# Full text of "Horizontal tail loads in maneuvering flight"

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o CASE FILE^ I tt 54078 ifJ^m^*i*a NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL NOTE 2078 HORIZONTAL TAIL LOADS IN MANEUVERING FLIGHT By Henry A. Pearson, William A. McGowan, and James j. Donegan Langley Aeronautical Laboratory Langley Air Force Base, Va. Washington April 1950 BATIOML ADYlSmi GCmaTHm FOR AERomncics TECBHICAl NOTE 2078 EORlZOmiAL TAIL LOADS IS mimwmiSQ fligbi By Heiirj A- Pearson^ William. A. McGowan^ and James J. Donegaxt SI3MMAHY A method is glTen for determining the horizontal tail loads in maneuTerlng flight- The method is "based upon the assignment of a load- factor Yariation with time and the determination of a minimitQi time to reach peak load factor- The tail load is separated into Tarious comr- ponents. Examination of these coiDponents indicated that one of the components was so small that it could he neglected for most conYentional airplanes^ therehy reducing to a minimum the numher of aerodynamic parameters needed in this computation of tail loads . In order to illustrate the method,, as well as to show the effect of the main yariahles^ a number of examples are giren. Some discussion is given regarding the determination of maximum tall loads ^ maximum pitching accelerations^ and maxiincum pitching toIoc— ities ohtainable* im^OOTCTION The subject of maneuTerlng tail loads has receiyed considerable attention both experimentally and theoretically. Theoretically^ methods and solutions have been derived for determining the horizontal tall load following either a prescribed elevator motion (references 1 to 3) or an assigned load— factor variation (reference k-). The first approach has been adopted into some of the load require- ments where the type of elevator movement specified consists of linear segments whose magnitudes and rates of movement are governed 'bj the assigpuent of a maximum initial elevator movement consistent with the pilot Vs strength. The rates of movement and the time the elevator is held before reversing are so adjusted that the design load factor will not be exceeded. 2 HACA TN 2078 The results of reference 5 sho^y as is to "be expected^ that only vh^::i the aerod^iaaiiilc force eoef f iciants are accxErately loio^wn from wind— txmnel tests can good agreeuaant he ohtajned hetwean measxired and caloulated tail loads. At the design stage^ however^ only general aerodynamio and geoisetrio g^xiantities are arailahle and some of the more iBoportant stahility parameters are not toowx acourately, Thxis ^ the work inTolTed in the solution for the tail load following a giT^n elevator motion is not oonsidered to he in keeping with the accuracy of the results ohtained. Conseq,uentlyj there appears to he a need for an ahhreTiated desiga method of coiii)utiEg tail loads which, although incorporating approxl nations , will neyertheless he hased on the theoretical considera- tions of the prohlem. If the load— factor Yariation with time is specified and the corresponding tail load, eleyator angles, and load dlstrihutions are suhseguently determined, a simpler and equally rational approach to the tail— load prohlem can he loade. Although this approach has heen used to a limited degree (referenoe ^), several shortcomings hare limited its use* The pxirpose of this paper is to develop further the load— factor or inverse approach and to present a method of computing horizontal tail loads which is coi!i)rehensive and generally sinrple- To this end, (1) the shape of the load— factor curve and the mLnlmum time req.uired to reach the peak load factor have heen determined from an analysis of pull— up maneuvers that were a-vailahle, (2) the minimum time required to reach the peak load factor has heen determined from a theoretical analysis which is supported in some measure hy statistical data ohtained from a number of flight tests with airplanes of widely varying sizes, and (3) the equations relating the various quantities are presented. SIMBOLS h wing span, feet; also shape factor in equation (13) ht tail span, feet c chord, feet c mean aerodynamic wing chord, feet Gl lift coefficient (L/qS) C^ pitchingHiioment coefficient of airplane without horizontal tail (Mb/qS^) NAGA TH 2078 Guj. pitchingHmoment Goefflcieirt; of Isoiatad Jaorlzoaa^ tail surface g acceleiration of grayity^ feet par secoad per sacoBfl. 2 1 pitohing iQomeixt of inartla^ sl-ug-f eat ky radlias of gyration g-bout pitcJxLog axis^ feat K ©i!5)lrical constant danotlrig ratio of daii5)JLng xaoiaent of coii5)lata airplane to daTTipiixg moioant of tail alone L llf t^ pounds 2 local lift at aoj spanwlsa station m airplane mass, sliags (W/g) M moment , footr-pounds n airplane load factor at any instant N loaziiiinni increment in load factor q dynamic press^ore, potuads per sq.'uare foot (rP'S^j S wing area^ sgioara feet S^ horizontal—tail area, square feat t time, seconds t^L time to reach peak of eleirator daflection, seconds Y airplane true Telocity, feet per second W airplane weight, pouads x^ l6ngi3h l^om center of graTlty of airplane to aerodynamic center of tail (positiTa for conrentionyal airplanes), feet y* noxidiznensloml spaavise dimensioa (^) h ' HACA TH 2078 a^b^c 1 cozistaats occurring in equatioxxB (13)., (23)^ (26)., and A,B,C,D,e/ (30) Kj^yT^yKo po3astants occurring in basic differentla.1 eq.