NASA Technical Memorandum101573

TIME-CORRELATED GUST LOADSUSING MATCHED-FILTER THEORY ANDRANDOM-PROCESS THEORY -A NEW WAY OF LOOKING AT THINGS

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Anthony S. Pototzky, Thomas A. Zeiler, and Boyd Perry III

1

APRIL 1989

NationalAeronautics andSpace Administration

Langley ResearchCenterHampton, Virginia23665-5225

https://ntrs.nasa.gov/search.jsp?R=19890015861 2018-09-05T21:26:39+00:00Z

P

_r

Time - Correlated Gust Loads Using Matched Filter Theory anti

Random Process Theory - A New Way of Looking at Things

by

Anthony S. l'nlolTky _'and Thomas A. Zeiler**

Pla,mlng Research Corporalion

Aerospace Technologies Division

Ilanption, Virginia 23666

and Boyd Perry, !!I***

Langley Research Cenler

National Aeronautics and Space Administration

Ilampton, Virginia 23665

Abstract

This paper describes and illustrates two ways ofperforming time-correhued gust-load calculations. The first isbased on Matched Filter Theory; the second on Random ProcessTheory. Both approaches yield theoretically identical results andrepresent novel applications of the theories, are computationallyfast, and may be applied to other dynamic-response problems.A theoretical development and example calculations ttsing bothMatched Filter Theory and Random Process Theory approachesare presented.

Nomerlclatttre

Rol?tal, l

G(s) transfer function of vertical velocity component

of yon Kannan turbulence (gus0

hi(t) time response of output variable i to unit im-pulse

[li(to) frequency response function of output variable i

K arbitrary constant

L scale of turbulence in G(s), equation ( I I)

Rij('t) cross-correlation ftmction between quantities iand j; auto-correlation ftmction if i equals j

s Laplace variable

Sij((n) cross-power spectral density function betweenuantities i and j; auto-power spectral density

unction if i equals j

t time

t,, arbitrary time shift

V velocity of aircraft in eqoation (I I)

* Engineering Specialist. Member AIAA

** Project Structures Engineer. Member AIAA

***' Aerospace Engineer

wg[t)

W,(o_)

x(t)

X(o_)

y(t)

yy(t)

yz(t)

Y(o_)

z(t)

zy(t)

Zz(t)

z(o)

(3"i

't

O3

g

h

X

vertical velocity component of gust

Fourier transform of wg(t)

"matched" excitation wavefonn

Fourier transform of x(t)

time response of output quantity y

time response of output quantity y to excitationmatched to y

time response of output quantity y to excitationmatched to z

Fourier transform of y(tl

time response of output quantity z

time response of output quantity z to excitationmatched to y

time response of output qnandty z to excitationmatched to z

Fourier transform of z(t)

Greek

root-mean-square of quantity i

time argument for at,to- and cross-correlationfimctions

circular frequency (rad/sec)

Superscripts

complex conjugate

Sub._cripts

gust

unit impulse time response

matched excitation waveform

y responsequantityy

z responseqtmntityz

lnlroduclion

The primary structure of ;|n aircraft II]USt withMand allthe static and dynamic loads it is expected to encounter during itsllfe. Svch load's include landing and taxi loads, maneuver loads,

and gust loads. There are several methods used by airframemanufacturers to compute gust loads to satisfy certificationreqttirements and include both stochastic methods anddeterministic methods. The U.S. Federal Aviation

Administration (FAA) recently asked the National Aeronauticsand Space Administration (NASA) for assistance in evaluatingthe Statistical Discrete Gust (SDG) Method [ilas a candidate

gust-loads analysis method for complying with FAA certificationrequirenlents. The SDG method is a time-domain approach and

yields time-correlated gust loads by employing a searchprocedure.

Duri,ag the course of the NASA evaluation of the SDGmethod, the authors recognized that Matched Filter Theory(MFT) [2] could be applied to the linear gust problem tocompute time-correlated gust loads. Computing time-correlatedgust loads in this manner has the twin advantages of beingcomputationally fast and of solving the problem directly.Historically, MFT was first utilized in the detection of returningradar signals. Papoulis [31 showed that the principles of MFTcan be used to obtain maximized responses without the use ofcalculus of variations and in applications other than signaldetection. The first purpose of this paper is to demonstrate thatMFT is also applicable to aeroelastic systems, and specificallyapplicable to the computation of time-correlated gust loads.

