7 A-A92 057 AIR FORCE EN61EEING AND SERVICES CENTER TYNDALL AF-ETC F/6 1/3MODAL ANALYSIS FOR AIRCRAFT RESPONSE TO RUNWAY SURFACE ROU404NES--ETC(U)JUN 80 R R GA.JEWSKI

UNCLASSIFIED AFESC/ESLTR8032 N

E hEEEEI muuuuuuuuuuLEEEEEEh

LEVMODAL ANALYSIS FOR AIRCRAFT RESPONSETO RUNWAY SURFACE ROUBIBESS

0 RALPH R. GAJEWSKI, CAPT, USAF

lit DEPARTMENT OF ENGINEERING MECHANICS

UNITED STATES AIR FORCE ACADEMYCOLORADO 80840

JUNE 1980

FINAL REPORT

1 JUNE 1978-30 JUNE 1980 FDTICECTE

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I 0 ,Ralph R./Gajewskil Capt, USAF /'I

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I9. KEY WORDS (Continue on reverse aide linecesaary and identify by block number)

F-4E TAXI TestsRunway RoughnessAM-2 MatsRapid Runway RepairModal Analysis

2 . a ~RACT (Continue on reverse aide It neceesary aind identify by block number)

This report develops one and three degree-of-freedom linear vibrationmodels for the prediction of aircraft response to runway surface roughness.The equations of motion are integrated in principal coordinates using modalanalysis. The modal parameters required are natural frequency, damping ratio,and mode shape for each degree of freedom. Comparison of results is made withthe TAXI code that has a nonlinear strut model. Results are presented forasymmetric motion due to spall profiles in the runway. i

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PREFACE

This report was prepared by the Department of Engineering Mechanics, UnitedStates Air Force Academy, Colorado, 80840, for the Engineering and ServicesLaboratory at Tyndall AFB, Florida, under Job Order Number 21042B30 during theperiod June 1978 through June 1980. The Project Officer was Capt David Salz,AFESC/RDCR.

This report has been reviewed by the Public Affairs Office (PA) and isreleasable to the National Technical Information Service (NTIS). At NTIS, itwill be available to the general public including foreign nationals.

This technical report has been reviewed and is approved for publication.

XL Dt ap t U SA F LAPSLEY R. CALDWELL, Lt Col, USAFPr ctOf HAVE BOUNCE Program Manager

OBERT BOYER, on USAF FRANCIS B. CROWLEY III, C nel, USAFDir, Engineering & Services Laboratory

Accession For

NTIS CRA IDTIC TAB ElUnnnuounce d F1Justification.,

By

Dist ribut ion/

Avail-ibilit Codes

(The rvs of this g is b

(The reverse of this page is blank)

k 4

TABLE OF CONT'ENTSI

Section Title Paqe

I INTRODUCTION ..................

II EQUATIONS OF MOTION ............... 2

III SOUTIONS .................... 8

IV RESULTS .. . . . . . . .. ... .. . .. ... 12

V CONCLUSIONS AND RECONMENDATIONS .................... 32

17

LIST OF FIGURES i

Figure Title Page I:

1 Aircraft Geometry and Coordinates ..................... 32 Piecewise Linear Runway Profile ...................... 83 Half Mat Profile ............. ..*.. .............. 144 TAXI Results for Half Mat Profile ................ 165 SDOF Results for Half Mat Profile .................... 176 3DOF Results for Half Mat Profile ..................... 187 Full Mat Profile ............................... 198 TAXI Code Response to Full Mat Profile ................ 209 SDOF Code Response to Full Mat Profile ................ 21 v

10 3DOF Code Response to Full Mat Profile ................ 22IU SDOF Code Response with Frequency = 1.80 Hertz ........ 2312 3DOF Code Response with Modified Spring Constants ..... 24"13 Spall Models ........ o ..... .. .......... . ............... 25

14 Response to Two Foot Spall for Left Main Gear Input ... 2615 Response to Two Foot Spall for Left Main and Nose

Gear Input ................................... 2716 Response to Two Foot Spall for Three Gear Input ....... 2817 Response to Double Spall for Left Main Gear Input ..... 2918 Response to Double Spall for Left Main and Nose Gear

Input ........................... . ..... ........... . . . 3019 Response to Double Spall for Three Gear Input ......... 31

iv

'Ja.

