RESPONSE OF A NONLINEAR TWO~DEGREE-OF+FREEDOM SYSTEM

SUBJECTED TO AN IMPACT LOADING

by 7:

Jerome G+ ‘Theisen

Thesis submitted to the Graduate Faculty of the

Virginia Polytechnic Institute

in candidacy for the degree of

MASTER OF SCIENCE

in

Applied Mechanics

APPROVED: APPROVED:

Director of Graduate Studies — "Head of Department

— Supervisor

August 10, 1956

Blacksburg, Virginia

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TABLE OF CONTENTS

LIST OF FIGURES

INTRODUCTION

NCVENCLATURE

PHYSICAL CHARACTERISTICS OF THE SYSTE

EQUATIONS OF MOTION

Derivation of the Equations of “otion

Introduction of Dimensionless Variables

Transformation of the “quations

SOLUTION OF THE TRANSFORMED EQUATION

Considerations Necessary to Obtain a Solution

Presentation of the Solution

Qualifications of the Solution

EVALUATION OF THE ANALYTICAL SOLUTION

Comparison with Solutions by the Runge-Kutta Method

Comparison with Experimental Data

CONCLUSIONS

BIBLIOGRAPHY

VITA

APPENDICES

A. The Derivation of the Solution to the Transformed

Equations

Re The Determination of an Approximate Value for Un

C. Constants of the Physical System

Page No.

10

22

22

25

27

29

29

31

3h

Nd

16

51

53

55

55

61

-~3-

I. LIST OF FIGURES

Figure No. Page No.

1. ‘Two degree of freedom system considered in analysis. 11

2a. Schematic representation of shock strut, wheel and

tire with balance of forces on lower mass. 12

2b. Schematic representation of shock strut and balance

of forces on upper mass. 13

3. Comparison of the true force-deflection curve of the

tire and a linear approximation to the tire spring. 20

he Plot of the integration of the incomplete elliptic

integral of the second kind, E(v), with respect to

the argument v. : 35

5. Comparison of the actual dimensionless spring-force

curve and the approximate spring-force curves for

various initial conditions. 36

6. Comparison of the true maximum dimensionless lower mss

displacement, u,, and the approximation of Uy for

the full range of the initial velocity parameter, u's 37

7+ Plot of the ratio of energy which the linear spring is

eapable of absorbing to the total energy which both

the linear and cubic springs can absorb, as a function

of the initial velocity parameter, ust ho

Ficure No. Page No.

8. Comparison of solutions for lower-mass displacement

and upper-mass acceleration obtained by the Runge~

Kutta procedure and from the analytical equation, 2

9, Comparison of solutions for upper-mass displacement

obtained by the Runge-Kutta procedure and from the

analytical equation. 13

10. Comparison of solutions for the lower-mass velocity

obtained from the Runge-Kutta procedure and from

the analytical equation. bb

1l. Comparison of solutions for the upper-mass velocity

obtained from the Runge-Kutta procedure and from

the analytical equation. LS

i2. Comparisons between theoretical results and experi-

mental data on upper and lower-mass velocities and

displacements for a typical impact. u,' = 8.86

feet per second. h?

13. Comparison between theoretical results and experi-

mental data on the upper-mass acceleration for a

typical impact. 48

1. Comparison between theoretical results and experi-

mental data on the ground vertical force for a

typical impact. 50

~ 5.

II. INTRODUCTION

The solution of the equations of motion of an aircraft fuselage-

landing gear configuration during landings is of interest to the designer

who must predict the landing loads which an airplane encounters in serv~

ice. In general such solutions are difficult because of the highly non-

linear characteristics of the oleo~pneumatic shock strut which couples

the lower mass of the landing gear to the fuselage.

In the past, attempts to obtain solutions by linearization of these

shock strut characteristics have resulted in unrealistic predictions of

landing gear motions. Therefore, it has been necessary to carry out

most of the theoretical analysis associated with landing gears by means

of numerical integration procedures. These numerical methods are tedious,

and as a result a large portion of design work has been carried out by

means of trial and error drop testing of a system of masses representa-

tive of an airplane and landing gear. This in turn has proved to be

time consuming and expensive.

This paper presents a method for obtaining an analytical solution

of the equations of motion for a basically nonlinear system which closely .

resembles an actual airplane and landing gear configrrotion. The non+

linear system considered has two degrees of freedom and is composed of a

large mass representative of the fuselage~wing combination connected by

an oleo~pneumatic shock strut to a wheel. The shock strut is assumed te

have velocity~squared hydraulic damping and coulomb friction forces on

m & =

the strut bearings. The nonlinear spring characteristic of the tire is

represented by a sectionally linear spring.

In the first part of this paper the equations of motion for this

nonlinear system are derived making use of a few simplifications which

previous papers have shown to be justified. Also, the degree to which

these assumptions limit the results is discussed. Next these equations

of motion are solved in analytical form by a method which may be called

"equivalent nonlinearisation." It is shown that this solution is exact

only for a specific combination of impact parameters, but that for a

wide range of parameters the solution describes the motion of the system

adequately for design purposes. Finally, a few analytical solutions are

compared with solutions obtained by numerical integration methods; and

the results are compared with experimental data for a typical impact.

-7~

III. NOMENCLATURE

pa, 3 coefficient of quadratic damping tern, —_t,

2(CgAo) pneumatic area

hydraulic area

area of opening in orifice plate

coefficient of friction~force tern, aya

orifice discharge coefficient

=: E(v) Legendre's incomplete elliptic integral of the second kind

ineomplete elliptic integral of the first kind,

v= 3 \ 2Hu,'(1 + e7*/3)

air compression force in strut

axial friction force in strut

hydraulie resistance force in the strut

total axial shock-strut force

vertical force, applied at the ground

normal force on upper bearing (attached to inner cylinder)

normal force on lower bearing (attached to outer cylinder )

constant satisfying equal energy condition, \2 ~- Rig ~ X;)

nondimensional constant dependent on the wing lift,

ate - ea Ko 1l-B We

Slope of the assumed tire deflection curve

wing-lift force

~ 6 =

approximation to the average distance between bearing

surfaces during the impact, (a, + 3) ~ (4 ~ v»)|

axial distance between upper and lower bearings, for fully

extended shock strut

total weight of system, W, + Wy

weight of the upper mass above strut

weight of the lower mass below strut

outer radius of upper bearing, attached to inner cylinder

outer radius of lower bearing, attached to outer cylinder

distance from the axle to the strut centerline

base of natural logarithms (2: 2.71828 .. .)

