1 Introduction

The aircraft chosen, named Phoenix Jet, is a long range intercontinental

business jet. The Phoenix Jet's primary design characteristics are as follows,

Range (R)= 12,000 km.

Cruise Mach number= 0.87

Number of passengers= 9

Number of crew members= 3

Cruise Altitude= 12,200 m (40,000 ft)

In the previous report, the weights and the center of gravity of dierent

aircraft components of Phoenix Jet were evaluated and tabulated. The cen-

ter of gravity locations of Phoenix Jet were determined at dierent stages of

mission prole. The X-center of gravity lies in the range 14.923 m - 15.34 m

from the nose of Phoenix Jet. The Y-center of gravity lies in the range 0.904

m - 1.446 m from the bottom of the fuselage. The CG envelope of Phoenix

Jet was plotted.

In the current report, the neutral point and the static margin of Phoenix

Jet are estimated and the location of components of the aircraft are modied

if the aircraft is found to be unstable. The trim analysis of Phoenix Jet is

also performed to estimate the stability characteristics of the aircraft. Finally,

modied three view layout of Phoenix Jet is created using the data obtained

in the previous and current report.

2 Neutral Point and Static Margin

The magnitude of the pitching moment derivative changes with the center of

gravity location. For any aircraft there is a center of gravity location about

which there is no change in the pitching moment as the angle of attack is

varied. This point is the neutral point and is the most aft center of gravity

location of the aircraft beyond which the aircraft becomes unstable. The

neutral point can be interpreted as the aircraft's aerodynamic center.

Static Margin is the distance, in percentage of the mean aerodynamic

chord of the wing, from the neutral point to the center of gravity.

1

Figure 2.1: Illustration of forces and moments acting on Phoenix Jet.

[2]

2.1 Fuselage Moment Coecient Slope

Figure 2.2: Plot of K

f

vs J

[2]

J is the quarter chord position of the wing's root chord in terms of % fuselage length from

the nose. J=42.59% for Phoenix Jet

The variation of the pitching moment coecient of the fuselage with angle

of attack is given by the following expression

[2]

,

CM(fus) =KfW

2fLf

cSw(2.1)

2

where W

f

is the maximum fuselage width, L

f

is the length of the fuselage

and K

f

is the empirical pitching moment factor.

W

f

=2.4 m

L

f

=28 m and K

f

=0.02 (obtained from Figure 2.2)

CM(fus)=0.3358 per radian

2.2 Horizontal Tail Lift Coecient Slope

The slope of the lift coecient curve of the horizontal tail is given by the

following expressions

[2]

,

CL(ht) =2piAR(ht)

2 +

4 + AR22

2(1 +

tan2 max(t)2

)

SexposedSref(2.2)

2 = 1M2 (2.3)where max(t) represents the sweep angle of the horizontal tail at the maxi-mum thickness position of the airfoil.

M=0.87 (Cruise Mach Number )

2=0.2431=0.95 (Airfoil Eciency)The airfoil for horizontal tail was chosen to be NASA/Langley LS(1)-0013.

The maximum thickness is found to occur at 40 % of chord

[3]

.

max(t)=33.990

AR(ht)=5

S

exposed

/S

ref

=1 for horizontal tail used in Phoenix Jet.

Therefore, CL(ht)=4.5992 per radian.

2.3 Variation of Tail angle of attack

The horizontal tail of Phoenix Jet is present behind the wing. The formulas

used for the calculation of h/ are given below[2][1]

,

h

= 1 (2.4)

=2CLpiAR(2.5)

3

d

d=

2CLpiAR(2.6)

h is the eective angle of attack of the horizontal tail. is the downwash angle.AR=7.5 (Aspect Ratio of the wing)

CL=6.4644 per radian (Slope of the lift coecient curve of Phoenix Jet)The value of h/=0.45128

2.4 Neutral Point

The formula for estimating the neutral point location in the power o con-

dition is given by

[2]

,

hnp =CLhacw CM(Fus) + h ShtSw CL(ht) ht hacht

CL + hShtSwCL(ht)

ht

(2.7)

where h represents X/c where X is the location from the nose and c is the

mean aerodynamic chord length and h represents the ratio of dynamic pres-sure at the tail to the freestream dynamic pressure

h=0.9[2]

From the previous report, the following values were obtained.

c=4.59 m (mean aerodynamic chord length)

h

acw

=X

acw

/c X

acw

=15.195 m h

acw

= 3.31

S

ht

= 36.13 m

2

(Horizontal tail planform area)

S

w

= 119.88 m

2

(Wing planform area)

h

acht

=X

acht

/c X

acht

=30.01 m h

acht

=6.5381

Using the above values, the neutral point of the Phoenix Jet is found to be

located at a distance of X

NP

=16.16 m (h

NP

=3.52)from the nose of Phoenix

Jet.

