Analysis of Landing-Gear Behavior

AKRO. & ASTRO. LIBRARY

~

NATIONAL ADVISORY COMMITTEE

FOR AERONAUTICS

REPORT 1154

c. 3

....

ANALYSIS OF LANDING-GEAR BEHAVIOR

By BENJAMIN MILWITZKY and FRANCIS E. COOK ,.

i: :.

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REPORT 1154

ANALYSIS OF LANDING-GEAR BEHAVIOR

By BENJAMIN MILWITZKY and FRANCIS E. COOK Langley Aeronautical Laboratory Langley Field, Va .

r

National Advi sory Committee for Aeronautic

Headquarters, 1724 F Street NW, Washington 25, D. O.

Created by act of Congress approved March 3, 1915, for the supervision and direction of the scientific study of the problems of flight (U. . Code, title 50, sec. 151 ). Its membership was increased from 12 to 15 by act approved March 2,1929, and to 17 by act approved May 25,194 The members are appointed by the President, and serve as such without compensation.

J E ROME C. HUNSAKER, Sc. D ., Massachusetts Institute of Te chnology, Chairman D ETLEV W. BRONK, PH. D ., Pre ident, Rocl{efeller Institute for Medical R esea rch, Vic e Chairman HON. ROB ERT B. MURRAY, JR., nder eeretary of Commerce HON. JOSEPH P. ADAMS, member, Civil Aeronautics Board.

for Transportation.

ALLEN V. ASTIN, PH. D. , Director, National Bureau of Standards.

RALPH A. OFSTIE, Vice Admiral, United tates Navy, D eputy LEONARD CARMICHAEL, PH. D., Secretary, mithsonian In stitu- Chief of Naval Operations (Air).

tion.

DONALD L. PUTT, Lieutenant General, United taie Air Force, LAURENCE C. CRAIGIE, Lieutenant General, United States Air Commander, Air Research and Development Command.

Force, Deputy Chief of Staff ( Development ).

JAME S H. DOOLITTL E, Sc. D., Vice President, Shell Oil Co. ARTHUR E. RAYMOND, Sc. D., Vice President-Engineering, LLOYD H ARRISON, R ea r Admiral, United tates Navy, Deputy Dougla s Aircraft Co., Inc.

FRANCIS \V. REICHELDERFER, C. D., Chief, United Stat es and Assistant Chief of the Bureau of Aeronautic.

R. M. HAZEN, B. S., Dir ector of Engineering, Allison Division, \Veath er Bureau.

THEODOR E P. WRIGHT, Sc. D., Vice Pre sident for Research.

General Motors Corp.

\VILLIAM LITTLEWOOD, 1. E., Vice Pre ident-Engineering, Cornell University.

American Airlines, In c.

JOHN F. VICTORY, LL. D., Executive Secretary HUGH L. DRYDEN, PH. D ., Director EDWARD B. CHAMIlERI,IN, Ex ec utive Officer JOHN W. CROWLEY, JR ., B. S., Associate Director Jor Research HENRY J. E. REID, D. Eng., Director , Langley Aeronautical Laboratory , Langley Field , Va.

SMITH J. DEFRANCE, D. Eng., Director , Ames Aeronautical Laboratory. Moffett Field, Calif.

EDWARD R. SHARP, Sc. D., Director , Lewis Flight Propulsion Laboratory, Cleveland Airport, Cleveland, Ohio LEWIS FLIGHT PROP ULSION LABORATORY, LANGLEY AERONAUTICAL LABORATORY, AMES AERONAUTICAL LABORATORY , Moffett Field, Calif. Cleveland Airport, Cleveland, Ohio Langley Field , Va.

Conduct, under unified control, Jor all agencies. of scientific research on the Jundamental problems of flight II CONTENTS Page SUj\ Ij\IARY _ _ _ _____ __ __ __ ___ _____ ____ __________ ____ _____ __ ___ _ ___ __ ______ _ 1 INTROm ; CTION _ _ _ _______ __ _____________ __ ________________ __ _______ _____ 2 YlIfBOLS ______ ____________________ __ __ ______________ __ ___________ __ ______ 2 :\lECHAKICS OF LA:-rDIXG GEAR _______________________________________ 3 D y namic of y ste m____________________________________________________ 3 Fo rces in Shock t l'u L _________________________________ ____________ ___ 4 H ydraulic [ol' ce ____ __ __ __ ___ __________ _ __ ____ __ ___ _ __ __ ______ __ __ ___ 6 Pn eumati c fOl' ce _____________________ _ _________________________ __ 7 Int ernal fricti on for ce _______________________________________________ 7 For ces on Tir e__________________________ _ _ __________ _ __________ _ EQUATIONS OF ?lIOTlO N ________________________________ _______________ 9 Motion Pri or to Shock- tru t D e fi eclion _______ __________ _________________ 10 MoLion Subse quent to B ginn in g of Shock-St rut De fi ect ion______________ __ __ 12 'OLU TlOK OF EQUA TIO NS OF :\IOTIO N ______ ___ ________________ _______ 12 ~Llme ri ca l In teg ration Pro ce dlll' ____ __________ ____________________ __ ___ 12 U e of Tir e Force-De fi ection Chara cte ri L ic ________________________________ 13 Effect of Dra g Load. ____ ___ _____________________ _________________ 13 EVALUA TIO N OF A.\TALYSIS BY COMPARISON OF ALCU LATED RE U LTS WITH EX P E RDlE :-rTAL DATA ___________ ______________ __ 13 :-rormal Impa ct __ _________________________ _________ _________________ 14 Impa ct \Y ith Tir e Bo ttom ing __________________________ _________________ 14 PARAj\lETER STUDIE _ ____ _____ __ __________ __ __ __ __ _ ___ __ __ ____ __ __ __ _ __ 14 Re pr esen ta tion of Tir e For ce -D e fl ection Chara cte ri ·t i cs ___ ___________________ 17 Norma l impa cL __ __ _ __ __ __ __ __ ___ __ __ ________ __ _ __ __ ____ __ __ ____ __ 20 Impa ct with t ire bottoming__________________________________________ 20 Effect of Orifi ce Discharge Co e ffi cienL _______ ____________________________ 20 Effect of Air-Compression Pro ce _ ___ _____________________________________ 22 nIPLIFICATION OF EQ ATIO NS OF :\IOTIO :-r _________________________ 24 Eva lu at ion of implifi cat ion s_ ______________ _____________________________ 25 Gene ralized Tr eatment _ _ __ _________ ______ ____ __ _ __ __ ___ __ __ __ ____ ___ __ __ 25 EquaLions and so lu tions______________________ ____ ___________________ 25 Appli cab ili ty of so lu tion s_ _ _ _ __________ ______________________________ 29 S :\DIARY OF RES LTS AND CONCL ,'IO:-r ___________________________ 31 APPENDIX A- N i \1 ERICAL I ~ TEGRATION PROCED RES _____________ 36 Li near Pr oced u re_ _ _ ______________________ ______________________________ 36 Quadratic Pr oce dur e_ _ _ ____ __ ________ __ ___ ____ _ __ __ __ __ ____ __ _ ___ ___ __ __ 38 Rung e-Ku tta Pr ocedu re_ _ _ _____ ______ __ __ ________ __ __ ______ __ ___ __ _____ _ 41 APPENDIX B- OURCE OF EXPER D'IE I TAL DATA ____________________ 42 Eq ui pmen L __ _____________ _____________________________________________ 42 T · t peci ln en _____________ ___________________________________________ 42 In trumentation ______________ _________________________________________ 42 R EFE RE NCES ___ _________________________________________________________ 44 BIBLIOGRAPHY __ ___ ___ _____ __ _ ___ __ __ _____ ___ ___ __ _ __ __ _ ____ _ _____ ___ __ _ 44 III

REPORT 1154

A NALYSIS OF LANDING-GEAR BEHAVIOR 1

By BE N JA~IlN :'I T ILWl'l'ZKY and FR ANC 1S E. COOK SU MM ARY egment appro xim at ion.s to the actual tire jOl'ce-deflection characteristic, which ne gl ect the effects oj tire bottom in g, Thi s report present a th eo retical turly oj the behavior of the although adequate up to the instcLn t of botiom in g,ja ils to indicate conventional typ e oj oleo-pneumatic lcmd in g g ar during the til, pronounced increase in land in g-gear load that result from pl'oc e oj land in g impact. The basic analysis is pre ented in bottom in g of the tire. The u e oj exponential and linear- a generat jorm and treats the motion oj the land in g gear prior egment approximations to the tire characteristic which take to and su b equent to the beg innin g oj shock-strut de fl ec tion. I n into account the i7 crea e el stiff n ss oj the tire which re ults j rom the analy si oj the fir st pha se oj the i mpact the land in g g ea r is bottoming, how eve r, yields go od l' ults.

treated a a sin gl e- degree - oj - jr eedom system in order to deter- 17 M tudy oj the im portance oj the discharge coefficient oj the mine the conditions of motion at the in ·tant of ini tial shock- ·trut orifice indicate that the magnitude of the discharge coe ffi cient d~flection, aj ter which instant the land in g gear ic on 'ider ec l as ha a mW'ked effect on the calculated behavior oj the land in g a sy tem with two de gr ees of jr ee dom. The equations jor the gear; a decrease in the discharge coefficient (or the product of the two- degree -oj-freedom syst em consider such jactor as the discharge coe ffi cient (t'rul the net ol'ijice area) 1'esult in an hydraulic (v loc i ty squar e) resi tance oj the orijic , the jo rces approximat ly proportional increa e in the ma~ ;i mum upper - due to ai r compression and inte mal jriction in the shocle strut

. . '

ma s acc l ration.

the nonltnear jorce-de fl ecti on characteri tics oj the ti r e, the wing Th e study oj the im po l' tan ce of the air-compression proces' L ~ ft , the in cl in ation oj the land in g gear, and the e..t f ec ls of wheel in the ho ck strut in dicate that the ail' springing is oj only 8pin-up dl'ag l oa d s.

minor ignijicance th1'oughout most oj the impact and that The applicability oj the analy is to actual land in g g al" ha variation in the e.tfective polytropic expon lit n between the b een inve t ~ gat ed jor the particular ca'e oj a vertical land in g isothermal va lu e oj 1.0 and the near-adia batic value oj 1.3 ha ve gear in the ab ence oj ctrag loads by comparing calculated ollly CL secondary effect on the calculat r 1 behavior oj the landing results with exper im ntal drop-test data jor im pacts with a nd gear. Even the as umption of constant ail' pressure in the trut wi thout tire bottom in g. The calculated behavior oj the landing equal to the initial pres ure, that is, n=O, yield jairly good gear was jound to be in good agreement with the drop-test data.

1'esults which may be ad quate jor many practical pUl'pO es.

tudies ha ve al 0 been made to determine the effects of varia- I n addition to the more exact treatment, an inve tigation has tion in such parameter as the dynamic jorce-deflection been made to determine the ntent to which the ba sic equations characteri tics oj the ti r e, the orijice di charge coefficient, and the of motion can be simp liji ed and till yi eld acceptable re ults .

polytropic ex po nent jor the air-compre sion pr oces, which Thi s study indicates that, jor mCLnY practical purpo e , the might not be l mown accurately in practical design problems.

air-pressure jol'c in the s ho ck strut can be completely neglected, Th e study of the effect oj va riation in the tire characteristic the tire jorce-deflection relation h ip can be assumed to b linear, indicat es thcLt in the case oj a normal im pact wi thout tire and the lower 01' un pl'ung ma can be talcen equal to zer o.

bottom in g reasonable val'iations in the jOl'ce-deflection chal'act er - General iza t ion oj the equations oj motion jol' this impZijied istics ha ve on ly a relatively 'mall .tfeet on the c7.lculated beha vio r system shows that the b havior oj the s Yl>tem is completely oj the land ing geal'. A1Jpl'01'imating the rather complicated d terminal by the mCLgnitud of one parameter, namely the jorce-deflection characteri.·tic ('j the actual tire by 'implijied dim ensionless ini tial-velocity pammeter. olutions oj the e e; rponential or l inecLr - egment VCL r ia tions CL PP ar to be adequate generalized equat ion are pre ented in terms oj climensionles for practical purpo es. Tire hy teresi' was jound to be varia bl e jor a wide range oj la7Uling-gear and impact parameters relatively unimportant. I n the case oj CL . vue im pad involving which may be useful jol' rapidly est~matin g la7Uling-gear perjormance in preliminary design.

tire bottom in g, the u e oj < imp liji ed J'ponential cuv! linear- I Supersedes NACA TN 2755, "Analy sis of Landin g-Gear Be havior" by B enjamin ]\filwitzk y lind I'r !'lnds E. Cook, 1952.

287 -16 -54- 1 1 REPOR 'r 1154 - NA'rIONAL A DVISORY C OMMITTEE FOR AERONAUT I CS INTRODUCTION S in ce some par amete r s, such a the dyn am ic force- de fl eclion characte ri tic of th e Lire , the orifice eli ch arge Th e hock-ab orbing character i tic of airpl ane landin g coefficient, and th e poly tropic ex pon en tIol' the air-compre ion gears are normally developed lar gely by mean of extensive pro ce s, may no t be accurate ly known in pracLical de ign trial-and-error drop te ting. Th e de ire to reduce the ex- problems, a tudy is made to assess the effect of var iations pense and time required by such methods , a well a to pro- in these param eters on Lhe calculaLed landing-gear behavior.

vide a more rationa l ba is for th e pr edi ct io n of wheel- inertia tudie are also pre en te d to evalu ate the extent to which drag loads and dynami c stres e in fl exible airframes during the dyn am ical y tem can be implified with out gr eat ly im~ landing, emp ha ize the n eed for suitable th eoretical method pairing the va lieli t), of the calc ul ate d re nIts. In add ition to for the ana lysi of l and ing-gear behavior. uch theoretical the in ve Ligations fo r pecific case, generalized so lution for methods sh ould find appli cat ion in the de ign of la ndin g th e behavior of a implifieel y tem are pre ented for a w ide gear a nd co mplete airpl ane Lru cL ures by permiLting range of landing-gear and impa ct par amete l which ma y be (a) the dete rmination of the behavior of a given landin o- - useful in preliminary de ign.

gear co nfig uration under vary ing impa ct co ndition (vel ocity at conLact, weight, wing lift, etc.) SYMBOLS (b ) th e developme nt of a landing-gear co nfigur ation to pn e um atic area obtain a pec ifi ed be h av ior und er given imp act conditions hydraulic area (c) a more ratio nal a ppr oach to th e d ete rmina tion of wh eel ar ea of opening in orifice plat e pin-up and spring-back loads which ta ke into accolmt the interna l cros - ec tional area of sh ock- tru t inner shock-absorbing ch aracte ri st ic of the parti cular landing gear c)'lindcr under consid erat ion A2 external c ro -secti ona l area of sho ck- str ut inn ee (d) improved determinat ion of dy nami c loads infl exi ble cylinder airplan e st ru ct ur es during la ndin g. Till problem ma :v be Ap C 1' Oss- ec tional area of meterino' pin or rod in treated eith er by calc ulating th e re ponse of the ela tic sys- plane of ori fice te m to landing-gea r forcing fun c Lion determined wlder th e A" net orifi ce area a sumpLion th aL L h e airplan e i a rigid body or by th e simul- Cd orifi ce discha rge c oeffi cip nL taneous olution of L h e equation of motion for the l anding cl overall diame Ler of Lie e gear cou pl ed " 'i th the equations repre en ting the ndcl i Liona l Fa pn eumatic for ce in shock tmt degrezs of freedom of Lhe str u cLL u ·e. In man.,- case the former Fh h.Hlmulic for ce i.n hock stru t approach hould be s uffi cientl.v accuraLe, bu L in ome F fri eL ion for ce in hock tru t f in sLa n ces, parti cularly when Lhe landing-gear atLac bm en L Fs Lotal axial shock- tru t fo r ce poin ts experien ce large eli pla ce menL s relative to lh e nodal FJ normal for ce on upp er bearin g (a ttach ed Lo inner point s of L h e fl e :-. :ible .\ ' Lem, L he l aLLe r approach, which take c.dincler) in to account I he interact ion bet ,,- ee n th e defonnations of the F2 norm al for ce on 10\\ -p [" bparing (attac hed to ou tel' tructu re and the landin g gear, ma.,- be requi red in order to (' ., -linder) repre e nL the sy Lem adequaLel ., -.

FN for ce normal to axi of hock st rut , app lied at a S ill ce many aspect of Lhe landing- impact problem are so axle inLima tely co nn ecLed with the mechanics of Lhe landing gear.

F Va yCt"ti cal fOI"CC, app lied al axle L h e subj ect, of landing-gear be h ay ior ha s receiyed ana l ytica l FH a h orizontal fo r ce , applied aL axle tr eatme n t at variou. tim es ( ee bibliography). 1Iany of FRa re ultant forc e, applied aL axle L h e earlier investigations, in order Lo r educe the math ema ti cal Fs for ce pa rallel to axis of sho ck trut , applied to g co mpl exity of th e ana lysi s, were limit ed to co nside ration of tire a t gr ound highly simplified linear ystems whi ch h ave lillIe relation to FN for ce normal lo axis of hock st rut , applied Lo g practical landing gears. ome of the m ore r ecent paper tire at gr ound co n ici er, wi th different degrees of implifL cat ion, more r eal- F V yertical forcl' , app li ed Lo tire at gr ound g

i Lic nonlinear .v Lems. Th e pre ent r eport r eprese nL s an

Fllg h o ri zonta l fo rc e, app li ed to tir e at gro und attempt at a mor e co mplete analysis of the mechanics of FR g res ultant force, app li ecllo lire at gro uncl pracLica ll anding gears and, in addilion, ilw esL igates the im- g grav itational co n tant pOltance of lhe various cl em e nt which m ake up the l and i ng K L lift fa ctor, L I lT" gear, as ,, -e ll a the exte nt to whi ch the sy lem can be reason- L lift for ce ably simplified for the purpose of rapid ana lysis .

lJ axial di tan ce hel l \"('en up per and l o\\'Ct" braring , Th e basic an alysis i presenLed in a general f or m and Lakes for full y extended shock stru t into acco unt uch Jaclors as the h .H l rau lic (yeloc il y sq uare) l2 ax ial el i tance between axle an d lower b ~ arino- re is lance oJ the orifice, the force du e to air co mpr ession and (a tt ac hed to outpr (" ~ ' lind er ) , for fully ex- internal friction in the hock trut, th e nonlinear force- tended sh oc k st r11 t de fl ec tion ch aracteri sl i cs of th e Lire, the wing lift , th e in lina- a,b,m,r 'on tant s C'OlTC pondin o- (0 the .-urioll reglmes o' tion of th e laneling gear and lh e e ff ecls of ,dI ce! spin-up d ra of lh e (ire -cl e fle ction pro ce load s. An eva luation of the app li cabiliL )' of the ana ly i to a' co mbin ecl constant, ad actual landing gear i pre e nl ed for the ca e of a vertica l m' comb inecl C'o n stanl. mriT landing gear in th e ab ence of drag load b. \- comparing cal- polytropiC' expon ent for air-compre ion process n cula t eel r es ults wi tll drop-l est. dota .

in hock stru t A AL YSIS OF LA DI:\fG- GEAR BEHA VlOR MECHANICS OF LA DING GEAR Reyno ld numbrl' R air pr e ure in upp er c hamb er of hock trut DY AMTCS OF SYSTEM Pa h yd raulic pre ure in lower cha mb er of sho ck In view of the fa t t hat landing-gear performance appears PII st ru t to be relatively unafl'ected by the ela. tic deformation of volumetric rate of discharge through orifice Q the au'plane tructure (sec, for exampl e, ref. 1 and 2) par- radius of deflected tire 1'a ticularly s in ce in many cases the main gear are lo cated shock-strut axial troke S fairly clo e to the nodal pOUlts of the fundamental bending wheel inertia torque reaction T mode of the wing, t hat part of the airp lan e which act on 11 time after co nta ct given gear can generally be considered a a rigid rna. s.

time after beginning of ho ck- trut deflection A a result, Ian ling-gear drop Le t are often onc1uctec1 in air volume of hock trut a jig where the ma of t he au'plane is repre ented by a polar moment of inertia for wheel a sembly concent rat ed weight, In particular instance , ho\\'ever, such about axle as in the ca e of airplanes having lar ge concent ra ted mas es vertical velocity di posed in an outboard po ition in the wing, espccially horizon ta l velocity airplanes equipped with bicycle landing gear, con ideration total dropping weight of the interact ion between the deformation of Lhe airp lan e weight of upp er mas above trut tructul'C and the landing gear may be nece ary Lo repre- weight of lower mass below trut ent the system adequatel)' , horizontal Ii placement of lower mas from ince the prc cnt report is concerned primarily with the po i~ion at initial c ontact mechanic of the lan ding gear, it is a umed in th e ana ly i vertical di placement of upp er mass from posi- that Lhe landing gear i a tLached to a rigid ma \\ 'hich ha tion at initial contact freedDIn only in vertical tran lation. The O'ear is a sumed vertica l di placement of lower rna from po i- infinitely rigid in bending. The combinalion of airp lan e tion at initial on ta ct and landing gear con idered therefore con tiLuLes a system dimensionle upper-ma displacement from ha ving two degree of freedom ( ee fig. 1 (a) ) a defined by position at init ial contact the vertical di placement of the upper ma s and the ver tical dimensionle s lower-mass di placement from di placement of Lhe lower or unsprung mas , which i al 0 position at initial contact the tire deAe ction, The tl'ut stroke s i detcrmined by climen ionIc s hock- trut trok e, U\-U2 the difference between Lh e eli pla ce ment Zt and Z2 and , in dimrn ionIc time after contact the ea e of illclinecL gear, by the angle ep b tween tll axi angle between sho k- trut axis and vertical of lhe tl'Ut and Lhe vertical. For inclined gears, comprc ion of the hock strut produce' a horizontal di placcmenL of th e

1 ":;;;11 du

axle 1'2 . From (;o nsideration of the kinemati s of thc system hork - tru t efIe tivene , Ul /I m.az CT max Z z it can be sen that S =Z I- 2 and J'2 =S in ep = (Zt- 2) tan ep.

