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NASA MEMO 12-2-58A

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MEMORANDUM

ESTIMATION OF DIRECTIONAL STABILITY DERIVATIVES AT

SMALL ANGLES AND SUBSONIC AND SUPERSONIC SPEEDS

By Frederick K. Goodwin and George E. Kaattari

Ames Research Center

Moffett Field, Calif.

NATIONALSPACE

AERONAUTICS ANDADMINISTRATION

WASHI NGTONDecember 1958

Declassified December 8, 1961

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

NASA MEMO 12-2-58A

ESTIMATION OF DIRECTIONAL STABILITY DERIVATIVES AT

SMALL ANGLES AND SUBSONIC AND SUPERSONIC SPEEDS

By Frederick K. Goodwin and George E. Kaattari

SUMMARY

Methods are presented for estimating the directional stability

derivative increments contributed by the stabilizing surfaces of sub-

sonic and supersonic aircraft. These methods are strictly applicable at

zero angle of attack and small angles of sideslip. The procedure of

totaling the incremental coefficients to obtain an estimation of the

total empennage side-force and yawing-moment coefficient derivatives is

also shown, together _th numerical examples. A correlation is presented

between estimated and experimental incremental coefficients which indi-

cates that the methods of this report generally estimate the increment

of side force gained by the addition of a panel to within ±i0 percent of

the experimental value while the yawing-moment increment is generally

estimated to within ±20 percent. This is true for both subsonic and

supersonic Mach numbers.

An example application of the methods to one of the problems in

directional stability, that of minimizing the effect of Mach number on

the side-force coefficient derivative of the empennage, is discussed.

INTRODUCTION

Flight through an increasingly large supersonic Mach number range

and at the large angles of attack required for high altitude maneuvering

has introduced a significant problem in maintaining directional stability.

With increasing Mach number the side-force coefficients of the vertical

stabilizing surfaces decrease. The high angles of attack introduce the

nonlinear destabilizing effects of body vortices and reduced dynamic pres-

sure in the region of the upper vertical stabilizing surface. _]ese

effects are briefly discussed by Nielsen and Kaattari in reference i.

2

In this reference it is pointed out that the problem of determiningthe variation of the side-force and yawing-momentcoefficients with angleof attack and angle of sideslip can be considered one of first determiningthe derivatives of these coefficients at zero angle of attack and smallangles of sideslip and then determining the nonlinear effects of thevortices and the dynamic pressure associated with large angles of attackand sideslip. The purpose of this report is to consider in detail thefirst of these steps, and to present methods for estimating the deriva-tives of the side-force and yawing-momentcoefficients at both subsonicand supersonic speeds. The second step, that of estimating the non-linearities introduced at high angles, is considered by Kaattari inreference 2.

SYMBOLS

a

b

B

c r

Cn

Cn_

Cy

Cy_

h

H

K

KH(B)

KV(B)

vertical semiaxis of body in region of the tail panels, in.

horizontal se_miaxis of body in region of the tail panels, in.

body

exposed root chord of a panel, in.

yawing-moment coefficient,yawing moment

2swq_Sw

rate of change of yawing-moment coefficient :_ith angle of

sideslip (_ = 0°), _Cn/_ , per radian

side-force coefficient,side force

q_Sw

rate of change of side-force coefficient _th angle of side-

slip (_ _ 0°), _Cy/_, per radian

vertical distance of wing or horizontal tail above or below

body center line, in.

horizontal tail

apparent mass ratio

apparent mass ratio due to adding a horizontal tail to a body

apparent mass ratio due to adding an upper vertical stabilizing

surface to a body

KV(BH )

KV(BW)

Y

_m

P

%o

s

SH

sU

sv

sw

SU

Sv

SW

U

apparent mass ratio due to adding an upper vertical stabilizing

surface to a combination of a body_ horizontal tail_ and

lower vertical stabilizing surface

apparent mass ratio due to adding an upper vertical stabilizing

surface to a combination of a body and lower vertical stabili-

zing surface

apparent mass ratio due to adding an upper vertical stabilizing

surface to a combination of a body and wing

apparent mass ratio due to adding a _ng to a body

effective apparent mass ratio

distance from the nose to the leading edge, body j_cture of

a panel, in.

distance from the center of moments to the center of pressure

of a panel, in.

distance from the nose to the center of moments, in.

free-stream Mach number

added panel

free-stream dynamic pressure, ib/sq in.

panel span measured from the body center line, in.

horizontal-tail semispan measured from the body center line,

in.

span of lower vertical stabilizing surface measured from the

body center line, in.

span of upper vertical stabilizing surface measured from the

body center line, in.

wing semispan measured from the body center line, in.

exposed lower vertical stabilizing surface area, sq in.

exposed upper vertical stabilizing surface area, sq. in.

total wing area, sq. in.

lower vertical stabilizing surface

4

V

W

Y

CL

A

k

ALE

upper vertical stabilizing surface

wing

side force, ib

angle of attack, radians

angle of sideslip, radians

incremental coefficient due to the addition of a panel

taper ratio

sweep angle of leading edge of a panel, deg

THEORETICAL CONSIDERATIONS

The method presented here to estimate incremental values of the

side-force coefficient derivative, 2_Cy_, is similar to that which is

used to predict the lift characteristics of wing-body combinations in

reference 3. Briefly, as stated in reference i, the principle underlying

the method is that the ratio of the change in side force due to the

addition of a vertical stabilizing panel to the side force developed by

the panel alone is the same as the ratio of the change in apparent mass

of the cross section at the base of the empennage when the panel is added

to the apparent mass of the panel alone. (The panel alone is taken as

the exposed panel mounted on a reflection plane. ) This principle is true

only when the crossflow is two-dimensional, implying the configuration is

"slender," and when the trailing edges of all the panels in the empennage

lie in the base plane of the configuration (sketch (a)).

Sketch (a)

From the principle previously stated, the problem of estimating

_Cy_ resolves itself into the determination of two quantities, theapparent mass ratio, K, and the side-force curve slope of the added panel,(Cy_)p; that is,

ACy_ = K(Cy_)p (1)

The value of (Cy_)p6 can be determined theoretically using, for example,reference 4, 5, or . For the greatest accuracy, an experimental valueshould be used if it is available.

To determine the apparent mass ratio, the work of Bryson, reference 7,

has been used. In this work, a solution of the symmetric inertia coeffi-

cient tensor of the cross section of a body for the six apparent mass and

apparent moment of inertia coefficients is presented. The only apparent

mass that is necessary in determining Z_Cy_ is that which is solely

dependent on the potential due to a unit velocity in the lateral direction

in the base plane of the model. In reference 7 this is referred to as

m22. Since K is the ratio of the change in apparent mass of the crosssection at the base of the empennage when the panel is added to the

apparent mass of the added panel alone, three apparent masses must be

determined. These are: the apparent mass of the cross section of the

complete empennage, the apparent mass of the cross section of the empen-

nage without the panel, and the apparent mass of the exposed portion of

the added panel mounted on an infinite reflection plane. The desired

ratio, K, is then obtained by taking the difference of the first two of

these three apparent masses and dividing by the third. Sketch (b) shows

///// z///

Sketoh(b)

6

the cross sections whose apparent masses must be determined to obtain

the value for K that would be used in estimating the increment, ACyB,

occurring when the upper vertical stabilizing surface is added to the

combination of body, wing, and lower vertical stabilizing surface shown

in sketch (a).

K Charts

The methods of references 7, 8, and 9 have been used to compute

apparent mass ratios for various families of configurations and are

presented as design charts in figures i through 13. The charts presented

in figures i through 12 show apparent mass ratio as a function of a/s

of the added vertical panel for various values of a/s of the existing

vertical panel. Each chart is for a specific body cross-sectional shape,

a/b ratio, and a specific body horizontal semiaxis - horizontal surface

semispan ratio, b/s H or b/s W (i.e., the horizontal surface can be either

the horizontal tail or the wing). A sketch of the typical cross section

involved is shown on each chart with the added panel indicated by a dashed

line. In utilizing these charts, it should be remembered that the added

panel can be either the upper or lower vertical stabilizing surface;

however, for convenience the sketches on the charts show only the upper

vertical surface as the added panel. Special mention should be made of

the K charts presented in figure 3. These charts are used in estimating

ACy_ due to adding an upper vertical stabilizing surface to a body with

a horizontal surface located in the high, tangent position. These charts

also apply in estimating ACy_ due to adding a lower vertical stabilizing

surface to a body with a horizontal surface located in the low, tangent

position. Presented in figure 4 are K charts which are utilized in the

same manner as figure 3, except that the horizontal surface is tangent to

the body on the side opposite that to which the panel is added. An index

of the K charts and the range of variables covered by each is presented

in table I.

Estimation of Side-Force-Coefficient

Derivative Increment aCy_

Before presenting methods for estimating ACy_, let us again look at

the conditions of the method that were mentioned previously. They are

that the crossflow be two-dimensional, that the configuration be "slender,"

and that the trailing edges of all the panels in the empennage be in the

base plane of the model. A method that only applies to configurations

which meet these conditions is quite limited in application. Therefore,

to present a general method, it is necessary to determine some manner

of handling cases which violate any or all of the conditions. In general,

it is found, as will be shownlater, that good accuracy can be obtainedfor nonslender configurations where the crossflow is not strictly two-dimensional. It is also found that at subsonic speeds good accuracycan be obtained by determining the apparent mass ratio on the basis ofthe spans of the panels in the empennageregardless of longitudinallocation.

In supersonic flow the componentsof a configuration give rise tonumerousleading- and trailing-edge shock waves along its length. _q'_echaracteristics of the longitudinal flow vary discontinuously in passingthrough these waves, thus creating a series of zones, each being charac-terized by a different apparent mass ratio. The supersonic interferenceproblem thus requires a three-dimensional consideration. This three-dimensional effect is taken into account by determining the meanvalueof the apparent mass ratios from the various interference zones involved.This meanvalue, or the effective apparent massratio, K', gives goodaccuracy in determining the incremental directional coefficients.

Twomethods for estimating ACyB will now be presented in detail.The first is for subsonic speeds and the second is for _upersonic speeds.

Subsonic speeds.- The method of determining (CyB) P will not be

discussed in detail except to repeat that for the greatest accuracy the

experimental value should be used. If this is not available, methods

exist for detenmining this theoretically at subsonic speeds (e.g.,

refs. 4 and 5). To determine the apparent mass ratio, K, it is necessaryto determine the ratios of body sermiaxis to panel span for all the panels

present in the empennage. For elliptical bodies_ the body vertical semi-

axis is used in conjunction _ith vertical stabilizing surface span, and

the body horizontal semiaxis in conjunction with the horizontal surface

semispan. If these semiaxes change length in the region of the panels

in the empennage, the average lengths should be used. The ratio of the

body vertical semiaxis to the body horizontal semiaxis, a/b, and the

horizontal surface vertical position with respect to the body axis are

also required. The value of K is determined by means of K charts

appropriate to these parameters (or interpolation between charts if

necessary). Then Z_Cy_ is the product of these two quantities, (Cy_)p

and K. Appendix A presents a numerical example for a subsonic Mach

number.

Supersonic speeds.- To clarify the method for estimating Z_Cy_ at

supersonic speeds, let us consider the empennage shown in sketch (c).

//

//

/ / //

//

//

/

Sketch (c)

It is required to determine ACy_ due to adding the upper vertical

stabilizing surface to the combination of body, wing, horizontal tail,

and lower vertical stabilizing surface. To determine the effective

apparent mass ratio, K', it is first necessary to draw in the Mach lines

and trailing-edge shock waves emanating from the panels at zero angle

of attack, as shown in sketch (c). Any point lying forward of the Mach

line drawn from the leading-edge body juncture of a panel does not feel

the presence of the panel, and it is assumed that any point lying behind

the shock wave drawn from the trailing-edge body juncture of a panel does

not feel the presence of the panel. With this in mind, it can be seen

from sketch (c) that the upper vertical stabilizing surface is divided

into three areas. Area S l senses only the presence of the body; area

$2, the body and lower vertical stabilizing surface; and area $3, thebody, horizontal tail, and lower vertical stabilizing surface. Since

area S I senses only the body, the apparent mass ratio due to adding

the upper vertical stabilizing surface to the body, Kv(_ is determined--X_/ ,

from the K charts. In a similar manner, the apparen_ mass ratios due

to adding the upper vertical stabilizing surface to the combinations of

body and panels, S2 and $3, are also determined. The question arises asto whether the maximum spans of the areas affected by the different

combinations or the total span should be used in determining these K's.

For the sake of uniformity and simplicity in the calculations, the total

spans are used in all cases. As will be shown later, this simplified

procedure is justified on the basis of the degree of correlation which

was obtained between the estimated and experimental values of _CYB. Itis assumed further that the contribution of each of these three apparent

mass ratios, KV(B), KV(BU), and KV(BHU), to K' is proportional to thepercent of the total upper vertical stabilizing surface area, SV, whichsenses each combination of body and panels. For the example showninsketch (e), the expression for K' is

SI S2 S3= -- + KV(BU) -- + KV(BHU) (2)K' Kv(B)Sv sv

This value of K' and the side-force curve slope of the added panel

(Cy_)p are used to determine ACy_ from equation (i) with K replaced

by K'. In using this method for determining K', one can see from

sketch (c) that for a sufficiently high Mach number it is possible for

the wing, even though located forward of the upper vertical stabilizing

surfacej to affect this panel. This would add an additional term, the

apparent mass ratio due to adding the upper vertical stabilizing surface

to the body-wing combination, KV(BW), to the expression for K'

Appendix B presents a numerical example of the estimation of ACy_ fora supersonic Mach number.

Estimation of the Incremental Directional Coefficient £Cn_

The method of estimating ACn_ is simply that of multiplying the

estimated value of ACym by the moment arm that equals the distance

between the center of moments of the configuration and the center of

pressure of the added vertical panel. Methods for estimating this center

of pressure are given in reference 3 for plan forms having straight or

sweptforward trailing edges. Additional considerations are necessary

for configurations having sweptback trailing edges. In the present calcu-

lations the center of pressure was considered to be at the intersection

of the mean aerodynamic chord and the quarter chord for subsonic speeds

and at the centroid of area for supersonic speeds. The numerical examples

presented in appendixes A and B show the determination of ACnm for thesetwo cases.

