Non-steady Supersonic Flow About Pointed Bodies of Revolution by Wi. H.i Dorrace...:..;..: i... Project MX -- 794 USAF Contract- W 33-038- ac-14222 Willow Run Research Center Engineering Research Institute University of Michigan UMM-80. November 1950

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WILLOVW RUN RESEARCH CENTER LUNIVERSITY OF MICHIGAN JMM-80 TABLE OF CONTENTS Section Page List of Figures ii Symbols and Nomenclature iii Greek Symbols iv I Summary 1 II Introduction 2 III Fundamental Solutions 3 A. The Non-steady Source-sink Potential 3 B. The Non-steady "Doublet" Potential 7 C. Boundary Conditions 8 IV Normal Force and Pitching Moment Coefficients 11 V The Low Frequency Oscillations and Steady Motion 13 A. Harmonic Pitching About the Point 14 B. Harmonic Normal Oscillations in Pitch 15 C. Steady Pitching About the. Point 16 D. Steady Angle of Attack 17 VI Example Results for Two Bodies of Revolution 19 Appendix A 24 Appendix B 26 Appendix C 28 References 30 Distribution 31

WILLOW' RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN -UMM-80 LIST OF FIGIRJES No. Page 1 Cylindrical Coordinates and Orientation of Body of Revolution 4 2a Harmonic Pitching About XO 9 2b Harmonic Normal Oscillations in Pitch 9 2c Steady Pitching About Xo 9 2d Steady Angle of Attack 9 3 Values of C and C for Body (a) 20 N. M 4 Values of CN and CM for Body (a) 20 w w 5 Values-of CN, CM for Body (a) 22 6 Values of CN, CM for Body (a) 22 M~.

W IL LOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-80 SYMBOLS AND NOMENCLATURE c = local velocity of sound Cn = normal force coefficient CmqoSL = pitching moment about xo (positive in stall) rCh 7Cn m dimensionless non-steady pitching stabilCh C6( ) ity derivatives CnCn m W m dimensionless normal acceleration stabil23(M) W(L L) ity derivatives C=Cn aZCm dimensionless steady pitching stability 0\($) clUo) derivatives Cach rC incremental lifting steady aerodynamic o-Tk a coefficients Cp ( o+ pressure coefficient (5)- { Sx~s fX, 4 J = Laplace transform of f(x) KO(z) = modified Bessel Function of second kind with modulus z M wr k Ur = reduced frequency Uo L = length of body of revolution M = free stream Mach number p = local static pressure qo = free stream dynamic pressure q = steady pitching velocity (Fig. 2c) R(x) = locus of surface of body of revolution S = reference area S(x) = cross-sectional area of body of revolution at station x t = time in seconds Uo = free stream speed V = local velocity w(x, t) = upwash velocity (Fig. 1)

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UmM-80 xo = center of rotation of pitching motion x, r, e; = cylindrical coordinates X, Y, Z = body stability axes through xo GREEK SYMBOLS CaY = angle of attack = F = - compressibility parameter y = ratio of specific heats to = free stream velocity potential 0I, 1, ~2 = perturbation velocity potentials = total velocity potential U = frequency of periodic motion

WNILLOW(V RUN RESE.ARCH CENTER UNIVERSITY OF MICHIGAN Umm-80 I. SUMMARY The general velocity potential for a slender pointed body of revolution oscillating periodically in a supersonic uniform stream is presented. The solution is shown to reduce in the limit to the sum of well-known stationary source-sink and doublet solutions when the frequency of oscillation approaches zero. Stability derivatives for low frequency oscillations are determined by making use of the approximate potential obtained by expanding the oscillating velocity potential in powers of reduced frequency, retaining only the first order terms in reduced frequency. In particular, the following four cases of steady and non-steady supersonic motion are treated: (1) harmonic pitching about a point xo; (2) harmonic normal oscillations in pitch; (3) steady pitching about a point xo; and (4) steady angle of attack. Equations for the slender body theory normal force and pitching moment stability derivatives are also determined. Two types of bodies of revolution are examined to demonstrate how the aerodynamic stability derivatives may be varied by changing the contour of the body. "Flaring" the aft end of the body increases the lift as well as increasing the aerodynamic damping moment of a finless body.

