A Study of the Dynamic Longitudinal Stability of Airplanes with Special Application to Design Citation Root, Leonard Eugene (1934) A Study of the Dynamic Longitudinal Stability of Airplanes with Special Application to Design. Master's thesis, California Institute of Technology. doi:10.7907/an7m-3j68. https://resolver.caltech.edu/CaltechTHESIS:09302024-220529723 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Aeronautical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Millikan, Clark Blanchard (advisor) von Kármán, Theodore (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1 January 1934 Record Number: CaltechTHESIS:09302024-220529723 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09302024-220529723 DOI: 10.7907/an7m-3j68 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16769 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 02 Oct 2024 16:47 Last Modified: 02 Oct 2024 16:53 Thesis Files PDF - Final Version See Usage Policy. 57MB Repository Staff Only: item control page

A STUDY OF THE DYNAMIC LONGITUDINAL .S TABILITY OF AIRPL1.NES WITH SPECIAL APPLICATION TO DESIGN THESIS by Leonard Eugene Root In partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE California Institute of Technology Pasadena, California 1934 CONTENTS I Summary o.o II Introduction 1.0 III Th eoretical Analysis 2.0 IV Graphical Representation of Stability 5.0 V Discussion of Parameters 8.0 VI Stability of Gliding Flight VII Dynamic Stability Diagrams, Case r-13.10 VIII Dynamic Stability Diagrams, Case II- 13.2 IX Conclusions 14.o X Procedure for Use i n Design 16.0 XI Range of Parameters 17 .o XII Accuracy of Diagrams 18.0 XIII Examples of Use of Diagrams 18.0 XIV • Acknowledgements 21 .. 0 xv Suggestions for Future Research 22.0 XVI References 23.0 •11.0 SUM.l\~ARY This thesis is a revision and extension of a previous work on longitudinal dynamic stability(l). In the present work, an attem~t is made to simplify the classical stability theory which is f a r too complicated a nd cumbersome to be of practica l use in design. By employing a no n-dimensional set of units, a:i.d by making use of a graphical method, the range of dynamic stability for different value s of fundamental parameters is defined graphically by means of diagrams, t h e coordinates of which a re conveni ent l y chosen as the position of the center of gravity and the s ize of the tail of the airplane. The coordinates , combined with two chara cterl stic parameters, completely define t he reginns of dynamic longitudinal stabi lity; while the diagrams thems e lves determin e the period a nd the de,mp:ing f actor of t h 8 longitudinal oscilJ.ation,thus bringing the complica ted theory into t he r a.nge of a pplicability to des i gn . {l) INTRODUCTION Because of the :tnherent complex.i ty of ih e classj_cal stabili.ty t'.heory, 1h e airplane designer has not found it practical to apply the theory to actual des ign. Instead, he provides for a small amount of static stability, dependirg on the precedence of previous satisfactory designs from which certain empirical rules have been developed, for satisfactory dynamic characteristics. Such procedure, however, does not guarantee proper dynamic stability; in fact, undesirable properties in dynamic stability may develop in a plane which has the proper static stabilit~. Thus, we see that in the process of a new design, it is