On the Non-uniform Torsion of Trapezoidal Thin Wings

Transcription

1 On he Non-uniform Torion of Trapeoidal Thin Wing Minghao Zhang College of Aeropace and Civil ngineering Harbin ngineering Univeriy Harbin, Heilongjiang, China Abrac The rapeoidal hin ing ih ymmerical airfoil (e.g. double edge--airfoil) i of coniderable imporance in aviaion indury. We conider a non-uniform orion iue ha arie in he udy of he orion analyi of a rapeoidal hin ing. Moivaed by hi ne engineering heory, i.e. he plaebeam heory pu forard by Prof. W.F. Zhang, hi paper ill firly bae upon he Kirchhoff hin plae heory and Vlaov rigid ecion aumpion o derive o mechanical model, i.e. he energy variaional model and differenial equaion model for he problem of he non-uniform orion for he rapeoidal hin ing ih double ymmerical airfoil. Then he approximae and exac analyical oluion of i angle for he canilever cae of a rapeoidal hin ing under oque applied a he free end are derived and preened by maing ue of he energy variaional model and differenial equaion model, repecively. Finally, he correcne of he analyical oluion obained from he non-uniform orion heory of he rapeoidal hin ing i verified by he FM numerical imulaion. Keyord: Trapeoidal Thin Wing; Non-Uniform Torion; nergy Variaional Model; Differenial quaion Model; Analyical Soluion I. INTRODUCTION The rapeoidal hin ing ih ymmerical airfoil (e.g. double edge-airfoil) ha been idely ued in aviaion indury. Therefore, he orion performance of hi ind of ing i of coniderable imporance in pracical aeropace engineering[,]. A i non o all, he heory of he non-uniform orion of a hin ing i he deign fundamenal of he fluer or aerodynamic inabiliy for high-peed aircraf and hence many reearcher have devoed hemelve o olving hee ind of problem. For inance, in he apec of recangular hin ing, Timoheno[,4] obained he decay index of re by energy varaional mehod baed upon he Foppl or on he re of he non-uniform orion, and derived he ip roaion angle of narro recangular plae under nonuniform orion; F.V. Chang[5] uilied he double riangle erie o derive he ip roaion angle and deflecion of a narro recangular plae under non-uniform orion baed upon he Kirchhoff plae heory and he principle of uperpoiion. Hoever, heir heoreical derivaion are no univeral, uch a he concep of decay index of re, and hence canno be ued in he cae of a non-recangular hin ing. Thi hinder he developmen of he heory of he nonuniform orion of hin ing. Moreover, from he lieraure revie, i i found ha fe or ha been publihed on he heory of non-uniform orion of a rapeoidal hin ing. The curren unaified au of reearch may be due o he complexiy of he problem of he non-uniform orion, coupled ih he grea difficuly of i mahemaic and mechanic. Therefore, he heoreical udy on non-uniform orion (i.e. rerained orion) member ha been developed loly[6]. In fac, from he poin of vie on he naure of mechanic, non-uniform orion exi in any ied member, namely, he non-uniform orion and uniform orion are co-exi in he orion problem of any hape of cro ecion, bu under exreme condiion, ome of he cro ecion, for example, he orion of a member ih circular ecion ill be dominaed by uniform orion. In oher ord, in he acual projec, "pure orion" member doe no exi. Conequenly, developing ne heory i of grea imporance for he heoreical reearch on he iue of he non-uniform orion. Recenly, a ne non-uniform orion heory, i.e. he plaebeam heory ha been pu forard by Prof. W.F. Zhang[7- ], hich i no only imple and eay-o-ue, bu alo poerful and univeral. Moivaed by hi ne heory, hi paper ill firly preen o mechanical model of he nonuniform orion for he rapeoidal hin ing ih double ymmerical airfoil, hen he exac and approximae analyical oluion are derived and verified by he FM imulaion. II. NON-UNIFORM TORSION THORY OF TRAPZOIDA THIN WING A. Problem Decripion and Aumpion ) Problem Decripion: Generally, he reearch objec in hi paper i a rapeoidal hin ing ih double ymmerical airfoil (e.g. double edge- airfoil). For impliciy, hi ype of ing may be hough of a a apered plae ih recangular cro ecion. One end of he apered plae i free and ubjeced o an applied orque M. The oher end i aumed o be fixed and canno arp. Moreover, he diribued orque m i alo applied along he cenroid of he apered plae. In hi cae he hin ing ill produce orion deformaion, and can be implified a a canilevered, apered plae ubjeced o a ip orque and diribued orque a hon in Fig.a. Knon: The lengh of rapeoidal hin ing i, he idh of roo ecion i h, he hicne i ; The elaic modulu of ing i, hear modulu i G, Poion raion i μ. When he apered plae i, he orion angle of cro ecion i aumed o be θ() a hon in Fig.b..ijer.org 88

