National Research Co t'nationalI* Council Ca a e recherches Canada

LEVETHEORETICAL ANALYSIS

OF THE TRANSIENT RESPONSEOF A WING

TO NON-STATIONARYBUFFETS LOADS

IDD I

by ~ 1

B.H.lK. Lee c. National Aeronautical Establishment

,.,i

OTTAWAAPRIL 1979

NRC NO. 17465 A OAIL

jo t: :, \(" REPORTLR-597

T.HEORETICAL ANALYSIS OF THE;RANSIENT RESPONSEOF AWVING TO PON-STATIONARY BUFFET LOADS

(JNALYSE THEORIQUE DE LA REACTION TRANSITIOIRE D'UNEAXLE S-OUMISE AU BUFFETING NON-STAT!O14NAIRE)S

by/par /(7

4. 0110,) Speed Aerodynamics Laboratory/ F.R. ThurstonLaboatoiro d'a~rodynamique i hautes vitewese Diretor/Directeuwz

SUMMARY

A method for predicting the response of a wing to non-stationarybuffet loads is presented. The wing is treated as a cantilever beam with /known mass distribution. Using generalized co-ordinates, the vibration ofthe wing is governed by the second order mass-spring-damper oscillator ~vequation. The buffet load on the wing is expressed as an integral of thesectional force, which is a function of the spanwise location and time. Thenon-stationary load is represented by the product of a deterministic time

function and a statistically stationary random function. The time history ofthe applied load is segmented into a number of time intervals. Analyticalexpressions for the mean square response of the wing displacement arederived using a power spectral density for the random part of the appliedload, similar to that used in the theory of isotropic turbulence. The effectsof damping, ratio of the undamped natural frequency of the system to thehalf power frequency of the power spectral density, length of time segment,and duration of applied load on the response of the wing have been investi-gated for three examples of the load versus time histories.

RESUME

La pr~sente communication porte sur tine m6thode do py~voireIn r6action d'une sule sourniise ai une charge tion-stationnaire do buffeting. Onlconsid~ro quo I'sile est une poutreoen porte-ti-faux dont In r6partition de Inmasse est connue. En faisant usage Ai des coordonn(~es gn&A-alis~es, Invibration die Iaile est gouvernk~ par I'tquation do l'oscillateur xnasse-resortanortic du deuxi~no ordre. La charge dun buffeting sur l'ailo est exprini~econine WICe int~grale do la force par unit6 do surface de In section ot qui estune fonction du temps et de Is position suivant l'onvorgure. La chargeP lrinstationnsire est repr~senthe par le produit d'une fonction d6torniine' dutenmps et une fonction stationnaire statistiquement Wi~atoirio. L'6volutionchronologique do In charge appliqu~e est fractionn~ie en un certain no"mbred'intervalles de temips. On de'rive des expressions analytiques do la re'ponseefficace du d~plaeceient de I'aile en utilisant, comnie dans Ia tht~orie de laturbulence isotropique, une densit6 spect~rale de'veloppi~e en puitssances pourla partie aIfstoire de In charge appliqu6e. On a 6tudi6 pour trois exorpende Ia charge en fonctioti de I'volution chronologique les effets sur hni rponsedo I'nile do I'aniortissernoiit, du rapport do In fr6quence propre non amortiedu syst~mc stir In fr~quence A derni-puissance de In densit6, spectrale di~velop-p~e on puissances, do In longuewr do 1'intervallo do temps et do In durt~ed'appllcation do la charge.

$M ra"..U

j- - 7V.....

CONTENTS

Page

SU M M A R Y .............................................................................................................. (iii)

SYM BO LS ............................................................................................................... (vii)1.0 INTRODUCTION ................................................................................................... . 1

2.0 ANALYSIS ..................................................... 2

2.1 Differential Equation of Wing Vibration ......................................................... 22.2 Response to Non-Stationary Input ................................................................... 62.3 Evaluation of Mean Square Response for a Given Power Spectral Density of

the Input .................................................................................................... 112.4 Computational Procedure ............................................................................... 14

3.0 RESULTS AND DISCUSSIONS ............................................................................ 15

4.0 CONCLUSIONS ...................................................................................................... 18

5.0 REFERENCES .................................................. 18

ILLUSTRATIONS

Figure Page

1 Schem atic of W ing ............................................................................................ . 21" 2 Schematic Illustrating Mean Square Loadiig on Wig Versus Time ............... 22

3 Schematic Showing Variation of c With Time ...................................................... 23

4 Contour Integration Paths for l(t, w) ................................................................ 245 Normalized Power Spectral Density ................................................................... . 25

6 Illustration of the e Versus Time Histories for the Three Cases Studied ............... 26

7 Response to a Step Modulated Input for w,,/Wc 0 ........................................... 27

8 Response to a Step Modulated Input for /cof 0.5 .......................................... 28

9 Response to a Step Modulated Input for o,,/w 1 ............................................ 29

10 Response to a Step Modulated Input for ,,Iwfk i 2 ........................................... 3011 Response to a Step Modulated Input for " 0.02 ................................................ 31

12 Response to a Step Modulated Input for " 0.04 ................................................ 32

"(iv)

ILLUSTRATIONS (Cont'd)

Figure Page

13 Response to a Step Modulated Input for 0.08 ................................................ 33

14 Decay to a Pulse Modulated Input for cl/wf 0, " -0.02 ................................ 34

15 Decay to a Pulse Modulated Input for /wf 0.5, t 0.02 ............................ 35

16 Decay to a Pulse Modulated Input for ww/f1 . 1, " 0.02 ................................ 36

17 Decay to a Pulse Modulated Input for /f= 2, 0.02 ................................ 37

18 Decay to a Pulse Modulated Input for o,/,o, - 0, t 0.04 ................................ 38

19 Decay to a Pulse Modulated Input for w,,Iw( = 0.5, t - 0.04 ............................ 39

20 Decay to a Pulse Modulated Input for w,./,r 1, " 0.04 ................................ 40

21 Decay to a Pulse Modulated Input for w/,,If - 2, t - 0.04 ................................ 41

22 Decay to a Pulse Modulated Input for wIco,' a 0, " 0.08 ................................ 42

23 Decay to a Pulse Modulated Input for iw, r 0.5, t 0.08 ................... 43

24 Decay to a Pulse Modulated Input for ,/w1 f - 1, " 0.08 ................................ 44r 25 Decay to a Pulse Modulated Input for ,/wf . 2, ' = 0.08 ................................ 4526 Mean Square Response for Case 1: T' 40, 0 ..................................... 4627 Effect of t on Mean Square Response for Case I ............................................ . 47

