Transcript Conic Sections Applied to Aircraft
Conic Sections Applied to Aircraft Dr. S.M. Malaek Assistant: M. Younesi
Conic Sections
Conic Sections
The Ellipse
Though not so simple as the circle, the ellipse is nevertheless the curve most often "seen" in everyday life. The reason is that every circle, viewed obliquely, appears elliptical.
The Ellipse
The early Greek astronomers thought that the planets moved in circular orbits about an unmoving earth, since the circle is the simplest mathematical curve. In the 17th century, Johannes Kepler eventually discovered that each planet travels around the sun in an elliptical orbit with the sun at one of its foci.
The Ellipse
The orbits of the moon and of artificial satellites of the earth are also elliptical as are the paths of comets in permanent orbit around the sun.
The Ellipse
On a far smaller scale, the electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus.
The Parabola
One of nature's best known approximations to parabolas is the path taken by a body projected upward and obliquely to the pull of gravity, as in the parabolic trajectory of a golf ball. The friction of air and the pull of gravity will change slightly the projectile's path from that of a true parabola, but in many cases the error is
The Parabola
The easiest way to visualize the path of a projectile is to observe a waterspout. Each molecule of water follows the same path and, therefore, reveals a picture of the curve.
The Parabola
Heat waves, as well as light and sound waves, are reflected to the focal point of a parabolic surface.
The Hyperbola
If a right circular cone is intersected by a plane parallel to its axis, part of a hyperbola is formed. Such an intersection can occur in physical situations as simple as sharpening a pencil that has a polygonal cross section or in the patterns formed on a wall by a lamp shade.
Conic Sections Applied to Aircraft
An Aircraft Fuselage
General Graphical Construction Technique
General Graphical Construction Technique
Curve equation:
F
(
x
,
y
)
A
(
x
2
B A xy
C A y
2
D A x
E A y
F A
) 0
General Graphical Construction Technique
The basic principle of the usual method of constructing the general conic section : The
B
tangents to the curve at the points of contact
o
and are
AO
and
AB
respectively.
The given control point ( fifth condition ) through which it is desired to pass the curve ids
D
.
General Graphical Construction Technique
The graphical procedure involves the location of a point
P
which lies on the curve determined by the tangents and points described.
General Graphical Construction Technique
The graphical method is as follows: 1.
2.
3.
4.
Draw the line
BE
and
OF
through
D .
Draw any radial line
AG
through
A
.
AG
intersects
OF
at
J
and
BE
at
H
.
Draw the line
OK
through
H
and the line
BL
through
J
.
5.
6.
The lines
OK
and
BL
intersect at
P
.
Then curve.
P
is the required point which lies on the specified
Analytic Approach
Analytic Approach Analytic Approach:
1.
Obvious Approach
2.
Control Conditions
Analytic Approach ) Obvious Approach (
A more obvious approach was to make the conic
Ax 2 +Bxy+Cy 2 +Dx+Ey+F=0
pass through five given points (no three collinear) by substituting in the equation the coordinates of the five points in turn.
This give five simultaneous equations to solves for the five essential ration.
Analytic Approach ) Control Conditions (
Conic control conditions most commonly used in designing an aircraft streamline shape.
Two point-slopes and a control point (for four conditions ) ( fifth condition (.
Analytic Approach ) Control Conditions (
All the conics of the family have the same tangents
t 1
respectively. and
t 2
at O and B
Analytic Approach ) Control Conditions (
Every conic which is tangent to
t 1
and
t 2
at B at O is uniquely determined by a third point
D
and therefore belongs to the family.
Analytic Approach ) Control Conditions (
There are
two
distinct degenerate conics in the family; the tangents
t 1
and
t 2
form the one , and the chord of contact (
t 3
) (taken twice) constitutes the other .
Analytic Approach ) Control Conditions (
If :
t 1 =0, t 2 =0 and t 3 =0
Then the equation of the family of conics becomes:
t
1
t
2
kt
3 2 0
k
t
1
t
2
t
3 2
Analytic Approach ) Control Conditions (
To develop the equation of any particular conic of the family, it is merely necessary to evaluate
k
for the coordinates of a point
D
(control point).
Analytic Approach ) Control Conditions (
O: (0,0), A: (a,0), B: (b,c), D: (d,e) Tangent OA: y=0 Tangent BA: (a-b)y+c(x-a)=0 Chord OB: cx-by=0
Analytic Approach ) Control Conditions (
Tangent OA: y=0 Tangent BA: (a-b)y+c(x-a)=0 Chord OB: cx-by=0
k
t
1
t
2
t
3 2 Through substitution of tangents and chord equations in k equation:
y
(
a
b
)
y
c
(
x
a
)
k
(
cx
by
) 2 0
k
(
c
e
)
a
(
cd
(
cd
be
) 2
be
)
e
Analytic Approach ) Control Conditions (
By a simple reduction, expressed as y=f(x):
y
Px
Q
R P S
0 .