2 ) Aumpion: a) Ignore he hear rain caued by ou-plane bending: Thi aumpion i imilar o ha of he "plane ecion" of uler beam. b) Ignore he deformaion in he mid-plane caued by ou-plane bending, namely, he mid-plane canno be reched: By hi aumpion, he bending and reching problem of a hin plae can be olved epraely. Thi i he deformaion decompoiion hypohei, hich i he heoreical bai of he plae-beam heory[7-]. c) normal re and rain in he normal direcion can ommied. d) Rigid ecion aumpion: Thi aumpion i imilar o ha of Vlaov[6], hich i idely ued in he heroical derivaion of hin-alled rucure. h (a) (b) Fig. Calculaion diagram and i deformaion of narro apered hin-plae, B. Mechanical Model of Trapeoidal Thin Wing ) nergy variaional model: To faciliae he decripion of he deformaion, he plae - beam heory inroduced o e of coordinae, i.e. he global coordinae yem xy and he local coordinae yem n, a hon in Fig.b. The cenroid coordinae of he cro ecion in he global coordinae yem are aumed o be (,). In he local coordinae yem, he coordinae of arbirary poin in he cro ecion are denoed a (n,). While in he global coordinae yem, he coordinae of arbirary poin on he cro ecion are denoed a (-n,-). According o he aumpion (d), he diplacemen of arbirary poin are[6] : x x n ; y y () in co r n n rn co in () co in n v n vn in co () Accordingly e can ee ha, hen conidering only he orion problem, he diplacemen of he cenroid of cro ecion are u ; v ; ; (4) Baed on deformaion decompoiion hypohei (b), hen he apered plae only occur ou-plane bending. A hi poin he laeral diplacemen of arbirary poin caued by ou-plane bending deformaion are ;, u, v n n (5) While he longiudinal diplacemen of he cro ecion hould be deermined baed on aumpion (a) [7-], ha i u n,, n n Geomeric equaion (i.e. linear rain) [6-] n v v n (6) (7) (8) (9).ijer.org 89

3 Phyical equaion (i.e. coniuive equaion) [7-] Baed on he aumpion () and (4), for he claical Kirchhoff plae model, here are ( ); G () Uing he above derived relaionhip, he orional rain energy of a rapeoidal hin ing can be obained eaily. Firly, ih he folloing expreion of rain energy U dndd () Vc and ubiuing q. (7) - q. () ino q.(), e can ge he orional rain energy of he rapeoidal hin ing a follo[7-] U Vc dndd n G n dndd V c If eing I I n dnd Ac c ( )an 44 h h () a he non-uniform orional rigidiy or arping orion rigidiy of he rapeoidal hin ing, and GJ G J 4G n dn Ac d h c ( )an h G () a he uniform orional rigidiy of he rapeoidal hin ing, no he orional rain energy of can be implified a[7-] U I GJ d (4) In he cae of a concenraed orque i applied a he free end and a diribued orque i applied along he lengh of a canilevered, rapeoidal hin ing, he correponding load poenial energy i V M m d (5) No he oal poenial energy of he orion problem of he rapeoidal hin ing can be expreed a[7-] I comp GJ m comp M d (6) Thi i he oal poenial energy for he non-uniform orion problem of he rapeoidal hin ing. I i conien ih he reul of radiional orion heory in he form. Hoever, he derivaion of hi paper i more naural, and only he Kirchhoff hin plae heory and Vlaov rigid ecion aumpion are ued. Uing expreion of he oal poenial energy (6), he non-uniform orion problem of he rapeoidal hin ing ubjeced o orion load can be ranlaed ino uch an energy variaional model: Wihin he range,, looing for a funcion θ() o mae i aify he pecified geomeric boundary condiion, i.e. he endpoin conrain, and he energy funcional defined by he folloing formula i minimum. Where F, d (7) F, I GJ (8) ) Differenial equaion model: The differenial equaion, along ih boundary condiion, can be readily obained by he principle of energy variaional, hich ae ha he rue ae of deformaion i diinguihed from all oher aically correc ae of deformaion by he condiion ha he energy funcional be a minimum, i.e. (9) Then e ge I d M GJ m Ue inegraion by par, e ge I d GJ m I GJ I M () ().ijer.org 9