Estimation of Directional Coefficient Derivatives

of the Complete Empennage

A necessary extension of the method presented to estimate increments

due to the addition of one panel to an empennage is to determine the total

empennage side-force at zero angle of attack and small angles of sideslip

which results from the addition of all the panels, both horizontal and

vertical, to the body. The simplest method of doing this is to total the

i0

increments gained by successive additions of the panels. For example,it is desired to find the side force gained by the addition of ahorizontal tail, upper vertical stabilizing surface, and lower verticalstabilizing surface to a body. The first step is to find the incrementgained by the addition of the horizontal tail to the body. Next, theincrement gained by the addition of the upper vertical stabilizingsurface to the combination of body and horizontal tail is found. Finally,the increment gained by the addition of the lower vertical stabilizingsurface to the combination of body, horizontal tail, and upper verticalstabilizing surface is determined. This order of configuration build-upis usually used. An exception to this is the case at supersonic speedswhere the vertical panels are staggered along the body so that theirleading edges intersect the body at quite different locations. In thiscase, the panel whoseleading edge intersects the body the farthest aftis added before the other panel, in order to obtain the most realisticestimate of the side force. If the panel lying the farthest forward isadded first, the interference effects will be overestimated by the methodpresented here.

Discussion in the previous sections has been limited to determiningincrements due to the addition of a vertical panel. The method of esti-mating the increment due to the addition of a horizontal surface requiresa somewhatdifferent approach. The addition of a horizontal surface,either a horizontal stabilizer or a wing, in a low or high tangentlocation to a body, defines a ratio KH(B) or KW(B) which is defined as

- YB (3a)KH(B) = YB

or

KW(B) = YB

In this apparent mass ratio_ the side force developed by the body alone

has been used to normalize the change in side force due to adding the

horizontal surface to the body. This quantity, rather than the side

force developed by the added surface which was used for the case of an

added vertical panel, has been used, since a horizontal surface by itself

develops no side force and would result in an infinite apparent mass

ratio and an indeterminate solution for side force.

To estimate the increment gained by the addition of a horizontal

surface, figure 13 is used to obtain the apparent mass ratio KH(B) or

KW_B). This is then multiplied by the side-force curve slope of thebody, as determined by experiment or slender-body theory. In coefficient

form this increment is, therefore,

ii

= B (4a)

or

Acy = (4b)

depending on whether it is the horizontal stabilizer or the wing whichis added.

At subsonic speeds this increment is used as presented. However,

at supersonic speeds it is necessary to use the Mach lines and shock

waves from the horizontal surface to determine what portion of this

incremental gain is realized. It is assumed that this gain through inter-

ference is felt over the area on the body defined by the Mach lines and

shock waves from the leading- and trailing-edge body junctures. Sketch (d)

shows two possibilities and the region of interference for each. The

,

Regions of

interference

_Sketch (d)

entire incremental gain is realized in the case of the top configuration.

For the case presented in the lower part of sketch (d), the portion of

the incremental gain that is realized is proportional to the ratio of the

area of the actual region of interference to the area for a body extended

indefinitely.

To determine the yawing-moment coefficient derivative of the complete

empennage, the appropriate moment arms are applied to each of the incre-

mental side-force quantities. The centroid of area of the region of

interference approximates the moment arm in the case of the increment

gained by adding the horizontal surface. For the other increments the

methods presented previously are used.

12

Appendix C presents a numerical example showing the calculation ofthe side-force and yawing-momentcoefficient derivatives of the completeempennagefor a supersonic Machnumber.

RESULTSANDDISCUSSION

ComparisonBetweenEstimated and ExperimentalValues of the Stability Derivatives

To evaluate the methods presented in this report for estimating thedirectional coefficient derivative increment gained by the addition of avertical stabilizing panel to an empennage,a series of calculations havebeen madefor configurations for which experimental data exist. Thegeometric and aerodynamic characteristics of these configurations arepresented in table II for the subsonic cases, and in table III for thesupersonic cases. The correlation between the estimated and the experi-mental values of ACy_ is shownin figure 14. A 45° line denotingperfect agreement is shown, as are ±lO-percent deviation lines. Atsubsonic Machnumbers (fig. 14(a)), the use of K to estimate ACy_generally results in accuracy that is within ±i0 percent of experiment.The samedegree of correlation is also obtained when K' is used toestimate ACy_ for the supersonic Machnumbercases (fig. 14(b)). Thescatter in both correlations appears to be randomabout the line ofperfect agreement.

The question arises as to whether sufficient accuracy could beobtained if K, at subsonic speeds, or K', at supersonic speeds, wereignored and the side-force curve slope of the added panel were used as anestimation of _Cy_. If the cases presented in tables II and III areexamined, it can be seen that there are somewhere the value of K orK' is close to 1.00. If this quantity were ignored in these cases littleeffect on the correlation would result. However, there are a numberofcases where K or K' is significantly larger than 1.00, so that ignoringthis quantity in these cases would result in a considerable underestimationof ACy_. Figure 15 is presented to show graphically the improvement inthe over-all correlation through the use of K or K'. The open symbolsshow the comparison between estimated and experimental values when K orK' is ignored. Generally, ACys is underestimated. When K or K' isused to account for interference effects, the correlation shownby thesolid symbols results.

The method presented for estimating the stability derivative incre-ments at supersonic speeds requires the determination of an effectiveapparent mass ratio, K', to account for the three-dimensional nature ofthe flow. To justify the use of this effective apparent massratio,rather than the apparent mass ratio applying at subsonic speeds, let us

13

look at table III and figure 16. In table III the values of both K and

K' are given, along with the stability derivatives determined by each.

In a number of the cases K and K' are equal. This is true for cases

where a vertical stabilizing surface is added to a body alone or to a

body-wing combination where the wing is located far enough for_ard that

its presence is not sensed by the added panel. The cases where K and

K' are not equal are shown in figure 16. The open symbols show the

correlation when K is used, and the solid symbols, the correlation when

K' is used. As can be seen from this figure, there is a marked improve-

ment in the correlation when K' is used in estimating Z_Cy_ atsupersonic speeds.

Figure 17 presents the correlation between the estimated and experi-

mental values of _Cn_ for subsonic speeds (fig. 17(a)) and supersonic

speeds (fig. 17(b)). =For both speed ranges the method presented previ-

ously, that of applying a moment arm to the estimated value of ACy^,

generally estimated _Cn_ to v_thin ±20 percent of experiment. At D

supersonic speeds there are some cases which are off by more than this

amount. Since _CYB includes, in addition to the side force developed

on the added panel,- the side force developed on the body and other panels

in the empennage through interference with the added panel, the use of

the center of pressure of the added panel does not result in a realistic

moment arm. The method of this report for estimating ACy_ does not permit

a breakdown of this increment into the portions developed on each of the

surfaces present in the empennage. If this could be done, and the appro-

priate moment arm applied to each of these portions, a better estimate of

_CnB should be obtained.

Effect of Stabilizing Surface Arrangement on the Side-Force

Coefficient Derivative of the Complete Empennage

A useful application of the method for estimating the side-force

coefficient derivative of the complete empennage at zero angle of attack

is in comparing the effectiveness of various stabilizing surface arrange-

ments on minimizing the effect of Mach number on Cy_. Two families of

configurations have been investigated and the results are presented in

figures 18 and 19. The two configurations chosen have the same total

wing area and the same total exposed vertical stabilizing surface area.

The example of figure 18 has an original upper vertical stabilizing

surface of aspect ratio 4.5, while that of figure 19, an aspect ratio

of 2.0. The necessary dimensions for making the calculations are presented

in fibres 18(_) and 19(a).

Let us first consider the effect of shifting a part of the original

upper vertical stabilizing surface area to a lower vertical stabilizing

surface with the same exposed root chord. Figures 18(b) and 19(b) show

14

the results of shifting approximately 29 percent of this area. The shift

of area of the configuration of figure 18(a) results in a gain in side

force throughout the Mach number range considered (fig. 18(b)). At a

Mach number of 4.0, even though the aspect ratios of the panels of the

Su/S V = 0.4 configuration are less than that of the panel of the

Su/S V = 0.0 configuration, the leading edges remain supersonic and the

lift per unit area is the same so that the gain in side force is broughtabout by an increase in interference. The area that is shifted from the

original upper vertical stabilizing surface is moved from a region of low

sidewash velocity to a region of higher sidewash velocity which increases

the effectiveness of this area as a lifting surface. At a Mach number of

1.5, the percentage gain over the Su/S V = 0.0 configuration is not nearly

as large as it is at a Mach number of 4.0. This lower percentage increase

can be explained by comparing the average lift per unit area of the ver-

tical panels of the two configurations. At this Mach number the leading

edge of the panel on the Su/S V = 0.0 configuration is supersonic, while

reducing the aspect ratio by shifting the area causes the leading edges

of the two panels on the Su/S V = 0.4 configuration to be subsonic.

This results in a lower lift per unit area for the Su/S V = 0.4 case

which partially offsets the gain in sidewash interference obtained byshifting the area.

The result of shifting the same amount of area of the original upper

vertical stabilizing surface of the configuration shown in figure 19(a)

to a lower vertical stabilizing surface is shown in figure 19(b). A gain

is realized at the higher Mach numbers, while there is a loss at the

lower Mach numbers. As was the case with the aspect ratio 4.5 configu-

ration, there is an increase in interference throughout the Mach number

range because of the shifting of the area to a region of higher sidewash

velocity. At a Mach number of 4.0, this gain is partially lost because

of a lower average lift per unit stabilizer area. The lower vertical

stabilizing surface of the Su/S V = 0.4 configuration has a subsonic

leading edge, while the upper vertical stabilizing surfaces of the two

configurations have supersonic leading edges. At a Mach number of 1.5,

the vertical panels of both configurations have subsonic leading edges,

but the average lift per unit stabilizer area of the Su/S V = 0.4 caseis so much lower than that of the Su/S V = 0.0 case that the gain in

interference by shifting the area is more than offset and the net result

is a decrease in the side-force coefficient derivative of the empennage.

If figures 18(b) and 19(b) are now compared, it can be seen that

the SU/S V = 0.4 configuration of figure 19(b) is desirable for several

reasons in the design of aircraft for a Mach number of 4.0. Among these

are that the value of this coefficient changes the least with Mach number,

the wave drag can be reduced, and with the smaller spans involved, lighterstructures can be used.

15

Figures 18(c) and 19(c) show the effect of horizontal-surface

position on the side-force coefficient derivative of the complete empen-

nage. The Su/S V = 0.4 configurations from figures 18(b) and 19(b) have

been used as the basic configurations, since their characteristics are

more desirable than those of the Su/S V = 0.0 cases (i.e., the percent

decrease with increasing Mach number is less). Since the side-force

curve slopes of the vertical panels alone do not change with the addition

of a horizontal surface, the changes in side force result from theinterference effects.

Let us first consider the case of adding a horizontal surface in the

mid-position. Figures 18(c) and 19(c) both show the same result when this

is done, a slight gain in side force throughout the Mach number range.

This is brought about by small changes in sidewash over the two vertical

panels. When the horizontal surface is added, a small amount of sidewash

is diverted from the lower vertical tail to the upper vertical tail.

Since the upper panel is a more effective lifting surface than the lower

panel, a gain in side force results. When the horizontal surface is added

in the high tangent position, a gain in side force throughout the Mach

number range considered is again obtained. %::is gain is the net result

of a gain in the side force developed by the body, a decrease in that

developed by the addition of the upper vertical panel, and a_: increase

in that developed by the addition of the lower vertical panel. Favorable

wing-body interference, taken into account by KW(B), causes the increaseon the body. The change in the side force developed by the addition of

the panels is caused by the horizontal surface diverting sidewash from

the upper panel to the lower panel. Also, the effect of Mach n_uuber is

less on this configuration since a larger portion of the side force is

developed on the body and the low-aspect-ratio lower vertical stabilizing

surface. The side-force curve slopes of these surfaces are affected less

by Mach n_::ber than that of the upper vertical stabilizing surface.

When the horizontal surface is located in the low tangent position,

there is also a gain in side force throughout the Mach number range

considered in figures 18(c) and 19(c). The side force developed by the

body is not affected by shifting the horizontal surface from the hi_: to

the low tangent position. If the sidewash over the empennage of this

configuration is compared to that over the configuration :cithout the

horizontal surface, it is found that when the surface is added, sidewash

is diverted from the lower vertical stabilizing surface to the upper

vertical stabilizing surface. This causes an increase in the side force

developed by the upper panel and a decrease in that developed by the lower

panel. Since the side-force curve slope of the upper panel is the most

sensitive to Mach nm'_:ber changes_ the variation in side force ,cith Mach

number is the greatest for this case, as the greater portion of the total

side force is developed by this panel.

16

The yawing-moment coefficient derivative curves of the complete

empennages for the configurations of figures 18 and 19 would generally

show the same qualitative effects of shifting vertical stabilizing

surface area and adding a horizontal surface as the side-force curves

show. The percentage decrease in Cn_ with increasing Mach number wouldbe approximately the same since the mSment arms used to convert side force

to yawing moment are affected very little by Mach number. The high or low

tangent horizontal surface cases would not show the gain over the hori-

zontal surface off or mid horizontal surface cases that is shown in side

force. The region of wing-body interference is generally disposed in

longitudinal location so that its centroid of area is relatively close

to the center of moments of the configuration. The side-force increment

gained by the addition of the wing to the body would contribute little

to the total moment. The high horizontal surface configuration gives the

yawing-moment curve which is the least sensitive to Mach number changes.

CONCLUSIONS

The methods presented for estimating the directional stability

derivative increments gained by the addition of vertical stabilizing

surfaces to airplane and missile configurations are utilized in a series

of calculations for configurations for which experimental data exist.

Comparisons are presented, between the calculated and the experimental

increments, which indicate the following:

i. The methods presented generally predict the side-force coeffi-

cient derivative increment, _Cy_, at zero angle of attack and small

angles of sideslip, to within ±i0 percent of that of experiment.

2. The yawing-moment coefficient derivative increment_ ACn_ _generally can be estimated to within ±20 percent of that of experiment

by the application of a moment arm determined by the center of pressure

of the added panel to the estimated value of ACy_.

An example application to one of the problems in directional stabil-

ity, that of minimizing the effect of Mach number on the side-force

coefficient derivative of the complete empennage_ demonstrates that the

method is useful for preliminary design studies.

