Effect of interaction on landing-gear behavior and dynamic loads in a flexible airplane structure

APPENDIX A CONDITIONS AT BEGINNING OF SHOCK-STRUT MOTION Since the shock stxut does not begin to deflect until the preloading force imposed by-the internal air pressure is overcome by . the inertia forces, the shock shut is essentially rigid during the interval between the instant of initial contact with the ground During this interval, since the deflection of the tire is essentially and the beginning of shock-strut motion at some time t=&.

the same as the displacement of the landing-gear attachment point, the system used in the numerieal calculations to represen t the airplane and landing-gear combination has only two degrees of freedom, namely, the rigid-body or zero-mode displacement The purpose of this appendix is to consider and the deflection in the first flexible mode, the higher modes being neglected.

the motions of the system prior to the beginning of shock-strut deflection in order to determine the conditions which exist at the instant the shock strut first begins to move; these motions then serve as the initial conditions for the equations of motion For this purpose it may be reasonably assumed that the tire forc.c- of the system during the main part of the impact.

deflection relationship is linear for the relatively small range of deflection prior to the beginning of shock-strut motion ancl In order to avoid a step jump in the time-history solution at the time G, the constant m’ that, therefore, Frz(z,) =m’efi should be determined so that m’ Zfr= m zf,r (Al) DISTBIBUTBD SYSTEM Prior to time G the equations of motion for the airplane and landing ge~ are given by equations (18) with initial conditions:

Zf(o)=au(o)=al(o)=o

Zf(o)=&(o) =Vvo

Cil(o)=o

Since fi= ~a~ equations (18) can be written as & (A20)

Mo&= –m’z,–m.ZJ– ‘JV.,(K.– 1)

(A2b) The exact solution of equations (A2) can be shown to be

A2–C ~osAt B’–C

1 AZ–C Sk At B2–C

—7. sin Bt +D ~ -TCOSB,+C($-+)]} @3,

V.. ~

‘J(t)==

{( )[

M’,~,2(Mo+m.)

c= ~

D=(li?.-l)g E=m’(M1+M&2) + c Q m’M1q2 F= ~ EFFECT OF INTER4C71’ION ON LANDING-GEAR BEHAVIOR By successive differentiation of equation (A3), the higher dmivatives of zJ(t) are found to be V,O[(A’–C) Cos At–(B’–c) Cos m]+l)

w=~~i (A4)

[FF)-(%shl’t]}

{

VVO[B(132-C) sin Bt–A(A’–C) sin At] +D[(B’–C) cos B-(.42-C) cos At]

w=~~ (A5) { }

“zJ(t)=&* VVO[B2(B’–C) cos Bt–A’(A2–C) cos At]+D[A(A’-C?) sin At–B(B’–C) sin Bt]

(A6) { } At the time G, the equation of motion of the unsprung muw of the landing gear m a free body is given by equation (2) which, with ~v~=m’z~, may be written as mu~f,+m’zf,=p.oA. cos e+ w.

(A7) Substituting for z~, and ZJ, in equation (A7) gives a relational.ip between t. and m’: $C (m’–mtiB2) cos Bt+m’C ~— =p%& co’ e+m. (A8) (Aa J)]} Becnuse equation (AS) is transcendental in both k and m’ (m’ being involved in the constants A and B), in order to obtain an explicit solution for G or m’, some approximation to the trigonometric terms is necessary, the order of the approximation depending on the accuracy required.

For the determination of G and m’ it will generally be su.flicient to assume ii.mkrder approximations for the trigonometric terms where only the first terms of their series expansions are used. With these approxi- mations the solution of equation (AS) for G is

Q(nOAa cos o+KLW.)

t,=

(A9)

m’MIMOVVO

As indicated previously, m’ cannot be chosen arbitrarily but must be determined in accordance with ;quation (Al), which may be written as m’=mzfr’-l The tit-order approximation for Zfr, obtained from equation (A3), is

Zf,= Vvok

(A1o) WMh these substitutions equation (A9) may be written as

Q(P.OA cos O+KLW.) lfI

(All)

“=CO miw,lkfo

[ and the equation for m’ becomes r–1

Q(p%f4 co’ O+ K-w.) T

mf=m,lfi

(A12)

MJ40

[ 1

—--- —-—— . . . ..