desirable to know the dynamic longi tud:ln al stability characteristics of an al rplane in order to foresee and to avoid possible undesirable properties. Before this can be done, however, it is necessary to reduce as much as possible the complexity of the theory to the extent thct 1t can be u s ed practically. In this paper, following an idea presented by Mr. S.B. Gates (l), an attempt is made to simplify the theory by reducing the number of variables thEt appea r, and by applying a graphic a l method of representa tion of the stability chara cteristj_cs. By refe rence to the ch a rts that have been de- veloped, it i s possible to determi ne immedia tely the dynamic st ability cha.r a cteristi c s of an a.rpl a ne, knO'.'r i ng it s fundamental paramet er s . It i s ho:9ed that t he deve l opment of. these charts vri l l cont r i but e t o design procedure b_y determining the longitudin8.l dynamic stability of an airplane. ( 2) THEORETICAL ANALYSIS 2 From classical aerodynamics< : we know that in the most general case, the longitudinal stability of an airplane is determined by the roots of the que,rtic, ,,,\-Xt1 --- (1.0) -Xw -z ,t\-Zw -1Y7 'Cf -Mw t.f j?CO.Sf%-~ +A'% 7;\ -~~ !J'S/N~ - = 0 ' AZ-~~ where: X, Y, Z are the components of the resultant aerodyna..mic forces along the axes -;the mass of the al rple.ne L,M,N are the moments of the external f orce s about the axes+ the corresponding moments of inertia, A,B,C, u 0 ,v0 ,w0 are the component s of the resultant velocj_t,y in the undistu rbed motion. (v0 = O) ::: the upwqrd incJ.ination of the longltudina l axis to the horizontal. For simplicity, wind axes are used in which the ste,bility derivatives Xq and Zq = 0, and u 0 = U, w0 = O. 80 becomes the angle of climb in the undisturbed motion. Since it is very desirable from an analytic standpoint to have the stability criteria form airplane expressed in non-dimensional form, t he dimensionless system devised by Mr. Glauert( 3 ) is introduced. Thi s syste m is based on the fact that an aerodynamic force can be expressed in t he form X= -Gf- SC/.z ; from which the force derivat ive is of the form, = n?dX Xw C7t,/ or - ?l~ 6pi7 1177 - - A Xu . (3) Thus , in. t he ·1evr nota tion, the 9 re vious cla ssic a l deri va tj_ves Xu , Xq, Mu, r1c1 , etc., 9,re re ::7l a ced by tbe dimensionle ss derivative s , ~,xq,mu,mq, etc., definea. by the equa tions, Xq A - mt 1 , where A u 1 By subs tituting these non-dimensio nal deriva tives M = into the classical stability equation, j_t, is found thg t, al l the parameters which occur are dimensionles s . fro m the derivA., tives themselves, the only parameter vrhj_ch affects the stability of the plane is V n? ~ = ..-/A pS./ Apart /4 13.08_/ ( For sea level). In the dimensionless - sy ci tem, t.h8 unit of time is ,,-J-= ~7 and the unit of length j_s,,/, tak en 8,s the distarne from t".1.e center of gravity of the plane to the tail post. From this, the unit of velocity becomes-/:!