Ames Research Center

National Aeronautics and Space Administration

Moffett Field, Calif., Sept. 12, 1958

17

APPENDIX A

A NUMERICAL EXAMPLE OF THE DETERMINATION OF ACy_ AND ACn_

FOR A CONFIGURATION OPERATING AT A SUBSONIC MACH NUMBER

As a numerical example of this type, let us consider configuration

number 3 of table II and determine _Cy_ and _Cn_ due to adding the

upper vertical stabilizing surface to the combination of body 3 wing_ and

horizontal tail. The Mach number considered here is 0.06. Table II

presents the dimensions necessary to make the calculations.

The first step is to determine the side-force curve slope of the

added panel. The exposed upper vertical stabilizing surface is approxi-

mated by one which is trapezoidal in plan form. From reference 4, (CY_)vis found to be

(CY@)v = -0.522 per rad2an

referenced to the total _ng area, SW.

Next, the apparent mass ratio must be found. The empennage in this

example consists of the body, the horizontal tail, and the upper vertical

stabilizing surface. The apparent mass ratio to be found is that due

to adding the upper vertical stabilizer to the combination of body and

horizontal tail_ KV(BH ). The sketch of the configuration shown in

table II shows that the %ody is nearly square in cross section. Since

apparent mass solutions for bodies of other than circular or elliptic

cross section are not available_ the ass_nption is made that the ratio

of height to width_ a/b_ is the important parameter and that the details

of the body contour are of secondary importance. This assumption was

used for the cases of tables ii and III _ere the body is other than

circular or elliptic. As can be seen from these tables, the resulting

correlation with experiment is as good as for the cases to _ich the

K charts apply directly.

From table II the necessary ratios for this configuration are:

a/b = 1.000, (b/S)H = 0.275 , and (a/s)v = 0.206. The horizontal tail is

located in the mid-position_ h/a = 0.000. Since this example does not

have a lower vertical stabilizing surface, (a/s)u is equal to 1.000.

Table I indicates that for these values of the parameters an interpolation

must be made to obtain KV(BH ). From figure 2(b), (b/S)H = 0.200, KV(BH )

is found to be 1.35. Figure 2(c), (b/S)H = 0.400, gives a value of 1.24

and figure 2(d), (b/S)H = 0.600, a value of 1.17. Interpolating graph-

ically gives, for (b/S)H = 0.275 ,

KV(B_ ) = 1.30

18

For these values of (CY_)v and KV(BH), the increment of side force

gained by adding the upper vertical stabilizing surface is

_Cx_ : KV(BH)(Cy_) v

= -0.678 per radian

This estimate corresponds to an experimental value of -0.64 per radian.

Since the aaded upper vertical stabilizing surface has a sweptback

trailing edge, the intersection of the quarter chord and the mean aero-

dynamic chord is used in determining ACn_. This is found to lie at

_6.1 percent of the exposed root chord. Therefore, the moment arm, made

dimensionless by dividing by the total wing span, is

i : Zm - ZV - 0.561 cr

2s W 2s W

and

: -0.451

i (_y_)_Cn_ = 2s W

: 0.306 per radian

This estimate corresponds to an experimental value of 0.28 per radian.

19

APPENDIX B

A NUMERICALEXAMPLE OF THE DETERMINATION OF ACy_ AND Z_Cn_

FOR A CONFIGURATION OPERATING AT A SUPERSONIC MACH NUMBER

As a numerical example of this type 3 let us consider configuration

number 3 of table IIl and determine ACy_ and ACn_ due to adding the

upper vertical stabilizing surface to the combination of body, wing,

horizontal tail, and lower vertical stabilizing surface. The Mach number

for this example is 2.01. Table III presents the dimensions necessaryto make the calculations.

If the exposed upper vertical stabilizing surface is approximated

by one which is trapezoidal in plan fom, reference 6 can be use.& to

determine the side-force curve slope of the panel. This is found to be

with

(Cy_) v = -0.398 per radian

SW as the reference area.

The next step is to take the approximate planar view of this config-

uration, sketch (e), and draw in the shock wave from the wing trailing edge,

//

Total upper vertical stobilizino surface / / /area, Sv=19.20 inz // /

/ /i I// I /

f f

f If

fI

M®=2.01

Sketch (e)

2O

body juncture and the Machlines from the leading edge, body juncturesof the horizontal tail and the lower vertical stabilizing surface. Thishas been done in this sketch. Area SI senses the presence of the bodyand the _ing. For this area the apparent mass ratio due to the additionof the upper vertical stabilizing surface to the body-_ing combination,KV(BW), must be determined. Area S2 senses only the body so that forthis area KVIB) has to be determined. The contribution of each ofthese apparent massratios to the effective apparent massratio, K', isproportional to the percent of the total upper vertical stabilizingsurface area which senses each combination. Therefore, the expressionto be evaluated in order to determine K' is

$I S2- __ + KV(B)K' Kv(Bw)Sv

To determine KV(BW), the following ratios have to be determined:

a/b, (b/s)w , and (a/S_v. From table III a/b = 1.170, (b/s) W = 0.157,

and (a/s) V --0.247. The lower vertical stabilizing surface is not felt

in this region, so that (a/S)u = 1.000. Since the wing is located in

the mid-position, table I shows that a double interpolation will have to

be made in order to obtain KV(BW ) for a/b = 1.170 and (b/s)w = 0.157.

Figures 8(a), 8(b), and 8(c) are used to determine KV(BW ) for a/b = 0.667

and (b/s)w = 0.157. A graphical interpolation between these three figures

gives KV(BW ) = 1.48. For a/b = 1.O00 figures 2(a), 2(b), and 2(c) are

used and KV(BW ) for (b/s)w : 0.157 is 1.49. Figures 10(a), 10(b), and

10(c) are used to determine KV(BW ) for a/b = 1.500. This is found to

be 1.45. Interpolating between these three values determines KV(BW )

for a/b : 1.170, which is

KV(BW)= 1.49

To determin_ KV(B) it is necessary to interpolate between figures i,

7, and 9, since the region we are considering here does not sense the

wing and, therefore, (b/S)w = 1.000. This interpolation gives

KV(B)--1.27

These apparent mass ratios, KV(BW ) and KV(B) , and the areas shown

in sketch (e), determine K'.

1.49(4.63) 1.27(14.57)K' = +

19.20 19.20

= 1.32

21

Cy_ and K' are now known_we findSince ( )V

ACy_= K' (Cy_)V = -0.525 per radian

This estimate corresponds to an experimental value of -0.52 per radian.

Since the upper vertical stabilizing surface has a sweptback trail-ing edge and we are considering a supersonic Machnumber, the centroidof area of this panel is used in estimating ACn_. This is found to lieat 93.3 percent of the exposed root chord. Therefore, the momentarmmadedimensionless by dividing by the total wing span is

Im - IV - 0.933 Cr

2s W 2sw

= -0.575

and

_Cns _ _ (Z_Cy_) = 0.302 per radian2s W

This estimate corresponds to an experimental value of 0.30 per radian.

As another example in determining K' _ consider the case shown in

sketch (f). This is the same configuration as was just considered;

/ / /

Totalupper vertical stobilizingsurfo //j_//cearea, Sv= 19.20 in= _/

/ ////{, ,Z , :'_'39i°t/'7. s,:281in /." II

//

r

/ /

i / /I/

/

M==I.30

Sketch (f)

22

however, the Mach number is 1.30. If this sketch is compared with

sketch (e), it can be seen that the upper vertical stabilizing surface

now feels the presence of the horizontal tail and the lower vertical

stabilizing surface over part of its area, and the wing is not sensed

by any portion of the panel. The expression for the effective apparent

mass ratio, K', for this case is,

$I S2 $3

The apparent mass ratio KV(B) has the same value as before; that is,

KV(B) = 1.27

An interpolation between figures i, 7, and 9 is made to determine KV(BU ).

From table III the necessary ratios are: a/b = 1.170, (a/S)u = 0.684,

and (a/S)v = 0.247. A graphical interpolation gives

KV(BU) = 1.30

The values of the ratios necessary to determine KV(BHU) are:

a/b = 1.170, (b/S)H = 0.328, (a/s) U = 0.684, and (a/s)v = 0.247. Since

the horizontal tail is located below the body center line, h/a = -0.480,

and the body is elliptical, we have a cross section for which K charts

are not presented. To handle this case, either the body has to be

considered circular in cross section and an interpolation made for

horizontal-tail height, or the horizontal tail has to be assumed located

in the mid-position and an interpolation performed between a/b ratios.A comparison of figures 2(c), 8(c), and lO(c) shows that a small variation

from a body which is circular in cross section has little effect on the

apparent mass ratio in the region in which this case falls, (a/S)v = 0.247.Therefore, the interpolation for tail height will be made rather than

for body shape. First, let us determine KV(BHU) for a combination of a

body, high tangent horizontal tail (h/a = 1.O00), and upper and lower

vertical stabilizing surfaces. Figures 3(b), 3(c), and 3(d) are used

and KV(BHU) for (b/S)H : 0.328 is found to be 1.06. With the horizontal

tail located in the mid-position (h/a = 0.000), figures 2(b), 2(c), and

2(d) give an interpolated value of KV(BHU) = 1.41. Figures 4(b), 4(c),

and 4(d) give a value of 1.68 for a low tangent horizontal tail

(h/a = -1.O00). An interpolation between these three values of KV(BHU)

gives, for h/a = 0.480,

KV(BHU) : 1.55

23

The values determined for KV(B) , KV(BU), and KV(BHU) and the areas

from sketch (f) yield

= . 1.30(14.39) 1.55(2.00)K' I 27(2.81) + +19.20 19.20 19.20

= 1.3 2

24

APPENDIX C

A NUMERICAL EXAMPLE OF THE DETERMINATION OF THE SIDE-FORCE

AND YAWING-MOMENT DERIVATIVES OF THE COMPLETE EMPENNAGE

As a numerical example of this type, let us consider configuration

number 3 of table III and determine Cy_ and Cn_ due to adding the

horizontal tail, upper vertical stabilizing surface, and lower verticalstabilizing surface to the body-wing combination. The Mach number for

this case is 2.01 and the dimensions necessary for the calculations are

presented in table IIl.

The first surface to be added to the body is the horizontal tail.

The side-force curve slope of the body, as determined by slender-body

theory and referenced to the total wing area, SW_ is

= -0.147 per radian

From table Ill the ratios necessary to determine KH_B) area/b = 1.170, (b/s)H = 0.328, and h/a = -0.480. Since K dharts are not

presented for the case where a horizontal surface is added to an ellipti-

cal body, the body will be considered circular and an interpolation made

between the curves of figure 13 for horizontal surface height. From

figure 13 we find that for (b/S)H= 0.328, a surface mounted tangent to

the body in either a high or low position gives a value of K equal to1.06, and in the mid-position K is zero. In lieu of apparent mass

solutions for adding a horizontal surface at an intermediate point on

the body, it is assumed that small changes from a mid-position result in

little additional interference. Therefore, rather than interpolating

linearly a higher order interpolation is to be made. For the sake of

simplicity the equation of an ellipse was used.

25

K

1.0

Low

J.

0 I0

Highh5

Sketch (g)

From sketch (g) the equation for KH(B) when h/a : -0.480 is

--I

KH(B): K_w [_- ,/i - (h/_)_J:io6 [_-& - (o _8o2]

= o._3

26

_ °

Regions of interference

/

.I//

Mm=Z.OI

Sketch (h)

Sketch (h) shows that not all _f the side-force increment gained by the

addition of the horizontal surface to the body is realized, since a

portion of the area determined by the shock waves and the body, if it

were extended, lies behind the base of the actual body. Of the total

area 87.5 percent lies on the body so that this percentage of the incre-

ment estimated by KH(B) and (CY_)B is realized.

(CY_)BW H - (CYz)BW = 0.875 KH(B)(Cy_) B

= -0.017 per radian

The second increment to be detenmined is that due to adding the upper

vertical stabilizing surface to the combination o£ body, wing, and hori-

zontal tail. From reference 6 the side-force curve slope of this panel,

referenced to the total wing area, SW, is

(Cy_) V = -0.398 per radian

Sketch (i) shows that part of this panel, area Sl, senses only the

body-wing combination, while area $2 senses only the body_ so that the

K' when this panel isexpression for the effective apparent mass ratio, ,

added_ is

K'V(Sm) -- KV(BW) "%--Sv+ _:V(s) _vS_

27

ff

s463Total upper vertical stabilizing surface //f

area, Sv=i9.20 in_ ._/" .

._'/$2=14.57 in2/

Total tower vertical stabilizing surface _,_-'_ _ _ _ $4:0"50 inz

area, Su=3.24 in2 _ \$3=2.74 in2 -

M®=2.01

Sketch (i)

The use of the appropriate K charts for the dimensions given by

table III, results in,

K,V(BWH) = 1.49(4.63) + 1.27(14.57)19.20 19.20

= 1.32

and_ therefore_

(CY})BWHV - (CY@)BW H : K'V(BWH)(Cy_) V

= -0.529 per radian

The third increment which must be found to determine the total

empennage side force is that due to adding the lower vertical stabilizingsurface to the combination of body_ _ng_ horizontal tail_ and upper

vertical stabilizing surface. The slender-body value of the side-force

curve slope of this panel_ referenced to the total _ng area_ is deter-

mined and found to be

(Cy_) = -0.021 per radi_U

28

From sketch (i) it can be seen that area S3 of the lower vertical

stabilizing surface senses only the body_ and area $4, the body and

horizontal tail. The expression for the effective apparent mass ratio

is, therefore

Ss $4

The use of the appropriate K charts gives

2.78(2.74) 2.18(0.50)= @

K'U(BW_V) 3.24 3.24

= 2.68

and, therefore,

(CY_)BWHV U - (CY_)BWHV : K'U(BWHv)(CY_)u

= -0.056 per radian

The total of these three increments of side force due to adding the three

surfaces gives us the side-force coefficient derivative of the complete

empennage

(Cy_)Bw_vU - (cy_)Bw= -0.598 per radian

This estimate corresponds to an experimental value of -0.57 per radian.

To determine the yawing-moment coefficient derivative of the complete

empennage, the appropriate moment arm is applied to each of the three

incremental values of side force which make up the total. For the incre-

ment gained by adding the horizontal tail to the body, the centroid of

area of the region of interference on the body is used. _is is found

to lie at 81.0 percent of the exposed root chord of the horizontal tail

and_ therefore,

= Zm - ZH - 0.810 Cr

2s W 2s W

= -0. 600

£9

and

(Cns)sW= (cY )Bw]

= 0.010 per radian

'I_e centroid of area of the upper vertical stabilizing surface lies

at 93.0 percent of the exposed root chord, so that

= -0.5752s W

and

(Cnf)BWHV - (Cnf)BW H : 0.304 per radian

._he centroid of area of the lower vertical stabilizing surface lies

at 43.7 percent of the exposed root chord, so that

= -0.4392s W

and

(Cn_)B_gH_j - (Cn_)BWHV = 0.025 per radian

The addition of these three increments of yawing moment gives the

yawing-moment coefficient derivative of the complete empennage gained by

the addition of the horizontal tail, upper vertical stabilizing surface,

and lower vertical stabilizing surface to the body-_ing combination.