–.——. ..

642 REPORT 1278—NATIoNAL ADVISORY COMWTTY3E FOR AERONAUTICS are Tho fket-orderapproximations for the derivatives of ZJat time L, from equations (A4) to (A6),

&f,= VvO–Dtr (A13)

Zf,=VrOIC– (A’+ IP)]L-D (A14)

and

Z,7=(VV0–DQ[C– (A’+ P)] (A15)

With the values of G and m’ calculated from equations (Al 1) and (A12), the values of zf,= zU,, if,= ~U7, and Zfr= z ~, con be calculated from equations (A1O), (A13), and (A14), respectively. These values provide two-thirds of the initial conditions for the process subsequent to the beginning of shock-strut deflection (eq. 19). The remaining initial conditions, for example, aor, d,, and h,, can be obtained by manipulation of the differential equations (A2). From equation (A2b) it can be seen thnt (A16) “=&2[($+m~)5f,+(w+m’)”f,-$~+wa(K’-1)l By differentiation, (Al?)

~=~[(g+mx)’f.+(w+m’)’f,-%~l where, from equation (A2a.), tn’zf7+mti5Jr+W,0,(K.– 1) ~=– M, (A18)

PA COS e+W,JKrQ+Wx

=—

Mo } Differentiating equation (A2a) gives m’2fr+mU2 f, -.

aO,= — (A19) - M, The substitution of equations (A18) and (A19) and the initial conditions previously determined (z,,, 2J,, ~f,, and ‘;,,) into equations (A16) to (A18) provides the remaining initial conditions for the second phase of the impact.

EQUIVALW THREELMASS SYSTEM The equations of motion for the equivalent. three-maw system prior to the time G are equations (20) with initial conclit ions

Zf(o)= 2,(o)= o

and

if(o) = i.(o)= Vvo

Since it has been &own that, equations (20) are identically equivalent to equations (18) for the distributed system when the relationships between the constmts of the two systems are as defied by equations (27) to (34), it follows that equations (A3) to (A15) are equally valid for the three-mass system when the constants are redefined in accordance with equations (27) to (34). The redefined ponstants, in terms of the properties of the three-mass system, maybe written as ~=w&(M+@ Mo(mr+mJ

D=(KL–l)g

E=

mf’lm=+c m,m’*2

3’=

iK0(7n,+mJ EFFECEl?OF lNTERAC1’ION ON LANDINQ-GEAR BEHAVIOR Wlllm MO=mJ+m, (Al 1) and (A12), become The equations for G and m’ , equations

1 H(PaOAa GOS 19+KLWJ ‘fl

(A20)

‘7T0

[ m 1

and r-l (A21)

m’=mlfi H(paO& cm o+KJ’7J ~

[ 1 whore ==mf+m.

mr The values of G and m’ given by these equatio~ permit the calculation of z~r= z.,, if,= ~tir,and Zr7= z., by means of equa- tions (A1O), (A13), and (A14). The remaining initial conditions for the second phaae of the impact, z, and its derivatives at the time G, can be obtained by manipulation of the diiIerential equations (2o). Solving equation (20a) for z, at time f. gives

z,r=~ (m,+m~z f7+(k+m’)zJ,+.Lf + Wf—W= (A22)

[ Differentiating equation (20a) and substituting FVa(zf) =m’zr gives @23) An expression for Z% cm easily be obtained from equation (20b) as follom: (AM) ––~ (mf+m=)Zfr+m’zf,+ (L,+Z,)–(W,+Wf+W=) z8,— 1 s [ Equations (A22) to (A24), in conjunction with the values of zf,, 2fr, and Zf, previously determined, supply all the initial con~tions for the second phase of the impact of the equivalent thee-mass system.

APPENDIX B DYNAMIC LOADS IN AIRPLANE STRUCTURE Along station mass centers.—At the mass center of any The equations of motion of the airplane have been pre- station the displacement is designated ~ and z= e so that viously presented in several forms so that solutions for the equation @l) becomes mot ions of the structure cm be obtained in terms of the variables % and al, % and zf, or Z~and z~. The purpose of (B7) f=%+wl this appendix is to present equations horn which the accel- erations, bending moments, and shears at any point on the where ~1 is the modal function for the station mass center nirplane structure can be calculated once the time-history and is equal to m+ e%.

