MISSILE COMMAND REDSTONE ARSENAL -AL … · -Fatr a Are , I labMM M 35898 I Approved for public release; distribution unlimited. ... Assuming7,4 that the longitudinal wind, u , is

AD-All 348 ARMY MISSILE COMMAND REDSTONE ARSENAL -AL SYSTFMS SI--ETC F/S 19/4AN ATMOSPHERIC TURBULENCE MODEL FOR THE DYNAMICALLY AIMED FREE-ETC(U)TAREDIKN

UNCLASSIFIED ORSMI/RD-82-1-TR SBI-AD-E950 196 N.-EEEEEEEEIIIIEEEhEEIhhIhEIIEEEu~

I__________________

13

MICROCOPY- ROT ESCH

MICROCOPY RESOLUTION. TEST CHART

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14 TECHNICAL REPORT RD-82-1

AN ATMOSPHERIC TURBULENCE MODEL FOR THEDYNAMICALLYAIMED FREE-FLIGHT ROCKET (DAFR)ERROR ANALYSIS

Richard E. DicksonSystems Simulation and Development Directorate

US Army Missile Laboratory

OTIC

1 October 1981

-Fatr a Are I M , labMM 35898

I Approved for public release; distribution unlimited.

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READ INSTRUCTIONSREPORT DOCUMENTATION PAGE BEFORE COMPLETING FORMI. REPORT NUMBER f2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

TR-RD-82-1 44.-A-ILL. ?etY4. TITLE (end Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

An Atmospheric Turbulence Model for the Technical ReportDynamically Aimed Free-flight Rocket (DAFR) 6. PERFORMING ORG. REPORT NUMBER

Error Analysis

7. AUTHOR(e) 8. CONTRACT OR GRANT NUMBER(.)

Richard E. Dickson

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASKCommander, US Army Missile Command AREA & WORK UNIT NUMBERSATTN: DRSMI-RDRedstone Arsenal, AL 35898

It. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATECommander, US Army Missile Command I October 1981ATTN: DRSMI-RPT 13. NUMBER OF PAGES

Redstone Arsenal, AL 35898 2514. MONITORING AGENCY NAME & ADDRESS(l different from Controlling Office) 15. SECURITY CLASS. (of this report)

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17. DISTRIBUTION STATEMENT (of the abetract entered In Black 20, It different from Report)

IS. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reveree side It necessary and Identify by block number)

Turbulence, atmospheric Roughness lengthMarkov process Isotropic turbulence al scaleDryden correlation coefficientArtillery RocketArtillery Rocket, error analysis

20 A 9STRACT eCont hwe n revrg oft 0f necefemny and Idenitfy by block mu.et)

Previous error analysis of the Dynamically Aimed Free-flight Rocket (DAFR)utilized a linear model for the wind uncertainty. Also, the data used in themodel had been filtered.

This effort utilized a first-order Markov process to model the wind longi-tudinal turbulence uncertainty. The effect of terrain roughness is considered.

DO ?AM 1473 EOIOOF INOVSISOSOLETE UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE (lhen Data Entered)

PREFACE

The author wishes to acknowledge the literature search

done by Dr. Dorathy A. Stewart, Aerophysics Branch, Research

Directorate, US Army Missile Laboratory, US Army Missile

Commiand.

TOC

"or- -

- ,,- e

.j*.r--

1 1' 'L

TABLE OF CONTENTS

Page

I. INTRODUCTION .. .. ... ..... ...... ..... ..... 3

II. WIND AND TURBULENCE .. .. ..... ...... ..... .... 4

III. THE ERROR ANALYSIS - REVISITED .. .. ... ...... ...... 13

IV. CONCLUSIONS AND RECOMENDATIONS. .. .... ..... ...... 21

itREFERENCES. .. .. ..... ..... ..... ...... ...... 23

DISTRIBUTION .. .. ... ...... ..... ..... ........ 24

kL1

LIST OF ILLUSTRATIONS

Figure Title Page

1 Power spectrum of horizontal wind speed .... ........... 6

2 Roughness length for various terrains .... ............ 7

3 Intensity versus height ....... ................... 9

4 Turbulence integral scale versus height .... ........... 11

5 Turbulence standard deviation versus time delay ... ....... 12

6 Number of rounds already fired ...... ............... 15



9 Number of rounds already fired . .................. 18


11 Number of rounds already fired ..... .. ... ... .. 20

2

h. INTRODUCTION

Law of Revelation: "The Hidden Flaw Never Remains Hidden."