W ILLOW RUN RESE ARCH CENTENR -UNIVERSITY OF MICHIGAN UMM-80 II. INTRODUCTION Attention has been directed lately to the problem of dynamic stability of low-aspect-ratio high-speed aircraft (Ref. 1 and 2). The problem of determining the static and dynamic aerodynamic force and moment stability derivatives of low-aspect-ratio wings at supersonic speeds has been treated by many investigators (Ref. 3, 4, 5, 6, 7, 8, and 9). Little consideration beyond first approximations has been given to the corresponding problems for bodies of revolution (Ref. 10). The dynamic stability derivatives for such bodies become of prime importance for finless missiles and missile configurations having relatively low area, low-aspect-ratio surfaces. It is the purpose of this paper to present the general potential for the harmonic motion of a slender pointed body of revolution in a steady supersonic stream. Following this, the doublet potential for low frequency harmonic oscillations is demonstrated. The slender body theory doublet potentials are also determined. Example solutions for a body of revolution are calculated to illustrate the variation of the stability derivatives with Mach number in a limited supersonic Mach number range. It is shown how the stability derivatives can be modified considerably by varying the contour of the body of revolution. The solutions for steady pitching and steady angle of attack are also presented for comparison purposes.

W IL LONW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-80 III. FUNDAMENTAL SOLUTIONS The well-known non-linear potential equation for compressible non-steady flow in cylindrical coordinates is (Ref. 11); figure 1, | +ift -+ — +1-~'eZ t a + V v = 0 (1) Where, from the time dependent energy equation for compressible flow,'t+ ~+: F(t) 2t Ft),1 (2) We will proceed to linearize equation (1) in the customary manner by assuming that the desired solution to equation (1) can be represented by' = (',,e) + (,het) (9) where go? (',x ) is the free stream potential f (11,eijGt) is the perturbation potential representing the unsteady body of revolution (considered small with respect to co) Should equation (3) be substituted into equations (1) and (2) and the products and second and higher powers of the derivatives of * be neglected as being of secondary order of magnitude, we arrive at the Prandtl-Glauert equation for non-steady compressible flow. B 4r-% ff ~nL~ge-e~-o ffi + B it <4~ BZ A+ (4) The solutions with which this paper is concerned will satisfy equation (4). As such, these solutions are all subject to the usual limitations of the linearized theory. The solution sought in this

0 C) o-o =R-: O0x1 M.>1 Z (X(x,t) FIG' _I OF FIG. 1 CYLINDRICAL COORDINATES & ORIENTATION OF BODY OF REVOLUTION' 0' O1 I,

W ILLO4W RUN RESE ARCH CENTER - UNVERSITY OF MICHIGAN UMM-80 paper is __~=.cose 7ae (5) where 4- is the solution to the equation IZS~~-~P~h- I B 7X- kl C t+ t (6) Direct substitution of equation (5) into equation (4) will verify this. Here, the perturbation potential representing the body is composed of Y — f=,+P <(7) The potential ~- is axially symmetric with respect to the x-axis and cannot yield "lifting" velocity perturbations (except as a product with the ~2 potential velocity perturbations). The potential B2 being odd with respect to the plane 0 = will yield the desired lifting velocity perturbations. A. The Non-steady Source-sink Potential Let | ~ —N('.h) t itw~f-~C I ( C )3 (8) in equation (6). There results NFh,+ IN Ne ue- BNac1 N o l Further, make use of the Laplace transformation, and let

WILLOWX RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-80 | N(S,'f)= {SX N(%,)I (10) If N(ol)= (,h) appropriate to pointed bodies, operating upon equation (9) with equation (10) yields, d + I CB = (11) The desired solution to equation (11) is the modified Bessel function of the second kind. j (st)- Ko(Bk Ls -B] B ) (12) Using transform pair (871.5) of reference 12, we obtain Nt j [(%^- B"~) t 8 (13) l The point source solution to equation (6) is obtained through equations (8) and (13). Ill _ _.$. I II II ]Xh] ~ ^C Bt}t] 4 (14) If the sources are distributed along the x-axis at points | according to f(~), then equation (14) becomes (X-)>B- (15)

WIlLLOW'0 RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-80 if w = O as in the steady source-sink solution, then equation (15) becomes the easily recognized form (16) x-: > B& B. The Non-steady "Doublet" Potential Equations (5) and (15) are now employed to obtain the desired supersonic non-steady doublet potential. The differentiation indicated in equation (5) involves differentiating under the integral sign of an improper integral. This differentiation is presented in Appendix A and results in the doublet potential below. X-BA. 0O,t e- Q- 8^M] m (17).. 2 -iwm x-~> $im If, in equation (17), w = 0, the recognized steady doublet potential results. X-B Qb aE ( W xx-B>. If in equation (17), w 0, the recognized (18)

