wih igh no howing h noa ipacmn componn an h oca coorina rfrnia ax. Figur - Pinch cyinr howing h icri pan. Figur Naura frqunci in ra. - for a amina cyinrica pan wih py ayou [-5/5] impy uppor bounary coniion an /h5 for icr vau of b/h. Figur Variaion of h criica bucing xrna prur in KPa vru py ang for a [θ/-θ] amina camp conica pan. Pino Corria a. 7

28 LIT OF TALE Tab aia ipacmn (in) of h poin of appicaion of h forc P for an ioopic pinch cyinr. Tab Normai cnra fcion w of a camp [9º/º/9º] amina conica ET h h pan w w. pl Tab Normai raia fcion of a h poin of appicaion of h forc for a amina orhoropic pinch cyinr (L/). Tab Convrgnc of h funamna frqunci wih mh rfining for a amina cyinrica pan wih /h5 b/h an py ayou [-5 /5 ]. Pino Corria a. 8