Landing-gear impact

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NATIONALADVISORYCOMMIITEE

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FOR AERONAUTICS

TECHNICAL NOTE 2743 MNDING-GEAR IMPACT By W. Fliigge Stanford University Washi&ton October 1952 TECHLIBRARY KAFB. NM

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lC anb57a7 NATIONAL ADVIS03Y COMMITTEE FOR AERONAUTICS .

TECHNICAL NOTE 2743 MNDINGGEAR IMPACT By W. Fl&ge SUMMARY This report deals with the impact forces in landing gears. Both the landing impact and the taxying impact have been considered, but drag forces have been so far excluded. The differential equations are devel- oped and their numerical integration is shown, considering the nonlinear properties of the oleo shock strut.

A way is shown how the dimensions of the metering pin may be determined from a given load-time diagram.

A review of German literature on landing-gear impact is also presented.

INTRODUCTION The objective of this report is to study the impact forces acting on the wheels and shock struts of an airplane. For practical reasons .

the investigation has been limited to the vertical forces and does,not consider the effect of the drag load which acts on the wheels during .

the spin-up time. Within the limits drawn by this restriction, an attempt has been made to develop a method for numerical computations which, it is hoped, will be useful in practical design work.

The oleo-pneumatic shock strut which is now in general use snd which has attained a high degree.of perfection exhibits-a rather com- plicated relation between the force, the stroke, and the rate of stroke.

For practical work, it is imperative to express this relation in mathe- matical form and to develop a method for the numerical solution of the ensuing differential equations.. A detailed discussion of this subject will be found in the section “Intentional Nonlinearities.” Nevertheless it is sometimes useful to consider a highly idealized type of landing gear which has linear differential equations. Although such a model will never correctly reproduce the details of the real landing impact, it admits of easy mathematical treatment and permits study of questions of a more general character. This has been done in the section “Linear Spring-Damper Systems” and the usefulness of the results obtained there lies in the fact that they do not depend on the more or less incidental details of real landing gears which unavoidably enter the computations of the nonlinear theory.

NACA TN 2743 .

The later sections of the report are devoted to refinements of the — method. Except for the simple mass correctio~., they will not be used In daily routine work, but they may be of particular importance when * __ airplanes of unusual design are”built. ‘-:- —.

This.work was conducted at StanfordUniversity under the sponsorship and with the financial assistance of the National Advisory Committee for Aeronautics.

While prepari~ this report the author has received valuable information on curre~t American.practice throw Mr. J. F. McBrearty, Lockheed Aircraft Carp., Mr. K. E. Van Every, Douglas Aircraft Co., Inc., and Mr. A. I. Sibila, Chance Vought Aircraft, for which he wishes to express his thanks. — He also wishes to thank Mr. C. W. Coale for his active help throughout the preparation and the writing of this report.

SYMBOIS An) Bn coefficients inner cross section of barrel at oil level Ao Al total cross section of piston l -.

A2 inner cross section of piston A3 area of gap between metering pin and edge of orifice 0-- a, b distances of landing gears from center of gravity (used only in section “The Airplane as a Whole”) b damping constant for one shock strut (used only in section “Linear Spring-Damper Systems”) — F. “forcein strut when strut is fully expanded and at rest F1 compressive force in shock strut F2 compressive force between wheel and ground (if different from Fl) — F3 force in auxiliary landing ge~ acceleration due.to grqvity g i– NACA TN 2743 height of obstacle encountered during taxying - moment of inertia of airplane with reslect to longitudinal and lateralraxis, respectively radii of gyration kl spring constant for one shock strut spring constant for one tire % m mass of airplane ml that part of mass m attributed to one landing gear unsprung mass for one landing gear .

%2 pressure in both chsnbers when strut is fully expanded Po and at rest pressure in upper chamber of strut l pressure $n lower chamber of strut Pa reference time t time v vertical velocity of landing gear when it first touches ground oil velocity in orifice vertical force, other than impact force, acting on airplane (weight minus lift) WI that part of force W attributed to one landi~ gesr x stroke of shock strut vertical displacement of mass ml vertical displacement of unsprung mass m2 w .

NACA TN 2743 .

— -.

used instead of xl when a second landing gear must be ‘3 w considered simultaneous Q z displacement of center of gravity of airplane a angular displacement (angle of pitch) of airplane in its — plane of symmetry ratio of specific heats — — 6 static deflection of mass ml — density of oil P All displ~cements are zero when the wheels touch the ground without .— — pressure.

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..

LINEAR SPRINGDAMPER SYSTEMS Differential Equations — Essentially, a landing gear consists of-a shock strut and a wheel with a tire. The shock strut may be compressed considerably. It opposes this deformation with an elastic force increasing with increasing stroke and with a damping force which depends on the rate of stroke and which ,dissipates mechanical ener~. This shock strut may be represented by a l spring and a damper arranged in parallel (fig. 1). The tire is for the present purposes a simple spring whose deformation is more or less pro- — portional to the applied force.

Between these two deformable elements there is the mass ~ of the wheel, including those parts of the shock strut which participate in the motion of the wheel. On top of the whole landing gear there is the airplane mass or, more exactly, that portion ml of the airplane mass which belongs to the landi~ gear under cons~deration.

When the airplane lands, this system approaches the ground with a considerable velocity. As long as the spinning up’of the wheels is not considered, only the vertical component V of this velocity is of — interest. The impact begins when the lower end of the landing gear — touches the ground. This instant is designated t = 0, and the vibra- are studied’ which follow for t > 0 - tions of the masses -ml ‘and ~ d- when the motion of the lower end of’the spring-mass system is suddenQ stopped.

Y NACA TN 2743 .

Linear differential equations are obtained when the shock strut is replaced by a simple spring and a viscous damyer (dashpot) and when the tire is assumed to be a linear spring. There might also be a lipear damper coupled with the tire but, compared with the shock-strut damping, the contribution of the tire to damping is so small that it does not seem worth while to include it in the equations.

The differential equations of the landing-gear tipact will now be formulated. Figure 2 shows the mechanical system intwo positions, one for t = O and the other for some later t“tie. The displacements of the masses ml and m2, measured from their positions at t = Oj ~e called xl and ~, respectively. Their difference x.x 1-%2 is the stroke of the shock strut.

constant of the tire, then the force trans- If Q is the spring mitted from the ground to the unsprung mass ~ is (la) On the other hand, the force in the shock strut is the sum of an elastic force klx and of a damping force which,,in linear theory, must be assumed proportional to the velocity ~ = dx/dt with which the masses ml and ~ * approach each other: F1 = klx + b; (lb) The third force is the load W1 which acts as an external force .onthe mass ml . It is a part of the weight of the airplane minus a corre- “The spondi& part of the wing lift. It willbe shown in the section be Airplane as a Whole” what part of the total weight and lift must attributed to each landing gear.

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determine the motion of the The three forces Wl, Flj and F2 masses ml and m2, according to the equations b m&= F1-F2 b“ 6 NACA TN 274.3 s .— When F1 and F2 are expressed here by x and ~ according to b equations (la) and (lb), or better still by x-l and X2, the differ- ‘- .- ential equations ofithe linear landing gear exe ob.tain.ed: — The problem is of the fowth order and req~ir$.s fo~ initial conditions..

for t = O. Before-its solution is given, a simplified version will be ..— —- considered which is sufficient in most cases.

Solution Neglecting UnsprW MWS — -— —. -- —.

me unsprung mass ~ is,rather small, tisually between 2 and 5 percent ~f the mass ml. Under certain coni-itions it has a very “ definite influence .onthe force in the landing gear. But it will be seen that it is only of minor importance for the early phase of the landing impact, up to and b~yo’ndthe maximum of the impact force. One may therefore beginyith a simplified Set Of :.differential ?quations~ ., obtained from equations (2) by dropping the te~ with m2: (3a) (3b)

+ - d -+, -‘J+ %X2=0

Since xl and ~ are the highest derivatives occ~ring @ these .- — equations, the problem is of the third order, and there must bethree initial conditions.. .- fact that the displacements xl and ~ Two of them follow from the are counted from the position of the system at t = O. Therefore .

t=o: xl = 0, Xp=o .

NACA TN 2743 .

The third condition is that the mass ml has at this time the veloc- ity V: D It is useful to replace the second conditionby an equation for xl.

This may easily be done by introduci’& all three conditions in the dif- ferential equations. Equation (3b) yields ~ = V, and it is seen here that this is not an independent fourth condition, as one might feel inclined to think. Equation (Sa) yields now: and this ifitial relation may be used instead of any one of the other three, preferably instead of x2 = O.

One may easily find a psrticul~ solution of differential equations (3): ~ (4) wl X2== It describes the position in which the system is in equilibrium under the load W1. Besides this, the solution of the homogeneous equations is needed. Since all coefficients of the equations are constant, the homogeneous solutions are exponential functions of time, say xl = AeAt When this solution is introduced into equations (Sa) and (~b) after .

dropping there the term Wl, two linear equ@tions are found for A and B: u NACA TN 2743 A m1X2 + bk + kl

)++kj)=o

(

(5) A(bk + kl) - B(bX+kl+~)=O J Since these equations are homogeneous, they will not have a solution — A ~ 0,- B #“O tiess the determinant of the coefficients yanishes, and” this condition yields the characteristic”equation of the problem:

‘1% o

h+k2#+k2A+-=

~3 .,.

(6) mlb b ‘1 — — One of its three roots must be real and, : .

It is of the third degree.

..- positive, this rogt must necessarily be since all c-befficients are negative, s~ The other two-roots may also be real and negative, , .

and v — .-.

or they may be conjugate complex: It may easily be shown that in this case the real pan must be negative, -k l For the rest of the formal treatment the cases of real and of .

complex roots X must be separated.

NACA TN 2743 2C m In the case of high damping with High damping, all roots real.- differential equations (3) is all roots real the general solution of kl+~ -L3t -L~t -?yt + A3e xl = WI + Ale + A2e ‘1% (7) -L3t, ~=wl&+Ble ‘A2t + B3e ‘Alt + B2e .- Al, ~, and A3 may be chosen In these formulas only the constants B2, and B3 depend on them through equa- arbitrarily, while Bl, tions (5), in which in each case the appropriate k must be inserted’.

When t is set equal to zero and then xl and its derivatives are introduced in the initial conditions, a set of three linesr equations is A2, and A3 . They are: obtained for Al, -v

‘1

=—

k12Al + L22A2 + L32A3 m, L then.the displacements may be They must be solved numerically, and found for any time t.

and The most interesting quantities are the stroke x = Xl - X2 B’s” are needed in terms of the A’s: For both the the impact force F1 .

ki - bXn (8) n=l,2j3 ‘n= Ankl+~-hAn and then -X.st *3%2 e-~2t + kl+~-bA3e NACA TN 2’743 .

and F1 = F2 = k2x2 need not necessarily be positive. Then there The coefficients Bn may occur a time t >0, where F2 = 0. If this hwpem~ it would termi~te the domain of validity of the formulas. For greater values of t, the force F2 would not become negative (i.e., tensile), but .- the wheel would leave the ground; the airphne Wmld rebomd.

Because of the force Wl, the airplane would soon return to the Meanwhile its horizontal speed or the angle of attack might ground. — The vertical velocity at the have decreased and hence W1 @creased.

second impact would be, on the other hand, co~iderably smaller than V.

The new impact would therefore be”less violent, but not necessarily uninteresting, because it would find the shock strut in a less favorable .

x >() and, perhaps, close to the possible limit.

condition, with .— Whether rebounding will occux and how Strong the second impact will a+ be can be determined only from detailed numerical computations in each particular case. But one may say quite generally that the probability of a zero of F2 .Ss greater the more solutions the homogeneous equations Since each additional mass and each additional spring increases have.

the order of the equations, one should avoid mechanical complexity if rebounding is undesirable.

LOW dsmping, One p air of roots complex.- The complex exponenti.als which appear in the case of low damping with one pair of roots complex - may be expressed in real form by exponential and trigonometric functions: .

kl+~ + A3e-%t + e-Vt(A1 cos Vt + A2 sin Vt ) kl~ ‘1 ‘ ‘1 (9)

r

-J@

l+ e-~t B1cos W+B2 @

n Vt) + B3e ‘2

.(

‘wl~ NACA TN .2743 .

The relations between the A’s and the B’s are here more involved because equation (8) can be applied only before the trigonometric .

functions are introduced.

When this is done the following relations sre found: 133= (1 +UG)A3 with ~2(kl - b4 - ‘2(kl + b.