Simply stated, the dynamically aimed free-flight rocket (DAFR or DAFFR)concept is to decrease the sensitivity of an artillery rocket to other errorsources by increasing wind sensitivity. The wind effect is reduced duringfiring by round-to-round correction in aiming based upon measurements of thepreviously fired rounds. The concept has been described in detail by McCorkleand Lilly ', and some refinements to the error analysis were later made byGibson.2

It was assumed 1,2 that the measurements,. z , would be averaged toobtain an unbiased minimum variance estimate, x

= N Zk (1)

The variance of the estimate is

oR P k (2)2

and, assuming the variance, k is constant,

2 2

= N (3)

The variance of the wind is not constant and was modeled as

au N2 y2 (4)

where N is the number of rounds fired and y is the sensitivity slope which

William C. McCorkle, Jr., and J. A. Lilly, "An Adjusted Fire Techniquefor a Highly Accurate Free Flight Rocket Artillery System," US Army MissileCommand, Tech. Rpt. RD-74-13, Redstone Arsenal, AL, 25 June 1974 (AD B007539L).

2 J. D. Gibson, "Error Analysis of a Dynamically Aimed Free Rocket,"Texas A&M University, College Station, TX (Unnumbered and undated, not heldby DTIC, formerly DDC, or RSIC).

3

was derived from data in the Tactical Artillery Rocket Environment (TARE)

report.'

Since

N

k2 , N(N + 1) (2N + 1) (5)k=O

the resulting variance is

2 +(1)12 [1 + (N +1)(2N +1) 1y2_f + + + 1 +j(6)N Op N 6N

where a2 is the sum of the squares of all Gaussian white plant noises, andp

av2 is the variance of the observation noise. Equation (6) eventually diver-

ges as N becomes large because the linearity assumption, Equation (4), breaksdown. Though additional data may not improve an estimate, it should not makethe estimate worse.

The TARE report 3 states that "Readings were taken with aerovane typeanemometers located at heights of approximately 60 feet and using a smoothingcircuit of one (1) minute." The times between rounds previously considered','were around five to ten seconds. The time delay characteristics previouslyused were most likely those of the "1 minute time filter" not the wind!

This leaves us with two questions:

9 What are the (unfiltered) characteristics of the wind?

* What is their effect on the error analysis?

II. WIND AND TURBULENCE

Thumb's First Postulate: "It is better to solve a problem with a crude approximation

and know the truth, t 10% , than to demand an exact solution and not know the truth at all "

The wind model presented below was developed mainly from the results ofa literature search on atmnspheric turbulence done by Stewart.'

3 Tactical Artillery Rocket Environment (TARE) Committee, "TacticalEnvironment for Large Free Flight Rocket Systems," US Army Ballistic MissileAgency Report, R-S-61-1, Redstone Arsenal, Alabama, I June 1961(AD B952069L).

D. A. Stewart, "A Survey of Atmospheric Turbulence Characteristics,"US Army Missile Command, Technical Report RR-81-6, Redstone Arsenal, Alabama,19 August 1981.

4

Figure 1 is a plot of the horizontal wind power spectral density.'- 6 Thespectral gap between eight hours and ten minutes serves to explain why we mayspeak of an average wind. To reduce randomness one should operate as far aspossible on the other side of the peak due to turbulence.

