II

Abstract

AMS Annual Meeting

12 tbConference on Aviation, Range and Aerospace Meteorology

January 2008

Discrete Gust Model for Launch Vehicle Assessments

Frank B. Leahy

NASA/Marshall Space Flight Center, Natural Environments Branch

Analysis of spacecraft vehicle responses to atmospheric wind gusts during flight isimportant in the establishment of vehicle design structural requirements and operational

capability. Typically, wind gust models can be either a spectral type determined by arandom process having a wide range of wavelengths, or a discrete type having a singlegust of predetermined magnitude and shape. Classical discrete models used by NASA

during the Apollo and Space Shuttle Programs included a 9 m/see quasi-square-wave gustwith variable wavelength from 60 to 300 m. A later study derived discrete gust from amilitary specification (MIL-SPEC) document that used a "l-cosine" shape. The MIL-

SPEC document contains a curve of non-dimensional gust magnitude as a function ofnon-dimensional gust half-wavelength based on the Dryden spectral model, but fails to

list the equation necessary to reproduce the curve. Therefore, previous studies could onlyestimate a value of gust magnitude from the curve, or attempt to fit a function to it. This

paper presents the development of the MIL-SPEC curve, and provides the necessaryinformation to calculate discrete gust magnitudes as a function of both gust half-wavelength and the desired probability level of exceeding a specified gust magnitude.

H

P2.g DISCRETE GUST MODEL FOR LAUNCH VEHICLE ASSESSMENTS

Frank B. Leahy*NASA Marshall Space Flight Center, Huntsville,Alabama

[

1. INTRODUCTION

Analysis of spacecraft vehicle responses toatmospheric wind gusts during flight is important in theestablishment of vehicle design structural requirementsand operational capability. Typically, wind gust modelscan be either a spectral type determined by a randomprocess having a wide range ef wavelengths, or adiscrete type having a single gust of predeterminedmagnitude and shape. Classical discrete models usedby NASA during the Apollo and Space Shuttle Programsincluded a 9 m/sac quasi-square-wave gust withvanable wavelength from 60 to 300 m (NASA, 200(3). Alater study derived discrete gust from a military standard(MIL-STD) document that used a "l--cosine" shape(Adelfsng snd Smith, 1998). The MIL-STD documentcontains a curve of non-dimensional gust magnitude asa function of non-dimensional gust half-wavelengthbased on the Dryden spectral model, but fails to list theequaSon necessary to reproduce the curve (DAD, 1990).Therefore, previous studies could only estimate a valueof gust magnitude from the curve, or attempt to fit afunction to it (Adelfang and Smith, 1998). Furthermore,the MIL-STD curve is based on a 1% dsk gustmagnitude; so there was no way to determine gustmagnitudes for,other dsk levels.

This paper presents the development of the MIL-STD curve, and provides the necessary information tocalculate discrete gust magnitudes as a function of bothgust half-wavelength and the desired probability level ofexceeding a, specified gust magnitude. Thedevelopment background is an extension of thedescriptions provided in Chalk et al, 1969, and Jones,1967.

22L. *

2Lvt+3(Lv )2. [t+i.o)2].w t÷3(L..)2-[t+<o),]'

(1)

where Q is spatial frequency (wavenumber), a is theturbulence standard deviation, L is the turbulence scalelength, and the subscripts u, v, and w denote thelongitudinal, lateral, and vertical components,respectively (NASA, 2000). The longitudinal componentof turbulence is parallel to the steady-state wind vector,while the lateral and vertical components areperpendicular to it. The non-dimensional Dryden spectraare shown in Figure 1.

The spectra shown herein are in the spatial domain.The spectra can easily be transformed to the frequencydomain by use of the Jacabian transform Q = _/Vwhere_e is radial frequency and V is the magnitude of themean wind vector relative to the speed of the aerospacevehicle (NASA, 2000).

An important detail that will be needed later is thatthe autocarrelat_on (lag correlation) is determined bytaking the inverse Feuder transform of the Drydenspectra. The autocarrelagon describes the correlationbetween gusts separated by a distance or time interval.