LIST OF TABLES

Table Title Paqe1 Aircraft Model Data .................. 122 Equivalent Spring Stiffness ............................. 123 Mode Shapes and Frequencies................ ........ 134 Revised Mode Shapes and Frequencies ...................... 15

(The reverse of this page is blank)

SECTION I

INTRODUCTION

The United States Air Force requires the capability to operate aircrafton rough runways created by the rapid repair of bomb damage (Reference 1).The current rapid runway repair technique uses AM-2 matting and proceduresdescribed in Air Force Regulation 93-2 (Reference 2). Initial data for F-4Eperformance on AM-2 mats have been acquired in the HAVE BOUNCE test program(References 3 and 4). These data can also be used to validate computersimulations of aircraft response to runway surface roughness. These cam-puter simulations may then be used to develop surface roughness criteria(Reference 5).

The United States Air Force Academy (USAFA) has provided engineeringconsultation on surface roughness to the Air Force Engineering and ServicesCenter (AFESC) since 1977. During this reporting period, evaluations havebeen provided for using statistical analysis techniques and using astationary test facility for surface roughness evaluations. The purposeof this report is to develop a three-degree-of-freedom model for analysisof aircraft taxi response over repair mats and spalls. A one-degree-of-freedom model was used to predict the symmetric motion response for the HAVEBOUNCE Phase II program (Appendix C, Reference 4).

This report uses modal analysis to predict aircraft response to surfaceroughness. The rigid body aircraft model includes the bounce, pitch, androll degrees of freedom. The surface roughness for the nose and two maingear tracks is modeled as a sequence of straight line segments, such asramps and mats, placed on an originally flat runway surface. The equationsof motion are integrated using modal superposition and the piecewise exactsolution algorithm (Reference 6). The model will accept experimentallydetermin-d natural frequencies, damping ratios, and mode shapes. The modalsuperposition solution is suitable for programming on minicomputers. Theresults presented include a comparison with the TAXI computer code (Refer-ence 7) and predictions of asymmetric motion due to spalls in the runwaysurface.

L

SECTION II

EQUATIONS OF MOTION

1. GEOMETRY. The aircraft model geometry and coordinates are shownin Figure 1. The spring and damping constants represent the strut and tirecombined in series for vibrations about the equilibrium position. Theattachment point displacements are kinematically related to the center ofgravity displacements through a geometry matrix:

(u..) [GEOM][uG3 (i)

where

uN ULL

1 N

[GEOM] = -DM D1 -DM -DR

(UGI=

2

I

L P

Figure i. Aircraft Geometry and Coordinates

CXI Aircraft center of gravity

R,L,N Struit attachment points for right main, left main, andnose gears

D, DN Momnent arms from aircraft centerline

'RP

u~p, u~p, u~p Runway profiles for three gears, + upward

SDisplacement of CG, + upward

6 Pitch angle, + nose up

0Roll angle, + left wing up

MG, Ip I Mass, pitch and roll inertia

U uR, , uN Displacement of attachment points, + upward

kR, kL, kN Gear spring constants

CR, cL, 0 N Gear damping constants

IR

Fiur 1 Arcaf Goety ndCordnae

CG Arcrat ceter f grvit-,, Stu atahmn pont fo.igtman.lf.mi, .n

2. EQUATIONS OF MOTION. The equations of motion are derived assumingnose profile and attachment point displacements are ordered so that

UN?> UN > 0

and their slopes are ordered so that

> > 0.

