DEPARTMENT OF AERONAUTICAL ENGINEERING UNIVERSITY OF MICHIGAN PRELIMINARY STUDY OF TH E APPLICATION OF ELECTRONIC DIFFERENTIAL ANALYZERS TO AEROELASTIC PROBLEMS By M. A. Brull Assistant Professor of Ae ronautical Engineering R. M. Howe As:sociate Professor of Aeronautical Engineering AIR-9 August, 1954

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TABLE OF CONTENTS Page 1. Introduction 1 2. Summary and Purpose 1 3. Equations of Motion 2 4. The Aerodynamic Loads on the Wing 3 5. Analytical Solution for Untapered Wing 6 6. Representation of the Unsteady Airloads for Solution by the -Electronic Differential Analyzer 9 7. Principles of Operation of the Electronic Differential Analyzer 12 80 Electronic Differential Analyzer Circuit for Solving the Beam Equation 14 9. Electronic Differential Analyzer Circuit for Computing the Unsteady Airloads 18 10. Electronic Differential Analyzer Circuit for the Complete, Gus-t-Response Problem 18 1l. Example S.olutions of the Gust Response of a Tapered Wing 21 Discussion 24 Bibliography 25 and 26

LIST OF FIGURES Figure Page 1 Operational Amplifiers 13 2 Wing Divided into Stations 14 3 Analyzer Circuit. for.Solving the 4-Cell Cantilever Beam 1 7 4 Analyzer Circuit for Computing the Lift Am as a -Function of Wing-Displacement u.19 5 Analyzer Circuit for Computing the Lift A as a Function of Gust-Velocity w g 19 6 Analyzer-Solution for the Karman-Sears Function 20 7 Analyzer Solution for the Wagner Function 20 8.Step-Gust Response of Tapered.Wing 22 9 Step-Gust- Response of Tapered Wing with Tip-Tank.Equal to the Weight of the Half-Wing 22 10 Response of Tapered Wing to a Sinusoidal Gust of Unit Period 23 111

PRELIMINARY STUDY OF THE APPLICATION OF ELECTRONIC DIFFERENTIAL ANALYZERS TO AEROELASTIC PROBLEMS 1. Introduction In recent year-s the aero-elastic properties of wing:structures have proved to be a major factor in aircraft structural design and analysis [ 1]. The procedure of replacing time-dependent loads by static loads with a'"suitable factor of safety" leads to inefficient structures. This is due to. the implied assumption that the stresws -distribution is the same for static and dynamic problems..Furthermore, the term "suitable factor of safety" is itself nebulous, since the selection of any given factor.can only be based on previous experience and therefore is:useless when one deals with wings structures of a new type or shape. It seems more logical to express minimum requirements in terms: of the. dynamic characteristics of the structure in such a way that the structural weight can be safely reduced to a minimum. One of the.more important problems in aeroelasticity is the defterfmination of. the r...esponse of a:wing to gusts of given profile and intensity. This problem is the subject of the. report......Summary and Purpose'-,' _,. The purpose of this investigation is to determine whether the gust response problem can be solved.conveniently by means of an electronic differential analyzer. Since the greatest difficulty encountered in the problem is a suitable representation of the unsrteady aerodynamic forces, it was decided to consaidetr only a s~implified problem as far as the eelas'tic representatiorn of the wing is concerned but to investigate thoroughly the representation of all the aerodynamic terms. The wing was represented by a simple cantilever beam and only flexural oscillations were considered. In the actual case, of course, the wing should be considered as.a free-free beam or plate with a large *;Numbers in brackets refer to the bibliography 1

mass:co~ncenntrated at the center (representing the fuselage); both bending and twisting should be allowed, together with the rigid-body modes of motion [ 2]. It is expected, however, that the actual problem will not present any more fundamental difficulties than the simplified problem considered here. 3. Equations:of Motion Consider a:tapered cantilever beam of length b/2 which is subjected to a lift load L(x, t) per unit length of its span. The equation of motion for this beam is [ 3] 2 2- 2(EI ) + y = L(x, t) (1) ax aOx at where i(x, t) = lateral displacement of the beam x distance along the span EI = stiffness of wing cross-section t = time b /2 s- emi-span length L (i, t) = lift per unit span of wing y = mass per unit length of wing It will -be:convenient to put Equation (1) in dimensionless form. Because of the form of the lift functions [1 ] to be explained later, the new dimensionless:time variable is defined as s -= -- Ut (Z) where co = mean chord of wing U = airplane velocity. It is -clear from Equation (2) that s is the distance traveled by the wing in unit time expressed in terms of the half-chord. The following addi-tional dimensionless variables are defined: x = -/(bl2) u ='/l(b/2) 2

