Methods for Determining Supersonic Roll Damping Coefficients for Slender Cruciform Configurations

Transcription

1 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 09-1 January 01, Nashville, Tennessee AIAA Methods for Determining Supersonic Roll Damping Coefficients for Slender Cruciform Configurations Melissa A McDaniel * U.S. Army RDECOM, Redstone Arsenal, AL The computation of dynamic derivatives is crucial to the understanding of the stability of an air vehicle. This paper discusses the computation of the roll damping for generic missile configurations experiencing supersonic flow. A linearized potential approach using Evvard s theory is considered. In addition to the wing pressure loading characteristics, body carry-over loads are also considered. Comparisons are made with available experimental data and empirical correlations to assess the accuracy of the method. Nomenclature A = fin area b = exposed fin span C l = rolling moment coefficient C lp = roll damping coefficient, per radians, dc l /d(pd/v) C lδ = rolling moment due to cant of fin surfaces, dcl/dδ C N = normal force coefficient C p = pressure coefficient C r, C t = root and tip chord lengths d = reference length (body diameter) ds = incremental area I xx = moment of inertia about the longitudinal axis M = Mach number pd/v = reduced roll frequency p = roll rate p = rate of change of roll (rad/sec ) Q = dynamic pressure s = distance from body centerline to fin tip S ref = reference area Uo, V = free stream velocity δ = deflection angle α = angle of attack β = M 1 ρ = atmospheric density μ = Mach angle Λ = leading edge sweep angle I. Introduction VER the last 60 years, significant gains have been made in the area of aerodynamic predictions. Numerous Omethods have been developed to determine the aerodynamic characteristics of a variety of body shapes, fin designs, and body+fin configurations. However, the majority of this effort has focused on the static aerodynamics. To fully understand the aerodynamics of a flight vehicle, such as a missile or airplane, the dynamic properties of the vehicle must be understood. Such understanding requires knowledge of the vehicle s response to rotational motion; * Aerospace Engineer, RDMR-SSM-A, AIAA Senior Member 1 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

2 also known as the damping of the system. In other words, once disturbed from its initial flight orientation, how quickly does the vehicle return to steady state? Damping derivatives have often been considered a second order aerodynamics effect. For most airplane configurations, roll damping in particular, has been considered insignificant 1. As a result, there was little reason to expend significant resources to calculate these terms in most missile simulations. Very rarely is experimental testing done to estimate the derivatives of a real system. However, with the increase in low aspect ratio, highly swept, highly maneuverable airframes, complex configurations, and UAV launched missiles, dynamic stability is rapidly becoming an area of interest again. One of the inherent issues with the prediction of damping derivatives results from the fact that it is an unsteady effect 3. The majority of existing aerodynamic prediction codes do not have the capability to model unsteady aerodynamics while computational methods may be too time intensive in their calculations. Research has shown that some steady state theoretical methods may be expanded for the calculation of damping derivatives, most notably classic slender body theory 4,5. A notable feature of damping derivative calculations is that a build-up approach can be used for their calculation. That is, the body and lifting surfaces can be calculated separately with appropriate theories and the results synthesized as they are for static aerodynamics 6,7,8. This approach has not necessarily been fully exploited and is typically limited to empirical methods. Several methods exist for the calculation of roll damping of wings in supersonic flow. Many of these methods are empirical or semi-empirical in nature. Theoretical methods also exist for explicit configurations such as a rectangular or triangular wing and do not include the presence of body interactions. This effort explores the commonly used approaches for roll damping and builds on the approaches available for a specific wing alone in supersonic flow to calculate the roll damping of a generic configuration. Although no attempts are made to model the non-linearities that exist due to angle of attack, the method presented in this section represents a method useful for generic configurations. Comparisons with experimental or numerical data are provided as a means to evaluate this approach to the calculation of damping derivatives. II. Roll Damping and the Importance of Damping Derivatives A stability derivative provides the rate of change of a force or moment with respect to the motion that is creating it. With this in mind, the roll damping is the rolling moment created by a rolling motion about the longitudinal axis of the vehicle. It is defined as a measure of the vehicle s resistance to a rolling motion. Unless very large roll rates are considered, the primary contribution to the roll damping is through the lifting surfaces. As the vehicle rolls about its longitudinal axis, the spanwise angle of attack of the surface is altered. The angle of attack at any spanwise location is related to the roll rate through the relation py/v where y is distance from the body centerline to the span location of interest. Figure 1. Local change in angle of attack due to roll rate As a consequence of the angle of attack variation, there is a normal force variation as well. As long as the flow remains attached, this induced lift will always oppose the roll rate, thereby damping the motion 3. With knowledge of the lift force at each span wise location, the roll damping moment can be directly calculated. In practice, this is