Then the total upward nose strut force acting on the aircraft is

F k (2a)

Similarly, the upward main gear strut forces are

F SR 'R (uRP R) + cR (uRP -ul (2b)

FsL '=L (uLP ') )+ L (U Lp - 'L) (2c)

The equations of motion from summing forces vertically and summingmoments about the pitch and roll axes are

FSN + FSR + FSL MCG u 0

DNFSN - DM (FsR + FSL) = p (3)

DL SL RVSR = RDefine

CeG

[M] = IP

0 0 I J

'4

V

These equations can be written in matrix form as

wh r MH UG) + [C 1 3( u + [K 11fu,,. ] [K 1 M(u) + IC1( )(4)

whereUPN

U PR

[K,) = GOM3T 0 k

[C 1) [GEOM) T cL 0

Then using Equation (1) the left-hand side of Equation(4) an be re-written in center-of-gravity displacements as

+G [C]) G' + [jD](uG' [K.l)(UP, + [C1)(uJ (5

where

CKD] [GEOM]T L0 kL 0j[GEOM)

0 0k

[CD] [ GEOM]T 0 cL 0jEGEOM]

0 0 c R5

3. MODAL EQUATIONS. The modal equations of motion are derived usingthe natural frequencies and mode shapes for the undamped system which arecomputed from standard vibrations techniques as in Reference 8 or computertechniques as in Reference 9. The system characteristic matrix equationsare

-[ 2 [M) MODE,)= (0 (6)

where wi is the natural frequency and (MODEi) is the corresponding modeshape. Let [MODAL] be the modal matrix constructed using the mode shapesas columns. The mode shapes are orthonormalized so that

[MODAL]T [M][MODAL] = [I] (7)

where [I) is the identity matrix. Then

[MODAL] T[KD] [MODAL ] = [2] (8)ii

is a diagonal matrix with the square of the natural frequencies on thediagonals.

The same sequence of matrix products does not generally diagonalizethe damping matrix

[MODAL]T[cD][MODAL] = [Cp] (9)

It is very desirable that a diagonal matrix [Cp] be obtained so that theequations of motion will decouple with the introduction of principal co-ordinates (p) defined by

fuG = [MODAL(p) (1O)

For lightly damped systems such as seen in the HAVE BOUNCE Thase Itest results (Reference 3), it is reasonable to neglect the coupling ofmodes through the off-diagonal terms of [Cp] and retain only the diagonalterms (Reference 6). These diagonal terms are then rewritten as

c =2 C(1)pi ~ i "i

where Ci is the ratio of damping in mode i to the critical damping for thatmode.

The decoupled modal equations of motion are obtained through multi-plying Equation(5)by [MODAL]T and substituting Equation(lO),resulting in thesequence of equations of the form

6

- • -- _ - . . ..... . .. . .. . . . . . . I nBl io • Ii-

Pi + 2 ~wp + W2 pi =MODE 1]T ([K1)fuj1 + [C1]{u 1) (12)

Thus, the response of each mode can be solved as a single degree of freedomsystem and the total response found by superposition of the principal co-ordinate responses using Equation(1O).

SECTION III

SOUJTIONS

1. DIRECT INTEGRATION. There are several methods for directlyintegrating the linear matrix equations of motion [Equation 5] such as theNewmark Method and Wilson 8 Method (References 6 and 9). Some of thesemethods are conditionally stable,and each requires a small time step tominimize period elongation and amplitude decay of the results.

2. MODAL SUPERPOSITION. This method integrates the decoupledequations of motion [Equation(12)l separately and then adds each modalresponse for the total solution. If a direct integration method such asthe Wilson 9 Method with the same time step for all modes is used, the re-sults will be the same as directly integrating the system matrix equations.But modal superposition provides the opportunity to only integrate themodal response for selected dominant modes and to use the accurate piece-wise exact integration scheme which does not require a small time step forstability and accuracy as do the direct integration schemes.