28 Mean Square Respons for Case 1 . - 0.02, T 40, wtl/, 0.1............... 4829 Mean Square Response for Case i1 . 0.02, T1 40, /. 0.5 ................... 4930 Mean Square Response for Case I: " 0.02, T' 40, 1 /w1 1 ................ 50

31 Men Sqare espose fo Cas I: 0.0, T~ 40,......- ............... 531 Mean Square Rte nse for Case I: T0.02, T=40, 2 ...... ................ 51

; 2 Metl Squasre lWollse for Case 1: 1% kl 1 0, tOWtII, " 0 ............ 52

33 Mean Square Response for Case 1. 0.02, T- 10, WI/wf 0.1 .............. 53

34 Mean Square Response for Cas: 1 0.02, T4 10, w,,olr 0.5 .............. 54

35 Mean Square Rsponse for Caw I: 0.02, T'4 - 10, Wl/iw 1................ 55

36 Mean Square Response for Case 1: - 0.02, T4 10, w/w - 2 ...................... .6

37 Variation of Delay r with Duration of Applied Force T' ...................... 57

SoC')T 5

ILLUSTRATIONS (Cont'd)

Figure Page

38 Variation of Amplitude Function 6 with Duration of Applied Force TB for*'no /w f = 0 ........................................................................................................ 58

39 Variation of Amplitude Function 6 with Duration of Applied Force T3 for

w ,~ /c f = 0.1 .................................................................................................... 59

40 Variation of Amplitude Function 6 with Duration of Applied Force Tj for0.5 .................................................................................................... 60

41 Variation of Amplitude Function 8 with Duration of Applied Force T' for. . ..n/.................................................... 61

42 Variation of Amplitude Function 6 with Duration of Applied Force TB forn/w f = 2 ........................................................................................................ 62

43 Variation of Amplitude Function 6 with Duration of Applied Force T foro0.02 ................................................... 63

44 Variation of Amplitude Function 6 with Duration of Applied Force TB for0.04 ......................................................................................................... .. 64

45 Variation of Amplitude Function 6 with Duration of Applied Force T' forS0.08 ...................................................................................................... .... .. 65

46 Variation of Amplitude Function 6 with Duration of Applied Force T'H for'0.1 ....................... ....................... ...... ........... ............ 66

47 Effect of At on Mean Square Response for Case 11 ............................................. 6748 Mean Square Response for Case 11 and III: w,/Wf = 0 ....................... 68

49 Mean Square Response for Case 11 and 111: . 0,02, w/f. 0.1 .................... 69

60 Mean Square Response for Case II and Ill: - 0.02, "/oI.. 0.5 .................... 70

51 Mean Square Response for Case I1 and II: 0.02, h/wfo 1 ....................... 7162 Mean Square Response for Case I1 and Iil: - 0.02, wn/w( 2 ........................ 7253 Mean Square Response for Case 1 and Ill: w/ 0 ....................... 73

54 Mean Square Response for Case II and III: " 0.02, wlIw/f 0.i .................... 7456 Mean Square Response for Case II and I11: 1 0.02, w/f 0.57............... 7656 Mean Square Response for Case II and 11I: " 0.02, w,1/or 1 ................. 76

57 Mean Square Response for Case II and HI: 0,02, co/kof - 2 ................ 77

(vi)

4SYMBOLS

Symbols Definition

b wing span

c mean chord

cly) wing chord

cn(y,t) fluctuating sectional normal force coefficient

C141(y) effective slope of sectional lift curveC generalized damping coefficient

SCz 1(t, ,t ) covariance function

E Young's modulus

E[z2 (t)J expected value of z(t)

iI z2t)) defined in Equation (55)

f(t) stationary function"F(yt) sectional fluctuating force!i i ,forceH It(W) frequency respnso function

-I sectional mnoment of inertia

l(tW) tine dependent frequency respouse function

K generalized stiffnets

'L~t) golalized forcela(y) mass distribution

iI generalizod muss

q dynamic premure

RLl) auto-correlation function of input load

constant in expresion for power spectral density

SLL(EtaW2) power spectral density of applied load

t'T time

(vii)i:) 1 . .

SYMBOLS (Cont'd)

Symbols Definition

TB duration aircraft spent in buffet r~gime or duration of applied load

Tn undamped natural period

At, duration of time segment

u(t) unit step function

V aircraft velocity

w1 (y) mode shape function

y distance along span direction

z displacement

13- 1Ht u(t-tr1 ) - u(t-tr)F v( t) deterministic function

6 spatial decay coefficient; also amplitude function

. intensity of applied load

correlation length

damping ratio

Y2~

P (Y a2z ,r) cross correlation coefficient: r delay; also2t-t

w frequency

w widamped natural frequency

Wtd ddmped natural frequency

(f half power frequencySubscripts: 'T' denotes first fundamental bending mode

denotes rth time segment

VL' denotes applied load

SSuperscript: denotes nodimuenslonal time with respeit to T17

(viii)

-1-

TIIEORETICAL ANALYSIS OF TIlE TR.ANSIENT R.ESPONSE

OF A WING TO NON.STATIONARY BUFFET LOADS

1.0 INTRODUCTION

In recent years, increased attention has been given to the investigation of buffet characteristicsin order to meet the design requirements of future aircraft. This arises from the demand for greatermaneuverability and gust requirements in the transonic flight r1gime. Furthermore, the advent of thesupercritical wing held promise for expanding the buffet onset boundary at transonic speeds.

Historically, the term buffeting was first introduced in connection with the structural failurethat occurred to the tail of a Junkers monoplane in 1930. This led to wind tunnel testing of tail vibra-tions excited by the separated flow of the wing. The complexities of the buffet phenomenon, whichinvolves the solutions of the vibrations of the wing surface and the unsteady aerodynamics around it,made the problem not easily amenable to theoretical analysis. Most of the earlier investigations in-volved wind tunnel tests to predict buffet onset and to provide some indications of buffet intensities.A complete account of the earlier work on the study of buffeting is given by Pearcy (Ref. 1).