*uation (see eqiiation (3) and table I) A. time to reach peak load factor., seconds p ' loass density of air^ slxags per cubic foot T)^ tail effioiency factor (q^t/o.) a- ^ing angle of attack., radians a average angle of attack of horizontal stabilizer^ radians o^ tail angle of attack., radians 3 angle of sideslip, degrees 7 flight-path angle, radians attitude angle, radians (a + 7) 5 eleyator angle, radians e downwash angle, radians ( —O') i-j^ tail setting, radians Q?he notations a and a, a and 0*, and so forth, denote single and double differentiations with respect to t* buDScrlpt s: Initial or selected value t tail ■max Biaxiifflini value h zero lift geo geometric c ca.Tnber NACA TH 2078 MKEHODB Method of Deteirininlng the Dyna-mio Tall Load Basic equatlags of motion .— The siii5)le differential eq.'uatlons for the longitudinal lootioa of aja alrplaiie for any eleyator deflection (see method glTen. in reference 2) maj he written as ^rv -^^ q-s - (^ n^t ^ = , (1) IT "^ r - ^"t + -^ V ^M - rn^ 9^0- (2) Egiiatlons (l) aztd (2) represent sunTmations of forces perpendicular to the relative wind and of 3iioments about the center of graTlty. (See fig. 1 for direction of posltiTe qiuantities. ) Inopliclt in these equa- tions are the f ollowljag assuir5)tion^ : (1) In the interYal between the start of the maneuver and the attainment of maximum loads ^ the flight— path angle does not change materially^ therefore, the change in load factor due to flight—path chaoge is small. (2) At the Mach nuoiber for which computations are madej the aero- dyoajnic derlvatlYes are linear with angle of attack and elevator angle - (3) The yarlatlon of speed during the maaeuTer may he neglected. (4) ITiisteady lift effects may he neglected. By use of the relations Q = 7 + a, = 7+ d, and = 7+ a, equations (l) axtd (2) are reducihle to the eguiyalent second-order differential equation a + K^d + K2 Aa - K3 m (3) where K-l^ ^^ ^^^ ^^ ^^® constants for a given set of conditions (see table !)• mcA m 2078 IQ equations (l) aad (2), Aa^ y, Bj Zs6, aad AL^ vlll^ In a glTSD: i]3aa0UT0r^ Tary with time. Using the relations "between 0> 7, a^ and their deriYatiyes permits equation (2) to t)e rewritten as follows to gire the increment in tail load: ^t-loT^^lx^ xt X,, ^ d5 %^ -b^x^ ^ ^"^^ In a still shorter fona^ egxmtion (k) inaj be written as AL4. = i5L. + AL. + AL. + AL. (5) ^ % % ^7 ^c Equations {k) and (5) show that the tail— load increment (the increment above the steady-"f light datum -value) at any time is coii5)Osed of four parts: AL. ♦ associated with the angle--of-attaclc change; AL^,,^ asso— ciated with angular acceleration about the flight path; ALj. ,, associated with angular acceleration of the flight path; and Mj^ 3 reijuired to c coicpensate for the moment introduced by change in camber of the horizontal- tail surface. The load /iL^ is generally ^mall but In some extreme configurations may amount to 10 percent of the total increm^at and thus for the present it is retained in the development. If the load— factor—increment variation with time An is Imownj then by the usual definition so that dCr g = ft W/S da and .. / dCr WACA TH 2078 Tlie follo'wl.ns relation also exists between. An and. 7: Aa g = jfW (7) so that 7 = ^ (8) When equations (6) to (8) are substituted into equations (If) and (5), the four tail— load components then become ^t, = a:^i-Aa (9a) ^% = - ^ ° ■ (9b) ^t,=^^^ta^^ (9d) .Tiiiis^ If the Yarlation of the load factor with tliiie Aa aad the geoinetrlc and aerodyaamic ciiaracteristlcs of the airplane were fcaown^ the first three coirqponents of the tail load could he found .iTnmediately. The magnitude of the fourth ooir5)onentj that due to horizontal-tail caniber, would follow from equation (3) in which the eleTator angle is seen to be 1L3 £3 K3 Substitution into equation (lO) of the values of Ax^ a^ and a ftrom equation (6) yields the Talue of the elevator angle at any instant 8 „ mCA TN 2078 A5 = ^ (a + K^^n + K2 Ar^ (11) •^ da ^ so that, fjjmllj, the fotirth casaponent Is glTexx as AL. = r^ T]. ^i— . .y^ fn + Enfi + Kp An) (12) to ^3 The procediire outlined shows that the tail— load magnitude oaxi "be determined if the load— factorr variation is kno'wn. Types of load--Caotor Yariation *— The relation between the tall load, the geometric and aerodynamic characteristics, and the load factor haTing heen established, it is desirable to establish a load— factor yariatlon which is reasonable as well as critical insofar as loads are concerned. The maximum value of load factor is usually specified; however, there are many possible variations for the shape- Eegardless of the details of shape, the load factor may be considered to rise smoothly aad con- tinuously to a maximum, the rate of rise dependiaag upon several variables • Beyond the maximum value of the load factor the return to initial condl— tioixs caa, at the will of the pilot, be either gradual