2. BACKGROUND

For analyses of spacecraft vehicles, wind gusts canbe treated as either random (spectral turbulence) ordiscrete. For random gusts, typical spectral modelsinclude the Men Karman and Dryden turbulence models.The Von Karman model has widely been considered themore "realistic" model when it comes to definingturbulence spectra. However, due to the computationalcomplexity of the Von Karman model, the Dryden modelis typically used in aerospace vehicle analyses. Thelongitudinal,lateral, and vertical Dryden spectre are:

10 t

-- Lcngitudlnal.........LateraMen ca

_. 10110° ............................... """%" "'"%%10"2 ",,

1010-1 100 101 10_

L_

* Correeponding author address: Frank B. Leshy, Figure 1. Non-dimensional Dryden spectra for theNASA/MSFC, Mail Code EV44, Huntsville, AL 35812; longitudinal, lateral, and vertical components ofe-mail: frenk.b leahy@nasa.qov turbulence,

Forthe spatial domain, these autocerrelations are(Hogge, 2004):

d

R,,(.)=7

where d is the lag distance. The autocorrelations as afunction of non-dimensional values d/L are given inFigure 2.

-- LongitudinalI ..........LateratVertical

-00 I 2 3 4 5 6

d/L

Figure 2. Autocorrelation as a function of non-dimensional values d/L.

The discrete gust provides a "spike-type" inputwhose magnitude is based on information from thespectral modal. Along with the gust magnitude, the gustgradient over a specified temporal or spatial interval isalso important. The classical shape of the discrete gustis that of a "l-cesine" shape given in Equation 3 andshown in Figure 3 as a function of gust width (Chalk etal_ 1969).

V=O x<O

2L t<.)jV = 0 x > 2dm

(3)

Here, V is the gust magnitude at distance x, and Vm isthe gust magnitude at din, the gust half-width. Severalvalues of dr, can be chosen to tune the gust width toexcite desired vehicle responses. The next sectiondescribes the development of the methodology todetermine appropriate discrete gust magnitudes forgiven gust half-widths.

v

0_-o

o

d m 2d m Distance

Figure 3. Graphical depiction of the l-cosinediscrete gust.

3. DEVELOPMENTOF DISCRETE GUSTMAGNITUDES

Generally, random gusts about a mean wind ereconsidered to be normally distributed. This is true foreach of the gust components (longitudinal, lateral, andvertical). Therefore, the random gusts, V, will have aprobability density function (pdf) of the form:

L 2 )(4)

where ,Uvand _v are the mean and standard deviation ofthe gusts. An Initial gust, V_, will be related to a gustsome distance away, V2,by the conditional probability:

s(v ,v,)--f(Fi ) (5)

where f(Vr, V2) is the joint pdf of the two gusts. Themean values of the gust, VI and V2,are simply the valueof the mean wind. The important factor in thisdevelopment is the deviation of gusts around a mean,not the value of the mean itself. Therefore, these can beremoved by setfin9 them equal to zero. If the initial gust,V1, is assumed to start at zero, then Equation 5 willreduce to:

where p is the correlation between V1 and V2. Ifwe let

e= l /77p2 (-,'J

T]I

end substitute into Equation6, we get the common formof the normal pdf (see Equation4)

- - ! 1_(8)

The cumulative distribution function (cdf) then becomes:

V < ! v, 1 V2

V2m'= L e j(g)

Since Equation 9 can not be integrated in closed form,computer routines are relied upon to determine thenormal cdf, To determine the gust magnitude value forV2, computer reutlnes to calculate the inverse of thenormal cdf are needed, Here, one would input theprobability level (P), the mean of the gusts (g - 0), andthe standard deviation (e) To determine e, substitutethe appropriate value of R from Equation 2 for p inEquation 7.