=m al 1 kl - bp)2 +b%2 ( b~ ~2 + V2 - 2?LC1VV

)

(

!-31 =ml .

kl - bK)2 +b2V2 ( The boundary conditions will now yield the following set of three equations for Al) ~, and A3: A1+A3=-W1—.+L kl k2

()

LLA1- vA2 + A3A3 = -V — — NACA TN 2743 .

When these equations have been solved nurierically, Blj B2, k and B3 may .be found from the preceding formulas. Then -%2

‘1

=—

‘1

(lo) Fl = F2 ‘A3t + e-pt B1 cos vt + B2 sin W =Wl+~B3e

(

[ .

..

In this case rebounding is rather yrobable because of the trigonometric terms, and it will be inevitable when WI = O.

Solution Not Neglecting Unsprung Mass The results just described may be considered representative for the landing impact if it canbe shown that they are not seriously affected by the neglected mass ~ of the wheel. .— !l?hi~.side of the problem will — — now be investigated.

Instead of equations (3) set (2) must be used which still contains the term with m2. Four initial conditions are required: t=o: xl = o, Al =V, %2=V ‘2 = ‘~ NACA TN 2743 13 .

which now are all independent of each other. When they are introduced into differential equations (2), there are obtained .

“+=0 yip% once more and again everything known If equation (2a) is differentiated is introduced, another dependent condition is found: bW1 gl=.— t=o: “ ml For actual use, choose from all these conditions the set of four which refer to xl only: bWl ‘1 :.

q--- Xl=o, il=v, Y1-ml) t=o: (11) ml The general solution of equations (2) is e-V2t A3 COS v2t + A4 sin v2t (12a) )

(

‘Vlt B1 cos Vlt + B2

sin Vlt + X2=W~~+e ( ) % ‘v’2tB3 cos ~2t +B4 sin v2t) e (12%)

(

NACA TN 2743 , and V are obtained from the real and imaginary parts of the Here w four solutions l .— .. “- of the frequency .

and the B’s depend on the A’s by the relations B1 = (1 + O-IAI + ~1A2 . .

) .

B2 ==plA1 + (1 +al)A2 ( 14) .

‘3= (1+ ~2JA3+ “44.

B4 =-&~ + (1 + %)A4 with ~n2 k - b~n kl +_bwn - Vn 2 ) ( ) . ml an ( kl - bvn)2 + b%n2-- ( bvn Fn2 + Vn2 - 2klpnVn ) ( = 1,2 Pn =q (% ‘. bpn)2 ‘-b2”n2 n J . . , All these formulas are established in the same way as are the corresponding formula for ~ = O. ~ It may happen that one pati of h’s is real (or even both pairs). The formulas will then undergo similar changes, like those explained for the third-ordcm problem.

i When solution (12) is tatroduced into the initial conditions (equations (11)) the followir.u ~ — m equat~ons for Al to A4 will result: .-wl$+~ Al + A3 () -l-LIA1 + -k#3+v2A4=v ‘1% (16) .

WI {p 2. ~12 Al - ~lVlA2 + P22 - V22A3 - 21J2V2A4 ‘~

) )

( \l bWl ~1(~12 - 3H2) AI - h(W12 - VL2)A2 + W2(lJ22 - 3~22)Azj - v2(~22 - V22)A& = — ~12 (14) This set must be solved numerically and then one may find Bl to B4 from equations and (15), The stroke x = xl - X2 of the Bhock strut is then: - e+lt ~=g

+Pl@ COB vlt - (PIA1 - al% sin vlt -

) kl [ al% e+@ a@3 + 13@4) cos v2t - (p#3 - a.#4) sin v2t (17a) i G NACATN 2743 .

The force acting between the tire and the grofid is again — F2 = %X2 and may be computed from equation (12b). It is res~~~ible for the ~ stresses in the tire, and when it becomes zero the &plane will rebound.

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But F2 is not equal to the shock-strut force l?~ which in turn is responsible for the action of this part of the landing gear and for the — dynamic load on the airplane structure. This force must be found from equation (lb): bvlalA~ + 131A2)- (bVI - kl)(~lAl - cLlA2] sin vlt +

[(

) .

sin V2t (lp) bV2~2A3+ P@4) - (b~2- kl) (P@3 - ~ 2A4)]

[(

) l — From the formulas shown here it is clear that the final result is -. - connected with the data of the problem through-an algebraic equation of the fourth degree tid through a set of four linear equations. Solutions must be obtained numerically for a given set of data, so it is not pos- sible to discuss the features of the solution in general terms.

To find out how they look, a series of systematically chosen examples has been computed which willbe discussed in the sectiw “Discussion of Numerical Results.” Undamped System As a basis for this discussion, it is useful to consider the case when the damping b is zero and, additionally, The second ml >> m2.

assumption is certainly good but, if used alo~e, it would not give any substantial mathematical relief. The first assumption is, of course, 3C NACA TN 2743 a not very realistic in a landing ge~ whose essential ~urpose is damping, but the conclusions derived will help in understanding the more realistic cases.

When b = 0, the frequency equation (equation (13)) loses the terms with k3 and X and, when it is assumed that ml>> ~, the first psrt of the coefficient of X2 may be neglected.

The equation reads then ~2+w2-o ‘1% and its solutions are purely imaginary, say J.= % and & = *iv2 .

X2 is solved and everywhere When the equation for l/ml is neg- lected against l/m2, it follows that .

X2 = O and hence VI = The upper sign yields O; the lower sign yields .

kl+~ V2 = m2 r This indicates that V1 << V2, but evidently too many small terms were neglected to find a reasonable value “for VI. It may be obtained from — V2V2 must equal ~ p : the fact that the third term in the equation kl%

V12 ‘—

mlmpVp2 NACA TN 2743 smaller than v20 .

This is indeed much -p-t Since VI = w2 = 0, solution (12) loses the damping factors e and becomes: k,+~ Xl=wl ‘ ‘ + Al cos Vlt + ~ sin vlt + A3 cos V2t + A4 sin Vpt klb —n X2,= W1 ~ + B1 cos vlt + B2 sin Vlt + B3 cos V2t + B4 sin v2t !2 When this expression for xl is introduced in inttial conditions (11), a very simple set of equations is obtained: Al+A3= -W2F+G () V1A2 +V+4=V (18) ‘1 V12A1 + V22A3= - ~ V13A2+ V23A4= O /“ It consists of two independent pairs which will be solved and discussed — .- separately.

NACA TN 2743 From the first and third equations: ‘1 1 2%+k2 —.~2 Al = 2 q ‘1% V22 - V1 ) ( ‘1 m21

AX2&’

‘kl+~ml m2 kl% _ [ pkl+~

‘1 1

A=- —-v~ V22 - vf ‘1 ‘1% )

(

=0 Equations (15) yield 131= @2 = O and n= 1, 2 and equations (14), 13,= A=(.1 - V,2~) = -w&

‘3 ‘A4 - ‘22?)

20 NACA TN 2743 The last equation is not very convincing as it stands, since it gives as a product of A3, which is almost zero, and of the large !J ‘3 factor ml/m2. But one may check the result by first finding an exact expression for B3 and then neglecting ~ against ml.

,, From the results it is seen that only the low-frequency motion — Al, Bl is of importance and that the ratio_~: xl is at all times — ( ) uninfluenced by the presence — the same as for the static deflections, — of the unsprung mass.

(18) may be handled in the same The second and fourth of equations way. The result is this: “1/2 ‘2 ‘%% “( )

K@’

k1k2 .— .

(%+ %)2 .s Here again it is seen that for xl the low-frequency motion A2

()

is by far preponderant, A4 being smaller by a factor (@J3/2j but for the displacement of the wheel (B2, B4) the factor is only .

(m2/m-J 1/2. In the low-frequency motion the ;atio x2: xl is the same — as for the static deflection, but in the high-frequency motion x2 is xl, the wheel moving up and down between the ground much larger than and the almost unmovable airplane mass.

t– On the whole this analysis shows that through the presence of an “unsprung mass” a high-frequency motion is added to the low-frequency — motion of the airplane. This high-frequency motion does not affect XL, – i NACA TN 2~43 .

and but it makes a certain contribution to the forces in the shock strut .

in the tire. Since (m-Jm~) l/2 is still as small as 0.13, one may neglect the presence ~f the’mass m2 if the accuracy requirements are not too high. In view of the many arbitrary assumptions which enter the analysis (e.g., the value of V), one may think of neglecting m2 fOr design purposes, but one should keep an eye on it when evaluating tests.

This rule, of course, is derived from the behavior of the undsmped shock strut.

How far it is modified by the damping can be seen only from the systematic numerical work which will now be discussed.

Discussion of Numerical Results Dimensio?iless parameters.- The formulas developed in the precedhg sections have been used to compute some typical examples.

In order to draw maximum information from this work, it has been done in dimension- less form, and therefore the choice made for the dimensionless quantities must be discussed before the results may be discussed.

For the displacements xl and x2 and the stroke x a reference length is needed, and when they sre plotted against time a reference time is needed. Since the deflections start from zero at t = O and approach asymptotically definite values, the static deflections, it .

seems reasonable to adopt the static deflection of the mass ml as a standard of length: * A simple time standard may be found in the period of the vibrations which the mass ml can make on the springs kl and ~ in the absence of damping. This period is 2z~; drop the factor 2n and choose as time standard For the forces F1 and F2 the load. WI might be used as standard; but, since Wl depends on the horizontal speed of the airplane and on .

the angle of attack of its wings, it may have rather different values for different landing cases of the same airplane.

It is therefore better to use as a reference value.

mlg NACA TN 2743 b Besides these variables there is a set of constants which influence the impact.

They are the masses ml and ~, the spring constants kl k and ~, the damping b of the shock strut,~the load W1 (weight minus lift), and the vertical velocity V of the airplane. These constants may be combined to the following dimensionless parameters: m2/ml, kl vT/8. These parameters must be chosen .— kl +-~~ bT/ml~ w~m~g, and for each example.

InfIuence of ~prung “ilass.- Of most interest in a study of the —— linear spring-damper system is the influence~f the unsprung mass on the impact force: Since this influence is small in the undamped system, one may hope to find the same result when daai~ingis present.

To check whether this is true, a landing gear has been investigated analytically for two extreme values of-the mass parameter, ~/ml = O and m2/ml = 0.050. Values ”common in current practice”lie approximately halfwsy between, and the choice has been made in order to make the effects more clearly visible.

For the other parameters the following values were chosen: kl kl+~ = 0“25 .

bT/ml = 0.5 .

W1/mlg’= 0.2 _ VT@ = 2.0 “z The weight parameter lies halfway between a fully buffered landing with__ — W1 =-.0 and the usual asswnption of W1 = +Ing.

The other three figures 3 1. — are so chosen that they correspond to an actual airplane, at least so ‘ far as a correspondence letween a linear and a real shock strut is — — possible. — The result of the computations is seen in figure 3(a) which shows the shock-strut force. 1’1 against time t.

At t~e start there is a definite difference: The unsprung mass absorbs the fi~st impact.and the shock- .. __._..~ strut force-develops slowly; a slight overshofi follows; and then the NACA TN 2743 curves are practically parallel, the distance between them corresponding to the addition of the wheel mass ~ to the total mass ml + ~ which must be decelerated.

This result confirms the view that the maximum of the shock-strut force is not much influenced by the unsprung mass and that one may safely assume ~ = O wherithis leads to a simplification of the theoretical or rmunerical work. However, this simplification may not be admissible when the wheel spin-uy must be considered. Figure 3(b) shows the force F2 between the wheel and the ground, and here it appears that in the early stage there is a considerable difference between the two cases such that F2 and hence the drag load will increase with the unsprung mass.

Influence of damping.- High damping in the shock strut is desirable since it dissipates the kinetic ener~ of the airplane and thus prevents repeated rebounding.

The influence of damping on the landing imyact may be seen in figure 4. Here the force F2 on the wheel and the stroke x of the shock strut are plotted against time for the following set of parameters: m2/m~ = 0.025 kl = 0.25 kl+~ W1/mlg = 0.2 vT/6 = 2.0 and bT/ml = 0.5 and 1.0. In the initial stage there is not much dif- ference between the two curves {fig. J(a)), since the impact is caught by the tire, but then the rise of F2 is much faster for the case of higher demping; the maximum is reached more quickly but it is only 7 percent higher than that for the case tith half as much dsmping. The development of the stroke (fig. h(b)) is on the whole similar in both cases, but the maximun is lower for high damping. The figures show that, apart from its influence on rebounding (which occws much later), high damping has its pros and cons and that they must be balanced in each design.