or rapid. Experiments as well as theoretical studies have already Indicated that the maneuver that comblxies maximumL angular and linear accelerations causes critical loads in both the wing and tail. One such maneuver occurs when the maximum, load factor is reached as rapidlj as possible by using an initial elevator movement which is greater than that required to reach a given steady— trim value of the load factor. This initial elevator move- ment is followed by a rapid checking of the maneuver either by returning the elevator quickly to neutral or l:)j reversing the controls. The shape of the load— factor curve for such a maneuver may be expressed approximately by several analytic functions, one of which is An = at^e"^* (13) By way of illustration, figure 2 sliows details of the shape of the load- factor curve obtained with the use of equation (13) for which the constants have been adjusted so that an 8g peak is reached in 1 second. By further NACA m 20T8 9 adjiistmjsat of the coaa^tants the loa^L factor caa, withlii certatet liiriits> "be made to rise to amy specified peak aoad to diminish. ±xi any presorihed manner; Becaiise the positiye slopes obtained from, equation (I3) are always greater than the negative slopes , the positiTe angular acGelerations are greater than the negative ones. In. general ^ this condition is true for most high g critical maneuvers performed "by loost classes of airplanes, hut 3iianeuyers may occasionally he performed for vhich the reverse may he true, particularly for small airplanes. Determination of constants. — 'From equations (9)^ (il)j QJ^d (12) the required quantities relating to load factor are seen to he ^n, n, and n. Since the increm.ent Mi is to he given hy An = at^e^"^ (I3) then at maximum load factor 4-) n = 0=Ari^-c (ll^) is Thus t = — at maximum load factor. Let N = An^^y. Tlaen N = 4) ^'-b so that ^ ^ / t \ e (15) (16) Let - = X. Then c m.h^J'(}-j) (IT) Equation (17) is in nondimensional form vhere X is the time to reach the peak load factor and b is a constant. 10 lACA TN 2078 When eqiiatioii (17) Is dlff ereaatiated, the first and second derxva— tlYes became f^i^-') .(18) and £h - iV - 22. + 1 ^-^ ^2\ W t (19) In epilations (IT) to (19) the qtiantitles N^ X^ and "b are now regialred in order to determine the variation of Aa^ n^ and ii. The value of H is iDamediatelj availahle from the required loaneuTer load factor^ "Whereas the time to reach the peak load factor X can he oh- tained from examination of availahle records or by specification. The constaat h^ as may he seen from equation (l?)^ can best be described as a "shape" factor and has no particular physical sig^alficance. The values of A. and b should be associated with a maneuver which produces maximum tall loads. Therefore the time X to reach peak load factor should be the minimum possible consistent with possible pilot action and airplane response. The shape factor b should also be con- sistent with both of these. In connection with the determination of the minimum time to reach peak load factor^ the results shown in figure 3 for a typical airplane are informative. Figure 3(a) shows the load— factor variation following several abrupt Jud?) elevator movements. The load factor varies with the elevator position, but the time to reach peak load factor does not. Figiore 3(b) shows the load— factor variation for several abrupt hat- shape elevator Impulses. Again the load factor is seen to vary with the amount of elevator deflection but the time to reach the peak value reioalris constant. Although the time to reach the peak load factor shown In figure 3(b) remains constant, it is seen to be less than that shown in the previous .case; therefore, an lir5)ulse elevator motion pro^ duces a smaller value of X than the J-uir^? type. Because of Inertia and elasticity in the control system, the pilot cannot move the elevator instantaneously but requires some finite time tj_ to do so. A possible critical type of elevator iir5)ulse thus appears to be one which Increases llaearly to maxiJmiDi and decreases at the same rate to zero. In order to determine the minimum time, to reach peak load factor associated with such a variation, the equation of motion (equa- tion (3)) has been solved for the triangular elevator impulse for airplanes of various static stabilities and damping. mCA TN 2078 11 The results of the computations axe glTen in figure k in which the Hilnlmum tiine X to reach peak load factor is plotted against the time t] regulred to deflect the elevator. For coir5)leteness the curves of figure k are labeled for the actual values of E^ eiirplojed in the con5)utation as well as for relatiYe values of stahility. By a series of coii5)utatiQns the daiDping term K^ was founds as was to he expect ed^ to have onlj a secondarj effect on X- The curves apply to an average value of the dair5)lng constant. The upper curve, laheled "low stability j>" should he associated with rearward center-of-^avity positions (that is, low static margin) in coinbination with one or both of the following: low dynamic pressure or heavy air- planes. The lower curve, labeled "high stability," would be