During the course of the MFT investigation, the authorsfound that the maximized and as well as the other responses ofthe system looked similar to auto- and cross-correlation

functions of Random Process Theory (RPT) [41. From thisobservation and also from theoretical considerations given later,it was recognized that time-correlated gust loads computed byMFT are theoretically identical to those obtained using RPT. Tothe knowledge of the authors, the correlation functions of RPThave, t, ntil now, riot been interpreted as time-correlated gristloads. Some preliminary analyses demonstrating thisinterpretation were presented by the authors in reference 5. Thesecond purpose of this paper is to show the relationship betweenthe results ote MFT and RPT.

Both the MFT and the RPT procedures for computingtime-correlated gust loads involve novel applications of thetheories and unconventional interpretations of the intermediateand final results. This paper outlines the mathematicaldevelopments, recognizes a duality between MFT and RPT, andpresents example calculations using both MFT and RPT for

computing time-correlated gust loads.

Backr, round

This section of the paper defines time-correlated gustloads and the current analysis capabilities in this area.

Time-Correlated Loads

This paper de'ds specifically with time-correlated gustloads and figure I illustrates two types of such loads. Time-correlated loads are time histories of two or more different load

responses to the same disturbance quantity. In the figure thedisturbance quantity is the vertical component of one-dimensional atmospheric turbulence and the time-correlated

loads (the output quantities) are the resuhing [oad responses (inIhis case tl_e bending moments and torsion moments) at severallocations on Ihe airplane wing.

The first type of time-correlated load is illustrated on theright wing: loads (two bending moments in this illustration) at

twf_ different locations on the airplane. The second type isilhlstrated on the left wing: two different loads (bendingmoment and torsion moment in this illustration) at the samelocation on the airplane.

As indicated by the time histories in the figure, timecorrelation provides knowledge of the value (magnitude andsign) of one load when another is maximum (positive ornegative). In general, the time correlation principle allows thecomputations of values of all loads at all the stations with respectto the time when any one maximum load occurred. Such

information may be used directly during analyses and testing ofaircraft structures [61.

I - _M T

/l / _ ,

/ l" t, i.;...... r..oI

Pf..Z+ ....t,,'t.,,,,-,,,,

• w " ", BM 3

,_ TM3

Figure l. Time-Correlated Gust Loads

_tatistical Discrete Gust Method

The SDG method determines the response time histories

of "worst case" gust loads (such as shear forces, bendingmoments, and torsion moments) and the corresponding "critical

gust profiles" which produce them. These loads are timecorrelated and this feature is a major advantage of the SDGmethod over some other gust loads analysis methods.

Another advantage of the SDG method is its applicabilityto nonlinear, as well as to linear, systems. The SDG method

employs a search procedure to obtain its answers. For a linearsystem, by taking advantage of superposition, the searchprocedure may be simplified and reduced to a one-dimensionalsearch. For a nonlinear system, however, this is not the caseand the resulting search procedure remains muhi-dimensionaiand can become computationally expensive [71.

In light of the technical contributions provided by theSDG method as outlined above, this section states the conditionsunder which and the means by which this paper makes acontribution to the area of time-correlated gust-load calculations:

(a) the SDG method is capable of performing both linear andnonlinear analyses, whereas the present methods ,are restricted tolinear systems only; (b) the SDG method uses a searchprocedure, whereas the present methods obtain the samequantities directly and with a significant reduction in

computation time.

TbeorHical Development

In this section the theoretical background of time MFT and

RPT approaches will be presented. For this paper, the Fouriertransform of a function f(t) is defined by,

FF(to) = lift) e i,,. (it.

IV

and the inverse Fourier transform is given by,

Mulched Filler Theory ADoroach

The objective of MFT, as originally developed, is the

design of an electronic filter such that its response to a knowninput signal is maximum at a specific time [3,8]. It found earlyapplication to radar considering the "filter" to be a detector that.in response to a known input signal, produces an output signalfor further processing [21. In this ease the detector design is theoptimum design for maximizing the output signal-to-noise ratio.The matched filter is designed so that the unit impulse response,

hy(O, of the output y of the filter is proportional to the knownsngnal, shifted and reversed in time,

hy(t) = Kx(-t + to), (la)

where K is an arbitrary constant, to is an arbitrary time shift, andx(t) is the known input signal. As will be shown later, the time

shift may be selected through criteria of convenience only andwill be the time that the matched filter response is maximum.