gravitational constant

modulus of Jacobian elliptic functions and integrals

polytropic exponent

air pressure in upper strut

initial air pressure in upper strut

pressure in the lower chamber of the strut

shock-strut stroke (displacement), (2, - %9)

maximum possible stroke (fully closed strut)

time after contact

time after contact before the equations of motion apply

to the system, 920908

Be

dimensionless lower-~mass displacement from position at

initial contact

p

9

-~9-

maximum value of displacement during a given impact

dimensionless upper-mass displacement from position at

initial contact

dimensionless initial contact velocity

initial air volume within strut

vertical displacement of the upper mass

vertical displacement of the lower mass

initial contact velocity

dimensionless displacement in the transormed equation of

motion, + ® y/2

maximum value of displacement during a given impact

dimensionless time from time of initial contact

coefficient of friction for the bearing surfaces

mass density of hydraulic fluid

any w amplitude or v

The use of dots over symbols indicates differentiation with respect to

time,

Prime marks indicate differentiation with respect to dimensionless time,

8, in the text.

IV. PHYSICAL CHARACTERISTICS OF THE SYSTEM

Other investigations have indicated that flexibility of the fuselage

and wing structure of an airplane usually has only a small effect on

landing gear performance (refs. 1,2, and 3). For this reason in the

dynamical system to be investigated, the fuselage and empennages of the

airplane may be represented by a single large mass, Wa» which is assumed

to have freedom only in vertical translation designated by a coordinate

Z + The lower mass, Wy, which is considerably smiller lies below the

oleo strut and consists principally of the wheel, tire, axle, and strut

inner e¢ylinder assembly as shown in figure 1. The strut is assumed to

be infinitely rigid in bending. Horizontal forces at the axle, usually

ealled drag or spring back loads, will not be considered in this inves-

tigation. Therefore, the lower mass has freedom only in vertical trans-

lation designated by the coordinate Zoe Lift forces are externally

applied to the system to represent aerodynamic lift.

Figures 2a and 2b show a schematic representation of a typical oleo-

pneumatic shock strut. The lower portion of the cylinder is filled with

hydraulic fluid and above this is an air space in an enclosed chamber.

The outer cylinder contains a perforated tube which has a small plate at

its base containing an orifice. When the strut is compressed the fluid

is forced through the orifice at high velocity, and the pressure drop

across the orifice resists closure of the strut. The turbulence thus

created dissipates a large part of the impact energy. This motion of

the strut also increases the air pressure in the upper chamber causing a

force which resists closure of the strut.

LLL.

SON

ASS

Upper mass —- Wy ¢

©)

O aN

SO

LLLLLLLLLLLS.

Air compression spring

Outer cylinder——]

|_——>~ Bearing surfaces \

-45 Orifice

Hydraulic damper

(velocity square)

Lower mass

Tire spring, Ko

Figure l.- Two degree of freedom system considered in analysis.

-1-

Fa = Paka

po LA Cut section rf ji! eos faccirs |

lot TT) ope A | Py | ye i

| Th antl] pov nk

I Ho | | OF | |

|! Vy | ql |

edufou tot | — \ ! { | i! I | o

ao | 1 + a | | | | S

aa : | tH A + || m

ae ce | | | ———— Quter cylinder {| ; | i |

attached to upper mass I | | . |

iff i | | ofus

| | | | ee + foo oo

| | J

I~ Orifice and supporting tube

attacned to upper mass

Fh = (Ph ~ Pa An N. *\aner cylinder

S=Z,~Zp

Lower mass

FY Ground vertical force

Figure 2a.~ Schematic representation of shock strut, wheel and tire with balance of forces on lower mass.

-13-

L Wing lift force

Lj.

g ol

Wy

= ——| Fa = Pada |

in| re E H oF z

|, || Fluid i |, # 4H TI

| tI A U outer cylinder

nit S = 2, 7 29 I ||

| H | i | i!

<——————Inner cylinder attached || to lower mass lA Fo | tae . ae Ho] | | | --erifice and supporting tube

jy Fh = (py > Paap

Figure 2b.- Schematic representation of shock strut and balance of forces on upper mass.

~ 1h -

The hydraulic resistance force, fF), in the shock strut can be

derived from the equation for discharge through an orifice, namely,

Q @ ALS = Cady \ &(Py ~ Pa) (1)

where the dot refers to differentiation with respect to time and

Q volumetric rate of discharge

Ap area subjected to the hydraulic pressure drop where the smiil

difference between A, and (A, ~ A,) is neglected .

Ph pressure in the lower chamber

Py pressure in the air (upper) chamber

§ velocity of closure of the strut, (8, - %)

Ca discharge coefficient

A, orifice area

p mass density of the hydraulic fluid

Since

Fi - A, (P, ~ P,) (2)

equations (1) and (2) may be used to derive the expression for the

hydraulic resistance force:

pA?

« ——2 4/8 (3)

where

beh, - a,

In this equation the absolute value of one velocity factor is retained

-~15 ~

to indicate that this damping force is the usual quadratic damping which

retains the sign of the velocity. The orifice coefficient, Cg, may be

considered as a constant during the compression stroke, according to

recent experimental data (ref. );). Since maximum impact loads are

reached prior to maximum strut stroke, the designer is primarily inter-

ested in the strut compression process rather than the elongation

process. Most struts have a pressure operated check valve which comes

into operation when the strut velocity changes sign, thus closing off

the main orifice. Fluid is then forced to return to the lower chamber

through @ number of small passages the total area of which usually varies

with the stroke in an indeterminate manner. The orifice coefficient also

4s unknown for this phase of the impact. Therefore, solutions for the

motion are made only for the closure case, although in deriving the

equations of motion the sign of the velocity is indicated where appro-

priate for clarity.