The use of statistical theory in the investigation of buffeting was pioneered by Liepman(Ref. 2) who examined the problem of the lift force exerted on a two-dimensional thin airfoil movingin turbulent air. Later, he extended the method to wings of finite span (Ref. 3). The analysis wasgeneralized by Ribner (Ref. 4) using a model of turbulence represented by the superposition of planesinusoidal shear waves of all orientations and wavelengths. These analyses assumed the exciting forceto be statistically stationary, so that well known statistical concepts developed in other branches ofscience could be applied. Experimental studies involving model testings in wind tunnels and flight testswere carried out mainly-by NACA using strain gages developed for measuring the wing root bendingmoment. These investigations were reported in detail by Huston and Skopinski (Refs. 5, 6), Huston(Ref. 7), and Skopinski and Huston (Ref. 8). After this period of intense studies in the fifties, researchin this subject was continued into the sixties at a slower pace. Aside from wing buffeting, work ondynamic response of launch vehicles to transonic buffeting was carried out by Cole (Ref. 9), Cole,Robinson and Gambucci (Ref. 10), and Coe (Refs. 11, 12). In the seventies, the rapid advances intransonic aerodynamics generated numerous studies on the transonic buffet characteristics of windtunnel models as well as of actual aircraft. Hollingsworth and Cohen (Ref. 13) reported comparisonsof flight tests with wind tunnel data for the F-4 aircraft. Mayes, Lores and Barnard (Ref. 14) studiedthe transonic buffet characteristics of a 60 degrees swept wing fighter aircraft. The flight and windtunnel tests on the F-8D aircraft was carried out by Damstrom and Mayes (Ref. 15), and the buffetstudies of the F-8 supercritical wing aircraft was reported by DeAngelis and Monaghan (Ref. 16).Detailed investigations of the Northrop F-5A using an extensively instrumented aircraft to study thedynamic buffet pressure distributions on the wing surfaces and the responses during a-series oftransonic maneuvers were reported in References 17 and 18 by Hwang and Pi. A first generationmethod for predicting the buffet loads and responses was described by Mullans and Lemley (Ref. 19),while Cunningham et al (Ref. 20) used fluctuating pressure data obtained for a rigid wind tunnel modelas the forcing function to calculate the aircraft dynamic response and presented results for the F-111A.In general, methods employing wing fluctuating pressure data are found expensive due to the work andtime involved in model construction. Butler and Spavins (Ref. 21) used a more direct approach in pre-dicting buffet response in flight. The technique was based on a method proposed by Jones (Ref. 22)who suggested using a nondimensional representation of the overall aerodynamic excitation and damnp-ing parameters determined from model testing in a wind tunnel. Preliminary results for a combattrainer aircraft showed good agreement between predicted and measured response data.

All the prediction methods for the dynamic response of a wing to buffet loads assume theexciting force to be statistically stationary. This assumption is valid if at a certain buffet condition,the aircraft flight path is unchanged within a long analysis time. In wind tunnel testing, statisticalstationarity requires running time to be very long compared to some characteristic time, for example,the natural period of the first bending mode, assuming the first bending mode to be the dominant one.

.2-

However, in actual flight during buffet maneuvers, the condition of stationarity may not be met, or inwind tunnel testing, transients have to be considered if the running time is very short or the model ispitched during a test. To analyse the response of a wing under such conditions, nonstationary tech-niques to predict the dynamic response has to be developed. A theory of non-stationary response of alinear dynamic system has been given by Caughey (Ref, 23) and extended by Barnoski and Maurer(Ref. 24), and Holman mad Hart (Ref. 25). Using some of the concepts developed in these earlierstudies a method is given in this report to compute the non-stationary response of a wving for specifiedpower spectral density and time history of the exciting force. With this method, it is possible toinvestigate the continuous excursion of an aircraft into the buffc. . Information such as themaximum rms response and the time delay between tie maximunl response and the applied force canbe computed for known structural and inertial properties of the wing as a function of the duration theaircraft spent in the buffet r6gime.

2.0 ANALYSIS

2.1 Differential Equation of Wing Vibration

Figure 1 shows a schematic of the wing which is treated as a cantilever beam fixed at y 0.TThe differential equation of beamu bending relates the time and space derivatives of the displacement zto the load distribution on the beam which may include inertia forces, external or applied components,as well as internal components. The equation can be written as:

ry" l &yit) -- re (y~t) t + F(y,t) (1)

Let

ZYt (y10 (2)

where w, (y) is the motde shape finction., Hero z(y,t) is confined to a single tiormal mode appropriateto vibration in the first fundamiental boildilig motde, anid the subscript "1" is used to denote this mode.Normally. ,',qtation (2) is written as a sumanation of all the possible modes, but in almost all wingbuffeting tests, nIost of the energy in the power spevtrum of the huffet henlding moment is concentratedinl the vicinity of the natural frequemncy of the first mode. Usually, the first mode is well separated fromn-te higher modes, atnd as a result, the response due to the higher mode is small and Elquation (2)suffices in describing the vibrating system. To find z(y,t) at any point on the wing, it is newes'ry todetermine j (t) first. When tie wing is subjected to a time dependent loal, zl(t) is governed by theusual simple setcnd order mas,.pringdcuper oscillator etuation in generalized c)-ordinates.Equation (1) call now be written as:

Mil + Cz' + Kz1 - LMt) (3)

where the dots denote differentiation wift rtespct to time.

The generalized miass NM and stiffness K are defin'd as:

fI fi(y)wIj(ywy (4)

b d2 wv1(y) 1K fFEl11 (5)

Included in the terim F(y,t) in Equation (1) aye two aerodynamnic forces, namely, a fluctuatinglift force and a damping force associated with the fluctuating angle of attack (i/V) induced by therelative velocity of the wing. If the fluctuating sectional normal force coefficient c, (y,t) is known, thegeneralized force L(t) can be evaluated as:

bL(t) q fc(y)w (y)c,, (y,t)dy (6)

4M while the generalized damping coefficient C is

qbC f el (y)c(y)w2(y)dy(7

In the above equation, c1 (y) is the effective slope of the sectional lift curve. Define the undainpedr natural frequency Wn as:

and let

(9)

Usaing the above two equations. Equation (3) can livwritten as

il+ 2ow~i + -nz L~t) (10)

Equ~ationa (7) gives the aerodynamnic darNuinig coeffivientC Whttn structural damping is to beincluded, it can be added directly to C to give the ~.uwl damnping coeff icient (Refs.. 22. 26). Anotherapproach of introducing structural dampin~g is discus"e by Seatan and Rosenbaum (Rof. 27) andDavift and Worrioi (Ref. 28). The damping is assumed to be proportional to the displacement but inphas with the velocity of a harmonically oscillating system. The method applies strictly to harmionicmotion and to the complex exponential form of solution, For m'adotn motion, gwa~t caution has to beexertcised in using such a form of dampingj Let the fluctuating sectionall force be wrtitton as.

bly.t 0 Qciy) ca(y-t w1 (y) (1

.4-

and from Equation (6),

L(t) = b F(y,t)dy (12)

In the above equation, the buffet load L(t) on the wing is expressed as an integral of the sectional forceexpressed as a function of the spanwise location and time. If the sectional force is assumed to be inde-pendent of position along the span, the overall load is obtained by multiplying it by the wing span.However, the exciting force thus obtained will be overestimated since fluctuttions at one chord stationwill not be in phase with fluctuations at another station.