In the present application, the filter is considered to be asystem whose dynamics are known. Specifically, the system is

characterized by the combination, in series, of the dynamics ofatmospheric turbulence and the dynamics of aircraft loadresponse. Thus by(t) is known. The input signal, x(t), is to bedetermined from'the impulse response of the system, Fromequation (In),

x(t)=hy(to-t)/K. ([b)

The simple result of MFT allows direct determination of theinput signal, or excitation, that produces a maximum response ofthe system. The result, as will be shown, is the maximum loadresponse and the critical gust profile.

Unit Impulse Impulse Response

1/MalcheO Filler ]'beery ,[. hi. = K,c(- II. I))

Resull 1=l_h t -

i

Matched E:gcdalion WaveformResponse to MalchedExcflation Waveform

Figure 2. Mulched Filler Theory - Signal Flow Diagram

Maximum Response tO ",M_tqhed" Excijotion Waveform -Equado.s (In) and (lb) express the fundamental result ofMatched Filter Theory and figure 2 illustrates the result: theresponse of tile system "filter" to a unit impulse is proportionalIo the desired input waveform, x(0, when delayed by to andreversed in time. For convenience and to be consistent withreferences 3 and 8, the form given by equation (la) will be usedin the remainder of the theoretical development of the MFTapproach. It should be pointed out that either form of the theequation will lead to the same final results.

The system response to x(t) may be obtained from theinverse Fourier transform

m

(2a)

where

Y(m) = Hy(tO) X(m), (2b)

and Hy(to) and X(co) are the Fourier transforms of hy(t) and x(t).

Hy(o.)), which is the frequency response function of the outputy, may be determined from equation (la) as

tly(tO) = Ke i_. X(-m)

= KX'(_) e-i_. (3)

Thus the response y(t) is

.co

(4)

Note that the product X'((o) X(m) is the energy spectrum or

power spectral density (PSD), Sxx(to), of x(t), and is an even

function of frequency, so

y(t) = K 2"_/S_tx(m)c°s(°_t - t°)) dto. (Sa)

The maximum response occurs when t = to,

(5b)

It is convenient Io set the term in brackets in equation (Sb) equalto unity, which means that the root-mean-square (RMS) of theexcitation waveform, x(t), is unity. That is,

27t "(6)

It can be shown that a necessary result of this assumption is thatthe arbitrary constant. K, is equal to the RMS of the impulse

response, oh_ . The stipulation in equation (6) that thewaveform be of unit energy will be used later as a constraint onall excitation waveforms considered and will provide a basis forcomparison of their effects.

Finally, substituting for K in equations (5a) and (5b)gives,

y<t)- % cos(0)(,-to))

and

Ymax= Ytto) = Ohy •

(7a) and l[y(0)) is the frequency response function of an aeroe[asticsysten) Io turbulence. With the turbulence "pre-fiher" known,the critical gust profile can be obtained as an intermediate resultas

Wg(to) = G(o_)X(m)

(7b) and from equations (3) and (13),

The input x(t) is said to be 'qllatched" to y, since x(t) isdetermined from the impulse response of y , eqt,ation (1), andwill l',e referred to as the matched excitation waveform.

Resp_onse to Arbitrary Excitation Waveform - It can be shownthat the response y'(t), say, of y to any arbitrary excitation, sayx'(t), that satisfies the normalization constraint of equation (6),wilt never exceed the maximum, equation (7b), produced by thematched excitation, x(t). To see this, the Fourier transform of

this response may be written

Y'((o) = I-lyCm)X'(o). (8)

Substituting eqnation (3) for Hy(¢O) and taking the inverseFourier transform gives

y'(t) = -..9"_--Ja" X'(m)X'(m) e io_t-t,,) do). (9)

Applying Schw,"wtz's Inequality to equation (9) gives

Both bracketed terms equal unity because of the constraint,equation (6), imposed on the waveforms. Thus

ly'(t)l _ %,. (10b)

which states that the magnitude of the response of y to anyexcitation waveform (suitably normalized to satisfy equation (6))never exceeds the maximum response to the excitation waveform

that is matched to y.