WILLONW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-80 Equation- (17) is the desired non-steady doublet solution to equation (4). The balance of this paper will be devoted to presenting particular solutions to equation (4) appropriate to: (a) harmonic pitching about a point xo; (b) harmonic normal oscillations in pitch; (c) steady pitching about a point xo; and (d) steady angle of attack. All of these situations can be treated by making use of equation (17) and its derivatives. C. Boundary Conditions The boundary conditions for the steady or non-steady supersonic flow about a body of revolution specify that the velocity component normal to the surface of the body at the surface must be zero. That is, if k=R(X) = (19) is the locus of the surface of the body in all meridian planes, then the boundary condition is (Fig. 1) |( X)_ -Wg(,%t))cBeO (20) in general. Equation (20) takes a different form for each of the four steady and non-steady motions previously mentioned. 1. Harmonic pitching about the point xo In this case,((t)o4(0A4.L,'WJ, and from figure 2a and equation (20) ('21a ) (-a" -- olo -,,,t [U i(, o]

WILLOVW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-80 ~-'0~ ~x, ~)- Uodo ~st ru I-xo ) 4 o FIG. 2a HARMONIC PITCHING ABOUT Xo ALtie=00 tJ(t) w(t)= tlo e it FIG. 2b HARMONIC NORMAL OSCILLATIONS IN PITCH / (X)-, U,- x-xo) / \ FIG. 2c STEADY PITCHING ABOUT Xo FIG. 2d STEADY ANGLE OF ATTACK

W IL LOW R RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-80 2. Harmonic normal oscillations in pitch From figure 2b and equation (20) (a ):wo Gcp{%wt}'Ce (2lb) 3. Steady pitching about the point xo From figure 2c and equation 20) a8\ At) = _ aer~~ua~+9S(1L ~ (21c) \ Xk R-L The capital letters X and ~ in figure 2c denote body stability axes. The point xo is the center of gravity, and the steady pitching velocity is taken in its defined sense. 4. Steady angle of attack From figure 2d and equation (20) atL jR-+O (21d) This case has been treated in the literature before (Ref. 13 and 14) and will not be dwelt upon in any great detail in this paper, other than to compare it with the other solutions. Equations (17) and (18), together with equations (21), are now used to specify the function f( ). The value of'( -r / is obtained in Appendix B, where it is shown that R- 0 The function f(~) is now determined through equations (21) and (22) for all of the cases treated in this paper.

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-80 IV. NORMAL FORCE AND PITCHING MOMENT COEFFICIENTS The steady and non-steady pressure coefficients can be obtained from the general Bernoulli equation. Pt z + 9 -F(t) Equation (3) is substituted into the Bernoulli equation, and terms are dropped in keeping with the process used to linearize equation (1). Keeping only terms odd with respect to the plane 0 = 2 results in the approximate "lifting" pressure coefficient.'C)z-.0 t ~ 1Uo} (23) If normal force coefficient and moment coefficient are defined as Cn 5 Cf R coe J d and Cm-= J |L (~o-f) coe,tfAi because Cp is proportionate to cos e, it follows that on c, RdL (24)

WILLOW.V RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMm-80 and Cm=cnt+ o f[oe] t (25) We are now in a position to obtain the lift and pitching moment coeffiWe are now in a position to obtain the lift ad pitching moment. coefficients for the four motions discussed in Section IV,

WXILLOWX RUN RESE ARCH CENTER UNIVE/RSITY OF MICHIGAN UMM-80 V. THE LOW FREQUENCY OSCILLATIONS AND STEADY MOTION The task of obtaining solutions for the non-steady supersonic flow about bodies of revolution using equation (17) is simplified for the case of low frequency oscillations. It can be shown by expanding equation (17) into powers of reduced frequency that for harmonic frequencies of the order of the non-steady potential can L be represented by the first order terms in k with an error of about 5% or less. For bodies of the order of 20 to 30 feet in length in the Mach number range of 1.5 to 2.5, the first order terms in k in equation (17) would represent the non-steady potential for harmonic frequencies of the order of 35 to 115 radians per second within about 5%. Before proceeding to this expansion process circumvent the improper integrals in equation (17) by introducing the dummy variable | through the substitution lk Ad = $<(26) If equation (26) is substituted into equation (17),. there results (27) B_,,B +MB Lf(u) e M co4 t (i 4t het )nm X B CB. 411W M'cop~o, -A