3 difficult as the loading is heavily dependent on the aspect ratio, sweep angle, taper ratio, and Mach number of the surface. This limits the applicability of simplified approaches. In order to accurately determine a satisfactory method of calculation, it is necessary to understand the relative importance of the factor under consideration through a sensitivity analysis. The following equation of motion fully characterizes the roll motion as a function of the roll acceleration, yaw acceleration, sideslip orientation, and rolling and yawing velocities 3,9. pd I xx p I xz r = roll moments = QSd C lβ β + C + C lp V lr rd V + control terms (1) For the purpose of this analysis, a simplified solution neglecting the control terms will be utilized. For the fully symmetric configuration considered in this analysis, the value of I xz is neglected. Furthermore, for this analysis, the side slip angle, β, the yaw rate, r, and the change in yaw rate,r, are assumed to be negligible. As such, the change in rolling moment with sideslip, C lβ, and the change in rolling moment with yawing motion, C lr, are neglected. This leads to the simplified equation p = QSd pd C I lp () xx V It is evident that the roll acceleration (or deceleration for a stable system) is dependent on the vehicle size (d, S, and I xx ) and flight conditions (V, Q, and p). For this analysis, the Basic Finner configuration, shown in Fig. was used for analysis purposes. This configuration was chosen as it is a basic research model with a considerable amount of data, to include roll damping data, available in literature. The configuration is a 10 caliber long body with a caliber conical nose. Four fins with a one inch exposed semi-span, one inch root chord and one inch tip chord are located with the trailing edge at the back end of the body. A value of Ixx obtained from literature is slug-ft. Figure. Basic Finner configuration Figure 3 presents the solution of the p equation using the provided value of I xx. Additionally, a 3 Hz and 1 Hz roll rate were used to assess the importance of roll rate on C lp. The range of C lp values were chosen based on Missile Datcom 10 predictions, shown in Table 1. Although the focus of this research is supersonic roll damping predictions, a few subsonic values were included for illustrative purposes. Examining the equation, the roll rate is simply a 3

4 multiplier to the slope. Thus, the slope for a 3Hz roll rate will be exactly 3 times the slope for a 1 Hz roll rate. The variations in C lp under consideration represent the accuracy with which the value is predicted. As previously noted, except for rare cases, the values of a C lp for a finned body will always be negative 3. In other words, motion will be damped, not amplified. This is evident from the negative (or decelerating) values of p. At low Mach numbers the vehicle is relatively insensitive to the roll damping values. For the case of Mach 0.5 and a 1 Hz roll rate, the slope of the line is , or nearly zero. In contrast, as Mach number increases, the value of p becomes more sensitive to the C lp accuracy. At Mach and 1 Hz, the slope of the line is 0.361, considerable higher than at Mach 0.5. Additionally, as roll rate increases the vehicle sensitivity to C lp accuracy increases. Table 1. Missile Datcom predicted values of C lp Mach C lp (per radian) (a) 3 Hz roll rate (b) 1 Hz roll rate Figure 3. Roll deceleration variation with C LP While insightful, this analysis is relatively intuitive and contains little information about the vehicle aerodynamics. A more comprehensive approach analyzes the complete equations of motion without disregarding the roll terms. This is done through the analysis of the lateral stability matrix and provides insight into the effect of roll damping accuracy on the lateral stability modes of the vehicle. The lateral stability matrix is shown in Eq (3). The matrix requires knowledge of the side force (Y or C y ), rolling moment (L or C l ) and yawing moment (N or C n ) derivatives with respect to rolling motions (p), yawing motions (r), and sideslip angle (β). Additionally, the inertias of the vehicle (I xx, I xz ) must be known in order to determine the system responses to changes in sideslip (β ), roll rate, yaw rate, roll angle (φ ), and yaw angle (ψ ). In order to determine the modes of the system, the eigenvalues of the matrix must be calculated in the form λ=ω±ni, where λ is the eigenvalue in complex form. Note that for this analysis, a controls-free approach was again taken to simplify the equations. 4