U1mu co ep

r

J( ut dUl

In lhe analy i , cxte rnal lift force , cOl'l'es[)oncl in g Lo thc lanclin g-o 'eal' effect iv enes =--::.0-;:- __ _ Tl lo b J u/' maxU1max aerod .\ 'l1amic lifL , arc assumed Lo act on lhr y Lem Llu'ough- out the impact. In addition to the vcrtical forces , arbitrary time ulterval in numeri al integration procedure E drag lo ads are con idere'd to act bel wcen the tiJ:e and the coefficient of fri ction between tire and runwa~ T ground.

coefficient of friction for upper beal'inO' (att ached The ,, ' stem t reaLed in the analy i may therefore bc con- to inner c~ ' lind cr) idered to rcprc cnt either a landing-gear drop tc l in a jig c oeffi ci ent of fl icLion for lower bearing (a tLac hed J.L 2 where wing lift and drag loads are imulaLcel, or lhe landing to outer cylindel') impact of a rigid airplane if rOlational motion arc neglected.

mas clensi t ,\ ' of hydraulic fluid p Ro tat ional frecdom of lhe' airplane, whf'rc ignificant, may angular acceleration of wheel ex D e Laken into accounL approxinlalely by usc of an appro- A.X(' priate effcct iv e mass in th analys i.

vertical axis, positive downward z Figur e 1 (b) show a chematic represenlation of a typ ical horizontal axi , po itive rearward X oleo-pneumatic hock trlll u ed in American practice , Th e •'ub cr ipL

lower chamber of the trut contains h. nlraulic n u iel and the

at in tant of ini t ial contact

o

upp er c hamb er contain ail' und er pre sur e. The outer c,\'l- at in tant of initial hock- trut defleclion incler of Lhe trut , which i atLached to lhe uppcr mass, at in tant of \\ 'h ee l pm-up su conta in a perforaled lub e which upport a plale with a maximum yalue max sma ll orifice , th rou gh whi ch th e hydraulic nuid i forced to T otation: flow at high ve lo city a a re ult of the tdc oping of th e

I( )I ab olu te value of ( )

strut. Th e h yd rau li c pre ure drop across lhe orifice t hu ( )* e ti maL ed va lue of ( ) produceelre ists the clo m e of the strut, and thc tllrbulenca Th e use of dots over ~' mbols uld icatt·s differentiation wi tIt created provide a powerful means of ab orbing and dis- 1'e pect to time t 01' 7, ip ating a large parL of thc impa ct enel'g~ ' , In some struls Prime marks indicalc difl'c rcntia hon wi th respcct to lhe orifl ce area i co n tanl ; wherea ,in olher ca e a metering dimen ionless timc fl .

4 REPORT 11 :j4-NA ,] ' IONAL ADVI ORY COMMITTEE FOR AERONAUTICS I A2

U- .- -- --Pa

ri-

I

--==

- - - - - - ~ - ~I=:: - - -- - - - - - r- - - - Fluid - - - - - - - - - - - - - - - - -

n

Z'-Z2 5:-- COS <p - - Aa - -

~ 7 c-

Ap Ph --- ..

AI- r-- -----...

(

'\

I I (b) (0) (a) Sy . tem with L II"o degre es of fre eelom. (b) Sche matic re pre se ntation of Hhoek st rul.

Fl r.r R" l. - Dynamical s ~ 'ste m considered in analy. is.

pin or rod is usrd 10 ('ontrol the' sizt' of the orific(' and gOYt'rIl and the tir (' muluall.l - influence th e behayior of all(' anolher t il l' performance' of thr s trut. and mu (1)(' can idNed imultanroll sly in analyzing; the system.

TIt(, comprrs s ion of lllr slr ul product' an ilwl"('ast' in lht' FOR CES IN S H OC K ST R T nir pr(' SS UI"(' "'hieh al a n. 's i sts til(' do lIfr of l h (' trut. 1n From can iclrralion of the pressur s fl.cting in th e shock figun' I (b) Ph f rp r rsrnts lbe oil prrssu rr in lh e lo\\'e r chamlw r sl ru I i I can be' fraclil)T rr11. hom figure] (b ) Lhat the lolal and Pa ]"('presrut thr air pn' urr in lur upper ch ambrr.

axial £01" er clllr 10 h.nlraulie 1'0 i tance , air compn' ion, and In addition to lhr hyd raulic rr si tanee and air-prr s ur e bt'uring friction can br e xpfes rd by fOI ("rs, i nlrrna llwarin g friction aL 0 cont riblllr s f01"('r s ,, - hi eb (",111 appr r ciabl.Y aO'eeL lll(' brluwior of lhe lrut.

F s= P II(L l, - A p) + P a( A 2 - A ,)+ PaA p+ F f Till' for('rs c]"ratrd " 'it hin til(' s trut impart an accrlrralion wil e 1"(' to tllr upp('/" mass and aLa producr an acceleration of thl' .A, inl e rnal (Toss-srclional area of innrr cylindcl' 101\"('r mns and a dd l rction of tIl(' tirr. Figllrr 1 (e) sh ows ... h exlerna l era - ectiona l area of innrr c.dindcr tll(' balancl' of foret'S and rractions for the wh eel, thr imH'r ~ 1 7! cross-sc ctional area of mctrring pin or rod in plane cy lin(il-r, and the oull'r n -lind('r. It is drar that til(' trut of o ri fic(' ANALYSIS OF LA T DING - GEAR BEHAVIOR L W, ..

gZI Moment .-- ' / F, - - --1-- Ho ., / / / / / /

/

/ /

Fs /

/ / / /

/

Forces on ou ter cylinder / / / / / / Forces on inner cylinder ,------- -x Z (cl Forces on wheel (c) Balance of for ce and reaction s for landing-g e ar component .

FI GU RE I. -Co ncluded.

~87R46-54--2 REPORT 11 54- N A TIO l A L ADVISORY C OMMIT T EE FOR AERO I A U TI CS Thi s e xpr e ion c an al 0 be Wl'i Uen a P"- Pa and the ar ea AI! whi ch i subj ected t o t h e hy draulic pr e s ure , as pr eyiou sly noted . Thu Fs= (P h- P a) (A A v) + PaA 2+ F, 1 -

= ( P . ~- P a) A h+ p (L A a+ F , (2)

(1 ) Equation (2) c an be mad e appli c abl e to both th e c ompr es- whc1' e P h- Pa pre ur e drop ac ro th e orifi ce

ion and elongation trok es by introdu cing th e fa ctor I! I

Ah hydrauli c a1'ra (A A v for th e tru t s hown in fi g. 1) 1- to incl ic at e th e sign of th e hydrauli c r e is tan ce; t hu Aa pneumati c ar ea (A 2 for the tl'lI t s hown in fi g. 1) In thi s re port th e te rm s (P h- Pa )A and PaA a a re referred h ( 2a ) to a h ydrauli c for ce Fh and pn e umati c for ce F a, r espec- tively. F or th e tru t sh :) wn in fi g ure 1, th e h y dr auli c The n et orifice ar ea A " may be e ith er a c on t ant or , when a and pn e umati c ar ea ar c relat ed lo the trut dim en ion as met e rin g pin is u ed, c an vary with s tru t troke ; tha t is, pr eviou ly not ed . In the ea e of trllt s ha ving different A n= A o- A p= A ,,(s), wh ere A o is lhe ar ea of th e ope ning in internal configurations, th e h ydrauli c and pn e umati c ar eas th e o),ifi cc plate and A p is the ar ea of th e met erin g pin in th e mfty be ar ome wha t differen t relation to th e dim ensions plane of th e orifice. A t th e pr esen t tim e th ere a ppe ars to b e of th e tru t. In such eft es, howev er , c on ide l' a lion of th e orne te ndcn ey to elimina te t he m. etering pin and u e a con- pr es ur cs ac ting 0 ~1 th e variou s co mpon e nt of th e trut tant orifi ce a rea, parLicularly for la rge airplane , in which s hould p e]'mi t th esc ar c as to be readily defin cd.

case A n= A o. In the gen e ral ca e, lhe orifi ce di ch arge Hydraulic fo rce.- Tb e hydrauli c r e i tan ce in th e s hock coefficient mi O' hl be e xp ected to "aI',\" omrwhat during an s lrut 1' C ult s from th e pr ess ur e difl'er ence associa ted with t be impact becau e of chan ges in tb e ize and confi O' Ul' a tion of flow thr ough the orifice. In a landin g ge ar th e orifi ce area th e net or ifi ce ar ea , c han ges in th c e xit c ondition on the i usually m all enou gh in relation t o th e diam eter of th e clown ir('am fac (' of th e orifi ce du e to v ftrialion in th e amoun t tru t so t ha t th e jet velo cit ie and R ey nold number ar e of h.nlraulic fluid abov e the orifi ce pl a te , c hang e in th e e ntr y s uffi cie ntl y larg e tha t tbe fl9wi s fully turbul ent . As a c ondition due to variation s in. the len g th of the flow c hamb er r es ult th e dampin g for ce var ies as th e s quar e of thc tele- ups tr e am of th e orifi ce, and becau sc of ,T a riation in th e scoping veloc it y rath er th a n lin early with th e Veloci ty .

Re .nlolcls number of thc flo\\', 0 th a t , in general, Ca = Cd( s, R ).

ince th e hy dr auli c 1' e i lan ce i th e maj or co mp o nen t of Although th e indiyidu al e O' ec t of th e e fac tor s on th e dis- tb e tot al ho c k- s tru t for ce, vi scous cl a mpin g c anno t be c har ge coe ffi cie nt for orifi ce in s ho ck s trut hav e not b een rca onabl y a timed , eve n thou gh su ch an a umption c yaluated , th ere is ome cxperimental eviden ce to indi c at e " 'ould gr eat ly simplify th e anal ys is.

appre c iab le ya riation s of th e eli c har ge coe ffi cient durin g Th e hy dr auli c ]' e is lan ce c an be rc'tdily derive d by makin g impa ct , parti cularly in th e casc of trut with m e terin O' pin .

usc of th e well-known e quation for th e di c har ge throu gh It mig ht be e xp ect ed that sur h yariation s \\'oull be con- an orificc, namel y, s iderably , malIC!' for ge ar s haYing fl con tant orifi ce area.

In o]'dr r to evaluate the precis ion with which the or if ice di sc har gc coe ffi cic nt ha s to be kno\\,ll, a bri ef tucl~ ~ is wh ere pr e e nt ed in a sub eq ue nt ection wh ie ll ho \\' s th e e ff ect of Q Yo lum et]'ic rat eo fdi ch argc th e di . cha rge coeffi cie nt on the ca lculat ed beh av'o r of a Cd coe ffi cie nt of di sc har ge landin g gca r " ' ith a cons tfln.t orifi ce flr ea, under th e ass ump- An n et o rifi cc area tion that the eli c har ge coe ffi cie nt is c on s tanl durin g th e p" ll y dra1.l1ic pr c s ure in lo\\'er c hamb er impa ct.

Pa a il' pr e ur f' in upp cr c hamh er Th e forcgo ing di scus ion h as been co n c('rncd prim a rily ",ith p m ass cl e l1 s it~ ~ of hydrauli c fluid th e co mpr ess ion s trok e of the shock t ru t. :-10s1 s tru t Fr om co n i cic rati ons of co n t inuit~ " th c "olum et ric ]' a te of in c orpor ate ome form of pr ess ur e-ope ral l, d rebound check di schargc ca ll al 0 he c xp1' e scd a th c pro d uct of th e telc- v alvc , om c tim cs ca lled a s nubber v a ly (' , whi ch c om es into co ping Yelo c ity S i tnd th c hy drauli c ar ca A " ac tion a ft ..' I' th e m a ximum trok e ha b ee n a tt a ined and closes o ff th e m a in o rifi ce as soon as th e tl'ut begins to el ongate, so Q = A " ,~ that th e Ouid i fo ], ce d to rc turn to th e ]0 \\'l ' 1' ch a mb er t hr ough Equ a ti ng th c pr e ('( ' ciill g c:q)r eSS i Olls for t be el i (' 11 ar O'e pe r- small pa ssage . Th e a ct ion of th e s nuhb er \~ a l ve in tr od uce mit s wrilin g tll(' fo ll owing impl e c quation fo ), thc p)'e ur e g l' ea tl.\ ~ in C/ 'eased hycl/'l1uli c I' ('s is tan ec to dissip ate th e en er gy dr op acr oss ll w orifi cc tO l' ed in the trut in the form of ail' pr e's ul'e a nd to pr eyen t (' x('essi" e rebolllld. Th e pr o du ct CdA n to be used in e qu a tion ( 2a ) durin g th e elon gl1 t i on st roke i g m c rall . \~ un ce r ta in. Th l' x act a rea A n durin g cl onga ti on is ll s u a ll~ 7 some wh at cli[ft eul t Th e h~ ~ cl ra uli c r ('s islivl CP F" du c t o th e t el escoping of tb (' to d efine fr om th e gc oll1 cl r.\ ' of th e tru t in ce in m any cases s trut i glWll b~ ~ til c pr odu ct of th e clifT ercn t ial pr ess ur c th c numb cr of co nn c ctin g pa s ages v ari cs \\ ~ i t h s tr oke an d the AJ.'l"ALY IS OF L NDI G- GEAR BEHAVIOR le akag e area around the pi st on ma y be of the same ord er of case of dry fri ct ion , the r es i tance dep end s on th e ph y ical c hal'a cte ri tic of th e s lidin g s urface , i e ent ially propor- magnitud e as the area of th e r et urn pa ssage. Fur the rmor e, the magnitud e of the orifice di c har ge coe ffi cie nt , and even tional to th e normal force, and i a pproxim ate ly ind ependen t possibly the natme of th e re i sta nc e, arc qu est ionabl e du e to of the urfa ce area. Th e coe ffi cien t of friction Jl , defined a th e ratio of the fri ct ional r esi stance to th e normal force, is the foaming tate of th e r et urn i ng fluid. Fortunat ely, the o-enora11y s om e wh at greater uncle I' condi tions of r est (s tatic primar y interest is in the compression proc ess rath er than fri ct ion) th an und er con di Lion of lidin g ( kin et ic friction).

the elongation pro cess in ee Lh e maximum l oad alwa ys occurs Although th e coe ffi cien t of kin et ic friction genera lly de- before the maximum st rut stro ke is reached .

cr ease lightly wi th in cr ea 'ing ve loci ty, it i u ually co n- Pneumati c force.- Th e air -pr ess ur e force in the upper c hamb er is det e rmin ed by the ini tial st rut inflation pr es- id er ed, in fu· t approximation , to be ind e pendent of velocity .

s ur e, th e area subj ected to th air pre ss ur e ( pn e umati c ar e a), If , on the other h a nd , the uda ces arc co mpl ete ly sep a rat ed and th e in sta ntan eou e ompres ion ratio in accordance by a fluid film of lubrican t, p erf ect lubri cation i sa id to e xi st.

wi th th e polytropic l aw for eom pr e ion of ga es, nam ely Und er the e co nd it ions th e resistanc e to r e la tive motion P av n= Constant , or depend primarily on the ma gni t ud e of Lhe relative veloci ty, th e phy si ca l c hara cte ri st ic of th e lubri ant , th e area , and th e film thickne s, and is essent ially ind epe nd ent of the normal force and th e charac teri st ics of the slidin g urIace.

wher e P e rf ect lubri cat ion i rar ely found in practice but i mo t Pa air pre ssure in upper c hamb er of shock st ru t likely und er co ndi tion of hi gh voloci ty and relatively sma ll P aO air pr essure in upp er c hamb er for fully e xt e nd ed st ru t normal pr ess ur e, where the s hap e of th e lidin g sur faces i v ail' volum e of sho ck t ru t co ndu c iv e to th e gene ration of fluid pre ur e by hy dro- vo air v ol lUn e for fully exte nd ed str ut dynami c act ion . In mo t pra ctica l app lication in volv in g in ce th e in tantan eous air volume is eq ual to the differen ce lubri cat ion, a sLate of imp e rf ect lubri cat ion e: ,.i st and th e between the ini tial a ir volume and the produ ct of the s trok e l'C i sta nce ph e nom e non i in te rm e diat o b tw een that of dry

and pneumatic area A P a= P ao ( vo~~a ) n. Th e force due

a, fri ct ion and perfe ct lubri cat ion .