c = Chord of wing at any spanwise position 0c'(X) = c/co 0m (x)z ~/ o yo = mass per unit length of wing at the mean chord Of (x) = EI/EI~.EIo = stiffness of wing cros s-section at the mean chord. Substituting. the above expressions in Equation ( 1 ) gives 2 2 2 2. EIo l2 Fu bU a.3 0x:.f(xI) + 2 YO m( x) —' = L(x,s) (b/2) aax ~x' co as or, dividing by EIo/1(b/ 2)3 2 a u (b(3) -a0(X) + a 0ux = (WA)3 L(x, s) (3 ax ax Os xEIl where 2 2 [ b 4 U y0 XL21 El co The.boundary conditions for a cantilever.wing are u(O, S).(O, s).)= o ax.2 2 --- (1,s) =.(,s) = 0 Ox and the initial conditions are u(x, 0) = f(x) (6) a. (x, 0) = g(x) as 4, The Aerodynamic Loads on the Wing The aerodynamic loads on the wing can be divided into two parts. [ 1] L(x,. s) = Lg(X, s) + Lm(X s:) (7) 3~~~~~~~~~~7

where Lg (x, s) is the lift on the wing due to a gust of given intensity and profile and L m(x s) = lift due to disturbed motion of the wing. The change in lift coefficient due to sharp-edge (i. e., step-functiorn gust of intensity w- is [4] ACL - 2 f - U(s) (8) g U where q(s) is the-Karman-Sears function which is:approximated by 1 -*0.130s -s (s) = 1- 1 (e. 30s +e-) 9) 2 From Equation (8) it is:,lear that the lift per unit span due to a sharpedge gust is Ljg 1 p UC ACL 2 Lg irp Ucw ~.(s) For a gust of variable intensityr W(s) the lift per unit span is. therefore Lg = rpUc f w(e).(s-r) d'r (10) The lift due to disturbed motion L is the sum of three effects: the m apparent mass effect due to motion of acertain volume of air induced by the wing motion, the so-called quasi-steady effect which is due to the change in apparent angle of attack caused by the wing motion, and the effect of the wake reaching back on the wing. The wake effect is not important unless the frequency of motion is high. The lift due to dis:turbed motion is [ 51 2 2- S Lm - wpU 2 C''pc j (s-a(dr (11) c as 0 where a.(S) = angle of attack 0 (s) = the Wagner function which represents the quasi-steady and wake terms for a step change in angle of attack. 4

The Wagner function is.represented to sufficient accuracy by O(s.): 1 - o0. 165 e 455s- 0._ o 335 e 300s (12) The apparent angle of attack due:to wing motion is 1 au z a.u U8t =c as:hence' (s) = 2 C0Os and therefore the total lift due to disturbed motion of the wing is -In terms of the dimensionless variables -z2- a 2 -: Lm =TPU2[?J 0c (x) 8s 2 z'UL70 2C The total aerodynamic force acting -on the wing is therefore L(xs) = xp U c= iC(pU) J w'(f:) s s-. ) d' -zTnpU U [ 0c(x) U L=i Oc(X) f 0ds-o') where'W(es o - )t h ) = dimensionless gust intensity (15).(bL3.U 2(x) s= (x) (s-r)w'(r) do EIo R o STubstituting this expression in Equation (3) gives for the equation of ~~~i~~~~~xOn' L d o'(13)

L f(x) i C ~ [X~ 0(x)1J O aa L f{X) G28A lk2 m(X) + - "cX' ax a[x 2 as (16) - 0(X') (s) + 60 (x)i m(s)'R' g where g (S) = f (s- s) w'( r)d, (17) g 0 Q Lns) = - /S (s-'')' 2.do, (18) 2s 6 = 2wp U {b12) (19) EIo and b R = - = aspect ratio (9a) 0 The solution of Equation (10) together with the proper boundary and initial conditions (Equations (5) and (6))will yield sthe lateral motion of the wing under the unsteady aerodynamic forces. 5. Anlalytical Solution for Untape red Wing The solution of Equation (16) is in general impossible to carry out for any case of practical interest such as a tapered wing. In view of the complexity of the wing structure9 it is necessary to resort to s'ome approximate method of solution or to make use of automatic computation facilities. However, it will be of some interest to examine and solve the special case of an untapered benam as its solution will bring out an interesting property of Equation ( 16). For an untapered wing Of6(x)= 0 (S) W (X) 1 and Equation (16) reduces to a4 [ 2z+5]a2u = J @(sf) wt2 ~s ) d2 6 f 0(s )u ax4 2 as R o a2 6