Assume the time average value of F(y,t) to vanish, that is, F(y,t) 0. The auto-correlationfunction of L(t) is defined as:

tim 1 TR f L(t)L(t+r)dt (13)

Substituting Equations (11) mid (12) into (13), IL(r) c4a" now be expressed as:

... . bbIL L(T) a f f 1(y 1,y2;r)dyIdy 1 114)

where 111:(y 1 ,yj ;r) is the spaketime correlation function of the sectional force, whots wean is taken toWbe ro. Define a crossacorrelation Loofficient P1Y ,Yj,) such that

Rpy 11y 2,r) py,y;r) (5

where F&(y ,t), F&(y 2 ,t) O.e the mean squar values of F(y1 ,t) and l wy2 1t) respectively, Onve11 .-(y y12 ;r) is known, then from Equation (14) the meian square value of the load Lit) can be obtained

,L(t) a RL(O) (16)

For some spe.ial types of flow, Ij,(y ,y2, -r) has he sane shalpe at different values of y anddepends only on - y2 - yt' From Equations (14) to (16),

b b-yL3(t) F2(yt) f dy f pt()dt (17)

If p(t) is equal to a constant and has a value unity over the interval 0 , y < b, thenEquation (17) bemes

LE(t) F2(y,t) b2 (18)

which gives the mean square value of the load for perfect correlation. If p(Q) drops rapidly, thenb-yf p(t)dt hardly depends on y. Since this has the dimension of length, it can be termed as the correla-i -Ytion length X. From Equation (17),

L2(t) F2(y,t) X'b (19)

Since the sectional force is correlated only for a short distance, then X is small and hence L2 (t) issmaller than that obtained from Equation (18) for perfect correlation. In actual cases, the form of thecorrelation function RF(yI ,Y2 ,T) depends on the wing geometry (swept or unswept), presence of lead-ing edge flaps and other special characteristics in the design of the wing. However, in the preliminarystage of design, it is helpful to obtain a rough estimate of the buffet load. In order to do so, someapproximate form of the correlation function is needed since the exact form will not be known for thewing until it has been constructed and tested experimentally. Although it is not within the scope ofthis report to discuss this subject in great detail, two other forms of RF(Y1 ,Y2 ;r) are briefly consideredand the resulting values of L2 (t) are evaluated to give some idea of the form of the fluctuating load onthe wing.

For a straight wing with large aspect ratio, the influence of tip vortices on the fluctuatingload is unimportant. The buffet load will be predominantly determined by the fluctuating pressure inthe separated flow region. In Reference 2, Liepman considered isotropic turbulence to illustrate therandom excitation on a thin wing moving through turbulent air. If it is assumed that the buffet loadhas a similar form of power spectrum, the correlation function can be expressed as:

RF(t,O) F2 (t) e- 1/ ( - (20)

where Z" is the mean chord and F2 (t) is constant and independent of y. Substituting into Equation (17)gives

L2 (t' F2(t)-(l - Cb/')bi" (21)

Another form of the correlation function that is of some interest is given by

p(,) =Fim 0-. (22)

where 5 is the spatial decay constant which is determined empirically. This form of the correlationfunction is similar to that given by Mullan and Lemley (Ref. 19) based on data obtained from windtunnel tests of a 10% scale model of the F-4E aircraft. For highly swept wings, they found that wingtip vortices and snag vortices have an important influence on the fluctuating pressure on the wing. Thebuffet load for this correlation function, after substituting Equation (22) into Equation (17) and per-

k forming the integi,.tion, is

.. . .. .

1 W , -~

1~ L2(t) F2(t)'- - - (1 -6b/c) (23)

2.2 Response to Non-Stationary Input

For a given buffet maneuver, the mean square loading on the wing versus time may be repre-sented schematically as in Figure 2. The points A and B correspond to the onset and exit of buffetingrespectively. The duration of time the aircraft spends in the buffet r6gime is denoted by TB. The loadis non-stationary and time varying properties, such as the mean square of the response, can only bedetermined by instantaneous averaging over an ensemble.

The load L(t) can be represented as:

L(t) y(t)'f(t) (24)

where y(t) is a deterministic function of time and f(t) is stationary (Ref. 29). Furthermore, it isassumed that f(z) is Giussian such that the response is Gaussian as well. The non-stationary input canbe represented by a time segmentation technique whereby the time TI) the aircraft experiences buffet-ing is divided into a number of time segments. At any segment, L(t) is expressed as:

i ,. InL(t) 2 y,(t)'(t) (25)

where the subscript r denotes the rth time segment and the deterministic function y,(t) may be con-sidered as a shaping function written in the following form:

Y (t) CAM (26)

where 0,(t) * ult,. 1 ) - u(t-t,). Here u(t) is a unit stop function. c, is a constant at each time seg-ment and represents the intensity of the generalized input load. Figurv 3 illustrates the variation of cwith time and hence reprements the load intensity as a function of time. The formulation as describedin the above manner gives an appr-oximate description of a continuously varying load. However, such arepresentation can be exact in practice, for example, in wind tunnel testings where the model angle ofincidence advances ill steps with time so that the buffet load can be represented by E quation (25).

To determine z1(t), the response zl,(t) due to the rth time segment is first evaluated since

Iz~J ~ E[,(t] (27)

where m is the time segment where t is locatad, that is, t .. t < tIn and E denotes the expectedvalue, Knowing the load L(t), Equation (10) may be solved for z,(t). Introduce a frequency responsefunction HM(co as

14W) +1wit -

: -.

' , "i .xo~ (28), . ., , , . , , . ." ...

".7,

and

IH(w)I2 12( 22+422 2 (29)

The covariance CI z(t 1 AD2 can be expressed in terms of the power spectral density of the applied loadSL L(~lW I 2 ) as:

I(Wit 1-w2t2)

where H*(W.2) is the conjugate of H(W2). If z, (t) has zero moan, then

where R z I (t13,2) is the auto-correlation function. The mean square response can simply be obtainedby setting t~ I t2 -t, that is,

E~z1 2(t)I R~1 21(t't) (32)

In Equation (25), let f,(t) 0. Since RL .1.41t2) EFL(t 1 )L(t2 )I, it may be expressed in thefollowing form after using Equation (25) for L(t1 ) and L(t 2 ):

RLL~lltl 1;'y~l 'Ys(t 2)E~fr(ti~fs~t)1 3

Expressinig Ejfr(tt )fg(tj 12) - tRi,(tt ,t2), Equation (33) can be rewritten as;

RLL(tl,t2) to So Y/r(t1)Y~2R~%t2 (34

If f~l AD~ ,t)is known, t1He from1 the generalized Wioner-Khlnchino relations (Reof. 29),