NACA TN 2743 h In another example the influence of Influence of spring co”nstants.- and ~ is compared. The forces F1 and F2 “” ‘= the springconstants kl .— — .

have been computed–for m2/ml = 0.025 ..

bT/ml = 0.5 .- — -.

Wqlmlg = 0.2 . — L/ J- V@ =2.0 kl . .

- and for —= 0.15 and 0.25.

kl+~ The standards of length and time, b and T, will not be changed — if-the sum of the reciprocals l/kl + l/~ is kept constant, and under__ -- this condition an increase of the parameter “kl/(kl+ ~) simply means that the shock strut is made stiffer and the tire softer.

Consequently, a greater part of the total deformation will take place in the tire, and h since there is no damping the impact force will build up more slowly.

—.

This is clearly seen in figures 5(a) and 5(h), which show that the influence is the strongest on F2 . Of’course, the smaller the impact — force, the less the airplane will be decelera@d-”tid-”the higher the force .

must-rise at a later stage to bring the vertical motion to a stop. This is also seen in the diagrams, and in figure 5(c) one sees the consequences .- for the displacement xl of the airplane and the deformation ~ of — the tire.

Influence of weight and lift.- Remember _&at the notation W1 — represents the resultant static load on the l-&nding gear, essentially — the difference of the weight of the airplane ad the wing lift. In the — landing-gear literature one finds discussions of the whole gamut of .- Wl=o” possibilities from the buffered landing with to the pancake landing W1 = mlg. The present American regulations consider W1/mlg = O as a standard--assumption.The,cases where W/mlg = 0.333 — .

and 0.2 have-been computed, assuming l * NACA TN 27k3 4C .

m2/ml = O ,

‘1

kl + ~ ‘0”25 bT/ml = 0.5 V’@ = 2.0 F2 = F1 are shown in figure 6, and one The corresponding forces This is easily recognizes that the difference is not very pronounced.

of the air- understood when one considers the maximum displacement xl plane mass. It is 1.655 h the ftrst case and 1.825 in the second, while the final static deflections will be O and 0.28, respectively.

It is therefore essentially the kinetic energy of the airplane mass and not the weight that is responsible for the impact.

All the examples given here show the general trend of changes which a change of one of the parameters will induce. In the details much will be different when real shock struts with their essential nonlinearity are considered.

l l Taxying tires and the shock When an airplane taxies on the ground, its springs of an automobile.

struts have the same functions as the tires and forces depends on the Whether they willbe subject to serious dynamic of the taxying impact smoothness of the ground. In the investigation it has become customary to assume that the airplane rolls at moderate speed over a bump shaped after a sine curve. On a turf-covered airfield such a bump may represent a frozen molehill or a similar obstacle, but on a well-kept concrete runway it is difficult to discover an obstacle of this kind from which the length and height of the bump might be taken, and the same is true for the deck of an aircraft carrier.

It is preferred therefore to assume as a standard obstacle a step in the ground, as it is encountered in the joints between the runway slabs or if a wheel should get over the edge of the pavement (fig. 7).

NACA TN 2743 .

For the mathematical formulation of the.problem it is assumed that the airplane is taxying on the left part of the pavement and that the 4.

landing gear has the static deflection: kl + k2 xl = WI ‘lk2 X2 .@ %2 Since taxying is done at low speed, W1 in these formulas comes close “– to the total weight mlg to be carried by one shock strut.

—- At t = O the wheel hits the ste~ in the pavement, and for t > 0 the term %X2 in equation (2%), which represents the force in the tire, must be replaced by %(x2 + h).

The equilibrium is_then disturbed, and it is desired to know the resulting vibration.

It will be found by solving — the differential equations ( lga) b ( lgb) for the following set of initial conditions: .— L.

It will again be useful to write all initial conditions in terms of one variable.

Since it will be seen that in the present.case x2 is more important than Xl) X2 is chosen to formulate the conditions. The ,,-. -— procedure is almost the same as that described in the section “Solution Not Neglecting Unsprung Mass,” and the result is thiq: ‘f b~h — k2h .

t=o: X2 . W1 —, :2 = “–— Y2=— (20).

i ‘2=0’ ““:%‘ m22 v , . , The solution of equations (19) has, of course, almost the form of equations (12). Because of the new term on the right-hind side of equation (19b) it is: kl + k2 ‘Ppt A3 COB v2t + A4 sin V2t (21a) = W1 +~sinvlt +e ‘vlt Al COB Vlt -h-!-e xl ) ) ( ( %%?

‘p2t B3 cos v2t + B4 sin vat ( 21b) e ~ . WI ~ - h + e-wit (B1 cos Vlt + B2 sin V~t) +

)

(

equatiom (20), the followi~ set of When thie is introduced into the initial. conditions, will result: B1 to B4 equations for B1+B3=h w .— V12 - - @lVlB2 + P22 - V22 B3 - @2vr#4, = m2 YL2)%

)

( (

lql.q2 -

(

3~2)Bl - dw12 - ‘f)Bp ++ - ‘V4B’ - ‘+: - ‘4B4 =- ‘3

J

When these equations have been solved, the constants A must be computed. This is done W inverting equations (14): 1 +U1)B1 - 131B2 ‘1= (1+ CLl)a + ~12 28 .

NACA TN 2743 l 1 + al)B2 131B1+ J%= l+U1 2 + plz () ‘3 = .— — l+~2B4 Q3 + A4 = (1 +m’)’ + p: Now xl and x2 and their derivatives may be calculated as functions of time and from them,.the stroke x = xl - X2, the force on the wheel F2 = IL& + h), and the force in the shock strut / .

[@~vIA1+ (P,’ - vf’)~~ sin,,] -

-p’t mle 1122- v’2 A3

- @2v2-A4 COS V’t +

{K ) -1

E’2V$3 + h2 - ‘z’)Ail Sinv’g An example may illustrate the mechanical content of these formulas. The following set of dimensionless data is chosen: t .

NACA TN 2743 29 kl %+%=0”15 bT/ml = 0.5 W1/mlg = 1 h/b = 0.1 A~er solving frequency equation (13) one may find B1 to B4 from set (22): B1 = 0.01375 B2 = 0.0115 = 0.08635 ‘3 B4 = 0.06175 The precision of B2 is rather poor, but it is not possible to obtain a more accurate value unless the data of the problem are given with such accuracy that a computation tith more than slide-rule accuracy would be justified. However, is multiplied in equation (21b) with a factor ‘2 which increases rather slowly with t, and the term does not reach an important magnitude before the essential phase of the impact is passed.

The displacement x2 of the wheel is shown by the solid line in figure 8.

of the airplane has not been plotted because it is The displacement xl almost constant, and only after a considerably longer lapse of time does the airplane climb slowly to the new level givenby the step in the runway.

.- 30 NACA TN 2743 .

in the shock strut and F2 in the tire are — The forces F1 .- represented. by the solid lines in figure 9. me force F2 jumps *- instantaneously from its static value F2 = W1 to 1.67w1, the rise — being determined by’-theheight h of the step and by the stiffness ~ .— — of the tire. Actually this sudden increase of F2 is smoothened by - local defo~ation of the tire. The force Fl which through the shock strut acts-on the airplane structue rises smcmthly to a maximum and — — then returns in damped oscillations to its static value WI. In the present example ’the essential part of the imps’ctis passed at t = 0.15T.

The fact that the mass ml hardly moves.withi.nthat time which is of interest suggests simplifying the computations by putting ml = m.

When this IB done, equation (19a), must be dro~ped entirely (it simply yields xl = O) and in equation (19b) there must be put — ‘1 + ‘2- .- ..

X1=6=W1 kl$ ‘7 %1 =0 — .

The problem is then”reduced from the fourth to the second order and its -— solution is .

$h

‘1

+ e-pt + B2 sin Vt

B1 COS ti .-

)

( ‘2=~-k1+k2 with — Of the initial conditions (equations (20)) only the first two remain * -.

valid, and from them NACA TN 2743 .kB ’20 With these formulas it is rather easy to compute the deflection x2 and the forces

‘1$$

— - e-wt klB1 - bPB1 +b~2) COS ~t + ‘wl+kl+~ _

F

k1B2 - klvB1 - bI-LB2) sin W ( F2 = ~~ the dimensionless representation However, there is still a difficulty in to ~, the quantity which was i~ put equal of the results. When ml But and WI seems to be lost.

used as a reference basis for ~~ bTj ml is set equal to w in an this difficulty is only appsrent. When equation, this does not mean that the mass really is infinite but only that the inertia is intentionally overrated. Nevertheless, there is a certain weight mlg to which the load (weight minus lift) W1 and tie weight ~g of the wheel may be referred and which will produce a — NACA TN 2743 l certain static deflection ?3. To this deflection the height h of the step is referred, and the damping b is handled by writing l b =— ~g r mlg With this interpretation of the dimensionless quantities, the results of’the simplified theory may be plotted in the ssme diagrams which were used before. They are represented by the broken lines in .

,-.

figures 8 and 9, These curves show the following featuresfi (1) In the domain of interest they are so close to the exact curveE that they can hardly be distinguished. (2) For large values of t they have different asymptotes. This is easily explained. When xl = Constant, the springs will find themselves at last more compressed than they were before the impact. The wheel can therefore not rise by the full height h of the step, and the forces F1 = F2 will be higher than W1. However, this deviation between the two solutions is of,no practical importance, not *- so much because of its small magnitude but because ofiits late occurrence.

It is of some interest to study the extreme case that ml = m .

and~=O. Since the mass m2 is responsible for the difference -.

between the forces F1 and F2, it is seen from a glance at figure 9 . .

that this simplification of the problem goes too far to yield results of immediate practical value.

The formulas therefore will not be reproduced, but some yoints computed from them_.have been entered in the force diagram (fig. 9).

This line ofidots which represent both F1 and F2 shows approximately how the solution will be changed if the unsprung mass m2 is substantially decreased: The sudden rise of F2 is the same, but the following decrease is faster. The force F1 rises more rapidly (in the “limitingcase has the same discontinuous increase as F2), and its maximum will be higher the more ~ is decreased. This . .

shows that in taxying it is not advantageous to have the unsprung mass .-.

.- .

too small.

* .

NACA TN 2743 5C l INTENTIONAL NONLIMFARITIES .

The kind of shock strut considered in the section “Linesr Spring- Damper Systems” is the only one which leads to linear differential equations.

The spring terms in these equations correspond to the action of helical or other steel springs, and such springs were used to some extent in early shock struts.

Modern shock struts use air as an elastic material, and air does not show linesr elasticity unless there is time enough to dissipate the heat generated by compression. However, the nonlinearity introduced by a pneumatic spring is not severe, even in the extreme case of adiabatic compression.

Quite different is the situation with the dsmping term in equa- tion (lb). Viscous damping is never realized in shock struts, their damping being produced by the acceleration of oil squeezed through narrow orifices or slots. If the cross section of the orifice does not vary, one has a veloci@-square damping, and this already presents an essential nonlinearity. But more than this, the necessity of making the best use of the structural weight of the landing gear has led to the introduction of a metering pin which changes the width of the orifice in such a way as to make the impact force increase quickly to its peak value and then stay at this value for a considerable time.

The nonlinearity which the metering pin introduces into the differential equations is intentional and essential, and one has to study the equa- .

tions of motion with the corresponding dsmping term. This will be done in this part of the report and, since there is no additional difficulty .

connected with it, the nonlinear elastici~ of the air spring will also be included.

Differential Equation of Oleo Strut Oleo struts are built in different forms (fig. 10). They all have this in co?mon: A piston moves in a cylinder, and there are two chambers, separated by a diaphragm and connected by an orifice. The lower chamber is filled with Oilj the upper one, partly with oil and partly with air.

When the strut is compressed, oil must flow from the lower to the upper chamber, and there may or may not be a metering pin which fills part of the orifice and makes the remaining gap depend on the position of the piston.

The pressure pl in the upper chamber depends only on the air volume and hence on the position of the piston. The pressure p2 in s the lower chamber is greater by the pressure which is needed to squeeze oil through the orifice. The difference is proportional to the square of the piston velocity and thus produces a damping of the shock-strut .

motion.

34 NACA TN 2743 , .— .

between the force F1 transmitted through the shock The relation l x, and the rate of stroke strut, the stroke ~ will now b.eestablished.