4 Due o he arbirarine of in he above formula, e could ge he folloing differenial equaion I GJ m and he correponding boundary condiion Fixed end (cro ecion canno freely roae, nor freely arp) () ; () Free end (ih ip oque, cro ecion can freely arp) I GJ M I (4) Thu, he non-uniform orion iue of he rapeoidal hin ing can alo be expreed a follo: Wihin he range,, looing for a funcion θ (), i aifie he differenial equaion (i.e. he equilibrium equaion) (), and a he ame ime mee he boundary condiion (5) and (6). I hould be noed ha in he equilibrium equaion () given herein, he fir erm i he inernal iing momen caued by he non-uniform orion, and he econd erm i he inernal iing momen caued by he uniform orion, i.e. he free orion (S. Venan orion). I can be een ha, according o he propoed plae-beam heory, o ype of orion, i.e. non-uniform orion and uniform orion, could be inegraed in one mixed orion equaion naurally, namely, he "eparaed" radiional orion heory i included in one heoreical frameor, Therefore, he ne mixed orion heory ha imporan heoreical and pracical value. III. ANAYTICA SOUTION OF TRAPZOIDA THIN WING SUBJCTD TO TIP TORQU A. Approximae Analyical Soluion Baed on nergy Variaion Model In hi ecion, e ill bae on he energy variaion model o examine he non-uniform orion iue of he rapeoidal hin ing. For he purpoe of implifying hi dicuion, e conider here he cae here only concenraed orque i applied in he free end orque. In hi cae he oal poenial energy implifie o I GJ d M (5) In order o obain an approximae analyic oluion, e can chooe he rail funcion for he roaion of cro ecion a follo A x x (6) Obviouly, he above rial funcion aifie he folloing boundary condiion of he canilever plae ; (7) If q.(6) i ubiued ino q.(5), hen e ge c ch ( )an c 44 h A x x d ch Gc ( )an h A x x M and i inegraion reul i Where A x x 6A an A GJ 5 h A 9an A h 8an A 6an A 5h 5h ch c 44 c I I ; A M (8) h c GJ G c Baed upon energy variaional mehod, here mu be A (9) ().ijer.org 9

5 Afer arrangemen, e ge 8 an 5 h 9 an K h M A an 5 h 6 an 5 h Then he oluion can be obained here () 6 GJ A M GJ K () an an an 5 h h 5 h 6 an 5 h K I GJ () (4) (5) B. xac Analyical Soluion Baed on differenial equaion model In hi ecion, e ill bae on he differenial equaion model o examine he non-uniform orion iue of he rapeoidal hin ing. For he purpoe of implifying hi dicuion, e conider here he cae here only concenraed orque i applied in he free end orque. In hi cae he implified differenial equaion i I GJ comp comp (6) Uing he fir condiion of he boundary condiion (4), e can obain I GJ ( )an h ( )an M h (7) Thi i a more complex, hird-order differenial equaion ih variable coefficien. In hi paper, baed on i unique mahemaical rucure coniued, along ih he remaining hree boundary condiion, i exac analyical oluion i derived. Firly, q.(7) i rerien in dimenionle form. Seing Afer arrangemen, e ge ( )an (8) h Mh an GJ here K an ; K h I GJ (9) (4) Then, uing he heory of differenial equaion, he exac analyical oluion can be derived a follo here, Ah Ah Mh ln A an GJ ; (4) A, Ah and A h are hree conan of inegraion, hoe expreion are a follo, hich can be obained according o he end of he boundary condiion Mh co Ah ; ( ) GJ Ah ; Mh co A ( ) GJ (4) Finally, he maximum roaion (ip roaion in he free end) of he canilevered, apered plae can be obained.ijer.org 9