associated with forward center-^of --gravity positions in coirflDination with one or both of the following: high dynamic pressure or light airplanes. It is seen that X increases almost linearly with tj_ and also increases when the restoring forces are reduced, that is, when the stability is reduced. A preliiainary value of the shape factor b (required in eq.ua— tions (17) to (19)) was initially determined from flight records of typical in5)iilse maneuvers by fitting curves of the type given by equa- tion (13) through several points of the actual time histories and determining the constants- The results of this first step were then modified by the results of the same computations which had been made to determine X, QJ^d. the variation of b with tj_ given in figure 5 was obtained. Since the b factor is not found to be critical, an average value of 5.0 is suggested, although as a refinement the values from figure 5 ^J^y ^® used. The question of the value of t-^ to use is one which mast be solved either from experience or from a knowledge of the characteristics of the controls and the control system. For conv^ational airplanes having the usual amounts of boost and no rate restrictors, the following values of tj^ are suggested as representative: H Fighters or small civil aiirplanes with weight limit from about 500 to 12^000 poimds^ seconds ...••..••...« 0.20 Two^ngine airplanes with weight limit from 25,000 to ^5^000 poxxnds, seconds .•.••,.•.••••.«••. 0,25 Four— engine airplanes with weight limit from 50,000 to 80,000 pounds, seconds •..•«•.•.•«.••.••• O.3O Airplanes with weight limit above 100,000 pounds, seconds . • ; O.to 12 IWCA TN 20T8 The mlidjmm time X sXren In figure 4 was actimllj ©stal)lls]aad. *separat©lj from tlie adopted load— factor Tariatloni therefore^ to applying tlx© irnrerse matliod^ tlie deriTed elevator iicp-ulse wotild not "be expected to agree in detail with the "taat" type iirrpiilse iised in the derivation- The first three tail— load c6iii)aaents can now *be coirputed by insert- ing the Talues of 4n, n^ and ii from equations (17) to (19) toto e(iiiation (9) 3^d xising appropriate Talues of X from figure 4- In , order to facilitate this confutation, curTes of Aa/l, ixX/By and nX /H plotted against t/x are giyea in figure 6 for the suggested Talue of h = 5. Actually to apply the results of figure 6 it is conTeniaat to find first the coiii)onents AL , and so f orth_, in te3:ins of the nondi— <X meiisional time t/x and then to convert to time t in seconds. In order either to compute the fourth Gomponexit or to obtain the elevator angles for use in chord loading, the constants K^^ "^2, and Ko of equation {3) loust also he known. Thxis, in terms of t/x and the ordinates of figure 6, the various tail—load components are ^^ = ^ S/ordinate of fig. 6(a)) (20a) AL^. = ^^ S-fcrdinate of fig. 6(c)) (20I0) gScixt-^/- ^t; -Wk2 ^^ X, SfQrdiixate of fig. 6(b)) (20c) 1. A. \ / AL. - "^^ ^ ^t" ¥ M ^ 2 Orrdlpate of fig. 6(c) ^ X^ Kn (Ordinate of fig. 6(b)) , \1 v -ii — *~1- + K2(0rdinate of fig. 6(a)j|(20d) The constants K^^ Kg^ and Eo defined in tahle I are the same as those given in reference 2^ except for changed signs caused hy specifying x^ as positive. MCA fN 2078 13 The conversion to time t is made by m-ultlplyi33g values of the base scale t/X by X^ Saiople 'calculatlohs^ f^ incretaental tail loads >^ The results of several examples are given to illustrate nOt only the method btit also the effect of each of a number of variables on the incremental tail load of a typical fighter airplane^ the geometric and aerodynamic character- istics of which are gi'V'en in the following tables. In order to illustrate the effect of static stability^ results have been computed for three center— Gf--gravity poBitions with the assumption that ati 8g recovery is made at 19>100 feet from a vertical dive at an equivalent airspeed of 400 miles per hour* In order to illustrate the effect of the time of the elevator iMpulse on the tail load^ computations were carried out at one of the center— of -gravity positions for several values of t^. The cases considered and the airplane characteristics follow: GEOMETSIC GHARACTERISTICS Gross wing area^ S^ square feet «•.«•;..••..••«• 3OO Gross horizontal— tail area^ S-^^ square feet .•.••••••• 60 Airplane weighty W^ pounds «•..•••.•.••«.•••*, 12^000 Wing span^ b^ feet « « « • . «••««••> ••«••••••• 41 Tail span^ b^^ feet •.•#«•....•.««...••..* I6 Radius of gyration^ ky^ feet .•*...•..••••..•.• G.k Distance from aerodynamic center of airplane less tail to aerodynamic center of tall^ x^^ feet: Center of gravity^ 29 percent M.A.G. *«..•.•.•••# 20,0 Center of gravity, 24 percent M. A. C* «..«*•.,,..• 20,3 Center of gravity at aerodynamic center n , • . , , • • , , 21*0 AEROEOTMIC CEARACTEEISTICS Slope of airplane lift curve^ dC|./'da,^ radians • , . . • , . • . 4.87 Slope of tail lift curves dCj^/dcc^^ radians . , . . , , . . , • 3.15 Downwash factor^ dc/dcx ..*..*.., , . * •*•..•... 0.5^ Tail efficiency factor h /q\j r\ «••••«•#.,.«•«• 1,00 Empirical airplane daiirping factor^ K .*.*..•••••.• • 1,1 Elevator effectiveness factor^ dCj_/d8, radians .•.