{;, where the q_ue,nti ty,,,,,,t/ may be called the " rel:iti ve densi ty 11 of th'.3 airple.ne sinc e it is propor+,ione,l to the ma ss m d inver s ely proporti.onal to the cube of t he linear dimensi.ons and the density. The quant ity ~ 11 ~ exc e \Jtlng the deriva tive coeffic i ents, is the only :r,a.rameter which effects the d stabj_li ty ; and,, proves very useful in concisely summari.zing several important para1J1eters vrhich occu r in the theory. is the only q_uantity in the di mensionles 2. system i n whicb the linear sca19, tb.e 7ving l oading, and the d.enslty enter directly. Any v,g,riation in any one of these quantities is j_ncluded in the one v a riable ,~ . It (4) I 1 t erm s of t h e non-d.imensio ro,l system, neglecting the a.eri:atives Zq and xq and using wind. axes, the sta bil- i ty equation (1) can be expressed in the following manner: By making use of the equilibrium condition, I _z..C L = Wcos~ = Z\v from which Ls, r-sv C / -~ ~ c7 J" acos~-=;- J ~ • - 0:5/IY 0, -3::: "'7 <) = pSiT c;_ {,/r~N(9a \ ,2_ m .-e, ' we have ,A '-r:z# .X.-w Zq- ~+Zw ~r~TRN0o-~] 177(,./ .n'lw ,,/'2+nyJ' (2.0) _,,w~ ..z = o. Equati.on (2 .. 0) can be reduced. to the form, (3.0), i.e., • , 4- (3 .0 ) ,.!J J",2 .;1 -/- ~~ -/- c; /J I + °✓ -1 +£; =O where the ' coefficients E1 ,c 1 ,D 1 , and E1 are functions of the stability derivative s; namely, Bl = cw C1 ...,. -t-~ +x.,., ,nf{.2:-w+Xu)~/77w +~~w -Xw~ )~nz(~ +Xvv) D1 = n:,(Xw-2=\¥-Xvv¼)~n?w(Xq-~ ,A>N0 El = zc;_~.n7w. {i=t.t-><;;, T'9N',.0)- ~~~(zvv-XwT-9N~ It is noted thatt, below the sta lling speed of the a rplane, all the non-dimensional derivatives except mw, mu , and ~w are :::iositive. Longitudinal stability can thu s be described completely in terms of t hese derivatives and ;,u", vrhich is aJways associa ted with either rnvr or ffiu• Bec e,use the absolute value s of two root s are large in comparison to the va lues of tbe ot,h8r tvro, we use Bairstow' s e,pproximate method m d express ( 3 .o) in t he form of a, bi- (5) The fi rs t ou a dratjc re pre sentI s the s o-ca lled II short •• os cilla tior 11 whi ch has negl igi ble i nfluence on longitudinal stability below t he s t a ll si nc e it is h eav ily damped ru~d invari a bly, stable . however, re pr esents t h e II The second quadratic, Phugoid 11 mot ion, or a sli ght1y da mped periodic mode whi ch direct l y determines t he dyna□ ic stability of an a irplane. GRAPHICAL RE? R~SENTATI OH OF STABILITY From t h e c lassical t h eory, i t is known tha t a sys tem r epresent ed by ~~+ ~,;J3+ c;~:z.+ 4~+ E l = 0 i s s t a ble if the foll ov,r ing co nditi. m .s a r e satisfied: 1. 2. The coeffic ient s B1, c 1 , D1 , & E1 mu s t be 9o s i tive; 2 Z Routh ' s discri minant, R1 = 13;c;q-D-L3'E> o. If B1 , Cl' or D1 goe s through z er o, then the discrimi na nt, R1, reaches z ero f irst. i ncreases . However, if E1 goes through ze r o, R1 Hence, for s tability, vre must cons ider e spe c i a lly R1 and :13:: 1 . If R1 = O, t he condition of s t a biJ.i ty changes from a damper . osci l l ation to a di v e r gent one . If E1 = O, a subsidence change s into a divergence. Hence we see that the re g ions of sta bility m d i nstability can be sepa r a ted fro m one another by worki ng wi th the equations, E1 = R1 = O. For a graph ic a l re pr esentatio n, it is possible, by usi ng the two ec1u a tions E1 = o, R1 = 0, to es t abl·.sh botmda ri es for the regions of stability . O r /"'">if077t:7-'½ ~£~=:- (B°_} p ~ c/ .:JS. (6) If the variabl.es 11 where 11 X11 and "Y" are chosen as the coord.ine.tes, X11 involves the C.G. postion and "Y 11 involves the size of the tail of the airplane, it is possible to establish on the "XY" plane boundaries separating the regions of instability from those of stability. Two fund- amental parameters are varied, t he