The movement of the inner cylinder compresses the air in the upper

part of the strut and this results in an air-pressure force, Fas which

may be determined by means of the polytropic law for compression of gases

n ep, | ——S— () Pa ™ Pa,| vo - Ags

where

Pa air pressure in upper strut

Ps initial airy pressure in upper strut °

v initial air volume within strut

-~ 16 -

8 strut stroke (2, - %)

A, the pneumatic cross sectional area

n polytropic exponent

(vg ~ Ags) the air volume for any stroke

Then

n Yo

Fa @ Pata * Pa Aa ( “= Ags (5)

Other investigations have revealed, however, that the air conmpres~

gion process has only minor effect on the landing gear loads which are

developed (refs. 5 and 6). This is true principally because the air

pressure force becomes relatively large only for maximum values of strut

stroke, which always occur long after the principal part of the impact

is over. Therefore the air pressure force will be neglected in this

analysis.

Friction forces within the strut result from the relative sliding

of bearing surfacea on the inner and outer cylinders. For the type of

janding gear being considered, vertical loads applied at the axle are

eccentric to the strut center line and produce a bending moment which is

resisted by forces F, and F, normal to the upper and lower bearing 1 2

surfaces respectively as shown in figure 2a. Conditions of relatively

high normal pressures, slow sliding velocities of the bearing surfaces,

and the poor lubricating properties of the hydraulic fluid lead to the

conclusion that the internal friction forces my follow laws similar to

those of dry (coulomb) friction; that is, the friction force is

~17-

proportional to the normal force. Other landing gear studies have used

this same relationship (refs. 5 and 7), although at present no conclusive

experimental evidence in support of this assumption is available.

If it is assumed that the friction coefficient is the same on each

sliding surface, then the axial friction force, Fp, may be determined

from the equation

reo re (lrl + Pal (6)

or, since summation of horizontal forces (fig. 2a) indicates that

F, * F, for this strut,

3 Fos lr £ Tar 2 /Fal

where

LL coefficient of friction

He factor to indicate sien of friction force

Fy normal force on the upper bearing

Fy normal force on the lower bearing

It is desirable to obtain Fy in terms of the resultant axial force

within the strut expressed by F Fp + Fa + Fes If Ry is the

distance between the upper and lower bearing surfaces for the fully

extended strut, then this distance may be expressed as (Ry + 8) for any

other strut position. Considering the forces applied to the lower strut

piston as a free body, and taking moments about position M in figure 2a,

the equation for Fy is

~ 18 =

(Ry + 8)Fy ™ BF, (d + b,) + uF (d ca by) + a(F, ” oy ” WF) (8)

where b, and By are the outer radii of the upper and lower bearings,

respectively, and d is the distance from the axle to the strut center

line. Since Fy, * Fo, the equation for F, may be found as

d. "1." Ty *s) - nib; - by *s (9)

In order to obtain a solution as presented in this paper, it is

necessary to express the quantity [Ry +3 ~ u(by ~ oe) as a constant.

A similar assumption was made in reference (7). If R, is large with

respect to the maximum value of s which is possible, s,, then a reason~

able constant to use would be ( + 2) “u(b, - v2)] R where R is

approximately an average distance between the bearing surfaces. 8,

usually is quite well defined by the strut geometry, and the accuracy of

this assumtion for R depends on the particular strut being considered.

The friction forces are often quite small for the simple vertical impact

cases and the inclusion of friction, as will be shown, does not comli-~

cate the final solution obtained. Therefore it was felt that this term

should be included for possible future studies on the qualitative effects

of friction in the landing gear.

The normal bearing force from equation (9) may now be expressed as

and from equation (7)

Fp = ir a F.| (11)

~ 19 «

Note that since F, is the total axial strut force, it can be

expressed as the summation of these internal forces:

Fe * Fh * Fa + Fe (12)

For this analysis the portions of mass comprising the inner and

outer cylinder of the strut are considered to be weightless, and their

weights are included with the major portion of the masses to which they

are respectively attached, The assumption that the center of gravity of

the lower mass lies at the axle center line also is a usual stipulation

in such an analysis,

Recent data (refs, 5 and &) indicates that the dynamic force deflec«

tion characteristics of the tire my be sharply nonlinear for small

deflections, but can be represented by a linear approximation with only

minor loss in accuracy for the major remaining portion of the curve,

Thus a linear segment approximation to the true tire spring may be made

as in figure 3, For the 27-inch-diameter tire being considered, this

assumption gives no force for deflections less than Z%» * 0.0508 foot,

and a constant slope Ke thereafter, Such a representation seems to bea

entirely adequate for most practical ourvoses (ref. 5).

If F, is defined as the vertical force on the ground resulting

from deflection of the spring Kos then the force may be written

FL 20 for 0 ¢% ¢ 0.0508 foot (13a)

Fy = Kno for Bo > 00508 foot (13b)

lb

Spring

force,

- 20 -

True force-deflection curve of tire

Linear approximation to the tire spring, ko>= 21,300 lb/ft

| _ it ! L |

016 2h; 032 00 06

Tire deflection, ft

Figure 3.,- Comperison of the true force-deflection curve of the tire and a linear approximation to the tire spring.

Finally, the wing lift from aerodynamic forces is represented by

the force L acting through the center of gravity of the upper or fuse-

lage mass. Because of the very short time involved to complete the

impact, the wing lift can reasonably be assumed to be a constant. Recent

experimental data (ref. 9) indicates that for most landings the wing lift

at ground contact is very nearly equal to the weight of the airplane.

During the experimental droop tests used for comparisons with theory in

this analysis the wing lift force was simulated so that it was equal to

the weight of the test model.

Ve. BQUATIONS OF MOTION

Derivation of the Equations of Motion

With F, and F, defined as in the preceeding section, the equa-

tions of motion may be written in simple form. Considering the summation

of vertical forces on the lower mss, (fig. 2a), the mtion of the axle

is given by the equation

W Br F | 2

Py ~ Fy + g 29 (1)

Simllarily, the vertical motion of the upper mass (fig. 2b) is determined

by the equation

y Fg + LW) - 3 ay (15)

Where L is the wing-lift force as defined earlier. If equations (1))

and (15) are combined, the overall balance of forces can be expressed by

y We

Fy +t b= We ~ By =F 89 (16)

where

We Wy + Mo

Any two of these equations is sufficient to obtain a solution, since

there are two degrees of freedom, In this derivation, equations (1))

and (16) will be used.