Care must be taken when performing thiscalculation. Recall that gust magnitudes are normallydistributed and are two-sided. That is to say, theprobability of having a gust magnitude greater than V2 isthe same as the probability of having a gust less than-V2. Therefore, the probability level chosen must be thetwo-sided probability. Most computer routines computethe one-sided probability. To account for this, lotP=l-(riskJ2), where risk is the desired probability ofexceeding a given gust magnitude.

Appropriate turbulence standard deviations, _, andlength scales, L, are provided in Table 2-79b of NASA,2000 for vadous altitudes and turbulent cases (light,moderate, and severe). Table 1 below lists the valuesfrom 1 to 10 kilometers for the severe case.

Table 1. Severe turbulence standard deviations andlength scales for the longitudinal (U), lateral (V), andvertical {_1 turbulent components.

Standard DeviationAltitude Length Scale (m)

(km) (m/see)U V,W U V,W

1 5,70 4.67 832 624

2 5.80 4.75 902 831

4 6,24 5.13 1046 972

6 7.16 5.69 1040 1010

8 7 59 5.98 1040 980

10 7.72 6.00 1230 1100

4. EXAMPLE CALCULATION AND COMPARISONTO PREVIOUS STUDIES

As an example, e longitudinal gust magnitude farsevere turbulence at the 10 km level for a 500 m gusthalf-width is computed. The turbulence standard

deviation and length scale at this level are cr = 7.72 m/sand L _- 1230 m, The dsk of exceeding the gustmagnitude is 1%. Therefore, the probability to input intothe inverse normal computer routine isP=1-(0.01/2)=0.995. The resulting gust magnitude is14.83 m/sec. Figure 4 shows the longitudinal gustmagnitude for this example with gust half-widths in therange 1 to 10000 m.

2O

1°ooi

101 102 t03 !04GustHalf-Width(m)

Figure 4, Longitudinal gust magnitude for varyinggust half-width, o,= 7.72 m/sac and L = 1230 m.

A previous study (Adelfang and Smith, 1998)attempted to fit a function that repmduced the MIL-STDcurve, whmh is non-dimensional gust half-width (d_/L)vs. non-dimensional gust magnitude (Vm/o). The twocurves are shown in Figure 5. Notice that the Adelfangand Smith curve is slightly conservative compared to theMIL-STD curve. Figure 6 is the same as Figure 4, butwith the corresponding Adelfaog and Smith curveincluded. The percent difference between the twocurves is depicted in Figure 7. The Adelfang and Smithcurve over estimate the MIL-STD curve by as much as13% in the 200 to 300 m gust half-width range.

5. SUMMARY

A development method of the MIL-STD non-dimensional discrete gust magnitude curve has bepresented. The development allows for slight reductionin gust magnitudes determined by other studies(Adelfang and Smith, 1998). Also, the new techniqueallows for selection of various risk levels of exceedinggust magnitudes (see Figure 8).

6. ACKNOWLEDGEMENTS

The author wishes to thank Dr. Stanley I. Adelfangof Stanley Associates at Marshall Space Flight Centerfor his expertJse in the subject matter, and for hisindependent verificabon of the developmentmethodology.

25

2

1,5

1

o.5

i .,,,.,,"" ...........

f"

,,r'

10-1 10° 101

dm/L

Figure 5. Comparison of MIL.STD curve to Adelfangand Smith, 1998, curve.

12

10° 101 102 103 104Gust Half-Width(m}

Figure 7. Percent difference between the Adelfangand Smithcurveand the MIL-STDcurve.

2_=

"G2C

--M,L-STD I.........Adelfeng&&Srntth,1998 ..............

i ,.¢

, , ,Ihl,I I , I ,I,lll , , I,,,,ll , ,,,,

i0 ° 101 102 103 104

GustHalf-Width(m}

Figure 6, Same as Figure 4, but with Adelfang andSmith, 1998, curve included.

0

_-0-2, I , ,,,,,i , ,,,,I i , i ,F,I

10-1 10_ 10_ .d /L

m

Figure 8. Non-dimensional gust half-width (d_lL) vs.non-dimensional gust magnitude (VJcr} for variousrisk levels (MIL-STD Is the 1% curve).