SLL(0 1 w1(W I -&I It - W212)SL1W4 W2) f RLL(t1,) AD dtd (35)

Combining Equations (34) and (35) yields:

la8w

-( I I -w 1t- 2 t 2)SLL((&)1,(2) 2; f f ^i,(t1)^Ys(t2)Rfrf(tj~t2) e dtjdt2 (36)

If f, (t) and f, (t) are stationary, then Rf rf s(t, ,t2) Rf, f(t, t2) Rrz.f s(r), that is, the corre-l~ation depends only on r t1 - t2. Since r

Ry fr Sy,(co e1' dc (37)

and substituting into Equation (36) gives

InSLL(W~I,WO2) 1; 2 f S(&)A,(wo-w.) A (w-w,.))dw 38

where

21r

and

1 - -i(w-w 2 )t2-f y,(t 2) e dt2 (0

Fronm Equation (30), the mean square value of z, (t) is given as:

VE Iz2() f f SLL~(At,WO)H(c11 ) Ir(c 2)e dwjdw2 (41)

Substituting Equation (38) inito Equation (41) yields

E [ZI(t)J I; E [Z, 2(t)] 1; 2 ' S~ (&)Itt())1(,()d (42)

where

IW1 t

it-----.T (tw Aw-w-.) 14.--. o dw (4 .

-9.

and

-iw2t

I(t,w) is the time dependent frequency response function. Equation (42) gives the general formulationfor amplitude modulated stationary inputs and is a function of the spectral density Syrf('i.) and crossspectral density Sf f,(w~~).

To evaluate I(t,wo), consider the time segment denoted by t 1 -I t < t, (Figure 3). It can beshown that in Equation (39)

7 A~(~-~) I {rs(wi -W) + {ei -1tre (45)t

Substituting this expression into Equation (43) gives

er iwtr H(o 1)) eiwl(t-tr)Ir(t,wa) e fe -e i(w, 1

-e f -- e d4., 1 (46)

The integrals f e dow and f e dw, cant be evaluated bycontour integration. The function H(w1 I (w- has poles at a 1 1 , a2 co. + I W11 and03 ~ - ,Where CWd is the damped natural frequency and is defined as:

(4 Wu (4'7)

Considering the first integral, the Integration path for t > tr is taken above the real axis asdepicted in Figure 4a, while for t < t,, the integration is below the real axis. Similar paths of Integra-tion are taken for the second integral. Upon substitution of these integrals into Equation (46), thefinal form becomes:

1w r wr- Ww

U(t-tr) [e etp)t+t,)+ -t (48)

-10-

where

S08e Cos dt +"sinC.d (49)

and

(t) e sin Wdt (50)

In buffet maneuvers or in wind tunnel testing of a pitched model whose incidence changeswith time like a step function, the loading on the wing can be approximated by Equation (25) withfr(t) being the same in all time segments. The only changes in L(t) being the intensity. Th-s may be afair approrimation since buffeting is primarily due to the fluctuating pressure forces of the separatedflow on the upper wing surface. The frequency structure of the separated flow may not change by alarge extent as the incidence varies. If it is assumed that there is no correlation between differentsegments, then from Equation (42),

E [z(t)] = f Sf r(W)Ilr(ts)1 2 dwA (51)

where after some algebra, IIr(tW)12 is given as follows:

for t,_ < t < tr,

+ 0 2 (t-t,.) + i 0 2(t-tr ) (52)CWdd

fort> tri

* ~~~~II(t,w)I 2 = r2 IH((O1 ' 2 tw~ + OP(tdtr)

-+ W 2(tt r-O) + -- 0(t-tr) (53)c2 c2W + _ .2ttr-+_.tt)

2cos((tt-t)_) 44t 0(t-tr)

S~+

+ 2-sin w(t,-4- 04044Wd

- *(tY.O~i~t-~)]}(53)

2.3 Evaluation of Mean Square Response for a Given Power Spectral Density of the Input

Equation (51) can be integrated to give the mean square response, provided the form of thepower spectral density Sf r f(w) of the input is specified. Cole (Ref. 26) gives a form of the spectraldensity which has been found to adequately describe the buffeting input to launch vehicles. In thepresent notations, this can be written as

SoSf, f (W) (54

Sfrfr( W) (where wf is the half power frequency and S0 is a constant. In Figure 5, is plotted versus

so nOn

for different values of -. For - 0, Sf f (w) constant So, which is the power spectral density(On

for white noise. As - increases, the energy of the exciting force near the natural frequency w,,(Af

diminishes and this will have an appreciable effect on the response of the system.

Consider the rth time segment and using Equation (52) or (53) for II(t,w)12 together withthe expression for IH(w)J 2 given in Equation (29), the mean square value E[zIt (t)] can be evaluatedanalytically since the form of the power spectral density is known. Let

2rM2o1 3&Czi[ 2(t) - i E[ZI12(t)] (55)irS

Using contour integration and after much algebra the final expression for & (Zi 2(t)z are as follows:

where for tr.. tt,

-122

[F4-t-Y.i + ~ -e

1+- 2-+ ~~~ 2 A' ~ 5 6

and for t> t1,

E 2t

+

(57IW

with

11(t) 11+( )+(58)

H ~t) +~ Ot)- G(-0)

KQ +(1 -4 Pt2) - +2t(1

for2

GH1t

-14-

and for t> t,,

W. &Z, 2 (t)] C~{~2 (t~)+i 2 t

+ (64)

In the limit for -0, these two expressions give the response to white noise.(A) f

2.4 Computational Procedure

The steps involved in computing the mean square response is fairly simple and straight-Vforward. For a given value of, a~ [z2 (t)] can be evaluated from Equations (56) and (57) (or

bWfEquations (63) and (64) for white noise) once the c,'6 are given as functions of time. Referring toFigure 3, at any specified time t, the procedure is to establish the time segment where t lies. If it fallsbetween t~and t.n, then &(z1 2(t)] is obtained as follows.

8[2(~ &Iz2(t)] + Z 8[z1 (t)J (65)Tel

where the first term on the right hand side of the equation is determnend from Equation (56) while thesecond term from Equation (57).

Three cases are considered where e varies with time in the form of the following functions.

Case: 1: SinusoidalCase 11" RampCaweIII: Tringular.

These functions are shown schematically in Figure 6 and shall be referred to the case number in thefuture. F'or convenience, the maximum value of e is taken to be unity and this does not affect the

-15 -

generality of the solutions. The number of time segments in TB will be specified later on after theeffect of varying the segment duration (tr-tr..i) has been investigated. The variables involved in the

Wmnanalysis are -, , and TB, and their influence on the mean square response are studied in some detail.

3.0 RESULTS AND DISCUSSIONS

Since Equations (56) and (57) are the principle expressions to be used to compute the meansquare value of z, (t) for specified non-stationary loads, their characteristics are first investigated insome detail. Equation (56) determines the response or rise to a step modulated stationary excitingforce, while Equation (57) governs the decay when the applied force is withdrawn. Taking er to beunity, Equation (56) is plotted in Figures 7 to 10 as a function of time t'-t',_1 for three values of the

Wn Wndamping ratio (' = 0.02, 0.04 and 0.08) and four values of -, namely, - = 0, 0.5, 1 and 2. The

of 0Jr

superscript ' is used to denote that time is non-dimensionalized with respect to the undamped naturalpctiod Tn - 21r/w.. From these figures, it can be seen that for larger values of , the mean square ofz1 (t') reaches an asymptotic value for t'-t41 -b 0o sooner than that with smaller values of . Increasingthe value of cjn/Wf tends to diminish the asymptotic value, This can be explained from Figure 5 wherethe normalized power spectral density is plotted versus w/n-. The energy in the exciting force nearthe natural frequency cn diminishes for increasing Wn/WOf, and hence it can be expected that the valueof the mean square response will decrease as wn/Wf increases. From Equations (49) and (50), it is seenthat as t'-tr_1 -o- o,(t'-t4_) - 0 and 0(t'-t4-1 ) -- 0. Hence Equation (56) can be written as (takingEr = 1):

W 2 .n31 +(1-42) - + 2S2 Wf3(66)

I+ (on 2)2- 4t n2Wf2 / 4 2 f

FQ [z 2 (t')] decreases as wn/wf increases. For Wn/,f >> 1,

g [ (67)r WnWf

In the limit wn/Wrf P co [z 1 (t')] -+ 0.

In the computations, no significant changes in P [z, 2(t')] compared to that at w,/Iw = 0 aredetected for values of on/Wf less than 0.1. These curves are replotted in Figures 11 to 13 for fixedwith wn/wf as the variable. The oscillations observed are due to the sine and cosine terms in theexpressions for 4(t'-tr_) and 0(t'-t.r_1 ), but they are rapidly damped out for increasing t'-t>.

Equation (57) is used to compute the decay of the mean square of zl_(t') after the excitingforce has been withdrawn at time t. Using the same values of and wn/Wf as before, the decay to apulse modulated statiolqary exciting force are shown in Figures 14 to 25 for different values of the

16-

pulse duration (t-t_ - 0.5, 1, 2, 4 and 8). Since the curves are plotted with t'-t'-. as the absicca,then t-t41 is simply measured from 0 along the axis to the time t'-t'.., when the decay curve starts.

For large values of the pulse duration, Equations (49) and (50) give

( t', -t ,_ ) - 0T T

ast' - tr"! o

Also from Equations (58), (60) and (62),

F(4 - t',_1)- 0

H(tr' - t- 1) 0

Q(4; - t4-1)- 0

Equation (57) cAn be written as (taking er = 1)

Pz.Z2(t')] = {K0,2(t'-t') + 61(t'-)) (68)1-

(62

which is independent of tr-t',_1 . In other words, the decay is independent of the step duration forlarge values of tr-t'r 1. From the figures, it is also observed that for constant , the decay is more rapidas W,, fOf decreases, especially for cases with the larger values of t'-t - . This can be shown fromEquation (68).

For wn/wr

-17 -

Also of interest in those figures are the oscillations in the decay curves. For small damping,the oscillations are very clearly shown. The amplitudes increase with con/wr while the decay is lessrapid. This is even more significant for smaller values of t;-t;, and will be shown later to have quitean effect on non-stationary loads.

For an input load with sinusoidal variation of e with time (Case I in Figure 6), . [z 12 (t')] hasbeen computed using Equation (65). The duration of the applied load T' is forty times the naturalperiod, and it is divided into eighty equal time segments giving a value Atr = tr-t'r 1 of 0.5. Figure 26shows the response for = 0.02, 0.04 and 0.08 with wo,/cof = 0. Also plotted in the same figure is the eversus time curve for comparison purposes. Similar to Figures 7 to 10, the rise time is seen to besmaller for the larger values of . Shown in the figure is the delay r, which is the time where the maxi-mum value of . [zI 2 (t')I lags behind the applied force, and the amplitude function 5 defined as