Taking the Laplace transform of Equation (20) with respect to the:time variable s gives 4 d u + +X +- +60 *) p2u*= 6- pW* w ( 2 + 6 + [)*Epflx)~+ g~x)] dx 2 aR z (21) where the starred quantities indicate the Laplace transform of a function which is defined as X*(p) = L{X(s)} = f epsX(s)ds (22) 0 If we now consider the problem of a similar wing subjected.to the same gust but with the unsteady aerodynamic forces.neglected, the equation to be solved is 4 2 au-1 = - _ d 4is —) w' (-)d- o (23) ax4 as2 o and:the Laplace transform is 4 du' 22 * 6 _ + X p I - pj w (24) 4 dx R Comparison between Equation (21) and (24) reveals that these equations are identical except for the parameter Xk which is replaced by X + 6 /2+ 60* in Equation (21). Since the quantity (K2 + 6/2 + 608) is not a -function of x, it simply plays the role of a parameter in Equation (21). Thus, if we know a solution of Equation (21) subject to given boundary conditions, we can obtain the corresponding solution of Equation (24) by simply replacing X 2 by ( 2 + 612 + 60*) in this solution. Then we can obtain the solution including the unsteady aerodynamic terms for the wing by modifying the classical solution for a beam subjected to an arbitrary load. This is done by 1) taking the Laplace transform of the classical solution with respect to time, 2) replacing 2 by (k2 + 6/2 + 60 ) in this transform and 3) performing the itverse Laplace transformation. ~7

For the present case, the classical solution would be 00 X(x)[D En (2:5) +Cn s cos Wn((s-) f _ (O-T) w(T)dTdofj 0 0 where Xnx) = coshnx oss a - (sinh a, - sin anx) n n n n n n sinh an a sin an 0 (26) coshn n+ cos n n R 0 4 and an are characteristic numbers determined from the equation 1 +cosh a coS, = 0 The Laplace transform of Equation (25) is nul ~ l (= 1 Xn(~n) n + P + co (P +6 4. so that ince the parameter appears oltionly including the usteady forcesdify this u xp) = 128) n=l p +G(p) p +G(p) CnIn... p +Gn(P)

Hence the solution will clearly be of the form o00 u(X,s) = X (x) [Dn Fn(s) + En, Hn(s) +tc- n -W d d ] where Fn(S) = L-1 { p + Gn(P) (3.0) Hn({) = L{ Z = f Fn(5) do %. + Gn(p O With the function 0(s) defined by Equation (12) the expression in Equation (30) turns out to be the quotient of two polynomials in p and the inverse Laplace transform of such a function is easily obtained by the use of the Heavyside partial fraction expansion. It is interesting to note that the unsteady aerodynamic loads, which represent a special type of damping, do not affect the mode-shapes of the vibration. This is due to the fact that the aerodynamic loads were calculated from a strip theory which neglects the finite wing aerodynamic effects. Even though this assumption is mos:t likely in error, the results of this analysis constitute an approximate solution which may be of value in obtaining rough estimates of response characteristics for design purposes. For a more accurate analysis it is believed that the -only practical method of solution is by the use of some automatic computation method. 6. Representation of the Unsteady Airloads for Solution by the Electronic!Differential Analyzer The solution of classical beam vibration problems:on the electronic differential analyzer has been taken up in previous -studies [ 6, 7]. In the present case the flexural beam equations are satisfied in exactly the same manner as in this previous work, namely, by dividing the beam inrto stations spanwise anrd by replacing xderivatives by finite differences. The main difficulty in the present problem is to find a suitable method of computing the unsteady airloads. Only the 9

