318 J. SPACECRAFT VOL. 21, NO. 3

cross-sectional idealization (no overlap) used. BI-STEM tubesinevitably have some overlap in reality (Fig. 1) and are in-terlocked in some fashion, both circumstances providingadded strength. Furthermore, there is experimental evidence6

which also shows that Eq. (9) is conservative. The fact that theactual cross section is typically greater than the one used in thepresent theory can be taken into account by introducing thesection modulus Z as basis, rather than the radius r. Thesection modulus for the cross section used to derive Eq. (9) is

(13)

(14)

such thatEq. (9), with *> = 0.3, can be expressed as

Mcr=0.38£Z-a

where d = 2r is the tube diameter. In this form, with thesection modulus Z of the actual cross section, the equation issuitable for critical bending moment prediction for double-slittubing with overlapping and interlocking.

References1 Rimrott, F.P. J., "Two Secondary Effects in Bending of Slit Thin-

Walled Tubes," ASME Journal of Applied Mechanics, Vol. 33, No.1, 1966, pp. 75-78.

2 Rimrott, F.P.J. and lyer, K.R.P., "Nonlinear Bending Behaviourof Slit, Thin-Walled Tubes with an Arbitrary Slit Location," CASITransactions, Vol. 1, No. 2, 1968, pp. 78-82.

3 Brazier, L.G., "On the Flexure of Thin Cylindrical Shells andOther 'Thin' Sections," Proceedings of the Royal Society of London,Series A., Vol. 116, 1927, pp. 104-114.

4Reissner, E., "On Finite Pure Bending of Cylindrical Tubes,"Oesterreichisches Ingenieur-Archiv, Vol. 15, No. 1-4, 1961, pp. 165-172.

G.A., "Critical Bending Moment of Circular Cylin-' Journal of Applied Mechanics, Vol. 44, No. 1, 1977,

Thurston,drical Tubes,pp.173-175.

6Borduas, H. and Sachdev, S.S., "DIR Project E126 STEMResearch Progress Report," SPAR-TM 630/13, 1969, App. C and D.

A Simple Estimation Procedureof Roll-Rate Derivatives

for Finned Vehicles

choice of the proper fin-body interference factor, and 3) thecruciform configuration effects. Barrowmann et al.1 havebrought out a simple and logical approach for the estimationof Clp and C/8. However, their approach considers a uniformload, CNap over the fin in the subsonic regime which maylead to overprediction of the roll-rate derivatives. Even in thesupersonic regime the use1 of Busemann's higher-order theoryfor small incidence may lead to slight overprediction asstudied by Oberkampf.2 Further, many investigators1'4 donot appear to have considered the cruciform effects, asbrought out by Adams and Dugan.5

The present Note briefly outlines a more generalizedformulation for computing C/6 and Clp) discusses theadaption of proper fin-body and fin-fin interference effects,and presents a comparison of the present calculations withexperimental data which are found to be in good agreement.However, the present approach overpredicts Clp at subsonicspeeds in certain cases.

AnalysisRoll Moment Calculation

The roll moment coefficient for planar fin (suffix-) can bederived as

Fig. la Roll parameters.

Sheo Prakash*Institute de Atividades Espaciais

Sdo Jos£ dos Campos, Braziland

D. D. KhuranatVikram Sarabhai Space Centre, Trivandrum, India

Introduction

SOME of the important aspects that warrant considerationin the prediction of the roll-rate derivatives (Clp and C/5)

are 1) normal-load distribution over the fin planform, 2)

Submitted March 2, 1983; revision received Oct. 1, 1983. Copyright© American Institute of Aeronautics and Astronautics, Inc., 1983.All rights reserved.

* Researcher and Adviser, Divisao de Projetos; presently on leavefrom VSSC.

tEngineer-in-Charge, Aerodynamic Coefficients, AerodynamicsDivision.

0.2 0.4 0.6 0.8 1.0Body d.o/.pon «f ' r/• )

Fig. Ib Variation of factors K, and K2.

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MAY-JUNE 1984 ENGINEERING NOTES 319

where the symbols have the usual meaning (Fig. la) and A' isthe area of half of the fin-planform. For the supersonic case,ACp is known from the linearized theory for planar wings andcompiled6 for arbitrary planforms. The integration in Eq. (1)can, therefore, be performed numerically for proper con-ditions of o?(£) to obtain Cld_ or Cfp_ . For the subsonic case,an elliptic load distribution along the span is considered.Thus, from Eq. (1) one obtains

where A is the aspect ratio, A is the sweep-back angle, /3= (1-M2)' /2, and M is the subsonic freestream Machnumber.

and

-=—rSD)r

SD2 (2)

where the products S(£) (dCTV^/d?)^ represent the ellipticspanwise loading over the fin such that its spanwise in-tegration yields CNaf based on the fin area (2A'). CNaf isobtained6 as

Fin-Body Interference and Cruciform EffectsInterference Factor for Cip and C15

Pitts et al.7 have derived various formulae for calculatingKWB ( = kWB + ksw) based on the slender body and upwashtheories. In the computation of Clp\ Barrowmann et al.1 usingupwash theory, have derived the following interference factor

r-\-logr

KClp =( r + 7 ) ( r - X )

+ 1 (3)

Fig. 2 Comparison of Cn.

——————— PRESENT METHO^

——— ••———BARROWMANN et al [RE

x EXPERIMENT [REF.sl

-OBERKAMPFBRL BALLISTIC RANGE \ [~REF 2~J

-LINER THEORY J

CIS(BASIC-FINNER)

4.0—^BASIC-F INNER

Fig. 3 Basic finner roll dampingderivate.