5 Yv m β Lv b + I xz L p pˆ I x = r ˆ N v b + I L xz p φ I x ψ 0 0 L I N I p x x p Y p m b + I + I u o b 0 xz xz L L p p Yr uo m b Lr + I xz Lr I xp N r + I xz Lr I x u o tan Θ o b u o b cos Θ o g cos Θ o / u o 0 0 β pˆ 0 rˆ φ 0 ψ 0 (3) With the eigenvalues calculated, the effect of a given derivative on the stability of a system may be determined by analyzing the undamped natural frequency (ω n ), damping ratios (ζ), and/or time properties (time to half or double) of a system as a function of varying values of the selected stability derivative. The equations for ω n, ζ, and t half or t double are shown in the equations below. ω n = (ω + n ) 1 (4) ζ = n/ω n (5) t half or t double = n (6) The complete derivation of the matrix used for this analysis is found in Etkin 9. An eigen analysis of the lateral stability matrix results in three modes of motion for a typical system. These modes are: the dutch roll mode, the spiral mode, and the rolling mode. The dutch roll mode is oscillatory motion consisting of roll and yaw coupling. The other two modes are non-oscillatory in nature which the spiral mode characterized by yawing at negligible sideslip and the rolling mode characterized by rotation about the longitudinal (i.e. x) axis. A complete definition of these modes is provided in Etkin 9 or Roskam 3. As previously mentioned, the generic configuration analyzed for this sensitivity study was the basic finner missile. Unlike with the acceleration analysis, a considerable amount of aerodynamic data is needed to perform the complete matrix analysis of the airframe. These aerodynamic coefficients were again obtained from Missile Datcom 10. Table lists the aerodynamic coefficients used for the analysis. Table. Lateral aerodynamics of the basic finner Mach C lβ C lp C lr C nβ C np C nr C yβ C yp C yr Analysis of the lateral stability matrix showed no variation on the dutch roll or spiral modes of the vehicle. This trend corresponds to other results in literature that indicate the C lp will have a noticeable impact on the rolling mode only 9,11. Since the eigenvalues for this mode are real numbers, the mode is non-oscillatory. It is also a stable mode that converges. As a result, analyzing the time to half provides the most insight in to the sensitivity to the damping derivative values. The rolling mode times to half for variations in C lp are shown in Fig. 4. In plots (a) through (c) the red vertical lines correspond to the Missile Datcom predicted values of C lp. In plot (d), the colored vertical lines correspond to the C lp values of the matching Mach numbers. 5

6 (a) Mach 1.5, varied Ixx (b) Mach.0, varied Ixx (c) Mach 3.0 varied Ixx (d) Ixx= slug-ft, varied Mach Figure 4. Basic finner rolling mode time constant variation with C lp As expected, variance in the rolling mode time to half is a function of both the Mach number and the roll axis moment of inertia. As Mach number increases, the time to half of the system begins to increase for a given inertia. As a result, at larger Mach numbers, a system responds slower to changes. Likewise, as the inertia increases for a given Mach number the time to half also increases. However, in terms of sensitivity to accuracy in C lp, there is little effect due to the result in changing inertia or Mach number. If the C lp is varied by ±0 percent about its nominal value for a given Mach number, t half may vary percent regardless of Mach number or inertia. Regardless of the lack of dependency on Mach number a 50 percent variance in time to half is quite large. In order to eliminate the uncertainties that may result, it is necessary to know the roll damping with some degree of accuracy. These results are important as they indicate how much effort is required to obtain the values of C lp. At low Mach numbers and low roll rates where the vehicle is relatively insensitive to C lp, a full Computational Fluid Dynamics (CFD) solution may not be required and lower order methods are sufficient. However, as Mach number increases, the methods used to calculate C lp may need to be more complicated as accuracy is more important. III. Traditional Approaches Several different empirical or semi-empirical methods are available for the prediction of the roll damping derivatives. The two most common are slender body theory and methods based on the roll control, C lδ. Both methods are described in this section to provide a brief overview of the methods available. A. Slender Body Theory Slender body theory refers to the general method governing the loads on slender configurations over a wide Mach range 5. A slender body can be defined in several ways and is not limited solely to bodies of revolution. In the theoretical sense, a slender body is defined as a configuration where the changes in velocity due to the presence 6