In the case ot landing - o-ear ho ck st rut s, L h e condi t ion s to th e air pr es ur e is s impl y the produ ct of the pr e' urr and und er which interna l fri ction is of co n cern us ually in vo lv e the pn eu ma tic arra: r e lativel y hi gh norm al pr e ures a nd r el at ively sma ll liding vcloci Lies. ::'Iore over , the usual types of h yd rauli c .fl uid (3) u ed in s ho ck truLs ha ve raLher poor lubri cating prop ertie , a nd the sh a pe of th e b ea rin g urfaces i ge n era lly not con- In th e pr ece ding e quation s, the e ff ect ive polytropic du cive to th e o-ene ration of h yd rodynami c pre m es . It e xpon en t n depe nd s on th e ra te of c ompr es ion and th e ra te woull th erefore a pp ea l' tha t th e lubri cation of h ock trut of h eat tran fer from the air to th e s urrounding e nvironm en t.

b earing is, at best , imp erfect; in fact , the c ondition appear Low rat e of c ompre ion would be expected to r es ul t in to approach closely those for dr y friction. In th e pr e e nt va lu e of n a pproa c hing the i sot h e rm al value of l.0; wh er ea analysi , th erefor e, it is as um ed, in nr t app roxim at ion, Lhat high er value of n, limit ed by th e adiabatic va lu e of 1.4, th e in te rnal friction b etween L h e be arin g a nd the cy lind er would b e ex pected for hi gh er rat es of eo mpre s ion . Th e walls fo llow laws similar to t ho se for dry friction ; t ha t i , actua l thermodynamic proc e is co mpli cated by the violent Lil e fri ct ion force is given by I, he prod LI ce of the normal force mixing of th e hi g hl y t urbul en t efflux of hydraulic fluid an ancl a su i tab ly e ho en coe ffi cien t of fri ct ion.

th e air in the upp er c hamb er durin g impa ct. On the one Wi th these a umption s the int e rnal fri ct ion forces pro- hand , th e di s ipation of ener gy in th e production of turbu- du ce d in the st ru t depe nd on the magni t ud e of the for ce on I nce ge ner ates h eat; on the oth er h an d, h eat i ab orbed by th e a.x le, the in cl in atio n of the gear , the pa cing of L he bear- th e aer at ion a nd vapo riz at ion of th e f1uid. Th e efI'ec l of lhis ing , and the c oeffi ci ent of fri tion bet\\T oen the bearin g and mi.:·,ing phenom e non on th e polytropi c ex pon ent or on th e the cy lind er wall s. Fi gure 1 (c) sch ematica lly illu trate the quivalent air vo lum e is not clear. A limi ted amount of balan ce of force acting on the va rious co mpon e nt s of Lhe exp e rim en ta l data obtained in drop tests (r efs. 3 and 4), l and ing gear. Th e total axial f"i ct i on in the sho ck st ru t is how ever, indicate s t ha L th e effective polytropic ex ponent the urn of th e fri ct ion for ce co n t ribu ted by eac h of the may be in th e n eig hborhood of 1. 1 for pra ct ical ca es . A bea rings : brief tudy of th e impolt ance of the air-com pr essi on process and th e e ffe cL s which d iff er ent va lu es of n may h ave on th e ca lcul ated beh av ior of the la ndin g gear i pre se n ted in a where F axial fri ct i on force ubseq uent ect ion.

J I nt ern al friction force.- In the literatul'C on machine de- /-II coe ffi cie nt of fricLion for upper bearing (attac h ed to ign th e wide rang e of conditions u nd er whi ch frictional inner cylinder) rest tance an occur b et ween lidin g urface is generally FJ normal fo rce on upper be aring (attac h ed to inner cIa s ifi ed in L hr ee major catego ri e , nam ely, fri ct ion b eL ween cy linder ) dry s mfa ces, fri c tion betwe en imp erfecLly lubl'i ated urface , /-1 2 coe ffi ci ent of fri ct ion for low er bearing (attached to ouLer cylinder) and fri ct ion b ct \ een perf ect ly lubri cate d urf ace. In Lhe REPOR'l' 115-! - NA'l'roNAL ADV ISORY COMMITTEE FOR AEROI A TICS masses by th e kin e mati c r e lationship X2=(ZI- z2)tan <p , as F2 normaJ force on lower bearing (attac hed to outer previousl y no te d . Doubl e differe ntiation of this r e lation- c ~T lin de l' ) ship gives X2= (Zl- 22) tan <po Substitution of this e:\l)ress ion fa ct or to indi c at e sign of fri ct ion force into e quation (4b) g ives

m

Durin g the interyal prior to the beginning of shocl\:-st1'ut ' F +W ... W' motion the fri ct ion fo],ce s dep e nd on the coefficients of s tati c F F IN = IV SIn <p- ' H cos <p - Z I SIn <p- 2 SIn <p (4c) a g g g fri ct ion ; after thl' s trut begi ns to telescopc th e coe ffi cie nt s of kineLi c fri c tion a ppl y .

In e quation (4c) th e quantity 21 sin <p re pr ese nt s th e ac- From con sid erat ion of the balan c of mom ents it can be cel erat ion of th e lower mass normal to the str ut axis when the seen from fi g ure 1 (c) that gear is rigid in be ndin g, In the case of a gear fl exible in bendin g, the normal accele ration of tbe lower ma ss is not

l 2-8 )

Fl=FN --

(

a ll+ 8 completely d ete rmin ed by th e vertical acceleration of the a nd upp er mass and the an gle of inclination of the gear ', If it shou ld be n ecess ar y to ta k e in to account, in parti c ular ca8es, the effects of gear fl e xibility on the relation s hip be tween the so that normal force on th e axle and the g round rea ct ions, tbe q uan- (4) tity 21 sin <p in equation (4c) may b e re pla ced by estimated "alues of the actual normal acceleration of the lo wer ma ss as where ( 4a ) d ete rmin ed from consid erat ion of the b e ndin g res pons e of the gear to th e appli ed forces normal to the gea r axis. Th e effects n nd of ge ar fleAibilit)T ar e not co nsid ered in mor e d et ail in the FN forcl' norma l Lo s trut app lil'll at axle a pr ese nt analysis.

P nrlica l foree applied at axle Va FI/ horizont al force applied at axle a FORCES ON T i llE <p ang le belween st ru t ax is and Yl'l'tiral Fi g ure 2 (a) s how s dynami c force-defl ect ion characteris- /1 ax ial di sta n cl' bel "-e('n upper and 10w('I' bearin gs, for tics for a 27-inch smooth-contour ( t) rpe 1) tire inflat ed to 32 fully extended s tru t pounds per square in ch. The se c hara c tc ri st ics were de ter- L 2 ax ial di st an ce hel ween axIl' ancllo\\-e l' b en ring (a t tncllecl min ed from time-history m eas urem e nt s of vertical g round Lo outer c,\"lind('l') , for full.,- ('xtel1ded s trut force and tire d e fl ect ion in l anding-gear drop tests with a The quantiLi('s F , F , and FH a are [or cl's a pplied at the va N a nonrotatin g wh eel at severa l vertica l ve locities. As can be axle and cliffeI' from th e g round rl'a ct ions h.r am ount s equal seen, the tire co mpr esses along one c urv e and unload al ong Lo the inel'lin fo rces co rres pondin g to th e re s pect i,'e accelera- another, the h yste resis loop indi cat in g appreciable en erg)' t ion com pOlwnt of the 10\l-er m.ass. Since the inne r cy lin der diss ipation in the ti re . Th ere is som e que tion as to whether genc rall." r cp r esents only a rel at ive l." s mall fr adio n of the th e a moun t of hyst eresis would be a gr eat if the tire \\'e 1' e 10 wc1' ma s, th e lower ma ss ma) ' re asona bl ." be assumed to be rotatin g, as in a landin g ,,- ith forward sp eed . Th e force- conce ntrat ed at th e axle. With this ass umption , the rela- defl ect ion c urv e for a ve loci ty of 11.63 f eet pe l' sec ond is for tionships between the f01' ces at th e axle a nd the force at · the a severe impa ct in which tire bo tto min g occurs and s how s gro und are gi ven b.v the s harp in crease in force with d e fl ectio n. s ub sequent to TT' . .. ) bottoming.

IH= I - - -X'2

F

ll

(F

" g g In fi g ure 2 (b ) the sa m e force-de fl ect ion c hara cterist ics arc s hown plott ed on l ogarithmic coo rdinat es. A ca n be The normal force at the axk can therefore be ex pr essed in seen , the force ex hibit s an ex pon enLial var iation with de fl ec- te rm s of the g round reactions nnd the com ponen t accelera- t ion , A systematized r epr esen ta tion of the force -d e fl ect ion tions 01' th e low er ma s h."

r el ationship ca n therefol'e be obtained by mean s of s impl e eq uation s havin g the form (5) wher e Fl ' g vertica l force a pplied to tire at grou nd whe1'e FI/ hori zonta l force ap plicd to t ire at g round J( F v vert.ical force, applied to t ire at g round

n '2 effectiye mass 1)('10\1' s ho ck st rl! t, ass\l mecl conce ntratr ,cl

g

"2 vertical displacement of 10\\'(' 1' ma ss from position at

at llxh' g initial co n tact (ra dial deO ecL ion of tire) i ; 2 h or izo nl al aC'cdl' l'ali on of axIl' d overall cli ameLer of tire Z2 vertical accelerat ion of axle m, l' co n sta nL s co rrespond in g to the various regimes of the In the case of all in cl ined landin g gea r ha v in g infinite sl iO'- Lire-deflection process ness in bending, th e h orizon tal displ aceme nt of the 10"- e1' mass m' co mbin ed co n sta nt , mdT J: 2 is rela ted to the ve rt i ca l cii s pla cem ellts of the upper and l ower AI ALYSIS OF LANDING-GEAR BEHAVIOR 9

I

13xI03 12.6 II 10.8

'l

: ~

6 - Linear -segment approximation : :e _ 5 ~- Linear-segment approximation : VVo ~"" l( c o 7.2 OJ . ~- ~ c .2 OJ u ~ .E ~ 5.4 . ~ ~ 3.6 1.8 (0) (b) .2 .3 o .5 .4 .5 .6 Tire deflection, z2, ft (a) Uniform coordinatc s. (b) Logarithmic coordinat c~ .

FIGUHE 2.-Dynamic force-d eflection characteri tics of tire.

hydraulic, pneumatic, and friction force , a gIven by It may be nol ed from figure 2 that e sentia ll)' the same equation (1). lnce L h ese force act along the axi of the force-deflection em've hold during compre sion for all impact lrut , which may be inclined Lo the verLical by an angle cp, velocities, up to the occurrence of tire bottoming, and that in figul'e 2 (b ) the slopes of the CLllTe in each of the several the vertical component of the axial hock- trut force i given ]'('gimes of tbe tire -d eflectio n process arc also independent of b~' Fs cos cpo The vert ical component of the force normal veloc it y, except in the compression regime following tir(' to the shock st ru t j given by FN in cpo 'rhesc fOl'ces act a bottoming. in conjunct ion wilh the lifL force and weight Lo produce an Figure 2 al 0 show imple approximation to tile tire acceleration of the upper mass. The equation of moLion cha1'acte ]'i tic which W01'O obtaine 1 b., ' fitting straig ht-lin e For the upper mil IS egme nt s Oong-clash ed lin e) to the actual force-ci eflection curves in figure 2 (a) for impacts at . 6 and 11.63 feet pel' econd. These approximation , hel' inaft e1' referred to as linear-segment approximation, arc included in a stud ~T , The Yerti al componeni of the axial and normal hock- presented in a ub equent section, to ('valuate the degree of st rut force al 0 act, in conjunction with the ,,' eight of the accuracy required for adequate repre entation of th e tire low el' mas, Lo produce a deformation of the lil'e and an characteristics. The variou repre e ntation s of the tiTe acceleration of the lower ma . The equat ion of motion characte ri tic con id er d and the pert in ent co n Lant for for the lower rna IS ('ach regime of t ire deflection arc s ho\\'11 in fiO'Ll1'e 3.

EQUATIONS OF MOTION (7) Th e inLernal axial force Fs prod uced by the ho ck trut where Lhe ve rtical ground reaction Fv i expr es ed a a was hown in a pl'eviou ection to be equal to the urn of the g 10 REPORT 11 5 4- NATIOKA L ADVISORY C OMMITTEE FOR AERONAUTICS 13XI03 12 ~ Exponential variation : Exponential variation: FV 0 m' (~2)' q II II r 10 Regime m' Regime m' r ~:. ..

3 3 CD 78.6 x 10 1 .34 78.6 X 10 1.34 CD . ----0) .89 813.6 X 10 9 ~ (g) 34.0 X 10 2.80 ® 8 <3l 445.0 x 10 9. 70 30.5 x 10 9.70 <3l @ 157. 1 X 10 1.73 @ 126.1 x 10 1.73

7 @ 65 .5 x 10 ®/ 3

1.34 652 X 10 1. 34 ® -- 0) 6 Linear - segm ent z approximation: FV 0 0' ( }) + b q .0 3 2 0'0479 x 10 bo -10.8 x 10 5 Regime CD

/

C2l ~- 4 j c c o ;) o ;) ._-- @ -- --@ j --- Exoct exponential - - - - - Exponential, regime CD extended (no hysteresis) - -- Linear - segment approximation (no hystereSIS) ., .... .. .... .. .. Linear-segment approximotlon , reg ime CD extended (0) (b) 1 ___ I I j I I I .2 .3 .4 .5 3 .4 .6 .1 2 .5 Tire deflection, z2' ft (b) Impa ct wi t h t ire bottoming, VVo= l J .63 f eet per second.

(a) ImparL \I ' ithoul l ire bottoming , \ 'v = . 6 feet per secon d .

o FIGURE 3.-Ti re cha r acte ri st ics considered in so lution s (logarithmic coordinates).

function of th e tire de f! ction Z z. Th e r ela ti onship b cL \\' een th e impact. Th e equation of molion for Lhe one-degr ee -of- F v and Z2 h as b ee n di scussed in th e pr evious section on tire freedom .,-stem arc de riv ed in order to p erm it determination g e haracl e l'i tics. of th e in itial condition s required fo], the ana ly is of th e l anding - o-ear b ehavior ubscquenL to Lhe b eginnin o- of shoc k- B y comb inin g equat ion (6) and (7), the ve rti cal g round st ru t defleclion.

force can be wl'itten in terms of th e inertia r eac ti ons of th e ince 21= 22= 2 dming thi fir st phase of the impact , upp er a nd lo \\' er ma ses, th e li ft force , and the tot al weig hl.

equation ( ) may be writt en as Th e over all d ynami c equ ilibrium is given b~ - ( ) (9) Mono PRIOR TO S HO C K- S TR l' DEFL EC TION Co nv e ntion al oleo-pneumaLic shock trut s arc inflat ed to som e finite pre sme in Ut e fully exte nd ed position. Th u the strut docs no t begin Lo de fl ecL in an impact u ntil uffic ient For Lhe general case of an expone nLi al relation hip be ween fo rce j developed to overcome th e initial pl'el oad in g imposed vertical o- ronnd forc a nd tire deflection, e quation (5) applie by th e ail' pre ure and intel'llal fri ct ion. ince the stru L is and th e equ at ion of motion b ecome e ff ectively rigid in co mpl'e s ion , as well a in b ending, prior Lo this in tant, L h e ystem m a ~T b e con idered to h ave ( 10 ) on ly one d eg ree of freedom during th e initial stage of ALYSIS OF LAl'l' DI G- GEAR BEHAVIOR Th e shock str u t beain to t el escope ,,-hen the urn of the in er tia, weigh t, a nd lif t forces b ecome equal to th vertical co mpon e nt of the axial and normal shock- trut for ce. At thi in tant t s= Fa o+ Flr and eq ua tion ( 6) can be written a r, F (11) where Fao initial air-pre ure pr eload forc e, PaoA a F Ir tatic fri ct ion at in tant tT At the in tant i T, = 0 and eq uation (4) become (11a ) wh ere and Ji.1 and Ji. 2 ar e c oeffi cie nt of tatie fri ct ion.

ince th e trut is a um ed es entially ri gid in compressi on (a nd also rigid in bend ing), there is no kin emat ic disp la ce- me nt of the lower ma s III the hori zontal dir ection up to the beginnina of shock- st ru t de fl ection, so th at 7 2= 0 and equ at ion (4 b) b ec ome (11 b) In co rporating e qua tion ( l1a ), ( lIb ), and (9) into e quation ( 11 ) gives (12) In e quation (12) wher ever the ± sign appears, the plus ign

where the general ex pr e sion for the va riable z is obtained

app ly when FN > 0 and the minu ign app ly when FN <0.

from equation (14 ) without the sub Cl 'jpts T . P er forming ~ ~ the indicated int eg ration give From e quation (10) th e vertical displaceme nt of th e y tern at the in ta n t iT is given in te rm s of the corre ponding acceleration by I [ tV .. ] } l I T Zr= m TV (I - KL )- g ZT (13) { where

o g

Int egrat ing e quation ( 10 ) and noting that zo= O provid es the re l at ion hip b et ween th e vert ical velocity and the

' / Z0 2 ~ f + [(1- KL)g]

ve rti cal di pla ce m e nt of the ystem at the b eg inning of s ho ck- st ru t deflection Th e co mpu tat ion of tT can be great ly implified by use of the followina approximation which as ume a linear relaLion-

. - /. 2 2g [ m T + 1+ " fT T ( T7 1 ) ]

( 14 ) zT - -y z o -1 V r + l ZT .fi. L- Zr hip between ve lo c it y and time: In view of the fa ct t ha t the tire force-deflection curve i (15a) es e ntiany line ar for small de fl ect ions, it may be rea onably a umed that 1' = 1 for the purpo e of determinin g the time after c onta ct at which the st ru t b eg ins to telescope. With Equation (15a) hould be a fairly good approxim at ion in thi a s umption ir can be determined from the r elationship view of the relativ ly shor t time in terva l between initial con- tact and the b eg inning of sho ck- st ru t motion.

Equation s ( 12 ), (13), an d ( 14 ) pmmit the determination of the vertical accel ration , di placement, and velocity, re- spectively, of the system (upper and lower ma f' ) at the REPORT 115-!-N AT I O~ AL ADYISORY COMMITTEE FOR AERONAUTICS beginn ing of sh ock-slruL deflection. Equation (15) or (15a) permits calcul ation of the time interva l between inilia l contact and lh i in tanto The e equation provide the initia l condition required for the analy is of the behavior of the landing gear a a y tem with two degree of freedom after Lhe h ock strut begin Lo deflecL.

If drag load are considered, l he solution of equation (12) require know ledge of the horizontal ground force FH at gT Lhe instant tT. Si nce Lhe p re ent analysis doe noL explici lly treat th e deLermi nation of drag l oads, va l ue of F H h ave 8 T Lo be estimated, either from oLher ana l ytica l consideraLion , expe r imental daLa, or on Lhe basi of experience .

B SEClUE T TO BEG I N I NG OF S H OCK-STUUT DEF L ECTION MO TJ O Once lhe sum of the inertia, weight , and lift force heeome sufficienLly large to overcome the prcloading force in the hock strut clur to initial ail' pre sme and internal friclion, lhe shock tl'ut can deflect and the ystem become one haying Lwo degrees of freedom. Incorporating the expl'e ions for the hydraulic , pneumatic, and friction forces (eqs. ( 2a ) , (:3 ), and (4» into equation (6) permit the equation of motion fot" the upper ma to be wrillen as follows: ",hl'I"(' Z I- Z, 8= -- - CO cp cos cp z

and , s ineC' Fv = F /: , C'quation ( 4c ) becomes

2) V where Fv (Z 2) is determined from the force-deflection Chtll'aclel"i tics of lhe tire'. 1"i'01' the usual type of pneumalic tire, g • FI' (Z2)= JI 1ZzT, as prenoLlsly noted.

g Simihll"ly, the e quation of molion for the' lo,,' er mas s follo,," from equalion (7) : ( 17 ) The o\"erall dynamic e'quilibl'ium equation i lill , of cour e, as gi, · en hy equation ( l

rr .. +1 1 "2 :':' + I F ( T.- 1) +1 ' ( )- 0

2 1 "2 II 1\. !.- - ' I' 22- g g g SOLUTIO OF EQUATIO S OF MOTION ~ \.ny t,, 'o o[ the preceding equations (eq . ( lG ), ( 17 ) , and » arl' suffici e nt to de scribe the behayioJ' of thl' landin g In til(' general ca () th e analy si o[ a landing gl'ar inyolycs gear ub s equent to the beginning o[ shock- trut motion.

the solution o[ the equation o[ motion o-iven in the eetion The e e quations may be used to calculate the belul"iol" of 11 l'ntitled ":- [otion :-lu bseq uent to the Beo-inning of Shock- g iven landing- g ear configmation or to deyelop orifLcc and :-ltmt Defl ec tion ," with the initial condition tak en a the metering-pin characteris tic I"eq u i red to prod uc e a s pl'cifLed conditions of motion at the beginning of hock- trut deflec- bdtayior for given impact conditions. The.\" ma.\" al 0 be tion , as determined in accorclance with th initial impact used as a basi [or the calculation of dynamic load in fll'xible con clition and the equations given in th e ection entitlClI airplane s trllcturl ' either b., - (a) determinin g the landino-- ":-Iotion Prior to Shock-Strut Deflection. " g ear forcing function uncler the a sumption that th e upper NUMERICAL INTEGUATION PHOCEDURES mas is a rigid body and tben u ing this forcing function to In vie\\" o[ the fact that the equations o[ motion for th e cHl c ulate the re spon e of the clastic ystem or (b) combining landing gear ubs equent to the beginning of hock- trut the prl'ce ding eq uation with tlle eq uations rrpresenting the ddlection al"l' bighl.\- nonlinear , anal.," tical olution o[ the e additional degree of frl'eelom o[ the structure ; the imul- eq uation s docs not appear feasible. In the pre ent report , t,uleou s solution or the l'quations [or uch a )" tem ,, "ould therefore , finite-difference method arc 1"e ortecl to for the then take into account the intcraetion bet\\ "een the deforma- s tep-by- tep integration o[ the equation of motion. 1\.1- lion of tlte tructlll"e and the landing gear.