6 max co h M ( ) GJ ( )an h h co h ( ) ln ( )an an h IV. FM SIMUATION AND VRIFICATION (4) A. FM Model In order o verify he correcne of he above analyical oluion, he finie elemen ofare ANSYS i ued o eablih he model for he analyi of he non-uniform orion of he canilevered, apered plae ubjeced o a ip orque. The elaic hell elemen SH6 ih 4 node i choen o imulae he rapeoidal hin ing. The elaic modulu =.6 5MPa, Poion raio μ =.. When he model i buil, he ey poin of he lef and righ end are eablihed, and he apered plae i formed hrough hee ey poin, uing he SIZ command o conrol he lengh of elemen, hich i 5mm. Then, he freedom of all node a he fixed end are rerained o imulae he fixed conrain condiion, a uni of orque i applied a he cenroid of he cro ecion a he free end. In addiion, he CRIG command i ued in he finie elemen imulaion in order o define rigid region for each cro ecion. a, he orion angle a he free end i exraced afer he analyi. The correponding modeling, mehing, load and conrain and orion deformaion are hon in Fig.. B. Comparion of heoreical and FM oluion There are differen ie of he rapeoidal hin ing, uing he above heoreical mehod and finie elemen mehod (FM) o calculae he orion angle for all ing. The comparion reul are hon in Table I and Table II. From he analyi of daa of Table I and Table II, i can be een ha: () The reul given by he exac analyical oluion are almo he ame a hoe obained from FM imulaion. Thi prove he correcne of he heory of nonuniform orion preened herein. () The reul given by he approximae analyical oluion and hoe given by he finie elemen analyi are baically conien, and he error i ihin he range of -6.45% and -.76%. Hoever, he expreion of he approximae analyical oluion i more imple han he exac one. Therefore, he approximae analyical oluion i uiable for he engineering deign peronnel in he approximae calculaion or eimaion. Fig. FM model and i deformaion TAB I. COMPARISON OF TORSION ANGS BTWN XACT SOUTIONS AND FM SOUTIONS Number (m) h (m) FM oluion ( - rad) xac oluion ( - rad) rror (%) TAB II. COMPARISON OF TORSION ANGS BTWN APPROXIMAT SOUTIONS AND FM SOUTIONS Number (m) h (m) FM oluion ( - rad) Approximae oluion ( - rad) rror a (%) a. rro=(approximae oluion- FM oluion)/ FM oluion %.ijer.org 9

7 V. CONCUTION Theoreical udie and numerical imulaion pracice prove ha: () The plae-beam heory propoed by Prof. W.F. Zhang [7-] i univeral, and can be eaily ued o olve he nonuniform orion problem of he rapeoidal hin ing ih double ymmerical airfoil; () Deriving he differenial equaion model from he energy variaion model ha a clear concep of mechanic. () Approximae analyical oluion preened here i imple and pracical, and i baically agree ih he FM analyi reul. While he exac analyical oluion i almo he ame a he reul obained from he FM analyi. Thi prove he correcne of he oluion of differenial equaion model. RFRNC [] Y.C. Fung, An Inroducion o he Theory of Aeroelaiciy, Dover Publicaion, Mineola,. [] R..Biplinghoff and H. Ahley. Pringciple of Aeroelaiciy, Dover Publicaion, Mineola,. [] S.P.Timoheno and J. N. Goodier, Theory of elaiciy, Ne-Yor: McGra-Hill,97. [4] S.P.Timoheno, "On he orion of a prim, one of he cro-ecion of hich remain plane", Proceeding of he london mahemaical ociey, Vol., no.,9,pp: [5] F.V.Chang, On he rericed orion of narro recangular cro ecion by Kirchhoff hin plae heory, Applied Mahemaic and Mechanic, vol., no., Aug. 98, pp:57-5. [6] V. Z. Vlaov, Thin-alled elaic beam, Irael program for cienific ranlaion, Jerualem, 96. [7] Wenfu Zhang, "Unified heory of free and rericed orion for narro recangular plae", China Science and echnology paper online, hp://.paper.edu.cn/hml /releaepaper/4/4/.4. (In Chinee) [8] Wenfu Zhang, "Ne heory of orion and flexure-orional bucling for narro plae",proc. of he 5h Naional Sympoium on Modern Srucure ngineering,5, pp:78-74(in Chinee) [9] Wenfu Zhang, "Ne heory of elaic orional bucling for axial loaded box-hape eel column", Proc. of he 5h Naional Sympoium on Modern Srucure ngineering, 5, pp:79-84(in Chinee) [] Wenfu Zhang, "Ne heory of elaic orional bucling for axial loaded box-hape eel column", Proc. of he 5h Naional Sympoium on Modern Srucure ngineering, 5, pp:79-84(in Chinee [] Wenfu Zhang, "Ne ngineering Theory for Torional Bucling of Seel-concree Compoie I-column", Proc. of h Inernaional Conference on Advance in Seel and Concree Compoie Srucure. Beijing, China, 5, pp: 5-. [] Wenfu Zhang, "nergy Variaional Model and i Analyical Soluion for he laic Flexural-orional Bucling of I-Beam ih Concreefilled Seel Tubular Flang", Proc. of he 8h Inernaional Sympoium on Seel Srucure, Jeju, Korea, 5, pp: -8. [] Wenfu Zhang, "Ne ngineering Theory for Mixed Torion of Seelconcree-eel Compoie Wall", Proc. of h Inernaional Conference on Advance in Seel and Concree Compoie Srucure, Beijing, China. 5, pp: ijer.org 94