7 of the body are small. 6 The standard conceptual definition of a slender body is one whose length is sufficiently long compared to its diameter and the rate of change in dimensions is small 4,5. As a result, the theory can be applied to slender bodies, low aspect ratio wings, or low aspect ratio wing body combinations as long as the thickness dimension is small compared to the length 1. Furthermore, as long as the rate of change in cross sectional dimensions is small, the theory may be valid for thicker bodies 1. Slender Body Theory is based on the solution of the potential flow equation for steady, inviscid, irrotational flow. β φ xx + φ yy + φ zz = 0 (7) By applying the definition of a slender body (φ xx 0) or using a Mach number near unity (β 0) the equation reduces to φ yy + φ zz = 0 (8) To obtain a solution, a distribution of sources is used to determine the flowfield and subsequently the pressure distribution at zero degrees angle of attack. Similarly, a doublet distribution is used for an inclined body. A rigorous derivation is outside of the scope of this work, however, various approaches can be found in Ashley 1, Nielsen 5, or Moore 6. The most commonly known result from this theory is the normal force and center of pressure equations that follow. C Nα = A base S ref (9) X cp l n = 1 Vol n πr n l n (10) The above equations do not account for the ability to determine damping derivatives. However, the theory can be applied to such effects by altering the form of the velocity potential. Slender Body Theory as utilized for damping derivatives was developed by Bryson in the 1950 s 3. Although previous work by Munk 7, Jones 8, and Spreiter 13 had been used to determine the forces on slender bodies, wings, and combinations thereof, Bryson expanded the analysis to include all six vehicle inertia or apparent mass coefficients. This enabled the calculation of all static and dynamic stability derivatives with the exception of the axial force derivatives. To define the apparent mass coefficients, a crossflow plane fixed in the axis is considered with the coordinates ξ, η, and ζ defined parallel to x, y, and z as shown in Fig. 5. In this plane, the potential depends only on the normal velocities of the cross section at the instant under consideration. Figure 5. Coordinate system for slender body theory 4 Defining the normal velocities in the η and ζ directions as v 1 and v, respectively, and the angular velocity about the ξ axis as p, the velocity potential can be written as φ = v 1 φ 1 + v φ + pφ 3 (11) 7

8 Bryson then makes use of the kinetic energy per unit length, T, in the cross flow plane which is defined as T = 1 ρ C φ φ η ds (1) This definition of the kinetic energy is necessary as the rolling moment per unit length can be written in terms of the kinetic energy by use of the following equation. dl = d dx dt T + v T T p v v 1 (13) 1 v Substituting the velocity potential into the kinetic energy equation yields a total of nine integrals that are referred to as the apparent mass coefficients. They are given the form Aij and written as follows A 11 A 1 A 13 A 1 A A 3 = 1 A 31 A 3 A 33 φ φ 1 1 ds φ φ n 1 φ φ ds S 1 φ ref n 1 φ φ 3 D 1 ds 1 φ n D ds 1 n φ ds 1 n φ φ 3 D 1 φ D φ 3 ds 1 φ n D 3 ds n φ 3 ds n φ 3 ds n (14) Nielsen 5 provides a comprehensive explanation of the uses of Bryson s slender body theory for the computation of several of the damping derivative terms. For the purpose of this analysis, only the roll damping for a cruciform missile with a single set of identical fins is of interest. For this configuration, Bryson s version of slender body theory defines the roll damping derivative as follows for a configuration with no body present. 8s4 C lp = 4A 33 = (15) πd S ref In order to account for the presence of the body, the following relation is used, with the ratio obtained from figure 6. C lp = C lp 8s 4 C lp rb=0 πd S ref (16) Figure 6. Slender body theory C lp ratio 5 The result is not a function of Mach number and is only dependent on the size of the vehicle, in particular the relation between the fin span to the body diameter as s is defined as the exposed semi-span plus the body radius. While basic slender body theory is limited only to configurations where the change in thickness is small in 8

