Conic Sections Applied to Aircraft

Dr. S.M. MalaekAssistant: M. Younesi

Conic Sections

Conic Sections

The EllipseThough 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 EllipseThe 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 EllipseThe 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 EllipseOn a far smaller scale, the electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus.

The ParabolaOne 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 ParabolaThe 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 ParabolaHeat waves, as well as light and sound waves, are reflected to the focal point of a parabolic surface.

The HyperbolaIf 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 AircraftAn Aircraft Fuselage

General GraphicalConstruction

Technique

General Graphical Construction Technique

Curve equation:

0)(),( 22 =+++++=AFy

AEx

ADy

ACxy

ABxAyxF

General Graphical Construction Technique The basic principle of the usual method of constructing the general conic section :The tangents to the curve at the points of contact o and B 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 pointsdescribed.

General Graphical Construction Technique The graphical method is as follows:1. Draw the line BE and OF through D.2. Draw any radial line AG through A.3. AG intersects OF at J and BE at H.4. Draw the line OK through H and the line BL through J.5. The lines OK and BL intersect at P.6. Then P is the required point which lies on the specified

curve.

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 Ax2+Bxy+Cy2+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 (for four conditions) and a control point (fifth condition).

Analytic Approach ( Control Conditions)All the conics of the family have the same tangents t1 and t2 at O and Brespectively.

Analytic Approach ( Control Conditions)Every conic which is tangent to t1 at O and t2 at B 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 t1 andt2 form the one, and the chord of contact (t3) (taken twice) constitutes the other.

Analytic Approach ( Control Conditions)If : t1=0, t2=0 and t3=0Then the equation of the family of conicsbecomes:

02

321 =+ kttt

2

3

21

tttk −=

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=0Tangent BA: (a-b)y+c(x-a)=0Chord OB: cx-by=0

Analytic Approach ( Control Conditions)

Through substitution of tangents and chord equations in k equation:

2

3

21

tttk −=

Tangent OA: y=0Tangent BA: (a-b)y+c(x-a)=0Chord OB: cx-by=0

[ ] 0)()()( 2 =−+−+− bycxkaxcybay

[ ]2)(

)()(becd

ebecdaeck−

−−−=

Analytic Approach ( Control Conditions)By a simple reduction, expressed as y=f(x):

TSxRxQPxy ++±+= 2

ckbbam

QTmaQ

maPS

mbkP

makR

)(

5.0

5.0

25.0

2

2

2

+−=

=

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛=

−=

−=

Analytic Approach ( Control Conditions)

TSxRxSRxP

dxdy

+++

−=22

2

Slope equation:

Analytic Approach ( Control Conditions)The invert form of equation: the form x=f(y):

SyRyPyx ++= 2

[ ]2)(

)()(becd

ebecdaeck−

−−−=

ckn

naS

nakR

nbkP

=

=

−=

−=

2)2(41

212

Analytic Approach ( Control Conditions)Example: O: (0,0), A: (5,0), B: (6,3), D: (4,1)

42565361125

61)( 2 ++−−+== xxxxfy

425,

25,

65,

61,

3611,,1,

91

====−=== TQSPRmk

15,4

11,21,

31,

91

=−==== SRPnk

yyyyfx 154

1121)( 2 +−+==

Application of Large Scale Digital Computer Technique

Two TestsTwo 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 originat one of the end points.Assuming that neither tangent is parallel (or perpendicular ) to either coordinate axis.

Application of Large Scale Digital Computer Technique

TSxRxQPxy ++±+= 2

decfJbeafEbcadC

−=−=−=

2

)(J

EJECk +−=

Let:

Then:

Application of Large Scale Digital Computer Technique

2

)(J

EJECk +−=

CcaakcN

NCCdbacdkP

NackS

QTN

aQ

NkR

)(

5.0)(

5.0

5.0

25.0

2

2

2

2

−+=

+−+=

−=

=

=

−=

Application of Large Scale Digital Computer Technique

acbdP

acCS

QTac

CaQ

R

22

)2(4

)2(2

0

3

2

2

2

−−

=

−=

=−

=

=

If the program tests and finds the conic to be a parabola:

TSxQPxy +±+=

Application of Large Scale Digital Computer Technique

abS

acbcadR

=

−=

2

If c=2a :

SxRxy += 2

Application of Large Scale Digital Computer Technique

)()2(

)(

122

acCacacP

abQS

acCcaQ

abPR

−−

=

=

−=

+=

QPxSxRxy

++

=2

If N=0, , hyperbola section:)0( ≠a

Application to a specific Fuselage

Application to a specific Fuselage

Application to a specific Fuselage

O: (0,0), A: (a,0), B: (a,c), D: (d,e)

Tangent OA: y=0Tangent AB: c(x-a)=0Chord OB: cx-ay=0

The lower right hand quadrant of the fuselage cross section:

Application to a specific Fuselage

2)()(

aecddaeck

−−

= TSxRxQPxy ++−+= 2

2

22

22

2

2

2)(4

42

22

QTPQS

kakaccR

kaacQ

kacackP

==

−=

=

−=

TSxRxSRxP

dxdy

+++

−=2

2

Actual Numeric:

O: (0,0), A: (31,0), B: (31,36), D: (15,2.34662)

426980.87966782776.349

126534465.2789269.9386416997.1

0061909552.0

=−=

==−=

=

TSRQPk

42698.8796678278.3491265345.27893.9386417.1 2 +−−+−= xxxy

TSxRxQPxy ++−+= 2

2)()(

aecddaeck

−−

=

2

22

22

2

2

2)(4

42

22

QTPQS

kakaccR

kaacQ

kacackP

==

−=

=

−=

Application to a specific Fuselage