Intermediate Results - Any given frequency response functioncan be built up from several separate ones and thus represent thecombined effects of several subsystems. In the context of the

present paper, a yon Kan'nan turbulence "pre-filter", representedby a transfer function, is implemented ahead of the aircraftresponse fimction. The turbulence tr:msfer function used in thisstudy is adapted from that obtained from reference 9,

[ 1(l + z._ls_ ,)(t +o.ms..--+,)

,,. ,.o,, ,t-.o.,2,¢-,,(,.-o.o,,

Tht, s, the transform of the response y, as expressed above intemls equation (2b), would be rewritten as,

Y((o) -- ]ly(co)G(o)) X(tu)

Hy((o)G(to) = Hy({o)

where

_ __ * --* .i_t nW_{o.))- I G(to)G (_)Hyft..o)e ,

Ohy

(14a)

and taking the inverse Fourier Transform,

%(t) = / fwx(o,) e i_do,

fhy -M

(14b)

This theoretical development was carried out in thecontext of the response of an aircraft to atmospheric turbulence,where the pre-filter represented the atmospheric turbulence.Theoretically, other pre-filters can be used to analyze responses

to some other disturbance quantity. Potential alternative pre-filters inchtde runway roughness, pilot dynamics, and control-surface deflections. The range of ahematives is limited only by'the ingenuity of the analyst in developing appropriate pre-filters.

Dualilv Between MFT and RPT

Consider now two excitation waveforrns Xy(t) and Xz(t)that ,are matched to twooutputs, y and z, of a given system. Theresponse of y to xz(t) is then

I " * t")do). (15)yz(t) = Oh, 9-_ f Xy (_)xz(_) e i,at

From equation (3),

H;(m) _i_oXy(O_) = _--E---e (16a)

and

Thus,

[4:(o)) - i,,,,.X,(m) = _ e

(16b)

w

Yz(t) = _-_7-2-_- I HY(°_)H:(m)ei('_t t'O din.(17)

The product Hy(to) Hz'(Co) is the cross-PSD function. Sy,(¢o),for outputs y and z, thus,

"" ,_[.'-- o"*" ]Yz(t) = Oh, L Z)t._ " do) . (lg)

The term in the brackets is the cross-correlation function, Ryz(t -to), between outputs y and z {10], so,

(12)

yz(t) = & Ryz(t - to) . (19a)

(13)

Similarly, the response of z to xy(t) would be

where

Zy(t) =' l Rzy(t - In), (19h)O'hy

w

1 [" , ito(t - t,)

Rj:y(t - to) = _ | flz(m)l ly(o)) e din

=l f s.,,o) ¢20)Note also that these two cross-correlation functions satisfy the

relation (where t = t - to ),

Rij('t ) = Rji(-t ) .

Notice that if the response of y (or z) to its own matched

excitation, xy(t) (or xz(t) ) were being considered, auto-PSD's

(Syy or S,_) and auto-correlation functions (Ryy or R,.z) areinvolved. Thus, dynamic responses to vartous matchedexcitations are proportional to auto- and cross-correlationfunctions between the outputs. Further, when t = to,

_ 1 Ryy(O). (21)yy(t o) - Oh----_

The attto-correlation function is an even function of its time

argument, % and its value is maximum at I: = 0 and is equal to

the mean square of the PSD, Syy. Thus,

Yy(to) = Ymax = O'hy, (22)

which is the result obtained from the MFT approach, equation(7b).

The duality between MFT and RPT is apparent from

comparison of equation (15); which results from the applicationof MFT, with equation (19), and from comparison of equation

(7b) with equation (22). These equations show that thecorrelation functions of RPT can be interpreted as scaled timeresponses to the "matched" excitation waveforms of MFT; and,

necessarily, that the value of the autocorrelation function at 't =0, R (0), is proportional to the maximum response, Ymst, ofM[_. y The duality between MFT and RPT also allows a

relationship to be established [11] between MFT and PhasedDesign Loads Analysis 1"6]. Further, since the product

G(m)G°(o_) in equation (14b) is equiwdent to the van Karman

turbulence PSD, the critical gust profile resulting from thematched excitation waveform can also be determined from

f _.-;" . i¢o(t - to) ._ I _g(a_) tty¢c.o)e aco.wg(t) - 2nOb--------_ .-

(23)

Thus, RPT, normally employed to determine the statisticalproperties of random processes, may also be employed to obtaincertain deterministic responses provided that they are interpretedas responses to matched excitation waveforms as described by

MFT. In circumstances for which no PSD is available (e.g.maneuvers) the MFT approach would have to be used. Actualimplementation of the two approaches will be discussedpresently.