(Cn_)BWHV U - (Cn_)BW = 0.339 per radian

This estimate corresponds to an experimental value of 0.37 per radian.

3o

REFERENCES

lo Nielsen, Jack N., and Kaattari, George E.: The Effect of Vortex and

Shock-Expansion Fields on Pitch and Yaw Stabilities of Supersonic

Airplanes. IAS preprint 743, June 1957.

2. Kaattari, George E.: Estimation of Directional Stability Derivatives

at Moderate Angles and Supersonic Speeds. NASA MEMO 12-I-58A, 1958.

.

e

e

.

o

,

o

i0.

Pitts, William C., Nielsen, Jack N., and Kaattari, George E.: Lift

and Center of Pressure of Wing-Body-Tail Combinations at Subsonic,

Transonic, and Supersonic Speeds. NACA Rep. 1307, 1957.

DeYoung, John, and Harper, Charles W.: Theoretical Symmetric Span

Loading at Subsonic Speeds for Wings Having Arbitrary Plan Fom.

NACA Rep. 921, 1948.

ii.

Lawrence_ H. R.: The Lift Distribution on Low Aspect Ratio Wings

at Subsonic Speeds. Jour. Aero. Sci., vol. 18, no. i0, Oct. 1951,

pp. 683-695. (Also available as IAS preprint 313, Feb. 1951, and

Rep. AF-673-A-I, Cornell Aero. Lab., Inc., Aug. 1950)

Lapin, Ellis: Charts for the Computation of Lift and Drag of Finite

Wings at Supersonic Speeds. Rep. SM-13480, Douglas Aircraft Co.,

1949.

Bryson, Arthur E., Jr.: Evaluation of the Inertia Coefficients of

the Cross Section of a Slender Body. Jour. Aero. Sci., vol. 21,

no. 6, June 1994, pp. 424-427.

Bryson, Arthur E., Jr.: Stability Derivatives for a Slender Missile

with Application to a Wing-Body-Vertical Tail Configuration. Jour.

Aero. Sci., vol. 20, no. 5, May 1953, PP. 297-308.

Bryson, Arthur E., Jr.: The Aerodynamic Forces on a Slender Low

(or High) Wing, Circular Body, Vertical Tail Configuration. Jour.

Aero. Sci., vol. 21, no. 8, Aug. 1954, pp. 574-575.

Beam, Benjamin H., Reed, Verlin D., and Lopez, Armando E. : Wind-

Tunnel Measurements at Subsonic Speeds of the Static and Dynamic-

Rotary Stability Derivatives of a Triangular-Wing Airplane Model

Having a Triangular Vertical Tail. NACA RM A55A28, 1955.

Buell, Donald A., and Tinling, Bruce E.: The Subsonic Lateral and

Longitudinal Static Stability Characteristics Up to Large Angles

of Sideslip for a Triangular-Wing Airplane Model Having a Ventral

Fin. NACA RMA56HO6, 1956.

31

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

Few, Albert G.j Jr.: Investigation at High Subsonic Speeds of the

Static Lateral and Directional Stability and Tail-Loads Character-

istics of a Model Having a Highly Tapered Swept Wing of Aspect

Ratio 3 and Two Horizontal-Tail Positions. NACA RM L56E29, 1956.

Paulson_ John W._ and Boisseau_ Peter C.: Low-Speed Investigationof the Effect of Small Canard Surfaces on the Directional Stability

of a Sweptback-Wing Fighter-Airplane Model. NACA RM L56FI9a_ 1956.

Sleeman_ William C., Jr.: An Experimental Study at High Subsonic

Speeds of Several Tail Configurations on a Model Having a 45°

Sweptback Wing. NACARM L57C08, 1957.

Moseley, William C._ Jr.: Static Stability Characteristics of aCambered-Delta-Wing Model at High Subsonic Speeds. NACA RM L56HI3_

1956.

Wiggins, James W., Kuhn, Richard E. and Fournier, Paul G.: Wind-

Tunnel Investigation to Determine the Horizontal- and Vertical-

Tail Contributions to the Static Lateral Stability Characteristics

of a Complete-Model Cwept-Wing Configuration at High Subsonic

Speeds. NACA TN 3818, 1956. (Supersedes NACA RM L53EI9)

Goodman, Alex, and Thomas_ David F._ Jr.: Effects of Wing Position

and Fuselage Size on the Low-Speed Static and Rolling Stability

Characteristics of a Delta-Wing Model. NACA Rep. 1224, 1955.

(Supersedes NACA TN 3063)

Letko_ William; a_d Williams_ James L.: Experimental Investigation

at Low Speed of Effects of Fuselage Cross Section on Static

Longitudinal and Lateral Stability Characteristics of Models

Having 0° and 45° Sweptback Surfaces. NACA TN 3551, 1955.

Savage; Howard F., and Tinling, Bruce E.: The Static Lateral and

Directional Subsonic Aerodynar_c Characteristics of an Airplane

Model Having a Triangular Wing of Aspect Ratio 3. NACA TN 4042,

1957. (Supersedes NACA RMA55BII)

Spea_aan, M. Leroy_ and Driver_ Cornelius: Longitudinal and Lateral

Stability Characteristics of a Low-Aspect-Ratio Unswept-Wing

ALrplane Model at Mach Numbers of 1.82 and 2.01. NACA RM L56H06,

1957.

Spearman_ M. Leroy: Static Lateral and Directional Stability and

Effective Sidewash Characteristics of a Model of a 35° Swept-Wing

Airplane at a Mach Number of 1.61. NACA RM L56E23, 1956.

32

22.

23.

24.

Spearman, M. Leroy, Robinson, Ross B., and Driver, Cornelius: The

Effects of the Addition of Small Fuselage-Mounted Fins on the

Static Directional Stability Characteristics of a Model of a 45 °

Swept-WingAirplane at Angles of Attack Up to 15.3 ° at a Mach

Number of 2.01. NACA RM L56DI6a, 1956.

Spearman, M. Leroy, and Robinson, Ross B.: Static Lateral Stability

and Control Characteristics of a Model of a 45 ° Swept-Wing Fighter

Airplane With Various Vertical Tails at Mach N_nbers of 1.41, 1.61,

and 2.01. NACA RM L56DOS, 1956.

Spearman, M. Leroy, and Robinson, Ross B.: Investigation of the

Aerodynamic Characteristics in Pitch and Sideslip of a 45 ° Swept-

Wing Airplane Configuration With Various Vertical Locations of

the Wing and Horizontal Tail. Static Lateral and Directional

Stability; Mach Numbers of 1.41 and 2.01. NACA RM L57J25a, 1957.

33

TABLE I.- SUMMARY OF THE K CHARTS AND THE RANGE OF VARIABLES

COVERED BY EACH

Horizontal (b/S)H (a/s) Added (a/s) ExistingFigurea_ surface or

no, position _/jtbls_wvertical panel vertical panel

i i.ooo2(a)

(b)(o) 1.ooo(a)

(b)(_) 1.ooo(a)

(b)(o) 1.ooo(a)(e

5 .333

6(a)

(b)(o)(d)(e).

7 .6678(a_(b)(c) ,667

(_)(e)

9

0.0

.2

Mid .4 0 to 1.0

.6

.8Tangent to _,O "

body on ,2

same side .4 0 to 1.0

as added ,6

panel ."8Tangent to ,0

body on ,2

side oppo- ,4 0 to 1,0

site added ,6

panel .8

.0

.2

.4 0 to 1.0

,6

.8

,0

,2

Mid ,4 0 to 1,0

.6

.8

.0

,2

,4 0 to 1,0

.6

.8--- 0 to 1,O

,0

,2

.4 0 to 1.0

.6

.8Mid or .0

to 1,0

OO

•333 Mid

lO(a)

(b)(o)(_)(e)

ll12(a)(b)(o)(_)(e)

13

i.500 ---

I. 500 Mid

3..ooO.. ---

3.000 Mid

1.000 tangent

position

0 to 1,O 0,1 to 1,0

.0 to 1.0

.i to 1.0

.I to 1.0

.i to _.0

.i to 1.O

.0 to 1.0

.i to 1.0

.i to 1.0

.i to 1.0

.i to 1.0

,0 to 1,O

.i to 1.0

,i to 1,0

,i to 1,0

,i to 1,0

.i to i.o0 to 1.0

.0 to 1.0

,i to 1,0

,i to 1,0

,i to 1,0

,1 to 1,0

0 to 1,0 .I to 1,0

,0 to 1,O

,1 to 1.0

,1 to 1,0

,1 to 1,0

,i to 1,0

0 to i.0 .i to 1.0

,0 to 1,0

.I to 1,0

,1 to 1,0

,i to 1,0

,1 to 1,O

.i t_ i.o,0 to 1.O

,i to 1,0

,i to 1.0

,i to 1,0

,i to 1,0

1,0

34

TABLE II.- SUMMARY OF THE GEOMETRIC AND AERODYNAMIC CHARACTERISTICS

OF THE CONFIGURATIONS STUDIED AT SUBSONIC SPEEDS

No •Configuration SW, 2Sw, _m_ a3 a/b Surface s, h, I_ Cr,

Sketch in. 2 in. in. in. in. in. in. in. (leg

-_ /_ W 16.66 0.00 17.78 25.89 0.030 60.0

-__D536.0 34.32 28.65 2.94 1.O00 V 10.66 --- 30.91 13.091 .028 60.0

432 - 2865115 35 I .0001 83.4

316.8 30.86 31.22 1.85 1.000 V 10.04 --- 42.45 10.94 .461 35.8

23.78 3.32 23.70 21.80 .28O 47.0

666.5 47.56 40.03 3.32 1.00o

36.0 12.00 .64 1.O00-"

lO. 42

6324.0 27.10 28.94 2.25 1.000

324.0 36.00 30.00 1.85 1.000 V

976.7 36.50 27.00 1.46 1.0001

576.7 36.50 27.00 z.95 1.O00

12.06 .CO 58.51 8.72 .204 _0. 5H

V 16.11 --- 52.35 16.30 .280 50.9

w 6.00 .00 7.71 4.17 .332 48.6

K 2.40 .00 16.00 1.80 1.000 .0

V 2.40 --- 16.00 1.80 1.0001 .0

U 2.40 --- 16.00 1.80 1.000: .0

w 13.55 .80 2_1.52 19.12 .000 6o.0

v 9.3o --- 42.51 3.2.21 .0oo 6o.o

10.19 --- 39.75 9.49 .61o 55.2

Vz 10.37 --- 44.3/ 8.50 .000 i 42.5

V3 13.12 --- 43.00 ii.00 .000 42.5

V2 3x.74 --- _3.79 9.40 .0OO _2.5

Vs 13.49 --- 43,00 11.00 .000 42.5

I

C

3P

TABLE II.- SUMMARY OF THE GEOMETRIC AND AERODYNAMIC CHARACTERISTICS

OF THE CONFIGURATIONS STUDIED AT SUBSONIC SPEEDS - Continued

Configuration

No. Panel added

BWV-_

_WV-_

1 BWV-_

_4VU-BWV

BWVU- BWV

Estimation

_/2s W (Cy_)p K using K Experiment Ref.

ACY B _n_ ACy B ACn_ no.

0.25 -0.276 -0.244 i.)8 -0.386 0.107 -0.42 O.ll i0

.60 -.279 -.257 1.58 -.406 .i13 -.46 .12 10

.80 -.281 -.276 1.58 -.436 .123 -.47 .13 i0

.25 -.156 -.011 3.04 -.033 .005 -.046 .004 ll

.80 -.156 -.Oll 3.04 -.033 .005 -.046 .007 ii

2 BV-B .80 -.508 -._85 1.07 -.626 .313 --55 .29 12

3 BWHV-BWH .06 -.451 -.522 1.30 -.678 .306 -.64 .28 13

BWV-BW

_4V-_

4

BWHVU-BWHV

_4HVU-BWHV

BWV-BW

5

BWV-BW

BV-B

B¥-B

6 BV-B

BV-B

BV-B

BV_-B

7

BV_-B

.60 -.5021 -.2241 1.34 -.300 .151 -.31 .15 14

.80 -.499 -.239 1.34 -.320 .160 -.34 .14 14

.60 -.502 -.224 1.74 -.390 .196 -.35 .17 14

.801 -.499 -.239 1.74 -.416 .208 -.38 .18 14

.60 -.751 -.344 1.26 -.433 .325 -.40 .30 15

.80 -.754 -.365 1.26 -.460! .347 -.46 .31 15

.40 a-.425 -.450 1.07 -.482 .205 -.44 .23 16

•50 a-.425 -.453 1.07 -.485: .206 -.45 .23 16

.60 a-.425 -.460 1.07 -.492 .209 -.46 .24 16

•70 a-.425 -.464 1.07 -.497 .211 -.47 .24 16

.80 a-.425 -.467 1.07 -._00 .213 -.48 .25 16

.17 -.6OO -.239 .94 -.225 -135 -.25 .14 17

•17 -.601 -.401 .85 -.341 .205 -.35 .21 17

BV2-B .17 -.598 -.293 1.01 -.296 .177 -.30 .18 17

8

BV_-B .17 -.601 -.401 -95 -.381 .229 -.37 .24 17

alntersccLicn of me_ aer_do-n_nic chord _ul quarter cl_crd cf added p_el used in

determining moment arm.