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-80 carrying out the expansion into powers of k we obtain _yk AL, Bf (%LBt by? Il~d.(28) 15 C&r;. i.tL aLAA 0c B9 o If w = 0 in equation (28), the well-known steady potential results. t~e~88F. =11i 4t e ~ Ji (~tu 1L ~(29) BW Equations (28) and (29) will be used to obtain the four non-steady and steady solutions with which we are concerned. A. Harmonic Pitching About the Point xo The equation for the distribution f () is obtained from equations (21a) and (22). f(~) D (g) [OO)[u0 0+ W(%-o)o%] (30) Equation (30) is substituted into equation (28) and equation (28) substituted into equation (23) to obtain the pressure coefficient on the body. =Ico 8's) a + B I2 (31) U0 J. U),~MR I~v a iouo

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMI-80 The integrals In appearing in equation (31) are presented in Appendix C. These integrals will also appear in the remaiing expressions for the steady and non-steady pressure coefficients to be determined. Equation (31) is now substituted into equations (24) and (25) to obtain the desired derivatives. C J M')ay,2-B$O3Il]R"C (32) C —- L - — o — Kl' The slender body theory expression for pressure coefficient is obtained by substituting equation (21a) into equation (B-6) in Appendix B and substituting the resulting equation into equation (23). This equation is then substituted into equations (24) and (25) to obtain the following: s (L) ~ o\ VOL. 1 C =Z 5 — IL) LS(L)J (34) and Cma=<2 S(L), (35) B. Harmonic Normal Oscillations in Pitch From equations (21b) and (22), we have s(V) (36)

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-80 Equation (36) is used in equations (28), (23), (24), and (25), respectively to yield L Cxnt SL 61, NO +d Ad (37) and c c o l t s is (8) The corresponding slender body theory expressions are: VOL. Cn= 2 SL (39) and s~(L r S( VOL - s (-) c- 1 C^W= S l L L)L 5 (L)L ( ) C.. Steady Pitching About the Point Equations (21c) and (23) yield.d s)(~)) (41) l ~~~~~~~~~~(I

WILLXOWV RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-80 Substitution of equation (41) into equation (29) and the resulting equation into equation (23) and thence to equations (24) and (25) gives ~,+ 1 0 (42) and C c + fL, - rL ]jRd (43) The corresponding slender body theory expressions are: C'~ — 2 S I L -(44) and L S(L) f/0\'- to V. _s___ 1 (45) C S LL L LS(L) Lo' S(L) D. Steady Angle of Attack From equations (21d) and (23) s(,) (46)

MWILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN umnM-8o Then, from equations (46), (29), (23), (24), and (25) L Cea~X S | | 2 ogI R < (47) 0 L -Cn-=Cnd L + SL PIyd& (48) 0 The slender body theory results are: s(L) (49) and s(L) (/o VOL. Cdz =-2 $ (- LS(L)-I) (5~) This completes the list of steady and non-steady stability derivatives. These equations will be demonstrated in the next section for two different bodies of revolution.

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-80 VI. EXAMPLE RESULTS FOR TWO BODIES OF REVOLUTION The equations obtained in the previous section will now be applied to the two bodies shown in figure 1. The stability derivatives for body (a) of figure 1 will be obtained using both the first order in frequency equations and the slender body theory. Following this, it will be shown how modifying the shape of body (a) to body (b) changes the stability coefficients. Body (a) of figure 1 is generated by revolving a parabolic arc about the x-axis. The equation of this parabolic arc is 0z.O 0(/0rr — d -6 _ L) _ 8(51) Body (b) incorporates the nose of body (a) with a short cylindrical section and a slightly "flaring" rear section. The equation for the locus of the surface of body (b) is =; (/~/ -AT~) O:' a - (52a) r-.o =~ -O J..- 6 (52b),= ~ og /s -._ -, o6z;9 ~ g (52c) (52c) The point of rotation x0 for the steady and non-steady pitching derivative was taken to be Xo = 4.78 for both bodies. This point was chosen as being typical for certain rocket-propelled missiles. Figure 3 presents the values of Cr~ and Cm~ for body (a) using equations (32), (33), (34), and (35). The limitations of the slender body theory in predicting the magnitude and variation with Mach number of these stability coefficients is apparent. This depends, of course, on the validity of the first order in frequency theory.