In order to obtain equation (1),) in a more desirable form we mst

express the strut axial force more explicitly. From equation (12)

Fo =F, + Fy + Pp

or, since the air pressure force is neglected for the reasons given in

the preceeding section,

sy ae erage ba hl on Some simplifications become obvious at this ooint. Since the strut

compression process only is of interest, the atrut velocity and strut

axial force will always be positive. Therefore, the absolute value

Signs may be removed so that

F, = Ad? + BF, (18)

pL?

(Cho)

ao Equation (18) can be solved for F, giving

A =

A 2 Fe" 773 3 (19)

Using the definition for 8, substitute F, and F, from equations (13b)

and (19) into equations (1),) and (16) with the result

W 22 -~ —A__[ 3, - 25)° + Koay - Wy = 0 20 2 aay (4 2|° + Koay ~ Wy (20)

- 2h -

Wr. | Wo, zt ee? Koy +L = W= 0 (21)

Note that the net effect of including the friction force in the

equations of mtion is only to change the constant coefficient of the

quadratic damping term. In references (5 and &) it has been determined

that since the lower mass is a relatively small fraction (less than 1/18)

of the total mass, the system my be sinzslified even further. Caleula-

tions made in these references indicate that the lower mass may be

assumed to be equal to sero without greatly modifying the results.

Using this assumption, equations (20) and (21) may be written as

(i, - i,)? - rae | Zp = 0 (22) A

Eko w \ £ By + WL Zo» + (L - uy) a = 0 (23)

These are the equations to be solved. The initial conditions for the

vertical impact are given by

t #0

Zy * 29 - 0 (2h)

fi) * Zp * 2

However, the assumption of equation (13a) implies that the system mst

move 2 distance equal to 0.0508 foot after initial contact before any

finite ground reaction can develop. Since equations (22) and (23) assume

that the ground reaction increases linearly with deflection, these

~ 25 -

equations do not apply until some time by after contact. The initial

velocity remains essentially constant over this very short interval of

time, 8G %, can be found from

i, : (25) °

The initial conditions (2);) still apply to the equations of motion in

this case with the coordinate system transformed go that the tire force-

deflection curve passes through zero. In plotting the solutions, all

results must be displaced in time by t; seconds, and the displacements

will have 0.0508 foot added, to indicate the actual distance through

which the model has moved.

Introduction of Dimensionless Variables

The solution of equations (22) and (23) depends on six parameters,

namely, A, B, 2, Koy L, and Z + It is desirable to reduce the number

of parameters, and this can be done by introducing generalized variables

wu and u, in the following transformations:

A nu = * 4 W (2 - B) “2

(26) Ag

“1 Wy = F)72

and fia

9 = eT t (27)

- 26 -

so that:

ui aa A de 2|(i = 8B)

nit moe eG A 1 ds 1}(iT -'8)

and

a2 te ote A wa" * age 7 ALT)

With these new variables, equations (22) and (23) become

[vy -w)?-u=0

uy" +u+ Ky #0

where

. A(L - Weg

Ky, Kol ~ BYW,

Of special interest in this anzlysis is the case where the wing

(28)

(29)

lift is equal to the weight of the system, as indicated earlier in this

paper. When L#* Wy, then K, © 03; and we see that equations (22) and

(23) will have a single parameter only, the initial nondimensional

contact velocity, which is

. Ae || eB Yo! ta = 3) a | (30)

- 27 -

Equations (28) and (29) can be solved simultaneously to give a single

equation in one variable. Rewrite equation (28) as

uy! ~ yt yl/2 (31)

where the positive root must be used since the equations of motion are

applicable only for positive strut-stroke velocity, and this implies

that (u,! - ut) mst be positive. ifferentiate equation (31) with

respect to 6. Then

uy" ~ yt = 5 wrt/2y1 (32)

Solve equation (29) for u," and substitute in equation (32). This

gives

ul +a +s uvl/2yt + K, = 0 (33)

Transformation of the Equations

In order to obtain equation (33) in a form more amenable to solution,

use the following transformation:

use (34)

Then equation (33) may be written

d la(r2 1 a(x? glace). 2) as 02 2 (35)

Aovly the transformation

6 ele

oo | se (368)

or

dx ® BEB) (36b)

The final equation obtained is

2 3 aS< . at a “ZS +a + 2K, + Qe 0 (37)

To determine the initial conditions for this equation, rewrite equation

(36b) as

2 ao = MD ax

By substituting equation (3h), it can be seen that

du . dt Ge & (38)

Therefore, the initial conditions are:

for x20

vs u,/? a Q

(39)

eu!

- 29 -

Vie SOLUTION OF THE TRANSFORMED EQUATION

Considerations Necessary to Obtain a Solution

An exact analytical solution for equation (37) is obtainable only

for a specific value of K;. An equation similar in form to equation (37)

for which a solution is known, is

5+ dee Bas 2itrd =o (ho) -

where H is an arbitrary constant. This is a special case of an equa~

tion studied in reference (10). It is apparent, then, that the solution

of equation (37) is identical to the solution of equation (1,0) with the

values K, ® 1/9 and H®1l1.

For any given landing gear configuration, this value of Ls implies

that only one value of wing lift, L# Wy, will suffice. As was indicated

earlier in this paper, for most landings L = W,3; and in general the value

of wing lift will not satisfy the condition that K, = 1/9. However, it

will be shown that with other values of K» equation (0) has solutions

reasonably representative of the solutions of equation (37) for certain

values of initial conditions and the parameter H.

A device sometimes used in mechanics is to replace a nonlinear

spring or damper system by an equivalent system which will have an equal

absorption of energy within the range of the variable which is of inter~-

est (refs. 11 and 12). Usually this method is used only when the non-

linear equation is a small perturbation from a corresponding linear

- 30 ~

differential equation, so that the nonlinear terms my be replaced by

equivalent linear expressions which may then be handled analytically.

