1、NASAO_r,.r_lZCONTRACTORREPORTNASA CR-2779FINITE STATE MODELINGOF AEROELASTIC SYSTEMSRa.ja. W?aPrepared bySTANFORD UNIVERSITYStanford, Calif. 94305for Langley Research CenterNATIONAL AERONAUTICSAND SPACE ADMINISTRATION WASHINGTON, D. C. FEBRUARY 1977Provided by IHSNot for ResaleNo reproduction or net
2、working permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1. Report No. 2. Government Accession No. 3, Recipients Catalog No,NASA CR-27794. Title and SubtitleFINITE STATE MODELING OF AEROELASTIC SYSTEMS7. Authorls
3、)Ran jan Vepa9, Performing Organization Name and AddressStanford UniversityStanford, California 9430512. S_nsoring Agency Name and AddressNational Aeronautics and Space AdministrationWashington, DO 205465. RePort DateFebruary 19776. Performing Organization Code8. Performing Organtzation Report No,10
4、. Work Unit No,11. Contract or Grant No.NGL 05-020-24313. Type of Report and Period CoveredContractor Report14, Sponsoring Agency Code15. Supplementary NotesAdapted from Ph.D. Dissertation, May 1975Langley technical monitor: Robert V. Doggett, Jr. Topical report.i 16. AbstractA general theory of fin
5、ite state modeling of aerodynamic loadson thin airfoils and lifting surfaces performing completelyarbitrary, small, time-dependent motions in an airstream issystematically developed and presented. In particular, the natureof the behavior of the unsteady airloads in the frequency domain isexplained.
6、This scheme employs as raw materials any of theunsteady linearized theories that have been mechanized for simpleharmonic oscillations. Each desired aerodynamic transfer functionis approximated by means of an appropriate Pad_ approximant, thatis, a rational function of finite degree polynomials in th
7、e Laplacetransform variable.The modeling technique is applied to several two-dimensionaland three-dimensional airfoils. Circular, elliptic, rectangularand tapered planforms are considered as examples. Identicalfunctions are also obtained for control surfaces for two- andthree-dimensional airfoils,11
8、7.Key Words (Suggested by Authorls)Unsteady aerodynamicsAeroelasticityActive controls19. _urity Classif. (of this re_rtlUnclassified18. Distribution StatementUnclass i f i ed-Unl im i ted20. Security Classif. (of this page)Unclass i f iedSubject Cateqory 3921, No. of Pages 22. Price“188 $7.00* For s
9、ale by the National Technical InformationService, Springfield, Virginia 22161Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-INTRODUCTI ON .General l)cs
10、_-r-il_t io_. ,_! !i_. lr_,t)tem Rev icw _) Pc_-ctincnt i _:,ill t . .,qllnl!)lnF,/ i (kn_,t C_lt i_.i S SYNBOIff- FINLTE SfAF, W)l)l!,!h,; c_i ; l!:i:_ t _2h;.Finite ; .dealizati(m u_; a th:_m K,M FINITE SIAII blI)l)_l_I2q(: OV :_tR_IP,N:H(- l,(-!Al)_q .wo I)imellsulllll A 1 l(_i :; n(t_mprt_sl_ e
11、t tow S!t)_:()Ili, _ I(_.,.: ,qlltS(_V-,)iC _dt kt_llta;_,llJ(“ f(IW Tllree 1)_mensioilttl i,ii k in,4 Snvfcwes SUGGESTIONS FOR F:IirRt“ kl is explained. This scheme employs asraw materials any of the unsteady l inearized theories that have beenmechanized for simple harmonic oscillations. Each desir
12、ed aerodynamictransfer function is approximated by means of an appropriate Pad6approximant, that is, a rational function of finite degree polynomialsin the Laplace transform variable.The modeling technique is applied to several two-dimensional andthree-dimensional airfoils. Circular, elliptic, recta
13、ngular and taperedplanforms are considered as examples. Identical functions are also ob-tained for contro surfaces for two- and three-dimensional airfoils. N TR(H)UCTI (?NIn the last decade r,tpid advances have taken place in the area ofautomatic control of practicg_l engfneering systems. The vast t
14、ech-nologicai developments in autopilot: design and in the design of aircrafttake-off ,and landing systems hn_ led to the possibility of applyingthis technology to control the vibration modes of aircraft wing structuresand the elimination of aer(,clastic instnbitities in the flight envelopeProvided
15、by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-of the aircraft. Although the mathematical theory of distributed parametersystems has made rapid advances recently, it seems more expedient, from apractical point of view, to approximate aeroelastic systems, mai
16、nly aircraftwings and control surfaces, by finite state models. The techniques ofapproximating aircraft wing structures by finite state models which make useof finite elements and other structural idealizations are well known. Nosystematic techniques exist, however, for approximating the aerodynamic