25
ak
bk m
2
m
P
a m
0 .
5
Q
0 .
5
a m T
Q
2
Rx
2
Sx
T m
(
a
b
kb
2 )
c
Analytic Approach ) Control Conditions (
Slope equation:
dy dx
P
2 2
Rx
Rx
2
S Sx
T
Analytic Approach ) Control Conditions (
The invert form of equation: the form x=f(y):
x
Py
Ry
2
Sy k
(
c
e
)
a
(
cd
(
cd
be
) 2
be
)
e P R
2
bk
1 2
n
4
ak
1 ( 2
n
) 2
S
a n n
ck
Analytic Approach ) Control Conditions (
Example: O: (0,0), A: (5,0), B: (6,3), D: (4,1)
k
1 9 ,
m
1 , ,
R
11 36 ,
P
1 6 ,
S
5 6 ,
Q
5 2 ,
T
25 4
y
f
(
x
) 1 6
x
5 2 11 36
x
2 5 6
x
25 4
k
1 9 ,
x
f
(
n
1 3 ,
y
) 1 2
P
y
1 2 ,
R
11 , 4 11 4
y
2 15
S y
15
Application of Large Scale Digital Computer Technique
Two Tests
Two special tests are significant enough to have warranted programming development.
1.
For a
parabolic section
, awareness of which “flag” the computing process thru a greatly simplified, time saving subroutine 2.
For a
hyperbolic section
with vertical asymptote; loss of significance is avoided by diverting the algebraic through a special set of formulas.
Application of Large Scale Digital Computer Technique
We note five conditions defining the conic: two point slopes (for 4 conditions ) and a control point ( fifth condition ) To minimize algebraic complexities: the equation origin at one of the end points .
Assuming that neither tangent is parallel (or perpendicular ) to either coordinate axis.
Let: Then:
Application of Large Scale Digital Computer Technique
y
Px
Q
Rx
2
Sx
T C
ad
bc E
af J
cf
be
de k
(
C
E
J
)
E J
2
Application of Large Scale Digital Computer Technique
k
(
C
E J
2
J
)
E R
0 .
25
k N
2
Q T
0 .
5
Q
2
N a S P
ck
0 .
5
a cdk N
2
a
(
b
d
) 0 .
5
C NC N
c
2
k
a
(
a
c
)
C
Application of Large Scale Digital Computer Technique
If the program tests and finds the conic to be a
parabola
:
R
0
Q T
(
c
2
Ca
2
a
) 2
Q
2
S
P
(
c
4
C
2 2
a
) 3
d c
2
b
2
a y
Px
Q
Sx
T
Application of Large Scale Digital Computer Technique
If
c=2a
:
R
ad
bc ac
2
y S
b a Rx
2
Sx
Application of Large Scale
If
Digital Computer Technique N=0
(
a
0 )
hyperbola section
:
R
1
bP a Q
C
(
a
2
c
2
c
a
)
S P
bQ
a ac
(
c C
(
c
2
a
)
a
)
y
Rx
2
Sx Px
Q
Application to a specific Fuselage
Application to a specific Fuselage
Application to a specific Fuselage
The lower right hand quadrant of the fuselage cross section: O: (0,0), A: (a,0), B: (a,c), D: (d,e) Tangent OA: y=0 Tangent AB: c(x-a)=0 Chord OB: cx-ay=0
k
Application to a specific Fuselage
ec
(
a
d
) (
cd
ae
) 2
y
Px
Q
Rx
2
Sx
T P
2
ack
c
2
a
2
k Q
R
ac
2
a
2
k c
2 4
ac
2
k
4 (
a
2
k
) 2
S T
2
PQ
Q
2
dy dx
P
Rx Rx
2
S Sx
2
T
Application to a specific Fuselage
Actual Numeric: O: (0,0), A: (31,0), B: (31,36), D: (15,2.34662)
k
(
ec
(
a cd
d ae
) ) 2
P
2
ack
c
2
a
2
k Q
R
ac
2
a c
2 2
k
4
ac
2
k
4 (
a
2
k
) 2
S
2
PQ T
Q
2
k
0 .
0061909552
P
1 .
86416997
Q
93 .
789269
R
2 .
126534465
S T
349 .
6782776 8796 .
426980
y
Px
Q
Rx
2
Sx
T y
1 .
86417
x
93 .
7893 2 .
1265345
x
2 349 .
678278
x
8796 .
42698