WILLOW RUN RESEARCH CENTER ~ UNIVERSITY OF MICHIGAN UM M-80 1.80.... -.35 u'1.75 -.30 I 2 1.70 - -.02 >U c~ac t. 1.65 -.20 Z c /;< z: 1.60 -.15 SLENDER BODY THEORY,.5 -...........1! 1.4 1.6 1.8 2.0 2.2 MACH NUMBER FIG. 3 VALUES OF CN AND CM&W FOR BODY (a) 10 > c u I.4mom.08 I1.30.04 LU,",____~'____ /.02 SLENDER BODY THEORY CM, 1.20,, 0 1.4 1.6 1.8 2.0 2.2..02 MACH NUMBER.02 FIG. 4 VALUES OF CNj,CM FOR BODY (a)

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UvMM-80 Figures 4, 5, and 6 present values of Cnr, Cm*, Cna, Cmc, Cnq, and Cm for body (a) using both theories. The remarks above relative to the two theories apply to these results also. An interesting result of these calculations is obtained by comparing figures 3 and 6 which present the non-steady and steady pitching derivatives, respectively. In dynamic stability analyses the steady pitching derivatives Cn and Cm are often used rather than q q the non-steady derivatives Cnr and Cod. If this procedure were followed for body (a), both the magnitudes of the derivatives and the variation of the pitching moment derivative with Mach number would be in error in the Mach number range of figures 3 and 6. In both cases, however, the sign of the pitching moment derivative is such as to indicate a damping moment due to a pitching angular velocity. The slender body theory derivatives for both bodies are given in Table 1, and indicate qualitatively the effects of changing the shape of body (a) to that of body (b). All coefficients are based on the same reference area; the cross-sectional area at x = 5. TABLE 1 Slender Body Theory Stability Derivatives body Cna C Cmm Cnm Cnm Ci C C (a).82.93.33 -.13 1.58 -.13 1.25 -.002 (b) 3.63.15 1.47 -.48 3.08 -.59 1.61 -.114 Table 1 shows that body (b) has increased lift over that of body (a). It is also apparent from the values of Cm that the static center of pressure has moved rearward with changing body (a) to body (b). Thus, body (b) has improved margin of static stability over that of body (a). Both bodies possess steady and non-steady damping in pitching about xo. However, body (b) shows an increase in both Cm and Cm;'

STEADY DERIVATIVES ~ ~c~~~~~U..q.4~. ~STEADY DERIVATIVES O O O O 0, 0 0e 0 bo uo b 01 Cn o, zC Y7 <~~~~ C __~~~~~~~~~~~~~~~~~~C m~~~~~~~~~'n zu, z _ _ 0~~~~~~~~ rtoo z ~ ~~~~~~~~~~ _____ > -n rn, m, 0 ocw0 <;3 mm__ -o _ _'' \ cII 5 _ - - g ) cc STo AD DERIVATIVES ~~n b.STEADY DERIVATIVES L z,,

WILLOW' RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-8o over those values appropriate to body (a). Thus, body (b) has greater damping in pitch than body (a). The above discussion indicates briefly how the aerodynamic properties of bodies of revolution can be modified by varying the shape of the body. In general, it can be stated that increasing the base area of a body of revolution will improve its steady and non-steady lift and pitching moment characteristics in the supersonic range of Mach numbers.

NXILLOAW RUN RE SE.ARCH CENTER - UNIVERSITY OF MICHIGAN UMM-80 APPE]NDIX A Derivation of the Non-steady Supersonic Doublet Potential Equations (5) and (15) can be combined to obtain the supersonic doublet potential for bodies of revolution. However, the differentiation demanded'by equation (5) involves differentiating the rimproper integral in equation (15). The differentiation is simplified if the dummy variable is first introduced into equation (15) through the substitution Co4&Y AsA (A-l) If equation (A-l) is substituted into equation (15), there results 0 ),_ s -W t C- 4kk AA., (A92) 13)1 1 The differentiation required in equation (5) is easily performed to give (A-3) CBnv -I to

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-80 where f(o) is taken to be zero for pointed bodies. Substitution of equation (A-i) into equation (A-3) yields equation (17).

