A procedure for designing a fuselage having a prescribed effective area distribution computed from -90 deg Mach slices is described. This type of calculation is an essential tool in designing a complete configuration with an effective area distribution that corresponds to a desired sonic boom signature shape. Sample calculations are given for M=2 and M=3 designs. The examples include fuselages constrained to have circular cross sections and fuselages having cross sections of arbitrary shape. It is found that, for a prescribed effective area distribution having sharp variations, the iterative procedure converges to a smoothed approximation to the prescribed distribution. For a smooth prescribed area distribution, the solution is not unique

Adams, Mary S.

Barger, Raymond L.

English

NASA Technical Reports Server

NASA
Technical
Paper
2975
1990
National Aeronautics and
Space Administration
Office of Management
Scientific and Technical
Information Division
Fuselage Design for a
Specified Mach-Sliced
Area Distribution
Raymond L. Barger
and Mary S. Adams
Langley Research Center
Hampton, Virginia
https://ntrs.nasa.gov/search.jsp?R=19900009069 2020-03-20T00:03:19+00:00Z

Abstract
A procedure for designing a fuselage having a
prescribed effective area distribution computed from
-90 ° Mach slices is described. This type of calcu-
lation is an essential tool in designing a complete
configuration with an effective area distribution that
corresponds to a desired sonic boom signature shape.
Sample calculations are given for Mach 2 and Mach 3
designs. The examples include fuselages constrained
to have circular cross sections and fuselages having
cross sections of arbitrary shape. For a prescribed
effective area distribution having sharp variations,
the iterative procedure converges to a smoothed ap-
proximation to the prescribed distribution. For a
smooth prescribed area distribution, the solution is
not unique.
Introduction
One approach to minimizing the effect of sonic
boom noise is to attempt to design the aircraft con-
figuration so that its ground level signature has a
"low boom" shape. A class of such low-shock and
low-maximum-overpressure signatures has been stud-
ied in references 1 and 2. Reference 2 also includes a
systematic procedure for generating the effective area
distribution for a configuration that produces the de-
sired ground level signature along the flight ground
track. This effective area distribution includes the
distribution of areas of projected -90 ° Mach slice
cuts (cuts by planes swept at the Mach angle and
perpendicular to the xz-plane) through the config-
uration. It also includes a significant contribution
determined from the longitudinal lift distribution of
the aircraft and a contribution due to nacelle-wing
interference.
In order to design a configuration that has the de-
sired effective area distribution, one procedure that
has been adopted is to start with an aerodynamically
feasible configuration, approximate its total effective
area distribution with a low-boom distribution, then
redesign the fuselage only so that the desired distri-
bution is obtained exactly. The problem is thereby
reduced to that of designing a fuselage having a pre-
scribed effective area distribution. This problem is
treated in the present investigation in a general con-
text. The fuselage may be cambered, it may be con-
strained to have circular cross-section shapes, or it
may be allowed to vary in cross-section shape.
Symbols
A(x) effective area distribution; area
of projection on x = 0 plane of
sector cut by -90 ° Mach slice
that intersects z -- 0 plane at x
Ct
Ib
tp
M
P
r
x, y, z
Subscripts:
a
z-coordinate of fuselage camber
line at terminal point
body length
length of prescribed area
distribution
flight Mach number
iteration parameter
body radius for bodies having
circular cross sections
Cartesian coordinates; x-axis
taken through fuselage nose in
flight direction
= v/-_-_ _ 1
average of maximum and mini-
mum values
d design
i iteration number
s slicing plane
0, 1 0th and 1st iteration, respectively
Procedure and Sample Calculations
General Considerations
For this investigation, the general approach is to
gradually, by iteration, correct the fuselage shape to
yield the desired effective area distribution, using a
known analysis method. The fuselage equivalent area
distribution calculations utilized a Langley version of
a code originally written to compute wave drag from
volume-effect Mach-sliced area distributions (ref. 3).
The effective area associated with a particular Mach
slice is assigned at the x-location where the slicing
plane intersects the x-axis (see fig. 1). This effective,
or projected, area is calculated by computing the
coordinates of points of intersection of the slicing
plane with the body shell, setting the x-coordinates