————PRESENT METHOD

—X—BARROWMANN et al. [«EF l]

------OBERKAMPF |

O BRL BALLISTIC RANGE >[REF2]

A NSWC WIND TUNNEL J

AEDC WIND TUNNEL| (?EF 3]-----MOORE & SW4NPSONJ

EXPERIMENT [fREF. 9J

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320 J. SPACECRAFT VOL. 21, NO. 3

where r=s/r and X is the taper ratio. In the present method,KCip is used for $A >2 and KwB (Ref. 7) for 0A <2, as it hasbeen found6 that upwash theory gives a better CNa predictiononlyfor/L4>2.

A judicious choice for the interference factor in thecalculation of C/6 is necessary. Oberkampf2 has not con-sidered such effects explicitly in his analysis at small in-cidence. Barrowmann et al. * have considered only kwB in theircalculation procedure. One should also consider kBvv since thefin at incidence does influence the body to affect the moment.It can also be seen that for a similar situation in thecalculation of Ctp9 Eq. (3) is obtained from upwash theoryand is equivalent to KwB. Rollstin4 has used KwB as the in-terference in his analysis. In the present method, KwB is usedas the interference factor and, depending upon whether0A <2 or 13A >2, KwB is computed7 from the slender body orupwash theory, respectively.

Effect of Cruciform ConfigurationAdams and Dugan,5 using slender body theory, in-

vestigated such effects on Clp and C/d and concluded that

Clp+=K1(o)Clp_ and C/6+ =K2(a)Cld_ (4)

The factors Kt and K2 depend upon the body diameter-to-span ratio, cr, and are presented in Fig. Ib, where thisdependence is rather weak (up to a maximum of 6%) for0< a<0.3 or 0.4. Generally, the value of a is less than 0.4 formost of the sounding rockets, and Kl (a = 0) = 1.62 and K2(a = 0) =1.524 can be used for a fin-body cruciform con-figuration. However, in the case of configurations with largera (small span) or otherwise, appropriate values of K1 and K2from Fig. Ib can be used. Now the total C/6+ and Clp+ may becomputed with the help of Eqs. (1-4).

Results and ConclusionsThe present calculation method has been compared and

validated6 against a large set of experimental data, but due tospace limitations, only a few sample comparisons arepresented here in Figs. 2 and 3. In Fig. 2, Cld comparisons areshown for both Basic Finner and Tomahawk.1'8 The presentmethod can be seen to agree well with experiment in both thecases. The approach of Barrowmann et al.1 appears to over-predict C/6 highly for the Basic Finner in the subsonic region;that of Oberkampf also slightly overpredicts in the supersonicregion.

CIP results from the present method show very goodagreement with experimental and flight data2'3'9 (Fig. 3) in thesupersonic regime. However, the present method doesoverpredict Clp in the subsonic region. As can be seen in Fig.3, the approach of Barrowmann et al.,1 of course, seems togive a still higher prediction. It appears that the Clpcalculation is sensitive to the spanwise load distribution in thesubsonic range, and that the present assumption of an ellipticdistribution could be poor. However, it may be worthwhileanalyzing experimentally whether the possibility of a lift losscan exist in the subsonic region in Clp tests due to rotation ofthe model and thereby reducing Clp appreciably.

Thus, it is brought out that the adaption of proper fin-bodyand fin-fin interference effects leads to a consistently betterprediction of roll-rate derivatives. The subsonic regionprobably requires further investigation of the roll-dampingderivative.

ReferencesBarrowmann, J. s!, Fan, D. N., Obusu, C, B., Vira, M. R., and

Yang, R. J., "An Improved Theoretical Aerodynamic DerivativesComputer Program for Sounding Rockets," AIAA Paper 79-0504,1979.

2Oberkampf, W. L., "Theoretical Prediction of Roll Moments onFinned Bodies in Supersonic Flow," AIAA Paper 74-111, 1974.

3Moore, F. G. and Swanson, B. C. Jr., "Dynamic Derivatives forMissile Configurations to Mach Number Three," Journal ofSpacecraft and Rockets, Vol. 15, March-April 1978, pp. 65-68.

4Rollstin, L. R., "Aeroballistic Design and Development of theTerrier-Recruit Rockets System with Flight Test Results," SandiaLabs., Albuquerque, N.M., SAND-74-015, 1975.

5 Adams, G. J. and Dugan, D. W., "Theoretical Damping in Rolland Rolling Moment Due to Differential Wing Incidence for SlenderCruciform Wings and Wing-Body Combination," NACA Rept. 1088,1958.

6Khuruna, D. D. and Prakash, S., "Theoretical Estimation ofRoll-Inducing and Roll-Damping Moment Coefficients Due to Fin-Cant for Rockets," Vikram Sarabhai Space Centre, Trivandrum,India, TN-02-040:80, Dec. 1980.

7Pitts, W. C., Nielson, J. N., andKaattari, G. E., "Lift and Centerof Pressure of Wing-Body-Tail Combinations at Subsonic, Transonicand Supersonic Speeds," NACA Rept. 1307, 1959.

8Barrowmann, J. S., "The Practical Calculations of theAerodynamic Characteristics of Slender Finned Vehicles," M. S.Dissertation, The Catholic University of America, Washington,D.C., March 1967.

9Murthy, H. S., "Roll Damping Measurements on RH-560S andRH 300 Models," National Aeronautical Laboratory, Bangalore,India, NALTWT 1-27, Jan. 1981.

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