ANALYSIS OF LANDING-GEAR BEHAVIOR 13 though uch num erical methods lack th e generality of ana- tire manufa ct urer' static or o-called impact load-deflection lytical solutions and ar e es p ec ially time consuming if the data are availabl e, a is u ually the case.

calculations are carried out manually, the in creasing availa- EFFECT OF DRAG LOADS bility of automatic calculating machines largely overcomes Although th e pre e nt analysis pe rmit taking into account these objections.

the e ff ect of wh ee l pin-up h'ag load on th behavior of th e Most of the solutions presented in this report we re obtained landing gear, the determ ination of th e drag-load time history with a procedure, her einafter referr d to as the "linear pro- i not treated eA rpliciLly. Thu s, if it is de ir ed to con ider the ce dur e ," which assumes changes in th e motion variabl es to e ff ect of the drag load on t he gear be ha vior, uch a in the be linear over finite time int ervals. A few of the solution case of a ch-op te t in which drag loads are simulated by pr ese nted were obtained with a pro ce dur e, hereinafter referred reverse wh ee l rotation or in a landing with forward peed, it to as the "quadratic procedure," which a sumes a quadratic is neces ary to est ima te the ch' ag load, either by mean of variation of displaceme nt with time for successive interval .

oth er ana l ytica l con iderations or by recour e to experime ntal Th e generalized solutions for the simplified equations di - data . As a first approximation the instantaneou c hag force cu sed in a subsequent section were obtained by means of may be as um ed to be equal to the vertical ground reaction the Run ge-Kutta procedure. The application of these multiplied by a suitable coe ffi cie nt of fri ct ion J.L; that i s, procedures is described in detail in appendix A.

FH g= F vg J.L , up to th e in st ant when the wheel tops skidding, USE OF TIRE FO R CE-DEFLECTION CHARACTERISTICS after which the drag for ce ma y be assumed e qual to zero.

In order to obtain olutions for particular ca es, it is, of ( Th e c urr e nt O' rotmd-load requirements specify a skidd in g course, nece al'y to have, in addition to information regard- coe ffi cie nt of friction J.L = 0.55 ; limit ed ex pe rim enta l evidence, ing the phy ical characteristics of the landing gear, ome on the oth er hand , in li cate that J.L may be as hi gh as 0.7 or knowledge of the force-de fl ection characteristics of the tire.

a low as 0.4.) In ome ca e expe rim enta l d ata indic at e If extensive data regarding the dynamic tire charactel'- t hat r e pr e e ntation of the dra g-load time hi story can be i tic, uch as shown in figures 2 and 3, are available, an implified even furth er by a suming a linear variation of the accurate solution can be obtained which takes into account ch 'ag force with time during the period of wheel skidding.

th e various breaks in the force-d e fl ection curves (logarithmic Th e instant at which the wh eel tops skidding c an be coordinates), a well as the effects of hysteresis. In view of es timat ed from the simple impulse-momentum relationship the fact th at the constants m' and r have the same valu es throughout practically the e ntir e tire compression proce s r egardless of the initial impact ve lo city or the maximum load attained, these values of m' and r, as determined from the force-d eflection curves, can be u ed in the calculation of where the motion subsequent to the beginning of shoc k-strut defl ec - l w polar mom ent of inertia of wheel assembly about axle tion until the first break in th e force-deflection c urv e is V initial horizontal velocity Ho reached prior to the attainment of the maximum force. If Td. radiu of de fl ect d tire the conditions for the calculations ar e the same a those for t .u time of wh ee l pin-up which force-d flection c urv e ar e available, th e va lu es of Wh en the ch -ag forc e i expressed in terms of the vertical t m' and r for each of th e several r eg imes subs equent to th e

forc e, the va lu e of the integral r FH dt can be deLermined as

fir t br e ak can also be det ermined directly from the force-

Jo g

deflection curve . In general, however, the condition will the step-by-step calculation proceed and the ch 'ag-force not be th e same and int erpolation will be n ece ary to term e liminat ed from the eq ua tion of motion after th e re- est imat e th e va lu es of m' for the ubse qu e nt r eg ime s. quired value of the integral at the in tant of pin-up i uch int erpolation is facilitated, parti cularly aft er the maxi- reached.

mum for ce -de fl ection point has b en calculated, by th e fact EVALUATION OF A ALYSIS BY COMPARISO OF CALCULATED that each subsequent r eg ime ha a fixed va lu e of r, regardle RESULTS WITH EXP ER I ME TAL DATA of the initial impact conditions.

The u e of the tire-defl ec tion ch aracteristics in th e calcula- In order to evalu ate th e applicability of the for eg oing tion is great ly simplified if hy ter es i i negl ec ted in ce th analytical treatment to actual landing gears, te t were va lu es of m' and r which apply prior to the first break in the conducted in the Lan gley impact ba sin w.ith a conventional for ce -de £l. ec tion curve are then u ed throughout the e ntir e ol eo- pn e umati c landing gear originally de igned for a small calculation, exce pt in the ca e of evere impa cts where tire military traininO' aU·plan. A de cripLion of the test pecimen bottoming occurs, in which ca e new va lu e of m' and rare and apparatu u cd is gi ven in a pp endix B .

employed in th e tire-bottoming r eg im e. A imilar ituation In this section calculated result are compared with ex- e xi ts with r es pect to the constants a' and b wh en the linea l' perim enta l data for a norma l impact and a evere impact approximations which n eg l ect hy tere i are u cd. Th e e with tire bottomin g. Th e vertical ve lo cities at L he instant implification would normally be employed when only th of ground contact u ed in the calculation CO LT e pond to 2 78 46-54-3 RE POR 1' 11 54 - NATIONAL ADV ISORY COMMI'lv fEE FOR AERONAUT I CS the verticltl velociti es measured in the tests. Equations co mpl'e sion ratios, the air -pr essure force becomes larger (12), (13), (14 ), and (15a) were used to calc ul ate the va lu es than the hydraulic force.

of the va riabl es at the instant of initial shock- st ru t deflection.

IMPA CT WJTH TIRE BOTTOM! G N um erical int eg ra tion of equat ion s (16) and (17 ) provided Figure 5 presents a co mp ari on of calculated and e>"'}leri - the calculated r es ults for the two-d eg r ee -of-freedom system men ta l results for a severe impact (V vo = 11.63 ft per ec) subsequ ent to the beginning of sho ck- str u t deflection.

in which tire bottoming OCCUlTed. The tire force-deflection In these calculations the discharge coeffic ient for the orifice ch aracter i st ic used in th e calcul at ions are shown by the and the pol ytro pic expon ent for the air-compression process so lid lin es in fi gure 3 (b). Region (1) of the tire force- were a um ed to ha ve co n st an t values throughout the impact.

deflection curve ha s the same values of the tire constants Consideration of the sha pe of the orifice and examin at ion Tn' and T as for the case previously discussed. Following of data for rounded ap pr oach orifi ce in pipes suggested the occurrence of tire bottoming, however, different va lu es a va lu e of C~ eq ual to 0.9 . Evalu at ion of data for other of m' and r app ly. These va lu es are given in figLU'e 3 (b).

landing gears indicated t hat t.he air-compression proce s It can be seen from fig LU 'e 5 that the agreem ent between could be repre ented fairly well by use of an average va lue the ca l cul ated and exper in1 enta l results for this case is of the effective polytropic ex ponen t n= 1.12. In view of simil ar to that for the comparison previously presented.

the fa ct t ha t the l an ding gear was mounted in a vert ical Th e calcu la ted instant of tir e bottoming i indicated in position and drag loads were absent in the tests, fri ction fi gure 5. Wh en tire bottoming occurs, the great ly increased forces in the shock stru t were assumed to be negligible st iffness of the tire cau es a m arked in crease in the shock- in the calculations. ince the weight was fully balanced trut telescoping ve lo city, as is hown in the right-hand by lift force in the tests, the lift factor K L was taken equa l to portion of figure 5 (b ). • ince the strut i suddenly forced to 1.0. The appropriate exact ti re cha ra cter i st i cs (see fig. 3) absorb energy at a much higher rate, an abrupt increase were us ed for each case. in the h ydra ulic re i stance take place. The fmther increa e in sho ck- strut for ce immediately following the occurrence of ORMAL IMPACT tire bottoming i evi lent from the left -h and portion of Figur e 4 presents a co mp ar i so n of calcul ated results wi L h fi gme 5 (a). Th e udden increa e in lo wer- mas acceleration experimental data for an impact without tire botto min g at at the i nstant of tire bottoming can al 0 be een.

a vertical ve lo ci ty of 8.86 feet per seco nd at the instant of In this severe imp act the h ydrau lic resi tance of th e orifice ground co n tact . Th e exact dynamic force-deflection c har ac- represents an even greater proportion of the tota l shock- teri st ic of the tire, including hysteresis, were used in the str u t force thaD was indicated by the calculated results for calcul at ion . Th ese ti re c har acteri ti cs arc shown by the an initial vertical v lo city of . 6 feet per secon d previou ly olid lin es in figure 2 (a) and va lu es for the tire co n sta n ts discussed.

m,' and r arc given in figure 3 (a). Th e foregoing comparison indicate that the ana lytical treatment presented, in con jun ction with rea onably traight - Ca lcul ated time hi stor i es of the total force on the upper ma ss and the acceleration of L he lower ma ss are co mpar ed forward as umptions regarding Lhe parameters involved in L he equations, provide a fairly aCCLU ' ate repre enLation of with experim enta l data in figLU'e 4 (a ). imil ar comparison the behavior of a conventional oleo-pneumatic landing gear.

for the upp el' -mass displacement, u ppe r-mass velocity, lowcr- mas displacement, trut Lroke , and tl'ut telescop ing P ARAMETER STUDIES ve lo ci ty arc presen ted in fi g ur e 4 (b ). As can be seen, the In the previou section comparisons of calculated results agreement between the calculated and experim enta l results with e).'}lerimenlal data howed that the e luations which is reasonably good through out mo t of th time history.

ha ve been developed provide a fairly cyood representation of orne of the minor di crepanci es during the later stages of tbe be.b av ior of Lhe landing gear for the impact conditions the imp act appear to be du e to e rr ors in measurement since considered. In view of the fact t hat the equat io ns are the d eviat ion s between the calcul ated and experimen tal some what complicated and require numerical va lu es for upper-mass accelerations (as repre en Lcd by the force on several parameters such as the tire force-deflection constants the upp er mass) arc incom pa tible ,~ ~ ith those for the upper - m,' and t, the orifice discharge coeffici ent Od, aDd the poly- ma s displacement , wh ereas the calc ul ated upper-mas dis- tropic ex p onent n, wbich may not be readily or accmate ly pla ce men ts arc nece sa rily d irectly co mpa tible with the known in the case of p ractica l engineering prob lems, it calculated upper-mass accelerations. Th e maximum value a pp ears desirable (a) to determine the relative accuracy of the ex p er im ental accel erat ion of the lower mass m ay be wit h w111ch th ese var i us parameter h ave to be lmown and some what high because of overshoot of the accelerom eter .

(b ) Lo inve tig at e the extent to which the equations can be In add ition to the total for ce on the upp er mass, figure simp lifi ed and st ill yield useful results. In order to accom- 4 (a) presents calcu lat ed time hi tories of the hydraulic plish these objectives, cal culat ion have been made to and pn eumatic co mpone nt s of the sho ck-struL force, a eva lu aLe th e effect of in1plifying the force-defiection charac- dete rmin ed from equa tions (2 ) nnd (3), re pectively. It te ri tics of L he Lire, as we ll as to determine the effects which c an be seen th at throughout mo st of the impact the force diff erenL valu es of the orifice discharge coefficient and the developed in the sho ck st ru t ari es prim ar ily from the h y- e ff ectivc polytropic exponent ha ve on t.he calc ul ated behavior.

Th e re lUtS of the e calc ul ations are discussed in the pre ent draulic resistance of th e orifi ce. Toward L be en d of the section. The question of simplification of the equation of impact, however, because of the cl ecreased Lele co pin g motion is considered in more detail in a sub equent section.

ve lo citi es and fai rly large sL rokes which corresp,:md to high A "ALYSI OF L DI G-GEAR BEHAVIOR -7 o -6 -5 o ~ -- Colculated o Exper im ental : ~N

"'"

5 ,,; -4 u .!"!

<1'- c o '" ·2

:4

~ -3 ., Q) 0- n ~ :J u o c ::l -2 ., o u E , ~

2 ~-I

(0) LL-L-.~0 ~4~L-.0~8~J-- . ~12~J--.~16~i-~. 20 10 .04 .12 . 16 .20 .0 8 Ti me after cantoct, sec (a) Tim hi sto rie s of forc es on upp er ma ss and low er -ma ss acceleration.

FIGURE 4. -C ompal'i sons be t w een calculated re ult s and ex perim en ta l data for normal impac t; olution with e xa ct ex pon ent ial t ire characteri t ic . V = 8.86 f eet per econ d ; C d =O.9; n=1. 12.

vo 1.0 10 .6 6 Upper' mass .8 8 .5 5 displacement - _, Strut stroke-, 0 0 0 V Up per-mass 0 0 0 velocity-- , __ Strut v velacity - - ____ 0 v .6 6 4 - .4 '" .9- ;:: '" .9- .,- >: c Q) x ~4 ]3 ~.4 ~ ,3 Q) u 'u 0 0 > A 0.

A ~ 2 A '" 0 '2 is if, 0 A (j) A A .2 2 .2 2 E!l .1 0 0 --- Calcula ted ODAOV Ex per im e ntal (b) -2 0 .04 . 08 .12 .16 . 20 0 .0 4 . 08 . 12 . 16 .20 T im e after con tact, sec Lie and dis plac ements.

(b) Tim e h isto ri es of landing-g c ar v loci RE4 .- oncluded .

Fw ----- ----- - - 16 REPOR T 11 54-NAT IO AL ADV I SORY C OMMI TT EE FOR AERO AUTICS -12 -- - Calculaled -10 o Experimenlal ~ :~ L- -8 ] c ~ - 6 OJ u u '" '" E 4 o ci; -4 ~ o I .J f- - - - - Tlre I ballamlng I 2 -2 I I Tire I I o ball oming __ I I /, pneumallc o I ( a) I 0 I o . 04 .0 8 .1 2 .16 .20 o . 04 .0 8 .12 . 16 . 20 Ti me afler conta ct, sec (a) Tim e hi to ries of forces on upp er ma ss and 10\\ 'e r-ma s acc ele ra tion.

FI GURE 5. -Co mpari ons betw een calcu la ted r es ults and exp eri menta l data for impac t wit h t ire bottom in g; sol ution with exact exponentia l t ire cha ra cte ri st ics. Y = 1l.63 f eet p er se cond; C = O.9 ; n= 1.12.

vo d 1.2 12 o 1.0 .6 6 o Upper- mass d is placement _ __ , Upper-mass Strut stroke ·.

velacity -'-- Strut velocity· --_ .

.8

8 °

°

°

.:: '" .9- c OJ

°

~6 ~6 .4 u ·u 0 0 Ci ~ '" .9- '" <5 I> OJ- >- .>< I> I> .4 4

:B3 °

~ . 3 OJ I> > I> "2 - - Lower- mass "2 I> if,

°

displacement if> .2 2 .2 2

°

-- - Calcula ted 001>0'(1 Experimental 0 0 .I __ - Tire _- -Tire ( bottoming ~ -- bottoming I I I I (b) I I -2 . 12 16 . 20 0 . 12 0 . 04 .08 08 . 16 .20 Time after con tact, se c (b) Time hi tori es of landing-g ea r Yelocitie and dis placem ents .

FlGUHE 5. -Co ncluded.

ANALYSIS OF LA DII G- GEAR BEHAVIOR 17

REPRESENT AT IO OF TIRE FORCE-DEFLECTION CHARA CTE RISTr CS for a severe impact, involving tire bottoming, at a vertical velocity of 11.63 f eet per econd. In figmes 6 and 7 the In order to eva luat e the degree of accmacy r equired for olid-line cmve represent solution of the landing-gear adequate representation of the ti re force-deflection c hara c- equations when the exac t exponential relat ionship between teri tics, compari ons are made of the calcul ated behavior force and tire deflection are con idered. Since these so lu- of the landing gear for norma l impa ct and impacts 'with tion were pr eviou ly hown to be in fairly good agreement tire bottoming when the tire c haract eri tics arc represented with expe rim en ta l data (fig . 4 and 5), th ey arc u ed a a in various way. Fir t, the force-deflection c haract eri tic basis for eva lu ating the re ult obtained when tire hyst eresis will be a umed to be exactly a hown by the olid-line is n eg lected and the force-de fle ction characteristics are re pr e- curve in figme 2 (b), including the various br eaks in the sented by e ith er simplified ex ,})on ent ial or lin ea1'- egment curve and the e ff ect of hysteresis. Th ese characteristi cs relation hips.

are referred to h ereinafter as the exact ex pon entia l tire As in the calculations previou ly de cribed, the o lu tions characteristic. Th e eff ct of implifying the representa- in two part. Dming the flrst tage of the were obtained tion of the tire c hara cteri tics will then be investigated by impact the shock strut ,vas considered to be ri. gid until considering (a) the ec>.: pon e ntial c hara cteri st i cs without uflicient for ce was developed to overcome the init ial air- hyster es is; that i s, the tire will be assumed to deflect and pr e m e force. The calculations for the landing-gear behav- unload along the same exponential C LU'v e, (b) Lhe linear- ior sub e qu e nt to this in tant were ba ed on the equat ions egme nt approximation to the tire c hara cte ri st i cs (long- which con ider the gear to have two degree of freedom.

da hed line ), which al 0 neglect hy tere i , and (c) e rror Time historic of the upper- mas accel erat ion calculated on introduced by neglecting the effect of tire bottoming in the the basi of a rigid shock tr u t are hown by th e dott ed case of evere impa cts. The calc ulat ed r es ults pr esented in curve in figme 6 and 7. Th e e solutions show the greatest this tudy mak e use of the relation hip betwee.n vert ical rate of increa e of upper-mass acceleration pos ible with force on the tire and tire deflection, a hown ill fi gur es the e).. ,})on ential tire force-deflection c haract eristic con- 3 (a) and 3 (b).

idered. Comparison of the e olution with tho e for the Figure 6 pr esent a co mparison of the calc ul ated 1' e ults two -degree-of-fr ee dom ystem indicate the e ff ect of the for a normal impact at a vertical ve lo city of . 6 feet per ec ond, whereas fi gLU'e 7 permits comparison of the so lutions hock tr ut in atten uating the severity of the impact.

-3.0 -10 -2 .5 -8 ~ N 'I" ~5 -6 ti .2 c c .Q o

e

~ ~ - 1.5 Q! -4 Q!

~ tl u o o V> V> V> V> o o E

E "-

L - 1.0 ~ ~ - 2 Q!

Q!

Q.

~ ~

Q.