Equation (B2) becomes solutions for the basic variables have been obtained.

I f=uo+(z,–ao) : (138) ACCELERATION At any point.-The absolute displacement at, any point on Equation (B3) becomes the structure (see fig. 2) is (B9) F=+, [mA+mA+mWf-Z) & Z= ’W+XQ Sinco BSNDING MOMENTS w=ao+alwl Outboard of landing gear.-The bending moment at fmy and spantise station Vj outboard of the landing-gear station q=al~ summing up the inertia moments y= is readily determined by produced by the accelerations of the mass centers of all where WI and n are the modal functions for bending and stations i between station yj and the tip. Thus, torsion, respectively, ti

z=ao+al(wl+zpl) M miff(yf —yj) (B1o)

%izv>= -,

$

and Inboard of landing gear,-The bending moment at any Z=do+(il(wl+xp,) (III) spanwise station yj inboard of the landing-gear station y~ is Since equal to the sum of the inertia moments produced by tho ~l_zf—f% _— accelerations of the mass centers of all stations i betvmon .!3 station yj and the tip plus the moment produced by tim landing-gear force. Thus, the acceleration at any point< may also be written as

wl+ Xqq M (Bll)

=s~iii(Yi–Yj)+~(v.–u,)

%isrd

2= (%+ (z,— do) — (B2)

f=j ‘5 where Sinc13,from equation (36), F= – [Al&%+ W,O,(K..– l)+ W.]

SHE- Outboard of landing gear.-The vertical shear at any spnn- wise station yj outboard of the landing-gear station y~ is the acceleration can also be written as simply the sum of the inertia reactions due to the accelera- tions of the mass centers of all stations i between station ~, and the tip. Thus, Along elastio axis.-Atl the elastic axis, the displacement

s (B12)

=3 miff

(J’izv#l

i-j is designated w and z= O so that equation (B 1) becomes simply Inboard of landing gear.-The vertical shear at any span- (B4) ti=do+iilwl wise station yj inboard of the landing-gear station y~ is the sum of the inertia reactions due to the accelerations of the Equation (132) becomes mass centers of all stations i between station ~j ancl the tip, plus the landing-gear force. Thus, ?3= do+ (Zf— do)~ (B5) fl =~mti,+F

s

Equation (B3) becomes (Visua f -j (B6) =3 mifi–[M~ti~+Wt.t(K.-l)+M’.l (1310 i-j}

APPENDIX c RESPONSE TO GIVEN FORCING FUNCTIONS In this appendix equations are presented for the acceleration response of the airplane structure to predetermined forcing The cases considered me the arbitiary forcing function, the sine pulse, and a pulse functions applied by the landing gear.

For the particular case where the landing-gear forcing function can be represented bv a mfide up of sine and cosine segments.

single sine pulse, F(t)=Fm= sin Qt where $2is the circular frequency of the applied sine pulse and is expressed by T ‘=~T where T is the time to reach F-.

If the forcing pulse is not symmetrical in time about its maximum value, it may be represented by a combined pulse con- sisting of a sine function up to the time T and a cosine function subsequent to the time T. This latter function may be written as

(t’20)

F(t’) =3.= Cos Qlt’ where t’=t— T and G is the circular frequency of the cosine pulse; the initial conditions are the same as the conditions at the time t= T deter- mined from the response to the sine-function segment of the pulse.

The solutions are presented for the distributed system of the airplane (sketch a) and for the equivalent concentrated-mass system (sketch b)

%

k m, *

(a) 6

(b) FV) DISTRIBUTED SYSTEM The acceleration response of the rigid body or zero mode is immediately evident from the equation of motion~for n=O, namelv, W, F(t)+%t($-l)+w.

&=– The response of the deflection modes follows.

Arbitrary forcing funotion.-When the landing-gear forcing function is predetermined and arbitrary, the equation of motion for the nth mode (eq. (15a)) can be written as

(cl)

(n#O) where I’(t) is an arbitrary function of time and as is the genemlized coordinate of the nth mode.