equal to zero, and then integrating the resulting
closed contour in the x -- 0 plane. This projected
area is :_ times the actual area cut by the Mach
slice.
Circular Cross-Section Fuselage
If the fuselage is cambered, cross sections are
usually defined, for the sake of convenience, in planes
perpendicular to the x-axis (x = Constant planes)
rather than in planes perpendicular to the camber
line. Forsomedesignsthesecross-sectionshapesare
constrainedto becircular. In thiscase,a changein
theradiusat anystationaffectstheMach-slicedareas
cut by all slicingplanesthat intersecthat circular
section(fig. 2). Thus,it is not possibleto change
theeffectiveareadistributionof thefuselagelocally,
that is, at only onepoint. However,if the specified
areadistributionis smoothwith gradualvariations,
it is generallypossibleto designthe corresponding
fuselagewhilemaintainingthecircularcross-section
constraint.
For sucha fuselage,the designcalculationpro-
ceedsaccordingto thefollowingsteps.
1. Beginwith aninitial trial configurationhaving
aneffectiveareadistributionthat roughlyapproxi-
matesthedesireddistribution.This is usuallytaken
to bea bodythat hasa length
Ib = lp + 13Ct (1)
where lp is the final value of x for which the effective
area is prescribed, and Ct is the z-coordinate of the
camber line at the terminal point. The radii for this
initial body are then computed by the formula
(2)
2. Compute the actual effective area distribution
A0 for this initial body, and the distribution of the
error
AA(x) = Ad(x) - Ao(x) (3)
3. For each Mach-slicing plane, determine the
minimum and maximum x-locations of the intersec-
tion of the Mach plane with the fuselage.
4. For each input x-station within this interval,
change the corresponding radius in such a way that
the error AA is reduced.
5. Iterate step 4 until the desired effective area
distribution is attained.
The crucial part of this procedure is step 4. As
was mentioned earlier, the problem is that changing
the radius at any input location changes the effective
area for every Mach slice that intersects that circular
cross section. For low supersonic Mach numbers--
for which the slicing planes are nearly vertical--this
problem is avoided if the input x-locations are suf-
ficiently sparse. For higher Mach numbers, we rec-
ognize the effect of overlapping intervals but rely on
iteration and smoothing to yield the desired result.
We denote the initial radius distribution in equa-
tion (2) by r0(x). The new effective area distribution
Al(x) should vary approximately according to
(4)
Now if Al(x) is taken to be the desired distribu-
tion Ad(x), equation 4 should yield a radius distri-
bution closer to the required distribution ro(x). An
iteration equation is obtained from equation 4 by first
subtracting 1 from both sides,
_00 1 = - 1 (5)
then inserting an iteration parameter p, multiplying
through by r0, and switching to iteration indices:
[r i=ri_ 1 l+p Ai_l 1 (6)
The iteration parameter p is used to control the
size of the step taken at each iteration. It can be
permitted to vary with x and/or with iteration num-
ber, but the sample calculations in this study used
constant values for p. A large value of p accelerates
the iteration process, but too large a value may re-
sult in an instability in the iteration. Consequently,
small values (0.025-0.1) are often used, but for some
relatively low Mach number cases, values as large as
2.5 have been used effectively.
Sample Calculations
The computer code that implements this proce-
dure utilized iteratively the analysis procedure of ref-
erence 3. Its use is illustrated by the example shown
in figure 3. Figure 3(a) shows the prescribed effective
area distribution, as well as the effective area dis-
tribution for the initial shape computed from equa-
tion (2). Also shown are the results for the final
design.
In all the sample calculations, there is some devi-
ation of the effective area distribution from the spec-
ified distribution near the aft end of the fuselage.
This deviation occurs because the value of Ai-1 is
not permitted to fall below a certain positive value
since it occurs in the denominator in equation (6),
and consequently, the iteration cannot converge in
this region.
Figure 3(b) shows the fuselage shape resulting
from the design calculation of figure 3(a). Figure 3(c)
shows the design with the vertical coordinates ex-
panded by a factor of 6 to emphasize the variations.
To illustrate the procedure for an M = 3 condi-
tion, the same design distribution Ad(x ) was taken
2
asfor theM = 2 case. Figure 4(a) compares the ob-
tained effective areas with the prescribed distribution
Ad(x ). The resulting shape is shown in figure 4(b).
Figure 4(c) shows the design with the vertical coor-
dinates stretched by a factor of 6.
Smoothing, Solution Existence, and
Uniqueness
The computer code contains a provision for