Assuming7 ,4 that the longitudinal wind, u , is the sum of the longitudi-nal average wind, u , and the turbulent wind, u' , then

u(t) = TJ + u'(t) (7)

The turbulent wind is related to the turbulent wind at some previous timeby '

u'(t + T) = p(T) u(t) + u"(t) (8)

whrre p(T) is the autocorrelation coefficient for a time delay, T , andu"(t) is the random component of the turbulence.

The variance, a'., of the random components is defined by the relation-

ship7'4

y21 o2(1(9

uuso that the turbulent energy, u2 be conserved with time.

The standard deviation due to turbulence, au , is related to the average

wind, -U , by the approximation' 4

u (10)

n (h/z0)

where h is the height and z is the roughness length8 , Figure 2. The

5 A. J. McDonald, Wind Loadin -on Buildings, New York: John Wiley & Sons,

1975.

6 J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence,

New York: John Wiley & Sons, 1964.

7 S. R. Hanna, "Some Statistics of Lagnangian and Eulerian Wind Fluctu-

ations," Journal of Applied Meteorology, Vol 18, April 1979, pp. 518-525.

8 J. W. Kaufman (ed.), "Terrestrial Environment (Climate) Criteria Guide-

lines for Use in Aerospace Vehicle Development," 1977 Revision, Second Edition,NASA Technical Memorandum 78118, June 1979.

5

dI

Ei ~- - _

00

ICU0

0000

E.

p" poM eenbSuv0.

6 i1C

101

10 0i rg

TreesI Palmetto1O-1

(A,- High Grass

EU Low Grass

Steppe

wV) Mown Gras

0 Snow0-

Sand

10-4~

Mud Flats

Ice

10-5~

Figure 2. Roughness length for various terrains.

7

lateral components are approximately 9

av = (0.64)au (11)

and

= (O.52)o u - (12)

Of course, V = w = 0 by definition. These approximations hold best for aneutral atmosphere, which implies strong winds.

Figure 3 is a plot of the intensity, au /tr , against height for various

roughness lengths.

The Dryden correlation coefficients9 are as follow:

* Longitudinal

-U T/L u

p = e u (13)

o Transverse

-UT-/L

Pv(T) e v(1 - UT/2Lv) (14)

Pw(T)= Pv (T) (15)

where L is the isotropic turbulence integral scale.

In the atmospheric boundary layer9 ,

Lu - 2h c (16)

0

where

-0.025(zn z ) + 0.17 n z - 0.8c = e 0 (17)

9 W. Frost, B. H. Long, and R. E. Turner, "Engineering Handbook on theAtmospheric Environmental Gudielines for Use in Wind Turbine Generator Devel-opment," NASA Technical Paper 1359, December 1978.

8

-

4-

C1C

InS4

dS

Y~illn~n

and

Lv = Lw = (0.4)h , h < 250 meters. (18)

For h > 250 meters

L = Lv = Lw = 250 meters (19)U

Figure 4 is a plot of Lu against height for various roughness lengths. To

illustrate how the formula/figures may be used, consider a height of 5 metersand a roughness length of 0.1 meter. From Figure 3

= (o.25) (20)

and from Figure 4

Lu = 100 meters (21)

Substituting into Equation (13) and then Equation (9) results in

S1/2

u1, = (0.25) UI - e2T/l0) (22)

A plot of Equation (22) is shown in Figure 5. L and Lw are so short,

Equation (18), that the transverse components are essentially white noise andcorrection is not feasible, though they would contribute to the error budget.

The above model for the turbulent wind corresponds to a Gauss-Markovprocess, that is, Gaussian white noise passed through a low pass filter wherethe order of the process is the order of the filter". This model is onlyreasonable for the most right-hand portion of Figure 1, but for time delays ofless than one minute it is within 20 percent of observed autocorrelationcoefficients.' An in-depth discussion of the bases for the model will befound in the report by Stewart.

4

"0A. Gelb (ed.), Applied Optimal Estimation, Cambridge: MIT Press, 1974.

10

mo S

S

U0~So 4-

- C

z UCS

6

SL.