approximate strip theory will be considered here. It is in general difficult to mechanize directly an analog computatiosn of convolutLion integrals such as those appearing in Equations (10) and (11). For this reason it is convenient to derive differential equations having as their solutions the convolution integrals in Equations (10) and {1l1). The first terrm in Equation (1l1) presents no difficulty since it simply plays the role of an additional inertia term which simply increases the effec= tive mass per unit length of!.lthe beam. The second term on the righthand side of Equation (16), which involves the Wagner function, is defined in Equation (18) and can be computed inr the following manner: substituting for 0 from Ecupation (12) and taking Laplace transforms with respect to the time variable s gives,* _[1 0.165 0 335 p2u 32) p p+ 0. 0455 p +.30 - where I m is the Laplace transform of Im. Note that Equation (32) assumes -that the initial deflection and velocity of the beam are identically zero. This:is permissible:since under a s~teady lift the velocity will actually be zero and this initial deflected position can be chosen as, a new origin for deflectiono It should be remembered, however, that when the values for bending nmoments and stre-sses are calculated, the constant values due to the steady flight loads must be added to the values resulting fromn the present analysis. Equation (32) can easily be put in the form [p3+ 0.34552 + 1361 P+ 0. 2808p + 0. 01365 p u Applying the inverse Laplace transformation now yields 3+ 0,3455 + =0. 5 0O.2808- + 0.01365p 1 ds3 ds ds L as4 3as2 *s (3 3) Equation (33) is an ordinary differential equation with tm as the unknown and a-forecig function depending on quantities r~elated to the beam displacement. This equation can be solved with a differential arnalyzer circuit 10

having as voltage inputs the displacement quantities available from another analyzer circuit representing the beam equationrs of motion. It will be necessary to have a separate lift-comnputing circuit for each station of -the beam. For convenience in establishing the computer circuit, Equation (33) is integrated three times with respect to s with the result m + 0. 3455 fm ds + 0. 01365 firm ds ds u(34) - i_[0. 5 u + 0. 2, 8 u +0o, 01365 f uds] as Rearranging the terms gives m'= fli 0 3455 1m - 001365 u - 0,01365 fm ds- ds (35) - 0. 5 0. 2808 u as The equation is.now in convenient:form for solution by the electronic differential - analyzer, The first term on the right,-hand side -of Equation (616), which involves -the gust velocity profile and the KarmanaSears -function,. can be computed in a similar manner. Applying -the Laplace transformation and -substituting for 4 gives * - -"L-' 9.pwea* (36) g p 2 p +013 2 p+ 1 which can be pu:t in the form (p3 +1.13p2 +. 13p)Ig* [. 565p +. 13] pw or (p2 + 1 13p +0. 13p) Ig*g [. 56 + 0. 13 w (37) Applying to this last expression the inverse Laplace.transformation gives 2 ds —:+ 1 13 +- O. 13.1g [0.565 dw +0.3 wl {38) ~ds Zds." ds1 *11

Integrating Equation (38) with respect to the time variable gives 0. 565 w - 1. 13 f - g w ds (39) g ds g which, again, is now in convenient form for solution by the electronic differential analyzer. Before examining the analyzer circuits for solving the complete gust-response problem, i.e., Equations (16), (35), and 39), let us consider briefly the basic theory of the electronic differential analyz e r. 7. Principles of Operation of the Electronic Differential Analyzer For the reader unfamiliar with the basic operating principles of the electronic differential analyzer we have included here a brief description. Those who wish to become more familiar with this type of computer are directed to other references [8, 9] The basic unit of the electronic differential analyzer is the operational amplifier, which consists of a high-gain dc amplifier having a feedback impedance Zf and an input impedance Zi, as shown in Figure la. To a high degree of approximation the output voltage eo of the operational amplifier is equal to the input voltage multiplied by the ratio of feedback to input impedance, with a reversal of sign. If several input resistors are used, the output voltage is proportional to the sum of the input voltages (Figure lb). If an input resistor and feedback capacitor are used, the output voltage is proportional to the time integral of the input voltage (Figure lc). The operational amplifiers shown in Figure 1 can therefore be used to multiple a voltage by a constant factor, invert signs, sum voltages, and integrate a voltage with respect to time. These are the only functions necessary to solve ordinary linear differential equations with constant coefficients. Thus voltages are the physical quantities representing input functions and dependent variables in solving equations with this type of computer, while time represents the independent variable. The way in which operational amplifiers are connected together to actually solve differential equations will become clearfrom the schematic circuit diagrams in the following sections. 12