3. PIECEWISE EXACT SOLUTION METHOD. Many types of loading, includingthe runway mat profile, can be closely approximated as a sequence ofstraight-line segments with unequal time intervals. For example, Figure 2shows a runway mat profile with upheaval and settlement as constructed forthe HAVE BOUNCE Phase II tests (Reference 4). Each straight-line segment,such as the second up-ramp illustrated in Figure 2, has its elevationwritten as a linear equation of the form e (t) = a i + bit where the sub-script i denotes the segment number. The time axis values t are the productof multiplying the runway distance x and the constant ground speed of theaircraft V.

e(t)e4(t) = a + bt

e(tf) - e(tS) t s Itf =4 tf - s

Figure 2. Piecewise Linear Runway Profile

For any segment of time ts < t < tf, the one-dimensional vibrationequation to be integrated is

2 2x + 2 C wx + W x = W (a + bt) = A + Bt (13)

where a + bt is the equation for the mat profile in this time interval. The

8

exact displacement solution to Equation(13)is (Reference 6)

x(t) =A + A, + A2e-C Wt cos wdt + A3e' cAt sin wdt (14)

where

AO A - 2 CB

2 3

A BA1 =2B

A2 = x(t0) - AO

1

A +=-(x3 W 0 (to)+ A 2 -A 1)

Wd = W0 -

The solution is exact so using a small time step for stability andaccuracy is not required.

4. ONE-DIMENSIONAL MODAL ANALYSIS. The HAVE BOUNCE Phase I testresults indicate the aircraft responds in a dominant bounce mode which islightly damped (Reference 3). Based on these results, it appears that asingle degree of freedom solution based on Equation(14)would give goodestimates for main gear compression loads. The one-dimensional equationof motion is

u + 2 + 2u + u = 2 (t) + 2 C Wk'(t) (15)

where ug(t) is the main gear track profile. The frequency W and dampingfactor r may be obtained experimentally from a power spectral densityanalysis. The second term on the right-hand side of Equation(15)may beneglected due to the small amount of damping in the system. The main gearcompression load is the sum of the dynamic forces and the weight (W).

M WM - u(t)) + M0G 2 w ( (t) - u(t)) + W (16)

For lightly damped systems, the second term may be neglected and the re-sulting FM will be a conservative estimate of the gear load.

This model has been programmed at the Air Force Flight Test Center(AFFTC) and shows good correlation with the HAVE BOUNCE Phase II test re-sults (Reference 4).

9

5. THEE-DIMENSIONAL MODAL ANALYSIS. The success of the one-dimensional model indicates that a three-dimensional model could providegood results for the bounce, pitch, and roll degrees of freedom. Thismodel would estimate nose gear forces as well as compute asymmetric rollresponse.

The modal analysis is based on integrating Equation (13) using Equation(14) for each mode and each gear track profile. For three modes and threetracks, this solution will require integrating Equation(14)up to nineseparate times and adding the results.

Since Equation (14)is being integrated in principal coordinates, thegear loads are not entered directly. Instead, the gear loads are parti-tioned using the mode shape vector. This partitioning accounts for howmuch the gear load is exciting a particular mode shape.

For example, consider the modal solution due to left main gear profileonly. The gear load is partitioned using the orthonormalized mode shapesthat satisfy Equation(7). The first modal equation is

pl(t) + 2 IC31wpl(t) + W2 p (t) = P (t) (17)

where the load participation is

Ptjt) :tMOlEI 13 )K(t) (18)

For each segment of time where Pih(t) = A + Bt, the solution forp (t) is found using Equation(l)with w = wj. The load participation fort~e other modes is found using the appropriate mode shape in Equation(18)and the response found using the appropriate frequency and damping factorin Equation(lh) The center-of-gravity displacements are found using themodal matrix

= [MODAL] p (19)

P3

and the attachment point displacements are found using the geometry matrix

uL [GEOM] 9 (20)

The gear compression loads are ccmputed using

10

~V.~~$4 -~

FN -kN u. + FNST

FL -k (UL - ULP) + FMT (21)

FR -kR uR + FMST

where FNST and FMST are nose and main gear static loads. The response forthe nose and right main tracks follow in a similar manner.

i1

i*

SECTION IV

RESULTS

Calculational results for the single-degree-of-freedom model (SDOF)and the three-degree-of-freedom model (3DOF) will be compared with resultsfrom the TAXI computer code (Reference 7). Then spall results using the3DOF model will be presented.