In addition to the notations explained in figures lo(a) and 10(b), the following symbols ‘forthe different cross-sectional areas are used: A. -inner cross section of barrel at oil level total cross section of piston _ Al -innercross section of piston in figure 10(b) %2 ““ . . ,- — When the strut is fully expmded and at rest, there will be a certain pressure p.

in both chambers and a force up to the limit — F. = poA1 may be applied without displacing the piston.

When the piston is displaced, the content of the chambers is decreased by XAl and, since the oil is incompressible, the air ,volume decrease by this amount: must .

xl-1~ = (20 - z)& .

The collapse of.a shock strut under the landing impact takes less than 1 second, and one might think that this time would be too short to allow for much heat transfer. Then the compression of the air would follow the adiabatic law plzy = pozoy with Y = 1.4.

There is, however, a very efficient cooling of the air through the jet of cool oil which is shot vigorously through the orifice and scattered on the cylinder walls.

It may therefore be justified to — — y = 1.1 or even assume a much lower value for the exponent, say isothermal compression with Y = 1. To decide this point, temperature I n measurements in the air chamber would be needed.

.

NACATN 2743 .

Elimination of z from the last two equations yields the following expression for the pressure in the upper chsziber: l P,= PO(Z:;;:X)’ When the piston is dis~laced, the volume of the lower chamber is decreased and oil must flow through the orifice into the upper chs.mber.

In the case of figure 10(a) the rate of the oil flow is Al~; in the case of figure 10(b) it is A2~, and the oil velocity in tie orifice is or

v_A2~

-T Here A3 represents essential~ the area of the gap between the .$ metering pin and the edge of the orifice, inclusive of an orifice coefficient, if necessary. This gap area depends in a known way on .

the stroke ‘x. But A3 -includes-a~so any other leakage between the two chambers, and such additional gaps may depend on elastic deforma- tions and hence on the pressure p2. Complications are avoided by is known or sufficiently disregarding this fact and assuming that A3 estimated as a function of x alone.

The oil is accelerated to the velocity v by the difference between the pressures p2 and pl in the two chmibers according to Bernoulli’s equation

-PI+(3V2

P2 where p is the density (mass per unit volume) of the oil. Taking the & last two equations together, a relation is obtained between the pres- sures and ?: .

NACA TN 2743 PA12 ;2 --: P2 -Pl=— 2A22 J (24) or PA22 ~2 .-.

P2 -Pl=— 2A32 :.

respectively, for figures 10(a) and 10(b).

—.

In this form, the relation holds only for the upward-stroke.

During the recoil motion of the piston the oil moves in the oPPosite direction through the orifice and the pressure in the upper chamber is — the higher one. To cover this motion, one mq8t at least write pl - P2 instead of p2 - pl in equations (24). However, even this will not ,-- really describe the recoil motion, for the following reason: The oil jet which is shot in the upper chamber during the upstroke is so vigorous that air and oil get thoroughly mixed, and this foam is squeezed .- It is therefore scarcely possible to calcu- back during the downstroke.

n“ late the details of the downstroke until exp&rimental information becomes available concerning the degree of mixing &d the density p which should be used in equations (24) for this phase of the motion.

.

In figure 10(a) the force acting downwar~ on the piston is simply F1 = A1P2 - = AIP1 + Al P2 - @

(

part of the piston prot~des into the upper chaniber ; In figure 10(b) and is there exposed to the pressure PI. The total force is therefore — in this case - A2 P1 + A2P2

‘1

‘1 =

( )

=

.

NACA TN 2743 37 With equations (23) and (24) this yields for figure 10(b): (25) and for figure 10(a) the relation is the same except that A2 must be put equal to Al. Equation (25) is the equation of the oleo-pneuna.tic shock strut.

For obtious reasons real shock struts differ from the idealized fi~es 10(a) and 10(b) in that the upper end of the yiston is so shaped that it touches the wall of the barrel. In this way a separate annular space is created between the plunger piston and the wall of the barrel which is usually connected by good-sized holes with the upper chamber.

The oil flow through these holes may add some damping.

It is easily possible to take care of this effect by a correction of the factor of the second term in equation (25).

_ic Equations for Landing Impact For this study it will be assumed that the force in the tire follows a linear law: ‘2 = %?% but that the force F1 in the shock strut depends nonlinearly on the stroke x =d on the rate of stroke ;: the function F1 being givenby equation (25).

The equations of motion are essentially ”thesame as equations (2) except that the terms % - ~2) + %(.1 -%)= bi + ‘lx NACATN 2743 .

must be replaced by F1(~, x). The equations are therefore these: .— .

(26a) mlxl + Fl(ij X) = W1 . .

( 26b)” m2x2 - Fl(~j x) + ~X2,= O — Since these equations contain three unknown variables, a third equation is needed, the relation x.x.x (26c) _ 12- The initial conditions for these equations will be different from Because of the prestressing of the oleo those used with equations (2).

strut there will be a short but finite time at the beginning of the impact when the piston does not move and the.total of the deflection .— comes from the tire. During this interval the motion is governed by the differential equation , l and to it the initial conditions t=o: X2=0, ;2=li — must be applied. The solution is WI xl=x2=@l -cosut)+~sinuh .

NACA TN 2743 l It iS valid Up to the time t = to when for the first time , k2~ = F.

the prestress force of the shock strut. From

‘1

Z(l - cos u)to) + ~ sin cut.= F.

one finds to and then V

~.~l-

cos cmto) + ~ sin uto w. u) – — sin u)to + v Cos (.Dto

‘o - ;2

.

velocity at the end of the initial interval. With the displacement and these quantities the initial conditions for equations (26)may now be .

written.

They are (27) These conditions ought to be imposed at t = to. For the numerical solution it is more convenient to start a new time scale in which equations (27) are to be satisfied at t = O.

When plotting the results, one should of course convert the time so that the zero is at the moment of first contact.

Equations (26a) to (26c) with boundary conditions (27) must be solved by a step-by-step integration. As is well-known, this is done in the following way: At a certain time t, the differential equations are used to compute the numerical value of the highest derivative of each Unknomj then one of various integration methods is used to find NACA TN 2743 .

values of the unknowns themselves for the time t + At. The choice of the integration method determines the exactness of the result and the amount of computation work needed. This point will be discussed in the next section.

The other part of the process, the computation of ~1 and & is done in table 1.

TABLE 1 .

(1) (2) (3) (4) (11) (12) (9)1(10) (5)I(6)1(7)I (8)

I

I 1 m , , 1 1 1 I I t xl ;1 WI - F1 xl x2 o In the first line of this table, columns (2) to (5) are filled from the initial conditions. Then columns (6) and (7) are filled with the help of equation .(26c)and columns (8) to (10) and (11) and (12), with the help of equations (26b) and (26a). ‘The integration process _: F.

will then yield values for the second line of columns (2) to (5), and —— then the whole procedure may be repeated. .: 1.

Simplified Equations, Neglecting .UnsprungMass As has been seen before, the unsprung rnissdoes not essentially “- ‘— influence the load-stroke curve. It is therefore of interest to — reconsider equations (26a) to (26c) after dr&pping the term with m2: ml;l = W1- (28a) F1 (28b) Fl(i, x) = ~x2j x.x - X2 (28c) + As seen already in the linear case, neglecting the unsprung mass decreases the order of the problem by one and the initial condition for ~2 must .— be dropped.

“ NACA TN 2743 41 6C l Equations (28) may be handled numerically with the help of table 2, combined with a table for the numerical integrations.

.

TABLE 2 (lo) (1) (2) (3) (4) (5) (6) (7) (8) (9) F1 t xl ?1 X2 x i $ W1 - F1 “il o Here in the first line columns (2) to (k) are filled from the initial conditions and the other columns, with the help of equations (28c), (28b), and (28a). The first step of the numerical integration se&es to get the second line started, and so forth.

Methods of Numerical Integration There exists much literature on the subject of numerical integra- .

tions and it does not seem necessary to develop here new methods or to describe the old ones in detail. But it appears to be useful to recommend methods which have sufficient accuracy without being too .

laborious, to explain their background, and to present the necessary This working formulas in a notation adapted to the present purposes.

will be done here, and for further details the reader is referred to the literature.l The functions which have to be integrated with respect to time are “21, ~1, & and, if ~ is hot neglected, “~.

Let t be any one of them and assume that, for a certain time t = tnj A first approximation may then be in and Yn are known.

$und for the value Yn+l of y at t = tn + At by assuming ths.t y = ~n is constant throughout the time interval: (29) Yn+l =yn+&At Numerical Mathematical Analysis.

lSee, e.g., Scarborough, James B.: The Johns Hopkins Press (Baltimore), 1930, p. 227j second cd., 1950, .

p. 244.

NACATN 2743 .

.

When this value yn+l (in this case values of xl, ~1, and X2) is introduced in the-differential found, and now the ‘quations) ~n+l ‘s .

integration may be imyroved hy using the average of ~n and ~n+l: .

At (30)

+ in+l

Yn+l = Yn + ~ Yn

)

-(

In figure 11 the first formula uses the shaded rectangle as increment & and the second formula uses the shaded trapezoid, but this result is still — not final, because with the improved yn+l the differential equation will yield another and better value for ~n+l which now should be used for the average. The procedure must be repeated unt$l yn+l no longer and ~n+l change. When the time step At is well-chosen, this should occux after the first or second repetition.

An example is shown in tables 3 and 4. Table 3 is identical with — table 2 and corresponds to the simplified igalysis with ~ = O. In table 4 there are several consecutive lines-for each time t, each of them resulting from one complete cycle of iteration. One sees that *L and xl are practically settled after the first cycle and that x2 ““– — required most of the effort. The computation may be speeded up if proper advantage is taken of this situation. One begins the decond line + of table .kwith the last three columns. With x2 = 1.287 inches and ,— equations (28b) and (28a) the columns Fl, W1 - Fl, and %1 of the l second line of table 3 may be filled. It~s then possible to enter .

..

-155.2 inches per second squared in the second line of table 4 and to ‘1 = perform the two integrations leading to ~1 and xl at once with the .

trapezoid formula, equation (30). The values so obtained for &l and xl will be close to the final ones, and one may now run as many cycles as necessary in the ~ integration and finally check il and Xl again. Great care should be taken that the-next step is not started before a perfect result has been obtained, because otherwise avoidable errors would accumulate from step to step in the integration.

.

.

NACA TN 2743 43 h 9’ TAmE3 .

(1) (4) (6) (2) (3) (8) (lo) (5) (7) (9) .

W1-F1 “~1 (S:c)(J) (J.=) (h:) (1. ) (::) (lb) (in.;sec) ( J.e.) (fn./sec2) 1 12,393 0 -119.2 0.988 U9.5 O.* o 3-U.5 -lq 350 .CG25 LS36 llg.2 1.2870 16,0go 15.0 104.2 -16, ox -1.55.2 . W@ 1.* U9.2 1.269 .01.8 15,8Ea 14.2 105.0 -15,850 -153.0 . Oox 1.933 IJ.8.8 1.530 .053 W,1= 18.6 100.2 -lg, 1.20 -184.7 .m 1.593 U8.7 1.524 .059 19,050 18.4 lm.3 -19, c&J -184.o .007’5 1.879 u8.3 1.775 .104 22,203 2U.5 97.8 -Z2, 2(XI -214.2 .0075L879 118.2 1.772 .107 22,170 20.3 -22,170 -214.o 97.9 . Om 2.174 11’7.7 2.017 .l!x 25, 2CCI =.3 %.4 -25,200 -243.4 .0100 2.174 rL7.7 2.015 .m -25>200 -243.4 25, m 2L3 %.4 . .

. .

. .

TAME4 (1) (2) (4) (6) (8) (3) (5) (7) (9) fi~ “a % (S:c) (in./se&) (in./t3ec) (in. /8ec) (2) (2) (J...) (2) (2) o -U9.2 119.5 0.989 0.988 WJ.5 -0.298 0.299 O.m -155.2 Ilg. !al . 00Z5 1.* 104.2 1.287 -.343 .@ .& .0325 -153.0 llg. ti 105.0 1.286 1.2Ea -.340 .280 .- .Ooa ug.16 1.286 “ 1.263 .’%2 -.382 .297 -184.7 u8.-/f3 100.2 . Olm 1.583 1.53a -.422 .256 .@-r -1.84.O u8.7k 1.583 100.3 1.524 .CGW -.421 .297 .256 U8.74 1.5-33 1.524 .0050 .251 -.45U .296 -214.2 U8.28 1.879 97.8 .C075 1.77’5 -.498 .248 .& -214.o u8.24 1.879 1.772 .0075 97.9 -.498 .248 .296 U8.24 1.879 1.772 .@J75 .245 -.535 .295 .0100 -243. k 117.70 2.174 96.4 2.017 -.572 .243 .295 . Olccl 117.67 2.174 2.015 . . .