Input Output el -----.,e-, -. Zf E ier Zf a) Operational Amplifier R aia f - Rf ea -b lb -Output, O a-pm —-— al mRf a Rf m mf'eb- f e - e + e +e e) o" a Rb Ib Cb'W R, i e c b.) Operational Amplifier as a Summer Fua ia if i C ea1 -:: Rb ib_ I l \ Output = R a" ea+' a eb R a d. R i eC C'~ c) Operational Amplifier as a Sumnmer and Integrator. Figure 1. Operational Amplifiers 13

8. Electronic Differential Analyzer Circuit for-Solving -the. Beam Equation We have already seen from (16) that the equation representing.transverse motion of a beam (and hence, we assume, transverse motion of the aircraft wing) is given by x) + 0m(X) + (x) = _ 0 (x)A + 60 (x).a2-? fx I2 m yc R c c( m,aLx ax 2. (40) Here u is:the dimensionless transverse displacement of the wing, x is dimensionless distance along the wing, and s is dimensionless time. Equation (40) above is a partial differential equation, and if we are going.to solve it with the electronic differential analyzer, we must convert it to one or more ordinary differential equations. This can be done by measuring the wing displacement u(x, s) not at all values of x along the wing but just at certain stations along x, as shown in Figure 2. Let ul(s) equal the displacement at the first x station, u2(s) equal the displacement at the second x.station, etc. Furthermore, let the separation between stations be a constant Ax. Clearly a good approximation to au ax at the n + 1 /2 s tation can be written as + 1 u) (41) Ox n+ 1 A TIM - n+ 1/21 - Figure 2. Wing Divided into Stations 14

In fact, in the limit of infinitely small Ax this is just the definition of au/ax. Similarly, at the nth station a =_ I au au Ox Axax [x 1 n x ax n+1/2 ax n- 1/2 or f' | = 1 2 f [n + 1- u + u 42 af'(Ax)( n -n Using the same difference approximation we can write from Equation (40) the force equilibrium equation at the nth station. Thus F2~ d~un 1 1 u m 2 2c d [ ff Z 2 U. n6 j + 0 s (s)( (43) x f ni']R n n n n Equations (42) and (4:3) are iterated for each station. Note that un is a -function only of the time variable s. Thus we have converted the original partial differential Equation (40) into a set of simultaneously second-order -ordinrary differential equations which can be solved by the analyzer. Note that Ax = 1/N, where N is the number of stations into which the beam is divided. Equation (43) can be- rewritten as -n 6 n= _- mn +!1 A +2+0 m dn~ ds2'~ n-i-i n n- N4 cn m r~B 2 n (44) where 1 [x2 +6 02 "d T + c dn N m 2 n Here mn is proportional to the bending moment at the nth station and is given by mn = (un -2u + u 1) (45) The built-in boundary condition at x = 0 requires that u(0, s) = u/ax(O0, s) = 0. For the cellular beam this implies that 15

uo U1 = 0 where the actual built-in end occurs at the I/Z station. Similarly at the free end the bending moment and shear force vanish. For the cellular beam this implies that mN = mN + 1 =, where the actual free end of an N-cell beam occurs at the N + 1/2 station. For -the 4-cell beam shown in Figure (2) and considered.in this report, the following equations are usedo du2 m3 + 2mm2 03 d2 d3 ds RN 3 93 3R 3 4 dsZ i m3 4+ 94 g4 N 4 m4 where m = Of u2 m2= f (u3 - 2u2) (47) = f 3 u4 2u3 2) The electronic differential analyzer circuit for solving these equations is.shown in Figure 3. Note that only 9 operational amplifiers are required and that voltage inputs and outputs represent the aerodynamic lifting force and wing displacement u, respectively, at each s.tation. Although only 4-cells may seem like too crude an approxima-tion to the wing,, theoretical solutions for a uniform 4-cell carntilever beam have.shown that the mode shapes and frequenclies for the first two normal mode.s of vibration agree within several percent with the exact 16 there~ ~ ~ g ia.oeton crepnecbe eN rsso ausi h whercutadmsadsifesscefcet tec tto.Tu~t