1. TAXI CODE. This code was provided by the Air Force WrightAeronautical Laboratory (AFWAL). Additional FORTRAN coding was added tointerface with the USAFA computer plotting system and to evaluate slope ofthe strut air curve formulas to obtain spring constants for the linearvibrations model.

2. COMPARISON MODEL. The heavy-weight aircraft data of Table 1 wasobtained from AFFTC. The initial sprinq constants for thevibrations models were obtained from the TAXI code by computing the slopeof the strut air curve at the static equilibrium position. This slopegives an overall stiffness for the strut which is then added to the tirestiffness in series (Reference 8) to obtain an overall gear stiffness.

TABLE 1. AIRCRAFT MODEL DATA

MCG = 145.5 lb.sec2 /in

Ip = 1.92(106) lb-sec2.in

IR = 6.56(105) lb.sec2 .in

DI =D R = 109.25 in

DN = 234.9 in

DM = 44.35 in

TABLE 2. EQUVALENT SPRING STIFFNESS

Main Gear

Strut Stiffness 27809 lb/inTire Stiffness 13530 lb/inSeries Stiffness 9102 lb/in

Nose Gear

Strut Stiffness 1662 lb/inTire Stiffness 15600 lb/inCeries Stiffness 1502 lb/in

12

3. TI]REE-DEGREE-OF-FREEDOM (3DOF) MODEL. The linear vibrationsmodel for the data of Tables 1 and 2 has the mode shapes and frequen-ci,,i shown in Tabl 3.

TABLE 3. MODE SHAPES AND FREQUFCIES

Mode 1 Mode 2 Mode 3

w 7.271 rad/s 12.016 rad/s 18.198 rad/s

X 0.025931 -0.078743 0g

9 0.000685 0.000226 0

0 0 0 0.001235

The mode shapes indicate relative motion within a mode and are orthonormalizedaccording to Equation (7). The first mode has its node 37.9 inches aft of theCG so its primary motion is pitching at the nose gear. The second mode hasits node 348.4 inches forward of the CG so its primary motior. is bouncing.The third mode is the roll mode which is decoupled from the bounce and pitchmodes due to symmetry of the aircraft.

4. SINGLE-DEGREE-OF-FREEDOM (SDOF) MODEL. This model is based on

the bounce mode so it uses w = 12.016 rad/s and M = 145 .5 sec2/in. Both

SDOF and 3DOF models were programmed usinq CAL (References 6 and 10), a com-

puter analysis language developed for educational use at the University ofCalifornia at Berkeley. The basic programming of Equation (14) is simpleand can be done in almost any computer language.

5. RESPONSE TO HALF MAT PROFILE. The half mat profile used for thefollowing calculations is shown in Figure 3. The TAXI code response fora constant velocity of 20 knots is shown in Figure 4. The frequency ofvibration after coming off the mat is 1.80 Hertz,and the ratio of dampingto critical damping is about 1%.

The SDOF response for a natural bounce mode frequency of 1.91 Hertz

(Mode 2) and dampinq factor of 1 percent is shown in Fiqure 5. The straiqhtline through the plot is the static equilibrium load. The maximum compres-sion load is sliqhtly larger than the TAXI response, but the minimum valuesare much smaller including unloadinq at the fourth minimum.

The 3DOF response for a 20-knot constant run with 1 Percent dampinq

in all three modes is shown in Figure 6. The straight lines through theplots are static equilibrium values. The maximum main gear compressionloads are slightly smaller than both the TAXI and SDOF results, and the

minimum values lie in between the other two runs. The nose gear responseis smaller to the TAXI response but does not contain the hiqhest frequencymotion.