36

TABLE II.- SUMMARY OF THE GEOMETRIC AND AERODYNAMIC CHARACTERISTICS

OF THE CONFIGURATIONS STUDIED AT SUBSONIC SPEEDS - Continued

Configuration SW, 2SW, Ira, a, a/b SurfaceNo. Sketch in. 2 in. in. in.

i0

iI

12

13

14

15

16

576.7 36.50 27.00 2.93 1.000 V s

324.0 36.00 23.98 1.82 1.000

324.0 36.00 23.98 2.70 2.210

324.0 36.00 23.98 1.22 .451

324.0 36.00 25.$4 1.82 1.000

324.0 36.00 25.44 2.70 2.210

324.O 36.00 25.44 !1.22 .451

576.0 41.56 39.58 2.80 1.000

s, h, _" Cr' k ALE'in. in. in. in. deg

14.23 --- 43.00 ll.O0 0.000 42.5

8,o5 0.0o 39.94 4,581 ,66o 3,6

9.90 --- 39.26 5.69 .647 2.8

8.05 .co 39.50 4.73i .638 3.6

9.90 --- 39.32 9.43 .678 2.8

8.09 .co 39.59 4,36 .693 3.6

9.90 --- 39.2-2 5,84 .630 2.8

6.70 .CO 39.55 5.39 .673 47.9

V

V 8.25 --- 38.50 6.72 .658 47.9

H 6.70 .CO 38.89 9.61 .647 47.5

V 8.25 --- 39.46 6.41 .690 47.5

H 6.70 .CO 40.51 5.08 .714 47.5

V 8.25 --- 37.85 6.93 .638 47.5

H 11.20 .CO 64.24 7.05 .396 14.2

V s 13.28 --- 59.99 12.48 .195 54.0

V L 15.23 --- 97.75 14.72 .189 54.0

37

TABLE II.- SUMMARY OF THE GEOMETRIC Ah_D AERODYNAMIC CHARACTERISTICS

OF THE CONFIGURATIONS STUDIED AT SUBSONIC SPEEDS - Concluded

Estimation

Configuration Moo T/2s W (Cy_)p K using K Experiment Ref.

No. IPanel added ZS.Cy_ _n13 ZN2yl3 _Cnt 3 no.

9 BVa-B O.17 -0.601 -0.401 1.15 -0.461 0.277 -0.42 0.26 17

i0 BHV-BH .15 -.464! -.400 1.26 -.504 .234 -.53 .22 18

ll BHV-BH .13 -.464 -.331 1.53 -.506 .235 -.53 .21 18

3_2 BEV-BH .13 -.46_ -.448 1.11 -.497 .230 -.49 .20 18

13 BHV-BH .13 a-.464 -.267 1.35 -.361 .168 -.39[ .18 18

14 BHV-BH .13 a-.464 -.208 1.71 -.356 .165 -.40 .17 18

15 BHV-BH .13 a-.464 -.310 1.15 -.356 .165 -.39 .16 18

BVs-B

BVs-B

BV s-B

BVL-B

BVL-B

16 BVL-B

BHV s -BH

BHVs-BH

BHV_-BH

BHVL-BH

BHVL-BH

BHVL-BH

.25 a-.620 -.370 1.16 -.429 .266 -.4g .31 19

.60 a-.620 -.389 1.16 -.451 .280 -.4_ .32 19

.80 a-.620 -.415 1.16 -.482 .299 -.46 .34 19

.25 a-.599 -.51_ 1.07 -.550 .329 -.56 .39 19

.60 a-.599 -.541 1.07 -.579 -347 -.58 .41 19

.80 a-.599 -.577 1.07 -.617 .370 -.60 .44 19

.25 Ia-.620 -.370 1.34 -.496 .308 -.47 .32 19

.60 a-.620 -.389 1.34 -.522 .324 -.49 .33 19

.80 a-.620 -.415 1.34 -.556 .345 -.50 .37 19

.25 a-.599 -.514 1.25 -.642 .385 -.63 .41 19

.60 a-.599 -.541 1.25 -.676 .405 -.65 .44 19

.80 a-.599 -.577 1.25 -.721 .432 -.68 .48 19

38

TABLE III.- SUMMARY OF THE GEOMETRIC AND AERODYNAMIC CHARACTERISTICS

OF THE CONFIGURATIONS STUDIED AT SUPERSONIC SPEEDS

Confi&n/ration

No. Sketch

1

..&.

_[ 2SW, Zm, a, a/b Surface s, h, _' Cr' k ALE'2 in o in. in. in. in. in. in. deg

45.2 i 10.48 15.30 1.00 1.000

5.24 0.00 13.40 5.50 0.427 27.1

3.80 --- 20.67 4.50 -378; 44.0

160.4 25.32 19.11 1.78 1.000

W 12.66 .00 14.33 7.83 .538 38.1

H 6.06 .CO 27.80 6.46 .[9_i 38.5

---<>----

3 _ 114.5 19.08 18.00 1.75 1.170

272.2 32.41 24.91 '2.Oh .940

___--475 25.3 5.621 7.90 .63 1.000_

g6

6

,---.._3

-4-

25.3 5.62 7.90 .63 1.000

14.4 6.73 6.24 .54 1.000

See footnotes to table, p. 41.

V 7.48 --- 26.15 3.66 .192 50.4

W 9.94 .00 12.49 7.92 .4831 48.0

H 4.56 -.84 26.53 3.60 .517 48.0

V 7.08 --- 24,12 5.20 .392 49.2

U 2.56 --- 22.89 8.00 .000 70.2

W 16.21 -i,18 16.51 13,20 ,313 49,7

H 7.87 -1.37 33.62 5.89 .402! 49'.6

V o 7.20 --- 32.80 6.69 .416 23.5

Vex t 8.66 --- 32.80 6.69 .260 23.5

V_7 % 8.74 --- 32.37 7.20 .290 23.5

W 2.81 .00 .00:11.25 .000 76.0

W

H

V

aU

3.12 --- 10.36 2.10 .679 32.4

2.81 .00 .00 11.25 .CO0 76.0

1.88 ---!10.36 2.10 .833 32.4

3-37 -.54 3.81 4.22 .000 55.O

1.8o -.18 lO.73 1.44 .coo 53.0

2.75 --- 9.03 3.42 .000 60.8

1.57 --- 9.03 3.36 .CO0 76.1

39

TABLE llI.- SUMMARY OF THE GEOMETRIC AND AERODY_,_AMIC CHARACTERISTICS

OF THE CONFIGURATIONS STUDIED AT SUPERSONIC SPEEDS - Continued

Configuration Estimation Estimation Experiment

M_ T/2s W (Cy_)p K using K K' using K' Ref.no.

No. Panel added f_y_ ACn_ Z_y_ f_Cn_ Z_y_ _Cn_

i m_V-_ 1.82 -0.768 -O.418 1.33 -0.956 0.427 1.41 -0.589 0.493 -O.56 0.43 20

1.61 b-.471 -.429 1.25 -.536 .253 1.25 --536 .253 -.52

1.61 b-.471 -.429 1.4O -.600 .283 1.29 -.953 .262 -.)6

.22 21

.23 21

BWHU-BWld 2.01 b_.439 -.023_ 2.14 -.C&5 .020 2.76 -.056 .025 --051 .034 22

Ibgwmrc-_4_ru 2.Ol -.575 -.398 1.55 -.6z7 .355 1.32 -.525 .302! -.52 .3o 22

BWVo-_ 1.61 5-.4o3 -.247 1.39 -.343 .138 1.49 -.358 .lh/_ --33

BWVo-_4 2.01 b-.403 -.195 1.39 -.271 .109 1-72 -.335 .135 --30

5WHVo-BWH 1.61 b-.403 -.247 1.78 -.44O .177 1.49 -.368 .148 --37

BWEVo-BWH 2.01 b-.403 -.195 1.78 -.347 .140 1.72 -.335 .135 -.30

BWHVext-_4]{ 1.61 b-.414 -.321 1.57 -.504 .209 1.32 -.424 .176 -.41

BWHV_27%-BWH 1.61 b-.410 -.349 1.57 -.548 .225 1.35 -.471 .193 -.44

BV-B 2.94 b-.860 -.248 1.13 -.280 .241 1.13 -.280 .241 -.29

5WV-BW 2.94 b-.860 -.248 1.31 -.325 .279 1.311 -.325 .279 -.32

.14 23

.12 23

•15 23

.12 23

.17 23

.17 23

.26 (c)

.26 (c)

BV-B

BWV-BW

BV-B

BU-B

2.94 b-.780 -.12/ 1-55 -.i_ .147 1.55 -.18_3 .147 -.18

2.94 b-.780 -.IP3. 1.71 -.207 .161 1.71; -.207 .161 --19

.13 (c)

.13 (c)

2.9_ b-.781 -.388 1.10: -.427 .333 i.i0 -.427 .333 -._6

1.97 b-.810 -.183 1.P7 -.287 .230 1.}7 -.287 .230 -.30

BWI_VU-BWhV 2.94 b-.SlO -.164 2,071 -.339 .275 1.60 -.262 .212 -.23J

See footnotes to table, p. 41.

.14 (e)

4o

TABLE III.- SUMMARY OF THE GEOMETRIC AND AERODYNAMIC CHARACTERISTICS

OF THE CONFIGURATIONS STUDIED AT SUPERSONIC SPEEDS - Continued

Configuration SW, 2SW, _m, a, a/b Surface s, h, _, Cr, _ ALE ,No. Sketch in. 2 in. in. in. in. in. in. in. deg

i0

11

/2

See footnotes to table, p. 41.

14.4 6.73 6.24 0.541 1.000 V 2.76 --- 9.77 2.56 0.547 49.1

14.1 6.87 6.24 .55 1.170

14.1 6.87 6.24 .55 1.17o

144.0 24.00 20.81 1.47 1.000

144.0 24.00 20.81 1.47 1.0OO

144.0 24.00 20.81 1.47 1.000

144.0 24.00 20.81 1.47 1.000

H 1.50 -0.55 10.73 1.56 .000 53.0

V 2.76 --- 9.77 2.56 .5&7 49.1

aU 1.58 --- 9.03 3.36 .000 76.1

H 1.90 -55 10.73 1.56 .CO0 53.0

V 2.76 --- 9.77 2.56 .541 49.1

aU 1.58 --- 9.03 3.36 .000 76.1

w 12.oo .oo 15.6o 8.88 .z25 49.4

v 8.59 --- 28.66 6.94 .235 20.6

W /2.00 .00 15.60 8.88 .225 49.4

V 8.59 --- 28.66 6.94 .235 41.6

W 12.00 .CO 15.60 8.88 .22549.4

V 8.59 --- 28.66 6.94 .235 52.1

w ]2.oo .co 15.6o 8,88 .225 49.4

V 8.59 ---128.66 6.94 .235 62.5

41

TABLE III.- SUMMARY OF THE GEOMETRIC A}[D AERODIC_AMIC CHARACTERISTICS

OF THE C01fFiGURATIONS STUDIED AT SUPERSONIC SPEEDS - Concluded

Configuration

No. panel added

BV-B

BV-B

M_ Y/2s W (Cy_)p K

Estimation Estimation

using K K' using K' Experiment Ref.no.

z_Cy_ Z_Cn_ _Cy_ _Cn_ ZkCy_ Z_Cn_

1.97 b-0.852 -0.672 1.09 -0.732 0.624 1.09 -0.732 0.624 -0.71 0.63 (c)

2.9h b-.852 -.444 1.09 -.48A .412 1.09 -.484 .4].2 -.48 .39 (c)

i0

BV-B 1.97 b-.834 -.685 1.12 -.767 .640 1.12 -.767 .640 -.72 .60 (c)

BU-B 1.97 b-.794 -.187 1.61 -.301 .239 1.61 -.301 .239 -.32 .23 (c)

BVU-BV 1.97 b-.794 -.187 2.73 -.510 .405 1.88 -.352 .279 -.36 .13 (c)

BUV-BU 1.97 b-.834 -.685 1.36 -.932 .776 1.18 -.808 .674 -.76 .50 (c)

BI[U-BH 1.97 b-.794 -.187 I.i0 -.206 .164 1.38 -.258 .205 -.34 .27 (c)

BHV-BH 1.97 b-.834 -.685 1.45 -.993 .828 1.12 -.767 .640 -.77 .64 (c)

BHVU-BHV 1.97 b-.794 -.187 1.48 -.277 .220 1.52 -.28_ .226 -.31 .19 (c)

BHUV-BHU 1.97 b-.834 -.685 1.55 -1.062 .885 1.18 -.808 .674 --75 .58 (c)

BHU-BH 1.97 b-.794 -.187 2.11 -.395 .314 1.61 -.301 .239 -.37 .23 (c)

BHV-BH 1.97 b-.834 -.685 1.00 -.685 .571 1.09 -.746 .622 -.80 .60 (c)

BHVU-BHV 1.97 b-.794 -.187 2.66 -.497 .395 1.79 -.335 .266 -.29 .14 (c)

BHUV-BHU 1.97 b-.834 -.685 l.ll i -.760 .634 1./i -.760 .634 -.71 .52 (c)

LI

BWV-}_4

BWV-BW

BWV-BW

BWV-BW

BWV-BW

ggV-SW

1.41 -.465 -.788 1.o41 -.820 .381 1.o6 -.836 .389 -.92 -37 24

2.01 -.469 -.470 1.041 -.489 .229 1.16 -.545 .256 -.65 .31 24

12

1.41 b-.564 -.810 1.04 -.8_2 .475 1.04 -.842 .475 -.88 .42 24

2.01 b-.56_ -.482 1.04 -.502 .283 1.12 -.540 .305 -.60 .30 24

13

1.41 b-.630 -.761 1.04 -.792 .499 1.04 -.792 .499 -.85 .45 24

2.01 b-.630 -.512 I.O4 -.532 -335 1.06 -.542 .342 --53 .30 24

BWV-_4 1.41 b-.748 -.679 1.O4 -.706 .528 1.O4 -.706 .528 -.67 .36 24

14

BWV-BW 2.01 b-.748 -.554 1.O4 -.586 .438 1.04 -.586 .438 -.50 .32 24

aApproximated by triangular plan-form tail.

bcentrold of area of added panel used in determining moment arm.

Cunpublished data from Ames i- by 3-foot supersonic wind tunnels.

