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NASA
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
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