.’ . .

.

NACATN 2743 A @ In practical computation work it is more convenient not to write all the lines shown in tables 3 and 4 but to erase each figure as soon .- as it can be replaced by a better one. All that is then left of table 4 is shown in table k(a).

As one may see, this table has the great advan- tage that the f~gures needed for averaging always stand close together.

TABLE k(a) (1) (2) (4) (3) (5) (6) (7) (8) (9) ~1 % &~ $1 ‘2 %2 ‘2 % ‘1 see) (tie/sec2) (in./sec) (in./sec) “(in. ) (in.) (in./sec) (in.) (in.)

-119.2 119 l 5 0.988 119.5 0.988 -0.340 0.298 0.280 .0025 -153,0 119.16 1.286 105.0 1.268 -.421 .297 .256 .0050 -184.o 118.74 1.583 100.3 1.524 -.498 .296 .248 .0075 -214.0 118.24 1.879 1.772 97.9 -.572 .295 .243 .0100 -243.4 117.67 2.174 96.4 2.015 . . .

.

. . . . .

. . .

l . . . .

, .

. .

. . .

. . .

The procedure may be accelerated considerably if at the start of a new step a good guess is made for the new increment instead of first computing a poor approximation with equation (29). If this is done and if the step At is chosen small enough, the method works rapidly and nevertheless develops good accuracy.

Most of the criticism which this method has received in the litera- ture applies only to its use in problems which require a much higher accuracy than does,the landing-impact problem.

In this case slide-rule accuracy will always be sufficient, and this can be obtained by the trapezoid integration without resorting to painfully small steps.

However, the method has the disadvantage that one never knows exactly .

how large the error is. This drawback willbe avoided if the straight line in figure 11 is replaced by an interpolation parabola. This may be done as soon as four o’rfive successive values of y have been determined, .

NACATN,2743 45 Through five consecutive points (fig. 12) a parabola of the fourth degree may be fitted, and the coefficients of the corresponding polynomial in t may be written in terms of the ordinates tn-4j tn-3~ . . . in or) snd of a set of differences of increasing order better, in terms of tn ~~n ‘ jn - ;n-~ A&n=A1~n -Al~n-l and so forth which may be computed in the following scheme: t tn.k ‘n-s .

tn_2 Yn_2 tin-2 .

‘n-l

tn .

l The polynomial may then be integrated over any one of the intervals At and in this way improved values for the increments & may be obtained.

They are computed from the following formulas: &n=Yn - Yn-1 1.

(31a)

= At ;n

-&&l- &A3il-&&4k” “ “ - 5Alyn

( )

(31~) (31C ) NACA TN 2743 are obtained. When improved values of the y’s From these Ly’s they are introduced in the differential equations, better valUeS Of the .

derivatives will be found, and this procedure must be repeated until .

This should occur after two or three the results become stationary.

cycles. If it takes longer this indicates that the time step At was chosen too long, and one should at once make “anew start with shorter intervals. On the other hand, if the final values are hit at the first stroke, this generally indicates that the time step was chosen too short, At or cqntinue until eight and one should start again with a greater lines are completed and then double the step by dropping every other one.

When this polynomial method is applied to the landing-gear problem, columns (1) to (6) of table 4 must be replaced by table 5, and the columns — TABLE5 — :11) : 12) (13) ( 14) :4) [10) 1) (6) (7) (8) (9)

E

— Axl xl A4iil &l !+] 14?1 t kl — . .

. . . .

.

. .

. . .

. .

. .

. . . . .

. l . . . . .

. .

I

referriw to xo by a similar table or an abridged version, depending ‘-me results of the trapezoid on whether equa%ions (26) or (28) are used. — integration are introduced into column (2), differences in columns (3) to (6) are computed, and then column (~) is filled with the help of _. D --.= equations (31), identifying j with xlx FYom the increments in in column (8) may be found which are already column (7) values of il better than those of the trapezoid integrating. They may at once be — used for comyuting the differences in colunns (9) to (12), the increments xl, again using equations (31).

Axly and the values 4, it is found that When this,is done with the figures of table nor the xl~s are capable of im~rovement, but ~ neither the xl’s v_ .

NACA TN 2743 is changed appreciably. The final state of the integration for x2 is shown in table 6. Frantically all the correction is due to the first time interval, TABLE6 (6) (8) (2) (4) (5) (7) (1) (3) t + Aliiz ++ ’322 %% & . (i:) (see) (in./sec) 0.988 119.5 o 1.265 0.277 . 002s 105.2 -14.3 .256 I. 521 100.5 -4.7 9.6 .0050 .248 1.769 -2.6 2.1 .0075 97.9 -7.5 2.011 -1.4 1.2 -.9 6.6 .242 .0100 96.5 . . .

. . . .

.

.

. . . . . .

.

. .

. . . . .

.

When this polynomial method is applied - and the example demonstrates that it may be ~or~h while to do so --~hen it will be reasonable to use it not only for checking and correcting but also for integrating ahead.

To do this, one must extrapolate the polynomial in figure 12 beyond tn tn to tn+l.

integrate y from through the next interval and then used before: set of differences The result may be expressed by the 4Yn+l ‘Yn+l - Yn .

251 w - ~“ .At~n+~AlYn+12~n + ; A3& + ~AJ-@n* “ “ ) ( \ NACA TN 2743 .

With the-help of this formula one might find~in table 5 the values of Ail, Al, Axl, and xl in the next line. But actually it is neces- .

sary to use equation (32) only once, preferably for integrating ~2 \ lf ~ = O), and then approximate ~lues for the other deriva- (or +2>.:.

tives may be proc&ed in time to do all other integrations at once with equation (31a) which is mare exact and less influencedby the higher differences.

— When thus a new line in all tables (table 1 or.2 and the integra- tion tables) has been filled, equation (31a).is used repeatedly to improve and xl as long as they are cayable of improvement.

‘2 .- .- As soon as the columns for the derivatives (columns (2) and (8) in table 5) fill up, one might extend the difference scheme toward — — differences of higher order, but the farther one goes to the right, the smaller and the more erratic the differences—will become and they will not he able to influence the increments computed fmm equations (31) In general the time step At . .

and (32). should be chosen such that the fourth-order difference may be neglected.

Exce@ for the start of the computationwhich is always a little irregular, the higher differences should be rather small before they become erratic; otherwise one must either increase the accuracy of the derivatives by carrying more digits or decrease the step At. If it iS = .–.

intended to carry more significant figures,gne should keep in mind,that a many-digit machine computation is a wastedA_effortl if somewhere in the process a figure Wst be read from a grayh, for example, the effective =- orifice area A as a function of the stroke x.

In order to check the accuracy of the two method= - trapezoid and polynomial - an exam@e of a linear shock strut has been computed with the following data: 103.6 lb sec2/in.

‘1 = ~=() b = 500 lb see/in. - kl = 2800 lb/in.

k2 = 12,500 lb/in.

W=o .

V = 120 ino/sec NACA TN 2743 ‘7C .)

The results for this exam@e csm be compared with the exact solution, equations (9) and (10). for the The following values were obtained .

impact force Fl: Fl (lb) (S:c) At = 0.02 sec At = 0.01 sec Exact Trapezoid Polynomial Trapezoid 0.04 38.1 x 103 38.13 x 103 38.03 x 103 38.47 x 1.03 52.1 52.0 52.3 .08 52.0 56.2 56.2 56.4 .12 56.2 55.8 .16 55.8 55*7 55.7 .20 52.5 52.6, 52.6 52.6 .24 47.6 47.6 l Evidently, under these conditions the trapezoid method with At = 0.01 second is good enough. Encouraged by this result, the step At has been doubled. me results of the trapezoid integration .

are shown in the table. The polynomial method proved to be extremely tedious and was not pursued further when after several hours of computa- tion the first four lines had not yet stabilized. Hcwever, it was found practical to start with the small interval and double the step as soon as possible. The polynomial method with At = O.01.second was carried tot= 0.08 second, and then the results for t = 0.02, 0.04j 0.06, and 0.08 second were used to start the polynomial method with the double interval. This computation was carried up to t = 0.20 second and yielded results identical with those obtained for the shorter steps.

NuJnericai Example As an illustration of the methods just described an example has been worked out. The data chosen and the metering pin correspond closely to those of a recent American airplane. The data are these: * AI = ~ = 39.8 sq in.

.

50 NACA TN 2743 w Aozo . .

—=”23.5 in; —- ‘1 .

P. = 310 lb/sq in.

p = 8.42 x10-5 lb sec21in.4 7 = 1.1 ~ = 12,500 lb/in.

v= 10 ft/sec ..

The effective orifice srea is shownby the heavy ling in figure 13 as a function of the stroke x. The low paj% at the left-hand side of the diagram represents the bulbous end of the metering pin.

s The shock strut is prestressed with the.force — .

F. = po~ = 12,350 lb Until the impact force has reached this vslue, only the tire is deformed and the simple formulas mentioned after equations (26) a~ply. They yield to = 0.0088 second, V. = 119.5 inches per X. = 0.988 inch, and second.

These are the initial conditions for the numerical integration .— of equations (25) and (28). This integration was started by the trapezoid method, using equations (29) and (30), and the time step At was so chosetithat at least a few intervals would pass before the first break in the curve A3 = A3(x) was reached. This-is possible with .- At = 0.0025 second, and the first lines of this computation are shown in tables 3 and k (where t is counted from the beginning of this integra- tion, not from the first contact between tire and runway). When four steps were completed, the polynomial method was started and the results . of these steps were improved.

The computation was carried on to x = 0.571 inch, xl = 3.907 inches, and t = 0.0250 second with X2 = 3.336 tithes. This is sufficiently far past the first break in .

NACA TN 2743 51 A3(x) that it was possible to dotile the step. A new integration table .

was started with the results for t = 0.010, 0.015, 0.020, and 0.025 second and it was carried on with At = 0.005 second until t = 0.060 second with x= 1.939 inches. The next step would have led beyond the second break and hence to large values in the difference schemes.

in the curve A3(x) Therefore, it was necessary to return to the shorter time step At = 0.0025 second. This makes it necessary to interpolate values for the half intervals. To keep up with the accuracy of the integration, this must be done with the help of the same interpolation parabolas from which equations (31) and (32) are derived. With the notations used the following formula holds: in which the last term is often negligibly small.

With the help of this formula a new integration table was started, beginning with t = 0.0450, 0.0475, . . . second. when it c~e to A3 diagram was t = 0.0775 second, the next and last break in the reached and the higher differences rose so high that it becme necessary .

Eight lines beyond the discon- to reduce the step to 0.00125 second.

tinuity the step was increased to At = 0.0025 second and soon thereafter At t = 0.14 second it was realized that the higher to 0.005 second.

.

differences had become so small that the interval could againbe doubled, and with At = 0.01 second the computation was carried until t = 0.27 second, when * became negative.

The example which was chosen here as a test spectien for the Most of numerical integration is one of the most irregular possible.

the computation effort was spent on the bulbous end of the metering pin.

As soon as the last corner in figure 13 was passed, the work proceeded rather quickly to its end. When the pin is shaped more gently, or when there is no pin at all, it will be possible to start, say, with At = 0.005 second and to change after some time to At = 0.01 second, without the many tedious changes which”were necessary in the present case.

The results of the computation are shown in figures 14 and 15. There is a double time scale in the diagrams, one beginning at the first contact and one at the time to when the numerical integration begins.

Figure 14 shows the stroke x and the displacement xl of the airplane. There is a first, short phase during which only the tire is NACA TN 2743 b deformed and x =“0. Then the shock strut begins to work but, because of the bulbous end of the metering pin, the strut collapses, at first rather slowly.

Later it catchesup, and the curves x and xl approach . ._ - each other, indicating that the load maximum is passed and that the tire expands.

Figure 15 shows the load and its breakdown into the damping force and the elastic (air) force according to the two terms of equation (25).

Because of the bulbous end of the metering pln the damping force builds up rapidly, but then the orifice opens up and the increasing air pres- .- sure in the shock strut cannot compensate the decline of the damping _ force.