other parameters re- -~ maining constant, thus giving a family of boundaries for stability. It has been found convenient to take as the variables 11 x" and "Y", the following: __/z x = ~ h Y= 1:/z ~ z k; .S where: length from the C.G. to the tm. 1 post k 8 = radius of gyration about the Y axis Sc = area of horizonta. l tail surfaces S = area of wing d~ = slope of lift curve of tail surface (horiz.) dclt: h = dista,nce of C.G. from the aerodynamic center ,, of the wing section , exPRESsEo Rs ,q ~R,;cr✓oN or=- t-. The "XY" plane is filled with two families of curves;namely, Cl ~E, ~ -z c; 2. -p Z1t -- c; - 41(4 -£, - - B,EUZ - c;z C, C, which define the ve,lues of t h e roots of the quadra tic, t( ana_ f? in the non- d. :i.mens i onal s ys t em r1ay be con- vertea. into the damping factor and t he period, pc,vr7d ft .... sec. system, · by use of the appropriate factors: (7) .,. D.F. = R zr -~ Since W= ~ SC"'- from which ~ CL 7'~ 4 »ff 0.3 0.247 2.023 0.5 0.319 1.568 1.0 0.451 1.108 1.2 0.494 1.012 To get the damping factor and the period, we note the following relations: O.F = . X-(},r~)fF ~)= (rff)ff P To get the time to damp to one-half amplitude, we have the relation: 77 -= z o.e;93 • . D.F. The family of period curves in the form, 2ff = //£, _ _!_(4 _ 8✓ E,)Z P -c, 4 ~ c;:z. are very difficult to use for plotting. It has been r found sufficienty accurate, eY.cept when E1 becomes very • _L(g_ - a,4)Z sma.11, to neglect the term, 4 ( c, c;Z) in comparison with the term £, c; . period curves are reduced to the simpler form, Thus, the ( 8) Of the damping curves, I:(= 0 , is approximatel:r the boundary defining the transition of the phugoid into an increasing oscilJation. ~ z R1 We note that 2 = 8,CO, - qz- B, £; = 0 . is usually small-in comparison to the other terms, hence we h9.ve as an approximation .to R,= o, the two = O, and c;q-B,,s = 0, which is identically. K = 0 , since w= o,c; C, - ~£, ~curves, B, ~ As the above appr•oximation breaks down only where and c, are s rm.11, K B1 = 0 is a very good approximation to the R, = 0 boundary curve for all practica1 since B1 is large for any actual C 8. Se. pnrposes DISCUSSION OF PARAit ETERS In keeping with the purpose of this work, it was found necess a ry to include two sets of parameters; ' those which correspond to the more modern type of airplane, and those wh:i..ch refer to the older type still being built. Thus, two groups of curves have been plotted; one group ( Oas e I) using the other group ( Ca s e II) c:IC:.. =48 and clol using G= = o. OZ ~ , ~= 4.0 and ~ =0.05. In order t o repres ent conveniently the t wo families of curves, period and damping factor, it is necessary to J.irni t the number of variables occuning, ana. to give constant values to those parameters which do not vary greatly or are not of great importance in sta bility. In order to (9) a rrive at satisfactory average parameters, it was necessary to collec t flight test data, calculate the v ari ous parameters, <-md aveN.9e those whi ch seemea. t o lend themselves ·to such treatment. As a result., various s:i.mplific ati ons c1 were mad e as f o llows: 1. Lift Case I c:1c;__ Case II de;_ = 4.8 a'oC. c/Ol. == 4.0 ( These values are a s SUJ.med to hold up t o CL= 1. 