=> S ~, Tire characteristics co ns idered: ~ Exact exponen ti al -.5 O~--------------~------~~---- ---- Exponential (no hysteresis) Linear - segment (no hysteres is)

\

Exponentia l (r igid strut) (a) .20 20~ ---"L- -.;;'0 4 -;;-- --L--+::---.L ----:l= -....L-..L- ..L..:.:...-J 16 o .0 4 08 .12 .08 .12 .16 .2 0 Time after co nta ct, sec (a) Tim e historie of upp er-rna s acceleration and lower-mas acceleration.

FIG RID 6. - Effect of ti.re characteri tic on calculated landing-gear behavior in normal impact. VVo = ' 6 feet per second; C = O .9; n=1.12.

d 1 REPORT 1154-NATIO AL ADVI ORY COMl\1I'l'TEE FOR AERO AUTICS B 12 ~ 10 Upper moss .6 .5 c Q) Lower moss ~ .4 u o a.

OJ) .3 2 .2 Or---------~---- ~~ ------ T ire characteristics considered: Exa ct exponential Exponential (no hysteresis) -2 .1 Linear - segment (no hysteres is) _4~ ~~~L- ~~--~-L~--~( b ~ ) .16 o .04 .08 .12 .16 .20 o .04 .08 .12 .20 Time after con tacl, sec (b) Time histories of landing-gear di placement and velocities.

F1 ,URE G. - Cont inu ed .

.5 5 .4 4

= .3

"2 Vi .2 Tire choroclerlstlcs considered: .1 Exact exponential Exponential (no hysteresis) Linear - seg m ent (no hysteresis) I (c) . 12 . 16 . 20 . 16 . 20 0 . 04 . 08 . 12 o TIme of ler co ntact> se c (c) Time histories of shock-st ru t stroke and velocity.

FIGURE 6. - Concluded.

ANALYSI OF LANDII G-GEAR BEHAVIOR 19 -4 .5 -12 -4.0 -10 -3 .5 ~ -s :~ ~ : .... - u -3 .0 .E -6 u c: .E .9 c:

e

.9 -2.5 ~ -4 -0 u Q; u a:; o u u 0 '" o '" -2.0 E -2 '" , 0 ~ '" E I ~

(1) ~

a.

-1.5 a. O~----------------~~------ Tire choracteristics considered : ::J Exact exponential Exponentiol -1.0 2 (no bottoming, no hysteresis)

\

Linear - segment (with bottoming, no hysteresi s) Lineor - segment (no bottoming, no hysteresi s) Exponential (rigid strut) (0)

\

. 16 .12 .16 .20 o .08 .12 . 20 .04 . 08 Time after contact, sec (a) Ti me hi st ori es of upp e r-ma ss accele ra t io n an d lowe r-rna accele ra t ion.

FIG a U E 7.- E ff ect of t ire ch a ra cte rist ic. on calcula te d landin g-ge ar be havior for impact wit h t ire bot to ming .

VV o= 11.63 f eet p er eeond ; C =O.9 ; n = 1.12 .

d .9 12 .s .7 .6 6 _- .5 4 c: .9- (1) E o 5}.4 i:5 .3 o~--------~~----~--------- Tire choracteristi cs considered : Exact exponenti al - 2 .2 Exponentiol (no bottoming, no hysteresis) Linear - segment - 4 .1 (with bottoming, no hysteresis) Lineor- segment (b) ( no bottoming, no hysteresi s) -6L--L __ L--L __ ~~ __ ~-J __ ~~~ o . 04 . 08 .12 .20 . os . 12 . 16 .20 Ti me of ter contact, sec (b) Tim e hi t ories of landin g-ge ar dis plac emen ts and vel ocities.

FI GU R E 7 .-C on t il1u ed.

20 REPORT 11 54 - ATIONAL ADVISORY COMMITTEE FOR AERONA '. rICS

.6 .5

~

\

.4

~

Q) .:.: o i; .3

~

.~

\

.~ .2 Tire characteristics considered :

Exoct exponential \\

Exponential .~ (no bottoming, no h ys t eresis) .1 Lineor-segment (wi th bottoming, no hysteresis) Linear- segment (no bottoming, no hysteresis)

\

(c)

o . 04 .08 . 12 .16 . 20 o .0 4

.08 .12 . 16 . 20 Time after contact, sec (c) Tim e histories of s hock- st ru t st roke and velocity.

FIGURE 7. -Conclu ded.

Normal impact .- In the ca e of the normal impact at a Impact with tire bottoming.- In the case of tbe severe vertica l velocity of .86 feet per econd, figUl' e 6 shows that impact at a vertica l velo city of 1l.63 f ee t per second, the the solution ob tained with th e e}',' pon e ntial for ce-de fi ect ion e ff ects of tire b ott oming on th e upp er- ma accele ration , the variation which neglects h yste resis and the so lu tion with the lower- ma ss accele ration , and the st rut telescoping velocity linear-segme nt approximation to the tire characteri tics are are cl early indi c at ed in figUl' e 7 by the calc ulat ed r es ults in fairl y good agreeme nt with the re ult of the calculation based on th e exact tire characteristics. As can be seen, the based on the ex pon e ntial r e pr ese ntation of th e exact tire lin ear-segme nt appro}"'imation to the tire de fl ection character- c haract eristics . Th e greate t differences between the sol u- istics which takes into account the e ff ec t of tire bot toming tions are evid ent in the time histori es of upp er-mass and r es ul ted in a reasonably good r e pr e entation of the landing- lowe r-ma ss accele rations; considerably sma ller differences are gear be havi or throughout most of the time history. On the obtain ed for th e lower-order de rivativ es, a might be x- other hand , a might be expected, the calculations which pected. With regard to the upp e r-ma accele ration , th e neglected the e ff ec ts of bottoming on th e tire force-de fl ec tion c hara cteristics did not r eveal the marked increa e in the three solutions are in very good agreeme nt dUl'ing th e early stages of the impact. In the case of th e simplified e }''"ponen- upper -ma ss acceleration du e to th e increased st iffness of the tire sub e qu e nt to the occurrence of bottoming. It is also tial c har acteri tics, neglect of the decrea ed slope of the not ed that the disc1' epancie in the cal culated upp e r-ma ss force-de fi cction C lU'V e between the fir t br e ak and the maxi- accele ration due to n egl ect of h ystere i in the la ter stages

mum (reg im e ® in fig . 3 (a)) r es ult ed in the calculation of a

of th e impact are mor e pronounced in this case than in the somew ha t hi gh er va lu e of the maximum upp er-mass acceler- impa.ct without tir e bo ttoming pr eviously con ider ed, again ation than was obtained with th e e xact tire characteri tics.

as might be expected.

For the simplified e}" -rpon ential and line ar -segment c haract er- istics, neglec t of hy teresis res ult ed in tb e calculation of EFFECT OF ORIFI CE DISCHARGE COEFFIC I E T omew hat exce sive va lu e of upp er-mass accel erat ion ub - In view of th e fact that ther e i very li tt le information seque nt to the attainment of the maximum vertica l load .

ava ilable rega rding the magnitude of disc har ge coefficient It i of intere t to not e that the calcul at ed 1' e ult for the for orifices in landing gears, it app e ars d es irable to eva lua te expone nt ial and linear- eg me nt characteristic without hy - th e e ff ect which differen ce in the magnitud e of the orifice teresis were generally in quite good agreement with each coe ffici e nt can hav e on the calculated r es ult s. Figure other th roughout the e ntir e dUJ'ation of th e impa ct, al though pr e e nts compari on of calcul ated r es ults fo r a range of the assumption of linear-, egment tire force-de fl ect ion c har- va lu e of the orifice discharge coefficie nt Oil. betw een l.0 acteristics did re ult in somewhat excessive values for th e and 0.7. Th e foUl' olutions pr e en ted are for the sa me set of initial condition a the normal impa ct without tire maximum lower -ma s accele ration . On the whol e, the bottoming previou ly considered and are based on the implified tire for ce -de fl ecL ion characteristics con idered pe r- exponential til' force-d e fl ection c hara cteri tic which n eg lect mit calculated re ults to be obtain ed which re pr ese nt the hy teresis.

behavior of the landing gear in normal impact fairly we ll .

A AL YS IS OF LA J DI G- GEAR BEHAV IO R -3 .0 -1 0 ~ - 2 . 5 - 8 ~ ~ :~ u .2 c - 2.0 .Q ~ Q) Cii u u o ~-1.5 o E .!.

. ~~ :.:. .....

Q) a.

a.

~,:'" ............. : ..

::J , ............. '.

.........

Cd -1 .0 ..... '.

........

0: --- 1.0 -- . 9 "" " .

..... " ..... ".

---- .8 .... ............ . 7

\\"

,,".

- .5 Or-----------------------~~~----- 20 ~~L-~~~-~--L-~-~-~-L-~ .0 8 . 12 .1 6 .20 .0 4 .08 .12 o .04 T i me of ler con i oct, sec (a) Tim e hi to rie of upp e r-rna acceleration and lo we r-ma ss acceleration.

FI GU R E 8.-Effect of orifice di charge coefficient on landing-g ea r behavior; calculation with ex pon ent ial tire characte ri t ic s wit hou t hy tCl·esis. V =8. 6 f eet p er eco nd ; n=1.12 .

vo .8 .7 Up per moss .6 U pp er m os s 6 .5 CI) c .E- Q) ..... ......

::.

~ .4 2 '0 u o a.

~ <II (5 .3 0 Lower moss \.

.2 -2 .~ .

Cd .. ....... '--- 1.0

....... :::. --

-- - - .9 -4 .1 --- - - .8 .... .. .... .. ... .. .. .7 - 6 L_~-L __ L-~-L __ L-~-L __ L(_b)~ o .04 .08 .12 .16 .20 o .04 . 08 . 12 .1 6 . 20 Ti me of t er contact, se c (b) Tim e hi tor i es of landing-ge ar di placement and velocities.

FIG RE 8.-C o ntinu e d.

28i8 45-54--4 22 REPORT 11 54 - NATIONAL ADVISORY C OMMITTEE FOR AERO AUTICS .6 . 5

/

/ //

//

.4

4 ------ -------

1//

/' ""

/ //-----" ~

//

/ / ...... .. .......... ,,,~

Q) -" // ........ . ..... ... ",

//

e I .... . ... . \ tn.3

!;/

'/ f .... .. .... '\ "2 / / .. ' '.

/ / I ..'

Vi / I .: 11/ ....

(I I :' 1/ .: I I ·' /:.

//1 ....

. 2 // : Cd I . :' I.'

I.'

-- - 1.0 f; f : ' - -.9 III:' 'II.: -- - - .8 II;.: .. .. 7 .1 W·' ,/ .. ' /I."

Y" (cl o .04 .20 .08 .12 . 16 .20 .08 .1 2 . 16 .0 4 Time after contact, sec (c) Time hi , tor ies of s hock- · trut s troke and velocity.

FIGURE S. - Conc luded.

(i sot he rmal ), it appeared desirable to eva lua te th e im- Th ese cal cu la tions show that a decrease in the orifi ce dis- portance of the air-compre ion process and to determine t he e hf1.r ge coeffici ent l'esulls in an approximately proportional ('xtrnt to which diff erent va lu es of th e polytropic exponent lllerease in the upp er -ma s acceleration. Thi s vari- can influence the calc ulat ed results. Con equent ly , so lution ation is to be expected in ce the smaller coefficien ts cor- have been obtained for three different value of the poly- r espond to rcd uced efi'ective orifice areas which result in tropic exponen t; nam ely, 71 = l. 3 , l.12, and O.

gr eater shock- st ru t force due to increased hydraulic Th e value 71 = l. 3 correspond s to a very rapid compression r es i sta n ce . As a result of the increa e el bock- strut force in which an adiabatic process is almost attained. Th e act ing cio\\ 'nw arci on th e lower mass, th e maximum upward value 71 = 1.12 corre ponds to a rel ative ly slow compr es ion acceleration of the lower mass is r ed uced with decreasing in w b ich th e process is virtually i, 0 thermal. Tb e va l ue values of the discharge coe ffi cient. Th e increase in sbock- 1. = 0 is completely fictitious since it impli es constant air str ut force with decreasing cli scb ar o- e coe ffi ci ent al so re ults in a decrea e in the str ut troke an d telescop in g ve lo c ity bllt pressure wit hin the strut throughout the impa ct. Th e assumption 71 = 0 ha b ee n co ns id er ed sin ce it mak es one of an increase in the lower-mas velocity and di placement, as mig ht be expected. How ever , s in ce the increases in the terms in the quations of motion a co n sta n t and p erm it s simplifi cat ion of the calculations. Th e t hr ee so lution s lower-mas displacement and veloc it y are smaller than the decreases in strut st rok e and telescop ing ve lo city, the upp er- pr esen ted are for the same et of initial co ndition as t he mass displacement and velocity are redu ced with decreasing normal impa ct without tire bottoming previously con- sidered and are ba ed on the ex pon ential tire force-deflection orifice discbarge coe ffi ci ent .

Tb ese co mpari sons show that the m agnitud e of the orifice characteristics which negl ect h yste r e i .

Figure 9 show that the air pr es m e co ntribut es only a coe ffi ci ent ha s an important e fi' ect on the behavior of the re lativ ely ma ll portion of the total shock- str ut force through- landin g gear and indicate that a fairly accurate determi - nation of the num erical valuo of thi par am eter is necessary out mo t of the impa ct since the co mpr ession ratio is rela- to ob ta in good results.

tive ly small un t il th e la ter tages of the impact. Toward the end of the impa ct, however , the air-pressure force EFFECT OF A IR-COMP R ESSION P R OCESS becomes a lar ge part of tbe total force since the co mpr e ion . ince tbe nature of th e air-compression proce in a hock ratio bec om e large, wh ereas the hydrau lic resi stance de- st rut is not well-de fin ed and different investi gators have c rea cs rapidly a the trut telescoping velocity i re duc ed assumed values for the polytropi c exponen t ranging any- where b et ween the ex tr emes of 1.4 (adiabatic) and l.0 to zero.

Al AL YSIS OF LAN DI NG - GEAR BE H AVIOR 23

- 10 - 8 ~ N

:"

-6 .: ,9 E u vi .2 til c

E 4 -4

.2 ~

Q) e

Q) a. n a. Qj :0 u u 1. 3 c 0 -2 1. 12 <f) Q) til o ~ is? I, ~ Q) 0 ~

/ \

"- ...J "- ,

~

Pneumatic "

/

"- "- ~ \

~ \

~ \ \

\

--=~ " 4- ....... "' _"""'_=-=_= ______ ____ _______ , o .0 4 .08 .12 .16 . 20 0 .0 4 .08 .12 .16 . 20 Time after con tact, sec (a) Time hi to ries of upp r-ma ss force and lowe r-ma ss acc ele ra t ion.

FIGURE 9. -E ff ect of polytropic exp onent; ca lcula t ions with exp onent ial t ire cha r acte ri tic wit hout hy teresis.

VVQ=8.86 f eet per e cond; C = O.9 .

d .8 .7 Upper moss .6 .5 til c Q) ~ E ~.4 . ~4 u o a.

til OJ Lower moss > o ~ .3 ~ ~ ~

"

\, .2 \ :-.

n \ \ 1.3 , -- -1.12 \ \ -- - - 0 . 1 -2 \ - - ~ / \ / ", ,

"

- 4 . 04 . 12 o .08 .1 6 . 20 0 .0 4 . 08 .1 6 .12 Time after co ntact, sec (b) Tim e hi tories of landing-g ea r dis plac ement and velocitie .

FI GU R E 9. -Cont inued.

--- ---- ---- ---------- ---- ------ -------- -- -------- ---- ---- -- ---- ----------- REPORT 1154- JAT IO I AL ADVISORY COMMITTEE FOR AERONAUTICS .7 .6 6 ,..'" ,/ ,/ / / .5 // ~ / I.

/, h t h h !J D .2 n -- 1.3 --- 1.12 ---- 0 .1 tcl .20 .08 o .04 .0 8 .12 .16 Time after contact. sec (c) Time hi tories of. hock-strut t rok e and velocity.

FIGURE 9.-Concluded.

be Wl'i tten a follow for the co. e where the wing lift is equa l A a result, the calculalion s how that the magnitude of to the weight and the in terna l friction i neglected: Lhe polytropic exponcnt has only a very mall effect on the behavior of the landing gear tlu'oughout most of the impa ct.

WI z l+ A(ZI- 2)2+ W = O Z For th e practical range of polytropic e:;;.:ponents, variations g in the air-compression proce re ulL in only minor differ- ences in landing-gear behavior, even during the very late st (1 ) lV2 Z2- (ZI- 2)2+ 2+ b- -VT' 2= O A z a z g stages of the impact. The assumption of constant air pre UTe in the strut throughout the impact (n=O), however,

lVl .. +VV2 .. + + b- O

- ZI - z? a z? - doe l ead to the calcu la tion of excessive yalues of stroke and g g - - of the time to reach the maximum troke. Th e time hi story where of t.he hock-strut force calc ulat ed on the basis of this a ump tion i , on th e other hanel , in quite good agreement .

'with the results for th e pracLical range of ail'-compre ion and processes.

a slope of lin ear approximation to tire force-deflection On the whole it appears t hat the behavior of the landing c hara ctel'i tic gear is relatively in en itive to variations in Lhe ail'- b value of force corresponding to zero tire deflection, as compression proce . The foregoing result suO'O'e L tba t, in detennined from the linear- egment approximation to maIW case, fairly reasonable approximation for Lhe landing- the tire force-deflection c hara cteri tic gear force-time variation might be obtained even if the air- Th e motion va riabl es at the beginning of hoek-strut de- pre ure Lerm in th e equation of motion were completely fl ect ion can be readily determined in a manner similar to neglected.

t hat employed in th e more gene ral treatment previou ly eli - cussed. For the simplifi.ed equat ion the va riabl e at the SIMPLIFICATION OF EQUATlO S OF MOTlO instant tT are g iv en by The preceding studies hav e indicated that variation in the .. W Z T=-W g tire force-deflection character i tic and in th e air-compression I process indi, - idually h ave only a relatively minor effect on the ca lcul ated behavior of the l anding gear . These results sug - (19) O'e t that the equations of motion for the landing gear might be implified by comp l ete ly negl cting the internal ai.r-

. I . 2 ag 2

ZT= -y Zo -W ZT J

pressure forces in the sho ck st ru t and by c on sidering Lhe tire f01'e -defleetion characteristic to be lin ear . ,Vi th Lhe e as- In most cases the term ~fr z/ is small in comparison with Z02 sumpt ion s, the equat ion of motion for the upper ma s, lower mas, and complete sy tern (eqs. ( 16 ), (17), and ( )) can so that ZT:::< zo.

N AL YSIS OF LANDI G- GEAR BEH AV IOR -3 .0 -10 -2 .5 -8 t>, '-....

: '" ~ o· - 2 .0 : ~ -6 U o ' .2

U ~

c .2 .Q c

~

o .Q V [) Qj -1.5 Q; - 4 u

I \

Qj u o u g <f> <f>

<f> I ~ --

o <f> o E I E

I

;;; -1 .0 ~ -2 0.

0.

::J /

--- - So luti on of fi gu re 4 - .5 - - Simp li fied syste m, W2 = 13 1 Ib o ~--------------------------~ -------- - - - - Simplifi ed system, W2 = 0 (0) 2~ ~--~~~ __ ~~-L __ ~ __ ~ __ ~ __ L-~ o .04 . 08 .1 2 .16 . 20 0 . 04 .0 8 . 12 .1 6 .2 0 T i me after can ta ct, sec (a) Tim e histor ie of upp er-rna accel erat i on and lower-rna s acceler at ion.