7. REFERENCES

Adelfang S.I. and Smith O.E. 1998: Gust Models forth

Launch Vehicle Ascent. AIAA, 36 Aerospace SciencesMeeting and Exhibit, January 12-15, 1998, Reno, NV.

Chalk, C. R., Neal, T. P., Hania, T. M., Pdtchard, F. E.,and Woodcock, R. J., 1969: Background Informationand User Guide for MIL-F-8785B(ASG), MilitarySpecification - Flying Qualities of Piloted Airplanes.AEFDL-TR-69-72, August 1969.

DoD, 1990: Military Standard - Flying Qualities ofPiloted Aircraft. MIL-STD-!797A, January 30, 1990.

Hogge, E. F., Lockheed Martin Corp., 2004:B-737Linear Autoland Simuhnk Model. NASA/CR-2004-213021, May 2004.

Jones, J. G., 1967: Gradient Properties of a Model ofStationary Random Turbulence. Royal AircraftEstablishment Technical Report 67134, June 1967.

NASA, 2000: Terrastdal Environment and (Climatic)Criteria Handbook for Use in Aerospace Vehicledevelopment. NASA-HDBK-1901, August 11,2000.

Ill

PIt S e..rrro..TI 0(\

Discrete Gust Model for Launch Vehicle Assessments

Frank B. LeahyNASA Marshall Space Flight Center, Natural Environments Branch, Huntsville, AL

,,'".,,'

~"~~~05~

-",25 -5%

-""

Example CalculationAs an example. a longitudinal gust magnitude for s-evel'E' turbulencE' at the10 km level for a 500 m gust half·width is computed The tUl'bulencestandard deviation and length scale at thili level aN" 0 = "'I.~ m sand L :::1230 m. Th~ rilik of ~x:ceeding the gust magnitude is 1°0 Thereforl'. thl'probability to input into the inv~rse normal computer routine IS

P:::l-(O.01 2)=0.OQ5- The resulting gust magnitude is 14,83 m sec. Figure.. shows th~ longttudinal gust magnitude for this example With gulit half­widths in the range 1 to 10000 m.

Summary

The author wishes to th:mk Dr. Stanley I Adelfang of Stanley Associateliat Marshall Space Flight Center for his eXpt"I1:ise in thE' subjl'cl matter.and for his independent verification ofthl' dl'vE'lopment mt'thodology

Acknowledgemenls

~5,.::--":;o,--:::,,.;--:::,,;-,--'",Gu.th\l'.wodth(m)

Figure 4. Longlludlnal gUilt mllgnllude forval,>'lnggu,l hal(.wldlh. _ - 7·~::l: m/lec lind L • 1230 m.

d."-Fllun .a. NOD-dlmtn,lonai lUll haU·wldlh (d. L) ''I 1l0n.odlmen,Ionaiau,1 malDllud~(1'__) (or"arlous risk level' (loIIL·STO I, the l,"-cuI"e)

A developmt'nt method of thl' MIL-STD non-dimenslonal discrete gus-tmagnitudE' curve has be pl'f'sented. The deVt>lopm~nt allows for slightreduction IR gus-t magnitudl's determined by other studIes (Adelfang andSmith. 19Q8). Also. the new tl'chniqul' allows for sl'll'chon of varioulil risklevels of exceeding gust magnitudes (see Figlll'l' 5).

He'U.... E. F.• 1.ockIlMJ Mu'tul Carr.• ~'¥8-737t.lnNr A""..wad Suauhnk M.,...M, NASAJClI.·..«'-1·•• JO~.J.tn,~.

NASA.;C'OO'T~En~I...J(Chm.l1ClCn!..... tuDdtooltf«,L" ..~..v."~'.NASA·HalK"o)<'l.A"""u,~,

ReferencesM.Il......S.J.• iUldSIlll.h.O'E., ••G.-M,:..j,ft,;r~"1'LllU1dIv.bodeArft1I.AlAA.)t>th~... s.......__~~I1dE:aul;>< •• JM\I...,,~.S. ...R.-o.NV.