9 comparison to the length, Bryson s slender body theory method is further restricted to configurations with one set of fins whose aft end lies in the same plane as the base of the body. The geometry of each fin may vary so long as the trailing edge does not extend beyond the based. Additionally, to conform to the definition of a slender body the rate of change of the cross section in the longitudinal direction must be small. Note that due to the symmetry of a cruciform configuration, this application of slender body theory does not predict variation with angle of attack. B. C lδ Based Approaches The most common empirical approaches for calculating the roll damping of finned vehicles relate the values of C lp and C lδ. Initial theoretical results from Bolz and Nicolades 14 produced the following relation C lp C lδ = 0.67 d b o (17) where b o refers to the total span including the body. Subsequently, Adams and Dugan 15 produced a near identical relation, without the diameter to span relation. These approaches were never applied to any experimental data and subsequently were found insufficient for most configurations. Eastman 16 found the available theoretical approaches insufficient to match the experimental data available at the time. As a result, he expanded on these theoretical methods to create a fully empirical approach based on the relation of C lp and C lδ. His method requires knowledge of Y area centroid (Y cent ) of the fin. This distance is measured radially from the body centerline to the centroid of the exposed fin. This empirical correlation was determined from wind tunnel data on configurations shown in Figure 7. The graph used for his correlation is also shown in Fig. 7. Figure 7. Eastman s empirical derivation 16 9

10 His method presents the roll damping as follows C lp =.15 Ycent D C lδ (18) The simplicity of the calculation lends itself for use when wind tunnel data is available and the configuration is simple. There is no need to vary the method for different Mach regimes. However, its simplicity is also its limiting factor. Although the configurations used for correlation are widely varied, the available data appears to be small. Additionally, it is only applicable to configurations with a single fin set, thereby eliminating its use for a wide variety of missile variations. IV. Evvard s Theory The two approaches previously described offer a quick solution to the damping derivative problem for a limited set of configurations. However, the challenge in determining damping derivatives lies in the flow complexity. This complexity makes it difficult to determine simple solutions for roll damping. Although solution of the full Navier- Stokes equations is an accurate, viable, and often necessary method for the calculation of roll damping, a fast method is desired in many situations. The previous methods have shown the power of the potential flow equation and its ability to determine the damping derivatives when simplified. Once the velocity potential is determined, all other flow properties are known. However, the velocity potential equation in its full form is a non-linear partial differential equation for which no exact solution exists. Through the assumption of small perturbations, it is possible to linearize the potential equation and obtain a closed form solution 17. By establishing the proper boundary conditions, it is possible to obtain the pressure coefficient and loading for a rolling surface. Subsequently, it is possible to determine the roll damping from this information. This theoretical approach was developed by Evvard 18-0 and several other researchers 1-4 through the 1950 s. Evvard s theory computes the flow over lifting surfaces in supersonic flow by covering the wings with a source distribution 18,19. The wing is modeled as a flat plate and inviscid flow is assumed. Thickness effects are not considered and must be approximated by other theories if their inclusion into the damping derivative calculations is desired. As a result of this assumption, the linearized theory for compressible flow can be utilized for the calculation of a range of stability derivatives. In order to evaluate the roll damping derivative, the lifting pressure distribution over the wing for a rolling motion must be known. This distribution is dependent on the divisions created by the Mach lines. The theory takes advantage of the fact that in supersonic flow, Mach lines emanating from the leading edges of the fins provide a natural division of the fins into distinct sections, as shown in Fig. 8. Two types of divisions are possible, a supersonic leading edge and a subsonic leading edge. The type is dependent on the Mach number and leading edge sweep. If the Mach angle, μ, is greater than 90-Λ, the leading edge is supersonic. By definition, an upswept fin will always have a supersonic leading edge. Subsonic leading edges are typically seen only in highly swept wings or wings with moderate sweep and lower Mach numbers. 10