0

0 CN

Ca

n1

11

iniE LC4-

o]

Is

C14-

12R aI

(t/w) "n

12S

III. THE ERROR ANALYSIS - REVISITED

Grossman's Misquote of H. L. Menchen:

"Complex problemsa have sim ple, easy to understand wrofig answrs.

Instead of using

02 2 2 (4)

use the following in Equation (2):

a 2, C= 02 - e -21T/Lu) (6)

Since

N --a(N +1)~ ak 1-e

k0 e 1 ~ a (23)

for

T N NT (24)

N

2 T 2 + a2+ CY(25)

k= 1

becomes

= p2 + 2 + (i 2(N+1)U T/L£ NI+ [N + 1 ie TLu(26)

For large EUT/L u ,Equation (26) approaches

2 = p 2 + G)2 +cr u1 2(27)N

that is, the Gauss-Markov process "becomes" essentially Gaussian white noise.

13

Since 1,2

2 a 2 + c, 2 (28CXN+1 = u a 'N+1 (28)

substituting Equation (26) yields

2

A2 = 1+ V

XN+1 1 p N

+ a _2uT/L) +u e - ( -u/Lu (29)

Plots of Equation (29) are given in Figures 6 through 11. The standarddeviations have been arbitrarily expressed in mils; though representative, thevalues do not reflect an actual system. The quantity, u-/Lu requires some

explanation. For a given turbulence integral scale, for example,

L = 100 meters

u

and an average wind

= 5 meters per second

for

UT - 0.1

Lu

it follows that the time between rounds must be

T = 2 seconds

For a small precision error, Figure 6, the improvements from decreasingthe time between rounds, T , are dramatic. For a large precision error,Figure 11, there is not much improvement at all since that error dominates.

These figures may be used for other integral scales, Lu , and average

winds, 9 , but the time between rounds, T , would need to be recomputed.

14

C!J

E~zj E

(1II II I

15-

* q j- - .. U

EjEI I-

r

"-0 0

* . .

// -

. . . . .. . . It h ==, . . .. . .

EI

it

E E E CC

4D 0

odo

CD

.0

Emz

C.,

t+Nxfli!W) 0

17

N N 4 j-.

U, C!

o o d

inV,

0

ESz

(.4

di -- - - Ic

01w tNx

18

00

Lia.,

00 ir;d d d

12 M

M

In"

I +Nx.

Las 1

d d

BE

01

TCL

- E

|z

. . . . ..

q --

o [€o 08

l!)L+NxD

20

.. l .... ... ...... .. l ..... II .. ... ]1 .... . r " [[] -" . ... ...

IV. CONCLUSIONS AND RECOMMENDATIONS

Wetheru's Law of Suspended Judgement:

"Assumption is the mother of aU screw-Up,'

The results of incorporation of the revised wind (turbulence) model inthe error analysis supports the overall conclusions of the previous studies.''

2

Although it would appear that the time between rounds should be shorter, itdoes place those conclusions on a bit firmer foundation.

Probably the most significant contribution of this effort has been theincorporation of surface roughness via z Fichtl and McVehil11 report a

roughness of around 0.2 meter, where downwind of vegetation I to 2 meters inheight, but 0.3 meter when 450 meters downwind, and 0.4 to 0.6 meter when200 meters downwind of trees 10 to 15 meters tall. It would appear desirablenot to fire from positions downwind of trees or buildings if the tactical sit-uation permits. Deaves1 2

,13 has proposed a computational wind model which may

be of use in analyzing the effects downwind of changes in surface roughness.

Of course, if the average is taken over too long a time one would have

t , not u(t+T) . If the DAFR concept proceeds into development after demon-stration, it is recommended that a probabilistic rather than a statisticalfilter be developed, that is, a Kalman filter'" be utilized. The above turbu-lence model is suitable for Kalman filter formulation

uk1 1,0 a k0=+ (30)(uk:1) (::) ( k) +(k) (0

for the state equation, and

Zk+1 , + V k+1 (31)

k+ 1/

' G. H. Fichtl and G. E. McVehil, "Longitudinal and Lateral Spectra of

Turbulence in the Atmospheric Boundary at the Kennedy Space Center," Journal ofApplied Meteorology, Vol 9, February 1970, pp. 51-63.