The general solution of equation (Cl) maybe written as KLWJ.

t, (C2)

‘2(7) sin CO.(t-7’)dT+ Mm%, (Cos COJ--l)+as(o) CosCO”t+*) sin Cd

aa(~)= -M= ~ J The acceleration response is obtained by double differentiating equation (C2) as follows: (C3)

—an(o)%~ Cos4-U%(0)% sin d

‘a(t)=& @-%~; ~(~) ~ %(t–~)dT+KLWti COStixt 1 n . .

646 REPORT 1278—NATIONAL ADVISORY COMMITTEE FOR AERONAU’ITCS Equations (C2) and (C3) are general solutions to equation (Cl) and thus represent the response of any mode to an arbitrary forcing function J’(t). In the present study of landing impact, the initial conditions are

an(o)=o

and

a.(o)=o

Sine-pulse foroing fnncticm.-For the particular case where the forcing function is a sine pulse, the acceleration response, as determined from equation (C3), is

K.W=rf=

(C4)

f&(t)=F- *E Q2_@w2 ~ (Q sin @-u. sin Qt)–sin W – —M—+an (0)con’cos aJ-&(0) Onsin u.t

[ 1[ 1 n where, agg, a.(0)=0 and um(0)=O.

Half-sine-half-cosine pulse,-In this case the response up to time T is given by equation (CM). Subsequent to time

T the acceleration response, determined horn equation (C3), maybe written as

where t’=(t– T)>l

ax(0)=aq

u.(o) =unT

EQUIVALENT CONCRNTRATRD-MASS SYSTEM The equations of motion for the concentrated-mass system subject to an arbitiary forcing function are (see eqs. (2!.2)) nzfzf-lc (z.— z-f) +Lr W,= —F(t)

(cm

m,zf+mz~+ (Lj+L,) — (W,+ W~) = —F(t)

} Introducing the new variable u=z8—zf permits the combination of equationa (C6) into a single equation in one variable:

mfi+k 1-#$ u= F(i$)+J (C7)

()

where

J= LJ–Wf–~ (L.–W,)

The solution of equation (C7), by analo~ with equation (Cl), can be wmtten as

u(t)= & J-: F(T)Sin 4t-T)dT++2 (1–COS ult)+u.(o) Cos %t+~+) sin U,t (C8)

where M,

&.k —

m,m.

, EFFECT OF INTERACTION ON LANDING-GEAR BEHAVIOR 647 By substituting u(t) for z,— ZJin equations (C6) and combining, the following equations for the responses 2. and 2, can be obtained:

1’

F(7) sin ~(t–7)dT+:2 (1–COS qt)+?f.(o) Coscd.+ sin @] +-

z!8(t)= —~ — (C?9)

m, mfi o

s

[s

and

z~t)=+ –[F(t)*(~,–Wf)]+k [* J F(7) sin %(t–T)dr+&, (1– cos qi!)+u(0) cm qt+ ~ sin qt (Clo)

{ 1}

In equations (C9) and (C1O), u(O) =ti(0) =0 for the present application to landing impact.

the case where the forcing term is a sine pulse, equations (C9) and (C1O) become

Sine-pulse forcing funotion. —For k Fmm.(Q Sill qt–q SiIl ilt)+ J.

_ (1 –Cos qt)+u(o) Coscd+u+ sin 4]+

Z,(t)=—z (Cll)

m*(Q2—q2)

[

nnd whcm, again, U(O) =ti(0) = O.

Half-sine-half-cosine pulse. —The response up to time T is given by equations (Cll) and (C12). Subsequent to time

T, the responses (eqs. (C9) and (C1O)) become

(C13) and

k (COS @,t’–COs fl,t~_wa ~,t, _(Lf_wJ)+Jk(] ‘ME? @It’)

+k [u(O) COS CL@+%) sin qt’ (C14)

z~t’)=; F-

m~Q12—&) m~2

{[ 1 1} where t’=t— T=O

u(o) =%

ti(o)=ti~

APPENDIX D

AERODYNAMIC AND WEIGHT MOMENTS AND SHEARS In appendix B equations were presented for the bending moments and sheam due to the combination of the inertia forcos arising from the accelerations of the mssses distributed along the span and the landing-gear force. In the calculation of the total moments and shears, however, consideration must be given to the aerodynamic lift and weight forces. This appendix presents equations for estimating these aerodynamic and weight momenti and shears which, although only first approximations, are considered sufficiently accurate for the purposes of the present study.

