KB No. Li».FOT r I i NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ) ORIGINALLY ISSUED June 19^'<- as Eestricted Biilletlii IAF07 CALCTLATIOU OF STICK FORCES FOR AN ELEVATOR WITH A SPRING TAB By Harry Greenberg Langley Memorial Aeronautical Laboratory Langley Field, Va. "^Thaca WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. AU have been reproduced without change in order to expedite general distribution. 129 DOCUMENTS DEPARTMENT Digitized by tlie Internet Arcliive in 2011 witli funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/calculationofstiOOIang -71/ '^'7^ ^Y NACA RB No. li|F07 9^. ^^5-*^ NATIONAL ADVISORY COMMITTEE FOR AEROIIAOTICS RESTRICTED BULLETIN CALCTJLATION OF STICI' FORCES FOR AN ELEVATOR VTTH A SPRING TAB By Harry Crreenberg SUMMRY Forri'Ulas for the calculation of hin{^;e -moment characteristics of an elevator with a s-)ring tab have been developed in terms of basic aerodynamic parameters, spring stiffness, and airspeed. The formulas have been used in a study of the stick-force gradients on a P'ju'puit airplane equipped with an elevator with a spring tab. Charts are presented showing the variation of stick-foi'ce gradient in accelerated flight over a large range of speed and the complete range of spring stiffness for various center-of -gravity locations, altitudes, and airplane sizes. It is shown that the stick-force gradient for the elevator v.'ith spi-ing tab tends to decrease as the speed increases and for v/eak springs tends to approach the value corresponding to a pure servotab (no spring). This tenc'enoy is independent of altitude, size, or center-of- gravity location although the magnitudes vary with these parameters. The variation of stick-force gradient v/ith center-of-gravity location is less for the spring-tab than for a linked-tab type of balance. INTRODUCTION On most types of control surface, balanced or unbalanced, the control force per unit deflection of the surface increases approximately as the square of the speed. On a spring-tab type of balanced control (reference 1), the amount of aerodynamic balance increases with speed; this condition results in a control force that increases less rapidly than the square of the speed. This type of control can be used to advantage on ailerons since it reduces the difference between the control force per unit helix angle pb/2V at the high and low ends of the speed range . 2 NACA RB :io. lJ]FO^ The qxiestion has arisen as to whether ths Irnown advantages of the spring tab on the aileron could be realized for the elevator. The purpose of this report is to analyze the chaiacteristics of the spring-tab control used as an elevator. General expressions, by which cither the static or riianenvering stick forces for an ele-s.ator with a spring tab may be calculated, are developed and anplied to the calculation of maneuvering forces for a typical pursuit airplane. The maneuvering stic^-: forces for the same elevator arrangenent -.vith a servotab and with no tab are also presented for comparison, The effects of variation? in spring stiffness, airspeed, altitude, center-of-gravity location, airplane size, and tab size are considered. DESCRIFTIO?! OF SLj;VAT0R-TA3 SYSTEM In the spring-loaded elevator-tab arrangement referred to herein, the cor:trol is connected directly to the tab, as in a sei-votab, and to the elevator throiigh a spring. (See fig. 1.) Tliis arrangem.ent gives the ccntro]. system! characteristics that are betv/een those of a servocontr oiled elevator and an ordinary unbalanced elevator. A Vveak spring approaches the case of no spring, or pure servocontrcl. A stiff spring approaches the case of a rigid connection, or an ordinary unbalanced elevator. As the speed is increased, the aerodynamic forces increase vhlle the spring effect remains