*.••• 1,89 Rate of change of tail moment with camber due to elevator angle > QG^ Idby radian •*..•........•• -^.57 ik MCA TN 2078 Rate of change of moment coefficient with angle of attack for airplane less tail, dCj^/da, radians: Center of grayityj 29 percent M.A.C. • . . . . . • • • • •: 0.625 Center of gravity, 24 percent M»A.C, . ••••.. , • . • 0.^*03 Center of gravity at aerodynamic center . . . • • . . • • . 0,000 The specified conditions for the saiarple computations are given in tahle II, The computed results for tail components are given in fig- ures 7 and 8, :Figure 7 gives results for varying the center of gravity and figure 8 gives similar results for varying t-j- . The tail— load components are computed from equations (20) and the derived elevator angles, from equation (ll). If the increment in tail load due to camher and the incremental elevator angle are not required, the K values need not be computed and the computations are considerably shortened, -Fig- ures 7 and 8 show that a maximum error of only about h percent is introduced by this omission. Method of Determining the Total Tail Load The initial or steady— flight tail load and elevator angles to which the computed incremental values are to be added must also be determined. In steady flight, the horizontal tail furnishes the moment required to balance the moments from all other parts of the airplane so that the initial load may be written as . *"o "*-"'m wo / ^^ \ ^o = "-^^r- + 3n7 T^ =°^ ^o (21) Thus the total tail load at any time in a maneuver is composed of the four previously mentioned parts plus the components given in equa- tion (21), Only the first term of equation (2l) represents a new type of load because the second term is a load of the type given by equa- tion {9^) or equation (20a) and its effect may be imnediately included in the computations by multiplying the ordinates of figure 6 by N + cos 7^ instead of by N, The initial elevator angle required to balance the airplane in steady flight varies with airplane C^ and center-of— gravity position so that, in general, 6q must be obtained from wind— tunnel data. With- out results of wind— tunnel tests, a rough rule which can be used as a guide at the design stage in determining the elevator position is that NACA TN 2078 15 the final elevator setting will te so adjiisted by repositioning of the stabilizer setting dtiririg acceptance tests that it "will be near a zero position at the cruising speed and at the most preTalent center— of — graTit J position. Method of Detersdning Maximum Yalues Maximum tail loads and ani^ar acceleratiozis .— The method outlined, enables a point-by— point eTaluation to be made of the quantities that determine the tall load. Such detail may often be xinnecessary and the procedure may be shortened by eraluating only those points near the load peaks or J altematiYelyj by accepting an approximation to the results. One such approximation which may bq made is to balance the airplane at the combinations of load factor and angular acceleration which would result in maximum up and down tail loads. Figure 7 shows that the maximoxm. dowa tail load in a pull—up occurs near the start of the maneuver and before appreciable load factor is reached. This maximum load is practically coincident with the negative maximum in the Lj.., tail— load coiiqponent- Slnce^ for a given configu— a ration, this coir5)onent increases as the center of gravity is moved forward ' and since the steady— flight down load increases with speed, the maximum doWQ tail load in a pull-up occurs at the highest deslgp. spieed in combin- ation with the most forward center-of-igravity position. Figures 7 and 8 show that at the time of the maximum down-tall— load increment the elevator is near but has not q.uite reached its peak position. Also at the time of maximum up— tail— load increment the elevator is near its zero position, althotigh it may be on either side of this position depending upon the stability and the time t^. These results suggest that the maximum down load for the elevator and the hinge brackets would occur with the airplane center of gravity well forward and at the start of the maneuver. The maximum load for the stabilizer is likely to occur at the peak load factor. Figure 7 also shows that the up tail load occurs near the peak of the L-^;; component as well as near the positive maximum peak in the L-t- C03i5)0nent. Since the L-^ component increases as the center of gravity is moved rearward and since a decrease in speed generally reduces the initial down load, the maxlmzxm up tail load occurs at the upper left- hand corner of the T— n diagraju for the most rearward center— of~gravlty position. l6 NACA Tn 2078 The inaxiisum tail load In a puLl—up maneu'ver waj be written as G^ qSc r\n where the smn of the second and third terms is to *be a maximpm in the maneuTer. l^om the preTlo"U3 discuss ion the load— factor increment at maximum down load is nearly zero and at maximum up load it is^ nearly equal to IT so that if the positiye and negatiTe values of 6 can he determined^ a relatlTely simple method for dete3rminlng maximum loads is aTailahle. Since