14-2

43

6.0

4.0

i_i_]_iiiiiiiiiiii!iiiiiii:i:iiii_i_i_iii_i!: i_i iiii'i_

iiitii::L'_::::::l::ii_:li::it!ii/i:i/::!!ii_

3.o< :: i!!!,i!, ,t!i

;i_: :::::iiiii[iiii:::_!iiL_:,i i:i:::i_ii__<ii

_t

_ ,.. :::::

i::i:iiii iiiii!i

J::i111iti iiti::_i::t::i_:

vertical i_

verticoi ponel

1i:i: _ i_:iii!l _!::i

_._ , ._L ._;L;.5J I::;:ii;;!:i_!:i:, ;]ii_; _ iiVii;ii!t_i

: I :!_

ii_it ti::l! ii/I.B t

::i!titt I ::_,_

,1.0

i :/ ,/:./'ij_ _.lb _- '_ _i_

.s _i: _::.::ti_iit::iil:::.,i_f:i:.!i::iil: ii0 .2 .4 .6 .8 1.0

(-I)Added

Figure i.- Apparent mass ratio for various values of _/s of the

existing vertical panel in the presence of a circular body.

i;,_,:;,_::ii!ii:i!Ziiiliiii!!iii!!ii!!iiiilii.i_il I'i'i_i'!'Iii:iIi::iIiiii_.iii"ii::_:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::

i!ii_!i!!_iiii_i_i_ii!i_i_!!_i!i_?_ii_iii!_ii!iI!_!_iiii_!i:!_i_i!_ii!i_iiii_i!_!_!i!_!_i_:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

6.0

5 _i: ::_;:::I::::_::::_::_:I:_:;I::;_I_:_I:_:_'i:_t:::;I_:i:I_:_:_::_:_I:!i!_i:_:_:i:_::_I_i!_''I!i!!ti_i:t:_;il::Ii: :i::t!:!:ll;:!l!i! :i::i:ili:i:l!!:il!!;liil;_:!:ili:i:l:!:il!ii!ti:!:

4.0

5.0

2.5

::::::::::::::::::::::::::::::::: :: _:_ !:t::li:::ff,:l''i[iiitiii!tiiili!i!t!!:

K

2.0

1.5

I.O

.5

0 .2

Added ' I

'pan el-_l ' s

.4 .6 .8 1.0

(_) Added vertica, panel

(a) b/su or t/s w = 0.0

Figure 2.- Apparent mass ratio for various values of a/s of the

existing vertical panel in the presence of a circular body witha mid horizontal surface.

_5

8.0

7.0

6.0

5.0

2.5!i t! ' _di_i:;_l_,!ii i!!! !ii:--_i: .... +;-

K

2.0

1.5

/_ : ::_2 Added

: _i ] :.i!:!t;:f_ ponel_ dL.o_t_,_!_!li_ii1_:tiiiik ,v

Eililili:it!i:! :i:il _I i :_!/ guf

.5t :1: I :_.... t:_!_iil:: 1 :il I

0 .2 .4 .6 .8

('_) Added vertical panel

(b)b/_ or _/_w--02

I0

Figure 2.- Continued.

_6

K

8.o i!i iii,

7.0 _!i iilli!i i_ii

6.o:i' i!

5.0 _ii _:ii :l:i

:ii,, !!i,4.0 ;I :]:

:I:!

£::_ :!ii

!!i!

ii i!i!2.5 I: I:I:

_. 1.!.

2.0 ii ::I:i:tl

]:t:

": !!i!i't+

]i I:I:J; iLi;

:[

1.5 ;:

:,

ir t

1.0 _

_} !iii;!i

•5 i

0

l_!ii!!

i'i- "J"I:i- ;i

I_!!iii

_:i:illIiil:;:l't: :t+

I:I: :!7l:;: ;::

i i::: !i!

.... iiilil ;::

i,,i ii

i,i :.!

TI!!:UI I:1

[I[1

;t-4

-i:i i:i

:ill11:

-17

.:il

......ii!_', :.

,i ......

ili! i_ii

:I:! !il

: : :::

ii:iti:!:

!i41t!ii!Ii:it::"

:i:l l;i;ilill i!!

:!_!/!!

"#rff:t+!_i _

1"i'1"!1i

iliTii_i5t/::!i.l_l;lill

7117::

i.ttl.l+_

'ii7 !;: :GZ

g!!,ii!::

i_iI1!':i

::ii!!!!::i_iii[ii:.:_!:l:t:i:i::i:I:_: ',

.2

i!l iliiit_;i :]!+liiii llii;:I :i:IY;!'_ _71:!!i :!;Jliii! ii!i

ii] :_:llilil i:1:

rtt :: .....

tt.t • t, u_Ill:¢_!ili.;_ !:!lliiiiI:I:

u,t $!ti!i_

1ii !;_I

17 _i!i_:.i iJ!i

¢Ii::t:T_], i:t'trf'/

7 i!iilZ';!] i., Lil.

_i_77ti7i; 9

) i!!+ ,reiti!il i!il

itr'l: "!'!

i:i ti!i! :_;i,_ !:tt:li!: ;!;!

• 1- 1, ][!.!- ._.,

,,,1t ...... r

" i ;

ii: ........ !:!!1:! :[_!! "' II:

ii??l'ii! !iii

.4

:!J: :1:I I;:1

i:!::_:_]!il;-t; :i;i L. I

i:!; :!7; :i:li:l: ........ :i:l!:!; !!ii :!:1kl: :irt :l:l!:l: :l;i ;!:i

:i:: :[:!

.... ii:1: ' +J;; ;i-i

I:]: ....i:ii ,,,,'iii:i:i!:i: :!77 ...., , :P!

i{77 7]77 _7_l

1:1:!7i!7: :L '

_:i!I!:f; .... :i.

Iiiili:i:,! r::

',i:I!:: ::i ::::i:iii:il :!ii i!

iiiti:il ;:I _'.::i!!ti" -i:':I: :,.I iiii:!!!! 1'I i _

,iiiti!i! :',:I r;:;!tl: ;I: :t'! 1;If! T:-t'_ t::

ir :7: :" "qii .... : :!i;11 ;:;. :;:: :',:',I',;;; ;; ; :;-i

1111" ....... _ ..........

Added I-- ;i;

ponel_..._ :!i

r.

:i' i:i ii ;i !: ........!'! ::.................... , i..... I;; ......... ii ';i

.6 .8

1.1

i'!! ::

i:::.

_i_i!i,,ii ii:i:l!:i:! .I:[: :till

i:!: .;:4i ........

, ....... o

!'t + :_::!

!!ii2:1] -

_:_..:.a.q

:_:i ::i:|

:i :I

r:_:+:!:!!

I,i:, ]il

1.0

Si Added vertical panel

(o) b/_ o_ _l_w : o.4

Figure 2.- Continued.

8.0

7.0

6.0

5.0

4.0

3.0

_7

2.5i

.4 .6

(_')Added verticol ponel

(_) b/_ o_ b/sw = 0.6

Figure 2.- Continued.

8 I0

48

K

8.o!ilii!!i!!,_,i:r_!iiill

l;_il

6.0 :i:il

5.0 iti ;: Li:i_,,i!i

4.0 i

3.0_ _:_;i;;

'i'l

+i-i ]

2.5

r

i

"! iti!_..

7/i

5_!r; :! ii

0

'0 _

lS_Added vertical panel

(e) b/s H or b/s W = 0.8

Figure 2.- Concluded.

29

K

.2 .4 .6 .8 1.0

(a)Added vertical panel

(a) b/s. or b/s W-- 0.0

Figure 3.- Apparent mass ratio for various values of a/s of the

eY_isting vertical panel in the presence of a circular body _th

a horizontal surface mounted tangent to the body on the same

side as the added pane]_.

5o

!

PC

K

panel

.4 .6

(_)Added verticol ponel

(b) b/s H or b/s W = 0.2

Figure 3.- Continued.

8

4.0, _,, li!i!!_ii!_i_iti!iitli_i/!i_itlii¢ii!I!i!illii_I_

Added I--_-panel

.......iii!i!!iiiiii!i!!!iiiii!ii!i!i ili _S Sv iiiitiiiilii!ilii!!liiiiti:

iiiiiii!i,lilitiii!jiiiiiii!!iiii: , ...........................::::::::::::::::::::::::::_.5 ::l'::!:!::'!l!_l't! ....... _.L_ _. _ ..........................

::::!::i:!:i:!!::ii!i!ii!::_: .ili j_ o L

.:.;:::i:]i;;:1 :.;1:I11;;* ....I:::::;:::|:;::I:;;;|::::I::;:|;::l

:;:: :::::::::::::::::::::::::::::::::::::: l::::l;:::

3.o_ii!i,i!iitli!ili!iiliii!ti:iili!ilt!i!t!:i:t!!!iliiiif!i(})

ii!i i!!iliii!!ii!ili!iilliiiti!ii'i ;_i_!iii_!!i!iiii!!!!i!i_i!!iiii!_iii!ii!!_!7_!li_il_i_2.5

2.0

.i5

iil,iii_ i_#:ii i;_

t:,i/1.5 i!i /:

ii

4/t t ::),.o tlI/f,/_

.....: ::'.::,::::::::::::1::: : :1: :i ' i1::' : :I:: :1 I 7ti7I 7l;iif7i]1 i : ;0 .2 .4 .6 .8 1.0

(0) Added vertical ponel

(o) _l_I o_, b/_w -- o._

Figure 3.- Continued.

52

4.0

5.5

5.0

2,5

K

2.0

1.5

I.O

I _7_ _i,i!!i!i!!_!!iii!!i!!!iii!!ii!i

Added I _- ii!i,!!!!

panel ----I / --_--

0 .2 .4 .6

(°)gAdded verticol ponel

(4) b/s_ o_ b/sw = o.6

.8 1.0

Figure 3.- Continued.

53

4.0

3.5

panel .|

30

25

K

20

15

I0

5

0 .2 .4 .6

(_')Added vertical pane,

(_) b/_ x o_ b/%j = 0.8

.8 1.0

Figure 3.- Concluded.

2.0

1.0

.5

0 .2 .4 .6 .8 1.0

(°)S" Added verticol panel

(a) b/s H or b/s W = 0.0

Figure 4.- Apparemt mass ratio for various values of a/s of theexisting vertical panel in the presence of a circular body with

a horizontal surface mounted tangent to the body on the sideopposite the added panel.

K

2.0

1.0

.5

0 .2 .4 .6

(°)S' Added vertical panel

(b)b/_so_b/_w: o.2

Figure 4.- Continued.

.8 1.0

56

80ill_.oili6.o i_i ,,_ li!li_it!

!:i ili !:I :

5.0,i_:i_i_ili ii

4.0 !:!!,.!!i.,!J!, J!l ._,_ I _!:i: !'! :i: i:! :i: r!L

3.0 ;;;iiiiiii;ii_iiiiii

2.5

2.0

1.5

1.0

iiiiii!i!iiii

!ii_::'il

.il.

i!, !i:,

I:i l fl/

:!

ti':i/:::tl!i!t!:_i!i:i! '

!/i :i!J_ili;!::)t/r::/1:_:i

!!: 1.4

,'I"1:1

!:L_,

J ::;iii

//!!/i_ ili

_i iiiiliiiiI!iii'_!

i!i iii_ii;,L

_i: !! J"i !:!!_ ' '

_I:_!tiii!t!iiilii!i[iii!t_i ijiii_

iiifiiiiIiiiiliiiiliiiiliiiiIii!iIiiii0 !!!! !i!! !!!! !!!!iiili!iiIiii

r_#

f_!i_i_t_ii_ti!i!li!il_,_itiilili_Ji!i_li!!!ti!i!t_i_illilii_i_i_i_!+!iiiti!!!l!ii_liii!t!ilili!iijii!!l_i_ili_i_I_i_ili_ii

.5

o .2 .4 .6

i:l :1:: ll;1 :I : :I::: 1: ;:

iiti!::! :I::::_ :i_::: _ : :I: _::] : :!::I:" :::1; : 1:::;1::::

.8 1.0

(_') Added verlicol ponel

(c) b/s H or b/s W = 0.4

Figure 4.- Continued.

:>7

K

58

8.0

7.0

6.0

5.0

4.0

3.0

1.0

.2 .4 .6

(_),,°.,..,,0o,0oo.,

(e) b/s H or b/s W = 0.8

Figure 4.- Concluded.

8

4.0

59

3.5............... !........l........!.....................................

ilt"

illi

3.0

2,5iii!'iiilIiiiili!i!

2.0

1.5

1.0

.5

0

I J, . :_1_i_Added Ipanel _---I .......

_ _: :.!_..: !il !! :,i_,i! i Jb Sv . i,

.2 .4 .6 .8 1.0

('_) Added verticol ponel

Figure 5.- Apparent mass ratio for various values of a/s of the

existing vertical panel in the presence of an elliptical body

(a/b = 0.3S3).

6O

4.0

3.5

li_:i!tJ:i;l'li]ll!i:tii:L,t!!',:l;!!Ei,i.l: ', iii: I i!!:i:l:!l!:i:l:::il I ::

:', ,',' panel _----_1

II. I.: '"_, "_'-_SW SV .........

i:: !! i:ii _T_._ U i:::: : :::::,!: : ;-;-4. . .

3.0

2.5

2.0

J.5

panel

_if_ i_ :_:_ti!i̧

?:!il ..............' i

.2 .4 .6 .8 I.O

(a) Added vertical panel

<a) b/s_ or b/_w = 0.0

Figure 6.- A_paremt mass ratio for varlous values of a/s of the

existing vertical panel in the presence of an elliptical body

(a/b = 0.333) with a mid horizontal surface.

61

1.5

1.0

.5

0 .2 .4 .6 .8

(o) Added vertical panel

(_)_/s_{or_/_w--o_

FiG_ume 6.- Continued.

I0

62

_i_i::_i_l_'_l'_::_:l_:_:I:I_ :,:,I__:l:_,l I l

3,5

3,0

2,5

K

2,0

1.5

1.0

.5!

0

Added I _

panel-_ | -'-,,- S w _ , S.V

.2 .4 .6 .8 1.0

v.,,,co,oar,.,(o)b/s_orb/sw--o._

Figure 6.- Continued.

63

:iii:iiilii!iE:!:ili:!:]iili:l:ii:l:i:iE!ii![i:!ii:i:l:ilil!!!i!iii!!iii!!i:i:]!!iiiii!i!i.... iii" i...... i ,._ ..... ii..... iIi_i'Iiiiiliill ..... iii .................. _ .......

3.0

2.5

K

2.0

1.0

.5

0

•,2 .4 .6 .8 1.0

(O)_,_d_ ver,ico_ pone_

(a) v/_ or v/_. = o.6

Figure 6.- Continued.

64

4.0 l:il:i:ili:i:l:i:ili:i:,:!:!l::!:t:!::l:::!l! ::1:: :i::::l ::::::::::::::::::::::::::::::::::::::

3.5

3.0

2.5

iii i_.!_!

r.1,

_!ii"i!i_iiii:!_i!ii_!_i_!_i_i_iIiiii_iiiiI!iii_i!i!_i!ii_!i!!_!!!i_!!ii_!!ii

I!I!:1:!::

ii!i_

2.0

1.5

1.0 _i

.5

0

!i i_il

;:11

ii_ii!ii

iili;_l)ii!iiI_

tiLil!!i/!! !_

titli_i_l Y

!iI; _

!i_l!