C1Wk,c. R....u..I. T. P~HurilI,T.M.• F'n'~lr.ud.F. e.,.nJ W.x>4.~R.J.••_ &clqro:oo;ud W.:nIIl.IbOII

.Adl~GwJ. f"... MIL-F·8"'6.!AA9Jl. MW'U'1 ~lk.... F1TUlf.QlWIn. .{f'U.:,~Airr-4_. MFOL­TR·6<r-72.AlI$ust •..,.

(5)

(8)

(6)

(q)

.,,,.,

v.w..,."50'" 4-f.'7 S)2..", 4-i5 .."

6·'4 50" ''''''7·11> .."" '0<0

7·59 >0$ '04"

'."2 '.00 '2»

J(V,I")= [(V,.V,)-, J(V,)

I (I V')[(V,IV, =O)=/"""7ex. ---=-}oJ2m!2 2 e

Slandard DevlaUc'l'l

A1t1~ (m/Iee'

(l:m) U V.W

I v, I V'}F(V,)= P(XSv,)= c-> Jex --, IV"/211!2 _ 2 e

The cumulnti"l' di6tribution function (cdf) then becomes

Tllble I, Sev~re lurbule,u:t $Iandard devlatlon$ and lenlth $cale, (or the10nJItudinal (ll). Ialtrlll (V). and vertlclIl (W) lurbultnt componenu.

where p is the correlation bet......een \., and V... If we let

<=uH (-)and liubstitutl' mto Equation 6. we get the common form of the normal pdf(se~ Equation 4) :

wherej(\-·" ~") is thl' joint pdf of the two gu51s The mean values of the gust.\'., and VI" are simply the value of the mean wind The important factor inthii development is the de\-iation of gusts around a O1l'an, not the valut' ofthe mean itlielf. Therefore. these can be removed by setting th~m equal to2.ero. Uthe initial gust. V,. IS aSliumed to start at 2.ero, then Equation 5 willreduce to

where Pv and 0'1 are thl' mean and standard dl'viation of the gusts. Aninitial gust. ~',. will be rt'lated to a gust some distance away. 1:'" by theconditional probabilil)':

Generally. I'sndom gusts about 1\ mt'all wind are considered to be nOl'mallydistributed, This- is tnle for each of thE' gust components (longitudinal.lateral. and vertical). Therefore. the random gusts. \', will have a probabilitydensity function (pdf) of th~ form:

I J' <V-i'y)') (4)!(V):::7=fcx1.-2 ~

Development of Discrete Gust Magnitudes

Since Equation 9 can not be integrated in closed form, computer routinesart' I'elied upon to detel'minl' the normal cdf. To determine the gustmagnitude value for V... computer routineli to calculnte thl' invel"e of thenormal cdf are needed. Here. one would input the probability level (P), themean of the gusts (P = 0), and the standard dl'viation (e). To ddermine e.sublititute thl' appropriate "slue of R from Equation 2 for p in Equation ­Appropriate turbulenCf' standard deviations. 0, and length scalps. L. areprovided in Table 2--Qb of NASA, 2000 for various altitudt'S and turbul~nt

caRi (light, moderate, and severe). Table 1 be-Iow Iisu the values from I to10 kilometers for the se"en!' C8S~.

(2)

-LongJt.dW-L8IBrtlII.V8l1cal

An important detml ne~ded in the development is that theautocorrl'lation (lAg correlation) is determined by taking the inverseFourier tl'Ansform of thl' Dryden spl'ctra, The autocorrelation describesthe correlation bt'tween gusts separatt'd by a distance and art' gIVen by'

e:!.R,.(d)=<'

-i

R,.(d) =<T(I_ 2(~)-i

Rw(d)=<t:"(I- 2~)