13

4 0

6. RESPONSE TO FULL MAT PROFILE. The full mat profile used for thefollowing calculations is shown in Figure 7. The TAXI code response for a20-knot run is shown in Figure 8. The main gear compression load comingoff the mat (at three seconds on the figure) shows a reduction of the am-plitude of vibration. A comparison of the TAXI code responses of Figures 4and 8 indicates that the trailing edge of the mat can cause reinforcementor cancellation of the motion. To obtain accurate simulations of experi-mental results showing such reinforcements and cancellations, the computermodel must have the same period of vibration in its dominant mode as theaircraft.

The SDOF response for a frequency of 1.91 Hertz and damping factor of1 percent in shown in Figure 9. The initial response coming on the mat is

similar to the TAXI response, but the linearized frequency of 1.91 Hertzcauses the phasing of the trailing edge of the mat so that the motion isamplif ied.

The 3DOF response for a 20-knot run with 1 percent damping in all threemodes is shown in Figure 10. The main gear response is very similar to theSDOF response showing an amplification at the end of the mat. The nose gearresponse is very similar to that for the TAXI code response.

The difference in phasing at the end of the mat is due to a differenteffective frequency of vibration about the equilibrium position. The SDOFand 3DOF results used the slope of the air curve at equilibrium as a com-ponent value to compute the series stiffness of the gear assembly. TheTAXI code uses a nonlinear model for the compression air curve behavior ofthe strut as well as nonlinear damping.

11.5 in

-7 40 ft , . h- 4 ft ramps

Figure 3. Half Mat Profile

To evaluate the effects of the nonlinear TAXI model, the effectivespring constant, the change in pneumatic force divided by the change instrut compression was computed for each output value. The main gear springconstant ranges from a low of 4733 lb/in to a high of 11,362 lb/in. Thenose gear spring constant ranges from a low of 965 lb/in to a high of 266?lb/in. This variation in spring constants should be averaged in such a waythat the equivalent spring constant yields the same period of free vibrationwhile the aircraft is on the mat. Since the period is inversely proportionalto the square root of the spring constant, it is reasonable to compute theequivalent spring constant according to

1 =i n 11: (22)

eq

14

The revised spring constants are 8081 lb/in. for the main gear and 1508 lb/infor the nose gear. The revised mode shapes and frequencies are shown inTable 4.

TABLE 4. REVISED MODE SHAPES AND FREQUENCIES

Mode 1 Mode 2 Mode 3

7.282 11.328 17.148

Xg 0.025052 -0.079027

e o.0oo688 0.000218 0

0 0 0.001235

The bounce mode frequency is now 1.80 Hertz. The SDOF results usingthis fundamental frequency are shown in Figure 11. The 3DOF results usingthe revised mode shapes and frequencies of Table 4 are shown in Figure 12.Figures 11 and 12 now show much better agreement with the TAXI code resultsof Figure 8. in particular, the cancellation of motion at the trailingedge of the mat is consistent for each calculation.

7. SPALL MODEL RESULTS. The spall profiles used for the followingcalculations are shown in Figure 13. These profiles correspond to thespalls cut into the runway for testing at AFFTC. The spalls were cut acrossthe runway so that tests could be made with the left main gear only enteringthe spali and other tests could be made with both the nose gear and leftmain gear entering the same spall profile.

Figure 14 shows the response for the left main gear entering the two-

foot spall with a speed of 10 knots. Figure 15 shows the response for thenose gear and left main gear entering the two-foot spall cut across therunway with a speed of 10 knots. Figure 16 shows the symmetrical responsefor all three gears entering the two-foot spall. The main gear compressionloads are of about the same magnitude as shown for the mat encounters. Themaximum left main gear compression load is larger for the asymmetric spallinput (Figure 15) than for the symmetric spall input (Figure 16).