F I GU RE lO .-E valu at ion of ca lG ul ate d res ul ts for s im plified system . V = 8. 86 f eet pe r seco nd; C = O.9.

d vo Th e values dete rmin ed from e quation s ( 19 ) are u ed a Th e e ff ec t of negl ec ting the lower rna wa primarily t o ini t ia l condition in th e olution of quation (1 ). r educe th e lowe r-ma ss displacemen t (tire de fl ection), a a Th e fa t t ha t the lo we r rna is a rel at ively mall fr act ion of re ul t of the elimination of th e lower-m as iner tia re act ion.

he to ta l ma s suggests th at th e y tern migh t be implified On the whol e, it appear t ha t the assump tion con idered even fur ther wit hou t gr ea t;l y modifyin O' th e calcul ate d re ul ts pe rmit appr eciable simplification of the equ ati on of mo tion by a umin g th e lower mas to be equal to zero . Wi th this without g reat ly impairing th e va lidi ty of the calcula t edr es ult s.

a s ump t ion tT= O and th e ini tial value of the variables in GENE R AL IZ ED T R EATMENT e qu at ions (1 ) corre pond to th e conditions at initial c onta ct.

Equations and solutions.- By wri ting the simplified e qua - EVALUAT IO OF SI MP Ll FI CA TIO tions of motion in t e rm s of dimen ionle s variabl e, general- ized olution c an be obtai ne d for a ,vide range of la nd ing-gear In order to evalua te the a pp licability of th e e implifica- and impa ct pa ram et ers w hi ch m ay be u eful in pr e- tion s, the behavior of th e la ndin g ge ar ha b een calc ulat ed in liminary d es ign. If W is ta ken equal t o zero an d it i acc ordan ce wi th e qua tion (1 8) for an impa ct with an initial 2 further a um ed th at the tire fo rc e-de fl ection c ur ve i ver tical ve loci ty of . 6 feet per ec ond. A similar ca lcula- represen ted by a in gle traig ht line through th e origin t io n ha b een mad e wi th the ass ump tion W = 0. Th e e (b= O t hrou ghou t the impact), equ at ions (1 ) re du ce t o re ul t ar compared in fi g ur e 10 wi th th e mor e ex act olu- tions pr evi ou sly pr esen ted in fi gure 4, which include con- WI .. + A(' ') 2 0 - Z I ZI- Z2 = ider at ion of the air-compre sion s prin g in g and th e exact g exponen t ia l t ir e ch ara cteri tics. A time hi tory of th e lower- A( zI- )2- (20) z 2 a z 2= 0 rna s acc el e ration is not pr e en ted for the case where TiV is a tuned e qu al to zero in ce th e values of Z2/g ha ve no

WI .. + 0

- Z I a Z2= ignifi ca n ce in this case.

g Fi g ur e 10 hows th at the tw o implified sol ution ar e in wh ere W I, A , a nd a are c on tant ,a pr eviously defined, aod qui te good agr ee me nt with each ot her, a migh t be exp ec ted, g a nd are aloin fairly good agr ee me nt with t he mor e exact an y t wo of the foregoing e qua ti on are ufficient t o de cribe r ul t. eg lect in g th e air-pre ur e for ce and a uming a completely the behavior of the sy te rn. Wi th thi repre ent a- l in ear ti re for ce -de fl ection variation r e ult ed in the calcula- tion of th e ys tem , the ho ck trut b eg in to de fl ec t at the in - tion of ligh tly lower value fo r the maximum upp er -rna s sta nt of initial con tact (tT= O). Thu s, the initial conditions accele ration and somewhat higher value for th e maximum for equation ( 20 ) ar c the initial impact condition s; nam ely, troke t han wer e ob tained with the mor e e xa ct e qua tion .

z Zo= Z 2o= 0 a nd ZlO=Z 2 0= zo.

26 REPORT 1154 - ATIOI AL ADVISORY C OMMITTEE FOR AERO A TICS

.8

--

Upper mass§~ .7

j

j

.6

I

.5 6 c Q) ~.4 g a.

<f) c5 .3 2 .2 O~-------- --~ ----~ ~------ --- Salulian of figure 4 -2 - . 1 - -- - Simplified system , W2 = \31 Ib - - - - Simplified system, W2 = 0 (b) o .04 .08 . 12 .16 .20 .04 . 08 . 12 . 16 . 20 Time after contact, s ec (b) Time hi 'to l'i e of landing-g e ar di s placem e nt s an d velocities.

FIGU R E 10.- Continue d .

.6 .5 .4 .2 .1 ---- Soluti on of fi gu re 4 ---- Si mp li f i ed system, W2 = 13 1 Ib - - - - Si mpl if ied system, W2 = 0 (c) o . 04 . 08 . 12 .16 .2 0 0 . 04 . 08 .1 2 .1 6 . 20 Time after contact, sec (c) Tim e hi sto ries of s hock- s trut st roke and velocity.

FI GU RE lO. -Conc lud ed.

- - - - - - ---- - -- - -- - -------- ----

ANALYSIS OF LANDI I G-GEAR BEHAVIOR 27

and As can be een, with the e qua tions in this form, the solution depends on five param eters, nam ely, W I, A, a, and the g ini tia l co ndition Zo and zoo How ever, since zo= O in all where cases, the numb er of variable param eters is reduced to four.

In view of the fa ct that these param eters are indep e nd e nt of on e another and each ma y ts,ke on a large rang e of va lu es, a great ma ny solutions and a large numb er of graph 'would be With th e e n ew variables e quation s (2 0) can b e writ ten a req uir ed to cover the e ntir e rang e of landing-gear and impa ct par ameter with the equations in the form of e quation s (20).

Cut' -U/)2+Ut" =O }

Th e number of ind epende nt pa ra m eters which hav e to be ( 21 ) (UI'-U2' Y- U 2=0 considered may be great ly decrea ed by the introduction of generalized dimensionless variablc and thc corresponding UI"+U2=0 tran formations of e quation ( 20 ). In thi ca e, generalized variabl es can b e obtain ed which pe rmit transformation of the wh ere any two of the e e quation s are sufficie nt to describe e quati0n of motion to a form which do es not involve any the behavior of the y tem.

con t!1nts. With the equations in this form , there is oo1y Ina mu ch as equations (2 1) do not involve an y cons tant s, one variable param eter, namely, the initial ve lo ci ty param - their solutions ar e completely determilled by the initial val- eter . To dete rmin e the generalized variable which sat i f S- ues of the variab l es . ince th e displacements at initial the aforementioned requirem e nts , l et contact UI and U 20 are e qual to zero and the initial velociti es O UI ' and U20' ar e equal , the only p arameter is the initial O U=Zex dimensionless ve lo city and o= t(3 , . jA2g Thu , Uo =zo ", W 1a , du . ex u=d o =z 73 wh ere uo' =UI ' =U20'.

O and Generalized so lu tions of e quation s ( 21 ) are pr esented in II du' .. ex figure 11 for va lu es of Uo ' corr es pondin g to a wide range of U =Te=Z (3 2 landin g-gear and impact parameter. Par ts (a) to (e) of figure 11 show the variations of the c limensionle s va riabl es S ub stit utin g the e n ew variable pe rmit s equations ( 20 ) during the impact; part (f) a nd (g) show the maximum to b e wr itte n as va lu es of the mor e importan t va riabl es as functions of uo'.

Part (h) shows the hock- trut e ff ectiveness 7J . and the

(UI' -U2 ' )2+ (~ W I) ut" = O

g, landin g-gear e ff ect iv ene s 7J !g. Th e shock- str ut effective- ne s, om et im es called "effici ency" and, in Europ e, "p lani- (20a) metric ratio, " is defined as "+(a / (3 2) 0

l ~('~ du

UI W dg U2 = ut" maz u maz Th e numb er of ind epende nt param eter will be l'educed if all the combined con tants in e quation s (2 0a) ar e set where U= UI-U 2 is th e dimen ionless shock- st rut st roke.

e qual to one anoth er, that i , l et Since 7J a repre ents the r ati o of the energy actually absorbed ex W I a ex aj (3 2 by th e ho ck trut to the maA'imum ene rg y which the st rut A g= A (32 = W dg could pos ibly absorb for any combination of ma}rimum acceleration ( or load) and maximum stroke, it serves a a From this re la tionship, it can be seen that m easure of the ex tent to which a given combinat ion of maximum load and st rok e ha s b een utilized to absorb the A ex=W dg energy of an impact. A similar measure of the en e rg y absorption effectiveness of the landin g ge ar as a whole is u, nd given by 7Jl g which is defined by Thu s the generalized variabl es b ec om e 1] !g , Equations (21) may be reduced to a single equation in one variable by diiIerentiating the last equation and substituting for U2' in the fir st equation. This gives (u,'+u,m),+u,"=O.

By introducing tbe new variable w=uI', tbis equatlou may be reduced to the second· order eq uation (w+w"),+w'=O, subject to tbe initial conditions wO=UQ' ~nd WQ'=O.

REPORT 1154 - NATIONAL ADVISORY C OMMI TTEE FOR AERONAUT I CS -6 (jo' = 8

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F IGU RE ll .- Generalized solutions for implified system .

.5 (b) I 4.0 1 .6 2.4 2.8 3.2 3.6 o .4 .8 1. 2 2.0 Dimension l ess time , 8 (b) Rel at ion hip b et ween upp er-rna di spl aceme nt and t im e.

FI GU RE 11 .- Continu ed.

ALYSIS OF LANDI G- GEAR BEHAVIOR 1 .0 , Yo r.......

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FIGURE n .- Continued .

The generaliz d re ults pre ented in figure 11 3 can be it i fir t necessary to approximate the tire force-deflection u d to e timatc the performance of a given landing gear of characteristic by a imple linear variation. Two such lmown configuration for particular impact con lition or to lin em' appl'o.>..imation which might be considered suitable choose the dimension for a landing gear when the impact for this purpo e are shown in figure 12. Lin ear approxi- conditions and de ired performance are specified.

mation I i a traight line through the origin having a slope Applicability of solutions.- To illustrate the applicability a= l .5X 10 pounds per foot (a =ad= 41.6 X 10 lb) . This ' of the generalized olution , the curves of figure 11 hav e value of a and the oth er pe rtin e nt landing-gear and impact been applied to the previously con idered ca e of the normal paramet ers res ult in a value of the initial dimensionless impact at an initial vertical velocity of . 6 feet pel' econd ve locity paramet er uo' = 2.57. Linear approximation II is for compari on with the more exact so l ution presented in a straight line with lope a= 21.3 X 10 pounds per foot figure 4. In order to make u e of the O"eneralized so lutions (a' = 47.9 X 10 lb) which does not pa s tm:ough the origin 3 Although tlme·hlstory so lutions are presented C or va lu es of uo' as sma ll as 0.5, It will be but inter ec t the eli pla ceme nt axis at a value of / noted that values of 'alma ,., O" moz , '1" and '1h are not g iv en for values of uo '< 1. 5. It ca n be z\pv _o)=O.0508 foot. With this value of a, uo = 2.39.

see n from the time Ili sto ri es t l. at the cbamcteristics oC the so luti on in the later stages of the lm- g pact change as uo' becomes sma ll ; i.n p ar tic ul a r, 1.£1 increases and tu o c ur ve of 161 mo. as a Cunc- ince the so lutions of figure 11 have be en calculated only tion oC uo' appears to react. a min im um at some va lu e oC uo':;>J.5. Furtbermore, the l at~r stages oC tho so lutions gl'oatly st retch o ut in time and appear to be a lm ost asymptotic in for int eg ral value of uo', cu rv es for the for ego ing values of character. Several dlITerent analytical, num e ri ca l, and ana lo gue methods we re ap plied in all l uo wer e graphically inL e rpoiat ed by cros plotting . These a ttempt to s tndy this pbase of the problem Curth er but the extremely ~low rate of chan ge oC the variahl es in tbis region preve nt ed successlu] completion oC the so lutions. re ult were then converted to dim n ional value by multi- 2 7 46 -54---4

30 REPOR'l ' 1154 - NATIO AL ADVI ORY COMMITTEE FOR AERO AU TICS

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FI GU R E D. -C on tinued .

plying th e dim ensionles variable by th e appropriat e th e coordinate y L em originating at th e point of initial c onstant . Th e re ults obtained arc compal'ed in figure 13 c onta c L. It th ereforc follows Lhat th e upp er-mass an 1 with th e mor e exact so lution prese nted in figure 4. The lower-mass di placements dete rmin ed from th e generalized values based on linear appl'oximation I h ave been pl otted so lution for the ca e of lineal' approximaLion II mu t be exactly as determined from the generalized olution. Th e increased by a con tanL amounL equal to Z2(FVg~0)' in this 1' e lUts for linear approx i mation II , how vcr, have been ca e 0.050 foot , and all resulL rou t be displaced in displaced relative to the ol'igin of coordinates a indicated Z2(F =0) in the following di c ll ssion.

lime by u con lant in er me nL tlt = ;g ,in Lhi s ca

Th e a umption of lineal' approximation II implies that Vo Lhe system must move a di tance equal Lo Z2(Fv g =0 ) after

°8~5~ = 0.0057 econd, rdativ e to th e in Lant of initial

initial conLact ( at constanL yeloc iLy since th e wing lift is contact. The e c OlT ec Lion hav been inc orpora ted in plot- tak en eq ual to Lh e weight) before any finite gl'oundr eaction Ling th e curve for line ftr appl'oximation II bown in figure 13.

can develop. The lerivation of the equations of motion , As can be se en, the r es ult oblain ed by applicaLion of th e on the other hand , as ume that L he ground react ion in- generalizcd ol u tions, parli cularly b)' the method employing crea es lin early \\ ' iLh deflection from th e in stan t of initial lin e ar approximalion II , arc in fairly good agr ee me nt with th e c ontact. As are ulL , the equations of motion do not apply more exact oluLion. Th e di c l' epancies which exi tar e until aftrr the ystem h a attained a di plRcemrnt equal to atL ribuLable to the negl ec t of th e ho ck- truL pr eloading and Z\ F Vg=O)' which oc c urs at a time Rfter inilial conta ct pringing provided by th e air-pre ur e force, n eglect of th e lower mas s, and L v difference between tb e very simple tire Z2 (F V g-O ) In ocher words , [he equRlion of moL.ion fOl'c e- cle [l ec tion relation hips a umed anel the exact tire V Vo ch ara cteris ti c. On th e whol e, iL app e ar t ha t th e general- apply to a coordinate sy Lem transformed so that the tire ized result ofr er a mean for rapidly e stimating tb e behavior force-defl ection relationsh ip pa es through Lile origin; that of th e landing ge ar within reasonable limit s of accuracy and is , a coordinal e system di pla c rd by Z2(F V g =0) l'ela Civ e [0 may Lh ere fol' e b u eful for pr eliminary de ign purpo es.

ANALY IS OF L DI G-GEAR BEHAVIOR I.---- uo'=8 Y /"" V /

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FI GU RE] l. -Cont inu ed.

SUMM AR Y OF RESUL TS A D CO CLUSIO S c hara cteri tic of the tir e, the orifice di charge coefficie nt, and the effecti ve polytropic expo n ent for L he air-compre ion A theoretical st ud y ha been made of the bebavior of tbe pro ce ,w hi ch migh t no t be known accmate ly in pract ,ical conve ntional type of oleo-pne umati c landing gear dming t he de ign problems.

pro ce of landing impa ct . Th e basic analy is i pr esen ted In addition to the more exact treatment an inve tigation in a general form and treat the motion of the landing gear ha s also been mad e to determine the extenL 1.0 which the prior to and sub e qu en t to the beginning of ho ck- t rut basic equal.ions of mo tion can be implified and sti ll yie ld de fl ection. In the fil' t pha e of the impa ct the landing gear u eful results. Generalized so luti on of th e implified i treated a a sin O' le-d eg r e-of-freedom sy tern in order to quation obtained ar e pr e en ted for a wide range of landing- d eter min e the co ndi tions of motion at I.he in tant of ini I.ial gear and impact param eters.

ho ck-st.rut defl eetion, after which instant tb landing gear j On the basis of th foregoing sLudie th e following co n- con idered as a y tern with t wo d eg rees of freedom. Tb e clusion are indicated: e quation for the two-degree-of-freedom ystem co n id r 1. Th e behavior of th e landi ng gear a cal cu l ated from the uch factor as the hydraulic (ve lo c it y quar e) r es is tan ce of ba i. c eq u at ion of moLion \Va found Lo be in good agreement tbe orifice, th e for e du e to air c ompr e sion and interna l with ex perim enta l drop-te t data for the ca e of a ve rti cal fri ct ion in the hock trut , the nonlinear force-de fl ection landin g gear in the ab ence of drag l oad, for both a normal c hara cteri tic of the tire, tbe wing lift, the inclinal.ion of the impa ct and a severe impa ct in volving tire botLoming.

landing gear, and the e ff ect of wh cl spin-up drag load .

2. A st ud y of the effect of va riation s in the for ce -deflection The applicability of the analysis 1.0 act ual land ing gears c har acteri t i cs of Lhe ire in dicate that ha been investigated for the particu l ar ca e of a ve rLi cal landing gear in the absence of drag load by co mpar i ng a. In t. he case of a normal impa cL withouL Lire bo tt oming, calc ula ted r es ult s wi L h e ::\.']) erimen tal drop-test cI aLa for C01' 1' e- rea onable va ria tion in the force-deflection c hara cteri st ics ponding impact condition , for both a normal impact and a of Lhe tire ha ve only a relatively mall effect on Lhe calc ulat ed be ha vior of L h landing gear. Appl'oxirnaLinO' the rather evel'e impa tin oIving tiro bot t() min g.

tudi es hav e also b ee n mad e to dete rmin e the e ff ecL of C'o mplicated force-deflection cha ra cteristic of the actua l tire by simplified exponential or linear-segment var ia tion a pp ears variation in uch param eter a th e d yna mic force-d fl ection -- -- -- - - ---- - - - - - - - -- - - 32 REPOR'r 1154 - NATIO AL ADVISORY COMMI TTEE FOR AERONAUT I CS -6 I / /V

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FIGURE 11. -Co ntinu ed.

to be ad eq uate for pra ct ical purpos e. Tire hystere i was the landin g gear. Even the assumption of constant air found to be relatively unimportant. pressure in th e trut equal to the initial pr e Ul' e (n= O) b . In the case of a evere impact involving tire bottoming, yields fairly good r es ult , which may be adequate for many t he use of implified expone ntial and linear-segm ent approxi- practical pill· pO e .

mation s to the actua l tire force-deflection characteri tics 5. An inv estigation of th e xt ent to which the e quations of which negl ect the e ff ects of tire bottoming, althou o- h adequaLe motion for the lanelino- gear can be simplified and st ill yield up to the in tant of bottomin g, fail to indi cate the pro- accep table calculated r e ul ts indicat e that, for many prac - n01.ll ce d increase in landing-gear load which l' es ult from lical purpo es, the aiI· -pre ur e force in th hock trut can bottomin g of the tir e. The u e of exponential or lin ar- 1 e complete ly neglected, the tiI'e force-de fl ection relationship segme nt approximation to the ti re characteri tic which can be assumed to be lin ear , and the lower or un pnmg rna Lake into account the increa ed LifIness of th e Lire t ha t re- can be tak en equal to zero.

ult from bottoming , however , yields good r es ult . 6. Generalization of the e qu ations of motion for the 3. A st udy of the importan ce of th e discharge coefficient implifi ed system de cribed in th e pr eceding para g raph of the orifice indicates th at the magnitude of the eli c har ge hows that t he b e havior of this y tern i complete ly deter- coefficient ha a mark ed e ff ect on the calc ulat ed behav ior of mined by the magnitud e of on e param eter, nam ely, the th e landing ge ar; a decrea e in the discharge coefficie nt (0 1' dimensionles initial-velocity param eter. ,o lution of these the produ ct of the disc harg e coefficient and the net orifice generalized e quation in term of dimen ionle s va riable area) results in an approximately proportional incroa e in pe rmit compact r e pr e entat ion of the beh av ior of the sy tern the maximum upp e r-ma s acceleration.

for a wide range of l and in g-gear and impact parameter ) 4. A stu dy of the importan ce of th e air-compre sion proce which may be useful for rapieUy esti matin o- landing-g ea r in the shock st rut i ndi cate that th e ail' pl'inging i of only performance in preliminary de ign.

minor ignificance throug hou t mo t of the impa ct, and that variations in th e effective polytropic exponent n between the LANGL EY AERONA U TI CAL L AB OR AT ORY , isothermal va lu e of 1.0 and the ne ar - adiabatic value of 1. 3 N AT IONAL ADVI ORY COMMI'l'TEE FOR AERO I AUT I CS, hav e only a secondary effect on th e ca l culated behavior of L ANG L EY FIELD , VA ., May 1, 1952.