2.5 Static Margin

The following expression gives the static margin of the aircraft,

SM = hNP hCG (2.8)The most aft position of the center of gravity location of Phoenix Jet calcu-

lated in the previous report is found to occur at a distance of 15.34 m from

4

the nose. Therefore, the least value of the static margin of Phoenix Jet is

found to be 0.1788.

In the power on condition, the static margin is reduced by 3% for jet

aircraft

[2]

. The least static margin value becomes 0.1734.

A positive static margin implies that the aircraft is stable. Since the least

value of the static margin is positive, it is deduced that Phoenix Jet is stable

both in the static and cruise phase of the mission prole.

3 Trim Analysis

Trim condition refers to the state of the aircraft at which the net forces

and the moments about the center of gravity equals zero. Hence,the moment

coecient of the aircraft about the center of gravity (C

M

cg

) should be zero for

the aircraft at the trim condition.The expression for the moment coecient

of the aircraft about the center of gravity (C

M

cg

) is given by

CMCG = CLw(hcghacw)+CMw+CMfushShtSw

CLht(hachthcg)T

qSwZt+

FpqSw

(hcghp)(3.1)

The various terms in the above expression are determined in the following

sections.

3.1 Wing Lift Coecient(C

L

w

)

The expression for the wing lift coecient is as follows,

CLw = CL + CL0 (3.2)

C

L

0

=0.2507

C

L=0.112 per degree.

Hence C

L

w

=0.112+0.2507.

3.2 Wing Pitching Moment(C

M

V

)

The moment acting on the wing about its mean aerodynamic center is dened

as wing pitching moment. The wing pitching moment (C

M

w

) for a swept back

wing is given by the following formula

[2]

,

5

CMw = CM0(airfoil)(AR cos2

AR + 2 cos ) (3.3)

where =27.370 (sweep angle at the mean aerodynamic chord) for PhoenixJet.

C

M

0(airfoil)

=-0.14.

[3]

Therefore the value of wing pitching moment is -0.0931

3.3 Fuselage Pitching Moment (C

M

fus

)

The fuselage pitching moment is given by the following formula

[2]

,

CMfus = CM(fus) (3.4)

CM(fus) is calculated to be 0.00586/deg in the earlier section. Therefore the

fuselage pitching moment is given by 0.00586 at any given angle of attack()where is expressed in degrees.

3.4 Tail Lift coecient (C

L

ht

)

The expression for the lift coecient of the horizontal tail is given by

[2]

,

CLht = CL(ht) [( + iw)(1

) + (ih iw) OLh ] (3.5)where CL(ht)=4.5992 per radian/=0.54872i

w

=i

h

=0 (for Phoenix Jet)

OLh is given by the following expressions

OL = (OLh 0) = 1

CL

CLe

e

where

CLe

= 0.9KfCle

SflappedSref

cos H.L.

In the above expression, Kf is the empirical correction term for lift incre-ment determined from graph for (cf/c)=0.3, where cf is the elevator lengthand c is the horizontal tail chord and maximum elevator deection angle equal

to 30 degrees and

Cleis determined from graph for cf/c=0.3 and t/c=0.129(NACA 0013 airfoil).

6

Figure 3.1: Variation of K

f

with ap deection.

[2]

Figure 3.2: Variation of Cl/e with cf/c.[2]

7

Kf =0.62Cle=4.6/rad=0.0802/deg

H.L.(elevator hinge line sweep angle)=24.6degHence,

CLeis 0.0116/deg and OL is 0.147e where e is the elevator de-ection angle. Therefore,CLht is determined to be CLht=0.036+0.0188e.

3.5 Thrust Eects

The thrust has two main contributions to CMCG , namely the inlet normal

force due to the turning of the air and the direct moment of the thrust. The

F

p

(normal force due to turning of air at inlet front face of the engine) is given

by the following expressions obtained by momentum conservation

[2]

,

Fp =dm

dtV p

where p , V=258 m/s and dm/dt=110.612 kg/s. Therefore Fp isdetermined to be (28537.89)newtonThe direct contribution of the engine thrust to the moment coecient is

given by the following formula

[2]

,

CMengine = T

qSwZt (3.6)

T= 26.256 kN (Thrust required at cruise)

Z

t

=z

t

/c =0.362 (Height of the engine axis from the center of gravity)

Hence the C

M

engine

value was calculated to be 0.03637

3.6 Trim

The other parameters required to completely dene the C

M

cg

are given below,

h=0.9 (ratio of dynamic pressure at the tail to the free-stream dynamicpressure)

[2]

h

cg

=3.254

q=10057.82 N/m

2

h

acw

=X

acw

/c X

acw

=15.195 m h

acw

= 3.31

S

ht

= 36.13 m

2

(Horizontal tail planform area)

S

w

= 119.88 m

2

(Wing planform area)

h

acht

=X

acht

/c X

acht

=30.01 m h

acht

=6.5381

h

p

=5.403

8

The trim analysis of Phoenix Jet is performed for the cruise phase of the

mission prole.