constant; effec- tively, the spring becomes v'/eaker in comparison with the aerodynamiic forces and the condition of pure servocontrol is ar) or cached. In fig-'jre 1, BC is an idler that is free to pivot at the hinge of the elevator B. The control rod AC operates the tab through the linkage BCCS and operates the elevator through the spring and crank HO. The lengths of 30r and BG are assumed equal in the analvsis SYMBOLS A^^ v.dng aspect ratio Am ITACA HB No. Ti|F07 b wing span Cv, hinge -moment coefficient about elevator hinge / ~ — -__ \^ |pV-SeCe C]-^ hinge -moment coefficient about tab hinge ipv^s^c:^^ Ct lift coefficient of wing / — C-r lift coefficient of tall J-irn — P '.' Om Cjjj pitching-monent coeffic:^ent about airplane . „ .. /Pitching moment center of gravity / ^ c mean chord of vi'ing c^ mean chord of elevator cT" mean chord of tab F stick-force gradient in maneuvers fdF dn. P^ stick force F-j_ force in spring; positive when in compression F^ force in control rod aC at C; positive in sane sense as F^ F, force In control rod AG at A: positive as ^ in figure 1 g acceleration of gravity Hq hinge moment about elevator hinge k FACA RB No. L^FOJ 11. hinge ;noraent about tab liinge K linkage ratio (^^1/^2) k^ spring constant, pounds per foot Vi - m2 k;L'l^ 1- '-2 St ^iSe^ L lift of wing l_ leng-th of control stick I« distance between wing and tail L„ lift of tail l-y length of arn 30 Z.p length of arra DS M Mach n ■umber m macs of airplane n normal acceleration per g of airplane due to curvature of flight path; accelerometer reading minus component of gravity force q dynamic pressure r gearing ratio between control stick and rod S w area of v/mg S area of elevator s S,p area of tail surface area of tab V airspeed W -weight of airplane x/c distance bet-"een center of gravity of airplane and neutral point in fraction of mean wing chord ?TACA RB No. 1J4.FO7 a anp;le of attack at ving Crp angle of attack at tail 5^ dei'lecticn of elevator 6_ deflection cf elevator control arm BC s 5^ deflection of tab with respsct to elevator 9 anelG of pitch of airplane 6 pitching velocity DG nondimensional pitching velocity (c9/2V) ^j airplane -density parameter (ra/pS^,h^ p mass density of air "'^oenewcv 5 , 5^, am, 5^, a, and D9 are u?ed as subscripts, a derivative is indicated; for examolo, C™ = ~ — and C- = TTTT- ^^hen a derivative or coef- "la oa ^-D« CD 9 ficient is wTltten v/ith a bar above it - for example, C„, "'a the total derivative or coefficient is indicated, that is, the resultant or effective value \\hlch takes into account the floating tendency and spring action of the elevator wit.'i stick fixed. All angles are measured in radians. METHODS CF AilALYSlS The basic ass-oinptions involved in the analysis are as follows ; n. ) linkage ratio is constant (2) Aerodynamic derivatives are constant over the range of deflections involved (^) ■'^.ffect cf speed en the aerodynamic derivatives is given by the factor V /T- ,? n NACA RB So. LkPO? (lj.) i^rrect of power is neglected (5) Effect of changes in forwcrd speed during s pull-up is neglected (6) Effect of horizontal tail fle::ibillty is ignored Assrinption (1) is valid "oecaxise the link&ge ratio does not change appreciably within the sr-.all r-anp;s of deflections that occur in flight. Assumption (2) is valid, according to lo^.".'