0fl -m1 [ f m2 RN /60I -I -t 2, 02IIC\ I/d 0 5 N0.6 0 d' ~:.~m2 22 ds R /6 Al Caactos reMiroard 3 1 0.5 0 ~~~d.- 5 +'/g~~~~~~~~~~~3 I cj 3d 3 0.3l ~~b~~~S 3~d 3 du 1, 1 RN 4/c - C4 94 w ed n4 44 6 B C~C. du 4 -8 ~~~4 V "'M- ds ~Cc ds Initial-Condition and Ground Connections Omitted for Clarity. All Resistors iare Megohms All Capacitors-are Microfarads Figur~e 3..Analyzer -Circuit for Solving the 4-.Cell Cantilever Beam.r 17

one ~need only change by the appropriate amount the resistoar 0m which repres~ents t.he mass at station 4. 9. Electronic -Differential Analyzer:Circuit for Computing the Unsteady Airloads Equation (35) gives the lift term Am due to transverse wingdisplacement u. The electronic differential analyzer circuit for solving -this equation is shown in Figure 4. Voltage inputs to -the circuit are the wing displacement u and velocity 0d au/as. The voltage output is l m This circuit must be repeated at each station for the cellular wing representation. Equation (39) gives the lift term g due to the normalized gus t velocity w. The analyzer circuit for solving this equation is shown in Figure 5. Voltage input is the gust velocity w and voltage output is I If we assune that the gust velocity w is independent of the spanwise coordinate x (a reasonable assumption), then the single circuit in Figure 5 can be used for computing g at all stations. It will be remermbered that the Karman -Sears -function is;the change in lift coefficient corresponding to a step function in gust intensity, whille the Wagner function represents the change in lift coeffiiesnit due to a step change in angle of attack (or in au/as). Therefore the accuracy of the circuit in Figure 4 can be verified by supplyirg as an input a step function for au/as and a ramnp function for U. The output Am should then be the Wagner function. In the same manner, if a step function in w is imposed at the input of the circuit of Figure 5, the output of the circuit should be the Karman-Sears functiono The results of these verifications are shown in Figures 6 and 7. Compari= son of these curves with the actual @urves of the Karman-Shear and Wagner -functions given in Equations (9) and ( l2) shows excellent agreement. 10. Electronic Differential Analyzer Circuit for the Comple te, Gust-Response Problem The analyzer circuit for the 4~-cell cantilever beam repre-pre senting the aircraft wing is shown in Figure 3. Inputs to this circuit at the n th s:tatiorn are the lift terms lgrn arnd.nn. Outputs of the circult are the wing displacement un and velocity dun/ds. These outputs are used as inputs to the Wagner-function circuit at the nth station (see - Fgu-re 4). The output m of the Wagnaer-function circuit is, in turnD fed back 18

-0 3455 fmd 0.l365 ff ds ds- uds) m 0. 281u + 0. 3455 5Pm ds Figure 4.. Analyzer Circuit for Computing the Lift 7' as a- Function of Wing-Displacement u. 7, 7 as a Fnction ofGust-Velocity w 1;o7~1

: ~~~~~~~~~~~~~~~~.......:... li.................f - -—......... i j j ~ i: 1 I: I I 1 T fJl 4 111 Ii:: i':::'': T "''~ _-.:.~'!''-T' _:_L: 1.I..... ~..:.::::::.: ~........ 1 tjl 1;-I:................................................-.. i i i. li:: i --------- — 4~~~~~~~~~~~~~~~~~~~~~~~~~~~~:..:.]...:;::.: i[;;; i~~~~~~~~~~~~~~~~~~i;~ ~ ~ ~ ~~~+ 11 irr ~ -T ii-i ~::~:~~; ill!~~~~~~~~~~~~I- I:';~'.:i —~I' i-ti'j - -............. i:::~~~~~~~~~~...... I-i ~:I-:..iii i-i~~~~~~~~~~~~~~~~~~~i........... 4 LL......... T " k,...........; ~~Ti ~~~f ~ ~..... i i - Figure 6. Analyzer Solution for the Karman-Sears Function "'.It~~~~~~~~~~~~~~~. 1..;.. ~ ~ ~ ~ ~ ~ ~ ~....::..:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.: L~it~!;tieS il- ~~-:'I — ~ ~ ~- ~~~I~ ~ ~~-I~~ ~I ~ ~~ -I~ ~~I ~~ ~ ~I-~- I I~~- I ~ ~ I~:: ~ ~ ~ I:.......... ~ r~- ~-:- ~~~!1~~~1 ~1:.~1 ~~~( ~~1~~I 1-~~~~~~~~ ~~~~I ~ 1-~~1~~~1~~~1 ~~I- ~~I~~~i~~ 1~~1~~ I~~I~-...........!.-ii:..........~~~~~~~~~~~~~~~~~il iliL:I~~~~~~~~~~~~~~~~~~~~~~~~~m.:........ I-1-tti- 1-i..................'I.....i i I... - - - - - -'H MLi-; i-.i:..;.............:................... L iH-: it ----- T- t~~~~i'! I'cFT- - Figure 7. Analyzer Solution for the Wagner Function