1.0 iF_ii!]::ii::li:_::!ti::i_

.2 .4

..,,,oo, oo,,(e)b/s_or b/sw = 0.8

Figure 6.- Concluded.

.6 .8 1.0

65

8.0

7.0

6.0

5.0

4.0

3.0

K

I.O

:: !ii ........... ' :! : ! " ! i _!; .... : _! :_ !: ill:! ............... :_

.........(,).... - . ? O

_li Ex,,,,o-ii!i!iii:!i:!ii!i g.................. verticol

j_:::_.....

:!:]i:: -i: i iii'ii I "_

::: >.,,-.,

_.i:i!iIii_!fI _i"ii I:!: l_._:',.:::.e.I, I'_i_;'i:i !ii _. % i.0 !

_/! it:::i:it :IZ_i :ii'

I l:.i!!!V/._'::i I il

i!il tit Yi, il I: I I Added I--

_il:,!/_ ponel -_ " .......I/!ij::_:'::_!i:l!iIi I I I :: _b__ sv'/1/_ _.ii!::i::i:.::liilI ::Ii _+._,I_

fl/

0 .2

Figure 7.- Apparent mass

.4 .6 .8 1.0

Added vertlcol ponel

ratio for various values of a/s of the

existimg vertical panel im the presence of am elliptical body

(a/b = O.667).

66

8.0

7.0

6.0

5.0

4,0

3.0

2.5

2.0

1.5

1.0

.5

0

iii i',: :', ',:'_i= paneli:] !i'i;:i:-I I

,T_!t i!iil i _ii _ • -, t

i!!i! !iiti

_,.,=iiii! !

ii!i ii,i

::_:._.+iifiti!i!ti!i!i_Iiilliii_ii_=,!iH!.ii=_.:=,.,.i!ii,i!i!i!i_31iikliitiIi!i_:!3!i!

_!i!_!i_,!_li:!l!....iili! lilt _ i[i!i!l

i!/: iiit !ili:_, i;

SU

!!_i_i!iil i_

! !i!i!!!i:I:1;:iitiiiilii_i/::::t::i:l:;:!ll:::l:iii1:il;

!ili !ii _: i_! iti! _ !i!iiiiii,li_il i_i_

:_:_ !!! ii!i_ _i__'_,, ..... i!:! I!!:;;i:!I ;ii " iiil

................ !i!i

i! !il i_!!:t :ti

!!:

I;j: :14

.2

All volues of_,: !il

verticol

ponel ::' :::" ::": "

I_!_ii!!,ii!iIi!ii,:ii!i_ilii,i_ii

ill ....!I:i ,:i:[:i [:J:

.4 .6

!i!!iti!i!l!iliIi_ilt.8 1.0

(0) Added verticol ponel

(a) bls H or bls. = o.o

Figure 8.- Apparent mass ratio for various values of a/s of the

existing vertical panel in the presence of an elliptical body

(a/b = 0.667) with a mid horizontal surface.

67

8°iiii iiii,ii!i,li i!i!ii i....i!,i! i iii ,i i !iii!i!i!iiii!iiiiiiii!iii!!i!i !ii!i!iill

• : : 1.; ;i ; _;_, i , ;I, ;,i ; ,_ .... i L l i . i ; I _1 .... _ .... 1 .... _ .... i .... i

!_!i!i!i!!iiii!!i!iii!!!!iiiiIi:!!_i!i _[_!!!ii:!i!i!:!!i!ii!!i!:!!:i!_:i_!:!_!_i:i;i_:i

5.0 !ii:i:,i:ili:::T!i!l::::I::::I:: ili:I! :I :!Ii:i:l::ii:i::l:ii:Iii::ll:::l:i::li:I!i:: :I: :5::::5::::::::::::::::::::::::::::: :l: ::::::::::::::5::::5::::::::::::::::::::::

:i!![iiili?ii!i:!!ii!!!i!ii!!!!i!i[i!!i!!i!:!!!: !!:[i!!!i!!!!i2.!i!:!!i.ii!iiii!!ii:!:!ii!ii!!4.0 ;;:;;;i;;iil;ii:;iii;i!iii!i!i!:!:i!!!!i!!i;i!:;!:il; i:;:iiiiiil;!:;;;;;;ii!il;:;:;!!i!i;!

:ii3.0 :ii

,.5 ;i;!/!/;,_TiiiI::i:liiiti:__ :I; i: iii:::::::; G(:iiI!i!itililf{il:,_............... i ........._/_ _il!il}iiiilii!itiiiI! _ded , :_ :__

ii/_ ::ilii:]:iil i[:.iill; ponel--_ Sv

1.0 ii _, : ii: i:: ii: :;:! !! _::ll_:l::::f:::: ::;: :',:i :::: :::: : 0 :

.5

0 ,2 .4 .6 .8

(_) Added verticol ponel

1.0

(_) b/s_o_b/sw : o._

Figure 8.- Continued.

6S

K

_°iiiii!i_i!!--ii i ii !_ ili !ii i!!!!!(:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :1:''

_,0 :;!': !'i::!!')F!':_:i:::fl:!'_:'"! .... I:: ..... :1" : .......... ,;;I:i ;I ,'

5.0_!tlli!i_)!iliJi!ilt!i!iij_il)!iiiti_iitli;iliii::l_!ili:_l_i_iti_i!tli_

4.0iiilf_!_iliiTil!i!it_li_!li_iiti_i_!i_:ti!i!

-_ ts&,i_ti,_ ili _ t:.::l::.t ,,,,_co, .I/ilt:.i.lSt)):: :!:i_ ::)ti:/ :F:I F:',II,I)':.

1.5 fi/:!_

,,:L _:_:::_::_,,Lj_ili:!i: ¸

!) !.I i':

").i

:t_: :i!{'_::it!ii)!:.!:.'i if:_

! i !il :,

1.0

.5

0 .2 .4 .6

(0) Added verilcol ponel

(o)_/_ o__/_ = o._

Fig_e 8.- Continued.

.8 i.O

69

8°_::t:_i_I_t_l_,_f_!i_i_J!_iJ__,::_!iiL___J_!l_l_j_it_iiliiiit_!_Ii_::[; i!;iii:ii:ii::i_i;!i!:!iiiii::i ¸ i! ¸ ii :ii:;ii if! :ii:!ii :i: i i!:

7.0 ................... : .......... : ....... _........... :: : ...... _ ...... :

6.0 i :; : I : :t: : I:: :I ::; :: I ;:;1: ::I: : , I:: :1: ::t: ' :: :i: ::1 :::t::::k. _,

5.0

5.( a ..... _,

_.'_.l ., _

Prnel 7 -_:

1.5

2.(:

1.0

.5

0

i i!i:!i!:i, ri_!........ : .....

.t//_ -_Sw _ SV_b_

•2 .4 .6

...... ,, ponel---_ M

.8 1.0

(_) _/_ o= b/_ = 0.6

Figure 8.- Contimued.

7O

K

8.0_

7.0

6.0

5.0

4.0

2.5

2,0

1.5

1.0

.5

i! i!iiii__!ili!!i "_ fiii :_:_liiii ....

,_, i+t, *{'i i"

;i; Lii..... iiiJ ili !ii:, _I_E

i !' !f![ ;

:J !;i hi: "'_

_ i!!:,!-4- .

' 11 :xi,' Ilng[ ,

:I: verticol i"I._ panel !;

i]i !:.i I:_:: :, ili

i ili i_il_i_iI::,,_ ii', ilil il':,i_

,:] ::;:li:

i: r:i [l=li r I!

!!! ii I?I! #:

:I: iiii: I!

iij!1ii:::i/

ili ....:]:

r]_,,/ ii

.... I:'El :::, :I'.I ;:

iJ! _i!!iiii!ii_

:;: :;:; :t:! ]i

0 .2

ii:_:_...,_ ii!!ti_+,_,'I :i "_ :I_ , !;!

!i .......................iiii .... ;i_

i liii ii!it_iitt1_

.... :'!i ili!!i!i :!,_t....

_!ij5 :_!ii:i'i:i

-t_ :tt'l_,t

ii_i_ il;lii!_i:l

il iti _i_/!tiii!

17 ;;I: :];!

i! ; ;i?i:_i:/t:i:!

r;IrlH;!

ii ilil iiii_,,,

!__ i!!!.i!:!

: :i: _iii _iili_ iiil :I: :ii!l:i:l

ii :!41 :i: :;!!_:I:1fill.,.!!ii1!!!!

i ;H! :i:_I:]:ti;i1:I:I

:i]i_;i;!:]:i ;i: ii:]:]:lili! _i: !!!_!_!!!:]:; ::: ::!:t:]:!

.4

I!11! ....!!__!! !:!i_ii!7............. ili

_:_iiiilli!i i!i

!!i! i_!_I_i_I,;i;

_:',: :_:_

i!!t t!ii :_:_:::_ ....

t!i. !:! !I!:! i: i:',i!

.... ;:i: !!HI!!!I ;:i: :;:i:

:L._ ili !;iL_ ; i!: ! :; :

_:i ;:_:_:':' iii E:! i!!i iiiili!i! iil;

:r:! :

"g_ if!! iii::i:ii:i:i i:i: ::_::#f',il_ _i:l !:i: :;:11111!i::: ::::

i!;iiiiitiiii i:i:.:,, ........"_......ill; _._.,._._ _:_; ..... i.ii;.i .....

i!+t !'!:It1'! t:!F ;];l "!'!If'l" '_!"

li_i i!lil!lif i:i: :!:; :!:ii!:i: ::;: :_:::i!_i !:i:i:i:f i;ii iiii :::::::_:::i: ill:

!i!illiil ii! _;iil ii!!! :::I11;i: _:i: :!:4:

1111 :i:ll:;;: i:i: :i;] I

_;i;'_;_!ii!....i!!!_::::_i:ii__,ponu,Added__sw_Ir--sv_- i1;I_ ;:I:

I I ;.i

I:1:

!:_i ill! !_

..... !] .......6 .8 1.0

(O)_dded verticol ponel

(e) bls H or bls w = 0.8

Figure 8.- Concluded.

71

8.0

7.0

6.0 /i::i::!/:;i__2_i:;i/tZ/

3.0

2.5

2.o __;::[_i:,ti/ii!;_::i:;tiii/.:',t!:::::7

,: 7;:i[!_:!::!;iti!

![::7!!t_,::!!_;;::/;

;::: : _: : : :' _:;: ; : :i. :;:: ..... _/ ................ _ , • ¶ ..... : _,,

;_XL::__ r_:l:_ii:_l/ tjit_:!ll] t:ii:t::::t!{;

_ii ::I!:::!1!i:i: _:iI# t: ::if!;

{:_ :!{i!ii:_i i!l! [ t 1_/IZ_i : t_/E,,.ti._!i[ii:i

1.0

,.tiiiii_qi#iiii;itlili'iii!,:. : ._:.;.._.:.::iiiiiii!ii!il!ilili!i!!iilllili,:!:i:;: ' t/t:; ....... l , _ ;,

_:_::_/_/,!_!_1_iii i ii!iii!!ii!iliiii,i} }il

I IIi_ii_!t!ii!l!ii!li!iil;ililiil}li_:_t:iiiliiil!ili}ii;iti!i!I!iil

.5

0

:! :: :! .... :! :::: i:! : :_ ponel------I v ::::i::::

#k_ili_iiiiltlii_i_ii_i_!_t_i_;,ti:::lili!liiiili:iill]iiliiiililit!!i!_m::_,_

.2 .4 .6 .8 1.0

Added vertical panel

Figure 9.- Apparent mass ratio for various values of a/s of the

existing vertical pamel in the presence of an elliptical body

(a/b = 1.500).

72

K

8.0

7.0

6.0

5.0

4.0

3.0

2.5

2.0

1.5 J:i;

1.0 _

!.!i

.... i

.5 ili0

ki.l _:I:

ilil ili:!_i!iili

_,i; iili

.... _

:i :!!:

i; i:i:

11;2 2;1:

!ill;;;i !r

:::t

]:::

1-!]:t:

!:[:

it"

/

;i;i

:::]

ii

I1 ii

'"' i:tl

...... i.

i:ii ....i I_ :i:i

..... !-.

'!:: i;[

.i:_ iJii

i:!: :i:ii:i..i. :.]:

!i!_ ii!i i!ii

!:: ::'! I I

::i: i!ii I:i:.... I:]:..... r I

!! !1' ::!

gi t:l: ;i;i

: :. :',:r ::I

:I:; ]::',: i:ii

2_ 2 : .i

.2

fill !!11 :t:i i:i !i:i .;.i .i _. i._:.: _. : :: '-+-

.i.i :!!J

![ .i!ii iii_;i!i::i;iii):i!!!!! ::::f-!:':!:: !! _: i!_!:'.i',i 1:1" ':': :-i : i : ;'! : _,_ : ::: : : :::: : ; '; ;,; ;

...... .,_::: .... i it:i/:i)!i:_ii!!: :,_i!i_ii; !!ii ;I.! ....... :-.:_ #-: -: ..... : _ ....................

.,', :t :':it..................................................:ii! ........

:i:[ i:i ;i ; . r .................. ; :, :-; ;: ': .ib _ F _

i!1i :.i:.i !:.il ]!i All voluesofi ii iiiiiiitiiil

!:',: :::: /': i \'¢ E isti'_g :: :::::.::: : :. :

i.i ..... !:!: .,._ vertical ; ;]: ; :::;I::;,

................ ponel ii: ;:ii ........

I:_: == ::: i !ii!--

I; ii_] :ii:iil; _i_!_i_! i!iiliiiiiii_iiii !i!iiii_!i i:

:l:i : i: ;:;::::: :::i :.: : ::: ::: ':::: ii !!i::; ! i::i:::: ::i:i:: : : : ": : ::: t :': .:: ....

:!!Added

i: I Sv t

!ii s !i: :)::? ,u

.4 .6 .8 1.0

(O)Added vertical panel

(a) b/s H or b/s W = 0.0

Figure i0.- Apparent mass ratio for various values of a/s of the

existing vertical panel im the presence o£ an elliptical body

(a/b = 1.500) with a mid horizontal surface.

K

,4 .6

(°)Added vertical panel

(b) b/s_ or b/sw = o.2

Figure 10.- Continued.