hy definition = bo + 7^ an expression for angular acceler- ation can he derived from equations (6) and (7) and written in the form W/s •• q • The maximum angular acceleration can he approximated hy e. max '^I^^ ""^YX ^^^^ dHT ^ For the maximum positiye pitching acceleration, B is the maximum positive ordinate in figure 6(c) and C is the ordinate of figure 6(h) at a value of t/x for which B was determined. Thus, B is 6.5 and C is 0.95 for this exan^le- For the maximum negative pitching acceleration^ B is the maximum - negative ordinate in figure 6(c) and G is the ordinate of figure 6(h) at a value of t/x for which B was determined. Thus, B is —5.8 and C is 0.8G. For use in equation (23) the values of X for the maneuver are availahle from figure k- and the other quantities are avallahle from the conditions of the prohlem. The maximum loads can he given hy the following equations: NACA TH 2078 17 For BmxiiDMiii up tail load in tlie pTiLL-up: ^-fcinax+ ^r^ ^ 1%: tit ^^ "^ ^^ ^ X dC, T^^ ^^^^ For inaKimum down tail load in the pull— Tip: T - ^^ ° 4. ^^m WS „„„ ~ _ ^ N / 6.5 ¥/S . O.Q^g l /ph-KX For jpvBhr-d^omxB to limit load factor^ equations (2k-a) aod. (241)) still apply with Ghanged slgas for N axid chajoged directloBS- for L^^ ajad inax+ L^ . A question arises as to whether the maxinrum down tall load at inax— the start of a pull—up with forward center— of-gravlty position is greater than that which would occur when pulling up from a negatlTe load— factor condition^ with the center of gravity in the most rearward position. This can be determined only by computing both cases and seeing which is the larger. Maximum value of angular Telocity .— The maximum value of the pitching angular velocity in the pull-up may also be found in a manner similar to that used to obtain the maximum angular acceleration. Since = d + 7 and the relations involving these quantities in terms of load factor are given by equations (6) and (7), the following eq.uation may be written: The iraximum angular velocity may be approximated by a-i,|^..^ (.6) da 18 NACA Tn 2078 vhere D is the TO.x1.mm posltlYe ordixLate in f igwe 6(*b) «xid 1 is the ordixiate of figtire 6(a) at a value of t/X for which D was determiiied. Thxis Dy for this ^xmsple, is I.95 and E is 0.48. In the steady tixm or puLl—up at oan^tant g^ the ang:ular Telocity is Tisttallj given hy the expression Q = 1.0 ^. The difference "between the factor 1.0 of this expression and the factor 0.48 of equation (26) is more than made up by the angle-of--attack component of the angular velocity. Approii333ate Method of Determining Load Distrihution Symmetrical loading .— The spanwise distrihution of the total load can he fo3?mulated with various degrees of exactness. If information regarding details of the angle-of-ettack distribution across the span were available, then an exact solution could be obtained for the loading with the use of existing lifting-surface methods. The following method may be used as a first approximation to the solution. 'From the total tail load^ the total tail lift coefficient 0^ can readily be found. The average effective angle of attack a of the stabilizer portion is given in the definition \'-Jo %"§^^Vo %("°*^'5 t*^ (2T) where only a is assumed as unknown and cj and c? may be taken as the rates of change of section lift coefficient with a and 5, respectively. Thus, for constant elevator angle across the span. (28) HACA Tn 2078 19 In a practical casa "both integrals in eqixatlon (28) need "be eYaluated only once for a giren configuration and Mach nmiber. A plot of a against Gj_^ with 6 as a parameter woxild "be iis^fall in further ooinpvir^ t tatlons. with a kaowx as a function of CL- and 6. the local lift at any spanwlse station is then ohtained from the expression I = c^ge - [^cj^ a + oi^[% + A5j] qc (^9) IMsymmetrlcal loading *— Up to this point the total loads have "been assumed to he syoimetrical about the airplane center line, whereas^ in reality, the load may haTe an unsyimaetrical part. The sources of this dissyomietry may "be due to une'ven rigging, differences in elasticity hetweaa the tvo sides, or to effects of slipstreairL, rolling, and sideslip. The first two sources are usually Inadvertent ones while the last two are dlfficuilt to detemnlne without either wind— tunnel tests or a Imowledge of how the airplane will he operated. Present design rules regarding dis- symmetry of tail load are concerned laore with proYidlng adequate design conditions for the Te^T of the fuselage than with recognizing that at the maxiHEum critical tall load some dissymmetry may exist. Tests in the Langley full— scale tunnel (reference 6) and flight tests (reference 7) of a fighter-type airplane^ as well as unpublished flight tests of another fighter-type airplane, indicate that the tail- load dissymmetry varies linearly with angle of sideslip so that the difference in lift coefficient between the two sides of the tail can be given as ^L^ -% = AP ' (30) ^Right ^eft The average values of A per degree found for the two fighter— type airplanes are approximately 0.01. Ho similar values are available for larger airplanes nor for tail surfaces having appreciable dihedral. In 333aneuvers of the type considered herein it is doubtful that angles of sideslip larger than 3^ would be developed at the time the maximum tail load is reached. If the value of the sideslip angle at the time of maximum tall load can be established, equations (27) to (29) are easily modified to include this effect, provided the approximate value of A is loxown. 