11 Supersonic Leading Edges Subsonic Leading Edges Figure 8. Typical wing panel divisions from Evvard s theory 18,19 Per Evvard s theory the potential equation is defined for a singular flow type such as angle of attack, rolling motion, pitch motion, etc. A separate velocity potential must be derived for differing flow types. Super positioning may then be used to get combinations of loadings. In other words, the velocity potential for a rolling airframe at angle of attack may be written as 5,6 φ = φ α + φ p (19) In the above equation, only the term φ p will contribute the roll damping as it is the only term with the rolling motion included. The differential pressures and subsequently the roll damping values are found from the potential equation using the following equation 1. ΔC p,p = 4 φ p (0) V x C lp = 1 Sd pd ΔC p,pydxdy (1) V It is noted that the above C lp equation contains the term ΔC p,p, which is the differential pressure due to a rolling motion. In each section, the differential pressure coefficient may be determined in closed form and integration provides the fin loads and moments for a specified flight condition. Tables 3 and 4 provide the differential pressure equations for a supersonic and subsonic leading edge, respectively. In the equations below the following variables are used for simplification. The definitions of x, y, x a, and y a, are shown in Fig. 8. m = cot Λ () υ = y mx (3) Θ = tan (90 Λ) (4) 11

12 Table 3. Differential pressure equations due to rolling motion, supersonic leading edge 1 Region ΔCp 1 4pm x(β m υ 1) 4pm x πv(β m 1) 3 3 4pm x(β m υ 1) V(β m 1) 3 V(β m 1) 3 (1 + βmν) (1 + βmν) cos 1 βm(1 + ν) (1 βmν) cos 1 (1 βmν) βm(1 ν) + 4pm x(βmν 1) πv(β m 1) 3 mx a βmy a b (1 βmν mx cos 1 a y a (1 βm) + b mx a + y a + b βm y a (βm 1)(mx a + βmy a + b) 4 4pm x πv(β m 1) 3 (1 + βmν) (1 + βmν) cos 1 βm(1 + ν) 5 4pm x πv(β m 1) 3 (1 βmν) cos 1 (1 βmν) βm(1 ν) + 4pm x(βmν 1) πv(β m 1) 3 mx a βmy a b (1 βmν [mx cos 1 a + (βm + 1)] mx a y a βm my a (x a + βy a + b) mx a + β m y a + b (β m + 1) cos 1 mx a y a (1 βm) + b mx a + y a + b βm y a (βm 1)(mx a + βmy a + b) Table 4. Differential pressure equations due to rolling mmotion, subsonic leading edge Region ΔCp 1 pθ V xν (1 m ) 1 ν ( m ) m 8p θ 3θx + y(1 m) b (1 + m) b y Vπ 3(1 + m) x + y)(1 + m) Several authors have utilized Evvard s theory to calculate exact roll damping equations for various planforms such as rectangular, triangular, and trapezoidal 1-4. These authors have integrated the pressure equations presented here and in other references to obtain those solutions. However, those equations are often complex and only apply to limited configurations with several restrictions. The equations presented in Tables 3 and 4 apply to arbitrary wings with only a few restrictions. The wings may be of arbitrary planform with swept forward or aft leading and trailing edges and taper. However, it is necessary for the tips to be streamwise or in-line with the flow 1. The method can be applied numerically without the difficult integration. Practical application of Evvard s theory is presented in the following section. V. Practical Application of Evvard s Theory Direct application of Evvard s theory requires that the wings are separate panels joined at the root chord The resulting Mach lines will originate from the apex of the wing junction. However, for a missile configuration, the wings are not joined at the centerline and Mach lines originate from the wing body junction. This means that the wing must be extended into the body as shown in Figu Points that are contained within the body diameter are given a pressure loading of zero so that they do not affect the overall loading the fins. However, the Mach lines will 1