17、 loadson these structures by compatible finite state models for aeroelastic purposes.Thus a systematic theory for approximating aerodynamic loads on aircraft wingsby finite state models used along with well-known techniques of structuralidealization could be tremendously useful not only for understa
18、nding aero-elastic instabilities but also in the development of control systems forsuppressing aeroelastic instabilities. Such theories can also prove helpfulin the minimum weight design of aircraft structures.This paper is concerned with the finite state modeling of aeroelasticsystems. The well-kno
19、wn theories of modeling of aircraft wing structures arebriefly presented. A general theory is then developed for the modeling ofunsteady aerodynamic loads on wings and airfoils. These aerodynamic modelsmay be used in conjunction with structural models for aeroelastic purposes.General Description of
20、the ProblemThe general techniques of calculating unsteady aerodynamic loads forsimple harmonically oscillating airfoils and lifting surfaces are out-lined in _ , 2J and 3. With these techniques it is possible tocalculate the unsteady aerodynamic loads for different modes of oscilla-tion at a given f
21、requency of oscillation. Little is known about cheanalytical behavior of these loads in the frequency domain. Thus it is2Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-first essential to identify the behavior in the frequency domain of theaerodynami
22、c loads on airfoils and lifting surfaces.The next step is to approximate this behavior in a manner that willpermit the construction of aerodynamic models, which maybe used alongwith structural models for aeroelastic purposes. Also, for the workdescribed in this paper existing techniques of calculati
23、ng aerodynamicloads for airfoils and lifting surfaces were utilized whenever possible.Review of Pertinent LiteratureThe theory of finite state modeling of structures for dynamicanalysis is well known. The various methods of weighted residuals 4,finite element techniques 5 and variational techniques
24、4 have allproved extremely useful for analytical modeling purposes. Recentlysystem identification techniques 6 have been formulated for approxi-mating structures by finite state models from experimental data.The author is not aware of any systematic modeling procedures inunsteady aerodynamics. Howev
25、er, there were several related developmentsin the past fifty years. In 1925, Wagner1,3 first studied thegrowth of lift on a two dimensional airfoil in incompressible flow dueto an impulsive change in the vertical velocity of the airfoil. Garrick7 , later showedthe relationship betweenWagners solutio
26、n and Theo-solution 8 for the lift on an oscillating airfoil. Sears 9dorset sshowedthe relationship between solutions for a sharp edge gust and asinusodial gust. R. T. Jones I0 first considered the aerodynamic forceson finite wings of elliptic platform in non-uniform motion in incompressibleflow. W.
27、 P. Jones 11 calculated the lift on rectangular and taperedwings for impulsive motion in incompressible flow. Lomax, et al., 12solved the problem of obtaining the lift and momentfor impulsive motion,3Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ex
28、actly, at the starting instant i.8i (t) flap rotation of ith flapo(_,t) torsional rotationLaplace transform of Wagner s function._(s) : sm(_)Vt VtT or -b wCartesia_ coordinate systemProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FINITE STATE MODELIN
29、G OF STRUCTURESFinite Element ModelsThe finite element method is often regarded as generating a dis-crete model of a physical system. The method may also be viewed as avariational technique, where an attempt is made to minimize the demandfor ingenuity in the construction of trial functions. This dis
30、cretematrix technique for the formulation and solution of linear dynamicproblems in engineering mechanics has been widely discussed in theliterature 5. This approach has proved useful in obtaining approxi-mate analyses of complex structural configurations that are difficultto handle by exact mathema
31、tical formulations. The bookkeeping requiredfor solving the large number of linear simultaneous equations involvedis readily handled by matrix algebra techniques and the resultinganalytical formulations are tremendously simplified. Thus concurrentdevelopment or revision of different sections of larg
32、e digital computerprograms to perform the analysis is feasible. This feature has led toits use and acceptance as a basic tool for dynamic structural responsecalculations. The aim of this section is to briefly describe how thetechnique can be useful for finite state modeling of wing structures.The ef