Dimensioning of Metering Pin For reasons OY weight saving it is desirable that the shock-strut force rise quickly to a high value and then remain at this height for a sufficient time to bring the mass ml to rest. As a yractical means — for this purpose, the metering pin has beenintroduced into the design .- .

of shock struts. Now, since there is but one metering pin, it will not be possible to obtain ideal results for different impact conditions, but it is possible to pick out one landing case of particular importance and to shape the metering pin so that in this case a desired load history is obtained. The shape of the pin which has been found for this case must, of course, be subjected to a critical study in two respects: It must be — acceptable to the workshop, and it must yield at least tolerable load- time diagrams under other landing conditions. Tk? final compromise is, .— a true engineering decision which cannot be replaced by an analytical device.

There is no need to specify exactly how the impact force should rise from zero up to a certain level.

In”this first part of the load history the tire has an inrportant influence, and it wI1l be enough to choose the orifice opening A3 so that not too much stroke is lost while the force builds Up.

But when at a certain time t = t’ the force F1 has reached a certain value, say F1 =,F’, then it may be desirable to keep it constant If it is agreed to neglect the unsprung mass ~ equa- on this level.

tions (28) are simp~ ml~l + F’ = W1 (34a) .

Fl=~~ (34b) .

“53 NACATN 2743 d and it follows at once from equation (34b) that ~ = Constant, say, .

22=0 Equation (3&a) presents a simple integration problem and yields - F!

‘1 <t-t’)

‘1

:lF’(t -ty

X1=X1’+X “l’(t -t’) +

where k t and xl’ are the values which the variables have assumed .

at t=t’.

From equation (28c) .

k=k .X1-qf x and these values may now be introduced in equation (25) of the oleo strut, which then yields Aa: J (35) -1 This idea has been applied in two ways to the numerical exsmple of the preceding section.

# When looking at figure 15, one might think it useful to keep F1 some time on its peak level, thus decelerating for faster the vertical .

NACA TN 2743 & motion of the airplane without imposing a higher,dynamic load an it.

This would result in a saving in stroke and hericein weight of the A— shock strut. In this way the curves marked “I” in figures 16 and 17 have been obtained. The corresponding orifice area is shown in fig- .— —.

.- ure 13 by-”theline.I.

To keep the impact fgce at its peak level, the - orifice area must be decreased relative to tie original design, and it comes down to zero.when the motion of the airplane is stopped. The — steep descent at the end of this curve is, of comse, not acceptable for the design, since it means a complete-plugging of the orifice and — would lead to a high load peak in a case of-harder landing; but the upper pat of the curve may lead to an improvement of the design.

— One might think of another modification.,of the load-time curve, cutting away the peak and.keeping F1 as long as feasible on a medium level, say at F’ = 46,600 pOUlldS. When th~~ is done, the curves marked “II” in figures 13, 16, and 17 result. They show that in this case a slightly longer stroke is needed than in the original design, but there is a considerable saving in dynsmic load.

Since the rise of F1 is interrupted in this case, the orifice .— must be opened wider, and figure 13 shows that most of the bulbous end of the pin must be removed. The transition must, of course, be smoother ““” than that shown in the diagram, and this wo~d lead to a rounding of the corner in the load-time diagram (fig. 17). Except for this nece$sary modification and for the steep end of the A3 curve, the solution seems ‘“- * acceptable, provided that the pin shaped in,.~his way proves to be satis- -- factory in other landing cases.

.

But there is still one essential point that needs discussion.

Figure 18 shows the velocities *1 and * for all three cases. For the original pin heavier lines have been used and the two modifications are marked “I” and “II.” The first modification does not show anything — in particular, but for the second modified pin ~ jumps suddenly from .

~ = 21 .-;.

one value to another and so does Now, a sudden change of the velocity ~ :ill, of’course, meet with the inertia of the unsprung mass, and the metering pin II cannot be acc&ted without discussing this —.

.-.

influence.

starting from equations (26) and putting F1 = l?’: ml~l = WI ~ F’.= m& + %X2 = F ‘.. , -.

—— .

NACA TN 2743 55 The first of these equations is identical with equation (34a), and the second yields an undamped vibration: =~+Acosu(t-t’)+B sinu(t-t’) ‘2 ~ .=&/m2 Now, at t = t’, when this vibration begins, ~ . &21 and hence .

~=~’+~sinm(t -t’) .

When it is assumed in the example that m2/ml = 0.025, the circular .

‘1, that is, about frequency of these vibrations is u = 69.5 second 10 cycles per second.

The stroke x will show the same undulation as w and so will the metering pin.

Of course, nobody would think of bui;ding a metering pin of that shape, in particular since the length and location of these undulations would depend on the arbitrary choice of the conditions under As soon as a streamlined metering pin is which F1 is kept constant.

F1 will chosen corresponding to the simplified analysis, the force fluctuate slightly and thus provide the necessary damping for the transient vibrations of x2 and x.

There is still a better way of handling this last question. Since it is not feasible anywsy to make a metering pin with a sudden change of cross section, it is better to assume a force diagram on which the corner is well-rounded, say by a parabola F1 = Cl + c2t + c3t2 which is so chosen that there is no large discontinuity in dl?~dt.

NACA TN 2743 When this force-time”relationis introduced into the equations of the landing gear, they may easily be integrat-ed, and the resulting .“ expressions for x and A along with F niay be introduced into equa- tion (35) to find A3 and hence the cross section of the metering pin.

The short broken line at the outset of.the horizontal-line msrked “II” in figure 17 represents such a psrabolic rounding of a corner in .

the force diagram. The corresponding values of i and A? have been indicated by broken lines in figures 18 and”i3. “Onemay recognize that . . – no great change of the metering pin is needed to make the wheel motion .

much smoother, and the corner in the force diagram might still be rounded .

much more without a substantial loss of deceleration for the airplane.

.- ADDITIONAL NONLINEARITIES Tire - The elastic resistance of the tire depends only in small part on the elasticity of the rubber and is essentially due to the compression of the enclosed air. During the landing impact this compression is nearly adiabatic and therefore the relation between the tfie pressure and the deflection x2 is nonlinear. On the other hand, the relation between the pressure and the force F2 is nonlinear also because the —.

— tire flattens. On the whole, these and some--other influences seem to “- .

— compensateto some extent, and load-deflectfi curves from tests may be * fairly well approximated by a straight line.--This is illustrated by “- figure 19 which shows such a test result.

For design purposes it does not seem worth while to replace, under - these circumstances, the linear relation (equation (la)) by anything more complicated. However, for the.evaluation offiests it may be advisable = to use the lest available information on the behavior of the tire.

—.- The nonlinearity of the tire becomes sev&re when it comes to .—— bottoming. Then the force F2 may rise to high values without an — appreciable f~ther increase of x2. In general, bottoming should, of course, be avoided, but when it comes into consideration, then eqw- — tion (la) can no longer be applied, and it fist be replaced by the general .

relation F2 = Fp (x2) .

.

NACA TN 2743 8C .

which represents an empirical function determined from tests. The . equations of motion are then these: (36a) (36b ) instead of equations (26a) and (26b).

Because of the prestressing of the oleo strut these equations are not valid until the shock-strut force has reached the prestress value Fo.

For this initial phase of the impact the procedure described in the paragraph following equation (26c) must be applied. Since it covers but a small part of the whole impact, one may use there the linear law F2 = k2x2, the spring constant k2 being taken from the initial tangent of the load-cleflection curve of the tire: %= 1- ‘ -IX2* For equations (36a) and (36b) then the initial conditions (equa- tions (27)) are the same as those for equations (26). The equations are solved by numerical integration and table 1 may be adopted, changing only the heading of column (9) where F2 is written instead of ~~ and then using a graph of the function F2(x2) to fill this column.

Inmost cases it will be possible to neglect the mass ~. Then equations (36) are rewritten in the form . .

mlxl = W1 (37a) - F2(X2) (37b) F2(%) which corresponds to equations (28). For the numerical integration use at the top table 2, writing F2 = F1 of column (6) and filling this column with the help of the graph for F2 (X2)“ NACA TN 2743 b In both cases, equations (36) and (37), the integration step must- be decreased ap~ropriately when approaching-the region where the tire — . .

.

bottoms.

— Kinematic Nonlinearities In the equations of motion one needsthe second derivatives of the displacements xl and x2, the accelerations of the masses ml and m2, respectively.

For the shock strut, the stroke x is needed. Thus fsr it has always been assumed that x is equal to the difference xl - ~.

However, this relation holds only in the simple case, when the upper part of the shock strut (usually the barrel) is rigidly connected with the airframe and the wheel is atta-cheddirectly to the lower part (piston).

— A correction is already needed when the shotk strut is inclined from the vertical (fig. 20)-. In this case X1-X2 x= Cos a The changes which this relation requires in the integration schemes are obvious and there is no need to discuss them in detail.

x and &- However, there are cases in which the relation between xl-x is nonlinear.

Figure 21 illustrates what is meant. Most of 2) ( these devices have disappeared from current-~practicej but in a time of .

rapid development it is advisable t-odiscuss-briefly how similar cases may be handled. For all.these landing gears a nonlinear relation x = fxl - x2 ( 38)

( )

can be established by trigonometric methods.

By differentiating it with respect to time, the relation , .~(,, - +) (39) — l is derived.

The above two equations take the place of ecjuation (26c) and the corresponding relation for the velocities.

.

NACA TN 2743 Table 1 must now be replaced by table 7, in which columns (5a)j (6a), and (6b) have been added. Columns (~) and (6b) are filled from the preceding ones and column (6a) is filled from a formula or a graph TABLE 7 (1) (m) I (u) I (@ (9) t

r

re sed to fill for f’ xl - X2). Then equations (38) and (39)

(

columns (6) and (T). Everything else is done as explained for table 1.

of table 2. Again When ~ is neglected, table 8 is used instead COlumns for xl are protided.

additional

( -%+ (;, - ~), and ,1

column (Ta)j frcnq column (5) is filled with the help of equation (38); TABLE8 column (b) with a graph or formula for f’ xl - X2); and column (~)~

(

with equation (39).

These are very simple changes, the numerical integration being a very flexible instrument that can be adapted to almost every special requirement.

.

.

NACA TN 2743 1.

THE AIRPLANE AS A WHOLE .

— Introduction - In most investigations of landing gears”” the airplane is represented “.. . ...= — by a single mass ml riding on a system of-springs and dampers, with perhaps a small additional mass representing the wheel. All the ‘2 — — preceding sections of this report are exclusively concerned with this -- model.

However, the real airplane is a three-dimensional structure, and when one or more of its wheels hit the grdund, it may receive not only a vertical acceleration but also angular accelerations about different sxes. These angular accelerations and the rotatory motion resulting from them will, of coune, influence the landing impact.

A detailed study of this phenomenon leads into rather lengthy computations. Their quantitative results will depend on many details and may vary widely between different types “of airplanes. This section will therefore be restricted to some genera~.considerationsconcerning the best method of analysis. - “ — There are two principal Troblems, the symmetric case in which both b wheels of the main landing gesx strike the ground simultaneously and in identical conditions, either earlier or later than the auxiliary gear, and the asymmetric case in which the two wheels of the main gear touch r the ground one after the other.

Symmetric Impact ._ Figure 22 shows the side view of an airplane as far as it is of interest for the present purposes. The point-C is the center of gravity - where the mass m is located. To the right is the main gear; to the — left, the auxiliary gear which may be either-a nose gear or a bail gear.

— In figure 22 the airplane is shown in the position which it has at the time t = 0, when the main gear makes its first contact with the runway. From this time an impact force F1 of increasing magnitude will act in each main gear and it will cause both-a deceleration of the vertical movement of the center of gravity and a pitching motion about .

this point. The equations of motion are , .

NACA TN 2743 .

“ (40) where L. is the moment of inertia of the airplane with respect to its transverse axis.

The resultant acceleration at the upper end of the main landing gesx will be . .

=Y+ad.1 ‘1

‘1

= -—-

m a?~ Ill&+ w — =-

T’ m

) ‘%

(

.’ In the section “Linear Spring-Damper Systems” there was written .

F. W, yl=-~+d ‘1 ‘1 and the two expressions are equivalent if one chooses m ml .

21+?

() Y m %2 =.— 2a2+i2 Y .

.

62 NACA TN 2743 .

T- where These formulas show that ~=&/m istheradius ofgyraticm.

it is perfectly justifiable to study an isolated landing gear, provided .— one does not simply use as mass ml one-half of the airplane mass.