In equation (37) the nonlinearity is so large that equivalent lineari-

zation results in solutions that are greatly distorted. Since an

analytical solution to equation (0) is available, in this case the

energy considerations may be equally useful in replacing the linear and

cubic terms in equation (0) by equivalent linear and cubic terms with

coefficients adjusted to those in equation (37). Considering these

expressions as spring forces, the energy stored within the springs is

the integral of the force with respect to the displacement. For this

study the primary interest is in the transient response during the first

half cycle only; so if the maximum value of displacement in this interval

is designated by Tn? the energy relationship between the actual and

equivalent spring is

| Bb «+ 212e3] ac . / [2Kys + 2x3] de (442)

where

Ty ™ Un

The determination of the value of Wn is to be discussed later in the

section entitled "Qualifications of the Solution." Since the other

constants are fixed, H must be adjusted to satisfy the condition of

equal energy. By performing the integration, the following expression

~ 31-

is found

2/1 = | 2/1

A \ a(3 ) 2 (3 1} (2)

For a value of H thus determined, equations (0) and (37) may be

considered as equivalent.

Presentation of the Solution

The solution for equation (1:0) as derived in appendix A is

— \Pe" e*/ 35g E 2Hu,' (a ~ e*/ 3) (43)

Ken /2

where, for convenience, the modulus k* 1/2 is understood to apply to

all elliptic functions expressed herein. Since u * x", it can be seen

that

Del oo on

us ar e2X/3.gq2 3 \2Hu,* i ~ @*/ | an)

,

Keeping in mind that @ is defined implicitly as a function of x by

equation (36), then equation (36a) may be rewritten as

x 8 - | ar( Ede

0

where the limits on x are chosen to satisfy the initial conditions

for 6.

~ 32 =

Performing the indicated integration, it is seen that

#2 ginth de [3 fattag’ 1-3} - 2 8 H Sin | [2 ed 3 rUbey (Ll S ) oH (45)

Thus the solution for u(9) is in parametric form with equations (':},)

and (5). This form is convenient for obtaining the higher derivatives.

The solution may be expressed ia explicit form, however, as

Up ' 1 2 2 6 u(@) e el ~ 3 a! @ sd°@ (hy )

where

Qs eat] 2 sin(3 Q + is), 2 ow. 1/2| (47)

The nondimensional velocity cof the lower mass can be obtained by use of

equation (38) as

ut(@) * ustll - —sd ey od@ 1 i | @} sae (18) 3 ° JYemu' | dno ~ 3) 2H 3 om,"

For design purposes, the motion of the upper mass is also of inter~

est. The nondimensional acceleration can be obtained from equation (29).

$ 2 Ub "(@) om so. 1 + ne @ sd@g - (19) 1 2H 3 ehug! L

fhe nondimensional velocity of the upper mass may be derived by use of

equation (20).

~ 33 ~

2. 1 1 ed® , 2 || Yo 1

31 '(6) = Uo ( 3 \2Hu,! dne 3 5 | 3 \2Huy' 0

The dimensionless displacement of the upper mass, may be obtained

by graphical integration of equation (50), or more simply, perhaps, by

substituting uj '(@) from equaticn (31) in an integral relationship as

follows:

8 Q uy (8) 7 uz '(y)dy = [aren + at(7)| dy | ul/2(y)dy + u(8)

(Sla)

Then, using the transformation equations (31) and (36a), and selecting

limits of integration which satisfy the initial conditions, it ean be

shown that

x

u, (x) * 2 u(t as + u(x) ($1) 0

Then u, (8) is available in parametric form with equation (5). How

ever, using equations (51b) and (1) the following expression may be

derived:

uy (x) = u(x) + =, fav) ~ $ vo $ saveds| (3 (era, ' - ‘ + 3H

2 v In(dny) = + [ E(v)dv (Sic)

where

v= 3\(2Rugt(2 = e@X/3)

& F(?,k) incomplete elliptic integral of the first kind

P= amBamplitude of v

E(v) © E@,k) incomplete elliptic intezral of the second kind

The integral term of equation (5l1c) may be evaluated graphically,

and the result is presented for convenience in figure Ae

Since the solutions contain the parameter H which is a function

of uy, we mst find a velue for this maximum displacement before a

calculation can be attempted. It may be indicated thet the actual value

of u, which is used as the limit of integration in equation (41) is

not critics] with respect to t>e final solution. Any good approxima te

value for Uy, will suffice since the ecuivalent spring force-deflection

curves will be negligibly different for small variations in the limit

of integration. This may te seen from figure 5 where the effects on

these curves are very small for large changes in uge A method for

determining a value of u, for use as the above mentioned upper limit

of integration is presented in Appendix B, where a very good approxi-

mation for u, as a function of up! is derived using only equations

(42) and (44) ‘These values are compared with the true values for

Up in figure 6.

The degree to which the solution obtained by the method of "equiva~

lent neonlineariaation” represents the solution of the oririnal equation

(37) depends primarily on the proportion of energy stored by the linear

and cubic spring terms. Asa indicated before, if Ky, “3 then the

- 35 - °A

QUeumMBIe

043 04

YOodser

Witm

‘(A)H

SputTH puocoes

944 Jo

TeIZ9IAGT

OFAAMTTTe

syeTdmoouy

944 JO

UOT IBIZSAUY

914 JO

40%

-°F oindTyZ

sro

Dimensionless

spring

force

- 36-

10r

3 9 Actual spring force, 2T

r Approximate spring force, (26T+2ET }

f — _— —-~---- case uO =1, uy, = 0644 Where

8f ———— case uo=5, uy, = 1-82

—_—— / — case u | o

ae

3r

2r

le

ee

a “1 ! ! J i i a 0 02 04 26 28 1.0 1.2 1.4 1.6

Nondimensional displacement

Figure 5.- Comparison of the actual dimensionless spring-force curve

and the approximate spring-force curves for various

initial conditions.

-37-

°O n

Sr1ejyemBred

AYTOOTSA

TeTqTUy

944

JO esuel

T [nj

2q3

z0ozJ ™

go wot

pr xordde

eyuy pues

‘Mm ‘quameoetdstp

SSBmI Jaa0T

sseTucts

Tp wnuUTxem

oni

944 Jo

uosyredmoyn

<-°9 aINaTy

oO

‘

8 é

9 S

¥ g

3 T

f— T

1 TT

T T

T

n ‘rezyomered

AYLOOTOA

SSaeTUOTSUSTTID TeT}IUL

10BxG

Wy “gow temoT eyz JO

jusmsortTdstp SseTuctsuelltp wHutxen

original equation and the equivalent equation (1,0) are identical and the

proportion of energy capable of being stored by each spring mst be the

gama. Usually, however, the case K, =O is of the greatest interest

go that only a cubic spring term remains in equation (37). In equa~

tion (0) the coefficient of the cubic term is a function of u,3 80 a8

u, gets smaller, the coefficient gets smaller, with the result that the

linear spring stores a greater proportion of the total spring potential

and equation (0) becomes less representative of the original equation.