If it is assumed that the lift coefficient is constant along the span and equal to the average lift coefficient of the wing C~, the lift force at any station y, is equal to CL$ V~’A, where A, is the area assumed to be concentrated at the station, The moment at any station Y, due to the lift and weight forces at each station{ outboard of shtion ~j is

M&j)=c. ; v.’: h/,-w)-9 g ~f (w –w) , (IX)

If unsteady-state lift effects are neglected, the instantaneous lift coefEcient is related to the lift coe5cient at the instant of initial contact by the expression C.=c.o+ CL=(’Y-70)

= C.o+ C.a VL “

C-)

Inasmuch ss the total lift at the instan~ of contact is K~W~.~, ~ =KLW,O, Lo ~Av’~ so that

Map,)= Y+ L’. (Q,- (D2)

%mh-w) VVO)~ VL i?&&h-yh ,., a

[ 1

Similarly, the shear at any station y, is (D3) s=,,,= Kp’ ]~Ai-g~w ‘+ CL=(UO–VVO) ; VL [ REFERENCES 1. Fairthom~ R. A.: The Effects of Landing Shock on Wing and 7. Mayo, Wilbur L.: Hydrodynnmio Impact of a System With n Underwmiage Detlesion. IL I% M. NO. 1877, British A. R. C., Single Elastic Mode. I—Theory and Ckmeraffmd Solution Wth 1939.

an Application to an Elastio Airframe. NACA ROp. 1074, 1962.

2. Stowell, Elbridge Z., HoubolG John C., and Batdorf, S. B.: An (Supemedea NACA TN 1398.)

Evaluation of time Approximate Methoda of Computing Land- 8. Biot, M. A, and Bisplinghoff, R. L.: Dynamio Loads on Airplano ing St~ in Aircmft. NACA TN 1584, 1948. Structures During Landing. NACA WR W-92, 1944. (For- 3. MoPhemcm, Albert E., Evans, J., Jr., and Levy, Samuel: Intluence merly NACA ARR 4H1O.)

of Wiig Flexibility on Force-Time Relation in Shock Strut 9. Lawrence, H. R.: The Dynamics of a Swept Wing. Jour. Aoro.

Following Vertical Landing Impact. NACA TN 1995, 1949. Sci., vol. 14, no. 11, Nov. 1947, pp. 643-660.

4. Pirm, T. H. H., and Flomenhoft, H. I.: Analytical and Experi- 10. Ieakaon, G.: A Survey of Aualyticnl Methods for Determining mental Studim on Dynamic I.mads in Airplane Structww During Transient Stresses in Elnstio Struoturea. Contraot No. N6 Landing. Jour. Aero. Sci., vol. 17, no. 12, Dec. 1950, pp. ori-07833, Office Naval W& (Project NR-036-269), M.I.T.

.

766-774, 786. Mar. 3, 1950.

5. O’Brien, T. F., and pia~ T. H. H.: Effect of Structural Flexibility 11. Southwefl, R. V.: Relaxation Methods in Theoretical Physim, on Aircraft Lading. Part 1. Ground-Loads. AF Tech. Rep. The Clarendon Preae (Oxford), 1946.

No. 6358, pt. 1, WADC, U. S. Air Force, M. I. T., July 1951. 12. Scarborough, James B.: Numericnl Mathematical Amdysfe.

6. Mihvitzky, Benjamin, and Cook, Francis E.: Analysis of Landing- Second ecL, The Johns Hopkins Press (Baltimore), 1950.

Gear Behavior. NACA Rep. 11- 1953. (Supersedes NACA 13. Levy, H., and Baggott, E. A.: Numerical Solutions of Differential Equationa. First Am. cd., Dover Pub. Ino., 1960.

TN 2755.)