ANALYSIS OF LANDING-GEAR BEHAVIOR

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F w H E 11.- oncilided .

REPORT 1 1 5-!-c-.rATIOKAL AD "I SOR Y CO MMI TTEE FOR AERONAUTICS -3 .0 "'-- Exper imental : VVo = 8. 86 fps - 2 .5 c o ~ u '"

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=> --- Solut i on of figure 4 General i zed solutions - .5 --- - U ' = 2.57 (0 = 18 .5 XI0 ) t-- Li ne or appr oxi mat i on ]I : o --u ' = 2. 39 (0=21.3X I0 ) / 0 = 21 . 3x 1 0 Ibt lt o (0) .2 .3 A .5 .0 8 .1 2 .16 . 20 Tire deflection, z2' ft Time alter contact , s ec FIG URE 12.- Lin e ar app r oximations Lo t ire force-defiection character- (a ) T ime hi sto ry of upper-ma s acce leration.

i st ic s used in application of gene ralized olutions.

FIG li RE J3 .-Co mpa rison of generalized re s ults and more exact so lu- tion. \ "v =. 6 f eet per second ; d= O.9 .

o A.J.'<ALYSIS OF LANDI " G-GEAR BEHAVIOR .8 12 .7 10 Upper moss .6 .5 6 c Q) ~ .4 Lower moss u o a.

on o .3 2 Lower moss .2 O r--- ------~------~------ Solution of figure 4 -2 .1 Ge n era lized solutions vO' : 2.57 (0 : 1 8.5 X 10 ) VO' : 2.39 (0: 21.3 x 10 ) (b) . 04 . 08 .12 .16 .20 .08 . 12 .16 . 20 o Time after contact, sec (b) Tim e hi to ries of landing- gear di pl ac ements and velocitie .

FIG U RE 13.- ont i nued .

.6 .5 5 .4 ;:: oJ oX

e

.3 iii '2 \ i.n .2 Solution of figure 4 .1 Generolized solutions v ' : 2.57 (0 : 18 .5 x 10 ) o V ' : 2.39 (0 : 21.3 x 10 ) o (cl . 04 .16 . 20 . 08 . 12 . 16 . 20 o o . 04 Time after con tact, sec (c) Time hi to ri es of hoc k- t ru t t r oke and veloci ty.

FIG U RE 13.- Concluded.

APPEND IX A UMERr C AL I NTEG RATIO T PRO CE D R ES As pl"('\ ' iou s l~ ' notpd, 1110 t of thp s pP C' ifiC' so luti o ll prese nted tion. Ho\\'eyer , th e sam e gener al p r oe-edul"e can be u cd if in this J'('port \\'erp obtai IlPd \\'i t It a n 1I me ri cal int egra tion th e e, or other co mpli e-a tin g effects, arc included in th e pro ce (iul"( " tp rmed th e " linear pr oce dure ," whi ch as L1mc' s equatio n s.

c han ges in the yariab lp to be lin ea r oYerfin.ite tim l' int el'yals . For the e-ase under cons ider ation til equation of motion ,Yith thi" proced lll' e a tim e interyal f= O.OOJ econd wa s (eq . (16 ), (17 ), a nd ( )) ca n he writt en a follo\\' used in ord er to obta in th e de ired accllracyfor the particular cases co n ieiNed. A fe\\' of the s pecific o luti ons presented WI z l+ A(zl-z 2 )2 + B[l-0( z l -z2 )]- n+ D = 0 (AI) g \\'el'e o btain ed b y menns of a procpdurp, termed the "q uad- rati c procl'dul'l' ," whi ch a s ume a quadrati c \-al' iation of Jl'2 z2 -A (z l-z 2)2 -B[l-0( z l- z z)]- "+ F ( Z2) - 11 '2= 0 clisplacempnt \\ ' ith tim e for s Lic ce SlY(' interval. Thi pro- v g g ce ciure, a lth o ugh r c'q uirin g somc' \\ 'h at more comput in g time ( A2 ) prr int r n 'al, may prl'mit an increa e in the int rrya l i ze J or TVI .. + H '2;.:. + F ( )+ E - O z \ ~2 ' v Z2 1- (A3) a giYr n Hrcuracy, in om c' ca es allowing a l'e(\ucti o nin the g g g total ro mputin g timC' requir ed . In the case of the more \\'lI('1' e , A 3 exact pquatio ns of motion the acc ur acy of the CJuadratic pro- A =~-- P h ced urr wit lt a timc' intpryal of 0.002 eco nd appears to be

I ·~ I 2( C cl A n)2 cos <p

equa l to tltal of thC' lin ea r procrdure with an intetTal of 0.001 scco nd . Although tlte accuracy naturall y ckC1'ca C'S with increasing intl'rnll si ze , the 10 s in accuracy for pr o- portionatp inc reasc's ill intC'ITnl i zp appc'ars to 1 )(, smaller for 0 = ~ 1' 0 C OSIO th e quadratic than 1'0 1' the lin car procedure. In thp case of the simplifi ('( l C'ftllatio ns of motion rpasonably s,lti ",letoI'Y

D = Kl.. Tr - n'l

resu lt s \\ ' Pl"(' obtai ned in test comp ut ations \\ -itlt thp quadratic E = 1I' (K -l ) pl'ocC'ciul'e for int pl'Ya ls as la r gp as 0.01 scco nci , \\ ' I1('l' c'as til(' L linea l' pr oc('(1 LII' C' \\ 'a co n si ci c'l'c'cl quC's ti ona bl p for int c l'Y a ls

Soh -ing e quation ( A3 ) for z\ gi, ' cs

lal'grr titan 0.002 ('(·ond.

Th e ge npra li z('(l sol ut ions jH'psC'ntN[ , bccause of thc ZI=[F- G zo- IIF (z .. ) ].l (A4) v l'elati\'('ly s im plc f01'111 of til<' <'qllnt ions of mot ion , \\' r l' P g - g - g obta in c'd \\ 'itll the \\, p ll-kl1 o \\ 'n R u ngc'-Kutta pro('cdure .

A tu d.\ · of tI l(' a ll O \\ ' nble' int clTa l si zp l' Psl :lt ecl in tll(' l: sr of whcre all int('l'\'nl ~ 8 = 0 . 08 , \\ ' hidl ('o lT Pspon ds to a tinl(' int('l'nJl of al)01lt 0.00,'5 s (' c 'onel for t 1 1(' I and in~ gl'fll' under ('onsilil'rn t ion.

l.IN EA R PRO CE D U RE In this s tep-b.,·- t c'p procl'clul'e the yariation s in elis- placel1H' 1l t , \-('loci t~ ', and accele'l'a t ion a re ass11 med to be lincar O\ ' CI' each finite time int e lT al~. Th e l1wthod, as used , il woln 's onp stage of iteration. Line'ar pxtrapol ation of the Vl'loe it~ - at th c' end of an ., - intc'l'val i lIsc'cl to obta in est im atC'C [ \'nlues of \'('locit,\' and di s placement [01' lll(' 11 rx[ IntpgJ'ating cquatio n (A4 ) \\ 'i ll! l'C' pect to t b('(\\'een the interval. Th ese n1l1lPs are then u eel to calcllit1tp \' alues o[ limits l, and t and n ot ing that ZIT =zz,=z , g in s the accple'nlt ion in Hccorel ancl' \\'ith tl H' equa t ions of motio n .

In tegrnt ion of the accekrn t ion pro \' ielcs improH'el \-al ue of the w l ocity and , if de ired , the eli pla('emenL and accelcrn- tion. 1n thi pr o('ec!l11 ' c' a ll intc'grations Sf'(' j Wl'fo nn ecl b,' - where T= ( t-t, ).

app l ica t ion of the t rnpc'zoiclal rule'.

Integrating aO'ain and notinO' that ZI ,=ZZ,=z , g iy('s The following clel'iYation illustrates the app li cation o[ thc linear p1'oe('([u1'e to the equatio ns o[ motion [or the landin g gear, ",hidl app ly lib que nt to the beginning of sh oek-s tnl L ZI= (l + G)(Z , +Z ,T )+ ! I; l _ ( T ( ' F\ 'g(zz) dT dT G zz - lJ

- JoJo

d e fle ct ion at timc t In th e pxample p1'e cnt ecl intern al r .

friC'lion fol'c'cs arc Iw gk ctc'd in ol'clc'r to implify thp clcriya- (A6) ANALYSIS OF L Al DI NG - GEAR BEHAVIOR ub titu t inO' for ZI and ZI in eq ua tion ( A2 ) gives (A7) Th e motion of th e l anding gear ub equ ent to Lhe beginninO' appropriate inL egro li frerential equat ion for Lb e sys t em , of sho ck-s tru t deflecti on is determined by m eans of a ste p- e qu at ion ( A7 ) in the pr e en t case. Thu by- tep olution of equaLion ( A7 ). Tili num erical proce dur e (A1 3) yields t im hi sto rie of th e lowe r-ma ss mo tion va riabl es

22, zz, and i from which the mo L ion variab les for th e upp er

2, In equaLion (A7) th e in teg ra l ex pr essions can also be eva lu- mas ZI, ZI , and 21 can be ca lcul ated by means of eq uation s ate d by app Li cation of the trapezoidal rul e. For exa mpl e, (A4 ), ( AS ), and ( A6 ).

when FVg( z2)= m 22T , Th e initial c onditions for the step -b)T -s tep proce dur are (A ) ('\'14) a nd wh er e ZT, iT, and i T ar e th e co nditions of motion at th e b eg innin g of shock- trut de S. eet ion as determin ed from th e solu tion for th e one-degree-of-free dom syste m.

E timatecl va lu e of th e lower-rna veloc i t)- at th e en d of Hi e fir t time in cr em ent e following the b eg innin g of h ock- (AI S) st rut deflection can be obta in ed from the e"-'Pre si on An impro ved value for th e velocil)- is obta illl'd from th e ( A9 ) ex pr ess ion ( A1 6) or , as a fir t app roxima tion, Thi va lu e is u ed in the calcul at ion of the e Li m aLed velocity i: n+ 1 and displacement zi n+ 1 for the next interval.

Th e co rr e poneling displacement is given b)- If des ir ed, improved va lu es of th eli pla cemen t and acceleration for the nt h interval ub seq u ent to th e beginning of sho c k- st ru t de fl ect i on ca n be obta in ed as follow : ( AlO ) e ( . .)

Z = 2 - z? z?

211 2,,- 1+2 - n- I+ - l1 After th e ini tial co ndition s and L h e co ndition s at the e nd of L h e first time in cremen t are e tabE h ed, a tep -by- ste p 2 Z2 (AI7) =Z2n_ l+ ez Zn_ I+: ( Z2"_ I+ ,,) calculation of th e motion can be obtained by routine opera- tions as indica tcd by the following gener al procedure whi ch and (A I 8)

applie at any tim e T= n e aft er Lhe begiml in g of the proce . z·, = I ( Z2 z? T)

"n' n' - 11' 11 Th e operation indicaled ar e based on in t egrat ion b) T appli- wh ere '/( 22 ,22 , T,,) is a n a ppropri ate eq ua ti on for the s)' stem , cat ion of the trapezoid al rule: 11 11 uch as eq ua tion ( A7 ) .

W ith tbe values of 22 , Z2n ' and zz", the moLion va ri ables for

.", . + (. .). + e ( .. + .. )

z; = Z Z o - Z = Z - Z z? ( All)

- n 2 _ - n- I 2n_2 2n _ 2 2n _ 1 - 11-2 n 1 1 th e upp er rna s i l ", ZI ", and 21" ca n be calc ulat ed e para tely h om e qu at ion (A4 ), (AS ), and ( A6 ), a previou 1 :\' noted.

?

In sett in g up tlw num eri cal procedure u cd in obtaining the Z :n=Z2n_ I+~ ( Z2 t-Z:n)=Z211_ I+ eZ2 n_ l+ :- ( Z2n_ l +z~n_2) n_ 1 olutions pre ente d in this r epor t, an eva lu at ion of the e rror intro duce d by the pro ce dur e ind i cated th at it wo uld no t be (A12) nece sa !" y to calcul ate the improved va lu es of th e displace- Wi th the e timaLecl va lu es z: n and 2i n Lhe acceleration of ment 22" (eq. (AI 7) or the acceleration ZZn (eq . (AI ». How- the low ~r ma s can be determined by su b st itu tion in th e ever , improved va lu e of th e vcloc it:l- zZn were ca lcul ated by -- - --- -- -- - ---- ----- - - - - ---- - --- - -- -- - -- - --- 3 REPORT ] 15-l- NATlONAL AD YISORY CO MMIT TEE FOR AERONAUT I CS means of equa tion ( A16 ) f or th e purpo se of determ inin g motion variable from one in ter va l to Lhe n ext. Wi th thi s estimated yalue of the velocity ~ Z2 and the di splacement z~ a s umption the eli placf'ment va ri at ion over two uccessive (e qs. ( Al l) and (A12 )) for th e inc reme nt i mm e di a tcl~ T eq ual time int e rval s is co mpl ete ly ~ de termined by th e t hr ee fo ll owing. y alu es of di s pla ce ment at tbe beginning an d e od of each of In ord er to illustrate the app li cat ion of th e m et hod , a the two inL e rval . By wriLing t he q ua clr atic var i at ion in tabular com putin g pro ce dur e fo1' tb e solu tion of th e sy (e 111 difl'e1' en ce form , t he ve lo ci ty and acceleraLion at th e midpoin t represented b yequa ti on ( AI ), ( A2 ), a nd ( A3 ) is presented in of the doubl e interval ca n be ex pr e cd in terms of the three tab le 1. displ acement va lu es pr ev iou ly ~ m ent ioned. I llb L i tuting for Q ADRATI C PRO C EDURE the ve lo cit y ~ and acceler at i on in the differential equat ions for In th is s t ep -by- s tep proced ur e a q u acl r atic va ri at ion of th e ystem y-iel cls differen ce equat ion of motion in term of eli placem e nt is a sllm ed OYer uC'c'e s in equa l finite tim e successive d i placeme nt value "'h ich can be eva lu ated jnte n ~a l s for the purpose of extrapo l at ing Yalu('s o f t he int er va l by intel'val.

TABLE I LINEAR PROCED1.:RE Row Quantit., · Equation Procedur e t @ T

~--------------- I ----------~----- - -------~-~-

I

Z2 Z2 . _I+ ~ (Z2 ._ 2+ ._I) 0 " CD

1-

D eter mined from tirc forec- dcf i eet ion cha ract e ri s tic.

--------- Equation ( A14 ) Equation (.- \ 15) Eq ualion ( A 7) Ci" cn b~ ' e quation ( A 7 ).

CD g ® ® @ Eqllati on ( A'\ ) C:i\'cn by cq uation (. \ ·1 ) .

C:i\'(' n by c quation ( .\ ,5 ) .

@ Equati o n ( A5) @ Equati o n ( .-\6) C:iv en b," e quation (. \ 6) , t 0 . clcn o t c~ "alu c for pr c "ious tim c int cn ·al.

-- -- - ~ - ----- ------------ ------------- - - A JA LYSIS OF LANDI G- GEAR BEHAVIOR The following dcrivation shows how the procedure can be applied to the determination of the behavior of the landing gear ub equ ent to the beginning of hock- trut deflection at time tr. In order to simplify the derivation, int erna l frictioD forces are again neglected in ett ing-up th e eq uation of motion.

Th e assumption of a quadratic variation of displacement with time (constant accel erat ion) over two succes ive inter- z Zn+l va ls, each of dm-ation f , permits ex-pressing th e velocity and acceleration at th e midpoints of the double interva l (sec ketch ) in terms of the displacemen t va lu es at the beginning, midpoint , and end of the double int erval by the eq ua tions (sce ref. 5, p. 16 ): Z"+I - Z"_l in (A I9 ) 2f I I and (n + l).

0 2. (n - il . n.

Zn+ I-2z +z n- 1 r =t-tT n (A20) Zn= f2 where zn, zn, and Zn are the velocity, acceleration, and displaceme nt at the e nd of the nth int e rval (r =n f ) after the beginning of shock- strut deformation and Zl1-1 and Zn+l are th e displacements at the end of io tel'vals n-l and n+ 1, respectively.

ub stituting the difference relations for iI, zz, iI, and 22 into equations (Al) and (A3) permits writing the equation of motion for the landing gear in difference form as follows: (A21) and Z

Zln+l = 2 ZIn - Zln_l - G( zZn+l- 2 Z2 11 + Z2 _I)- I]f [ F vc ( zz .. ) +E ] ( A22)

n where the constant are as defined in the previous ection.

u b tituting for ZI +, in eq ua t ion (A21) O"ive n (A23) ZZn+l= .8 t'+ gA\lj/Z [ 2WtTifT z -,14 W1 Z W z(g A W.8 n+l+ W I W z )-g A l VZ(4 WI 2an+l+gA'Yn+I)]

n

where a,,+l = 2 liT z Z2n - 1 lj/2z21l_I-gfZ[ F v g( Z2,, ) + E] Z

.8 n+1 = 2 1r +( W,- W Z )Z2 _1 + 2WI( ZI" - Zln_l ) _gf [ FvcC Z2n) +E ]

zz 2n n and Equations ( A.22 ) and (A2 3) are e ent ially extrapolation formulas which permit the dcterm.ination of values for Lhe upp cr-mass and lower-rna displa ce ment s to co mc from the va ln es of displacemcnt all'eady cal ulated. Th e e equat.ion thus pe rmit tep -l y- sLep calcul ation of th e displa cc ment a thc impa ct progresses, sta r ting with tbe initial co ndition , from which thc upp er-rna and lower-mass velocities and acceleration an bc detc rmin ed by means of equations (A19 ) and (A20 ).

, ince the calculation of thc displaccment ZI and Zz at. any in tant by mcans of cquation (A22 ) and (A23) requires va lu e for Lhe displa ce mcnL s at t\\ ~ O pr cvious instant s, the routin c application of the c eq uation s can begin only at thc c nd of thc ec ond interval (7 =2 f) fo ll ow ing the bcginni ng of sho ck- trut deflection. Beforc thc di pla cemc nt s at thc cnd of the sccond int c rval can bc ca lcul ate d, howevcr , it i nc cessary to dete rmin c the displacement at thc end of the fir L int c rv a l. Th e c value' can bc obtained from the condition of motion at the in tant of initial sho ck- st ru t deflcction by applying equations ( AI9 ) and (A20 ) to thc in st ant t=tr.

REPORT 11 54 - NATIONAL A DVISORY C OMMI TTEE FOR AERONAUTICS the foregoing a ppl i cat ion of the difference eq uation res ult s .\ 1 lh e in stant of in iti al bock-s lrut deflection in id en l i ca l va lu es for L he upp er-rna eli placement and lowo r-m ass di splacem ent at the end of tbe fir t int er va l. Z, =Z? =Z T ( n=O · ,, = 0 Simultaneous so lution of equatio ns (A25) gi ve Lhe fo11owino- , , , ( A24 ) expl'es ion for lh e displacement at th end of the fiJ' t int erva l: z ' n= O= Z2,,=0= ZT , ,> Z =29 =2 1 = 0 - 11=0 'T (A26) 11

Z2"= I =Z ' '' = I =Z T + ~ Z T+ ~ ZT

Applicalion of the difl'erence equat i on ( A19 ) a nd ( A20 ) ' With the ya lu es for ZT and Z,,= I , eq nation (A22 ) an d lo th e in s tant l= lT (th aL i , n= O) g l YCS th e fo llo ,, - in g ( An ) p erm iL the tep-by- s tep ca lc ulation of Lhe uppe r-ma ss e quation: and lo\\- cr-ma s di s pla ce ment ub eq u enlto lh cfi r l interva l following th e beg innin g of hod ;: - [rut deflection, Th e co rr c poneling vclocitie and acceleration s of Lhc upp er and (A25) lowcr ma sses ca n bc dete rmin ed from lh e c alculated eli place- m c nt s b:v mean of eq u al ions ( 1'1..19 ) and ( 1'1..20 ), as prcv iousl)T no ted, A Lab ul a.l' c omputing proc e dw' e illu st r ati n o- Lho application Since the land ing gea r i co n id e l' ecl as a one -el eg l'e e-of- o f the method is pre se nLed in Lable II.

f J' cedom sy tern from ini tial con t act up to th e ins tan t 1= IT, TABLE II Ql AD TIATJ C P I WCED RE Qu ant it y Equation Pro ce dur e t ____ __ 1 r @ -------- -- --- --- ---------- \-- -- ---------------- --------- - Gi, 'en by equation ( A23 ) , Eqllat ion ( , \2 3) Equati on (, \22 ) Gi, 'en b,Y c quation ( A22 ) , 0 - (i) .