74

''iii I_IiIttt

7.0 :Iil I_!" 7TT1

1i:t i:1:_i_!iiii iii

6.0"

5.0 r;!r

..... :_:! !:I:_!!ii iiii iiil lii_

3.0 _;il;ii_,;i![!'_iil

_i_i_;iliilii .i:_'1:T'!rhlr'

ii?i_,iiiitilsi.li_ii:_::i :_ !;ii ill|,:i:L:,r: ,l:_ i,ll

Liili11!i!!!! ii'i_

2.0

i!!: _i_ _ lili:iil_ i ii if!! :__'_'_ ...... _.... __ .....

:'7"'*_ i_i ili!:: !.I, ........ "''1" _ .... _:l:

..... _ ............ ::::i[ iill i7 _i_i

i!!t _.}i:'i: !',!i _:_: _;:,.i}_ !i:i i!_ ,.,.1!I_ .....,..

:i!,!2i;i ill

film

i!il: ii::} 1!;,:. :.:,i_ 1:I;_i:[:'::,:i"ii I:, !;i; i iil]]! I:)onelVerlic°'_:;:]::::

1.5

1.0

.2

(o)

.4 .6

(O)Added verticol ponel

_/_ o_ _/_ = o._

.B 1.0

Figure i0.- Continued.

T5

8°i_i_ii_i_.ii_i.i_ii,iiil,iiii!!!i!!!i!ili!i!!iiii

7.0

6.0 i!

i!! i!ii5.0 _!i!f!i!ili!i!IiiiiI!!il,i!!_,ii!_ / :_: _:,,

_iiiiii_i!i!iii!!!iiiiii!iil;i!_' _:,

4.( :i:t!i!ili!!!tiii;ti!!!t!il)fli iii_i:i:

5.( ! ' _::t:::'_!iiL_-_ _ i_ilIi::i:Iii iit ::::::::l

i]tiiz.5 :: : i!ti {i:tii_r::i:::::::::::::::::::::::::::ilJi:,/:,!i

I!!iifi iI_K I::!if:' f::::

...............i ) :i i _:i:i

i t t:ill

:::_

JlH i/_:._

!_::t ,MI! t]I:: .5/L _-.6

i!ilil:i liVi :.e'

i:ti::!t t/,_._,:":__&x,,,,o_:it]//!:,ti:fit:- ve,.co_

r .... , ...... panel

!_:/::i! iiI! i::,:_/

Ii _ i_il_iiliii::l_,iI::i!li!::::ili:I:.ii!'f.

In_I_

i:li]i I iI!il!!:I!:.Ii! lli!' 'L

i:l '.i':il

_.o _Iii::lii::lii::iI:::.Ili Added I I

l/, _.iii I_iiti_iti!!ilii!t i t: b_l _- sv

i

0 .2 .4 .6 .8 1.0

(0)Added vertical panel

(d) b/s H or b/s W = 0.6

Figure i0.- Continued.

7'6

8.0

7.0

6.0

5.0 /;

4.0 /__ _/_i _r

/y/,ri!it_: :_

01/_'....,,++il rft i _,

1.0 ! ili

0 .2 .4 .6

t i ii!/ ti/ /_ .I 17

!i 1/Z/,i:ii ':/

i:. _!;_:iil i:':/J ii! ii_: :j,,:. :

/r, li_ ,:: : ..... ::_i:: ;

/i.<_7_L 11i j i t 77!7

=.f

_:_-I.o (g/ @

:i ;i. :j ponel

I ii::}:i::lli: !::_ i

I

b / .._ Sv

-@-

: : t

.8 1.0

(_) b/_ o_"_l_w = o._

Figure IO.- Concluded.

8.0

7.0

6.0

5.0

4.0

3.0

2.5

K

2.0

1.5

_':':I f!f/-

-l_iill

i-_ /)i,

!:/

/i_ _:

/JI/

7

/j

[ t:/

)_i/i/i //;i, z i:-i:/ ,/ _

/ Z/z

/ _iEt/t:

N:Utu ', : I-7,rll

--<7 != ii' i ii[ii.i

_.,,o(_I

:] !!!

77

1.0

.5

0 .2 .4 .6 .8 1.0

(o)Added vertical panel

Figure ll.- Apparent mass ratio for various values of a/s of theexisting vertical panel in the presence of an elliptical body

(a/b: 3.ooo).

78

8.0

7.0

6.0

5.0

4.0

3.0i

K

1.0

.5

0 .2 .4 .6 .8 1.0

,.,,;oo,.o°.,(a) b/s H or b/s W = 0.0

Figure 12.- Apparent mass ratio for various values of a/s of theexisting vertical panel in the presence of an elliptical body(a/b = 3.000) with a mid horizontal surface.

79

_O

K

8.0

7.0

6.0

5.0

4.o_:_ii_:!il .

3.0_,

2.5

2.0

1.5

I : :i: : :I i:::!::::I :i:t:i: I ::1::

iiii:_I;

: i:: i::2 !!::2!:i: :2:i' ....

! J-i:l:!i!ii!i:!:: :;

i!i:!i_:ii! : ; _l _I ; :,/!

liit_:.!::i!_iiit,

ii ! _;:_f::i;_7i l ii::it:/:;::_i!::77:',!/i::11:/th

i:.ii:t:_:'7_]i _,#7!

:::_f:t.i':t 71

.f,

1.0 ::::_',1

7ft

+-_

!iii_

0 .2

_.,,7 ii::i7/i::i;_//,_ t :i:tli!til :l:ii t ::

"+_'_ii:: :;!_:,:,ii ti: _:1!::1!iit : I;:i t ;:I;i',i

_:i:i _ /

/_!_ii_i_l_i!!li!!!t!!!!t_i_i!!''_'_'"_' ....'"=' ....'....'_:* ......... Ir 1i iiii !ii!!'!!!!_!!!i]!!!il! Added----_._l

:;:_ ::';:i';:::'!:;i"i:i'; panel Ib_ _- Sv :

/

i!i _ii!'::!;"iiI" Sw

:i:ti:1:t:[:i1;ii;_Fi:_1;;;iI;;;i1;ii;1!i!:

.4 .6 .8 1.0

(_)Added vertical panel

(c)_I_ o= _olsw: 0.4

Figure 12.- Continued.

8.0

7.0

6,0

5.0

4.0

3.0

81

K

2.0

1.0

.2 .4 .6

(_)Added verticol ponel

(d) b/s H or b/s W = 0.6

Figure 12.- Continued.

1.0

82

K

,0 :_:

:i;

i:t

zoili

6.O i[:i

5.0 i;

4.0 "_j,i

s.o il

:i

I:

2.5 i:

:1

;!i:

i:

2.0 :i

:i

1.5 ;;

:i

i.o il

!:

0

;i!!i ..... i I_I:llltti!i _!iii:i;Ui;_ll_ti:1:I:1:I;I _ 1;_

.....iil!i;:_i H:I:::I:I H!

i i_ .... iililit uli:i!l!

1:!" '!!! !!!I : :

!r!" : : tr_

_:, i,i, _,rJi i!.

r:t- -,:_ !:_! 1!!

_:t" ' " t;#t ....

....:,:.il /iii;!...

,,r..i!; .......

iji!i'il,.t., !="

Fii - -

""'........tt]il :l

': -" ;:;: f!_! f!

,!ii.... :_! t'.i: ,_,fill.! rJ

L ...

_,_:,i:i il} i

i:.!i i!i! ii!i :i:

.2

:I!!

r t_

ill

:;H!i!1:I:1=

i!ii

!:ii+.

:i!li:I

_.+,.r

i!iit+1.

. i

iii!_

t:II

fiiil

i iit

!i: i

0_T;i;t

:l:l

:1_,t

::iii.i.f

ii,,

r _

!:!i t

:i:i l.b :,,:, t/

:., i.:: ;i,,.+, ..... I:1:

I::: .... !i !i!i

!'! ii!i .... :_::: ....r_! j i :i']

i?!! .............:?:_ _r i:!!

:.ii! !:.!i i.,i ..,.,:!_Iix: ,,:: ..... i';;it?iil iii!:i: +*'_ 4.i+

• _ - *"" :l:l:iiii "" ..... !:'

:l_iii!!;!i ii! ........ _2; ;ti:]

.... :i: :i: ti:u :::'. !:i !i

l,g Exst#g_ !!i_verticol ' i:'.: :!i,,

ponel ii!i il}!!!1 iiI

_i_il:,!ii !!_!li_F !i:! iil:?:rl I : "r:]l![!: "_'+

ii!!!!i!i ::::;ii!! iiii !_!.,i_i:J:!:i i:iiiiiii ii;i !:i!!.... ,.,.r .... _-t'1 ....

::i:t.',:; ..... i!iJ }i{! :::u

!ii!!i!;........

iiiiliiii ....if!it!iilt !F!r ,

: Added I--panel

:il ii!i_ I:;I

!;i

iii i_!i

;:_: :;:;

• t:1 !-*.

.i.iI..I1! 1I:

:::I i!i!.f.4 ;;_.::E:I bl::

I;i_ :iI;

i.*-.,.ii.i ......

!2!2ii._i

ii?i i!

;):i i:!_

I:;: :;:;1

]:[: :qi2i,

:i:!

I s_

.6 .8 1.0

verticol ponel

(e) b/_ o1" _/_W = 0.8

Figure 12.- Concluded.

8_

K

15

I0

5

O

0 .5 1.0 .5 0

SH SW I b b

- -- _ I =_or_b °r b

Figure 13.- Apparent mass ratio due to the addition of a horizontal

surface to a circular body.

84

Symbol Configuration

V or U added to B

Vor U added to BHor BW

V or U added to BHU, BWU, BHV, or BWV

-I0

-.8

co

o--

o

$t'_ -.6

0<3

o

t-- -.4_D

E.--

t..

tn,

ILl

-.2

#

i¢

,/l ll_ l I+'° °i°

f

0 -2 -.4 -.6 -.8 -I0

Estimated ACy# , per radian

(a) Subsonic speeds.

Figure 14.- Correlation between estimated and experimental values of ACy_

-I.0

-.8

t-

"100

$-.6

mL)-

o<3

O

¢-- --.4 ¸

E.i

t,-

Q.x

UJ

-.2

0

Symbol

t

i

J

Configuration

V or U added to B

Vor U added to BU or BV

V or U added to BH, BW, or BWHVor U added to BHU, BWU, BWHU, BHV, BWV

or BWHV

-I0 °/_,

Line of perfect-_. Z =J J

ogreement _'_1._" .

I

-.4 -.6 -.8

Estimated AOy_, per rodion

-I.0

(b) Supersonic speeds.

Figure 14.- Concluded.

56

Lineof perfect-__ 7agreement

-.a ,_ "+Do%: / /

'_ eo __"-o Q

_-.4 _ _

JO 0

o. /x

U.I

/

./0 -.2 -.4 -.6 -.8 -I.0

Estimated AOya, per rodion

(a) Subsonic speeds.

Figure 15.- The improvement in the correlation between estimated and

experimental values of _ by the use of apparent mass ratios

to account for interference.

87

-I.0

-8

c-

O

0

-.6

)-

0

0

_" -.4a-

E

$Q.>(

L_

--.2

0

' /- I0 °/o//

O1,4_Line of perfect r:_='De/ /

acjreement_///

oo,o-,,/_'_e'"/

o ./, ,,o,_o/-.' / ./

i. --r

% _,'alr_, /

o /

-.2 -.4 -.6

Estimated AOyp, per radian

(b) Supersonic speeds.

-.8 -I.0

Figure 15.- Concluded.

_8

O AOy# : K (C,_plp

• AOyp = K'(OVp)p

- I.O I

Line of perfect--. // /agreement

'- _'+10 %.o_ /

e •/ ,/lb.

Q. --6

0

-g /

_-.4 _

¢_ /(9 "9

w Q/

O -.2 -.4, -.6 -.8 -I.0

Estimated /kCy#, per rodion

Figure 16.- The improvement in the supersonic Z_Cy# correlation through

the use of K' rather than K.

89

Symbol

e

ConfigurationV or U added to B

V or U added to BH or BW

V'or U added to BHU, BWU, BHV or BWV

.5

.4

0

"ooh=

a. .3

i,=

o<3

o

,- .2Q)

Eo_

G)

XLIJ

.I

0

/

/.

2o / /

Line Ofement__.perfect_l"/t'Q R _"

ag re /_e\

//< /i+ 20 °/o

.I .2 .3 .4 .5

Estimated ACn#, per radion

(a) Subsonic speeds.

Figure 17.- Correlation between estimated and experimental values of acm_.

9O

Symbol

&

1.0

Conficjurotion

V or U added to B

Vor U added to BUor BV

Vor U added to BH, BW or BWHV or U added to BHU, BWU, BWHU, BHV, BWV

or BW HV

-2o_, /Line of perfect _ /

ag reement---_ /

/.8

/

/ L"' -'_°: / 1/ O/o.6 J

///_-_, ,_'._

,j '"0

.2 .4 .6 .8

Estimated ACnp, per rodian

/

1.0

(b) Supersonic speeds.

Figure 17.- Concluded.

Dashed lines

y48;

iI 5.50"

&22"

1.00" ,[F

_5. '

'_ ]4.20" _ --

(a) Theoretical model.

91

O O

_'=" -o.2'..3

i

m-04

(.C,

-0.6

(b) Effect o£ shifting vertical stabilizing surface area.

O0

® -02 --

o

'_, -0.4

o-0.6

-0.81.0 1.5 2.0 2.5 5.0 5.5 4.0

M©

(c) Effect of adding a wing in various positions to Su/S V = 0.4

configuration.

Figure 18.- The effect of stabilizing surface arrangement on the total

empennage side force. Aspect ratio of the original upper vertical

tail is 4.5.

92

Doshed lines

S_"_U=o'4bvplonfor_

- m

_1%_ °°1.00 _[

14.20"- "'t

(a) Theoretical model.

0,0

-0.2

r_

O4

-OE

J - \_'_--.----.-' _v = 0.0 " "

,..........-----'

(b) Effect of shifting vertical stabilizing surface area.

0.0

-0,2 m_

>=

"-" -0.4 / -"

-0.6I0

. ,.,..,.,=-

I I1.5 20 2.5 3.0 3.5 4.0

(c) Effect of adding a wing in various positions to Su/S V = 0.4

configuration.

Figure 19.- The e_'_'ect o_' stabilizing surface arrangement on the total

empennage side force. Aspect ratio of the original upper vertical

tail is 2.0.

NASA-L_1_v n_l_, w. A-107