The nal simplied expression for C

M

cg

for a Phoenix Jet is given by,

CMCG = 0.13461 0.0167e 0.101 (3.7)At trim condition C

M

cg

=0. For a given value of e, the trim condition isachieved for specic value of . As e and changes the total lift on theaircraft also changes. However, since the total lift has to be equal to the

weight of the aircraft at trim, there are specic trim conditions(e(trim) andtrim) for the aircraft. To determine the trim conditions, CMcg

is determined

for arbitrarily assumed values of e and . For the same values of e and the total lift coecient ( CL

total

) is estimated using the expression given

below

[2]

,

CLtotal = CLw + CLhthShtSw(3.8)

The expression for C

L

total

for Phoenix Jet is shown below,

CLtotal = 0.119776 + 0.0051e + 0.2507 (3.9)

The total pitching moment coecient is plotted against total lift coe-

cient for various elevator deection angles and plot is shown in gure (3.3).

Additionally the plot of C

L

total

vs and CM

cg

vs for dierent values of e areplotted.

The values used for plotting the graphs are listed in table (3.1) and (3.2)

Table 3.1 : C

M

cg

values as a function of and eAlpha() CM

cg

(e=-100

) C

M

cg

(e=00

) C

M

cg

(e=100

)

-15 2.08515 1.91815 1.75115

-10 1.4121 1.2451 1.0781

-5 0.73905 0.57205 0.40505

0 0.066 -0.101 -0.268

5 -0.60705 -0.77405 -0.94105

10 -1.2801 -1.4471 -1.6141

15 -1.95315 -2.12015 -2.28715

9

Table 3.2: C

L

total

values as a function of and eAlpha() CL

t

(e=-100

) C

L

t

(e=00

) C

L

t

(e=100

)

-15 -1.59694 -1.54594 -1.49494

-10 -0.99806 -0.94706 -0.89606

-5 -0.39918 -0.34818 -0.29718

0 0.1997 0.2507 0.3017

5 0.79858 0.84958 0.90058

10 1.39746 1.44846 1.49946

15 1.99634 2.04734 2.09834

Figure 3.3: C

L

total

vs alpha

Table 3.3: Trim conditions of Phoenix Jet

Alpha() e CMcg

C

L

t

C

Di

trim

0.49

0

-10

0

0 0.2583 0.000869

-0.75

0

0

0

0 0.1608 0.000555

-1.99

0

10

0

0 0.0633 0.004092

The elevator deection at trim is determined from the graph in gure

(3.3) by interpolating for zero pitching moment at the required total lift

coecient. The trim condition parameters are listed in table (3.3)

The total induced drag (C

Di

trim

) including trim drag eects is determined

10

Figure 3.4: C

M

cg

vs alpha

Figure 3.5: C

M

cg

vs C

L

total

at trim angle of attack (trim) and elevator deection angle (e(trim) ) is givenby the following expression

[2]

,

CDitrim = K[CL()]2 + h

ShSwKh[CLht ]

2(3.10)

11

where K=1/pieAR and Kh

=1/pie(AR)htThe expression for C

Di

trim

obtained for Phoenix Jet is given by,

CDtrim = 0.0009582 + 0.022(0.036 + 0.0188e)

2(3.11)

4 Three View Layout

The three view diagrams of Phoenix Jet after stability considerations are

shown below. The three view diagram of the Phoenix Jet includes the

ailerons, rudders, aps, elevators and the landing gear of the airplane. The

sizing and position of these components was performed in the previous re-

ports. There is no change in the relative positions of the components of

Phoenix Jet after the stability analysis and static margin values were ob-

tained to be positive. The newly added component to the three view layout

are the aps.

12

Figure 4.1: Top View of Phoenix Jet

13

Figure 4.2: Side View OF Phoenix Jet

14

Figure 4.3: Front View of Phoenix Jet

15

Figure 4.4: Three View Layout of Phoenix Jet

16

5 Conclusion

The neutral point and the static margin of Phoenix Jet were estimated.

The neutral point of Phoenix is found to be at a location of 16.16 m from

the nose of the aircraft. The least static margin value (at the most aft

center of gravity) of Phoenix Jet is 0.1788 at power o condition and 0.1734

at power on condition. The positive value of static margin implies that

Phoenix Jet is stable in ight. The trim analysis of Phoenix Jet was also

performed to estimate the stability characteristics of the aircraft and graphs

between C

L

total

vs , CM

cg

vs and CL

total

vs C

M

cg

for dierent values of elevator

deection are plotted. Finally, modied three view layout of Phoenix Jet is

created using the data obtained in the previous and current report.

17

mainSticky Notein next report add analysis of CDo for landing gears and flaps

References

[1] Perkins and Hage(1967),Airplane Performance, Stability and Control,

Wiley and Sons

[2] D.P Raymer(1995),Aircraft Design: A Conceptual Approach, AIAA Ed-

ucation Series.

[3] www.airfoiltools.com

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