1、REPORT 1181STRUCTURAL RESPONSE TO DISCRETEAND CONTINUOUS GUSTS OF AN AIRPLANE HAVINGWING.BENDING FLEXIBILITY AND A CORRELATIONOF CALCULATED AND FLIGHT RESULTSBy JOHN C. HOUBOLT and ELDON E. KORDESLangley Aeronautical LaboratoryLangley Field, Va.Provided by IHSNot for ResaleNo reproduction or network
2、ing permitted without license from IHS-,-,-Q ,.,National Advisory Committee for AeronauticsHeadquarters, 151Y, H Street N_I:, Washington _5, D. C.Created by act of Congress approved March 3, 1915, for the supervision and direction of the scientific studyof the problems of flight (U. S. Code, title 5
3、0, sec. 151). Its membership was increased from 12 to 15 by actapproved March 2, 1929, and to 17 by act approved May 25, 1948. The members are appointed by the President,and serve as such without compensation.JEROME C. IIuNsAKEa. Sc. D., Massachusetts Institute of Technology, ChairmanDETLZV W. BROXK
4、. PH.D., President. Rockefeller Institute for .Medical Research, Vice Chairmal,Josher P. ADAMS, LL.D., member. Civil Aeronautic._ Board.ALLEN V. ASTJ.U, Ptl. D. Director, National Bttrettu of Standards.PRESTON R. BASSV-TT, M. A., President, Sperry Gyroscope Co.,Inc.LEONARD C?.RMICHAEL, PH D., Secret
5、ary, Smithsonian Insti-tution.RXLrH S. D IBy JoHN C. HOtBOLT and ELDON E. KORD_ZSSUMMARYAn analysis is made of the structural response to gusts ofan airplane hating the degrees of freedom of certical motionand wing bending flexibility and basic parameters are estab-lished. A conrenient and accurate
6、numerical solution of theresponse equ_tions is deceloped .for the case of discrete-gustencounter, an exact solution is made for the simpler case ofcontinuous-sinusoidal-gust encounter, and the procedure isoutlined for treating the more realistic conditi,m of continuousrandom atmospheric turbulence,
7、based on the methods ofgeneralized harmonic analysis.Correlation studies between flight and calculated results arethen gicen to ecaluate the influence of wing bending flexibilityon the structural response to gusts o.f two twin-engine transportsand one four-engine bomber.“ It is shown that calculated
8、results obtained by means of a discrete-gust approach reeeal thegeneral nature of the flexibility effects and lead to q_utlitativecorrelation with flight results. In contrast, calculations bymeans of the continuous-turbulence approach show goodquantitatice correlation with flight results and indicat
9、e a muchgreater degree _f resolution _ the flexibility effects.INTRODUCTIONIn the design of aircraft the condition of gust encounter hasbecome critical in more and more instances, mainly becauseof increased flight speeds and because of configurationchanges. Aircraft designers have therefore placed g
10、reateremphasis on obtaining more nearly applicable methods forpredicting the stresses that develop. ,ks a result, the numberof papers on this subject has significantly increase(. (See,for example, refs. I to 16.) .Many of the papers have treatedthe airplane as a rigid body and in so doing have dealt
11、 witheither the degree of freedom of vertical motion alone (refs.I to 4) or with the degrees of freedom of vertical motion andpitch (refs. 3, 5, and 6). In the main, these rigid-bodytreatments tacitl.v involve the concept of “discrete,“ “iso-late(l“ gusts, but more recently steps have been taken tot
12、reat the more realistic condition of (.ontinuous-turbulenceencounter in an explicit manner (see refs. 6 to 9).In addition to rigid-body effects, one of the more impor-tant items that has been of concern in the consideration ofgust penetration is tiw infhwme tilatwing fh, xibilit.v has onsllu_tllra r
13、esl)olse. This (OllCerll has Iwo nlaill aspects:(1) that including wing flexibility may lead to the calculationof higher stresses than wouhl be obtained by rigid-body treat-ment of the problem and (2) that wing flexibility may intro-duce some error when an airplane is used as an instrument formeasur
14、ing gust intensity. Thus, several papers have alsoappeared which treat the airplane as a flexible body. Inmost of these papers the approach used involves the tlevelop-ment of the structural response in terms of the natural modesof vibration of the airplane (refs. 10 to 15). In othex.,s theapproach i
15、s more unusual, as, for example, reference 16 whichdeals with the simultaneous treatment of the conditions ofequilibrium between aerodynamic forces and structural de-formation at a number of points along the wing span. What-ever the approach, however, these flexible-body analyses havetwo main shortc
16、omings. They too have adhered to the con-cept of simple-gust or discrete-gust encounter (ref. 10 is anexception) and also they are not very well suited for makingtrend studies without excessive computation time.The intent of the present report is to she( further lightupon the case of gust penetratio
17、n of an airplane having thedegrees of free(loin of vertical motion anti wing ben(ling. Ithas several objectives: (1) to establish some of the basic pa-rameters that are involved when wing bending flexibility isincluded, (2) to develop a method of solution which is fairlywell suited for trend studies
18、 without excessive computationtime, (3) to evolve methotls for treating continuous turt)u-lence as well as dis(rete gusts, and (4) to show tim degree ofcorrelation that can be obtained between flight-test and an-alytical results and, through this correlation, to assess howwell flexibility effects ma
19、y be analyzed. In effect, this reportis a composite of the discrete-gust studies made jointly bythe authors in references I I and 12 and of the contintlous-turbulence studies made by the fit_t atnthor in reference 1()and in nnpublished form.The report is developed as follows: The equations fi)r re-s
20、ponse (inchn,ling accelerations, displacements, and ben,lingmoments) are derived and the basic parameter.s outlined. Asimple solution of these equations follows for both discrete-gust encounter and for continuous-sinusoidaL-gust encounter.Next, the pro(edure for treating continuous atmospheric tur-h
21、uleuce is outlined. Then, the correlation studies involvinga comparison of llight-test results with the calculated resultsohtained for both discrete-gust and contimtous-turl)ulen(.econditions are -iven.fr,llt N .%f._ “IN 2 li:l I)3 Jolm (. loubcdt :tl_.,I Eht_rll F Kol I*-, lI,2. :tull N k “A !N 2“!