-- speed Vrf'ind-tunaei tests of the particular arrangement ccniiderej, for elevator and tab defleoticns up to 5°' -^ IO3 pull-up is not likely to involve deflections greater Vnsn y^ on the airplane considered herein. The metnod of accounting for the effect of speed (aasumpbion (5)), although approxlnately correct for factors involving lift of the v/ing and tail, is of doubtful valid! t;r for the factors Involving hinge moments. As pointed oat later, this correction for speed does not affect the coriparison of wlain elevator, sprinr- loaded tab, and servotab. The effect of pov/er 5.s to increase slij3htly the stick-force gradient. This increase would tend to counteract the effect of the spring, v.'hich decreases tiio stick-force gradient v;lth increase in speed. Pii^^ure 2 is therefore strictly applicable only to sliding flight. The error involved in approxiraation (5) is believed to be s:nall. The flexibility of the tail will Increase the stick I'orce gradients slightly at high speeds, according to reference y. On the basis of l;:iese assurap- tions, the relations developed herein hold for small deflections . From, the geometry of the olevator~tab arran.^cr.ent. It is obvious that 5^- = K (5_ -i- 6 ) and that + ? (1) Prom the condition for ociullibriun cf the elevator systeri. \ e t '^/ / I'ron bhe condition for equilibrivin of the tab, F2I2 = -/'S.-Cv,,, + 5..Cj^, + a^Gi,, \qS^c7 (3) Combinir.g expressions (1) to (3) gives q^gCg ^63 ^Ct ^a^ i The coiifressicn of th^ spring is '•■i(^^'s "^ ^e"^ ' hones, F- = ^■'1^1 (■''0 + f'e)' Substituting in e qua t ion ( q ) gives k,7. 2(0^ + 5 ) q-e^g -o^ ^cv ''am C .-, ! ij.j- C iij. J. .•-i.i. j If the ■"•alues of the aerodynamic coefficients on the right-hand side of equation (p) are obtained from io^v- speed data, they should be multiplied by — ^..t...^, to v'l - ^'^ apnly to hi,rh speeds, acccrdin?; to the C-lauert approxi- Tnation or, if / ', -> •./ i _ '■ V 7 - 1 , '- ^•'2 ~ " 8 NACA RB No. IJ.\F07 equation (5) may be wTltten as k 2 (3. ^- £^^ 5.,Cv ^'f^ii„ "^ "^T^h ii. a 71 ^e°e\^^ 5.U Og + ar :A., ^ (6) ar.-i / ^y 3y su'31'tituting for 5-^ in ter-.is of 5c. and 5q in GOf.ation (6) ani solving for o^, t'ae-^-e is attained kp - KC s+c; \ / 2 - ^^^'h. \ 64- ' C. J I4. ^5, i 1 ; t 1^ + T,-r< + 72: ^t^t ^h. '■^d ■.,„ + K— '^t^t ^.^1 ^e«e -t,^^ ^■^t (7) is/hich deteninines the auf le Cg at v;liic>: the elevator floats in response to a control deflection 5„ and anr'ls of attack am. Tl-.e tah angle is then determined hy the llnka n. - 3Ch^_ + K3Cii^ + Ch, arr. ■^t 'a, T (12) YLh.en the stick is fixed, the elevator moves v;ith changes in angla of attack in accordance with eqxiation (7) As a result, the stafcic-stabilicy derivative G™ and m. a the dam.pin,r- in pitching Cj^ ^ are affected. They may be calculated ov m. m. a 'in 5, ^c: ,iji dam a, IT. — - ■^m h ^"^ (15) and dan C m Da + c mc 'DG ^'^63 doe lrr\ U.Urn B + Cra IQ — - dan 5t dDe (1^) 10 NACA RE No. T..LPO7 TliO control ef f ec+:ivsnes?=' C-t^„ sir-iilarly depends on the r.iotlon of the elevator with respect to the control arm. ^e re 1st ion is -111; 'm A + C K(l + A) ^e h ( 15 ) After the five f and ament a 1 derivatives are obtained by equations (11) to (V-)), the stick force per unit normal acceleration in a pixll-up may be calcvJ.ated. The formula for this stick-f c ■'■OG gradient, which is taken from equations on pag'-3 ih of reference i, is Ct c cr e '-■^e e '■'j a "" ^ + H,v!'C 'D9 h. a 0, 35 ^m Dfi ^mg fl6) or a mass -balanced elevator. In f^orrnula (16 ), t otal and derivatives are used, "^ralues of C[- obtaixiod froiii Sa a a _T da,n Ox n. L-V- in e_ dl;0 Ch^ are ^ J (17) The effect of compressibilit'- must agajn be taken into account in using formula (16). All the derivatives in that expression should be malt ipl led by — ■', ■ : ■, If a/i - M^ the data used in computing these derivatives are based on lo'.v-speed measurements. The factor cancels out except in the second and fourth terms. The corrected formaila is /, S.