Al/Rikalion of Ihe Analysis Proce(htresIo Time-CorrelMilL_

MFT Approach

Figure 3 contains a signal flow diagram of the stepsnecessary to generate a maximum dynamic response at somepoint in the idrcraft _lructure. It is the mechanization of the

information presented in the Theoretical Development section ofthe paper. The signal flow diagram is presented in two parts;

the top half illustrates the generation of the system impulseresponse; the bottom half illustrates the generation of themaximum response of the system.

In the top half, the gust spectrum is excited by an

impulse of unit strength to generate an intermediate gust Impulseresponse which, in turn, is the excitation to the aircraft.Computationally, the time history of the impulse response iscarried out until its magnitude dies out to a small fraction of thelargest amplitude. Several output channels (y, zt .... ) areshown in the figure, however, only one response (y) is used.This impulse response is then normalized according to equation(6a).

The bottom half illustrates how all the time-correlated

responses are obtained including the maximum response of theoutput y, and the critical "gust" profile. The y response builds

to a maximum at t -- to at which point the excitation ends. Theresponse then decays to near zero. The maximum response,

Ymax, is equal to the RMS of the system impulse response. Itshould be mentioned that the responses shown for the otheroutputs are not maximized. The critical gust profile andmaximum response of the system are unique to only one load

output (the top output, y). For other maximum load responses aseparate but similar analysis needs to be performed.

An important detail illustrated in this figure is theintroduction of the a pre-filter. The effect of the pre-fiher is toprovide dynamics of the input disturbance which itselfcontributes to the shape and magnitude of "matched" excitationwaveform. In this example, the pre-filter is an s-planeapproximation of the yon Karman spectrum, but in otherapplications it could be landing or taxiing disturbance dynamicsor possibly "pilot" dynamics for maneuver loads.

RPT Aporoach

Figure 4 contains a signal flow diagram with two partsand is analogous to the diagram in figure 3. This figureillustrates the steps necessary to generate time-correlated gustloads using RPT. Whereas the signals in figure 3 were all in thetime domain, all but one of the signals in this figure are in thefrequency domain.

From the top half, the "Known Dynamics" box is thesame as that in figure 3. The input to this box is Gaussian whitenoise and the output includes atria- and the cross-power spectraldensity fimctions of some aircraft response, with an intermediateoutput being the van Karman power spectral density function ofatmospheric turbulence.

Time-correlated gust loads are obtained in the bottom

half of the figure by taking the inverse Fourier transforms of theauto- and cross-power spectral density functions obtained in thetop half. l't should be mentioned that to obtain precise

representations of the time-correlated loads it was necessary todeal numerically with two-sided spectra (that is, with both the

positive and negadve frequency components present).

Unit Impulse

(t) A

Known Dynamics

von Karman}Gust P'

(Pre-Filter)

AircraftLoads

Impulse Response

of Load y

v'

_J 1o

yon Karman Gust

_ Response

Matched Excitation

Wave form

von Karman|Gust

(Pre-Filter) J

Known DynamicsAircraft

Critical Gust Profile

t ""_J! a__'_ ,, '

0

Time-Correlated Responses to

Excitation Waveform Matched to y

Ymax """'h_.

y(t) I

0 to 210

Zl,('I

ny(t)^!

Figure 3. Implementalion of Malched Filler Theory Approach - Signal Flow Diagram

Gaussian White Noise

,1(f) •

N__y_ -ol

Auto-PSD

Cross-PSD

Cross-PSD

Ivon Karman,.-/ Gust

(Pre-Filter)

Known Dynamics

Aircraft

t'wg _ _nK_r_anum

_,. I),-

(11

I

,/inverseFourier

Translorm

v

(0

AuIo-PSD

S_(_o)

Cross-PSO

v

(o

v

(0

Cross-PSD

Auto-Correlation Function

/_ R_(_)_..,A A 111_ A,.._

vvv;vvv'"Cross-Correlation Function

A

___AA vv _..g

Cross-Correlation Function

Rz.y(_)