22、1“:4lllx.r_,ll._ A(A IN :llHal h_. ,.hll _. I.:ll,l,. I!l:l; :_.1:_o cqllthlill_ c,_.rnfi:tl Inat,Ti:t!t._. I I,h,r_ E K,r,l,._ :Lfl,t _ohn_ (“ II_,ilP,It. l_J.-_1Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2aa_AbCCOd,e,hEf(s)F9HIkLL,L_m31,SYMBOL
23、Sslope of lift curvedeflection coefficient for nth mode, function oftime aloneaspect ratio of wingspan of wingchord of wing :_-chord of wing midspansee equation (23b)Youngs modulus of elasticitynondimensional gust force, (s- e) dexternal applied load per unit spanacceleration due to gravitydistance
24、to gust peak, chordsbending moment of inertiareduced frequency, _-_.nondimensional bending-moment factor(34,=K_ 2 ,VUXf_o)wave lengthaerodynamic lift per unit span of wing due togustaerodynamic lift per unit span of wing due tovertical motion of airplanemass per unit span of wingnet incremental bend
25、ing moment at wingstation jimA.Prl, F2_ r:_St, rT(_), T(.q)/UVIVWREPORT l l81-NATIONAL ADVISORY COM_ilTTEEZ.tb12-1L =J_i cw.(y-y_)dy/_b/231, _ Jy t mw,(y-y_)dygeneralized mass of nth modeincremental number of g accelerationsee equation (58a)load intensity per unit spanwise lengthsee equations (I8),
26、(24), and (13)distance traveled, 2V t, half-chordsCowing areatimefrequency-response functionvertical velocity of gust or random disturbancemaximum vertical velocity of gustforward velocity of flighttotal weightof airplanedeflection of elastic axis of wing, positiveupwardw. deflection of elastic axis
27、 in nth mode, given interms of unit tip deflectiony distance along wing measured from airplanecenter line7/,Id, IPO“1-,Subscripts:expIFiJIgtnN0rtheoNotation:I l 1FOR AERONAUTICSVresponse coefficient based on a., _c0 asecond derivative of z0 with respect to ssecond derivative of zt with respect to sa
28、bsolute value of center-line deflection of fundamental mode in terms of unit tipdisplacementdistance interval, half-chords; also, strainnondimensional bending-moment parameter,8MapcoM, o_ICOreduced-frequency parameter, ff_nondimensional relative-density parameter,8M.a_oSmass density of airstandard d
29、eviation; also, distance traveled,2V- r, half-chordscofunction wlfich denotes growth of lift on anairfoil following a sudden change in angle ofattack (Wagner function)power-spectral-density functionsfunction which denotes growth of lift on rigidwing entering a sharp-edge gust (K0ssnerfunction)circul
30、ar frequencynatural circular frequency of vibration of n thmodefl=2_r _ 2kfrequency, -L-=_-=_.experimentalflexiblefuselageinputspanwise stationnumber of distance intervals travelednatural modes of vibrationnodaloutputrigidtheoreticalcolumn matrix when used in matrix equationssquare matrixDots are us
31、ed to denote derivatives with respect to time:primes denote derivatives with respect to s or or; a bar abovea quantity denotes the time average; and vertical bars abouta quantity denote the modulus.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-RESP
32、ONSE TO DISCRETE AND CONTINUOUS GUSTS OF AN AIRPLANE HAVING WIND-BENDING FLEXIBILITYANALYSIS OF RESPONSE TO ARBITRARY GUSTSEOVA_ONSroa ST_UCTUaALamPONSEThe following analysis treats the problem of determiningthe stresses that develop in an airplane flying through verticalgusts on the assumption that
33、 the airplane is free to respondonly in vertical motion and wing bending. The case of thetransient response to arbitrary gusts is considered first.A subsequent section is then devoted to the case of randomdisturbances in which explicit consideration is given thecontinuous nature of atmospheric turbu
34、lence.Equations of motion.-It is convenient to treat the problemsimply as one of determining tile elastic and translationalresponse of a free-free elastic beam subject to arbitrary dy-namic forces. For dynamic forces of intensity F per unit.length, the differential equation for wing bending is, if s
35、truc-tural damping is neglected,_2 _. _ _=-mwt_ O)where w is the deflection of the elastic axis referred to a fixedreference plane. The task of detelmMning the deflection thatresults from the applied forces F may be handled convenientlyby expressing the deflection in terms of the natural free-freevi