", 2d", DlSlNCeFIJure 3. Cnllpblc.1 dtplcllon o(the J.C~o,lnedl,cretelUll

-O'O~~----:----C3:-~--:----!"'-

The discl'ete gust provides a "spike-type" input whose magnitude isbased on information from the spt"ctral model Along with the gustmagnitude. the gust gradient over a spl'clfil'd temporal or s-patialinterval IS also important Thl' clas-sical shape of the discrete gust isthat of a ·l·cosint~ shape given in Equation 3 and shown in Figure 3 asa function ofgust width (Chalk et al. lo6Q)

v=o x<o

V=~[I-~~)] OS,fS2dM

V =0 :t>2d",

Hel'f', \. is- the gust magnitudl' at duotance x, lind \.-~m is tht' gustmagmtude at d,. the gus-t half·width Several values of d", can bechoRIl to tune the gust width to excite desil'f'd vt'hlc1e responSf's Thenl'Xl section dt'5cribes the development of the methodology todE'terminl' appropriatt' discrete gust magnitudes for given gust half­widths

Figure::l AulocorTt'I.Uon a, a (uncI Ion o( non­dlmen,lonal v.luud L

where d is the lag distance. The autocorrelations as a function of non­dimenSIonal values d L are gi"t'n III Figure 2

FlauN!' 1 Non-dlm~nsloDal DI1o'd~D spectra (or lh~

10nJiludinal, lateral•• nd ,·utkill compon~ntso(turbul~Ol:•.

~..(n):::qu2~;r I+{/.."O?-

.. (n).a '!:J<. ,+J(l"n)', 'x ~+(l"n)'f

() 'Lw '+J(L,Jl)'"w n =a

w -:;-['+(L,Jl)'f

where !J is spatial frequency (wavenumber), 0 is the turbulencestandard deviation. L is the turbult>nce scalt> It>ngth. and thesubscnpts u. l', and U' denote tht> longitudinal. lateral. and ,'erticalcomponents, respt>ctlvely (N.>\SA. 2000). Tht' longitudlllalcomponent of turbulence is paralJl'l to the steady-state wind "l'ctor.while the latt'r81 and v~rtic81 componellts are perpt"ndicular to itTh~ non-dimensional Dryden spectra are shown in Figure 1

IntroductionAnalysis of spset'craft vehicle responses to atmospheric wind gustsduring flight is important in the establishment of vehicle designstructural ~quirements and operational capability. Typically. windgu~ models C8n be l'ither 9 spl!'ctraJ type determined by a randomprocess having 11 wide range' of wavelengths. or a discrete typehaving 11 single gust. of predett"rmintd magnitude and shapeCl3lisical discrete models used by NASA dUring the Apollo and SpaceShuttle Programs included a C) m SPC quasi-square.wavt gust with\laMable wavelength from 60 to 300 m (N..o\SA. 2000) A later studyderived discrete gust from a military standard (MIL-STD) documentthat uSt>d a "J·cosine" shape (Adelfang and Smith, 1098). The MIL­STD document contains a curve of non-dimensional gUS1 magnitudeSI> a function of non·dimensional gust half-wa"elength bal>ed on theDryden I>pectral model. but faill> to list the equation n~e'SSary toreproduce the curve (000. 1990)· Therefore. pn'vioUI> studit'S couldonly t'Stimate 8 value of gust magnitude from the curve. or attemptto fit a function to It (Adelfang and Smith. 1908). Furthermon!'. theMIL·STD curve is based on 8 l' risk gust magnitude, so there wasno way to ddermine gust magrutudeli for other risk levels, Thedevelopment of the MIL·STD curve is provided herein. and thenecenary information to calculate discrete gust magnitudes as afunction of both gust half-wavelength and the desired probabilitylevel of exCt>eding a specified gulit magnitude are provided.

BackgroundFor anal)'les of spacecraft vehicles. wmd gusts can be treated aseither random (sp~tral turbulence) or discrete. For random gusts,typical spectral models include the Von Karman and Drydenturbulence models. The Von Karman model has Widely beenconSidered the more "realistIC" model when it comes to definingturbulence spectrll. However. due to the computational complexityof the Von Karman model, the Dryden model is typically used inaerospace vehicle anal)'ses, The longitudinal. lateral. and verticalDryden spectra are: