Aerospace Design Project Light Business Jet Family Design Challenge Authors: N. BOUVIER N. ESTEVES DE S OUSA G. GOFFARD P. LAFONTAINE K. MASROUR B. MOCKEL B. ROULETTE Faculty representatives: G. DIMITRIADIS L. NOELS A. CROVATO T. DOSSOGNE May 11, 2017
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Aerospace Design Project Light Business Jet Family Design Challenge
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Aerospace Design Project
Light Business Jet Family Design Challenge
Authors:N. BOUVIERN. ESTEVES DE SOUSAG. GOFFARDP. LAFONTAINEK. MASROURB. MOCKELB. ROULETTE
2.5, the request for proposal (RFP) will be analyzed to define the typical mission phases of the aircrafts under design, the
typical velocity of operation, and the necessities in terms of payload and performances.
Then, empirical expressions will be used to assess the gross take-off weight, and derive the requested lift, and therefore
to begin the wings sizing (a detailed methodology will be presented in Section 3.2.2 regarding the geometry of the wings).
The aircrafts geometries will be precised. Some parts, like the section of the fuselage, and the dimensions of the nose, the
body, and the aft, will be defined directly by the RFP, as well as the number of passengers that need to be considered. Some
other parts, such as the wings, the tail, and the propulsion, will have to be refined iteratively to reach optimal geometries and
performances. Once the whole geometry will be defined, a study of the location and the respective weights of all the parts,
using mostly empiric expressions, and estimates will be performed. This study will be the object of the Section 3.6.Using
those weights and locations, we will then be able, in Section 3.6.5 to compute the position of the centre of gravity of the plane,
and its evolution throughout the flight, as the fuel is being burnt by the engines. With the centre of gravity positioned, all the
tools are available to determine the stability conditions in the Section 5.1. Using conditions on the relative position of the
position xCG of the centre of gravity of the plane and the neutral point of the wing, we will optimize the wing position along
the fuselage, and size the tail in order to ensure both good manoeuvrability and stability throughout the flight.
As a first step, all computations will be made with respect to the a maximum takeoff mass configuration. Then, a trade-off
study will be performed by varying some principal parameters of 5% around their chosen value to confirm the suitability of
the chosen parameters. Afterward, additional details of design settings will be provided and performances assessments will
be carried out to finally end with an exhaustive cost analysis in the Section 6.
2.5 Mission requirements
The aim of the project is to develop the technology used for the category of light business jets to offer higher cruise speed,
larger cabins, and updated technology compatibility. The research work asks for a two-member aircraft family envisioned to
have a high level of part commonality between two family members to minimise the development and production costs.
The light business jets are designed to carry up to 6 and 8 passengers and the entry into service is 2020 for the first model
and 2022 for the second model.
The general requirements for both the families are:
• Maximum Cruise Speed of Mach 0.85 at 35,000 [ft];
• Rate of Climb of 3,500 [fpm];
• Service Ceiling of 45,000 [ft];
• Maximum Sea Level Takeoff Balanced Field Length of 4,000 [ft] at Maximum Gross Weight with dry pavement;
• Maximum Landing Field Length of 3,600 [ft] at Typical Landing Weight.
In the particular case of six seat family member the requirements to accomplish are:
2 MARKET, MISSION & DESIGN 9
Take-Off
Climb
Cruise/Approach
(Loiter)
Landing
2500 nm - M.85
35,000 ft.
Wto
W1
W2
W3
W4
W5
Figure 2.2: Schematic of a typical mission.
• Must meet FAA Federal Aviation Regulations Part 23 Airworthiness Standards for certification;
• Minimum range of 2,500 [nmi] at Long Range Cruise (LRC) assuming NBAA IFR Range with 100 [nm!] Alternate (1
pilot + 2 passengers);
• passenger/pilot at 200 [lbs] each;
• Baggage capacity of 500 pounds/30 cubic feet;
• 1 or 2 flight crew;
• 6 passengers, including 1 in the cockpit if there is no copilot.
The additional requirements to accomplish in the case of eight seat family member are instead:
• Must meet FAA Federal Aviation Regulations Part 25 Airworthiness;
• Standards for certification;
• Minimum range of 2,500 [nmi] at Long Range Cruise (LRC) assuming NBAA IFR Range with 100 nm Alternate (4
passengers; passenger/pilot at 200 [lbs] each);
• Baggage capacity of 1,000 pounds/60 cubic feet;
• 8 passengers.
A family of 2 aircraft is thus asked to be designed with a maximum of common components. The approach followed is
therefore to design the the 8 seats and then see what it can by done for the 6 seats. Basically, the major change between the
2 configurations is a certain length of the fuselage that will be removed (this approach will be validated in the trade-off study
section 4). Further details of the 6 seats settings will be specified in parallel with the 8 one.
3 AIRCRAFT DESIGN CHOICES 10
3 Aircraft design choices
3.1 Fuselage
To define the fuselage dimensions, the cabin section will first of all be established to ensure the optimum level of comfort, and
then the seats and compartments will be organized to derive the respective lengths of the fuselage body, nose, and aft.
3.1.1 Cabin design
The cabin section’s design is a key factor of the overall design process. Indeed, the goal is to balance the level of comfort
and equipment, and the induced drag of a higher section. In the proposed design, the cabin section has been optimized to the
highest, and the result is presented on the Fig. 3.1.
5 ft.
1.52 m.
5 ft. 3 in.
1.60 m
15.35 in.
0.39 m.
Figure 3.1: Cabin section.
This design is compliant with the FAR23 and FAR25 specifications (in terms of width of aisle). Moreover, the seats and
spaces are designed to ensure an optimal comfort to a statistical sample of American men up to the 95th percentile1. On Fig.
3.2, the cabin section is compared with competitors.
The Fig. 3.3 shows the seating diagonal space (24.8 in). The fuselage body has been designed to optimize the space within
the cabin. As the fuselage section is relatively small, the seat pitch has been increased to 47 inches to give a sensation of space
and freedom to the customers during the flight.
1The section design has been performed using CATIA v5’s ergonomic module, ensuring that the seating dimensions fit to the best the expected customers.
3 AIRCRAFT DESIGN CHOICES 11
Citation CJ4
(a) vs. Citation CJ4.
Embraer
Phenom 300
(b) vs. Phenom 300.
Learjet 40XR
(c) vs. Learjet 40XR.
Figure 3.2: Comparison of our section with some competitors.
3.1.2 Fuselage Length
The fuselage contains a large galley/minibar area, a wide lavatory/bathroom area, and two big luggage compartments (on both
parts of the rear-body/aft beginning). Both are pressurized. The Aft factor (AF ) has been set to 2, while the nose factor (NF ),
is 1.8. This gives us the definitive fuselage dimensions of the Fig. 3.3.
Most of the fuselage remains identical to the 8 seats version, excepted that the cabin’s length is reduced by about 2,13 feet.
The 6 seats version has therefore more space per occupant, as shown on the Fig. 3.4.
47 in.1.20m
Seat Pitch
Fuselage Body24.9 ft.7.59 m
Fuselage Length: 46.43 ft. / 14.15 m
Nose10.20 ft.3.11 m
Aft11.33 feet3.45 m
GalleyMinibar
Lugg.Comp.
63 cms
24.8 in
.
Figure 3.3: Fuselage dimensions and cabin.
3 AIRCRAFT DESIGN CHOICES 12
49,9 feet
47,7 feet
Figure 3.4: 8 seats vs. 6 seats fuselage lengths.
3.2 The wing
The conceptual design of the wing is a very complex and important part since it is strongly linked to every studied component.
First, design choices must be made in order to guarantee consistency throughout the wing design procedure:
1. CL = 0.3: here, low wing loading is preferred over optimal aerodynamics performance. Indeed, optimal lift coefficient
associated with maximum Lift-to-Drag ratio lies in the typical range of [0.4,0.6], as illustrated in Fig.3.5. This can be
shown using the statistical drag polar relationship,
CD =CD,0 +C2
Lπ ·e ·AR
, (3.1)
in which CD,0 and e are respectively the zero-lift drag coefficient and Oswald’s factor. However, such values lead to an
increase in wing loading, and so compel the structure designer to allow for higher structural strength. This is generally
performed using other more appropriate materials and is eventually more costly in terms of computation time. However,
a sufficient lift coefficient must be selected so that the jet is still able to fly. Thus, the typical interval which CL must lie
in is [0.2,0.4].
2. M = 0.85: The specification states that the plane has to be able to fly at maximum Mach of 0.85, which means that a
super-critical airfoil is required.
3. Another important choice that directly affects the selection of the airfoil is about the takeoff and landing field lengths.
Indeed, the family business jet should ensure really quick takeoffs and landings, which is suitable for a regular use and
enables to offer all the flexibility that customers of business jet may require.
In order to address the design of the wing, a preliminary weight assessment must be performed, using bold assumptions
3 AIRCRAFT DESIGN CHOICES 13
22
20
18
16
14
12
10
8
6
4
2
00 0.2 0.4 0.6
CL
CL/C
D
0.8 1
AR = 5
AR = 6
AR = 7
AR = 8
AR = 9
AR = 10
Figure 3.5: Optimal range for CL.
and empirical estimates (see section A for a description of the gross weight estimation method). As a result, the weight at
mid-cruise (i.e. when the fuel is half-consumed) is determined. This leads to
L =W =12·ρ ·V 2
∞ ·Cl ·S, (3.2)
where S is the gross wing area, CL is the 3D design lift coefficient of the wing, ρ is the air density, V∞ is the airspeed, L is the
total lift of the plane for mid-cruise conditions. On the other hand, an important quantity to be determined as soon as possible
is the aspect ratio, AR, the expression of which is given as follows
AR = b2/S, (3.3)
where b is the span of the wing. As the gross area is fixed by the choice of CL, the aspect ratio gives directly the span. Its
value is to be as high as possible since it is inversely proportional to the Lift to Drag ratio. Typically, it lies in [6,10], as it can
be seen in Tab. C.1 and C.2, in App. C. Here, to be consistent with what is found in actual business jets, namely in terms of
wing geometry, AR is imposed to 9.
3.2.1 Super-critical airfoil: NACA SC(2)-0714
Assessing several airfoils and considering the availability of the data, it has been decided that the NACA SC(2)-0714 is able to
fully satisfy the imposed requirements. Indeed, this airfoil belongs to the super-critical family, a category of transonic airfoils
that have the particularity to reduce dramatically the drag by increasing its drag divergence mach number. Although the design
3 AIRCRAFT DESIGN CHOICES 14
point of this airfoil is at 2D lift coefficient of 0.7, we can use this airfoil for a lower range of 2D lift coefficient, since we stay
in the low drag bucket region at cruise conditions. Furthermore, the high slope, cl,α , of the curve ensures a very quick rise of
lift during cruise if a change of altitude is needed.
This airfoil has a thickness to chord ratio (t/c) of 0.14 and is designed for a normal component of airspeed associated with
Mach 0.725. It means that the wing must have a sufficient sweep angle so that the airspeed in the normal direction is reduced
up to maximum Mach 0.72. The drag-rise Mach number associated with this airfoil is 0.79. Thus, going slightly beyond the
design Mach number do not cause a dramatic increase in drag provided this ultimate limit is not overcome.
The geometry of this airfoil is shown in Fig.3.6. Fig.D.1a and D.1b in App. D respectively shows the behaviour of the
airfoil at cruise and takeoff/landing configurations.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2
−0.1
0
0.1
0.2
x/c [−]
t/c [−]
Figure 3.6: Airfoil NACA SC(2)-0714.
Fig. D.1a and D.1b respectively shows the behaviour of the airfoil at cruise and takeoff or landing configuration.
It can directly be seen on those figures that the lift coefficient approachable by the airfoil is really high, which is also
important for optimization of the takeoff and landing field lengths.
3.2.2 Wing geometry
Here, the wing design is performed for the 8 seats configuration, which is almost identical to the 6 seats configuration (but a
fortiori less optimal). A discussion about the 6 seats configuration will be carried out later in Section 3.2.4.
Taper ratio lies statistically in the interval [0.2, 0.6] (see Tab. C.2 in App. C). It reduces the lift at the wing tips, so allows
to reduce the structural weight or rise the fuel capacity. Strong taper ratio increases occurring of tip stall. Thus, considering
those effects, a taper ratio of 0.3 is imposed. Stall is a serious issue that should be anticipated in the conceptual design phase.
In order to exhibit a good behaviour near the stall angle or velocity, the wing is twisted. Twisted wing permits to redistribute
the lift along the span. The purpose of lift redistribution is to ensure that the wing tip is the last part of the wing surface to
stall. Indeed, a twisted configuration allows the pilot to stabilize the aircraft in case of increasing stall, and so avoid losing
control of it. To this purpose, the geometric twist angle (εg,tip) is a washout angle typically in the range [-5 ◦, 0◦], as explained
in page 58, chapter 4 of reference book Aircraft Design: A Conceptual Approach [5].
As mentioned in the paragraph dedicated to airfoil, the maximum Mach number’s requirement makes necessary the wing
3 AIRCRAFT DESIGN CHOICES 15
to be swept. For the sake of simplicity, one decided to sweep it backwards (to the tail) so as to be in compliance with usual
aircraft designs. Seeing the design Mach number, the sweep angle at the quarter chord is determined as follows
Λ = arccos0.720.85
= 32◦. (3.4)
As for the dihedral angle, its value is fixed when studying lateral stability. It is worthwhile noticing that the maximum sweep
for an aircraft generally lies about 35◦, above which the performance might be reduced (see Lecture 4 of Mr. Noels’ course
notes, page 58 [6]). Further studies might be performed to determine the maximum acceptable sweep.
3.2.3 Flaps design
The last geometrical parameters to determine for the wing are about the flaps. Flaps design is strongly related to the perfor-
mance. Indeed, these are important to give an additional lift during the takeoff or enough drag for the approach and landing,
in order to reduce the takeoff field length and landing distance as much as possible. To determine the flaps area, the strategy
consists in estimating the CL,max necessary at takeoff and landing to meet the requirements in terms of field lengths. These
requirements are resumed here below:
• maximum takeoff field length: 4000 ft at maximum gross weight with dry pavement;
• maximum landing field length: 3600 ft at typical landing weight;
• rate of climb: 3500 fpm.
All thermodynamical parameters are given using ISA tables.
A simple way to estimate the 3D lift coefficient (CL) is to use the wing partition method, presented in the chapter 10, pages
437 to 441 from the reference book "The Anatomy of Lift Enhancement" [7]. The idea is to cut the wing into several sections
and compute the 3D lift coefficient of all the parts considering their respective properties (i.e., if this part is flapped or not and
other eventual aerodynamic properties of the considered part). References coming from the Ilan Kroo’s course notes, page
257 [4], state that "typical flaps extend over 65 % to 80 % of the exposed semi-span, with the outboard sections reserved for
ailerons. The resultant flapped areas ratio are generally in the range of 55 % to 70 % of the reference area". Considering this
observation, it is possible to dimension flaps such as presented in the figure 3.7.
In this figure Sflaps represents the total surface of the flaps and Sw f the flapped surface of the wing. In our case, the flapped
surface represents approximately 67.5 % of the gross area. The wing partition method applied to a swept wing allows to assess
the lift coefficient in 3D given the following formula (from the reference book "The Anatomy of Lift Enhancement", page 438
[7]):
CL,max ≈0.9S
{N f
∑i=1
cl,max,i ·Si ·cos(Λhingeline,i)+
N f
∑i=1
Cl,i ·Si · cos(
Λ 14
)}, (3.5)
3 AIRCRAFT DESIGN CHOICES 16
Λ1/4
= 32.1°
croot
= 7.81 ft
ctip
= 2.33 ft
I
II
III
IV
Λ = 34.5°
yAC
= 9.38 ft
xAC
=7.84 ft
MAC = 5.58 ft
AC
y
x
O
16.40 ft
1.98 ft 1.64 ft
45.7 ft
2.83 ft
Swf= 156.8 ft2
Swing
= 232.5 ft2
Sflaps
= 47.4 ft2
AR = 9
Figure 3.7: Left part: Flaps sizing and determination of the maximum lift coefficient: wing partition method. Right part:Representation of the wing’s main parameters.
where:
• Si is the reference surface of the considered part of the wing;
• N f is the number of the flapped surfaces;
• Cl,max,i is the maximum lift coefficient of the airfoil at takeoff or landing conditions considering the additional lift
provided by the flaps (flapped section);
• Si is the surface of the considered section of the wing;
• Λhingeline is the angle between the hinge line of the flap and the vector normal to the aircraft plane of symmetry;
• Cl,i is the lift coefficient of the considered unflapped section at takeoff or landing conditions considering a decrease of
the stall angle due to the deflection of the flaps;
• Λ 14
is the sweep at quarter chord.
The additional 2D lift coefficient of the flaps is given by the following empirical formula (presented in Ilan Kroo’s course
notes, page 258 [4]):
∆Cl,max,flaps = K1 ·K2 ·∆Cl,max,ref, (3.6)
3 AIRCRAFT DESIGN CHOICES 17
where ∆Cl,max,ref is the 2D increment in Cl,max for a reference configuration and its value is determined from an experimentally-
determined curve depending on the thickness of the considered airfoil. The constants K1 and K2 take into account the cor-
rection due to the variation between the reference configuration and the configuration of interest, respectively in terms of flap
chord extension and flap angle deflection. It is worthwhile noticing that this formula applies only for double slotted flaps. The
course notes Aircraft Design - Synthesis and Analysis at page 258 (see ref. [4]) states that it is possible to adapt the result
for different kind of flaps by multiplying it by a factor 0.98 for single slotted flaps and 1.08 for triple slotted flaps. In this
conceptual design, it has been decided to choose single slotted flaps for both 6-seats and 8-seats configurations. This choice
is motivated by several reasons:
• complexity of double and triple slotted flaps will certainly require a higher level of maintenance;
• the single slotted flap is cheaper to design and implement than both others;
• the single slotted flap is also lighter;
• the chosen airfoil has a high slope of the curve cl vs. α so able to provide a rapid increase in lift coefficient for the
takeoff. That is, a single slotted configuration is enough to meet the requirements.
Considering an airfoil with a thickness to chord ratio of 14%, a flap extension of about 30%, formula 3.6 gives the results
presented in the table here below (Fig.3.1).
Takeoff LandingAngle of deflection 20◦ 40◦
∆Cl,max 1.06 1.87
Table 3.1: Increment of 2D lift coefficient (∆Cl,max) due to a section of flap.
If needed, in case of emergency for instance, flaps can be eventually deflected up to 60◦. In that case, assessing the
increment in 2D lift coefficient becomes less precise but by extrapolation we can compute a value of approximately 2.12.
Using these values of ∆Cl,max,flaps in the formula 3.5 gives results presented in Tab. 3.2.
Phase of flight Flaps in Flaps outTakeoff 1.5 2.1Cruise 0.8 -
Landing 1.5 2.6
Table 3.2: Value of the 3D maximum lift coefficient (CL) for different configurations.
From these value, takeoff and landing field length, stall velocity and decision speed will be computed in the second part
of the report (Preliminary Design, section 5.3). The determination of the flight envelope (see section 5.4.1) will also strongly
depends on these values.
3 AIRCRAFT DESIGN CHOICES 18
All the chosen and computed parameters concerning the flaps are presented in Tab. 3.3. The dimensioning of the flaps is
drawn in Fig. 3.7
Flaps surface, Sflaps 47.4 [ft2]Flaps extension (relative to the span) 67.5%
Table 3.3: Summary of the principal parameters linked to the flaps.
3.2.4 Results
A sketch of the wing geometry with important geometrical parameters is presented in Fig 3.7. All the geometrical parameters
and results concerning the wing are listed in the Tab. E.1. The wing designed for the 8 seats configuration will be reused for
the 6 seats configuration in its entirety. Since the weight of the plane vary due to the fuselage shortening, the lift required for
the wing decreases. Therefore, since its geometry shall not change, the required wing lift coefficient reduces to CL6seats = 0.27
at cruise. This leads to a variation of the angle of the wing at root which decreases to αroot =−0.5 [◦].
3.3 Propulsion
3.3.1 Engine selection
3.3.1.1 Preliminary researches
In this section, a quick market analysis is performed. Indeed, a good indicator to the choice of the engine is the choices that
already have been made for aircrafts from the same class. Table 3.4 gathers data about the main competitor’s engines.2
Table 3.5: Comparison between results computed from conceptual design and approximations, with PW535-E as chosenengine.
3.3.1.3 Further comments about the chosen engine
Firstly, the manufacturer is the internationally located PRATT&WHITNEY. Meaning that support can be provided worldwide
very quickly. Moreover, the engines family of PW500 sums up 12 millions of flight hours and their reliability is such that their
time between overhauls (TBO) can go beyond 10,000 hours, directly translating into lower operational costs. Secondly, even
if the PW535-E suits the cruise conditions of the aircraft under study, additional information on other flying phases implying
its use were gathered.
• The Embraer Phenom 300 embeds 2 PW535-E, while having a comparable MTOW , and is yet able to take-off with a
3,138 ft long runway, which is clearly matching the RFP requirements.
3 AIRCRAFT DESIGN CHOICES 20
• In case of trouble for the climb, the PW535-A engine (same family) can provide a slightly higher thrust without any
change relative to the dry weight mass, BPR and SFC.
• In case of trouble with the landing field length, the same PW535-A engine is also certified with thrust reverser (it would
however induce some structural changes in the nacelle’s integration).
3.3.2 Placement and installation
Nacelle-engine set placement and incidence is determined using CFD and wind tunnel experiments, and is therefore out of the
scope of this study. However, the main characteristics are detailed in the following.
The chosen configuration for the engine placement is aft fuselage. Main reasons are aestheticism, and level of noise
comfort in the cabin. Some other advantages include
• less pylon interference (better lift and lesser drag),
• less yaw induced by an engine failure,
• shorter landing gear and ease of disembarking once arrived.
Nevertheless, some drawbacks are also to be taken into account.
• noise and vibration on the fuselage are severer (heavy insulators required and engines placed far away from passengers
and thus from the centre of gravity),
• supersonic flow throughout the fuselage-pylon-nacelle system is possible,
• structural advantage in the wing bending moment point of view are lost.
In the case of fuselage mounted nacelles, a gap of approximately one half of the nacelle’s diameter is recommended with
respect to the fuselage. In addition to that, the vertical position of the nacelle is an important (yet hard to quantify, at the early
stages of design) matter. As hot ejection gas exits the engine at a high velocity, a low-pressure zone is generated in the lower
part of the tail, creating a downward force, and generating a pitching moment.
Regarding the wings and V-tail implantation, the vertical position of the engine axis is set to 1.67 ft above the cabin’s axis.
Figure 3.8: Nacelles implantation.
3 AIRCRAFT DESIGN CHOICES 21
3.3.3 6 seats engine choice
As the 6 seats’ aircraft weight is lower than the 8 seats version, the suitability of the engine has to be re-assessed. As a result,
it is found that the FJ44-3AP model (WILLIAMS INTERNATIONAL) is also suitable to the RFP and FAR 25 requirements3.
As shown in Tab. 3.6, a slight fuel saving is noticed. However, for production cost minimizing reasons, the PW535-E will
be maintained from the 8 to the 6 seats versions, as demonstrated in the costs analysis.
PW535-E FJ44-3AP
Dry mass (lbs) 699 516
BPR (-) 2.55 2.2
SFC sls (lbs/(h lbf)) 0.44 0.46
Thrust sls (lbf) 3,360 3,052Resulting TO mass (lbs) 16,225 15,397
Fuel required (lbs) 4,702 4,474
Table 3.6: Engine smackdown for the 6 seat business jet (reaching 2,500 nmi at full payload).
3.4 Empennage
The empennage balance the moments applied on the plane by the different forces acting on it. Therefore the tail plays a
major role in both static and dynamic equilibrium. Among the different existing tail configurations , the most popular one
for business jets is clearly the "T" tail. This choice is partly aesthetic and allows place on the aft for the engines. But the
"T" configuration also have few drawback that it may be interesting to avoid : structural complexity (thus higher weight and
manufacturing costs) and propensity to be blanketed at high angles of attack, leading the plane into a deep stall from which it
is hard to recover.
A way to dodge these flaws is to choose a simpler, less common yet well known tail configuration : The "V" tail. These
tails present the asset to be structurally uncomplex since they are basically small, untwisted, symmetrical wings. They are
therefore expected to be lighter and easier to manufacture than other types of tails. Moreover the aerodynamic characteristics
of the "V" tails are interesting : the reduced number of junctions between surfaces is a favourable factor of reduction of the
induced and interference drag of the tail. Besides, originality and innovation are sure important aspects in the scope of putting
a new product on the market, and a jet equipped with a "V" tail is undoubtedly fresh.
3.4.1 V tail principle
A good method for designing a "V" tail is to start from a standard cruciform tail configuration. This design is conducted
through a statistical method stated in Aircraft Design : A conceptual Approach [3]. The V surfaces then should be sized so
that they provide the same total surface area 4:
3A complete performance study, as in section 5.3, was performed. However, in the interests of brevity, it won’t be detailed in this report.4As stated in Experimental verification of a simplified vee tail theory... [9].
3 AIRCRAFT DESIGN CHOICES 22
SVtail,exp = SH +SV , (3.8)
where SH and SV are the horizontal and vertical surfaces previously provided.
The tail dihedral then should be set as follows :
Vangle = tan−1
(√SV
SH
)(3.9)
leading to the reference area :
SVTail = SVtail,exp cos(Vangle). (3.10)
Note that requirement brought by Eq. 3.8 implies that once projected in the horizontal and vertical plane, the surfaces will
not be equal to the statistically provided ones.
3.4.2 Statistically prescribed horizontal and vertical surfaces
As stated previously, the surfaces of a cruciform tail can be sized statistically with respect to some dimensions of the plane. A
good approach is to express the effectiveness of the tail in a non dimensional way : through the tail volume coefficient. These
are defined as follows :
cvt =lVTSVT
bwSwfor the vertical tail (3.11)
and
cht =lHTSHT
CwSwfor the horizontal tail, (3.12)
with
• li the arm moment, commonly approximated as the distance between the tail quarter chord and the wing quarter chord.
• Cw is the wing mean chord.
• bw is the wing span.
Then typical values of these coefficient for business jet can be found in databases or as conservative averages based upon data.
Such values can be found in Aiplane Design ([10]), and typical values for business jet-like aircraft are cht = 1 and cvt = 0.09.
At this stage the moment arm for both surfaces can be estimated by a percentage of the fuselage length: for an aircraft with
aft-mounted engines, the tail arm is typically about 45-50% of the fuselage length. Inverting Eq. 3.11 and 3.12 provide
statistically prescribed horizontal and vertical surfaces. Note that these are quite raw approximations but since they are meant
to be modified by the projection in the V plane and in the scope of a preliminary design, the accuracy achieved on the surfaces
is estimated good enough.
3 AIRCRAFT DESIGN CHOICES 23
3.4.3 Design choices and geometry determination
Now that the surface of the "V" tail is determined, the other geometrical parameters are determined using empirical relation-
ships. These make use of main parameters of the wings and statistical range of values.
The choice of the airfoil for the tail is basically motivated by the same considerations as in the section 3.2.1. Tails usually
have symmetrical airfoils: since the lift to produce is small there is no need for a camber. Drag reduction is an everlasting
goal for the design, and a good method to achieve it for the tail is to choose a thin profile. On the other hand, thin profiles
are structurally weak and might need more structural reinforcement than thicker ones, resulting in an heavier tail. Therefore
a compromise has to be made between aerodynamic and structural/safety properties. The search is conducted in the NACA 4
digits series: these provides well known, easy to manufacture airfoils with good performances. Among the different available
symmetrical airfoils, the NACA 0010 is chosen: this thickness is a good trade-off between the drag the profile will generate
and the structural strength it provide to the tailplane.
Statistic data on subsonic jet from Aiplane Design ([10]) and Aircraft Design : A conceptual Approach ([3]) provide ranges
for the geometric parameters of horizontal and vertical surfaces. But since the "V" tail is neither horizontal nor vertical, these
ranges should be handled with caution. Typically, values in the mid range are chosen for some virtual horizontal and vertical
surfaces, then projected in the V plane. This way the horizontal and vertical projection of the parameters of the "V" tail remain
in the statistic range.
To ensure that the tailplane will stall after the wing, it is recommended for the horizontal tail to present a greater sweep
angle than the wing (commonly 5◦). The sweep angle of the vertical tail is typically between 35 and 55◦. Once projected back
in the V plane, the resulting sweep angle of the "V" tail equals 48.5◦ as resumed in the Tab.3.7.
The aspect ratio of the "V" tail can reasonably be approximated as a fraction of the one of the wing. Since the aspect ratio
of the horizontal tail is approximately one half of the aspect ratio of the wing, and since the aspect ratio of the fin is usually
lower than 2, it is clear that the factor for the AR of the "V" tail will be lower than 0.5. Indeed, after projection :
ARV Tail = 0.36ARWing = 3.24 [−]. (3.13)
Assuming the value of the taper ratio in the middle of the range prescribed for an horizontal tail: taperV Tail = 0.4 [−].
The remaining part of the geometrical parameters: span, mean chord, chord at the root, chord at the tip and mean aero-
dynamic chord, can be determined from these choices according to empirical relationships. The results are presented in Tab.
3.7.
3.5 Undercarriage
In this section will be discussed the process of selecting the undercarriage geometry such as height, wheel base, wheel track,
and the distance between main gear and aircraft center of gravity. In this selection several constraints have to be accomplished
to guarantee safe landing, takeoff and taxi operations.
Table 3.9: Weights of the different subsystems of the aircraft, provided in [lbs]
3.6.2 Payload weight
The payload weight is provided in the Request for Proposal, and is estimated to 3000 [lbs] (2 crew members weighing 200
[lbs] each, 8 passengers weighing 200 [lbs] each, and an additional 1000 [lbs] of luggage.
3.6.3 Weight of fuel
The maximum fuel weight to reach a given range is an important data, for both structural and financial reasons. In the
following, empirical relations will be used to estimate the fuel requirements of the 8 seats aircraft to perform a 2500 nmi trip
with full payload (2 crew + 8 pax + luggages).
Phase by phase, the following formulae, provided mainly by ref. [4] are used:
1. Taxi and takeoff:
Wfuel,T&TO = 0.0035 ·WTO (empirical). (3.25)
With WTO, the take-off weight.
2. Climb:
The empirical expression 3.26 is provided by [4], and is used for a climb angle of ' 10 ◦.
Wfuel,climb =WTO
100
(altcruise
31600+
12
M2cruise
). (3.26)
With Mcruise = 0.85 the design mach number in cruise, and altcruise = 35,000 ft the cruise altitude.
3. Cruise:
3 AIRCRAFT DESIGN CHOICES 30
The Bréguet range equation is used to assess the fuel consumption in cruise.
Wfuel,cruise =Wi
(1− 1
expξ
)where ξ =
R ·SFCcruise
a0 ·√
θ ·Mcruise·CD
CL(3.27)
Where Wi =WTO−Wfuel,T-TO−Wfuel,climb is the weight at the beginning of the cruise phase. R ft is the range, expressed
in feet. a0 is the speed of sound, and θ is the temperature ratio Talt,cruise/TSL, computed using the International Standard
Atmosphere as a reference. CD [-] and CL [-] are respectively the drag and lift coefficients, available in Tab. E.1.
SFCcruise is the specific fuel consumption, corrected with respect to the altitude.6
4. Reserve:
NBAA IFR Range with 100 nmi Alternate was demanded in the statement. One can approximate it by 100 nmi in cruise
condition. So, all other things being equal, Eq. 3.27 is re-used.
5. Descent:
As a first approximation, the descent is considered as a continuation of the cruise phase.
6. Landing and taxi:
Wfuel,L&T = 0.0035 WTO (empirical). (3.29)
3.6.4 Results
The different component’s masses are detailed on Fig. 3.13.
3.6.5 Center of gravity
The components of the planes and their respective weights have been estimated in the previous sections. Fixing the reference
frame at the nose of the plane, the longitudinal location of the center of gravity is computed as follow:
xCG =W−1total
n
∑i=1
xCGiWi, (3.30)
the i subscript designating the different components of the plane.
4 different configurations are considered:
• full plane with full fuel tank (corresponding to the maximum takeoff weight),
• full plane at mid-cruise,
6From [11], the following empirical formula (valid for engines with BPR > 2) is used, evaluating the SFC at cruise with respect to the TSLS:
SFCcruise(lb/(h · lb f )) = 0.8−0.00096√
Tsls. (3.28)
Computation relative to the SFC of example business-jets show an increase about 65% at cruise condition. In the specific case of the PW535-E, the SFC risesto 0.77. App. B also confirms such a number.
3 AIRCRAFT DESIGN CHOICES 31
Empty Weight
9977 lbs
Tai
l -
427 l
bs
Fuel
5295 lbs
Payload
3000 lbs
55% 29%16%
Fuse
lage
- 2452 l
bs
Engin
es -
2237 l
bs
Win
gs
- 1989 l
bs
Syst
ems
- 2872 l
bs
Cre
w -
400 l
bs
Pas
senger
s -
1600 l
bs
Luggag
es -
1000 l
bs
Tax
i/T
ake-
Off
/Lan
din
g -
392 l
bs
Cru
ise
- 4743 l
bs
Res
erve
- 160 l
bs
4% 25% 22% 20% 29% 33%53% 7%13% 3%90%
Take-Off Weight - 18272 lbs
Figure 3.13: Aircraft’s mass distribution for 8 seats business-jet at MTOW and covering a range in cruise of 2500 nmi.
• full plane on reserve,
• totally empty plane.
The shifting of the center of gravity during the cruise will be important in determining if the plane meet the stability
requirements at all time. The positioning of the components of the plane should therefore aim to reduce as much as possible
the variation of the center of gravity between the different flight phases. This is partly achieved by placing the centre of gravity
of the main source of mass variation, the fuel tank, at the centre of gravity of the plane. Solving the Eq. 3.30 gives:
xCGMTOW = 27.70 [ f t]
xCGmid-cruise = 27.73 [ f t]
xCGempty = 27.81 [ f t]
xCGreserve = 27.76 [ f t]
At it can be noticed the variation is very low ! However, this depict an idealized variation of the fuel weight, which assume
that the center of gravity of the fuel does not move and which does not take into account the actual repartition mechanisms
in the fuel tank Components locations following the same reference frame can be visualized in Fig. 3.14 and are numerically
defined.
3.7 Catia model
A 3D model of the plane has been realized by Catia software in order to have a realistic view but at the same time an other
tool to evaluate the CG location of the plane.
3 AIRCRAFT DESIGN CHOICES 32
1 2 34 5 6
9
10
8
7
MTOWEmpty
Mid-CruiseReserve
x
z
12 141311
12345
5bis
01.906.239.0723.6822.54
ACCrew
InstrumentsLanding gears
FuselageElectrical
24.0327.5628.1328.1328.6834.59
678
8bis910
FurnituresFuel
WingsWings hydr.
Control SurfacesLuggage (cabin)
38.4537.4644.7447.9647.96
11121314
14bis
PropulsionLuggage (aft)
APUTail
Tail hydr.
# xi [ft]Item xi [ft]# Item xi [ft]# Item
Figure 3.14: Determination of the CG of each constituent of the aircraft.
By the same software it is possible to have also an evaluation of the inertia contributions.
A whole 3D aircraft visualization is given in Fig. 3.15.
To compute the CG of the whole plane concentrated masses of the different components have been inserted in the Catia
model in correspondence of the CG location of each of them basing on the data given by Matlab. In such study two cases
have been analyzed, one corresponding to the MTOW and one corresponding to the ZFW.
The concentrated masses used and the corresponding CG location are shown in Tab.3.10.
In Tab.3.11 are instead shown the results obtained in the two cases.
The results obtained in Catia and Matlab are approximately the same and so we can conclude stating that we have a good
estimation of the CG location. In particular it is possible to notice that in both the cases the symmetry of distribution of the
weight with respect to the y axis is verified.
4 TRADE-OFF STUDY 33
Figure 3.15: CATIA 3D model
Component Mass[lbs] xCG[ f t] zCG[ f t]
Fuselage, nose and aft 2452.454 23.68 0.00Wing 1989.461 28.13 -1.31Tail 426.711 47.96 4.47
Table 3.10: Concentrated masses and CG location of the different aircraft components.
MTOW ZFWCATIA MATLAB CATIA MATLAB
Mass [lbs] 1.956e+04 1.827e+04 9.996e+03 9.977e+03xCG[ f t] 27.12 27.70 27.56 27.81yCG[ f t] 0 0 0 0zCG[ f t] -0.21 -0.36 -0.18 -0.22
Ix[lbs/ f t2] 4.606e+05 1.115e+04Iy[lbs/ f t2] 2.198e+06 8.586e+05Iz[lbs/ f t2] 1.813e+06 8.476e+05
Table 3.11: Masses, CG location and inertia
4 Trade-off study
The trade-off study is performed for the 8-seats configuration, which is the optimal one.
4 TRADE-OFF STUDY 34
4.1 Aspect ratio of the wing
In the section dedicated to the wing design, the conclusion was made that the optimal value of the Aspect ratio was 9. Here,
it is changed by 10% about this value in order to evaluate if this choice was optimal or if it needs to be changed.
An important parameter that is closely related to the aspect ratio is the aerodynamic wing loading, defined by the ratio L/S.
The variation of this quantity with respect to the aspect ratio is presented in Fig. 4.1 (Left). Despite the fact that the variation
of the magnitude seems small, the represented graph depicts a trend that might be surprising at first approach: the higher the
aspect ratio, the lower the wing loading. The small variation of the aspect ratio makes the magnitude of the aerodynamic
loading variations very small too. For bigger variation of the aspect ratio, significant variation in amplitude can be expected.
The unexpected trend depicted in this graph is explained by the fact that a small increase of the aspect ratio does not necessarily
imply an increase of the wing loading. Indeed, an increase in the aspect ratio can result in an increase of the exposed surface
if the span also increases as well as an increase in the aspect ratio can imply a decrease of the lift due to a smaller surface of
the wing, thus a smaller weight of the wing. The effect of the wing weight would seem negligible, but the depicted trend here
is also of negligible amplitudes. As a result, it can be conclude that, in our case, a small variation of the aspect ratio around
its initial value does not change the aerodynamic loading of the wing.
The influence of the aspect ratio on stability is also of big interest. Fig. 4.1 (Right) illustrates at what extent the Aspect ratio
has an effect on stability. This figure shows that the plane becomes more stable and less manoeuvrable as the stability margin,
Kn increases. For an aspect ratio of 10, the stability margin at takeoff is about 10 % which exhibit less manoeuvrability.
In conclusion, an aspect ratio of 9 is the best and the most reasonable choice of design considering the observations of Fig.
4.1 and discussions in section 3.2.
75.6176
75.6176
75.6176
75.6176
75.6176
75.6176
75.6176
75.6176
8 8.5 9 9.5 10
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
Aspect ratio [-]
8 8.5 9 9.5 10
Aspect ratio [-]
Aer
odynam
ic l
oad
ing [lbf]
Sta
bil
ity m
argin
Kn [%
]
Takeoff
CruiseLanding
Figure 4.1: Evolution of some important parameters with the aspect ratio AR for several flight configurations. Left: Evolutionof the aerodynamic wing loading, L/S. Right: Evolution of the stability margin Kn.
4 TRADE-OFF STUDY 35
4.2 3D lift coefficient (CL,w)
Four important parameters on which the variation of the 3D wing lift coefficient, CLw , has an impact are the takeoff weight,
the aerodynamic loading of the wing, the CL,plane/CD,plane and the stability margin, Kn. Their variations with respect to the 3D
lift coefficient are represented in the figures 4.2 and 4.3.
It can be seen in the figure 4.2a that the takeoff weight decreases with respect to an increase of the CLw parameter. This is
explained by the following formula:
L =12
CLρv2S, (4.1)
where L is the total lift of the plane expressed in lbf, ρ is the density expressed in lb/ft3, v is the free stream velocity seen
by the plane expressed in ft/s and S is the reference (or gross) area of the wing expressed in ft2. If the 3D wing lift coefficient
increases, the gross area of the wing has to decrease in order to keep the same value of the lift which is fixed. Indeed this
surface is the only free variable in Eq. 4.1. As the surface decreases, the weight of the wing also decrease as well as the
weight of the tailplane, even if this last has a negligible effect.
Another effect close to this result is the variation of the aerodynamic loading with respect to the CLw . Indeed, if the CLw
increase, the wing surface decreases in accordance with Eq. 4.1, while the lift remains the same. As a result, the aerodynamic
loading increases with the value of the lift coefficient. This trend is represented in the figure 4.2b.
0.27 0.28 0.29 0.3 0.31 0.32 0.334
6
8
10
12
14
16
18
20
Wto&
L[1000lbs]
CLw[-]
Takeoff weightFuel weight
(a) Takeoff weight and lift.
0.27 0.28 0.29 0.3 0.31 0.32 0.3368
70
72
74
76
78
80
82
84
Aero
dynamic
wingloadingL/S[lbf/ft
2]
CLw [-]
(b) Aerodynamic wing loading.
Figure 4.2: Evolution of the takeoff weight, lift and aerodynamic loading with the 3D lift coefficient, CL.
The figure 4.3a shows the increase of the ratio CL,plane/CD,plane with respect to the CLw . This behaviour can be explained
by the fact that the lift coefficient of the wing constitutes approximately 90 % of the total lift generated by the whole plane.
What is more, the low drag bucket region of the wing extends from values of the lift coefficient of approximately 0.2 to 0.7.
4 TRADE-OFF STUDY 36
Thus, an increase of the CLw in this range does not change the value of the drag generated by the wing, CDw . As a result, the
ratio CL,plane/CD,plane increases when the CLw increases. Needless to say, this is true only for values of the CLw ranging from
0.2 to 0.7.
The last parameter to be considered is the stability margin, Kn for each phase of flight. This is presented in figure 4.3. This
last figure shows that for an increase of 10 % of the CLw value, the stability margin is contained between 5 to 7 % which is
acceptable. In another hand, for a decrease of 10 % of the CLw value, the stability margin reaches values ranging from 12 to
15 % depending the phase of flight. These are still in the required range for both stability and maneuverability. Therefore
since the requirement on this parameter are met regardless of the lift coefficient variation, it is not critical in determining an
optimum value.
As a result the chosen value of 0.3 for the 3D lift coefficient of the wing seems optimal: although a higher value would
decrease the weight of the plane, it would drastically increase the aerodynamic loading of the wing. The structure of the wing
would need to be stronger, therefore heavier. It is complex at this point to measure at which point the decrease of the weight
brought by smaller wings is compensated by a heavier internal structure, as the latter should be designed multiple times.
However,
0.27 0.28 0.29 0.3 0.31 0.32 0.3311
11.5
12
12.5
13
13.5
CL,plane/CD,plane
CLw[-]
(a) CL,plane/CD,plane.
0.27 0.28 0.29 0.3 0.31 0.32 0.330.04
0.06
0.08
0.1
0.12
0.14
0.16
Sta
bilitymarg
inK
n[%
]
CLw[-]
TakeoffCruiseLanding
(b) Stability margin, Kn.
Figure 4.3: Evolution of the CL,plane/CD,plane with the 3D lift coefficient of the wing, CLw , for several flight configurations.
4.3 Fuselage length
A last important parameter will complete this trade-off: the fuselage length.The result of this last trade off is shown in Fig.
4.4. It can be observed that any variation of the fuselage length about its length of design, tend to decrease the surface of the
wing. That increases the aerodynamic loading and has a negligible effect on the weight as it can be seen in Fig. 4.4b. What
is more, the stability margin diverges between the different phases of fly and reach unacceptable values about 50 ft. On the
contrary, the stability margin is quite good below the length of design. However, the jet has been chosen to offer enough space
5 OPTIMIZATION 37
45 46 47 48 49 50 51230
230.5
231
231.5
232
232.5Winggross
surface,S
[ft2]
Fuselage length [ft]
(a) Wing gross area, S.
45 46 47 48 49 50 515
10
15
20
Wto&
L[1000lbs]
Fuselage length [ft]
Takeoff weightFuel weight
(b) Fuselage length.
45 46 47 48 49 50 510.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
Stabilitymargin,K
n[%
]
Fuselage length [ft]
TakeoffCruiseLanding
(c) Stability margin, Kn.
Figure 4.4: Evolution of the wing gross surface, takeoff weight, fuel weight and stability margin Kn with respect to the fuselagelength.
to meet expectations of passengers in terms of comfort. Changing the fuselage length of the 8 seats configuration is thus to
exclude. On the contrary, from the 6 seats point of view, such a results tend to enforce the fact that a shortage of the fuselage
length is the most suitable approach for its design.
5 Optimization
5.1 Stability
The stability of an aircraft is defined as its ability to remain or return in its equilibrium position after a perturbation. Indeed
although in flight the plane is in equilibrium, its flight dynamics must be such that unexpected phenomena do not lead to
5 OPTIMIZATION 38
durable nor violent perturbations of the flight. On the other hand a "too much" stable aircraft is hardly manoeuvrable; the
stability is therefore a compromise between safety and manoeuvrability.
5.1.1 Enforcing equilibrium
According to the methodology most of the geometry of the plane is now fixed: the location of the center of gravity and
therefore the torques caused by both the camber of the wing and the lift are known for different flight configurations.
Therefore defining ML as the force moment resulting from the aerodynamic moment of the wing and of the torque of the
lift from the wing in mid-cruise:
ML = Mw−L(xL,app− xCG05), (5.1)
the equilibrium of the plane is reached if the tail produce a lift counteracting this moment. The lever arm of the tail lift being
known, the longitudinal equilibrium is enforced with:
Ltail =−ML
armtail. (5.2)
To produce this lift, the V surfaces must have the incidence angle leading to the required lift coefficient. As a matter of
fact this incidence angle should not be high since an adverse effect is that the tail would produce a lot of drag. Typical range of
value is between 0 and 3◦. Note that since the chosen plane configuration is "CG forward", the tail has to produce a downlift.
Therefore this angle will be negative.
Aircraft Design - Synthesis and Analysis [4] provides an estimation of the lift coefficient taking account of the effect of the
wings wake on the flow perceived by the tail.
CL,tail =CL,α,tail ·(
CL,wing
CL,α,wing
(1− dε
dα
)−αL,0,root−ηT
). (5.3)
where
• dε
dαis the downwash gradient produced by the wing wake on the flow perceived by the tail. According to the definition
provided in ([4]), this gradient is equal to 0.29.
• ηT = iT − iw is the angle between the wing chord and the tail chord at the root.
Note that the V tail disposition may change the way the flow is affected by the downwash gradient since it is not located at
the same place in the wake as a conventional tail. At this stage of the design, without any clues to what extent the "V" tail is
differently affected, the full effect is taken into account. Further studies may have to be conducted to quantify the effective
effect of the wake on the tail.
A sweep of different values of the tail incidence angle allows to find the value for which the tail produces the requested
lift. This angle equals -2.9 ◦which as required is lower than 3◦.
5 OPTIMIZATION 39
5.1.2 Longitudinal stability
As stated previously, the longitudinal equilibrium is enforced by Eq. 5.2. This equilibrium can be rewritten in the form of a
residual momentum which has to be null:
Mres = ML +Ltail ·armtail
= Mw−L ·(xL,app− xCG05)+LTail ·armTail
= 0 (at equilibrium).
(5.4)
A good way to characterize the stability of the longitudinal equilibrium is by its derivative with respect to the angle of
attack, as set out in Aircraft Design - Synthesis and Analysis [4]. A stable equilibrium is then characterized bydMres
dα< 0.
Expressed in a more general way, this stability condition can be written as∂Cm,plane
∂α< 0, and since the lift coefficient CL is
proportional to α the stability limit can be approximated as
∂Cm,plane
∂CL,wing< 0. (5.5)
It is now possible to express the total momentum coefficient Cm as the sum of all its contributions, as proposed in [4] :
Cm,plane =Cm,0 +CL,wingxCG05 − xL,app
c+Cm,tail−CL,tail
Stail ·armtail
c ·Sw+Cm,fuselage. (5.6)
As it can be seen the second term depends on the position of the center of gravity of the plane. Therefore the variation of
the center of gravity also impact the stability and is a major factor. The neutral point xn = hn ·c is defined as the position of
the center of gravity for which the momentum derivative from Eq. 5.5 equals zero. Deriving Eq. 5.6 with respect to the lift
coefficient provides the expression of hn :
hn =xL,app
c+ armtail ·
STail
c ·Sw·
dCL,tail
dCL,w−
dCm,fus
dCL,w; (5.7)
Where
•dCL,tail
dCL,w=
(1− dε
dα
)·CL,α,tail
CL,α,w,
•dCm,fus
dCL,w=
kfus · fus2width · fuslength
Sw ·c ·CL,α,w,
• kfus is an empirical parameter, defined with respect to the fuselage geometry and the wing location.
A convenient way to quantify the stability of an aircraft is to express the distance between the center of gravity and the
neutral point, normalized by the mean aerodynamic chord. This define the stability margin:
5 OPTIMIZATION 40
Kn =xn− xCG
c(5.8)
According to the Federal Aviation Administration requirements, the stability margin of an aircraft must be greater than 5%
in order to ensure sufficient stability. Requirements on the upper limit vary depending on the tail configuration of the plane
since some tails provide more stability than others. Without specific requirements concerning the "V" tail, the acceptable range
of stability margin is assumed to be similar to the standard stated in the FAR, with a higher upper ceiling to favor stability:
from 5% to 15%.
As the center of gravity vary during the flight, the stability margin fluctuate as well. Computed for the different flight
configurations stated in the section 3.6.5, the stability margins are:
• Maximum takeoff weight : Kn = 9.24%.
• Mid cruise weight : Kn = 8.75 %.
• End cruise (on the fuel reserve) : Kn = 8.22 %.
• Empty weight : Kn = 7.3 %.
5.1.3 Lateral stability
The variation of the moment coefficient in yaw can be computed using the following formula presented in Aircraft Design -
Synthesis and Analysis [4])
CN,plane =dCN,plane
dβ·β , (5.9)
with,
dCN,plane
dβ=
dCN
dβ+
dCN,fus
dβ+
dCN,w
dβ, (5.10)
where,
• CN,plane is the total 3D moment coefficient of the whole plane in yaw;
• dCN,planedβ
is the variation of the total 3D moment coefficient of the whole plane in yaw with respect to the yaw angle.
Eq. 5.10 is nothing else than the derivative of the moment coefficient in yaw with respect to the yaw angle to which we
add respectively the effect of the fuselage and the wing. dCN,wdβ
is equal to 0.024 for low-mounted wing for low-mounted wing
according to lecture 4, page 57 from Pr. Noels’ course notes [6].
The results of the moment coefficient derivatives are presented in Fig. 5.1.
5 OPTIMIZATION 41
CL,w
of the wing [-]
CM
of
the
pla
ne
[-]
-0.2
-0.16
-0.12
-0.08
-0.04
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Yaw angle β [°]
CN o
f th
e pla
ne
[-]
-10
-8
-6
-4
-2
0
2
4
6
8x10-3
-3 -2 -1 0 1 2 3 4
Figure 5.1: Left: Variation of the moment coefficient of the overal plane in pitch with respect to the 3D lift coefficient of thewing. Right: Variation of the moment coefficient of the overall plane in yaw with respect to the yaw angle.
5.1.4 6 seats consideration
The "V" tail remain geometrically the same as for the 8 seats version. It still have to balance out the moments around the
center of gravity, it therefore imply a variation of the tailplane angle with respect of the fuselage axis, compared to the other
member of the plane family. To produce the adequate lift, the "V" tail must have an incidence angle of −3 [◦].
The stability of the 6 seats version is computed the same way as for the 8 seats: taking account of the diverse variations,
notably of lift coefficients of the tail and wing, the stability margins are respectively:
• Maximum takeoff weight : Kn = 8.84%.
• Mid cruise weight : Kn = 11.33 %.
• End cruise (on the fuel reserve) : Kn = 14 %.
• Empty weight : Kn = 8.5 %.
These are well in the stability range defined previously in section.
5.2 Aerodynamics
5.2.1 Computation of CL,plane and CD,plane
In the conceptual design, a first approximation was made to assess the lift coefficient of the plane by assuming that all the lift
was generated by the wing. Nevertheless, in the interest of rigour, lift of the tail had to be considered for stability computation.
A more accurate assessment of the lift coefficient is thus given by the following formula
CL,plane =CLw +CLTST
S, (5.11)
5 OPTIMIZATION 42
where,
• CL,plane is the total lift coefficient of the plane, neglecting the effect of the fuselage;
• CLw is the lift coefficient of the wing, set in section 3.2;
• CLT is the lift coefficient of the tail;
• ST is the reference surface of the tail;
• S is the reference (gross) area of the wing.
Like the lift coefficient of the plane, the drag coefficient of the plane was first estimated from the drag generated by the
wing on itself. However, to pursue the study more in details, we have to consider the interference drag due to interaction
between components. This induce a drag component, necessary to compute at the end the total drag of the whole plane. To
get a better idea of the drag coefficient, we can use the formula of the polar drag presented in the course notes in the slides
from Mr. Noels [6]:
CD,plane =CD0 +C2
L,plane
eπAR, (5.12)
where,
• CD,plane is the total drag coefficient of the plane, neglecting the effect of the fuselage;
• CD0 is the independent drag that takes the compressibility and 3D effects independent of the 3D lift into account;
• CL,plane is the total lift coefficient of the plane computed with the formula 5.11;
• e is the Oswald efficiency factor;
• AR is the Aspect ratio of the wing.
As well as the total lift coefficient of the whole plane, here the effect of the fuselage has been neglected. Although the
approximation for the total lift coefficient of the plane is not so important, it is no longer true when it comes to the drag. A
more accurate study has to be performed for the drag. This is the subject of section 5.2.3.
5.2.2 TRANAIR
TRANAIR is a software analogous to a typical panel method program developed to analyze compressible flow over complex
configurations at subsonic, transonic or supersonic freestream Mach numbers. The numerical method solves the full potential
equation subject to a set of boundary conditions. The solution is obtained on a sequence of successively refined grids which
are constructed adaptively based on estimate solution errors.[12]
5 OPTIMIZATION 43
Angle of attack α [°]
CL o
f th
e p
lan
e [-
]0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-3 -2 -1 0 1 2 3 4
0.04
0.035
0.03
0.025
0.02
0.015
-3 -2 -1 0 1 2 3 4
Angle of attack α [°]
CD o
f th
e p
lan
e [-
]
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.015 0.02 0.025 0.03 0.035 0.04
CL o
f th
e p
lan
e [-
]
CD of the plane [-]
Figure 5.2: Left: Variation of the lift coefficient of the plane with respect to the angle of attack. Center: Variation of the dragcoefficient of the plane with respect to the angle of attack. Right: Polar drag of the plane.
5.2.2.1 Structure mesh
The wing and tail elements being fully defined mathematically, it is quite easy to model them into a mesh. On the other hand,
the fuselage has a more "free hand" aspect to it, and has been meshed at a 100% true to what has been drawn in our CAD
models. This way, the defects and geometrical flaws will be easily tracked, not unlike what would be studied in a wind tunnel.
The full mesh is visible in Fig. 5.3 in a rendered version. Notice that only half the plane is there, Tranair resorting to symmetry
to spare computation time.
Figure 5.3: Rendered plane mesh.
Winglet contribution This device helps reducing induced drag caused by wingtip vortices. The design of those winglets
has been based on Whitcomb’s winglet design[13] and adapted to our geometry. As can be seen in Fig. 5.4, an optimum in
this drag reduction is reachable. The chosen parameters are labelled; notice that the ultimate minimum has not been selected
for the twist because it would have produced too steep of an angle at the junction with the wing.
5 OPTIMIZATION 44
λ [rad]1.32 1.34 1.36 1.38 1.4 1.42 1.44
CD
[-]
0.0426
0.0426
0.0427
0.0427
0.0427
0.0428
0.0429
λ: 1.37
CD: 0.0426
(a) Sweep angle.
CD
[-
]
α [degree]0 2 4 6 8 10 12 14 16 18
0.0426
0.0428
0.043
0.0432
0.0434
0.0436
0.0438
α : 14C
D: 0.04267
(b) Longitudinal geometrical twist.
Figure 5.4: Drag coefficient reduction with winglet parameters (wing element alone).
5.2.2.2 Cruise condition calculations
Launching a computation to simulate cruise condition, results are finally coming to life. As seen in Fig. 5.5, there is a strong
depression (corresponding to a shockwave) at the point of junction between the cabin and the aft. Indeed, this point has a
sharp angle and we have seen that the software struggles a bit with those. Nevertheless, it shows that the angle may be too
sharp and that the structure needs to be smoother at that junction. Moreover, we can see another depression point on top of
the nose, at the beginning of the cabin. Again, smoothing that connection with more flat curve would probably improve the
aerodynamics of the fuselage. Finally, a strong pressure concentration can be seen on the tip of the nose and at the angle
starting the bulge corresponding to the cockpit. Those area being constrained for practical reason, improving them may be
difficult but we would need more precise constraints to decree.
0.7456
0.5237
0.3019
0.08
-0.1418
-0.3637
-0.5856
-0.8074
-1.0293
-1.2511
-1.4730
Figure 5.5: Pressure distribution along the structure.
Other features of interest are the lift and pressure coefficient distribution. The latter, visible in Fig. 5.6 (left), shows a
rather strong and abnormally oblique shock. Two reasons may be causing this phenomenon:
• The study is made in non-viscous conditions, which means that in reality, the shock would be less strong and moved
forward along the chord.
5 OPTIMIZATION 45
• As the profile is cambered, the normal vector to the surface at the leading edge is also strongly leant backward. The
shock is oblique with respect to the absolute axis but is normal to the surface towards the trailing edge.
The lift coefficient distribution is also visible in Fig. 5.6 (right). Notice that the curve breaks at the wingtip due to the
connection with the winglet, the shape of this curve is otherwise expected as the 2D lift coefficient is inversely proportional to
the chord length.
y/span [-]
CL [
-]
1
0.9
0.8
0.7
0.6
0.5
0.4
0.30.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x/c [-]
Cp [
-]
1
0.9
0.8
0.7
0.6
0.5
0.4
0.30 0.2 0.4 0.6 0.8 1
Intrados Extrados
Figure 5.6: Left: Pressure coefficient around the mean aerodynamic chord. Right: Sectional lift coefficient distribution alongthe normalized span.
Finally, the moment distribution along the span can be seen in Fig. 5.7. The moment being extremely important at the
trailing edge and of negative sign, it shows that the plane undergo a nose-down moment, in this case compensated by the
downforce exerted by the tail.
5.2.2.3 Performances comparison with conceptual design
Graphs presented in Fig. 5.8 make the comparison between theoretical model and results obtained from the simulation with
Tranair. It is worthwhile noticing that the simulation was performed under the assumption of an inviscid flow of same free
stream velocity than considered in theory. Differences can thus be expected, not only due to variation from the theoretical
theory to the simulation of the flow around the plane, but also due to the modification of conditions.
It can be observed in left graph of the figure 5.8 that the theory predicts the same trend of the CL behaviour than the
simulation. Indeed, the red curve is simply an offset of the blue one, such that the linear relation is nearly conserved. However,
the differences in magnitude implies some changes in terms of stability. Taking into account that the lift generated by the wing
constitutes almost 90 % of the total lift generated by the plane, it can be stated that the variation of Kn with respect to CL,plane
follows the same trend than the variation of Kn with respect to CLw . As a result, it can be assumed that the graph presented in
5 OPTIMIZATION 46
1.8192
0.9534
0.0877
-0.7781
-1.6438
-2.5095
-3.3753
-4.2410
-5.1067
-5.9725
-6.8382
QMY
Figure 5.7: Pitching moment distribution along the span (extrados up, intrados down).
Fig. 4.3b depicts the the general behaviour of the stability with respect to the CL,plane. That is, an increase in the lift coefficient
to 0.4 would decrease the stability margin to an unacceptable value well below the stability margin, such that the plane would
not be stable.
Another issue concerns the wing loading. Such values of CL,plane would require a CLw bigger than 0.3 and considering Eq.
4.1, the wing surface would be too small implying a too high wing loading.
Fig. 5.8 shows that the value of the overall drag coefficient of the plane is the closest to the the value obtained theoretically
for an angle of attack of about -0.5 ◦. The more we increase the angle of attack the more both solutions diverge.
The right graph of Fig. 5.8 shows that for an angle of attack of −0.5 ◦, CL,plane/CD,plane is worth 16.53 which is much
more optimal than what theory predicts for an angle of attack of -0.17 ◦ (value of design). However as explained earlier in this
discussion, such a value of the CL,plane does not lead to optimal results in terms of stability and wing loading.
The new derivative a for the whole plane corresponding to the slope of the CL with regard to the angle of attack is obtained
by linear regression, which gives:
a = 0.098224 [deg−1], r = 0.99996.
The theoretical line gave a slope of a = 0.0898 [deg−1] which is a difference of 8.6%.
5.2.2.4 Further improvements
A rather important offset has been seen between the theoretical lift coefficient curve and the one computed in Tranair,
corresponding more or less to a difference of 2◦of angle of attack. This last value is perfectly valid when a viscous modeled
is compared with an inviscid one, and even though the design point is not met as shown by the inviscid output, building a
viscous CFD model would undoubtedly lead to a result very close to the theoretical result.
5 OPTIMIZATION 47
Angle of attack α [°]
CL o
f th
e pla
ne
[-]
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1 -0.5 0 0.5 1 1.5 2 2.5
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
-3 -2 -1 0 1 2 3 4
Angle of attack α [°]
CD o
f th
e pla
ne
[-]
Conceptual design Tranair
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.015 0.025 0.035 0.045
CL o
f th
e pla
ne
[-]
CD of the plane [-]
Figure 5.8: Left: Lift coefficient with regards to the angle of attack. Center: Drag coefficient with regards to the angle ofattack. Right: Performance comparison with the theory.
5.2.3 Drag analysis
In the following section, the total drag is evaluated in detail by using the collection of formulas presented in ref. [14]. Those
methods only hold in cruise configuration for subcritical speeds (M < 0.85). Thus, transonic effects such as wave drag are not
accounted for in this drag breakdown. Here, the drag study is performed for the 8-seat configuration.
When usually analyzing performance of aircrafts, the equation 3.1, introduced in a previous section is used. Instead, a
further breakdown is preferred.
1. Vortex-induced drag: associated with the system of trailing vortices and downwash which a wing generates. This
particular drag is computed for each of aircraft’s main components regarded in isolation.
2. Profile drag: skin friction and pressure (form) drag result respectively from boundary layers and the form of the bodies
(so the separation of flows). Here, bodies are considered as streamlined, and isolated. As the aircraft is not designed
in detail, it is often convenient to use the relation between a well-streamlined body and a flat plate providing accurate
results. A skin friction coefficient is determined for the flat plate associated with the aircraft’s part studied. Then
correction factors are applied to allow for the body’s thickness and difference in development of the boundary layer.
3. Interference corrections: The interaction of flow around main aircraft components lead to consider corrections. Indeed,
flows of the wing and fuselage interact about the root for instance. This drag component is not usually more than 5-10%
of the total drag, when the aircraft is well-designed.
4. Protuberances, surfaces imperfections, etc: Windshield, external fuel tank and other excrescences are accounted for
drag study to be exhaustive. Despite the fact that the design is not yet entirely defined, statistical formulas and correc-
tions must in some cases be considered.
The Catia model is used to provide volumes, wetted areas of main parts of the airplane. Furthermore, the 3D lift coef-
5 OPTIMIZATION 48
ficient used to determine the CD aforementioned is that of the aircraft which combines CL,w and CL,Vtail. Fig.5.9 depicts the
distribution of the total drag in terms of its components aforementioned .
The reader may notice that the profile drag contributes to 66% of the total drag. Table 5.1 reports the contributions of the
aircraft’s main components. It is clear that the fuselage accounts for more than 50% of the profile drag and the wing for at
least 25%. The weaker contribution of the tailplane and engine nacelles results largely from their smaller size. Moreover, the
lack of information about gas generator cowlings or the plug in the flow leads to regard this nacelle profile drag as minimum.
The second highest contribution is that associated with surface imperfections and protuberances, which the external fuel tank
(belly) accounts for 50%. Also, rough bodies are proven to experience more rapidly an increase in drag than smooth ones do.
This lead to allow for a certain amount of drag representing about 20% of the profile drag increment. Then, the vortex-induced
drag component is for the most part due to the plane wing and horizontal tail, since the fuselage generates weak amount of
lift. Eventually, the interference drag is negative due to the wing/tailplane interference. The downwash creates a tail upload
and so reduces for a given airplane lift the vortex-induced drag of the wing.
The total drag coefficient and the drag force, using the plane lift coefficient obtained with Eq.5.11, are therefore
CD = 0.0266 (5.13)
and,
D =12
ρv2SCD = 1,348 lbf. (5.14)
5 OPTIMIZATION 49
The lift-to-drag ratio considering the plane lift coefficient becomes,
CL,plane
CD= 9.89 (5.15)
5.3 Performance
The aim of this section is to assess the flexibility of use of the aircraft, and the suitability of the aircraft’s component for
each flying phase. In the last section, the drag study and the Tranair aerodynamic coefficients will be integrated to the
assessment.
5.3.1 Take-off
As requested by the RFP, the aircraft must be compatible with a 4000 ft, FAR Part 25 compliant Take-Off Balanced Field
Length7. The upcoming section will assess this requirement.
A determining parameter is the decision speed V1, defined as the speed at which the full stop distance is equal to the
obstacle clearance distance. Indeed, for a speed V >V1, the aircraft’s take-off is mandatory. In addition to that, the aircraft is
required to clear an obstacle of 35 ft at a speed V2 ≥Vstall at the end of the runway, as mentioned in the FAR Part 25.
5.3.1.1 Method
The following method, as well as some parameters are provided by ref. [15]. 5 distincs segments are identified as
A: all engines operating up to the decision speed V1,
B: one-engine inoperative acceleration from V1 to lift-off speed VLO,
C: one-engine inoperative acceleration from VLO to V2 by clearing the obstacle,
D: time interval separating the one-engine lost and the actual braking action,
E: deceleration from braking action speed VB to the stop.
The ground distances of each phase is then computed using Eq. 5.16.
xi =∫ Vi+1
Vi
Va
dv. (5.16)
As an approximation, the acceleration a is considered constant and equal to V07 = 0.7 · (Vi+1− vi).
F = T −µR−D , with R =W −L. (5.17)
7The term "Balanced Field Length", as stated by ref. [15] refers to the one engine operative condition of take-off.
5 OPTIMIZATION 50
The average acceleration during ground run is therefore:
a = g[(
TW−µ
)−(
1/2 ρ CL V 207
W
)(CD
CL−µ
)]. (5.18)
By translating the BFL definition in terms of those 5 segments,it follows that the decision speed corresponds to xB + xC =
xD + xE .
5.3.1.2 Results
First of all, as the parameter provided the manufacturer is the uninstalled static sea-level thrust, some corrections must be taken
into account. The loss due to the engine’s placement is chosen at its maximum statistical value (i.e. 8%), and the correction
due to the aircraft’s motion at take-off is given by Eq. 5.19 (source: ref. [6]).
TTO,available
TSLS= 1− 0.45M ·(1+BPR)√
1+0.75BPR+(0.6+0.11BPR) ·M2 (5.19)
• Segment A: In this segment, the aircraft speeds up from 0 velocity to the decision speed V1. The flaps settings are
assumed constant and equal to 20◦, the take-off weight is WTO = 18272 lbs, and the wing loading W/S = 78.7 lbs/ft2.
The stall velocity is Vstall = 166.5 f t/s, and the Thrust-to-Weight ratio is computed to T/W = 0.3 [-]8. The tires on the
dry pavement exert a friction coefficient on the aircraft of µ = 0.03 [-]. Finally, the lift coefficient and the drag to lift
ratio are respectively approximated by CL = 0.5[-]9 and CD/CL = 0.1
• Segment B: In segment B, an engine loss must be considered. Therefore, the Thrust-to-Weight ratio T/W = 0.15.
However, as the speed increases, the lift coefficient CL increases to 0.9 [-], and the drag to lift ratio slightly increases to
0.102 [-], due to the appearance of asymmetry (effect of the engine loss).
As the lift increases, the rolling friction coefficient lowers to 0.025.
Notice that the FAR imposes VLO ≥ 1.1Vstall . It is here chosen to respect the equality.
• Segment C: According to statistics, the time to flaring is set to 3 s. A uniform straight-line movement relation is thus
applied, using a velocity between of 1.2V stall (between VLO and V2).
• Segment D: Estimating a 1s time for pilot recognition10, and 2s for the brakes to enter in service, the segment D is
assumed to last 3s overall from V1 to VB.
• Segment E: The segment E marks the deceleration from V1 to 0 [ft/s]. This deceleration is characterized by a zero thrust
as the remaining engine is shut down, and a breaking friction coefficient µbreak = 0.4 [-]. The lift coefficient to consider,
as stated by ref. [15] is CL = 0.5 [-].
8As mentioned before, TSLS is corrected by a factor 0.89This value is advised by ref. [15]. It is hard to evaluate it without any wind-tunnel experiments taking the Reynolds and ground effects into account.
10A state of alert is assumed, as the take-off phase usually requires pilots to be focused.
5 OPTIMIZATION 51
The Fig. 5.10 represents the decision speed determination V1, and the Fig. 5.11 summarizes the main results of the take-
off performance assessment. Specifically, it is shown that the take-off balanced field length is 3602 ft, which matches
the RFP’s requests, and gives flexibility to the jet’s owner to take-off from small runways.
Take-off distance from V1 (segment A+B+C)
Stopping distance from V1 (segment A+D+E)
120
2000 2500 3000 3500 4000 4500 5000
130
140
150
160
170
180
Dec
isio
n s
pee
d V
1 [ft
/s]
Balanced Field Length [ft]
Figure 5.10: Balanced field length and related decision speed.
35 ft.
Recogniti-
on (13%)
Braking (47%)
Flare (16%) Acceleration - 1 engine out (44%)
Balanced Take-Off Field Length - 3602 ft.
Acceleration - all engines (40%)
V2
= 1.2 Vstall
V0
VLO
= 1.1 Vstall
Vstall
166 ft/s V1
156 ft/sV
B
V0
H
SL
Alt
itude
[ft
]
RoC [ft/s]
Figure 5.11: Left: Approached RoC evolution with the altitude. Right: Speed evolution and segment distances.
With the take-off parameters fully determined, the FAR 25’s second climb gradient must be assessed11. This design allows a
climb angle of 3.5◦, and therefore a climb gradient of 6.1%, complying with the certification.
11The FAR states that the climb gradient must be ≥ 2.4%
5 OPTIMIZATION 52
5.3.1.3 6 seats take-off performance
The RFP requests does not differ from 8 seats one but it is important to notice that the FAR 23 imposes a 50 ft obstacle height
to clear. As main results, it is obtained: V1 = 141 ft/s and BFL = 2936 ft.
5.3.2 Climb
The climb requirements are clearly stated in the request for proposal (Rate of climb (RoC) of 3,500 fpm and service ceiling of
45,000 ft). To assess the ability of the aircraft to reach the prescribed RoC, an approximate method for estimating the time to
climb is used (source: ref [4]). This method considers bold assumptions:
1. The thrust is considered independent of the airspeed.
2. The RoC evolution is approached by a straight line from a 0 ft altitude to the service ceiling H.
This yields the RoC linear expression given by
RoC = RoCSL
[1− h
H
](5.20)
5.3.2.1 Details of the approximate method
If the unaccelerated rate of climb condition is assumed, it follows:
t2− t1 =∫ h2
h1
1RoC
dh =∫ h2
h1
1V sinγ
dh =∫ h2
h1
1RoCSL
[1− h
H
] dh =H
RoCSLln(
H−h1
H−h2
)(5.21)
ROCSL is directly computed. Using the straight line properties, H can be found too. In fact, only the RoC of another point
is necessary. The 35,000 ft (cruise altitude) one is chosen. Computations are carried out according to the subsequent set of
equations:
1. The first assumption gives the following lift coefficient (associated to the maximum rate of climb):
CL =− T
W +
√( TW
)2+12 CD0L K
2K, with K =
1e π AR
. (5.22)
2. The corresponding airspeed and drag coefficient are thus equal to:
V =
√W
0.5 ρ S CLand CD =CD0L +K CL. (5.23)
3. Hence, the flying path angle can be estimate via:
sinγ =T −D
W=
TW− CD
CL. (5.24)
5 OPTIMIZATION 53
4. At last, the maximum rate of climb is given by:
R/C =V sinγ. (5.25)
5.3.2.2 Results
Using this method, it is possible to verify to the aircraft fulfils the RFP. As shown in Tab. 5.2, the time to climb up to the
cruise altitude is lower than 10 minutes. Other time intervals are also computed for typical altitudes.