to the corresponding input connection on the beam circuit. Also fed into the beam circuit, alt each s-tationris,the lift term Ig computed by the Karmen —Sears:function circuit in Figure 5, where w is'the gustlvelocity input. The entire representation of the gust-response problem,, as defined here,- requires 9 operational amplifiers for the 4-ell cantileverbeam. representation of the aircraft wing, 3 x 4 or 12 operational amnpli: fiers for the representation of the wake induced lift terms at each station, and 3 operational amnplifers for the lift caused by the gust itself. Thus the -total number of amplifiers is 24. 11. Example Solutions of the -Gust. ResRponse of a Tapered As an example problem an aircraft wing with a-: 1 taper rato in chord and thickness ad-an fundamental bending- mode frequency of 14 cycles:per second was set up on:the electronic differential analyzer. An. aircraft velocity of 500 feelt per second and -an altitude of sea level were used in the aerodynamic representation. Step functions of gust velocity of 30 feet per secod magnitude were applied and the wing response as computed by the electronic differential analyzer was. recorded. Figure 8 shows.the dimensionless wing displacement near the tip (u4) as a function of dimensionless time s, along with the velocity du2/ds near the -wing root. Note -how the fundamental bending mode is.damnped by the aerodynamic wake effect in the u4 recording; note also how the higher nodes.show up in the wing velocity near the root. Similar response curves for 30 feet per second step-gust inputs are shown in Figure 9, where -a wing tip tank equal to one.-half the weight of -the half-wing has been.added to the circuit. This was accomplished merely by increasing -the resistlor labeled dd4 in Figure 3 by the appropriate:amount. The longer period of the response curves is;clearly evident. In. Figure 10 is shown the wing response to a unit-sinusoidal gust input of 1.00 semi-chord lengths in period. Again the response has been computed both with and without the tip tank. 21

4 -f -H'.............. 4-.................:F........................................ T IFT......................... 7LE: % + N 77t 7 _LL-L I.:: w........................................................................................................... E............................................ th............................................ 4, w....... I T I........... 4z............................................ T.............................................................................................................. Cn +pT........... I...... - - -7 —---- --: .................................................................... 1!H 0;........... I.C +................. z t [-r, _14;4- 1101i" A................................................................................................................................... 4- k_............ 7 1: J...............4-4....... +........... _14 7 7 -44-H+t!t HI I................ T 44-F-~'LJ# 14- - 1...;;;.; ill iA H-1 I I - I ]j I'' - I i!;:;: 1;.......... 1, J, L.L.Lj -,Haiti T_ i TI-11- -L i-_ R Figure 8 Step-Gust Response of Tapered Wing 4+4+ It -444+ _JLLL __J_ _Q j~Z1 + + L [tj. T'I I Ht I I "t t AML: — t — lil -14 J_ +# + _1-_ <T 117 771, A-........... -L,-' r A -41T. Tt-i w all!! I, il RJ - tr# i T-1, I':J-1_1+ ~ H -T_Jtl i-'t -H _1 t1fliflil 1, 11 I I 14 +H d 0- Aim all.:j H Rt HIT V) + + I I I I I, I 111 11:1-N-1- 1-H-IH 111 -r7T-H;! H.[falls F- Ti- _:-III -[Joel[ J-t!:-r,- prT;Tt n _L4R -`- ~+ + _f_ f.P 4_4 -Pt mu F i7-.7 + t4 I t L1,+ 4- I H m rM I I -- ---- + — it F tthill-, Mt, +H+ I +H+ H++ 1 i Ell E ------- Figure 9. Step-Gust Response of Tapered Wing