12 D. M. Deaves, "Computations of Wind Flow Over Changes in Surface Rough-

ness," Journal of Wind Engineering and Industrial Aerodynamics, Vol 7, 1981,pp. 65-94.

13 0. M. Deaves, "Terrain - Dependence of Longitudinal R.M.S. Velocitiesin the Neutral Atmosphere," Journal of Wind Engineering and Industrial Aero-dynamics, Vol 8, 1981, pp. 259-275.

21

L.... kA. . .. ... ....

for the observation equation, where zk+ I is just Uk+ I with observation

noise, vk+1 ' added. Of course, these state and observation equations would

have to be expanded to include the missile and tracker dynamics. Besidesexplicit incorporation of the Markov process in the filter, an error analysiswould be generated in the filtering process.

There is no low except the low that there is no low."

-John Archibald WheelerPhysicist

22

REFERENCES

1. William C. McCorkle, Jr., and J. A. Lilly, "An Adjusted Fire Techniquefor a Highly Accurate Free Flight Rocket Artillery System," US ArmyMissile Command Technical Report, RD-74-13, Redstone Arsenal, AL,25 June 1974 (AD B007539L).

2. J. D. Gibson, "Error Analysis of a Dynamically Aimed Free Rocket," TexasA&M University, College Station, TX (Unnumbered and undated, not held byDTIC, formerly DDC, or RSIC).

3. Tactical Artillery Rocket Environment (TARE) Committee, "Tactical Environ-ment for Large Free Flight Rocket Systems," US Army Ballistic MissileAgency Report, R-S-61-1, Redstone Arsenal, AL, 1 June 1961 (AD B952069L).

4. D. A. Stewart, "A Survey of Atmospheric Turbulence Characteristics,"US Army Missile Command, Technical Report RR-81-6, Redstone Arsenal, AL,19 August 1981.

5. A. J. McDonald, Wind Loading on Buildings, New York: John Wiley & Sons,1975.

6. J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence,New York: John Wiley & Sons, 1964.

7. S. R. Hanna, "Some Statistics of Lagnangian and Eulerian Wind Fluctuations,"Journal of Applied Meteorology, Vol 18, April 1979, pp. 518-525.

8. J. W. Kaufman (Ed.), "Terrestrial Environment (Climate) Criteria Guide-lines for Use in Aerospace Vehicle Development," 1977 Revision, SecondEdition, NASA Technical Memorandum 78118, June 1979.

9. W. Frost, B. H. Long, and R. E. Turner, "Engineering Handbook on theAtmospheric Environmental Guidelines for Use in Wind Turbine GeneratorDevelopment," NASA Technical Paper 1359, December 1978.

10. A. Gelb (Ed.), Applied Optimal Estimation, Cambridge: MIT Press, 1974.

11. G. H. Fichtl and G. E. McVehil, "Longitudinal and Lateral Spectra of Tur-bulence in the Atmospheric Boundary at the Kennedy Space Center," Journalof Applied Meteorology, Vol 9, February 1970, pp. 51-63.

12. D. M. Deaves, "Computations of Wind Flow Over Changes in Surface Rough-ness," Journal of Wind Engineering and Industrial Aerodynamics, Vol 7,1981, pp. 65-94.

13. D. M. Deaves, "Terrain - Dependence of Longitudinal R.M.S. Velocities inthe Neutral Atmosphere," Journal of Wind Engineering and IndustrialAerodynamics, Vol 8, 1981, pp. 259-275.

23

DISTRIBUTION

No. ofCopies

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25

Documents

MISSILE COMMAND REDSTONE ARSENAL -AL … · -Fatr a Are , I labMM M 35898 I Approved for public release; distribution unlimited. ... Assuming7,4 that the longitudinal wind, u , is