WILLOWV RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN lMM-80 APPENDIX B Evaluating R t = From equation (20) we have (2) ->(Iw(%,I)tc&s (B-i) Now, from equation (27) ('R) -co te 4zwt 0 if we let BR (B-3) Then equation (B-2) becomes (T i cg 4rudiv 3 Roes R —o o (B-4) This gives (ti)voR -pivt) 6 (B-5)

WrILLOWV, RUN RESEARCH CENTER -UN.IVERSITY OF MICHIGAN UMMI-8o and so g) -+(X Wp tX } 7- (B-6) The combination of equations (B-1) and (B-6) yields S(mT, (,4p it (B-7) Equations (B-5) and (B-7) can be combined to yield the slender body theory non-steady potential for bodies of revolution. ct~ R.(B-8) 7Y~~~~~

WILLOWg RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMMv-80 APPENDIX C The Integrals In (n = 1, 2, 3, 4) Appearing in the Steady and Non-steady Pressure Coefficients Certain integrals appear in any or all of the expressions for pressure coefficient derived in this paper. These integrals are: r I,-( t -BR C44 ) C4 Ad.,cL& (c-l) BR II= t s'( -BRcOc(L-BRw) (C-2) (c-3) -BR (C -4) When these integrals are differentiated with respect to x, it should be remembered that. S(o) = sl(o) - 0 for pointed bodies of revolution.

W ILLOOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-8o REFERENCES No. Title 1. Donovan, A. F., Flax, A. H., and Cheilek, H. A., "Stability and Control of Supersonic Aircraft", Institute of the Aeronautical Sciences Preprint No. 136, January 1948. 2. UMM-37 - University of Michigan, "The Stability and Response of an Aerodynamic Missile in Plane Motion", by H. G. Mazurkiewicz and R. M. Spath, December 1949. 3. Ribner, H. S., "The Stability Derivatives of Low-aspect-ratio Triangular Wings at Subsonic and Supersonic Speeds", N.A.C.A. T.N. No. 1423, 1947. 4. Ribner, H. S. and Malvestuto, F. S., Jr., "Stability Derivatives of Triangular Wings at Supersonic Speeds", N.A.C.A. T.N. No. 1572, 1948. 5. Moeckel, W. E. and Evvard, J. C., "Load Distributions Due to Steady Roll and Pitch for Thin Wings at Supersonic Speeds", N.A.C.A. T.N. No. 1689, 1948. 6. Malvestuto, F. S., Jr., and Margolis, K., "Theoretical Stability Derivatives of Thin Sweptback Wings Tapered to a Point with Sweptback and Sweptforward Trailing Edges for a Limited Range of Supersonic Speeds", N.A.C.A. T.N. No. 1761, 1949. 7. Miles, J. W., "On Damping in Pitch for'Delta Wings", Journal of the Aeronautical Sciences, Vol. 16, No. 9, September 1949, PP. 574-576. 8. Moskowitz, B. and Moeckel, W. E., "First-Order Theory for Unsteady Motion of Thin Wings at Supersonic Speeds", N.A.C.A. T.N. No. 2034, 1950.

W ILLO W RUN RESEARCH CENTEBR UNIVFRSITY OF MICHIGAN UMM-80 REFERENCES (Continued) No. Title 9* Stewart, H. J., and Li, Ting-Yi, "Periodic Motions of a Rectangular Wing Moving at Supersonic Speed", Journal of the Aeronautical Sciences, Vol. 17, No. 9, September 1950, pp. 529-539. 10. Miles, J. W., "Unsteady Flow Theory in Dynamic Stability", Journal of the Aeronautical Sciences, Vol. 17, No. 1, January 1950, pp. 62-63. 11. Prandtl, L. and Tietjens, 0. G., "Fundamentals of Hydro- and Aeromechanics", 1st edition, McGraw-Hill, New York, 1934, Chapter 10. 12. Campbell, G. A. and Foster, R. M., "Fourier Integrals for Practical Applications", Bell Telephone System Technical Publications, Monograph B-584, September 1931, p. 113. 13. Tsien, H., "Supersonic Flow Over an Inclined Body of Revolution", Journal of the Aeronautical Sciences, Vol. 5, No. 12, October 1938, pp. 480-484. 14. Laitone, E. V., "The Linearized Subsonic and Supersonic Flow About Inclined Slender Bodies of Revolution", Journal of the Aeronautical Sciences, Vol. 14, No. 11, November 1947, pp. 631642.

WILLIOW/R RUN RESEARCH CENTER-UNIVEIRSITY OF MICHIGAN UMM-80 DISTRIBUTION Distribution of this report is made in accordance with ANAF-GCM Mailing List No. 14, dated January 15, 1951 to include Part A, Part B and Part C.

UNIVERSITY OF MICHIGAN l3 9015 02656 7423II I 3 9015 02656 7423