However, this procedure is subject to two essential limitations: It can be applied only until the auxiliary gear comes into action and must.

at least be modified when the rotation of the airplane leads to a sub- .- stantial change of the angle of incidence of the wing and hence to a change of the load W.

Consider the second point first.

The angular position of the air- .— plane is determined by the angle a between the ground and a reference line in the plane of symmetry of the airplane. This reference line is so chosen that u = O when all three wheels–of the airplane just touch ~ the ground without pressure.

The angle a .wlichis so defined is not identical with the angle of incidence of the wings, but the two differ — only by a constant which depends on the design of the airplane.

Since only small values of a need be c“onsi.dered, it may be asswned that the lift md hence W is a linear function of a, say: b W=wf + W’fu .

but since one must use numerical integration methods anyway an arbitrary function w =W(a) may be assumed when this appears to be necessary. .— The part of this — weight which must be attributed to one main landing gear is then Wl(a) =W(a) (41) .2 & ‘-3 .

NACA TN 2743 - Equations of motion (ho) are now written in the following form:

Yl=- +1 - WI)

(42a) ml “~ = -&Fl (42b) Additionally, there is a relation which connects with X

‘1

and ~, for example, equation (25) of the oleo strut, the relation x.x - X2, and the elastic equation of the tire F2 = F~ = %X2.

Table 2 which is used for the one-gear problem must now be “extended so that it may take care of equation (42b). It looks then as shown in table 9: !lYiBLE 9 (1) :2) :14) ., t xl a — o

F

Columns (2) to (5) and (8) to (10) are treated exactlv as &e the corresponding columns of table-2;”the first line in c&unns (6) and (7) is filled in from two additional boundary conditions (a given, & = O).

Also the starting value of W will be known and must, of course, check with column (6) and equation 41).

Column (12) is self-explsaa.tory, and t columms (13) and (14) follow from equations (42). Besides the tables for the integration of Yl, *1, and ~, an additional table is now needed to integrate “d and &.

With the results of these integrations “ the second line may be started.

Of course, this analysis does not consider the possibility that the pilot uses the controls to counteract the pitching movement of the air- plane. If he does so, a human element comes into play which is not easily incorporated in mathematical formulas.

This uncertainty may upset the usefulness of the procedure and will justify the application NACA TN 2743 .

of the simpler table given previously. This simplification, commonly — used in landing-gear analysis, is still more justified by the results .

represented in figure 6, which show that the.exact magnitude of the .— effective weight W1 is of secondary importance for the interesting portion of the impact.

Whether a simple one-gear analysis is made or the variability ,]f WI is taken into account, this computation ends at the moment when the auxiliary gear cornesinto action. The time t = t’ when this occurs is found in the following way: During the first phase of motion the .- — acceleration at the upper end of the auxiliary gear is: . .

.— ‘3=y-b& When it is assumed that the airplane approa~es the ground with the vertical velocity V, but without an angula..velocityj then the velocity fort>O is ‘3 b — “1

2* t

=v-- Fl dt +–– W dt (43) ‘3 m ~y2

J 0 ‘o *-

r

and W must be introduced as functions Under the integral signs F1 of -t according to the analysis of the main gear.

The displacement of the auxiliary gear is best counted from ‘3 the position inwhich the wheel just touches the ground. When the airplane lands at an angle a (fig. 22), then x s = -(a + b)u at t=o. For t >0.

t“ X3 =-(a+b)u+ 43 dt (44) o

J

.

and the time when this equation yields X3 = O is the time t = t’, .

I’?ACA TN 2743 9C .

When W =0 and ah<~2, theveloci~ *3 will decreasesmd the impact of the auxiliary gesr will be softer than it would be if .

this gear had hit the ground before the main gear.

If W ~ 0, there is an additional positive term in equation (43) and, since the atiplane is still falling under the influence of the force W, the velocity *3 may increase.

When ab > %2, then ~3 will certainly increase, pos- sibly even very much so, and the auxiliary gear may strike the runway rather forcibly.

When all wheels are in contact with the ground, the equations of motion are rather involved.

When all landing gesrs have a spring- damper unit as a shock strut, no demping in the tire, and no unsprung mass, the problem is of sixth order.

It is of little value to establish the formulas for the linear case, but it is useful to develo~ a numerical procedure which may be applied in linear as well as in nonlinear cases.

The equations of motion contain now the forces in main and auxilis.ry gears (fig. 23): &=-2yF3+W (45) -2Fla + F3b

w’=

)

Then there are two kinematic relations (46) Differentiating there and then and b from equations (45) yield .

.

NACA TN 2743 .

These equations may be used in the following way: For each landing gear a single-gear analysis is started according to the instructions given in the section “Intentional Nonli.nearities.”For the main gear it begins at t = O and runs exactly as explained there until t = t’.

For the auxiliary gear it begins at t = tl_.with X3 = O and the value of *3 which follows from equation ($3). When it is not desired to neglect the unsprung mass, table 1 is used, otherwise table 2. In either case the line for t = t’ may be filled up to the last two columns, but the last two columns are replaced by some columns which are adapted to equations (47). They yield. xl and X3 in terms of the forces F of both tables, and these values =e now integrated just as was done with ~1 in table 4.

A step-by-step integration of this kind requires twice as much time as a single-gear analysis and will yield everything needed for both gears.

One-Wheel Landing -- .— It is possible that a landing airplane may approach the runway with one wing low and that the wheels of the main gear do not hit the runway at the same time (fig. 24). There we then again two phases, a first one while only one wheel is in contact with the ground and a b second one when both wheels are.

— In the first phase there is only one force Fl, having the .

distances a and c from the lateral and longitudinal axes, respec- tively. It produces the following accelerations: — Vertical at center of gravity: ti=-Fl+W Angular with respect to lateral axis: -= with respect to longitudinal axis: .

!X4X=-FIC

-.

.

NACA TN 2W3 .

acceleration at the upper end of the active landing gear The resulting is .

and when use is made of the preceding equations there is obtained

‘1

“yl = .Yl+=f+$ +: x ‘Y

( )

Again it is useful to introduce the radii of ggcration by:

%’= %/”

and to write l .

‘1=+!+$+5N

When this is compared with the relation

‘1 W1

~l..—+_ q, ml used previously, it is seen that one must put m ml 2 C2 l+++—— x ‘Y ml Wl=w= NACA TN 2743 .

With these notations the’impact problem is again reduced to a one-gesr problem until the other wheel meets the ground. The time t = t“ at .

which this will occur may be found in the same way as in the case of a two-point landing.

The acceleration at the top of the second main gear is Integrating once yields the velocity: and integrating again, At the time t = t“ when X3 = O, the one-gesr problem ends and from then on both main gears must be dealt with simultaneously.

This is done as in the preceding section, but the formulas differ in details because there i-sstill one degree of freedom left, the During this phase rotation about the transverse axis of the airplane.

of the landing impact the equations of motion (fig. 25) are as follows: ti = -Fl -F3+W 1A = (-F1 +F3)C .

NACA TN 2743 69 .

Besides there are the kinematic relations .

and by combining both sets of equations the following equations are obtained which correspond to equations (47): Fl yl .

-Tl+ $+$)++$ -$)+: ( (48) %-~)-:~+fi+~)+: iy \ These equations may be handled exactly in the same way as equations (47), with, however, the restriction that the auxiliary ge& must-still be- ““” off the ground. As soon as it makes contact, the relations become more involved, but it seems at present not necessary to elaborate the details of the third phase of the impact which then will follow.

REVIIZWOF GERMAN LITERATURE ON IANDING-GEMR IMPACT Before the last war in Germany almost no theoretical work was done on landing-gear problems, and it seems also that in other countries interest was low.

During the war in Germany new and unexpected demands could fre- quently best be met by adapting an existing airplane type, with its well-established mass-production facilities.

Such modifications usually resulted in an increase of weight without supplying additional space into which a larger wheel could be retracted. Frequent tire troubles were the unavoidable consequence, resulting in a strong impetus to landing-gear research. All but one of the-papers revi~wed-here belong to this period of wartime research.

.

When studying this German wartime literature, one must keep in .

mind during what period and under what circumstances the work was done.

70 NACA TN 2743 .

All of these papers appeared in 1943 and 1944 and were thus the outcome of a rather short period of research, They represent an intensive attempt to tackle a-long-neglected problem. But before the work had _ . . . . - : yielded results of-final validity, it was cut off early in 1945 by — the national catastrophe. More than 6 years have elapsed and the — landing-load problem has undergone changes. ~ome ox the statements made in those papers have lost interest, others are no longer applicable without modification, and most of the analyt~cal methods are either — oversimplified or too complicated.

Nevertheless, it is still worth while to survey this literature briefly because it-~ontains”many of the ideas and methods which are ,.

— still the basis of.landing-gear analysis. IiiZie@d, fn writing this .

report the author has drawn much useful information from the German publications which.are reviewed on the following pages.

The goal of the early landing-gear rese~rchwas influenced by the .— attitude of official regulations. They required that a drop test be made in which the upper end of the shock strut was connected with a mass ml in the notation of this paper) and the two dropped on an ( anvil. At the instant when the anvil was struck, the weight W1 was — compensated by admitting compressed air to two cylinders.

The load- – stroke curve obtained by this test was then considered as “the” load- - stroke curve of the shock strut and was employed in all landing cases .— which had to be considered in the design of the airplane. _Consequently, the effort nf the early research was directed”toward the investigation .— .— of load-stroke diagrams of shock struts. ~~ .

The first paper that must be mentioned here, and the only one that appeared before the.war, was written by Michael (reference 1). It gives a detailed analysis of.the linear spring-damper system but pays only slight attention to the tire. A special feature of this paper is the use of spring diagrams in which the force is ‘~lottedeither against the” stroke with the rate of stroke as a parameter or, inversely, against the rate of stroke with the stroke as parameter. These diagrams are shown also for shock struts with dry friction or with velocity-square dampers, and they are used for a graphical solution of thediffkrential equation. Such diagrams are no longer possible when a second spring (the tire) is yresent, and therefore they have not been employed again in later ~apers.

.

— The first papers of the war period were .stillfocused on the load- stroke diagram.

Schlaefke (reference 2) criticized the drop-test method and suggested replacing the buffered drop test by an unbuffered test, that is, omitting the air cylinders and with them a possible source of + inaccuracy. His paper usesthe theory of the linear spring-damper system to establish some relations between the results of both tests.

.

NACA TN 2743 .

In a later paper (reference 3) the same author realizes that the damping in the oleo strut is far from proportional to the rate of .

stroke. He compares load-stroke curves for linear and for velocity- square dsmping and arrives at the strange conclusion that the former look more realistic.

‘I’&method used for the analysis of the nonlinear problem is of interest. A balance of kinetic and potential energy is established and from it, a differential equation between k as dependent and x as independent variable.

When it is solved, the damping force (proportional to k2) is also known in terms of x. However, this ayproach is not possible in the presence of a tire.

In the next group of papers the tire makes its a~pesrance. A paper by Kochanowsky (reference 4) gives a very detailed analysis of the oleo-tire combination as shown in figure 1. Kochanows@ finds that the unsprung mass is of no great importance for the landing impact and that the problem may readilybe simplifiedby assuming ~ = O. The study of this paper (and of many others) is rendered difficult by the author’s habit of using for all and everything dimensionless quantities so that the reader has to learn first a system of not very suggestive notations before he can follow the analysis or read the diagrams.

Another paper by Schlaefke (reference ~) covers approximately the same ground.

After having studied the linear oleo-tire system, the next logical .

step would have been to consider a nonlinear shock strut, but, inciden- tally, the few papers which did this were older than Kochanows~ts com- prehensive paper on the linear system.

One of them is by the same author .

(reference 6), and it was not ~hought to be a study of a nonlinear case.

It is concerned with a special type of spring which has long been used in railroad-car bumpers and was introduced in landing gears. It con- sists of a pile of ri~s with conical sides (fig. 26). When it is subjected to an axial compression, the hoop stresses in the rings are alternatively tensile and compressive.

During the elastic deformation, the rings slip on one afiotherand the pile becomes shorter. Because of the slip, there is -considerabledry friction, and when the load F is decreased, the deformation x is not immediately decreased but follows a law which is described by fi~re 27. ‘I’he area of the tri- angular loop represents a loss of energy and the ring pile may thus be used as a damped spring.

Kochanows@’s paper considers a shock-strut and tire combination in which the strut has no other elastic or damping element except such a ring-pile spring.