= 1- [jzi 2 (t')]m8x.

The effect of changing the number of time segments has been studied, and it is found that ifAt' < 1, the differences in the computed results are not very significant. Figure 27 shows the responsecurves for three values of At, and it is seen that the curves with At; = 2 and 0.5 are quite close. In allsubsequent computations, A6t' is taken to be 0.5.

The effect of wI,/cor on & [z, 2(t')] are shown in Figures 28 to 31 for wn/wf = 0.1, 0.5, 1and 2 with " = 0.2. The oscillations in the response curves increase with con/co, and the peak to peakvalues reach a maximum at wn/cof = 1. These oscillations arise mainly from the second term ofEquation (65) where previous results (Figures 14 to 25) show that for increasing values of w,/w,,the decay to a pulse modulated input has fairly large amplitude oscillations and diminishes ratherslowly with time. The summation in Equation (65) for a large number of segments help to accentuatethis effect, thus resulting in large regular oscillations.

To illustrate the effect of the duration of the applied force, results with T, = 10 are pre-sented keeping all other variables the same as before. Figure 32 shows P[z1 2 (t')] versus t' forW0n/wf = 0, " = 0.02, 0.04 and 0.08. Comparison with Figure 26 shows that the delay r is shorterwhile 5 increases. Comparing Figures 33 to 36 with Figures 28 to 31 gives the effect of T', on themean square response. Decreasing the value of T6 results not only in a smaller peak value of ' [z 12(t')],but also decreases the peak to peak value of the oscillations.

The delay r is a function of " and T%. Its variations with T' for " = 0.02, 0.04 and 0.1 areshown in Figure 37. The delay increases with T' in steps of half the natural period. For " 0.04,and 0.1, computations up to T = 60 show that r does not change from the values 1 and 2 whichcorrespond to those reached at T6 = 10 and 18 respectively. It thus appears that r has reached themaximum for these two values of '. For " = 0.02, computations have not proceeded for sufficientlylarge values of T to determine the maximum 7. Figures 38 to 42 show the variations of 8 with TB'for w,/cof 0, 0.1, 0.5, 1 and 2 with " as the parameter. The results indicate that 6 increases withdecreasing " for any fixed won/ol while increasing T6 results in smaller 6. For large values of T6, 6tends to an asymptotic value and the approach to this value is more rapid as increases. By increasingWn/wf, 6 is also increased as shown in Figures 43 to 46 where 5 is plotted against T' for fixed "with ,on/wf as the parameter. These curves again indicate that for larger values of , 6 approaches itsasymptotic value more rapidly.

Similar to Figure 27, the effect of At; on (, [zt 2(t')J is plotted in Figure 47 for Case IIwith " 0.08 and three values of At, (0.5, 2 and 4) are considered. The same conclusions are arrived atas before, that is, for At, < 1, the differences in the response curves are very small for different At;,and hence a value of 0.5 is again used throughout the computations.

The results for Case II (Th = 20) and Case III (T' = 40) are shown in Figure 48 whichserves to illustrate the effect of " at won,/wf = 0. Figures 49 to 52 give the response curves for these

-18.

two cases. The value of is 0.02 and n/%,= 0.1, 0.5,1 and 2. The effect of TB is demonstrated inFigures 53 to 57 where I = 5 for Case II and T - 10 for Case II1. These results are very similar tothose for Case 1. The most significant differences between Case I and Case III are the smaller values ofthe 9 (z, 2(t')1"';" for the latter case. This is to be expected since e reaches its maximum and decreasesat a much faster rate than in Case I.

4.0 CONCLUSIONS

A method for predicting the response of a wing to non-stationary buffet loads has been tdeveloped and applied successfully to a number of hypothetical examples. For buffet maneuvers in anaircraft, the fluctuating loads change continuously and hence the analysis presented herein whichmodels the load by a time segmentation technique, is to be treated as an approximation. However,this approximation has been found to be fairly good since a study of the effect of the duration of timesegments shows that if it is below a certain value, there is little change in the results when smaller timesegments are taken. In wind tunnel buffet tests, it is possible to control the incidence of the model bypitching it in a prescribed manner. The buffet load can be made to vary with time in the form of astep function. In this case, the method of representing the load described in this report can be consid-ered to be exact.

The form of the power spectral density of the input load used in this study is similar to thatencountered in the theory of isotropic turbulence, and analytical expressions for the mean squareresponse of the wing displacement has been derived. The effect of varying the ratio -of the undampednatural frequency to the half power frequency c.,/Iw- on the response of the system has been investi-gated and it is shown that if w,,/wI, is small, the maximum value of the response is greater than thatobtained for larger values of won/trf This is due to the distribution of energy in the input load since alarge value of w,/t-" implies that little energy is distributed near the undamped natural frequency. Itshould be noted that for white noise, equal energy is distributedL at all frequencies, and wj/W. in this!Lose is equal to zero.

The duration of the applied load T'D and the form of tile time history of the load have alsobeen studied. Detail results are presented for a sintusoidal variation of the force with time. ('oniputa-tions have been carried out showing the effect of danping and w,,1w, on thi delay r and amplitudefunction IS of the response curve. It is found that for larger values of , r rvaciv% a constant value lessthan those obtaimnd for smaller '. Also, the time I'l it takes to reach this comst'Wt value is muchshorter for larger values of . For fixed wjwfrt decreasing " increases 6, while for fixed ', increasingtonI , increases 6. Results for itiput Ioads expressed as raunp and trialtgular functions are also pre-senttcd, For the triangular input, fth results are quite similar to those for the sinusoidal cas; the aliostsignificant difference being the smaller values obtained in the rospon-se curves.

5.0 IEFERENCI4S

1. Pearcoy, H.H. A Method for the I'rediction of t(he Onset o( Buffeting and OtherSeparation Effects from Wind Tunnel Tsts on RTguz Models.North Atlantic Teaty Organization, Advisory Group for Aero-Snauticai Resarch mand Development Rcport 223, October, 1958.

2, Liepani. tI.W. On the Application of Statistical Concepts to the Buffeting Pxrblem.Journal Aeronautical Scisences, Vol. 19, No. 12, Dece.mber. 1952,pp. 793-800.