smoothing the fuselage after each iteration. However,
little, if any, smoothing is actually required because
the overlapping of intervals cut by different Mach
slices results in an inherent smoothing. This over-
lapping increases with Mach number, and so the con-
verged area distribution will be quite smooth. Con-
sequently, greater smoothness in Ad(x) is required
at the higher Mach numbers. In fact, if Ad(x ) con-
tains corners or regions of very high curvature, the
iteration cannot converge to Ad(x ). In this case, the
design calculation results in a smoothed approxima-
tion to Ad(x ).
In an attempt to obtain effective area distribu-
tions with regions of high curvature, a slightly differ-
ent procedure was used. Instead of altering all the
radii intercepted by a Mach slice, only the radii aft
of the intersection of the Mach slice with the camber
line were altered. This technique eliminated part of
the overlap, and consequently the iteration converged
rapidly except at very near the aft end.
However, for the same input variables as for
the previous example, the fuselage had a different
shape--one with cyclic radii variations (fig. 5). This
waviness occurs because the modified downstream
fraction of the interval cut by each Mach slice, in
attempting to correct for the entire interval, exag-
gerates the correction locally. For example, if a
particular Mach-sliced area is considerably less than
that specified, the radii for the downstream part of
the Mach-sliced interval would be increased signifi-
cantly, because no contribution to the increase comes
from the upstream radii. Now, however, downstream
Mach slices that intersect this greatly enlarged part
of the body have too great an effective area. The
required decrease is accomplished by decreasing the
downstream radii in an exaggerated manner, and
so this cyclic behavior is propagated downstream.
However, the desired effective area distribution is
obtained.
Thus, it is seen that, at least for some cases,
the fuselage corresponding to a given effective area
distribution is not unique.
Arbitrary Cross-Section Fuselages
When the cross section of the fuselage is not con-
strained to be circular, a similar iterative procedure
is used, but the design procedure is more straight-
forward. In theory, one needs only to adjust the co-
ordinates along each Mach slice in order to correct
the effective area for that slice. However, the dis-
crete input coordinates are located on x = Constant
stations and not along Mach slice sections. Inter-
polation could be used to compute the changes at
each of these coordinate points. However, since the
calculation must be iterated in any case, a simpler
method was used. The intersection of the Mach slice
with each lofting line was located, and the coordi-
nates were adjusted at the input point on the lofting
line that was nearest this intersection. (The jth loft-
ing line connects the jth points on each of the cross
sections.)
The z-coordinates are adjusted relative to the
central value of z
Zmax(X) -t- Zmin(X ) (7)
Za(X) = 2
These values of Za(X) represent, to a close approxi-
mation, the fuselage camber line. The iteration equa-
tion (6) is now replaced by two equations:
zi(x)=Za(X)+(zi(x)-za(x)) l+p Ai_l(X ) 1
yi(x) =Yi_l(X) [1-t-p(_ Ad(x)Ai_l(x) 1)1 (8b)
This iteration converges more rapidly than the circu-
lar cross-section calculation. A sample calculation is
shown in figure 6 for an M = 2 body.
One problem that can arise with the use of equa-
tion (Sa) is that the maximum and minimum z-
coordinates at each x-station are gradually changed
in the design process. Consequently, their average za
is changed, and the redesigned fuselage has a camber
line that is slightly different from that of the origi-
nal fuselage. This variation is normally slight and in
most cases does not represent a problem.
However, the problem is avoided altogether if
the z-coordinates are held constant and only the y-
coordinates are iterated (eq. (8b)). Figure 7 shows
the results of a calculation carried out in this way.
Comparing this fuselage with that of figure 6 reveals
somewhat greater lateral variation of the lofting lines.
Concluding Remarks
A procedure for designing a fuselage having a
prescribed effective area distribution computed from
-90 ° Mach slices has been described. This type of
calculation is an essential tool in designing a com-
plete configuration with an effective area distribution
that corresponds to a desired sonic boom signature
shape.Samplecalculationsweregivenfor Mach2
and Mach3 designs.Examplesincludedfuselages
constrainedto havecircularcrosssectionsandfuse-
lageshavingcrosssectionsof arbitrary shape.For
a prescribedeffectiveareadistributionhavingsharp
variations,the iterative procedureconvergesto a
smoothedapproximationto the prescribed istribu-
tion. Fora smoothprescribedareadistribution,the