20 MCA TN 2078 Ghortiwls e loadl3ag > — The chordwise MstributicxQ can be detaririlxted for axij one spaawise station in either of two ways. One ^aj for design work is outlined in reference 8* A knowledge of tke airfoil section and the q.xiantities contained in equation (29) suffices for this deter— mi nation* If pressure-distribution data are available for a similar section with flaps ^ ajx altei*nate way would, be to distribute the load chordwise according to the two-dimensional pressure diagrams with the use of the coinputed values of section lift coefficient and elevator aiagle. DISCDSSION The loethod presented is another approach to the determination of tail loads. 'From the results given in figures 7 s^^ 8^ it can be seen that the camber coir5)onent L^^ is so sjmll that for all practical cases it may be omitted with cojisiderable simplification in the coii5)utation of tail loads. This omission reduces to a minimum the nuiaber of aerodynamic parameters needed to compute the tail loads. It is possible^ in the application of the present miethod with the use of the suggested values of t-i_^ that the derived elevator angles may not be within the pilot's capabilities. Since it mxist be assumed that all airplanes^ to be satisfactory^ should have sufficient control to reach their design load bounda^riesj such an occurrence requires only that the time to reach elevator peak deflection t]_ be increased so as to reduce the elevator angle* The results of figure 8^ in which the time ti is varied^ furnish a useful guide for determining the in- crease ti that might be required. If sufficient information is available, it is recommended that existing lifting-surface methods be used in determining the spanwise distribution of the total load; however, if information of the angle- of-attack distribution across the span Is not fcaown, the method presented may be iised as a first approzl7nation> Along some of the boundaries of the V-na diagram, tail buffeting may occur. Measurements show that buffeting usually occurs along the line of maximum lift coefficient and again along a high— speed buffet line which is associated with a conrpressibility or force break on some major part of the airplane. All ai3rplanes are subject to buffeting at the design conditions associated with the left-hand comer of the V—n diagram. NACA TN 2078 21 Only Mgh-epaed aod/or highr-altltude airplanes are capable of reaclilag the other hoimdary. Measiireinieaats show that the oscillatory TD-uff etlng loads loay he so high that the designer shoxild at least he cognizant nf them at the. design stage. The maxlmma angular acceleration varies inTersely with airspeed and directly with the load factor^ with the contribution due to acceler- ation in angle of attack likely to be more iir^ortant than the angular acceleration of the flight path- A somewhat similar variation is indi- cated for the maximum angular Telocity (equation (26)) where it is seen by direct substitution that the part due to angle of attack is likely to be larger than the part due to the angular Telocity of the flight path. COUGLDDIHG EEMAEES A siii5)le method has been presented for determiniisg the horizontal tail loads in maneuTerlng flight with the use of a prescribed incremental load— factor Tariation- The incremental tail load was separated into four components repre- senting a, a^ yy and c The camber ■ coirqponent L^ is so small that c for most conTentional airplanes It may be neglected, thereby reducing to a minimum the number of aerodynamic parameters needed in this coxi5)utatlon of tail loads. An approximate method is presented for predicting maximum angular accelerations and maximum angular Telocities . The method indicates that maximum tail loads in a pull-^p occur at forward cante3X)f-graTity positions and early in the maneuTer. The maximum down tail loads in a pull-^p occur at the highest design speed in combination vith the most forward centei^of— gravity position. The maximum up tail load occurs at the upper left— hand comer of the V— n diagram for the most rearward cent er-of— gravity positions. Langley Aeronautical Laboratory National Advisory Committee for Aeronautics Langley Air Force Base, Ya. , February 9y 1950 22 NACA TH 2078 EEFERMCIS 1. Scimiidt^ W,^ and Clasen^ B.: Lioftla'afte auf Fliigel imd Holieiileitweirk GeradeatiBflug* Jahrl). 1937 <ie^ deutsclisa LuftfaJxTtforscifuzigj E» Qldeaaboxcrg (Munich), pp* I I69 - I 173* 2. Pearson^ Saury A.: Derl-VBtlon of Charts for Datermlxdjag the Horizontal Tall Load Tarlatloa with Any Elevator Motion. HACA Eep. 759, 19i|"3. 