It may be indicated, however, that this limitation is primarily

concerned with the frequency characteristics ‘and shape of the response

eurves. The energy considerations used in deriving the equivalent non~

linear spring imply that the maxim deflection of the approximate

spring system will be close to that of the original system, since the

work done in deflecting the springs to this maxim position is the same

for either spring system. |

The energy ratio spoken of indicates, then, to what extent the shape

of the response curves may be in error. This ratio of energy, r, can

be obtained from equation (1) for the case of K, * 0, as

5 tae

r« Q Ll.

=

Cnt a 2 2 [MB = + 2203] a 1+Z us,

0

or, by use of equations (3) and (2)

rs Fa, (52)

Since u, is known as a function of uo! (Fig. 6), this ratio of

energy may also be presented as a function of u,' as shown in figure 7.

Where the ratio of energy stored by the linear spring is a small fraction

of the total energy stored in the springs, the solution would be expected

to be very good. From figure 7 it can be seen that for fr > : and for

corresponding initial conditions u,! < 2, a more rigorous solution

would be desirable.

- O = °

Pn ‘194

emered

AYLOOTSA

TeT4YTUT

94q4 Jo

uotTyYOUNJ

B&B se

Qzrosqe

ueo

sButids

oTqno

pus

IBeUuTT

9804 YW10q

YOotum

ABISVUS TSO

94

04 Butqrosqe

Jo etTqedeo

st Sutads

reeutT

944 gota

ABr9Ua

JO OTWBI

901 JO

OTT

-*4 sInsTy

on

Sroq,ouered

AXYTOOTSA

SSeTUOTSUSMTIp

TeTr1yUT

i *kBr9au9 JO OT38yY

VIZ~e EVALUATION OF THE ANALYTICAL SOLUTION

Comparison with Solutions by the Runge-Kutta Method

Solutions for equations (28) and (29) have been obtained by numeri-

cal integration methods in reference (5). The two equations were solved

aimultaneously and the Runge-Kutta procedure applied as given in refer-

ence (13). These results were thoroughly checked for a few specific

values of the initial conditions and may be considered as nearly exact

solutions for the motion of the system. Therefore, comparisons made

between the results presented in reference (5) and the solutions obtained

in this paper should be a good indication of the accuracy of the analyti-

cal solutions.

The system analyzed in the reference did not have an eecentrically

located axle, so that d= 0 and no normal bearing forces were devel-

oped. Thus for these solutions there were no friction forees. Other

constants for the physical system are presented in Appendix C. Values

for the Jacobian elliptie functions were obtained from reference (1).

The dimensionless displacement and acceleration of the upper mass,

and the dimensionless displacement of the lower mass are presented as

functions of dimensionless time, 6, in figures 8 and 9. The ratios of

the nondimensional velocities of the upper and lower masses to the

initial dimensionless velocity are presented in figures 10 and 11. The

solutions are discontinued when (uz! ~- u!) « 0, since this corresponds

to the time after which 8 becomes negative, and the equations of

motion are no longer applicable. A wide range of u,! necessary for

design purposes is considered, and in most cases the agreement between

- 12 - *uoTzenda

TedoTyATBUB

344 Wor

pue

sanpsosoid

844ny-aeduny

93903 Aq

pautsyqo

uoTWWeleTeocoe

ssem

-reddn

pues yusmeoe[dstp

ssem-1amoT

10s suotantos

jo uostiedwop

-*g aun3Ty

@ ‘aut.

ssaTuoOTsusmtg

o°S

e

r°

Sr So

} I

aes

o

\

Ong

ONOMet

NS

SToqmAs

on SSUOTYNTOS

TBOTALATBUY

—— ———_

sanpeoolg

Be4yqny-e2uny

om

‘uOTYBISTAI00B ssemeleddn ssatTuoTsuemtq Ty uf

n ‘quomsceTds{p ssem-I9MOT ssoTUOTSsUEUTT

- }3-

suotTyenbs

TeoyTywATeue

971 Worl

pue einpaocord

eqyany-ssuny

94

fa peute1qgo

4yuemeoetTdstp

ssem-reddn

Joy

suotintos

Jo uostazedmog

-*6 sInNaTzZ

Q@ f‘eauty

sseTucTsueutqd

9°¢

3° B°S

b's 0°s

9°T o°*T

g° $°

I ]

I I

I |

I t

on ocd

V 4

© CO

©

On sToqudAs

¢

isuoTUNTOS

TeoTyYATeUY

—On

eInpss0aid

eyyn}-sdumy

Tn ‘quaemeostTdstp ssem-reddn sseTuotsueutg

'

Lowe

r-ma

ss

velocity

ratio,

Upg'/u

1.0

=} -

—_ RungeeeKutta imalytical

procedure solution

u.! Symbol

L | L LL

4 8 1.2 1.6 2,0

Dimensionless time 9

Figure 10.- Comparison of solutions for the lower-mass velocity obtained from the Runge-Kutta procedure and from the analytical equetion.

t

Upper-mass

velo

city

ra

tio,

v,'/uo

~h5-

—__—- Runge-Kutta Analytical

procedure solutions

' Symbols

oO

0

.o7

Vv

A

A

A

raN

A

oo

No , Vv

NO a ae

NN

-.2b \ NN Ro

Ne ne - NN 4 ore

4 | l l (Oy | ‘) 04 .8 1,2 1.6 2.0 2.4 2.8 3.2 3.6

Dimensionless time, @

Figure 11.~- Comparison of solutions for the upper-mass velocity obtained

from the Runge-Kutta procedure and from the analytical equation.

- hé =

the analytical solution and results obtained by means of the Runge-Kutta

procedure is good. Note that the acceleration of the lower mass is not

presented, since this quantity is of little importance to the designer,

and the corresponding inertia force was neglected in order to obtain

equations (28) and (29).

Some differences are apparent at the very lowest initial conditions

considered. As was indicated before, for u,'=1 the ratio r (fig. 7)

is rather large, indicating that the linear term in the equivalent spring

force is accounting for the major portion of the work done by the aprings.