~.o,f^-'^'^K ■'e e ^n^oe n 41 gr ^K ^1 . j2 ^DB Cl^C^,^^ V^ - l^ "^6< / (13) yACA RB I\To. riiFOY 11 Spring, elevator, and tab deflections corresponding to any aeceleration are calculated by the following formulas, derived b^' using equations (2) of reference 2: 0^ _ eg pA-j^iji ^ dOj ^^ 2v2\Cl^ dDn, A number of coKiOutations, based on typical airplane characteristics, have been made to lllustr&te the effect of spring stiffness on the characteristics of an elevator with a spring tao. The follovn.ng airplane dimensions and derivatives are used: W/S^j, pounds oer square foot 1|0 o, feet 7 Igr, feet 2 3t/Sw 0.18 Aw .6 At U.5 Lh/« 30 GLa ' ■ ^-3 daT/da , 1/2 darp/dDo 6.6 Cma (for e.g. location 0.05c ahead of neutral point) -0.252 CmDG -15 '5 Altitude, feet 20,000 [i- 23.5 The follov;ing elevator and tab dimensions and derivatives are used: S3, square feet 17»1|U cf, feet ^•^'^ ?rt/c- 0.336 Se/ST 0.52-5 S-b/Se O.llH 12 NACA RB ilo. LI1PO7 St^^/Se'c^ • O.Ol+li. Z-2_, fcot 0.5 K 1 Ch -O.kG? Cr 1.84 6e Cy,, -0.115 Cr . 0.115 J Ch. -0.115 Cl^r. 5.09 " am - a.-p Ch^-/ -0.115 Ora. -l.Okh Ch^ -o.3i+5 0,^,. -0.0615 -t O-m From equation ^7), kp + 0.130 C.II5 A d - 0.622 '" -ko - 0.622 ■^om equation (10), -O.kSykp - 0.0067 0.115k2 + 0.0025 '^'®Qs -'^2 - ^'-'^22 '^'Sa^ .v^ - 0.622 Prom equations (1?) to (15), ~n~ = -0-252 - Q.O655 -k^ - 0.b22 O.S38 Se ■ ■^■^•^ ■ -k^ - C.o22 k2 + 0.130 = -1.106- : 0.0615 6, -kp - O.o22 o m [■hese values can be substitute 1 in forir.ulas (I7) and (iS) to obtain Fj-. MCA RB Ko. I-ItJ'OY KESTJLTS AMD DISCUSSION The computed values of the stic]c-force gradient in nansuvers y^ for Via assixned airplane and elevator are plotted as a function of speed in figure 2, for varlOLiG values of the spring constant k]_. The top curve, for infinite soring ctiffness, applies to an ordinary unbalanced elevato-', for v.hich the spring is replaced by a ri^ild rod. The bottom c"i:u:-ve applies to a pxii'e servocontrol, for v/Iiich the spring is removed. Tiie intermediate curves are for the cases in vi'hich springs of various stiffnesses are connected between the control rod and elevator. The Increase in stick- force £;i'adlent v/ith speed for very hlj^h and very lov; values of spring stiffness is based on the ass-tunption of the effect of compressibility r.entioned previously and is net important for the pur:ooses of this report. The iiaportant fact is that the addition of a spring reduces the stick-force gradient in th.e manner shown. A very weak spring reduces the stick-force gradient at the hi^gh end of the speed range to a value only slightly higher than that of the pure servocontrol. In the case corresponding to com^plete servo-operaticn (no spring) in figure 2, the stick force is less than the mini.nnm value considered desirable, "i^his value could be increased by using a tab of increased chord. Increasing the span of the tab v.ould have no appreciable effect on the stick- force gradients because the increased forces on the tab are compensated by the reduced deflections needed. Other 'nethods of reducing the stick forcer, such as the linked balancing tab^ would result in a slight increase of stick-force gradient with speed ss is the case of the top and bottora curves of figixre 2. It is sometimes considered desirable to have direct control until a certain stick force is reacaed after wh.ich the tab control begins to function. This direct control is accomplished by preloading the spring by an amount that depends en the stick force at v.hich it is desired to have the tab com.e into action. The stick force varies v/ith acceleration in the manner shown in figrrre 3- The cxorve has the slope for infinite spring stiffness up to a certain point, as indicated by the solid line, and then has the slope corresponding to the spring stiffness used, as indicated by the dashed line. ll.L NACA RB No. LI1PO7 The point v/hers the slope changes of course depends on the preload in the sprang. Such an arrpngernent might he usefiil in maintaining reasonable control forces for piill- outs at very high speeds. The effect of increased airnlane size is shown in figure ].!. . The wing loading and control gearing Z.