36、brational modes of the wing.The wing deflection is thus assumed to be given by theequationw-acwo+atwl +oaw2+ . . (2)where the a_s are functions of time alone, and the w,srepresent the deflections of the various modes along theelastic axis of the wing, each being given in terms of a unittip deflectio
37、n. In equation (2), w0 represents the rigid-bodymode and has a constant unit displacement over the span;the other ws are elastic-body modes which satisfy the differ-ential equationand the orthogonality conditionn mw,w, dy=O (re#n) (4)t/2=M, (m-n) (5)In accordance with the Galerkin procedure for solv
38、ingdifferential equations, equation (2) is first snbstituted intoequation (l) to give, after use is made of equation (3),al_, 2mwt + a2ca22mw2+ . m (_owo+ fi,wl + .) + F(6)Now if this equation is multiplied through by w, then isintegrated over the wing span, and u_ is made of equations(4) and (5), t
39、he following basic equation results:/-b/2M.a. +o_.2M.a.= J_, Fw. (7)which allows for the solution of the coefficients a, if theapplied forces F are known. This equation applies for thetranslational mode n=O, for which ease oJ0=0, as well as forall the elastic-body modes. The quantity M, appearing in
40、the equation is commonly ealled the generalized mass ofthe nth mode.For the present ease of the airplane flying through a gust,the force F is composed of two parts: a part designated byL, due to the vertical motion of the airplane (including bothrigid-body and bending displacements) and a part L, re
41、sultingdirectly from the gust (this latter part is the gust force whichwould develop on the wing considered rigid and restrainedagainst vertical motion). On the basis of a strip type ofanalysis, these two parts are defined as follows:F=L.+ L,-2 ocV fo _b l-, (t-r)dr+ a ocl“ fwhere t-O is taken at th
42、e beginning of gust penetration.l-(t) is a function (commonly referred to as the Wagnerfunction) which denotes the growth of lift on a wing followinga sudden change in angle of attack and for two-dimensionalincompressible flow is given by the approximation1-_b (t)la - = 1-0.165e-_t-o.335e -6_t (9)an
43、d _(t) is a function (commonly referred to as the Ktissnerfunction) which denotes the growth of lift on a rigid wingpenetrating a sharp-edge gust anti for two-dimensionalincompressible flow is given by the approximation_ (t)a., = 1- 0.Se-_v- 0.Se -2v it0)Figure 1 is a plot of equations (9) anti (10)
44、.An additional term which involves the apparent air massshould be included in equation (8) ; this mass term is inertialin character and may be included with the structural mass(see ref. 16) althol,gh it is usually small in comparison. Thelift-curve slope a may be chosen so as to include approximateo
45、verall corrections for aspect ratio and compressibility effo_ts.The remainder of the analysis is restricted to uniformspanwise gusts and the assumption is nmde that the responsewill be given with sufficient accuracy by considering onlytwo degrees of freedom: vertical motion anti fundamt, ntalwing be
46、nding. On this basis, if w as given by the first twoterms in equation (2) is substituted into equation (S) antithe resulting equation for F is substituted into equation (7 .the following two resl)onsc equations result when rt is setProvided by IHSNot for ResaleNo reproduction or networking permitted
47、 without license from IHS-,-,-4I0REPORT 1181-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSjl._“ ii .I!“_ “-ii-I r-“ 7 _ i ! i6 8 I0$, half-chordsi :I! Ii iiIi iIii i! i :Ji i ii 1l i i i12 14FI6LnE 1 -Unstead.v-lift functions (see e(im (9) and (10) where, for o“=-2 _.:o Izl lalld-)J“ 1 -(._-allda4-. (
48、 “ It- . It,/_t “-I _ t “Jc #IX2 -“I = /l-O _-r2 -I )l)/1 _“Whl Il.q.ll, “t8.11_t=am.o, q_1 (*lX=_3qI. I1_S,S,12=al|d tl,prin(, tltno|ts a derivative with re:,;po(t ht a, l_(lll,“t-tions (16) aml (17) are the basiu vt,sponse e(lualiozis ill ttwi)rl,soltl atmly.,*i_. The live I)aLanteter_ alpl)t,_l, ring ill thest.eqll,alil)Its altd given by eqllalions 11,) (lel)t, nd upon th,forw.rd v(.ht(.ily. _lir donsity, lift-_ulve slope, aml tit(, air-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-RESPONSE TO DISCRETE AND CONTINUOUS GUSTSplane physica