Figure 17 shows the response for the left main gear entering thedouble spall with a speed of 10 knots. Figure 18 shows the response forthe nose gear and left main gear entering the double spall cut across therunway. Figure 19 shows the response for all three gears entering thedouble spall. The sharp downward plunge of the main gear compressionfollows from the assumption that the gear follows the runway profile. Theeffect of the tire footprint following such a spall profile is not in-cluded in these results but may be added by modifying the runway profileinput. The maximum main gear compression load is larger for the asymmetricinput (Figure 18) than for the symmetric input (Figure 19).

15

.. ... . : I

(-n)

rr

I 4->

4)

TH

I-I

LPU

- -- -T- -, -4

K 4-40

2I4

(14-

K 2

- I

00 I"WoCU

1-

I:

r rz r

) I- .- -.

I c~2

o-I -~ I

~i-. I 0)

4-'

K~ 1k?H

/1 )

(t w o

a -- (.n 4-)0)

2L C) 8.( ~-..

(I 1 0)

I- IT A-- KD 0CI)

\fl

-Jci -,

£7 NV- - ~Z4

(CE K rr

NK KI cv

CC)

B

~JI~i

- - S ~ ~

1.5 in

INN4 t 70 "" I

Figure 7. Full Mat Profile

19

CD r-q

0

'-4

Pr4

L- LH

Cc) C-1C)

z--L- - 7 r r --00 'Y1 00, 00 01 00'

J 1 )4

e.,

4

K. i

gw- L../- tJ

. .* 02 .

N0

!a)V. 0! °co(-

C_-D o)U

-I o

CC-00 Of-

21

4

.r4 r -T

A.,

) k

) 0)

IQ ( - '

_- / K. r3

.7_I'\ ° H

)0

I',

' C - .. -,i.

CD 4)

01

y0C a4

CD

(.nj

I-E

CD Vr

cJ

232

/ C

L 4J

0

Cl,

L F4

CD0

u4

1i~. H51 ntA F,. 6 in ramps

TWO-Foot Spall

1.5 in ~ I.~ ~ ~ ft-. k 6 in ramps

Double Spall

Figure 13. Spall Models

25

2 FOOT SPRLL 10 KNOTS

L-M4IN TNPUT

NOSF ', F q

cr:

1

-0.00 0.80 1.60 2. 40 3,20

SF C

o _-MqRN GFqR1-0_

cn;

T- F ........ r ' 1-0.00 0.80 1.60 2.40 3.20

o R-Mq!N GFqR

&-T-_

-0.00 0.80 1.60 2.40 3.20 I.X0

SEC

Figure 14. Response to Two-Foot Spall for Left Main Gear Input

26

"I'tL

2 FOOT SPRL.L 10 KNOTS IL-Mq!N +NOSE

NOSE GER9.

-0.00 0.80 1.02,40 3.20 4.00

SF C

cnCc L-MRIN GFRR

-0. 00 0.80 1.60 2. 40 3.20 1

~ -MRIN OERR

-0.00 0.80 1.60 2,40 3.20 405FE C

Figure 15. Response to Two-Foot Spall for Left Main and NoseGear Input

27

2 FOOT SPRLL 10 KNOTS

3 GERA INPUT

(n NOSE GFR

-0.00 0.80 1.60 2. q0 3.20 L.JSEC

~gL-MRIN GEER9

-0.00 0.30 160o 2.40 3 20 -o.

~JR-MqIN GR9

r- T- --0.00 0.80 t 60 2.40 3.?0 0

SE-C

Figure 16. Response to Nwo-Foot Spali for Three Gear Input

28

DOUBLE SPRL.L 10 KNOTS

L-MqRN INPUT

NOS- F R9

I5 c-0. 00 0.80 1.60 2. 40 3 20 .30SEC

L MN GFq9

C.)

C-

-0.00 0.80 1 60 2.'0i 3 20 30S Fl C

H-,F-M I N Q[ ,F8

00 .80 1.60 2. '.0 3.20 ".O

Figure 17. Response to Double Spall for Left Main Gear Input

29

. /I.