2) As i s cu stomary, the tota. drag , 1 s a,ssumed to be .z:. o f t "'E e f orm, Cn Cv -- S..,C + edR? C"• for a re: Ca s e II Case L CD== 0.02 + 0.065 cL 3. The values _ 2 Pitching Moment Due to wi ngs: The pitch ir1g moment abru t the C.G. is given a s; C-41 = CM0 + -ho) Q. _ J • (f. vrhere 1 h 0 is the aer odynamic cente~. ( 0.22.-::::.ho .-:::::.o . 25), rlt#l'/j!o,/V TRL- and " a 11 is the,,di s t ance from the leading edg e of the wing to the C . G. Due to tail : The pi tch lng moment due t o the t 8,i ili is assumed -; to be , C Mc -= --/~c t:- s z_-t;- . Due to fuselage : I f there 8,re wind tunnel tests on the planet _ { 10) then the effects of the f uselage can be Recounted for. If tests are no t ~vailab1e , then the f ollowing cor r ections may be a9plied to account f or ih e fuselage effect on the ➔~ moment curve : Change in /~SM) ( Small, l ow- wing monoplanes (.w 1,T o rthr op XFT- 1) + 0-0.25 Large, low- wing mo noplanes -r- 0 . 0 3 5 (v.>Dougl,g,s DC - 1) U'--LJW✓IVG The above correction can be absorbed by using an " effective " a/t in t he expression for the pitching moment for t he wing alone . In the sar.1.e manner, the vertic al displacement of the C.G. must be corrected for; i.e., ;,.. C7 ~ - 1 tb , vmere ' 'b' 10 ~ t to the chord line. be ve rt, J.· c a~7 di-~star1ce~ fro m the c.n-. Thus, the fin a l 11 effecti ve a/t 11 vr ill 1· s t'11,.., _. c;; ., , _ J , ft)erF and h 1:i 1.11 equal Dovn1wash a t th e tall j_sfa,aken as Ta il _1engthA gooa. average for 4 , or the ratlo of the length f rom the C. G. to the t ai 1 post to the chord of the wing, wa.s found to be 3 .. 00. Rotary Dertvative ..":.s in Gates .' work, m0 for the total :plane was t a ken k as 5/4 of that due to the tail. _ (11) Ta il Ef ficieric y 1?t , t he r a tio of the forc e on the horizontal. surf aces vrhen a ttached t o t he airple,ne to the force on the surface s when a t the same eff ec tive angle of a ttack in the fre e a ir s tream, = was f ound to be 0.72 ( Av e r a ge "tJalue.) LONGITUDINAL DYNA:•.HC STABILITY OF GLIDING FLIGHT The dimenfilonless derivatives in the case of gliding fli ght become: Case I Ca se II z. Xq = O.OZ-f- • 065C~ ~ = - o . 37,zsCL.., = Z . 41= rn. = -0-8X+0,5.36Y /,V ~~ y ~vv mg= ~ = - O.GG7X -/-0.3Z7Y '2:'Y ~ - The coefficients for the sta bility Qua r tic become: Cr1sE .Z =-::. ~ - ~ -t-i!::w -t-X,:,, y -i- o.o6.5c;_:z. +Z.4Z (/N=-=wo=Nr =~) _ C: = rnj(cw +X,,)~1?7w +- X,.,,..c.,,,,-XwZ:q =(ao.2 y r, 08/,zc;_z:.y + . 5 3 ~ y-o.~X _ -,..o. s-.z9C.._z + o . 048) ~ =°](Xt{~W- Xw2=q) ~n?,,,,.(Xq-t-Cj:) = • V(o.CA:6 r 0.66/ c;__Z +O.os~c:._~ +0. 0 / ~)-x/:;.oz~~o~c;_ E, ~ .-,/44,n,,,. ?( 2;,.,,. X, tJ 5!)/c;_'--ro,oz+ o a;;sc..'Jj , ':/4'(<>2€8 y - 0.4 Case II BI 5 = 4 y +0-06SC~ +:Z.a5 ~ ~ = (ZSG +.08/ZC:.,zr• 3,:i::z,0y--662,q'X +.3S7Sc;_ z.. + 0-/0 i7✓ = (o.1zs-ro.#66c;__ 2.-/-0.o3l~c;_ Z-1-;024-y,J Y-(a.o~rc}.o;;~c;_,jx "Jfc;_ "- E, ~( 6Jt-=>".?;;v y - o. 6G7 X o. 06.Sc;__"-_) r(o.os r j The family of non-dimensional damplng curves can be ✓ K= c;q- 8,£ cz written as O,R - KCI z = co-BE; 7 I / I / Letting B, = aY + b c, -- c Y + dX + e = fY + gX D, E, = hY + j X j Then: · K( czy z+c/z.xz..+zcdXY +zceY -t- .zale X +e2j z.. ~ -- ( cf -ah) Y + d gX + ( cg +df-a ~,) XY -i·(ef- bh)Y + ( eg - jb)X , or, r=J_= (~ K (.,,c;vz..-1- Bx~+CXY -1-Dy +EX+ ~ +(3X2.