Z2,1 + 1- Z2 _ n 1 - 2 - ~ - 2 ~

Z2 + - 2 Z2n + Z1 _

n 1 n 1 ® ~ 2 0 - 20 +0.

® ~ 2 t o. dcnoL cs \'alu e for pI' ,' ious lim e int c l'\ 'a l.

ANALYS IS OF LANDI G- GEAR BEHAVIOR R GE-KUTTA PRO CE O HE a, nd (A31 ) gives In tills tep -b y- tep pro ce dur e th e differences in L he de- p r nd e nL va riab les over any giv n int e rval of the indep e nd e nt vuriable are alc ulated from a definite set of f'J rmul a , t he sa m e et of formula s being used for all in cr ement . Thu th e (A32) vu lu es of the variab les aL th e end of any gi ve n in terva l arc co mpl et ely d eter min ed by t h e va lu e at the end of L h e pr e- ce ding int e rval. Unfortunately, h owever, unle s th e e qua- tio n to bc int eg ra ted arc relativel y s impl e, the m e thod ca n wh er e be c'J m e quit e le ngth y .

Th e followin g de riv ation illu st ra tes the app li cat ion of th e Ru nge -Ku tta m et hod to the generalized equatio ns of motion (e q . ( 21 )) for th e implified yste m co n ide red in Lhe sect ion on generaliz rd results. in ce these eq uation s can b e r ea dily ]'rdll ce d to the fir st order, th ey ca n b e integrat e 1 by tho

k 3= ( Wn _ t+~ ) t,O

tep-l y- top app li cat ion of th e genera l equati011 gi ven on p ages 3 01 and 302 of r eference 6 for fir t-order imultancou s

k = (w -t + l3)6O

4 n differ en tial e quation .

Th e generalizcd e qu at ions for Lhe implified 8ystcm pr e- v iously discus od (e qs. (21)) arc Ina mu ch !is any tw of th e e equal ion arc ufftcien t to describ e t he be h avior of t he y tern, only lh e la t two eq ua- tions ar c e mplo ye d in t hi s pro ce dur e. Th e e equ at ions ca n be l"CC lu ced to ~ fir t-O l'der ys lem by introdu cin g th e new variab le ( A27 ) w = u, ' o t h at w ' = Ut" (A2 ) and the eq ua tioll of mo tion b ecome

(W-U2 ' )2 _U 2= O}

(A29) W' +U2= 0 olvin O" equaLi ns (A29) for U2' and w', respecLively, g iv e 7n = [ (W n _l + l3 )- u2 n_t + m 3] /::"0 (A3 0) Wi th t hi s pro ce dure , Ut, W , fLnd Uz can be calcula te d in lep-by-slep fa hion from the yalue f or the preceding in ter- w' = -U z (A3 1) va l, the procedure beginning with the initia l condition.

From the e va lu e, Ut', u/', and U2' can be ca lc ulat ed by pply ing t h e ge n era l procedure presented in lh e r eferen e p)'evioll , ly ci te d to the imllitaneous eq llation (A27 ), (A3 0), mean of equ aL ion (A27), (A2 ), and (A30), respectively.

APPENDIX B SOUR CE OF EXPERIMENTAL DATA I STR ME TATION Follo,,-ing i a brief de cr ip tion of th e apparatus and t est specimen used in obLaining the experime ntal data pr e enLed A variety of time-hi story in st rum e nta t i on was u cd during th e tests. Th e ve rti ca l accele ra tion of t he upp er ma ss wa in thi report_ EQU I PMENT m eas ur ed by mean s of an oil- damp ed elec trical st rain -gage accelerometer having a range of ± g and a natural fre qu en cy Th e basi c pi ece of equ ipm nt employed in the te ts is th e of 5 cycles p er second. A low-frequen cy (16.5 cycles p er carriage of th e Langley imp act basin (ref. 7) whi ch provides econd ) I A A air-damped opti ca l-reco rding accele rom eter , m eans for e ff ect in g the con tro ll ed descen t of th e te t pcci- h aving a r ange of - l g to 6g, was u ed as a tand-b y in st ru- men _ In t hese tesls Lhe impa ct-basin c arria ge was u sed in me nt and a a ch eck against the Lrain-gage accele rom eter.

mu ch the am e mannrl' as a co nventional stationary landin O' - Anoth er oil-damped train -gage acceler ometer, having a gear Lest jig (see l' o f. ). In order to imulat e m ec ha ni ca lly

rang e of ± 12g and a natura l frequ en cy of 260 cycle p e l'

th e win g lift forces whi ch u sta in an airplane during landin g second, wa used to dete rmin e t he vertical accele ration of the th e pn e umati c c.dincler and cam s .\ -ste m in co rporal ed in lh e lower m ass. Th e vert ical di pla ce ment of th e lower m ass ca rria ge wa s use dlo app ly a con tant lift for ce to lhe dropping (tiro defl ection) and the shock-s trut troke were m ea ur ed ma ss and l anding gear during impact. Th c lif t force in th e e separate ly b:v me an of variab le-resi tan e lid e-wire l)ot en - le ts wa s e qual to the total dropping we ight of 2,542 pound s.

tiom eter . rrhe vertical di pla cem e nt of th e upper ma ss was TEST s PE c r ME N dete rmin e 1 by adcl ition of the s trut- st rok e a nd tire-de fl ection m ea urem e nL s. Th e ve rti cal velocity of th e landing gear at Th e landinO' gcar uscd in the te l was originall .Y designed th e in tant of O' l"ound contact \\ -as determin ed from th e output for a mall milit a ry traini ng airplane having a g ro s weight of of an elemental ele ct rom agnetic voltage generator. A time a pproximately 5,000 pound . Th e gear i of eonnntional history of th e vertical velocit.'- of the uppe r ma s wa ob- can tileyer constru ction and incorporate a sta ndard typ e of tained b. ,- m ec h a ni cally integ r at ing th o vert ical acce leration oleo-pneumatic ho ck s trut. Th e wh eel i filted with a 27- inch type I (smoot h- eon tour ) lir e, inflated to :32 pound per of th e upp er ma s ub e qu en t to the iostan!' of ground con- quare inch. Th e weigh t of th e landing gear i 150 pound . ta cl. El ec Lri cal clir1"el" e nti ati on of the cu rr e nt oULput of the Th e weigh t of t he 10ll'er mas s (un prun g weigh t) is 131 truL-s trok e circui t provid ed time -h i tory m ea urement of the shock- t ru t lelescoping velocity. Th e in tant of pound s.

ground con t ac L wa de!'ermined by mean of a micro- In th e pres e nt invc st io'ati on L he gear was some,,-h at modi- wit ch , rece cd into th e grou nd pl atform, whi ch closed fied in th at th e metcring pin \\ -a removed a nd th e or iginal a circ uit as long as th e tire \V a in con ta ot with th e pla tform .

orifice pl ate wa s re pla ced with one h avi ng a smaller or ifl ce Th e elect ri cal output of the in trument s wa s recorded on a diameter. Fi g ur e 14 lt O \\ ' th e int e rnal arrangement of thc 14 -cha nn cI 0 c il1o grap h . Th e ga lv anometer wore damp ed hock t ru t and pr ese nt detail of th e orifice. Oth er pelti- to approximately 0.7 critical dampin g and had natural fre- n c nt dim en ion are pre se nt ed in table III. Th e truL " 'as qu encies high enough to produce virtually un if ol'm r esponse fillcd with specification AX - YV- O -366 B h.\ -d raulie fl uid.

up to fr equenci es co mm ensu r ate ,,- ith those of th e m ea uring Th e infl ation pr e sure with th e s trut full.,- extended ,,-as 4:3.5 ins trum e nt ation. A ( ypica l oscillogra ph reco rd i 110W11 in pound p er quare ineil . In these tests t he landing gear wa s mount ed with thc sh ock-strut ax i vert ica l. Figure 15 figure 16 .

o It i believed th at th e measureme nts obtaine d in th e test is a photo g raph of th e la ndin g gear in sta lled for test in ·.

are accuraLe within th e fo ll owing limits: TABL E III Mea sure ment Accu ra cy nfP OllT. \. KT CII.\.ll.\ C'TEHI TICS OF L.-\.XDIXC CE AR Uppcr -ma ss accclc r ation, g_ _________________ ____________ ± 0 .2 l: SED I K TE STS Fo rcc 011 upper ma"s, Ib _________________ _____ _________ ± 500 Low c r-ma ss accclcralion, g____________________ __ ____ ___ ± 0.3 O. 05761 .!la, sq ft Vertical " clocily at gr ound contact, fps_____ _ ± O . l O. O~708

A ,,, sq ft

_ O. 00055 5 Uppcr -m ass "('locity during impact, fps _ ± 0 .5 I l o, sq fl O. 0 35-1-5 VO, Cll ft __ Upper-mass displaccment, fL __________ ___ _ _ _______ _ ± 0.05 6, 20-+ Poo, Ib/sq ft Lowcr-ma . " disp la ccmcnt, fL ______________ _______ _ ± 0. 03 O. 552 1 I" ft Shock-st r ut st rokc, fL ____________________ _________ ± 0.03 2. 2260-+ 1 ft 2, Shock- s trut, tc lcscop in f!; "('l oc ity , fps_____ __ ___ _______ ± 0 .5 2, "I J 1 11' " Ib __ T imc aflcr co nta ct, CC ________________________________ ± 0. 003 J 31 W , lb At ALYSIS OF LA NDI G-GEAR BEHAVIOR

_----0

~_ v_ - '_' - ---------@

CD Air valve

Lack screw Branze beari ng Cannecting hole Outer annular chamber @- ----- Pistan suppart i ng tube 0) ---- Fitti ng assembly flange @ ------ Outer cylinder

®

Inner chamber

®

®-----

Spacer @ Upper packi ng - ri ng spacer @

0 ------

Packi ng rings @ @- ----- 13 Lower pocking-ring spacer (J) ---- Bearing nut Wiper ring

!

Pistan @------ Orifice plate Lo wer chamber @------ - 19 Inner cyl inder @ End plate @- ---- l Yoke collar @ ------ .; 22 Yoke @ ----- 23 Filler pl ug

i

@ -----

@- -----

~ ---------~~--i ,kI" , '_<, ~~: '-'- / -"'-"~""'-"l.: ~ ------- @

~ -------- --'

~ ~~~~~~~~

@)---------

0 : --- @

@- --- -- i-

@---- : I I 0 : I .250 -' \- .2 50 R 1-0- ------- 2 . 936 ------ - --{ Orifice deta il s (Di men Sions in Inc he s) F roUlm 14, - hoc k t ru t of la n din g gea r teste d at L ang l ey im pact ba s i n, REPOR T 11 54- rA TIONAL ADV ISOR Y COM MI'l v!'EE FOR AEROI AUT I CS REFEREN CES 1. Mc Ph erso n, Alb r t E., Evans, J. , Jr. , and Levy, amuel: Influ ence of Wing Flexibilit y on Foree -Ti me R e la t ion in Shock Strut Folio\\"- ing Ve rtical Landing I mpact. NACA TN 1995, 1949.

2. towell, E lbr idge Z., Houbolt , John C., and Batdo rf, . B.: An Evaluation of Some Approximat e Methods of Computing Lan ling Stresses in Aircraft. N ACA TN 1584, 1948.

3. Wa ll , Jam e H .: Inv e. t i gat ion of the Air-Compr ession Pr ocess During Drop T e t of an Oleo -Pneumatic Lan li ng Gear. NACA TN 2477, 19 -1.

4. Hur ty, Wal ter C.: A t udy of the Re pon e of an Airplane Landing Gear Us in g the Differ ent ial Analy ze r. Jou !" . Aero. 'c i. , vol. 17 , no. 12, Dec. 19 50, pp. 756-764.

outhwe ll , R. V.: R el axation M ethod in Th eo r et ical Phy . ic s.

5.

Th e Clar endo n P ress (Ox ford ), 1946.

6. carborough , Jame s B.: Numerical Mathematical Analysi. econd ed., T he J ohns Hopkin s P ress ( Baltimore ), 1950.

7. Batt er on , idn ey A.: T he ACA Impact Basin and ,r ate r Landing T ests of a Float Mode l at Various Velocitie and ,, - ight. ~ A CA R ep. 795, 1944. (up rsedes JACA ACR L4H15. ) 8. i\Iilwitzky , Benj amin, and Lindquist, Dean C. : Evaluation of the R e duc e d-Ma ss l\Iethod of R epre enting W ing-Lift Effects in Fr e- Fa ll Drop T e ts of Landing Gear. NACA T X 2400, 1951.

BIBLIOGRAPHY Callerio, Pi et ro: Th e Shock-Absorbing Sy tem of the Airplane Land- ing Gear. JACA TM 93 , 1940.

Ko chano\\" ky, W.: Landing and Taxying hock With Oleo Leg Underca r riage. British Min i t ry of Supply, TPR 3/ TIB 2 Tran - lation No. GDC 10/ 5250 '1', Nov . 1944.

::ltowell, Elb ri dge Z., Houbol t, John C. , and Batdorf, . B.: An Eval- uat ion of Some Approximat e l\ I et hod of Computing Landing Stresse in Airc raft . NACA TN 1- 84, 194 .

Schlaefke, K.: Zur I\: enntn i der I\:raft\\"egdiagramm ,·on Flug- zeugfeder beinen. (On Force -D e fl ection Diagram of Airpl ane pring St ru ts.)

1. T cilbericht: Vergle ich von Diagramm en mit lincar er und CJua- dr ati s cher Dampfung. ( Par ti al R ep. No . 1: Compari on of Diagram W ith Linear and Quad.ratic Damping. ) T ech.

Berichte, Bd. 11 , I-l e ft 2, 1944, pp. 51-53. ( Tran s lation ava il able from CADO, \"\ ' right-Patterson Air Force Base, a AT! 27004. ) 2. T e il bericht: aherung s,·e rfahr en zum Berechnen der J\: raft- wegd iagr ammc mit nichtlinearer Federkellnlinie und line- arer oder quadrati s cher Dampfung. (P artial Rep. No.2: Approximation l\ I ethod for the' Calculation of Force-Deflec- tion Diagram \\' i lh a Kon-L in ear 'I ring Chart and Lin ear or Quadratic Dampin g.) T ech. Bc richle , Bd. 11 , I-l eft 4, 1944, pp . 105-109. ( Translation available from CADO, Wright-Patt e rson Air For ce Base , as ATI 27031.)

3. Teilbericht: Der Land estoss von Olluflfederbeinen. ( The Landi ng Impact of Air-Oil Shock Ab so rb e r. ) T ech.

FrcURE I5 . -Vie,,· of landing gea r and instrumentation. Berichte , Bel. 11 , HefL 5, l\fay 15, 1944, pp. 137-14l.

ANALYSIS OF LA.t'<DING-GEAR BEHAVIOR __ 0 ·

-~- '- I' ' "- -r- -' '1 --1-~ --

f' , I 1 I' Initial vert i cal velocity '\ ~"1'- # ' '"'+ ' . ........ I ~ ....... -+-_~ i . ~;'; ' . " : "" : -+ _""- ""t' - ., ,:': . l , I U pper- mass displaceme nt " I

,i I i I; '

l"IG Rl'; 16.-Typical o. cillograph record obta in ed dur in g te~t in Langley impact basin.

chlaefk e, E.: Zur I\: en nlni s der \\ 'e chse lwirku ngen zly ischen Fede r bein i\Iakov ski , S. A.: A i\ I et hod of Shock Abso rber Perfo rmanc e Pr ed i ct ion und Re if en beim Land e -toss Yon Fl u gzeugfah nl ' rk en . (On R ecip- ,\ ' ith a Note on It s Application Lo Tricycl e Ae ropl anes . T:\f ::\0.

ro c al Effects B et lY een Landing Gear and Tire in Landing Impact of . :'IL K 1 3, British R.A .K, Sept. 1943.

Airplan e Undercarriagcs.) T ech. Be ri chLe, Bd . 10, H eft 11 , Nov.

Frankland , J. i\I.: Gr ound Loads in Carri er Aircraft. R ep. Xo. 7992, 15 , 1943, pp. 363-367.

Chanc e Vought Aircraft, Di v. of Un i te d Airc raft Corp . ( Dalla s, i\[arquarcl, E ., and i\Ieycr zur Capcl\en, 'Y.: Na he rungs ll' eise Berec h- Tex. ), Oct. 20, 1949.

nung dcr zw i sc h en Fahrg este ll und Ru ml f beim Land en auftr e- R ead ey, W. B ., and LaFa vor , S. A.: An Analyti cal MeLhod f or Lhe tenden F ede rkraft e. ( Approximat e Cal culation of t he SJ10ck-Strut De s ign of M ete ring Pin . and Pr edict ion of Load 'L rok c Hi slo ry of Forc es Occ urring Betw ee n Landing G ea r and Fu selage in Landing. ) Landin g Gear . R ep. ~o. 1688, :'II cD on ne li Aircr aft Corp ., 1\Iay 12, FB Kr. 1737, D ut sche Luftfahr t forschung (Berli n- Adlershof) , 1942.

19-0 .

Marquard , K , and Mey r z ur Cap e ll en, , V .: Sahe ru ngs lI 'ei e Berec h- Be rr y, F. R. , a nd Frick , R . P .: Th eo r et ical D evel opment of Load nLlng del' zll'ischen Fahrge te ll und Rumpf b('im Land e n auft re- Factor V s. Tim e Curv es fo r Lhe DC -6 Main Gear. R ep. No. 21546, te nd en F ecler ung skrilfte. ( Appr ox im ate Cal culation of the hock- Dougl as Aircraft Co. , In c., Sept. 1, 1950.

St ru t For ces Occurring B et wee n La ndin g Gear and Fu se lag e in Hur ty , \\ T altcr C.: A Ludy of the R esponse of an Airplan e L anding L and ing.) FB N r. 1737/ 2, D e ut sche Luftfahrtfor sch ung (Berli n- Gea r sing th e Differe nt ial Anal.vzer. Jour. Aero. ci., voL 17, no.

Adler hof), 1943.

12, D ec. 19 50, pp. 756-764.

Yo r giad i , Al exan der J .: Gr aph i ca l Anal ysis of P erfo rm ance of H y- i \Ia sa ki , :'II amo ru , milg, Bcn, and M oore, C. I \: .: The Pr e diction of draulic Shock Ab so rb er in Air craft Landing Gears. J our. Aero .

ci ., vo l. 12, no. 4, Oct. 194 5, pp. 421 -4 28. Ver tical Two- Wheel L anding J Joad. 1\IR ~o. T SEAC5-4595- 2-1O, T e mple, G.: Pr e diction of Undercarr i age R e ac tion . R . & i\I. No. Air :'II aLe ri el Com.man d, Eng. Dil '., U . S. A il' Force, :'I1a y 28, 1946.

1927, Briti sh A .R. C., e pt . 194 4.

Fl i:t gge, W.: La nding-Gea r Impa ct . NACA T N 2743, 1952.

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