DOUBLF SPRLL 10 KNOTS

L--MqTN +NOS[

NOSF IIF R9-

-0. 00 0.80 1-02 Q .00

SF C

L-Mq N GLR9

0 -MII FFRF

-0.00 0.80 '1.60 2.40 3.20 -t.00

Q5F

Fiue1.Rsos oDobeSalfrLf ai n oeGa nu

03

DOLJILE 3PPLL IC) KNOTS

3 GEFqR INPUT

cnC. NOSE CEPR9

-0 00.80 1.2.60 3 ?0 i005F

L-MRIN GER9

-0.00 0.01CO2.o302

R F-MPIN CFRR

-0.00 0.80 1 60 2.4~0 T 20 03F C

Figure 19. Response to Double Spall for Three Gear input

31

SECTION V

CONCLUSIONS AND RECOMMENDATIONS

The one- and three-degree-of-freedom linear vibration models developed

in this report can reproduce the response of the essentially nonlinear air-

craft system to surface roughness. To correctly reproduce the response to

runway mats, it is important that the linear model have spring constants

that produce the same period of free vibration so that the phasing of the

trailing edge of the mat will cause the same reinforcement and cancellation

phenoanena.

The three-degree-of-freedom model can accept arbitrary runway profile

inputs for each gear. It can be used to predict results of asymmetric

motion due to spalls in the runway. The predictions of this model need to

be validated using the results of the recent spall tests at the AFFTC.

32

A ,VL

REFERENCES

1. Program Management Plan for Rapid Runway Repair, AFESC, 31 March 1980.

2. Rapid Runway Repair Procedures, Chapter 5, AFR 93-2, 29 July 1974.

3. Redd, L.T., and Borowski, R.A., HAVE BOUNCE Phase I Test Results,

AFFTC-TR-79-1, April 1979.

4. Lenzi, D.C., and Borowski, R.A., HAVE BOUNCE Phase II Test Results,AFFTC-TR-80-4, June 1980.

5. Caldwell, L.R., and Jacobsen, F.J., Interim Guidance for SurfaceRoughness Criteria, ESL-TR-79-37, October 1979.

6. Wilson, E.L.; CAL 78 User Information Manual, Report UC SESM 79-1,Department of Civil Engineering, University of California, Berkeley,November 1978.

7. Gerardi, A.G., and Lohwasser, A.K., Computer Program for the Predictionof Aircraft Response to Runway Roughness, AFWL-TR-73-109, Volume 1,September 1973.

8. Tse, F.S., Morse, I.E., and Hinkle, R.T., Mechanical Vibrations, Theoryand Applications, Second Edition, Allyn and Bacon, Inc., 1978.

9. Bathe, K.J., and Wilson, E.L., Numerical Methods in Finite ElementAnalysis, Prentice-Hall, Inc., 1976.

10. Wilson, E.L., "CAL-A Computer Analysis Language for Teaching StructuralAnalysis," Computers & Structures, Volume 10, Pergamon Press Ltd, 1979.

33(The reverse of this page is blank)

mt

Initial Distribution

HQ AFSC/DLWM 1HQ AFSC/SDNE 1

VHQ USAFE/DEX IAFFTC/TEOF 1AFFTC/ENAE 1

EOARD/LNS 1

HQ PACAF/DEM IHQ TAC/DRP 1

HQ TAC/DEM 1AUL/LSE 71-249 1HQ SAC/DEM 1HQ MAC/DEM 1HQ AFESC/RDCR 10HQ AFESC/TST 1HQ USAFA/DFEM 10USAE WES 4AFWAL/FIBE 1AFIT/DET 1AFIT (Tech Library) 1ASD./SD30MF 1TAC MEIST I550 CE CFE HQ 1DTIC-DDA- 2 2AFATL/DLODL 1

35

(The reverse of this page is blank.)