Figure 4. Implemenlalion or Random Process Theory A ppr.ach - Signal Flow Diagram

_[u olerical Examvles

Aeroelaslic Configuration

In the application o£ MFT and RPT, existing strtrcturaland aerodynamic models of the NASA DAST ARW-2 were

used. This configuration is a Firebee II target drone fitted withan Aeroelastic Research Wing (ARW) [121. This mt_el wasespecially well-suited for the study since it has structuralflexibility, a stable and dominant short period, and several loadoutputs. The load oulputs are comprised of shear forces, torsionmoments, and bending moments at several points along the spanof the wing. A detailed description of the dynamic loads modelis available from reference 13.

Figure 5 shows some of the more important data of thevehicle itself and of the analytical representation of atmosphericturbuleuce. The structural part of the model was derived from afinite element code and the unsteady aerodynamics (at a Machnumber of 0.7) from a doublet lattice code. Besides the tworigid-body modes, eight symmetric flexible modes were retainedfor this study. The final dynamics equations (the quadrupleequations), constructed with a matrix analysis code, consisted of97 first order equations, 9 output equations and one input.

These final equations contained the dynamics of the structure,the of the unsteady aerodynamics, the loads, and the yonKarman spectrum.

Time-Correlated Gust Ioad_ Using Matched FilterTheory

Figure 6 gives shows of the time-correlated loads at the

wing root and the wing tip stations. The top two plots on the

i ]"

lldy++i'5 Cnti(lilinns \ _ Vehicle D._criplinn

^,,,,,,,to71;_o;o,i __q -w;;,l,F_-i_io ,1,Math Ho = 0.7 Winq ,%_an = 227 I]4 ,_

rlefereltce Len+qlh : 23,1/ in

Aspocl I'lallo = 1(]3

^_mIylical Ii,,",in,eselzlatlon ot Vehicle

2 I.oltqlhtdinat fligid-nody Mrl(Ios

II Symm_lric flexible Mcwles

An._a_lylic__alR'e_p¢__es_en/afionol TurbulenceYOn Knt'tlfl_rl Sprc,rtlm

L =25n0II

Figure 5. Aeroelastie Configuration - NASA DAST ARW-2

righthand side contain the time-histories of bending moment andtorsion moment at the same root station on the structure. The topplot contains the bending moment response resulting from theexcitation matched to root bending moment. The critical gustprofile matched to the root bending moment is contained in the

left plot. Although their magnitudes and signs may be different,the general shape of waveform represented by this plot is verysiruilar for almost all the critical gust profiles of the system. Thecritical gust profile responses have the characteristic of being

dominated by the short period dynamics of the aircraft. Tiffsphenomenon can be explained by examining equation (14a).The higher frequency aeroelastic responses due to the elasticm(x:les are doubly attenuated by the high frequency roll-off ofeffectively two yon Karman "pre-fihers." The bottom righthandplot shows the time-history of bending montent response at aIc<'ation near the tip of the wing.

W<i

(in/see) =11

'=i Ae0 Root Bending

:I I/ "°m'°'R.. A{I A

_0_- U VCritical Gust Profile |or Excitation Waveform eo_=_-- ,-- •,..-, --._._._.__. L_

Matched to Root Bending Moment _R zc /_ /_ " "