solutionis not unique.That is, morethanonefuse-
lageshapemayhavetheprescribedeffectiveareadis-
tribution. This lackof uniquenessmaypermit some
freedominallowingforotherdesignconstraints,such
asminimizingwavedrag.
NASALangleyResearchCenter
Hampton,VA23665-5225
January9,1990
References
1. Seebass, R.; and George, A. R.: Sonic-Boom Minimiza-
tion. J. Acoust. Soc. America, vol. 51, no. 2, pt. 3,
Feb. 1972, pp. 686-694.
2. Darden, Christine M.: Sonic-Boom Minimization With
Nose-Bluntness Relaxation. NASA TP-1348, 1979.
3. Middleton, W. D.; Lundry, J. L.; and Coleman, R. G.:
A System for Aerodynamic Design and Analysis of Super-
sonic Aircraft. Part 2--User's Manual. NASA CR-3352,
1980.
Projection of section onto
x = Const
,'1 J_ _ Section cut from fuselage
/f _,,_ _ by slicing p._/_
Y _,,
Mach-slicing plane
Figure 1. Diagram to illustrate definition of effective area A(x).
[-_ Circular
cross section ._
Fuselage
Figure 2. Overlapping of Mach slices that cut a single cross section.
5
200 --
A, ft 2
160
120
80
(from eq. (2))
4O
3O
I I
60 90
I I I
120 150 180
x, ft
(a) Effective area distributions.
210 240 270
I
300
30 --
z, ft
-30
0
I I
3O 60
I I I I
90 120 150 180
x, ft
I I
210 240
I
27O
I
300
(b) Designed circular cross-section fuselage.
Figure 3. Example calculation; M -- 2 design.
6
z, ft
12
6
0
-6
-12
m
I
0 30 60 90 120 150 180 210 240 270 300
x, ft
(c) Designed fuselage with vertical coordinates expanded to emphasize variation.
Figure 3. Concluded.
7
20O
Prescribed
Obtained
A, ft 2
160
120
8O
Initial
4O
3O 6O
I I t I
90 120 150 180
x, ft
(a) Effective area distributions.
I I
210 240 270 30O
30 --
z, ft 0
-30
0
I
30
I
60
I I I I I
90 120 150 180 210
x, ft
(b) Designed circular cross-section fuselage.
Figure 4. Example calculations; M = 3 design.
I
240
I
270
I
300
8
12
6
z, ft 0
-6
-12
J. l J.
30 60 90 120 150 180 210 240 270 300
1 l
x, ft
(c) Designed fuselage with expanded z-scale.
Figure 4. Concluded.
g
30 --
z, ft
-30 I I
0 30 60
I I I I I I I
90 120 150 180 210 240 270
x, ft
(a) Actual scale.
I
300
z, ft
12
6
0
-6
-12 i i i i J
30 60 90 120 150 180 210 240 270 300
x, ft
(b) Expanded z-scale.
Figure 5. Fuselage designed for same conditions as in figure 4 but with different iterative modification technique.
10
A, ft 2
2OO
160
120
8O
Prescribed
Obtained
Initial
4O
0 3O
I I I I I
60 90 120 150 180 210
x, ft
(a) Effective area distributions.
1
240 270 300
Top
f
Side
(b) Fuselage design, actual scale.
Figure 6. Arbitrary cross-section fuselage; M = 2 design.
11
Top
Side
(c) Fuselage design, expanded y- and z-scales.
Figure 6. Concluded.
12
Top
Side
(a) Actual scale.
Top
Side
(b) Expanded y- and z-scales.
Figure 7. Fuselage designed for same conditions as for figure 6 but with y-variation only.
13
Report Documentation Page
Na[,onaJ Aeror, autlcs and
Space AdmlnlSt r at;o r_
1. ReportNAsANO.TP_2975 [ 2. Government Accession No. 3. Recipient's Catalog No.
4. Title and Subtitle
Fuselage Design for a Specified Mach-Sliced Area Distribution
7. Author(s)
Raymond L. Barger and Mary S. Adams
5. Report Date
February 1990
6. Performing Organization Code
8. Performing Organization Report No.
L-16651
Work Unit No.
505-60-01-02
Contract or Grant No.
9. Performing Organization Name and Address 10.
NASA Langley Research Center
Hampton, VA 23665-5225 11.
12. Sponsoring Agency Name and Address 13.
National Aeronautics and Space Administration
Washington, DC 20546-0001 14.
Type of Report and Period Covered
Technical Paper
Sponsoring Agency Code
15. Supplementary Notes
16. Abstract
A procedure for designing a fuselage having a prescribed effective area distribution computed
from -90 ° Mach slices is described. This type of calculation is an essential tool in designing
a complete configuration with an effective area distribution that corresponds to a desired sonic
boom signature shape. Sample calculations are given for Mach 2 and Mach 3 designs. The
examples include fuselages constrained to have circular cross sections and fuselages having cross
sections of arbitrary shape. It is found that, for a prescribed effective area distribution having
sharp variations, the iterative procedure converges to a smoothed approximation to the prescribed
distribution. For a smooth prescribed area distribution, the solution is not unique.
17. Key Words (Suggested by Authors(s))
Sonic boom minimization
Fuselage design
SST
Supersonic transport
18. Distribution Statement
Unclassified--Unlimited
Subject Category 04
19. Security Classif. (of this report)
Unclassified 20. Security Classif. (of this page)Unclassified 21. No. of Pages ] 22. Price14 A03
NASA FORM 1626 OCT 86
For sale by the National Technical Information Service, Springfield, Virginia 22161-2171
NASA-Langley, 1991)


Fuselage design for a specified Mach-sliced area distribution

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