3. Liepman, lI.W. Ext,.nsion of the Statistical Approach to Buffeting and GustRespon&, of Wings of Finite Span. .Journal Aeronautical Sciences, Vol. 22, No. 3, Mardi. 1955,

4 pp. 197-200.

I: t.

4. Ribner, H.S. Spectral Theory of Buffeting and Gust Response: Unification andExtension.Journal Aeronautical Sciences, Vol. 23, No. 12, December, 1956,pp. 1075-1077.

5. Huston, W.B. Measurement and Analysis of Wing and Tail Buffeting Loads on aSkopinski, Ti-I. Fighter Airplane.

National Advisory Committee for Aeronautics, Report 1219, 1955.

6. Huston, W.B. Probability and Frequency Characteristics of Some Flight BuffetSkopinski, T.H. Loads.

National Advisory Committee for Aeronautics, Technical Note 3733,August, 1956.

7. Huston, W.B. A Study of the Correlation Between Flight and Wind Tunnel BuffetLoads.North Atlantic Treaty Organization Advisory Group for AeronauticalResearch and Development, Report 111, An-ril-May, 1957.

8. Skopinski, T.H. A Semi-Empirical Procedure for Estimating Wing Buffet Loads in theHuston, W.B. Transonic Region.

National Advisory Committee for Aeronautics, 1tM L56201,September, 1956.

9. Cole, Jr. H.A. Dynamic Response of 1Ham merhead Launch Vehicles to 7ThansonicBuffeting.National Aeronautics and Space Administration, TN D-1982,October 1968.

10. Cole, Jr. H.A. Detection of Flow-F ield Instability in the Presence of Buffeting byRobinson, R.C. the Partial-Mode Techn~iq ue.Gambuoci, B..Y. National Aeronauticr. and Space Admninistration, TN D-2689,

Februiury, 1965.

11. Coe, C.F. Steady and Fluctuating Pressures at f'voicSpeeds on Two Space-Vehicle Payload Shapes.National Aeronautics and Space Administration, TM X-603, 1961.

12. Coe, C.F. The Effects of Some Variations in Launch- Vehicle Nose Shape onStea3dy and F luctuating Pressures at Transonic Speeds.National Aeronautics and Space Admainistration, TM X-646, 1962,

13. Hollingsworth, E.G. Determination of P-4 Aircraft Tra-isonic Buffet Characteristic&.Cohen, M. Journal of Aircraft, Vol. 8, No. 10, 1971, pp. 757-763.

14. Mayes, J.F. Transonic Buffet Characteristics of a 600 Swept Wing with DesignLores, M.E, Variations.Barnard, H.R. Journal of Aircraft, Vol. 7, No. 6, 1970, pp. 524-530,

15. Damstrorm, EXK Transonic Flight and Wind-Tunnel Buffet Onset Investigation of theMayes, J.F. F-SD A ircra ft.

Journal of Aircraft, Vol. 8, No. 4, 1971, pp. 263-270.

16. DeAngelis, V.M, Buffet Characteristici of the F-8 Supercritical Wintg A irplan,wMonaghan, R.C. National Aeronautics and Space Admniinst~ration, TM 66049, 197 7.

-20 -

17. H-wang, C. Investigation of Northrop F-5A WinW Buffet Intensity in TIransonicPi, W.S. Fligh t'

National Aeronautics and Space Administration, CR-2484, November,1974.

18, liwang, C. 'I-anson ic Buffet Behavior of Northrop F,-5A A ircraft.Pi, W.S. American Institute of Aeronautics and Astronautics, Paper 75-70,

January, 1975.

19. Mullans, W.E. Buffet Dynamic Loads 1)uri 'I)ansontic Manezvers,Lemley, C.E. Air Force Flight D~ynamics Laboratory, Technical Report AF'FDL-

TR-72-46, September, 1972.

20. Cuininghaim, A.B. Developmenit anid Evaluationi of a New Method for Predicting AircraftWaner, Jr. P.G. Bu~ffet Response,Watts, J.D. American Institute of Aeronautics anid Astronautics, Paper 7 5-69,Benepe, D.i. January, 1975.Riddle, D.W.

21. Biutler, CU.1" Prlinmimry Evaluationi of a 'Ieviniqw, for Predicting Buffet Load$ inlspavins, G.R. Flight fromt Wind-'1'unel Measuremenits on Alodc-is of Conventional

Conistruct iol.North Atlantie'T'reaty Organization, Advisory Group for AeroalpaceResearch and Development, Paper 23 of AG ARt C11-204, 1976.

22. Jonus, J.G. A Survey (if the Dynamic A nalysis of Buffetinig and RelatedRoyal Aircraft EsbIt',, 1'caia Report 72197, February,1973.

2. Catughey, T.K. Noit-'tatiotiory Ranidom Inputs and Responses, in ''RandomVillration'',Vol. 2 mi. Crandall, 8i11., MIIT Protis, Cambridge, Massachusetts,

24. IlarOtioki, R.L. Aleon-SquanRe UsJ)ori5& Of SUimt, jAl'echanical Sysit'llis to Non-Maurer, J.R. stat tollory Random I'.,xvitotionl,

'h'riisuctioii of the Amerivan Society of Mechimical Enigineersi,Series, le, Jotirnal of Applied Mechanics, Vol. 36, No. 12, June, 1969,

S221-227.25. 1lolmati, ICE.. Nonstationaryfv RsRipoom of Structual Systems.

I art, G.C. AS('l, Journal of Hngftiivering Mulwbanics Division, Vol. 100, No. EtM2,April, 1974,1qP.,4154.3 1,

26. Cole, Jr. WIA. On-Tlhe-Litit Analysis of Rtandomn Vibr-ations.Amnerican Inistitute of Aeronautics and Astronautics, Paper 68-288,April, 1968.

21. Scalan, Rtll. ItduiototeStudy of Aircraft Vibration anid F'lutter.Itosellbaum, It. Dover Publications Itic., Now York, 1968.

28. Davig, Di.U. luffe~t Tests of (tn A ttack-A irplaric Aodel ivith Emiphasis on AnalysisIWornom, D.E. of D~alto fromn lVitid-Twimnel Tests,

National Advisory Comwittee (or Aeironautics, HM 1671-113,February, 1958i.

29. Beundat, JXS Rattilomn Data: AnalysiN anid fttcasuremetnt ftocedures.Plorsol, A1.G. WiVley-Interscience 1971.

>0c (Y

KII

FIG. 1: SCHEMATIC OF WING

... . .77

-22-

L ft

BUFFET REGIME

0U.

twTa

w

TIME

FIG. 2: SCHEMATIC ILLUSTRATING MEAN SQUARE LOADING ON WING VERSUS TIME

fj vp~

-23.

E

4 E

tr tr T

TIME t

FIG. 3: SCHEMATIC SHOWING VARIATION OF cWITH TIME

-24-

t tr

w-PLANE

FIG. 4: CONTOUR INTEGRATION PATHS FOR Ir(t,w)

-25-

wnfA

1.0 .

0.90.05

$ 0.8

0.7

06.

q~0.5

0.4-

0.3s

00.0 0 1 2 3 4 5 6 7 8 9W/wn

FIG. 5: NORMALIZED POWER SPECTRAL DENSITY

-26-

CASE (I) SINUSOIDAL

0 TIME T

CASE (11) RAMP

0 T TIME

CASE (111) :TRIANGULAR

TIME Ta

FIG. 6, ILLUSTRATION OF THE c. VERSUS TIME HISTORIES FOR THE THREECASES STUDIED

-27-

[ i1.0'0..0

0.70

0.6-

0.7-

N-0.5-

04 0

0.4-

0.3 4

0.2-

0.1

0.00 1 2 3 4 5 6 7 8 9

FIG. 7: RESPONSE TO A STEP MODULATED INPUT FOR o,/~=0

-28.

0.9-

0.8-

0.7-

0.6-

- 0.52

0.45

0.4-

0.2-

0.1

00 I 2 3 4 5 6 7 8 9

/ FIG. 8: RESPONSE TO A STEP MODULATED INPUT FOR w,, =J 0.5avow

-29-

1.0-

0.9-

0.8

0.7

0.6-

0.5-

S0.5 .0

0.40

0.3-

0.2-

0.1

0.0 I0 I 2 3 4 5 6 7 6 9

FIG. 9: RESPONSE TO A STEP MODULATED INPUT FOR ~j~w

1.0

0.9

*1 0.8

1 0.7?

0.6-

I N. 0.5-

0.4 WIn2

1.0.3;

0.1

.' ~~~FIG. 10: RESPONSE TO A STEP MODULATED INPUT FOR (~,%O 2V

-31-

1.0

0.9 9a0.02

0.8-

0.7-

0.6-

0.4-

0.3-

L .. 0.2-

0.1

0.0

FIG. 11. RESPONSE TO A STEP MODULATED INPUT FOR 0.02

.32.-

1 .0 AIn

-0.04

0.9

0.8

0.75

0.6

0.4

0.3

1'.

I0.

-33.

1.00

0.9

0.8

0.7

0.6

C%~0.5

0.4

0.3

0.22

t tf FIG. 13: RESPONSE To A STEP MODULATED INPUT FOR 0.0841*

.. ~..

.34.

(O

ro N

0

co 4

o 0~

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IA LL

Ui.

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j -36-

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= 0

CL

A,, 0

Ile-co

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to

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w

LU

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--

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tco

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