+oxy+f,'Y+ex./ ·where: A= %. C ~ = cc~-oh) B= c1~ f3 = dg C = Zee/ y D = Zee ~ = ..... -- 1;' Zde F= e z = (Cy'+d,C-cy) (e~-bh) e - (eg7/'b) z (13) The f am ily of period curve s become: (Z77')z= £°J OR E;= 4P~ p Let Then or, 4-if~ = p~ JJ ½ ~ c:;. and substitute for E1 an:1 C: I (h-c!)Y -r-(J·-ad')x-Je ==o -= y bJ-JJx -1-(Je _7 . f h-cJi Lh-cJj It is now po s sible to represe nt these two familie s of curves on the" XY" plane by assignj_ng various va lues to the parameters,__,,u and ~ , at the same time varying :~ the c onstant values assigned to Kand .E· This has been done for the t wo groups of curves; i.e., for Cases I & II. There are sixteen curve s for each cas e, con~isting of fct..'!r' grou::,s, each gY'oup for a constant value of ~". The E, = O end K = 0 curves on each diagram represent the boundaries separating the sta ble from the unsta ble regions. Each diagram is covered with a series of curves of constant period and of constant damping factor. ( 1? ani _!) ( 13 .10) DTITAMIC STABILITY DIAGRI-\.MS CASE I (13.20) DYNAMIC STABILITY DIAGRAMS C_"i.SE II (14) CONCLUSIONS FROM STUDY OF DIAGR11.MB From study of the dynamic stabi lity diagrams, t h e folilio wing conclusions have been derived: ( 1) The a.am!") i ng factor and the p eriod, K and p, should be a s larg'3 as possible in order to avoid the region of increasing oscillation. (2) The damping factor decreases and t h e !)eriod increases with an incre9,se in a1titude ar.rl wing loading, other quantities remaining constant. ( 3) The period is increa sed by a backvra rd movement of t :r.e center of gravity of the airplane and by -:tn increa se j n the hori z.ontal tail su rface area, if the C .G. is behind th e :aerodynamic center. (4) Insta bil i ty may arise through an increasing oscill- at ion or 3, divergence. CL A. s the speed is dec r ea,sed, i.e., increa sed, the da.nger of instabilit y due to increasing oscillation i ncreases . ( 5) A large moment of inert iq, about t he lateral axis of tbe .gj_1.r:9la.ne i ncrease s the danger of instability; hence the dist r ibution of t :r.e weight glong the longlt1.1dinal axis should be as compact as possible. (6) When CL is small, t he damping is roughly independent of the coordinates , i.e . , the C.G . position and the tail area. ,?.:,..._ - - increases, the dependence of _!$ on X and Y ir!crea s es . The diagrams show that it is desirable to arrive at s tatic stability by means of a fairly l a rge horizontal tail sur f ace ra,ther t han by having the C. G. f a r forvrard ar}d using a sma lle r t a il surface. ( 15 ) (8) Rea s onable stat ic stability with a relatively large hor i zont a l tail s urface i n sures dynamic gliding s t 13,bil i ty. , Large s t a tic stability, howeve r, :-cake s tr.e :Jeriod ;:.; hort and the damp i ne: f ac to r small. ( X large negatively and Y s ma ll.) (9) An increas e in the slo pe of the lift curve j_ncreases the re gion of dynamic stability. ( Comy ,'3.:re Cases I & I I.) Conclusj__ons Important for Design (1 ) The plane should have a slight static stabi.lity ob- tained with a relatively large horizont al tail surface, C.G. to the rear. (2) The moment of inertia about the lateral ax is should bA as small as possible. (3) For assurance that the plane wi 11 be stable with :9ower-on, the boundaries of gliding stability should not be approached too closely. ( 16) PROCEDURE FOR t:SE OF CHlR TS IN DESIGN For practical use of the stability diagrams, the follm1 j_ng properties of the ai rplane must be known: W = weight of plane = wing area ( From this and the span, find AR).., S St -·· area of horizontal