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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
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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
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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
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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
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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
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Figure 3.1: Variation of K
f
with ap deection.
[2]
Figure 3.2: Variation of Cl/e with cf/c.[2]
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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
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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
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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
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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
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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.
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Figure 4.1: Top View of Phoenix Jet
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Figure 4.2: Side View OF Phoenix Jet
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Figure 4.3: Front View of Phoenix Jet
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Figure 4.4: Three View Layout of Phoenix Jet
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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.
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mainSticky Notein next report add analysis of CDo for landing gears and flaps
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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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