Therefore the method of equivalent nonlinearigation presented in this

paper is approaching the case of equivalent linearization for the vary

lowest initial conditions. Linearization is known to be impractical in

this problem, so the results may be expected to be less good in this

Tange»

Comparison with Experimental Data

Reference (5) describes experimental drop tests conducted with a

conventional oleo~pneumatic landing gear which closely represents the

system in figure 1. Data obtained from these tests, in which a value of

wing lift equal to the weight of the test model was simulated, are used

for comparisons with the theoretical solutions presented in this paper.

Application of the transformation equations (26) and (27) enables

the calculation of the displacements, velocities and acceleration in

dimensional terms. These quantities for the upper and lower masses are

presented in figures 12 and 13 along with the corresponding experimental

data for the impact initial condition of %, = 8.86 feet per second.

*puodes

zed

gesz

g9g°g =

Sn “°q4oudut

TeotTdséy B

soz SqyuUeuUa.BeTAsTp

pue

seTyPFooTesa

ssBuertemoT

pus

rzeddn

uo eyep

Teqyuewutzedxe

pue

sy[Nsal

TVOT},I9IOSIy

Usemyaq

suosTaeduoy)

-*gT

eainstg

des faut]

0¢°

9T°

oT°

80°

¥O°

0 03°

9T°

et’

80°

¥0°

i '

T

q T

F-

tC '

t t

4a-

oOO OG

O

CO 0

-h7 -

2

T°

42 ‘qgueusovrtdstad

Z‘*sseu

2TaemoqT O

oo 64 oO ht

° 3 CO

iM Oou

a 42

S‘sseu tamoT

p /

m+ ue)

% ‘sseu

aeddg

a

9 8 sseu

IemMoqT sseu

teddy

O O

: Tequowttaedxq

O O

O

TBeoTpoto9y], O000

40T

12 *sseu

azeddyg

Upper-mass

acceleration factor,

*,/g

-~ 18 -

Theoretical

O Experimental

Time after contact, sec

Figure 13.,- Comparison detween theoretical results and experimental

desta on the upper-mass acceleration for a typical impact.

This value is equivalent to the dimensionless parameter u,! m 2439.

The ground vertical reaction force F, is presented in figure 1) for

comparison with the experimentally obtained forces. In general the

results are in good agreement. The small differences that appear are

attributacvle to the neglect of the air pressure force, the omission of

the lower mass, to differences between the actual and the assumed tire

characteristics, and to experimental errors.

In view of the observed good agreement between theory and experiment

it seems that the solution presented will enable a designer to determine

with reasonable accuracy the loads a landing gear may experience and thus

aid him in preliminary design calculations.

1b

Ground

vertical

force,

Ty

- 50 =

7 - x10

O O Theoretical

a Experimental

6b

v /

O/ \ D

i O

‘ O 4b

O

O ar

O

ar

1b

| a | i _l to ! —___L J

0 OP 204 06 008 210 le 14 16

Tine after contact, sec

rigure 14,- Corparison between theoretical results and experimental data on the ground vertical force for a typical impact.

~ Sl -

VIII. CONCLUSIONS

An analytical solution has been presented for the equations of

motion of a two-degree~-of-freedom nonlinear system representing an air~

plane and landing gear configuration subjected to an impact load. The

following conelusions can be made from the evaluation:

1. The friction forces developed on the strut bearing surfaces

because of bending forces applied at the axle may be included in the

equations of motion without adding to the complexity of the final dif-

ferential equation. It is necessary to replace the variable distance

between bearing surfaces by an apvroximate mean distance.

2. The equations of motion for the system may be transformed to

an equation which closely resembles one having an exact elliptic solution.

For a given airplane, a certain value of wing lift during landing will

reault in the two equations of motion being identical, and the exact

solutions will deseribe the impact.

3. Since in general the wing lift force is approximately equal

to the weight of the airplane, a method of "equivalent nonlinearization"

mist be employed to obtain the equation of motion in a form amenable to

solution. The equivalent equation is derived on the basis of equal

energy considerations and the major nonlinear character is retained,

thereby presenting an improvement over the usual linearization methods

which have been unsuccessful in this problem.

he Comparison with more exact solutions obtained by numerical

integration methods reveals that the analytical solutions obtained herein

are very accurate within a wide range of initial conditions.

~ 32-

5. Comparison with experimental results indicates that notwith-

standing the numerous simplifications made, the solutions obtained

describe the motion of the system adequately for design purposes.

l.

2.

3e

6.

7

9

10.

il.

~ 53 -

IX. BIBLIOGRAPHY

“ePherson, Albert E., Evans, Je, Jv., and Levy, Samuel: Influence of Wing Flexibility on Force-Time Relation in Shock Strut Following Vertical Ianding Impact. NACA TN 1995, 19)9.

Pian, T. H. H., and Flomenhaft, H. I.: Analytical and Experimental Studies on Dynamic Loads in Airplane Structures During Landing. Jour. Aero. Seis, Yol. 17, No. 12, Dec. 1950, pp. 765-77), 786.

Stowel, Elbridge Z., Houbolt, John C., and Batdorf, $. B.: An Evaluation of Some Approximate Methods of Computing Landing Stresses in Aircraft. NACA TN 158), 1918.

Walls, James H.: An Experimental Study of Orifice Coefficients, Internal Strut Pressures, and Loads on 4 Small Oleo-Pneumatic Shock Strut. NACA TN 326, 1955.

Milwitzky, Benjamin, and Cook, Francis &.: Analysis of Landing Gear Behavior. NAGA Rep. 1151, 1953. (Supersedes NACA TN 2755.)

Yorgiadas, Alexander J.: Graphical Analysis of Performance of Hydraulic Shock Absorbers in Aircraft Landing Gears. Jour. Aero. Sele, Vol. 12, No. hy Oct. 1915, PD. 21-28.

Fliigge, We, and Coale, C. We: The Influence of Wheel Spin-up on landing Gear Impact. NACA TN 3217, Stanford University, 195).

Fliigee, We: Landing Gear Impact. NACA TN 273, Stanford University, 1952.

Iindguist, Dean C.: A Statistical Study of Wing Lift at Ground Contact for Four Transport Airplanes.