13 emanate from this joint at the center line of the body. As a result, for certain configurations, the discrepancy in the theoretical versus actual Mach lines may significantly alter the loading. This discrepancy may be alleviated if the wing is extended into the body but the Mach lines are shifted to the wing body junction. An example is shown in Fig. 10 for the basic finner wing. This essentially treats the wing itself as a single panel, but utilizes the coordinate system with the origin along the body centerline. This distinction is important for the determination of the proper value of the y coordinate. Since the roll loading varies with distance from the roll axis, in this case the body centerline, the y coordinates on the wing must originate from that point. Y 1 CR 0 C R C T b/ X Figure 9. Fin extension into the Body Mach Lines originating at wing body juncture Figure 10. Mach line orientations To apply Evvard s theory, a wing semi-span is divided into discrete panels with a control point located in each panel. The region in which a control point resides must be determined so that the proper pressure loading equation is utilized. This is done by defining the following variables, with y originating at the body centerline. x 1 = (y r b )β (5) x = tan Λ + β(1 y) (6) 13

14 x 3 = β( y r b ) (7) Next, the following equations are used to determine which region a control point is in based on the x location of the control point. This setup is shown graphically in Fig. 11 5,7. Region 1: x < x 1 and x < x Region : x > x 1 and x < x Region 3: x > x and x < x 1 Region 4: x > x 1 and x > x and x < x 3 Region 5: x > x 3 Figure 11. Control point determination With proper region determined, the value of C p is calculated using the equations in Table 3. The values of C p are summed over the entire wing and the process is repeated for all wing in a given fin set. The value of roll damping is then calculated from the following equation. C lp = Vb nfins C p da y 4p 1 (8) sref d In the above equation, da refers to the incremental area of the panel in which a control point is located. It should be noted that the Evvard equations derive C p and thus C lp as a function of the wing span. This effort converts the value of C lp to be normalized by the reference length as is common in missile configurations. To this point, one significant loading contribution has been neglected the body carryover load. Although the roll loading on the body itself is considering insignificant for the roll loads commonly experienced by missiles, the body still imparts a load to the wing that must be considered. The basic forms of Evvard s theory do not account for the presence of the body as they were developed for wings alone. However, other researchers have developed methods to handle the wing body carry-over by modifying the source distribution at the body. In particular, Tucker and Piland 8 have developed an alteration to the region pressure loading when the wing is in the presence of the body. To apply their alteration the following equation is used for control points in region in place of the equation in Table 3. 14

15 C p = 4p πv r b h 1 x β y β k + x β y β k + a kx β y β k cos 1 β(x ky) + kx +β y cos 1 β(x + ky) + β y kx kx β y (β k ) 3 cos 1 β(x ky) r b h 1 β y + kx kx +β y (β k ) 3 cos 1 β(x + ky) (9) k = tan Λ (30) h = C r tan μ + r b (31) The value of h in the above equation is the distance from the body centerline to the point where the root chord Mach line crosses the trailing edge. Using this carryover estimation places another restriction on the use of Evvard s theory for roll damping. In this case, the root chord Mach line may not intersect the tip chord. It must cross the trailing edge at some point along the span. Although this carryover equation was developed for rectangular and triangular wings, there is no limitation for its use with arbitrary planforms provided this restriction is met. VI. Comparisons with Experiments Unlike with static aerodynamic coefficients, there is not a significant amount of experimental data available for dynamic coefficients. However, through literature searches, data has been obtained for several configurations. The results presented in this section show comparisons between experimental data, Evvard s theory and empirical results. A. Basic Finner As previously discussed, Basic Finner is a standard research configuration for which extensive test data exists This makes it an ideal configuration for evaluating roll damping methodologies. For the comparisons shown in this section, experimental results from references 9 and 31 were used. Reference 9 provided roll damping uncertainties of ±.5-±3.5 percent for the data range considered and Mach number accuracy of ±1 percent. Data uncertainty was not available for reference 31. The slender body theory and Eastman equations presented in Section III were used to obtain their representative curves. In order to calculate the value of C lδ needed for the Eastman calculation, The 008 version of Missile Datcom 10 was used. A 5 degree roll deflection was applied to each fin and the value of C lδ was obtained using the following equation. C lδ = C l δ=5 C l δ=0 δ (3) The value of the y centroid of the fin was found from the Missilelab 3 analysis of the fin geometry. Since a plain hex cross section was used to model the fin, it is possible that the y centroid is slightly off and may skew the results slightly. It is noted in the comparisons that there is a slight scatter in the experimental data. Still it is possible to see a general trend of decreasing roll damping with increasing Mach number. With the exception of slender body theory, all of the methods presented in Figure 1 show this same trend. Slender body theory is by definition invariant with Mach number, but seems to provide the best match near Mach.5. Eastman s method follows the trend quite well, though the roll damping is under predicted by approximately percent across the Mach range. Given that basic finner is one of the test cases for Eastman s theory, the accuracy is not surprising. The Evvard methodology provides a slightly better approximation of the roll damping. The values are slightly under predicted if body upwash is excluded and near exact to slightly over predicted if the upwash is included. Since the effect of the upwash is to increase the pressure loading near the fin body junction, this increase in damping is to be expected. Additionally, as Mach number increases, the effect of adding the upwash decreases. This is due to the decrease in the Mach angle and therefore the decrease in the region over which the upwash is acting. 15