tail a1 rface a = distance,horizontally,from C. G. to leading edge of wing. b = vertical distance of C.G. from chord 1:i,ne t = mean aerodynamic chord ho = aerodynamic center of. wing Fi = [ft)erF-hj [tD fl i)P<'5C ±=(lo t-)/ (a/t)eff. = r = length from the C.G. to the tail post. 1 kB = radius of gyration abaut the lateral axis. ARt= aspect ratio of horizontal tat 1 surfaces. From this outline of Procedure (1) Determine which set of curves to use by referring to wind tunnel test or by calculatir1g ~~ and c;p• If dCZ =. 4.8 ~ If clC.. ~ (2) L = . and. CDp 4.0 = 0.02 use Case I. ' ' 0.05 use Case II. ' ani CD p From the given properties of the plane, calculate "X" and "Y", where ani (3) _j_ h;11__ X ·- y = 1-L& z 77 8 Determine A", the II k ' 7i = /z., . k; Ir- k.,4rAft- relative density 11 of the plane , ( 17) n7 = pS/ _,q = where~= w/s = 13.08 f ( For sea level) wing loading. Find f and .E ( non-dimensional units) (4) for the series of lift coefficients, o .3,0. 5 ,1.0,1 .2, using the ,, " factor~ calculateo. in step ( 3). Interpolate for the proper_.,.U if necessary. ( 5) By th e use of the factors appearing on each diagram, calculat e the Damping Factor a nd the Period in the proper units e The time to damp to ha lf amplitude is given by the formula, 7j z = 0.693 O.F. seconds. The shaded lines on each curve sheet represent the b(llundaries of dynamic longitudinal stability in gliding flight. The plane is dynamically stable if the point (X,Y) f a lls within the boundaries for all the lift coefficients. Any inherent insta bility should a ppear immediately by reference t o t he diagrams. POSSIBLE Ri'.\.NGE OF PAFU\.:•HETER VALUES Rana:e guantity ✓:1/k~ ____. S'c/s _________ __ __...... __ ___ ho ii de;, -"t: c7c;;f-t; X y .,./4/ .,._.,.. _ _ ..., -- 5 - 25 0. 10 - 0.20 0.20 - 0.25 - - - - - - - . -r - - -0 .. 05 - 0.20 -- ___ . ----- 2.00 - ---------------------------- -1 .00 ,_ .,. )1• • 00 - 1:.• 0 0 0.50 - 8. 00 5.0 - 20.0 (18) ACCURACY OF DI AGRAMS It has been found that the theoretical and measured longitudinal stability characteristics for gliding flight agree reasonably well if neutral stability is not approached too closely. A reasonably stable plane in gliding flight will~be stable with power on; but if gliding flight characteristics approach the boundaries of instability too closely, then dynamic stability with power-on is not assured. far as the accuracy of the charts ) themselves As a.re concerned, two practical examples of their use will serve to illustrate. EXAMPLES OF THE USE OF THE DIAGAAB/1'3 ( 1) Fo r t he first example illus t r et ing the use of the consj_ der t he Doyle 0-2 biplane. Its properties are as follows: (4) ch a r ts, W = 1315 # S = 159.5 sq. ft. b = 30.0 ft. st= 18.7 sq. ft. a/t = .34 b/t = .-. 36 h0 = .25 E = ( Estimated) (.34 - .036) - ( .25) = 0.054 1 = 11.8' kB=~ ARt = 4.33 ( Effect of fusel~ge neglecta (19) Pr oc edure : c0 p = ~04 -- G05 ( Estima t ed ) Us e Cha rts for Ca se II. 2. } = 13.08 /4 ✓ . 4. = I t wi ll be suff ic iently accura te t o use the cha rts for f l = 10 .. 00. Referr i ng to the cha rts, we have: CL -K :p 0.3 0 .043 32 . 