^ are assuuTiCd the same as in figure 2, biit all lengths are assumod doubled. The stick-force gradient for pure servo-operation (k-i = 0) is so'iiewhat higher than the value considered desirable and anj' ap"oreciable amount of sprlag stiffness v:ould raake the stick-force gradient too large. For this case, a tab vrith smaller chord could be used to give lower stick forces. The effect of altitude on the variation of ctick- force gradient with speed is shown in flgur-e 5* ■^'^- increase in altitude x'educes the stick-force gradient by an amoi.int that does not vary apnreciably with speed. The loss in stic;' foi-ce in pounds: decreases as the spring; stiffness is decreased. The effect of centei'-of-gravity location on the stick-force gradient for several types of balanced elevator is shown in figure 6. The elevator with spring tab shows the smallest change of stick-force gradients. The linked-tab balance ch.oeen for comparison was assumed to bo so linked as to give the same stick- force gradient as the elevator vi^ith spring tab for one particular center-of-gravity location (=: - O.O5). The variation of stick-force gradient with center-of-gravity location is less with the spring tab than with the linked tab becaiise fJv, is red\jced as well as Cv, : this condition loermits a smallej-' C-.^ f'or a gjven stick-f o"»''ce gradient. As sY'OWi in e'.[uation ( I8 ) , the variation in stick-force cradioiit with C-~. , which depends on cenber-of-gravity locetloji, is proportional to Cv, ^s KAGA RB No. Ll^PO? 1[ CONCLUSIONS Formulas have been developed for the calculation of hinf^je-rrornent characteristics of an elevator with a sprinii tab. The analysis included basic aerodynamic parameters, spring stiffness, and airspeed and indicated the following conclusions? 1, The stick-force gradients for an elevator v;ith a sprin^^ tab tend to decrease as the speed Increases, For a Y/eak spring at high t-peeds, the stick force approaches that of a pure servocontrol. 2. The variation of stick-force gradient with center-of -gravity location is less for an elevator with a spring tab than ;vith a li'iked tab. 5. Increase in altitude rediices the stick-fores gradients by a nearly constant amount over the speed range, for a c-iven soring; stiffness, Tb.e amount of reduction in the stick-fcrco gradient decreases as the spring stiffness decreases ^ Langloy Memorial Aeronautical Laboratory National Advisory Coriinittee for Aeronautics Langley Field, Va, REFERENCES 1, C-atos, S. B»; Notes on the Spring Tab, Rep, No. d.xi. 1665, R.A.E., April 19iil. 2. treenberg, Harry, and Sternficld, Leonards A Tlieoretical investigation of Longitudinal Stability of Airplanes v;ith Free Controls Including Effect of Friction in Control System. NACA ARR No. i^BOl, I'^kh , 5. Harmon, S, V.i Determ.inaticn of the Effect of Horizontal-Tall Plexibilit-y on Longitudinal Control Characteristics. NACA ACR No, L^^BOl, I9I4-5. NACR RB No. L4F07 Fig. 1 o •H •P 0} r-I (l> ■d 0) -d (d o I ^ ;^ a CO NACA RB No. L4F07 Fig. Z zo ^ l(o ^ O IZ ^ 8 ^ O / (Ib'/ft) z rsoo 2, 7 4- /O T" 20,OOO / zo / 1 1 -^ /i> (fb/ft) ^ 1 / Ot '\ ^ ■ /2 > X: ^ ,— - - " ■ ■^ 2 750<^ s. 6 V ^-- ~-^ _^ 4 ^■^ • ' -- ~ o O — — — — - _ ^^ — ^ "^ ZOO -foo True airspeedtj Vj mph ^OO NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS. Figure 5.- Variation of stick-force gradient with speed at sea level and 20,000 feet. k]_, spring stiffness; wing span, 42 feet. NACA RE No. L4F07 Fig. 6 U5 )b y^ y y y /2 Plain elei^a^or y y / A^ \ y 6 y y y A ElQ>vator ujith Storina ta.b t ^f^ ' ^ P- -^, ""^ -^ o \ ^ ^ ■ ■■ T ' ^ ^Linked • balance^ -tab ---^ -4 NA COMH TIONAL TTEE FOt ADVISOP AERONAl lY ITICS. .01 ,02 .05 .04 .OS Cen&er- o/- grai^jty /ocationj X'/