33、ficient utilization of the discrete element influence-coefficient approach in a digital computer would provide for con-struction of a stiffness matrix and a mass matrix for the entire wingI0Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-structure by
34、 simple superposition of each matrix, from correspondingmatricies for the discrete elements that modelthe wing in detail. A stiffnessand massmatrix generator program (SAMGEN)was written for this pur-pose. Several different types of elements were incorporated in theprogram. This program was found to
35、be extremely useful in structuraloptimization work 27.Idealization as a Beam-RodThe swept wing structure shownin Figure i is considered. Thewing is connected structurally to the fuselage, so that the root is notcompletely rigid. Further it is assumedto have a finite numberof trail-ing edge flaps, NF
36、 in number, which provide for the control torques.The flaps are assumedto function as rigid bodies which are good approx-imations for most control flaps. The wing is assumedto have a largeaspect ratio, at least, for structural purposes. This permits thewing to be modeled as a beam-rod. The actual ro
37、ot is replaced by aneffective root normal to the elastic axis of the wing. Rotationalinertia, shear deformations and sectional bending are neglected.Let h(x, t), e(x, t), _i(t) i = l, 2, 3, .- NF, be the verticaldisplacement of the wing along the elastic axis, the torsional dis-placement of the wing
38、 along the elastic axis and the angular displace-ments of the flaps respectively. The kinetic energy of the wing/flapsystem maybe written as11Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-i=l ii_- I .i )where _i(t) is as_-mr_!_,:ito be equal _,_: )
39、dxC“1 O, p .-Z.W :r_, j. _:( t “1A prime () w_ll b, use_i to rqu_-:en_ . -._,! . .37 to repr_sentThe aerodynamic fc_ ,_: and control L :u tv_,h;iauc_. _,. _,_,._of motion and boLLn_!ari ,malt +.i(:b_,ain !,he so.u%ionto the. pce_u_u _li_trJbuti,n on wing.- perfc_r;_:lng arbitrary oscillationsin two
40、dimens:ional incmlg_-_:_:_ibl_ Ylow. _-hem_thod is then generalizedand applJ_.d to two imun,_ real comp.-o:_ble flow p_obems.j_cO.pres_ihe !l_. - Aerody,lamic l_nds _m nrbitrarilyosilating rigid wiI_Hn in tw_-:iimeI_i,l_a incc,uq_sib flow can be syn-thesized f_or,_Wagne_ _ ;:oluton loz“ bhe lJ ft o_
41、l a wJn_ d_tci,oa sudden step-wise change in the do_,m_,a_h as _qeserJbed in Her. i. R. T. Jones I0has obtained an app_oJmate Laplace 1:ransfoz_r_ of Wagners indicialfunction. It can b,_ shown that eeuations or the pres,oure distributionand aerodyns.n_-i, loadf may b,_ obtaJrl= ! for eonw_rging or d
42、ivergingoscillatom: by _._:placJng C(k) by 6(s _6(s) where 9(s) is theLaplace transform of W_q._n,_r:, indicia! _imct:ion. Thus equations ofmotion of a wJng_ with a tr.a.!ing edge flap : or a wing with ana:,leron and _ Ilap _: , ; _ay b_ obtained by using the correspond-ing equ_t:iom_ for o_cilatot.
43、g moi, ior_. ,_“or converging oscillationsthe wake is asstm_e.d to b_: finit_ an,J dupendunt on initial conditions.In this s:etion_ Theodor_mn._ uircu!ation Izg function isan_lytical.ly e.oniJnu,-ci 7_- (:orl_,.!_iy_!_T _ “_, L_t ;_:_ ;iil o w_th no17Provided by IHSNot for ResaleNo reproduction or n
44、etworking permitted without license from IHS-,-,-oscillating components in two dimensional incompressible flow, andthe physical significance of the results explained. By representingTheodorsens function as a series of Kummer functions, an asymptoticexpression for Wagners indicial function is obtaine
45、d. The first twoterms of this approximation are identical to the approximation forWagners function obtained by Garrick. By applying the Padeapproximant theory, it is shown that Theodorsens function may berepresented by a sequence of rational functions which convergeuniformly. This method of approxim
46、ating Theodorsens function isgeneralized so one can construct rational function approximations notonly to Theodorsens function, but to gust response problems also.These rational function approximations may be easily inverted, usingLaplace inversion_ to obtain good approximations to various indicialf
47、unctions The lift and moment about the elastic axis on a rigid thin air-foil_ performing vertical translational or torsional oscillations canbe easily obtained by integrating the pressure and its moments (givenin Appendix B).These expressions are, 2 i s2+ 2s_(s) s-as +2$(s)(l+(_-a)s)Lb = 2wqb 2 -5 f. ,. l-f-5-Z . f . f.M as +2(a+_)(s).sls(a-_)-(7_+a )s +(a+_)2._5(s)(l+(_-a)sh/bC_18Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