-- -inl o

• "" .,o Root Tot slon• Moment

• • 6.0

OC

TBM

(Ib-in)o' Iol

-01

2 4 II II tO t2 14 111 18 L:_

rm@ (1eel

4 B B I0 12 14 I$ t8 2Q

Tirrl (r,+'¢'l

Figure 6. Time-Correlated Gust Loads - Matched Filler Theory Approach

100

8O

6OM,

Ib-in. 40

20

0

-20

-40

-600

Root Bending Moment

MFT...... RPT

2 4 6 8 10 12 14 16 18 20

Time, sec

4.0

2.0-

IbTiho-2.0 -

-4.0

-6.01

-8.0 •0

Root Torsion Moment

MFT...... RPT

I I I +-L .... I I I [

2 4 6 8 10 12 14 16 18 20

Time, sec

Figure 7. Comparison of MFT and RPT Calculalions

Comnarison of Resulls From MFT_agd_.R]_Calculalions Beferences

Figure 7 shows a comparison of wing root bending

moment and torsion moment time responses calculated with the

MFT and RPT approaches. The Matched Filter Theory results

are the same as those shown in figure 6. For purposes of

comparison, the s-plane approximation of the yon Kannan gust

power spectral density function is used for both the MFT and

RPT calculations. Except for some slight differences in the

peaks and troughs, results from the two approaches are

practically indistinguishable. This is in accordance with the

MFT/RPT duality described in the Theoretical Development

section of the paper,

Concluding Remar!fd

This paper has described and illustrated two approaches

for computing time-correlated gust loads. The two approaches

yield theoretically identical resuhs and ,are computationally fast.The first is based on Matched Filter Theory. and is a time-domain

approach; the second is based on Random Process Theory and is

a frequency-domain approach. These approaches represent

novel applications of the theories and unconventional

interpretations of the intermediate and final results.

An important contribution of this paper is theintroduction of a "pre-filter" in the application of Matched Filter

Theory. The pre-filter is the mechanism by which the dynamics

of the input disturbance (atmospheric turbulence) are

incorporated into the problem and the resulting matched

excitation waveforrn contains these dynamics. The output of the

pre-fiIter due to the matched excitation waveform is the critical

gust profile.

A second contribution of this paper is the interpretation

of scaled correlation functions from Random Process Theory as

time histories. It has been shown that the time-correlated gust

loads predicted by Matched Filter Theory are theoreticallyidentical to the attto- and cross-correlation functions,

appropriately scaled, predicted by Random Process Theory.

Further, the critical gust profile obtainable from the Matched

Filter Theory approach is also obtainable from the Random

Process Theory approach.

!. Jones, J. G.: Statistical Discrele Gust Tbeory for Aircraft Loads. RAETechnical Report 73167, 1973.

2. North, Dwight O.: Analysis of the Factors Which DetermineSignal/Noise Discrimination in Radm'. RCA Laboratories, Princeton,

New Jersey, Rept PTR+(-,C, June, 1943.

3. Papoulis, Athanasios: Maximum Respon_ With Input Energy Con-suaints and the Matched Fihcr Principle. IEEE Transactions on Circuitrheory, Vol. CT.17, No. 2, May, 1970. pp. 175 - 182.

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Although, in this paper, Matched Filter and Randont

Process Theories have been applied to the calculation of lime-

correlated gust loads, the approaches are general enough to he

applied a variety of dynamic-response problems, such as taxi

and landing loads, maneuver loads, and gust loads. The choice

of which to use depends on the intended application.

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Report Documentation Page

1. Report NO. 2. Government Accession No.

NASA TM- l 01573

4. Title and Subtitle

Time-Correlated Gust Loads Using Matched-FilterTheory and Random-Process Theory -A New Way of Looking at Things

7. Author(s)

Anthony S. PototzkyThomas A. Zeiler

Boyd Perry III

9. Performing Organization Name and Address

NASA Langley Research CenterHampton, VA 23665-5225

12. Sponsoring Agency Name and Address

National Aeronautics and Space AdministrationWashington, DC 20546-0001

15. Supplementary Notes

3. Recipient's Catalog No.

5. Report Date

April 1989

6 Performing Organization Code

8. Performing'Organization Report No.

10. Work Unit'No.

505-63-21-04

111 Contract or Grant No

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

Presented at the AIAA 30th Structures, Structural Dynamics and Materials Conference in Mobile,

Alabama, April 3-5, 1989.

16, Abstract

This paper describes and illustrates two ways of performing time-correlated gust-load calculations. Thefirst is based on Matched Filter Theory; the second on Random Process Theory. Both approaches yieldtheoretically identical results and represent novel applications of the theories, are computationally fast,and may be applied to other dynamic-response problems. A theoretical development and examplecalculations using both Matched Filter Theory and Random Process Theory approaches are presented.

17. Key Words (Suggested by Author(s))

Matched Filter TheoryRandom Process TheoryGust LoadsTime-Correlated Gust Loads

Maximum Response19. Security Classif. (of this report)

18. Distribution Statement

Unclassified - Unlimited

Subject Category - 05

20. O_ 'rlf" Classif. (of thi.', page) 21. No. of pages

Unclassified 10

22. Price

A02Unclassifiedk_IASA FORM 1626 OCT 86