16 Figure 1. Results for basic Finner roll damping B. Modified Basic Finner The modified basic finner is another configuration for which a significant amount of damping data exists The body of the modified finner configuration is identical to the basic finner; however the fin has been altered. The fin area of the modified finner is roughly half that of the basic finner. The dimensions of the fin are shown in Fig. 13. Experimental data for the modified finner were obtained from reference 31 and 33. As with the basic finner data from this reference, no indication of the data uncertainty was provided. Figure 13. Modified Finner geometry As seen with the basic finner configuration, slender body theory is the least accurate method for obtaining the roll damping. In this instance, Eastman s method tends to over predict the roll damping across the range evaluated in this effort. As this configuration was also one of Eastman s test cases, this difference could be 16

17 caused by inaccuracies in the value of C lδ or y cent, which were again obtained via Missile Datcom and MissileLab. Evvard s analysis provides the most accurate approximation of the roll damping given the methods under consideration. As Mach number increases, Evvard s theory tends to slightly over predict the roll damping values. This is slightly different than the trends seen for basic finner where Evvard s theory under predicted the damping if upwash was excluded. It is noted that body upwash is not considered for this particular configuration. Recalling that the root leading edge Mach line may not intersect the wing, the body upwash calculations are not valid until the Mach number exceeds.8. This value is well beyond the range of available experimental data for this configuration. Even at this Mach number, the root Mach line will intersect the fin very close to the tip. This trend is true of most low aspect ratio configurations. As such, the upwash characteristics need to be considered for fins of this nature. Even without the upwash, Evvard s theory provides a more accurate approximation to the damping than the other easily available methods. Figure 14. Roll damping comparisons for modified Finner VII. Conclusions and Recommendations The work presented in this paper is meant to present an approach to solving the supersonic roll damping of slender cruciform missile configurations. In order to understand the importance of accuracy when calculating damping derivatives, a sensitivity analysis involving both the equation of motion and the lateral stability matrix was considered. Slender body theory and Eastman s method were presented as common, rapid means for predicting the damping on slender configurations. Both of these theories are limited to configurations with only one finset. Evvard s theory has been used for generic fin configurations and body upwash has been included to obtain a more complete solution of the roll damping problem. The results are generally more accurate than the standard slender body theory and C lδ approaches that are commonly used. Additionally, the Evvard approach is applicable to a wider variety of configurations. The methods presented in the work do not include angle of attack effects due to the limitations of the methodologies presented. Future efforts should address at least modest angles of attack. Additionally, comparisons should be made for configurations with multiple finsets and non-cruciform configurations. Although this is a limit of slender body theory and the C lδ methodologies presented, it is not a limitation of Evvard s theory. However, available data is limited for comparisons. 17

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