3. Eelley, Joseph, Jr., an.d Mis sail, John ¥. : Maneuver log Horizontal Tall Loads. AAF TR No. 5185, Air Technical Service Conamnd, Anay Air Forces, Jan- 25, I945. k-^ Dickinson, H. B. : Maneuverahlllty and Control Surface Strength Criteria for Large Airplanes. Jour. Aero. Scl. , vol. 7, no. 11, Sept. 1940, pp. 1^69--^77- 5. Matheny, Cloyce E. : ConEparlson "between Calculated and Measured Loads on Wing and Horizontal Tall in Pall-4Jp Maneuvers. NACA ARR L5H11, 19^5- 6. Sweherg, Harold H. , and Dingeldeln, Eichard G. : Effects of Propeller Operation and Angle of Yaw on the Distribution of the Load on the Horizontal Tail Stirface of a Typical Pursuit Airplane. NACA ARR teio, 19^^. 7. Garvin, John B,: Flight Measurements of Aerodynamic Loads on the Horizontal Tall Surface of a Fighter-Type Airplane. NACA TN 1^83, 19^7- 8. Anon.: Chordwise Air-Load Distrlhution. AN0"l(2), Army-Jfavy-Clvil Committee on Aircraft Design Criteria, Oct. 28, 1942. MCA TW 2078 23 TABLE I COIBTAITS OGCnRRim UT BASIC DIEFEEEDIEIAL EQaflTIQII Gonstaxit Defliiitlcm K, Pi 2m dGx Q ^ 2 ^<H k^ -^1/% ^l./TTT da dot. E2 D?^ 2m iCm s2 •i^ k^^ fcv ■(' da/ da . /rf- 2 m \/\ 2m 'iGLt Stxt "^^ St^ ^% ^%,t KTlt^ D ^t^St^^ P 2 ™v 2 ky- -" - D^ky2 dot dS ^ ^ ^^ 21* mCA Tl 20^8 TCABLE H SPEQIPISa) eOjEDIHIOlS OF SMBM lECEBlIM Incremfint In. load factor • • . • Altxtude, feet ...,,... . Air density, sltig per eiibic foot 8.0 19,100 0. 001306 Case = eg (pereeait M. A, C. ) : *i ; % % 1 % i : (fig. k) 1 a.e. 0.2 : 4.93 \ 30.4 ■ -33 'k- 0.45 2 2k .2 : k.l2 16.2 ■ -32.2 { .50 3 29 .2 k.6l 8.45 : ^1.7 .56 \ 2h .4 : 4.72 ; 16.2 -32.2 .77 5 24 .6 4.72 ; 16.2 -32.2 ; 1.02 NACA TN 2078 25 (tag. /r'e/af/i'^e j^/nd Tanqenf fo- ilicjhi paih- C/pord hne path Figure 1.- Sign conventions employed. Positive directions shown. 26 NACA TN 2078 iT JO o ^ 6 I S -/ ^ ^ \ / \ s^ / / \ N, J / s NJ / N / f / / ■ / "^^^Z^nKa^ .Z .4 .6 .8 1.0 /^ 1.4 /.6 /.8 Figure 2.- Variation of load-factor increment. Zlsn = N250t5'53e-5.53. KACA TS 2078 27 AS . ^ : 1 1 :: "~ 1 1 1 1 fllg^^l^g^g^ i H i f t (b) Impulse. Figure 3*" Incremental-load-factor yariations folloving control movement* as NACA TU 2078 a.o KB 16 14 /Z LO B .4 * 5 — / / r IQ / / / Lo^ si-abi//fu / / / ZO J / / / / y / J // / / /, / / A / / A /- /y/ah sfoh/Z/fu / ^ V 7 u r // y r r / 1 1 \ \ o ,Z A .<5 .8 /.O //2 Figure 4.- Variation of X with t-^. NACA TH 2078 29 7 ^ 4 I J / (T / t / ^- "" - *-- *- . — .' -^ "-^' ^■i^- / / f / \ ^acjqesfed . value b=S / 1 / / / // / \', / > 1 1 \ t; 1 5 — 10 1 1 1 ,Z d .6 .& 1.0 /.Z Figxire 5»- Variation of shape factor b with t-^^. 30 HACA TN 2078 Z.0 IB IZ ^ a 14 -4 -3 -IZ ■JX^ tx _rX i ^ t- t t t t I J 4 2 3 ^ V X 4 \ / %^ N_^ 8 J 6 5 4 3 2 «2 -3 -5 -7 /\ \\ ■ 1 i ' ,^ «^ / / / 1 /. ' / J / / j / J ^=cr — ryr" -^^-^1^55^ ■ ^ .4 .6 , /£ /6 £.0 ' 4 B , IZ 1.6 dD t/X t/A ib) (c) Figure 6,- Variation of incremerxtal-load-f actor cuTYes with time ratio. mcA TN 2078 31 a x/(? >L_ ^ — — — r — 1 ^-i ~— 6 C/G^e / '/ ^^ K cq.Qfajz, 4 // \ i 1 ^ p. f ; \ ^\, c // \ \ -- ^ ^ --- \ •cn -- -.. \ "^""^ ^>v. / / ^ "'^ "^^h X ._ __ — 0-2 ■'/ "^ ''oc ~" ~ — — " \ * // --—Al!-- -4 \ 7 S / / -6 ij //^ V // :_____^,^c- -^ Vv- t 6 X/^ ^-^ — ■ — — — — ~~-\ -1 1 — 1 1 — 1 r- 1 1 a ^ / \ Case c: \ c.cj,afaa4MA.C, / \ 4 • N ^ \ \ ■^z / .^ ^- -^ \ M l^ \ T ^ —\t-\ -K^ S - •- ^ -. / ^ -_, ^. ^;__ - -• -- -2 H- '^ \ -4 \ ^ J -A \ ^' 10 x/(i ?^-| n — — — q ' — — 1 1 ! —\ ^ ,111 a / ,'' '" N uose t / \ aq.at a^9 MAQ a / \ 1 X ^^ / ^ ^^^ ^ --^ ^k k] / 4 /■ "N •^ "^. i^ / f N >--, / V s: ^ ^0 v^ ^ / ^_ — 2w — — — -^ ^•nH ^, I ^ - --^ "^ '-- ^ — -z \ rr \ / 1 -^ -4 \ \ v^ A .8 /jO 16 .6 c: id J C/£ .6 ^^ ./ llO .5 o ^6 .4 ■a s %Z ./ -e ./ iZ .6 J/^ .5 's ^ 5 T |(? ■t .1 I vUp^ "~~ ■""^ -™— -_ _— "^ "~1 "^ —\ "^ n ™ — Ca^e / ■ ■,-,■, An .^ / \ /'\ X / \ / \ >^ / \ \ / A \l / / \ s \ / / ^^ V ^ -- - .- _^ ^, — - ■ ^^ — 1 .y ■ r^ f^^] 1 1 t — — l ^ ^ s. ^~ ■\ / N / \ / \ f )i N 1 / \ s V 1 / \ ^N 1 1 / / \ y ^ N _ ^ _-.. _- ._ „ __ _ ^- ■ " " rUp- 111 Qq^^ 5 / ^ ^ V / / s N i^ s / / s J \ / N s ( / \ V i y / \ ^ 1 / / / \ rt/ ^ y \ " ' v^ __ ^ - -- ' ■— .- ■ "^ ,£ \6 Figure 7*- Effect of center-of -gravity position on incremental-tail-load coniponents. 32 lAGA TN 2078 §0 --J -4 -6 ^iU ^•s ^ "n "n ~n 2 — — \ Ca^se / \ > 2;=a^ 'X \ // \ \ ,f ^ -V ^ k V \ H V ^ ^ -' -S, ™-* ^T .,. r X y • s ^- " 1=^--] v A/., \ f ^ \ J L jif ^ zz /ii ^^ 1 /£ .6 ^10 .5 -z J Up^ — w__ — — - - — ~^ "1 n n ~1 n — — ___ Cqsq < ■V r ^ r \ '\ -^ __ ^ AH \ \ \ AfS \ \ \ 1 \ \ / f 1 \ K Ly s __^ -- -,_ "t: -^ X -r . 8 K/^ 1^-. r~ PI n PI ^ 1 1 1 1 —I 6; ' r tr0.d i 4 ^ ^ ^ t / / N ^^ ;^ ^ — 'n >^ J / "n ^ '^ § ^ y uL --- ^ _ - . ^ :-:-. =STa \ ^ "^ 'ir _ - -^ ^. 1 ^.__ *-. -- •- "*" „,.- -£ \ / ^c ^ ->< IZ .6 /^ .5 |4 I2 ^6> ^Z'] - — ■— ■~~ — — — II 1 1 __ Cose 4 / ^ N / s N / / ■\ N /I \ N > / ''s \ N ■ / / "^^ V N / / / *^ '^. X ^^ / ^ / ■*~"' -^ -^ -- — -^ ^ -- t3 A7(/ ill! Case 5 — t^0.6\ -^ ' "" ^ / <::- -. ^ ^ ^ y" =^ ^^ -rrT" ^ ^ / , -^ j: ^ ~Kr- .— , ^^ s -^.• 7^ ^ .- ^ -aI ~- "~ *" ^ -^ ^c IZ ,6 Up ^JO ■ .5 1> to « o -S4 <^.Z I 4 3 /2 TJmep^eo L6 ZD ^2 ^0 ^y J II II Co^e 5 — — / "^ N^ / N / \ : / / 'N s / N x* 7 -^ ^^ X N y / / ■— ^ -- ^^ ^ • / ■^^ — -- Figure 8-- Effect of t-^ on incremental-tail-load components. Center of gravity, 0.2^ c. NACA-Langley -4-21-50 -1050