During the first upstroke the analysis is extremely simple, since not even damping appears explicitly in the eq~tio~j but when the motion is followed beyond the force maximum, it is linear only in sections but nonlinear on the whole. The .

paper is an interesting study, but the ring-pile shock strut is not versatile enough to stsmd the competition with the modern oleo strut, and the problem is now obsolete.

-’ NACA TN 2743 .

The other paper %hich considers a nonlinear shock strut is a very serious and very detailed study by Marquard and Meyer zur CapelJ.en .— * (reference 7). The authors consider velocity-square damping and polytropic compression oethe air, formulate differential equations, Utiortunately, the authors overestimate and integrate them numerically.

the accuracy requirements of the analysis. .-In their tables values are given to six and even seven significant di~~ts, and consequently they — employ an exact but very tedious method of-step-by-step integration.

.- In addition to the detailed treatment o.fthe nonlinear shock strut, the paper is remarkable-for another reason. It not only considers an oleo-tire system with a very realistic shock strut, but it also considers .— the motion of the whole airplane in itsplane of symmetry. In a second paper (reference 8) the same authors extend their investigation to cases of unsymmetric landing. But here also the attempt at exactness goes too - far when the decrease of horizontal speed dining the short imPact time.

— is taken into consideration. This is pointed out in a paper by Scbmitz .— (reference 9). This author also considers the pitching motion of the airplane and includes the ensuing change of:the lift, but he falls back — to the old idea of “the” load-stroke curve fid fails to realize that the cooperation between the elastic reaction of the air and the dam@ng force caused by the orifice depends largely on the conditions of the impact.

Besides the landing impact, the taxying of the airplane has always met with interest. Michael’s paper (reference 1) pays attention to it$ I.

and Kochanowsky’s papers (references 4 and ~) both congider the taxying — In these papers the...statement. is made and proved impact in full d~tiail.

,.

mass ml * that when the airplane rolls over a sinusoidal ground swell, the travels ~actically on a level path and tha~ therefore the analysis may be made on the-assumption that ml = m.

Besides these papers there are two by Schlaefke in which taxying is considered. O“neof them (reference 10) covers the same ground as .- .— the corresponding part of Kochanows@’s paper (reference 4). The second (reference 11) is a short note concerning the im~ct during the take-off -.

run. It seems to. be the only paper devoted..to this subject, and not .“ much information is found in it.

Additionally> there are a number of reports on experiments. .Most of them were tests made by the airplane manufacturers and served essentially the ptipose.of improving a new,.qirplane modelto the Pint ,— where it was ready for production. Today, it is difficult, if.not impossible, to draw other than qualitative~@cymation from these reports since the-airplanes; shock struts, and tires used in these tests no - longer exist and details needed-for an anal@is may no longer be obtained readily.

.

.

NACATN 2743 10C However, one of the experimental papers must be mentioned in this review, a short report of H?5ke(reference 12) on the experiments he .

He measured, as made in the Deutsche Versuchsanstalt f%r Luftfahrt.

functions of time, the vertical velocity of the airplane immediately and lateral forces before and during the landing impact and the vertical velocity measure- on the wheel. The fine experimental technique of the ment is described in the paper.

Stanford University Stanford, Calif.j November 15, 1951 74 NACA TN 2743 .

BIBLIOGRAPHY OF GEW LITERATURE —— — * 1. Michael, Franz: Theoretische und experimentelle Grundlagen &r die Untersuchung und Entwicklung von FlugZ%ugfederungen (Theoretical and Experimental Principles of Landing Gear Research and 14, Lfg. 8, Aug. 20, 1937, Development). Luftfahrtforschung,Bd.

pp. 387-416.

2. Schlaefke~ K.: Zum Vergleich von gepufferten und ungepufferten Federst6ssen an Flugzeugfahrwerken (“Buffered” and “Unbuffered” Impact on Landing Gears). T. B., Bd._lO, Nr. 5, 1943, PP. 129-133.

— 3. Schlaefke, K.: Zur Kenntnis der Kraftwegdiagramme von Flugzeugfederbeinen (On Force-Deflection Diagrams of Shock Struts).

1. Teilbericht - Vergleich von Diagrammen mit- linearer und quadratischen D&hpfung (Comparison of Diagrams with Linear and Quadratic Damping). T. B., Ed. 11, Nr. 2, 1944, pp. 51-53.

2. Teilbericht - N6herungsverfahren zum Berechnen der Kraftwegdiagramme mit nichtline~er Federkennlinie und linearer oder quadratischer D&npflmg (Approximate Method for the Calculation of Force-Deflection Diagrams @th a Nonlinesr Spring Chart and Linear or Quadratic Damping). T.,B., Ed. 11, — .

Nr. h, 1944, pp. 105-109.

3. Teilbericht - Der Landestoss von blluftfederbeinen (The Landing , Impact of Oleo Legs). T. B., Bd. 11, Nr. 5, 1944, pp. 137-141. ‘ 4. Kochanowsky, W.: Landestoss und Rollstoss von Fahrwerken mit fliissigkeitsgedimp~enSchraubenfederbeinen (Landing and Taxying Impacts on Oleo Shock Struts).

Untersuchungen und Mitteilungen Nr. 1423, Deutsche Luftfahrtforschung;liov. 14, 1944.

5. Schlaefke, K.: Zur Kenntnis der Wechselwirkungen zwischen Federbein und Reifen beim Landestoss von Flugzeugfahrwerken (On Reciprocal Effects between Shock Strut and Tire in Landing Impact of Airplane — Undercarriages). T. B., Bd. 10, Nr. 11, 1943, pp. 363-367.

6. Kochsliowsky, W.: Iandestoss und Rollstoss von Fahrwerken mit Ringfederbeinen (Landing and Taxying Impacts on Landing Gears with Ring S-pringStruts). Forschungsbericht Nr. 1757, Deutsche — Iuftfahrtforschung,Yeb. 8, 1943.

.

NACA TN 2743 b 7. Marquard, E., and Meyer zur Capellen, W.: N~herungsweise Berechnung der zwischen Fahrgestell und Rumpf beim Landen auftretenden l Federungskr3ft e. Symmetrische Landung (Approximate Calculation of the Forces between Landing Gear and Fuselage of a Landing Forschungsbericht Nr. 1737/1, Aircraft. Symmetric Landing).

Tech. H. S. Aachen, 1943.

8. Marquard, E., and Meyer zur Capellen, W.: I%herungsweise Berechnung der zwischen Fahrgestell und Rumpf leim Landen auftretenden Federungskr&ft e. Unsymmetrische Landung (Approximate Calculation of the Forces between Landing Gear and Fuselage of a Landing Forschungsbericht Nr. 1737/2, Aircraft. Asymmetric Ianding).

Tech. H. S. Aachen, 1943.

Bewegungsvorgang, Stosskr~fte und Federwege bei der 9. Schmitz, G.: Landung eines Bugradflugzeuges (Motion, Impact Forces and Spring Strokes in the Landing of aNose-Wheel Airplane). T. B., Bd. 10, Nr. 12, 1943, pp. 389-392.

10. Schlaefke, K.: Zur Frage der Rollstossbeanspruchung von Flugzeugfahrwerken (On Rolling-Impact Load on Airplane Landing Gears). T. B., Bd. 11, Nr. 9, 1944, yp. 289-295.

Zur Ermittlung der Beanspruchung von Flugzeugfahrwerken 11. ScQlaefke, K.: beim Start (On the Forces in Landing Gears during Take-Off).

Bd. 10, Nr. 1, 1943, pp. 29-30.

T. B., .

12. H6ke, H.: Besnspruchungsmessungen sn Fahrwerken bei der Landung und l beim Rollen (Measurements of Landing-Gear Loads during Landing Bericht 169, L.G.L., 1943, pp. 28-37.

and Taxying).

v

..

— Figure l.- Representation of shock strut by spring and damper arrariged in psrallel.

Jw

I .

b I — -.

IF

.

time.

(b) At some later (a) At t = O.

.

system in two positions.

Figure 2.- Mechnical b ~/m, g b 2 ‘ / fn2/m, I “

=$9=

o i 1,0

t/T ‘“5

(a) Onshock-strut force. .

.

~ : I *S I .0

VT 1*5

(b) On impact force.

Figure 3.- Influence of unsprung mass.

F@t,g , I,o 2 ‘ 66’ \ / b T/m, I o 0 .5 100 ~,-r 1.5 (a) Force on wheel.

%/6 I .5 1.0 ~,T 1,5 .

(b) Development of stroke.

.

Figure 4.- Influence of damping--on landing @act.

l

F

b I o I o . 100

t/T 105

(a) Force on shock strut.

.

I

I o .

(b) Force on wheel.

. Figure ~.- Influence of spring constants on landing impact.

.

NACA TN 2743 ?

0,25 -0.15 A k, /(k,+ k,) al I - 0.?5 x,, 0.15 o ) 1.0 t/T 1.5 .5 (c) Displacement of airplane and deformation of tire.

.— Figure 5.- Concluded.

.

* I .—

+’

, , o ,5 1,0 t/T 1,5 Figure 6.- Influence of weight .mdJift on landing impact, NACA TN 27’43 llC \

Jw

.

I / —— _— .— — —— / /-- / /-

—— — —— — — I J——

I I ‘YF&Zg7

.3

Figure 7.- Landing gear encountering obstacle during taxying.

.05 ----- -“ /.

. ------ l-r ---m __* e l **’ l *

1-

+ ——

%/6

.10 .

.15 .1 .2 .3 *,T .4 .

Figure 8.- Displacement of wheel when obstacle is encountered during taxying.

82 .“ NACA TN 2743 f l I 0 .1 .2

“3 t/T l 4

Effect of encountering obstacle during tsxying on forces in Figure 9.- shock strut and in tire.

F I .

I

——

-T’

Z* z * —— P.

AT

‘-1=1=

—— ‘– P, —.—

--

—- -- — --

tt I

PI - i.

— — —

=

-1-_ L

— —

1--1-

P; — P;

—

r

B 9 I

[1 L L ——

i- x x .

—— -T

!

F

F

I

I

‘v

l (-b) (a) Figure 10.- Different forms of oleo struts.

NACATN 2743

Y

t in tn4 t Figure 11.- Illustration of trapezoid method of integration.

I I I I lw@Ji7 Figure 1.2.- Illustration of polynomial qethod of integration.

.

.

NACA TN 2743 I .0 c .-

“ I

1 II

c i I ) 15 x, in.

.

-“ l

Ii

I

I

J

*

El

Figure }3. - Cross section of metering pin.

NACA TN 2743 t .

l .1 t,sec ., 0 .1 .2 t -to,sec Figure 14. - Displacement xl and stroke x a%in~t time.

8oxlo3

I I I

6C / n ‘- 40 L- .

.1 t, sec .2 ,3 0 .1 .2 t -to,sec l Figure 15. - Force-time history of shock strut.

NACA TN 2743 i l n

x-

x /

‘=s=”

,1 f. sec .2 *3

o’ :1 :2 “

t-to,sec Figure 16.- Displacement xl and stroke x for different metering pins.

u 80 ‘-3

‘“ 40

L- .1 t, sec ,Z ,4 I I I I 1 I o .1 t-to,sec .2 Figure 17.- Force-time history for different metering pins. -- NACA TN 2743 15C I I a ,

R

it

Y .1 t, sec .2 l * r I I I I I t-to, sec ,2 o .1 Figure 18. - Vertical velocity i, and rate of stroke ~ for different L metering pins.

6C 4C 2C ( 2 4 6 8 10 I x2, in.

Figure 19.- Load-deflection curve of a tire.

88 NACA TN 2743 .- .- Figure 20. -’Example of landing gear with shock strut inclined from vertical.

////////////////// \ f — .— .— .— .

Figure 21. - Cases for which relation between stroke and displacements is nonlinear.

12C NACA TN 2743 l .

.

‘b

/

-~

~

‘aT ---

////////

//////////////27 Figure 22.- Schematic side view of airplane at time of first contact.

Figure 23.- Schematic side view of airplane at time of three-wheel contact.

NACA TN 2743

Kij&jj7 ~1

IF

I I Figure 24.- Schematic views of airplane during one-wheel landing.

. — Situation at first--contact.

w

(l o

/////~///////////J w/////

F

F

I 3

during one-wheel landing.

Figure 25.- Schematic views of airplane Situation when second wheel hits runway.

NACATN 2743 b .

I

KJi-xJti2j7’

I

I Figure 26. - Ring-pile spring.

tF

Figure 27. - Force-str~ke diagram of a ring-pile spring.

NACA - Langley Field, V&.