3 Oo5 0.036 25.6 l~O 0.,035 18.7 1.2 0.039 16.9 where K a.nd lJ a re in English uni ts~ Di scussion The above value s for K a nd :pare plotted in Fi g ~ lo (4_) The a 3 r e ement with experi ment a l da ta on the Doyle 0-2 very good for the da mping f actors. is For the :neriod, the agree ment is good for h i gh speed, i. e., lov, CL 1 s ; gives a high value a t the higher lift coefficients. but In the respect that the designer i s interest ed fundamentally /N THE ,q,-R/00 ,,,.9NO ( 20.0 ) damping factor in the range of high speed, the charts give good results fo r thi s particular case. (2) As a second problem for illu s tration, consider the new Dougla s Transport, the properties for which are the following: w = 17,500 # s = 939 sq. ft. st= 145.6 sq/ ft. a./t = .25 b/t = + .13 h0 = .23 h = ( .25 + .013 +.925) ~~ .23 ~ -0~058 1·= 36.62 ;==,.kB= /4-500 A¾= 4.75 c:ICZ = 4.7 ( From wind tunne l t e st) d::t, Procedure: 1. = 4.70 = 0.02 Use ch a rts for Ca se I. ,,1/ 2. _b._ =- • oseJ(3G,GZ) = X = ?l /. 74- 4S /45:G = 939.o 171.08 ~ 4. _/4 _/ = /3.0eJ (?a.1o 5 } = 6.GS 36.6~ Referri ng to the charts for Case •I, us'iing the cooro.inates X and Y from (2), we see that the pla.ne is well within the st able range . Thj_ s is due to the f a ct tha t Y i s very lar ge , ioe., the. tail a rea is l a rge. The va lues for K and pare : ( 20, t) ~ = 10 = 5 ~ p ~ = 6.65 K p p CL K 0 .3 ; OB,¢- 47,o .033~ 39-5 ,03¢- 44-;•' 5. 0.5 .05Z Zl!l.o .047 24-0 .050 ZG,7 1.0 I '. 1·:·'.30 1c . 3 /4,3 .1,,ZO /Z.o , 1/E6 lsk5 1.2 • /t!}O IZO ·/GO /0.0 .y·'y53 //.,:3 K - Final Values CL -K p 0.3 ,0/6 48-0 0.5 .0/8 37,.~ 1.0 . '.0 3,3 Z6.3 1. 2 , (5)c70 ,z4.o Discussion : in Figure 2. The above values of K ani p are plotted From actual fli ght tests, the Douglas Transport proved to be dynam-ically stable. indicate a similar resu lt. ·.The · r• data • f rom t he Charts From an observation at high speed, gliding flight, t he period of the phugoid was found to be about fifty(50) seconds. For a CL= 0.3, corresponding t o the high speed range, the value for the period i s which agree s v e ry well with the obs erved va lue. 46-0 .5.t:e. ( 21) ACKNOWLEDGEMENTS The author wishes to express sincere a ppreciation to Dr. C.B . M:illikan, Dr. T. von Karman, e.nd Dr. W.B. Oswald for assistance ana. helpful suggest i ons rendered during the progress of this thesis. He also wishes to express a ppreciation to Mr . J.E. Lipp for assistance in the cons truct ion of the stabj_lity diagrams. (22) SUGGESTIONS FOR FUTURE. RESEARCH The same type of analysis as that appearing in this thesis should be attempted for power-on flight, thus completely covering the r ange which should be trrvestigated f or stability characteristics. Before any degree of success in this further application can be attained~ however, there must be more research done on the subject of the effect of the propeller slipstream. h a s been made by Gates An analysis for power-on fligh t ( 1) , but for the same reason tha t the present work was done on gliding fli ght , that of making a vailable mode rn data, his analysis is a lit tle obsolete for use in modern design. (23) (1) Gates, S.B. , A Survey of Longitudinal Stability Below the Use, Stall, with an Abstract for Designers' R. - M. 1118. (2) Bairstow, Appli e d Ae rodynamics , Chapter 10. (3) Glauert , F.., A Non-dimensional Form of the Stability Equations of an Ai p plane, R. & M. 1093. (4) Soule, E.A. & 'tvheatley, J.B., A Comparison Between the Theoretical aid Measured Longi tudina1 Stability CbRracteristics of an Airplane, T . R. 442. (5) Hall, S.S. & England, T.H., Aircraft Performance Testing, page 138.