Painleve, P.: Acta Mathematica, Vol. 25, No. 53, 1902. {Original not seen. Secondary source: Kamke, £.: Differentialgleichungen. Third Edition, Chelsea Publishing Company, New York, 1918, p. 548.]

“elachlan, N. W.s Ordinary Non-Linear Tifferential Equations in

Engineering and Physical Sciences. Clarendon Press, Oxford, 1950,

Minorsky, N.: Introduction to Non-linear Mechanics. J. W. Edwards, Ann Arbor, 1917, pp» 233+238.

Scarborough, James B.: Numerical ‘Mathematical Analysis. Second Ed., the Johns Hopkins Press, Baltimore, 1950, pp. 301-302.

"ilne-Thomson, L. Mo: Jacobian Elliptic Function Tables. Dover Publications Inc., 1950, p. 19.

-~ 5 -

Xe VITA

The author was born in St. Cloud, Minnesota on July 1, 1928. He

attended Central High School, “inneapolis, Minnesota and graduated in

June 196. The following September he entered the University of

Vinnesota at Minneapolis and received the Degree of Bachelor of Civil

Engineering in March 1951. After graduation, he was called to active

military service with the 7th Infantry Division, of the Minnesota

National Guard and was honorably discharged in May 1952. Subsequently,

he accepted a position with the National Advisory Committee for

Aeronautics where he is presently an Aeronautical Research Selentist.

After some preliminary graduate studies at the University of Virginia,

he entered Virginia Polytechnie Institute in the sumer of 195) to begin

work toward the Degree of Master of Science in Applied Mechanics.

( OAOVINE aA - The cae bod

oo

-~ 5S «

XI. APPENDICES

Appendix A. The Derivation of the Solution

to the Transformed Equation

The solution to equation (23) may be determined as follows:

For this appendix only, let differentiation with respect to x be

denoted by prime marks above symbols. Then equation (1:0) becomes

u t+ t+ Sat 223 = 0 (AL)

Multiplying through by @2x/ 3 and arranging terms, equation (Al) may

be written as

(e + ; t)e2%/3 af «+ : 1) 02/3 2 ~2Hé_ e2x/3 (A2)

This equation may be rewritten

a @2x/3 ‘ + , *)| = +227 392x/3 (A3)

Multiply through by 2/3 (: + ; +) ay to obtain

Joa/3(t + : + ale2x/3 (« + ; *)| = 2H" (x0%/3}? (+ + 3 3] oX/3x (Aah)

Equation (Al) may be integrated, and the resulting expression is

1 ho (3) as ol/3 (4 + : x) +h - (2x3)

- 56 =

where c , is the constant of integration. Take the square root of

both sides and solve for He,

Hey? - = ef t +

Multiply through equation (A6) by (35 0 3ax) and rearrange factors to

a

obtain

4 eye”! “dx = — ox = (A7) 3 2° ( only" \2 ~ (3 3)?

cl “1

If a quantity » is defined as

. ein} (8 0/9 | (A8)

so that

wee] then substitution of equations (48) and (49) into equation (A7) results

in the expression

4 licje” * dx «= ——-du (A10)

- S7 -

| This may be written in elliptic integral form as follows

[3 Hee7*/3ax = oS

where k® = -] is the modulus. The result of the integration may be

(All)

ina

written in standard notation as

~Heye"*/3 + ¢p = Fo,k) = v (A12)

where co is a constant of integration and by definition

snv = sin o (A13)

Therefore, in terms of the Jacobian elliptic function equation (Al12) may

be written as

sn (ce - tie @7X/ 3) = 3 @x/3 (ALL)

k 2am] “L

The solution is, then,

cs $ 0je7%/3sn (eg - Heye"¥/ 3) (415)

2am]

Using the initial conditions from equation (39), the constants of

integration may be determined as

(A16)

- 68 -

By substitution of these constants, equation (A15) becomes

Ye f 4 - pai or en [3 fatugt(2 - 0/3] (417) k@=-]

This is a valid solution, but the modulus usually is tabulated for

positive, real values. By applying a transformation to the modulus

(ref. 1h), equation (417) may be written

Ts [$e o-v/304 [3 atta, (2 - ex/Z (a18) Kem} /2

~ 59 -

Appendix B. The Determination of an Approximate Value for uy,

To solve for the actual maximum value of ui it would be necessary to

evaluate equation (8) for u' = 0. The corresponding value of 98 would

be very difficult to obtain from this expression. For purposes of this

paper any good approximate value of u, is sufficient, and equation (1)

which is in the parametric form for u is much simpler to work with as

follows:

us oe e72%/ 3g 42 E | 2iu,' (a - ox/9| (51)

kal /2

Under the assumption that the exponential tera is relatively slowly

varying, the value of the argument vv corresponding to u, can be

found by differentiating sd(v,1/2) with respect to v, and equating the

result to zero as shom

gd. sd /,1/2) = S8.(¥,1/2) = 0 (B2) dv an

or, since the dn(.,1/2) is always finite

v = en7(0,1/2)

30 that

= K 21.8507... (B3)

where K is the value of the argument at the quarter period. Substitution

of this value in the argument of equation (B1) yields the result

vy = 3 \2iuyt(1 - 273/3) = K (Bh)

~ §9D <-

where the value of x now corresponds to the value of u = uqg in the

first approximation. From equation (Bl) it may be seen that

e7X/3 21 - —K (BS) 3 | eu,g!

Noting that

sd@(K,1/2) = 2 (B6)

expressions (B5) and (B6) may be substituted in equation (Bl) to obtain

i 2 Ug, = =o 1- LW

H h al (B7)

where, from equation (12)

2 | He \l- — for Ky, = 0 BB ou ( 1 * 0) (88)

From equations (B7) and (88) the desired approximate solution for u, may

be obtained. However, by combining the two equations and solving for u,!

a more explicit expression may be obtained as follows;

1. Syn, + (39) Up 3 pi 1

This equation is easier to use for obtaining u, as a function of wu,!'.

Aa, sq ft *

Ai, Sq ft 6 8

Koy $3 FL 7 # @

Voy cu ft ..

Ry, ft * @ @

Wy > lb * © # @

Ho, 1b o 8 «€

Appendix C.

e * » Sd °

Constants of the

*

*

*

o *

° «

«

oJ

* *

Physical System

0.05761

0.0708

0,0005585

0.03515

6,26)

0.5521

2,11

131