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Discussion it is realized that the problem treated in this:report does not represe:nt a practical.- case,, principally because.of the elastic representation.chosen. In. the.solution of.an actual problem it would be ecessary to allow bending and twisting of the wing together with rigid body pitching$ and plunging motions.:n the:case of a.st.raight wing:this problem would not present additional difficulties and the mechanization of the equations would simply require more amplifiers~ It; is probable that finite wing aerodynamic corrections can be incorporated into the appropriate circuits. I.n the case of swept and delta wings the elastic representation of the wing structure must be modified because of the elastic -coupling between bending and torsion present in swept wings and also because delta wingsbehave more like a plate than a beam. For the low aspect ratio swept or delta wing it will probably be necessary to use plate representation. The equations of motion for plates of variable thickness are very complex and it will be necessary.to resort to approxifate representations such as those suggested by.:Stein [ 10] or Sechler, Williams and Fung [ 1 1 ] It has been found -that in delta wings of low aspect ratio the chordwise bending deflection is not negligible compared to the other.deformations [12] ] Hence it will be necess:ary to include in the aero. dynamic terms the effect of this change of camber. It should be emphasized -that the method of soluhtion suggested here by means of an electronic differential analyzer has useful applications in the preliminary design of aircraft wings. Dynamic aeroelastic problems.are in general so complex that it does not seem practical to use the available analytical solutions to obtain design estimates. The computer representation suggested here has the advantage that once the circuits are connected up, it is extremely easy to change-the mass and -stiffness dis$tribution of the wing and to investigate the effect of such changes on the gust response. The differential analyzer used is of a type in common use in the aircraft industry. Considerable aeroelastic analysis has been carried out on the Caltech analog cormputer using similar equations [ 13]. Here:the beanam and torsional difference equations are represented by passive circui6ts. 24

BIBLIOGRAPHY 1. R. L. Bisplinghoff et al., An Investigation of Stresses in Aircraft Structures Under Dynamic Loading, Bureau of Aeronautics Contract No. NO.. as)....8790, Massachusetts Institute of Technology, 1949. 2. R. L. Bisplinghoff, Gust Loads onRigid and Elastic Airplanes, Bureau of Aeronautics, Contract No. NO a(s) 8790, Massachusetts Insltitute of Technology, 1950. 3. S. Timnoshenko, Vibration Problems in Engineering, Van- Nostrand Company, New York, 1937. 4. T. von Karman and W. R. Sears, Airfoil Theory for Non Uniform Motion, Journal of the Aeronautical.Sciences, vol S, No. 10, 1938. 5. W. R. Sears, Some Aspects of Non-Stationary Airfoil. Theorr and Its Practical Application, Journal of the Aeronautical Sciences, vol 8, No, 3, 1941. 6. R. M. Howe, V. S. Hanemanr Solution of Partial Differential Equations by Differernce Techniques Using the Electronic Differential Analyzer, Proceedings Institute Radio'Engineers, vol 41.(1953). 7. C. E. Howe, R. M, Howe, Application of Difference Techniques to the Lateral Vibration of Beams Using the Electronic Differential,,,..',,.......,.L.... Analyzer, Engineering -Research Institute Report 21 i 5 1 -T, OOR Contract No. DA-20- 018-ORD-21811, Feb. 1954. 8. G. A. Korn, T. M. Korn, Electronic Analog Computers, Mc"Graw= Hill (1952z). 9. C. E. Howe, R. M. Howe, and Do W. Hagelbarger, Investigation of the Utility of an Electronic Analog Computer in Engineering Problems, - Report UMM-28, Engineering Research Institute, University of Michigan, Ann Arbor, Michigan, April 1949. 25

BIBLIOGRAPHY (Continued) 10. M. Stein, J. E. Anderson, and J. M. Hedgepeth, Deflection and Stress Analysis of Thin Solid Wings of Arbitrary Planform with Particular Reference to Delta Wings 9 NACA._TN.2621., Feb. 1952. 11. E. E. Sechler, M. L. Williams and Y. C. Fung, Theoretical and Experimental Effect of Sweep upon the Stresses and Deflection Distribution in Aircraft Wings of High Solidity, AFTR 5761, Part 18, Jan. 1953. 12. S. O. Benscater and R. McNeal, Analysis of Multi-Cell Delta Wings on the Cal. Tech Analog Computer, NACA TN 3114, Dec. 1953. 13. R. H. Mac Neal, G. D. Cann, and C. H. Wilts, The Solution of Aeroelastic Problems by Means of Electrical Analogies, Technical Report, AnalysiS Laboratory, California Institute of Technologyo 26

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