In addition, the approximation of the absolute aircraft ceiling leads to a value of 47,201 ft and a subsequent service ceiling
Table 5.2: Summary time intervals to reach several altitudes.
altitude of 45,006 ft. Finally, mean climb angle is assessed to 6.5◦ (mean climb gradient equals 11.4%), all engines operative.
Such angle justifies therefore the employment of the empirical formula for the climb fuel consumption (see section 3.6.3). In
case of one engine failure, mean climb angle is assessed to 1.9◦ (mean climb gradient equals 3.4%).
5.3.2.3 6 seats climb performance
Since both FAR and market demands does not change, it will be unthinkable to obtain worst results than for the 8 seat. Indeed,
following main results are obtained: 8.3 minutes to reach 35,000 ft for a service ceiling at 45,478 ft.
5.3.2.4 Brief comparison
In order to assess the validity of the previous results, the comparison has been made with a business jet on Fig. 5.12. As stated
by the author of the ref. [15], "the bizjet carries out the quasi-steady-state climb at a constant equivalent airspeed VEAS until
it reaches Mach 0.7. Thereafter, it continues at a constant Mach number until it reaches the ceiling". Noting that the Tsls used
by the author is 3,310 lbf by engine, which is thus very close to the PW535-E one, one can conclude by the reliability of the
present aircraft results.
5 OPTIMIZATION 54
6,451 ft/min
47,201 ft
48,750 ft
7,450 ft/min
Figure 5.12: Comparison between computed results (orange for 8 seats and blue for 6 seats configuration) and a business jetexample coming from [15].
5.3.3 Cruise
In the previous sections, the engines have been determined to meet the RFP requirements. It has been shown that the available
thrust is 26% higher than necessary12.
5.3.3.1 6 seats cruise performance
It has been assessed that the available thrust is 28% higher than necessary. This slight change is largely due to a higher angle
of attack for the 6 seats configuration.
5.3.4 Turning rate
In this section, it is assumed that the turn occurs in a horizontal plane according to a perfectly circular curve. This case is
called by Raymer [3], a "sustained turn". Even if in most cases, a turn is accompanied by a change of altitude, this assumption
provide a strong indicator of the actual turning performance.
12The method was based on the polar drag of the wing. As it will be shown in the further sections, a more detailed drag analysis may lead to somerefinements.
5 OPTIMIZATION 55
Figure 5.13: Turning diagram.[6]
5.3.4.1 Method
Balancing the forces acting on the aircraft, as in figure 5.13, it appears a load factor n. Thinking in terms of load factor is
convenient to assess structural limits but the human factor must be taken into account. Indeed, human comfort limits cannot
range as far as structural one and for commercial transports, it is usually defined a nmax = 2.5. Due to this maximum value
and with the lift coefficient which cannot be higher than CLmax, the upper limit of the turning rate is:
(dψ
dt
)max
= g
√ρ S CLmax
2W
(nmax−
1nmax
). (5.26)
As recommended by [3], a simple way to evaluate the maximum load factor in such a configuration is to think that:
n W = L ⇐⇒ CL =n W
1/2 ρ V 2 S. (5.27)
Since
n =T LW D
(5.28)
must be verified at each time, this finally leads, expressing the drag by a drag polar, to:
n =
√1/2 ρ V 2
K (W/S)
(TW−
1/2 ρ V 2 CD0
W/S
), with K =
1e π AR
. (5.29)
Expressing Eq. 5.29 in terms of wing loading and thrust to weight ratio emphasises the multiple computation options it offers.
5.3.4.2 Results
Only the cruise condition results are introduced here. The CL,max for the computation of maximum turning rate is therefore
equal to 0.8. This comes from the integration on the whole wing of the 2D cl,max in cruise condition (Re ' 9.6 ∗ 106 with
respect to figure D.1), multiplied by a correction coefficient of 0.9 (recommended by [7]) to finally obtain the 3D CL,max.
Moreover, a wing loading and thrust to weight ratio of respectively 65.7477 lbs/ f t2 and 0.1 are chosen, corresponding to the
5 OPTIMIZATION 56
mid-cruise aircraft mass and 80% of the available thrust.
Table 5.6: Structural loading at the wing root for the different points of the flight envelope.
5.4.4 Materials selection
The materials selection can become a really tough task when it comes to aeronautics. Indeed many properties of the materials
have to be taken into account to provide both a sufficient level of performance for the aircraft and safety for passengers. For
5 OPTIMIZATION 69
the design of an aircraft, many families ans sub-families of materials can be selected, depending on the aircraft’s specifications
and material use.
Aircraft design and material selection require a deeper prospection in the following mechanical parameters:
• hardness;
• yield strength;
• density;
• ductility;
• toughness;
• fatigue strength at 107 cycles.
The strengths of non metallic materials are essentially the very low density and the possibility to produce the desire
structure in one piece. Carbon composite, like carbon reinforced fibers for instance, may be attractive in the sense that this
material exhibit a competitive yield strength to density ratio and Young modulus to density ratio. However, brittle fracture in
this kind of material represents a challenge for cracks detection. This becomes even more problematic for riveting process.
On the other hand, the new processes used to manufacture these structures are still on research and actual ones can take a long
time.
In the interest of safety and simplicity, it will be preferred to use metallic materials for which manufacturing processes are
well-known and mechanical properties of an excellent level. To prevent from using too high density materials, it is common to
use light alloy category, which consists mainly in the use of aluminum with alloying element, giving it mechanical properties
as good as metallic materials. The aircraft will be designed using aluminum alloy as the main material. For the structure and
the skin of the aircraft (the fuselage, the outer part of the belly and the wing), wrought aluminum alloy 7075 T6 has been
selected. Wrought aluminium 5052-H32 is one of the most suitable material to design the inner part of the belly due to its
excellent corrosion resistance and workability. The properties of these materials can be given using the software CES. These
are shown in App. F.
5.4.5 Structure preliminary design
In order to perform a preliminary design of the structure, the complex structure is idealized into a simpler mechanical model
under particular assumptions on the stress distribution, as defined in Aircraft Structures for engineering students ([18]).
Since the stringers are expected to have small cross sectional dimensions compared with the complete section of the wing
or fuselage, the variation in stress over this cross-section is small. Assuming that the direct stresses are constant over the
stringers cross-section, a good working hypothesis is therefore to idealize the layout of stringers and skin into a combination
of direct stress bearing booms13 and shear stress bearing skin. This model actually loose the actual shear distribution as it only13Concentrations of area.
5 OPTIMIZATION 70
depicts the average of the shear flow between two booms. Moreover, the expressions for direct stresses used in the following
sections are based on the Euler-Bernoulli assumption that the plane sections remain plane after bending. This is actually not
the case as the bending moments Mx,My and Mz are not constant along the fuselage nor the wing. In addition, these are
considered to be of uniform cross-sections, which is false considering the taper of the wing and the variation of the fuselage
cross section in the aft. Nevertheless, these assumptions provide a good first estimate of the structure design, although several
aspects of stress distribution are missed. Further studies using more sophisticated models should be conducted in order to take
account of these effects, eventually leading to a lighter and more effective structure design. However from the perspective
of providing a preliminary design of the structure and after acknowledging the simplifications made throughout the loads
computation, the following designs are limited to the basic model.
The sizing of the structure elements is based on a elastic design approach: as the material should deform in the elastic
region the yield strength σyield and the tensile strength τstrength are taken as reference values. Then these are multiplied by a
safety factor s = 1.5, providing the maximum value that the stresses shall reach:
σmax =σyield
s=
73 [ksi]1.5
= 48.5 [ksi] (5.41)
τmax =τstrength
s=
48 [ksi]1.5
= 32 [ksi] (5.42)
defining the maximum stresses allowable in the elements.
5.4.5.1 Fuselage section
The fuselage section is circular, therefore a simple stringers configuration is to arrange them symmetrically and equally spaced.
For the sake of a quick preliminary design, all the stringers have the same cross-sectional area, which remains constant along
the longitudinal axis (no sectional variation with respect to the x axis). As shown in Fig. 5.24, the stringers are replaced by
booms located on mid-line of the skin. The number of stringers placed on the section is not to be chosen randomly as it is
linked to the ability of the structure to avoid unstable failure such as buckling. Indeed, closely spaced stringers associated to
the frames better maintain the fuselage shape. Moreover, ribs and stringers spacing play a major role in achieving a lighter and
more efficient structure. As stated in Effect of Ribs and Stringer Spacings on the Weight of Aircraft Structure for Aluminum
Material ([19]), stringers spacing of 5.9 [in] is found to be stabilizing for the weight of the structure, i.e more smaller stringers
would not make the structure weight less. This correspond to 36 stringers disposed around the fuselage section. The frames
are mainly required to maintain the fuselage shape, therefore they are nominal in size and spaced 11.8 [in] apart.
As the structure must resist to the loading at all the points of the flight envelope, the worst case of loading have to be
considered. The maximal value of bending moment is reached during dive, with My = 105,305 [lbf · ft].
The direct stresses in the fuselage section are induced by the moment My and the normal reaction force Nx. As the second
5 OPTIMIZATION 71
z
y
z
y
Dfus
Figure 5.24: Representation of the fuselage cross section modelling.
moment of area can be expressed in terms of the booms area B: Iyy = Iterm ·B ·Dfus, the stresses are written:
σxx =My
Iyyz− Mz
Izzy+
Nx
B
=My
Iyyz+
Nx
B
=My
Iterm ·B ·D2fus
z+Nx
B.
(5.43)
Considering booms of identical area, it can be deducted from Eq. 5.43 that the higher stresses are located on upper and lower
booms. Therefore, inverting the equation for z = Dfus and σxx = σmax provides the required booms area: B = 0.0736 [in2].
The skin must resist to the shear flow due to the shear load Tz. As for the direct stresses, the worst case is considered for
the design loading: the design shear load is reached during the dive, with Tz = 3,836 [lbf].
As stated in Aircraft Structures for engineering students [18], the average shear flow between two booms i and i+1 is:
qi+1−qi =− Tz
IyyBizi−−
Ty
IzzBiyi
=− Tz
IyyBizi
=− Tz
ItermD2fus
zi
(5.44)
From this equation it is possible to express by recurrence the shear flows on each panel with respect to the closure shear
flow value:
qi+1 = qs,0−Tz
ItermD2fus
(5.45)
Since the loading Tz is applied through the shear center of the cross-section of the fuselage, no torque is induced by the
shear force and the shear flow is symmetrical. Therefore the shear flows on the lower panels of the fuselage are opposite:
5 OPTIMIZATION 72
q6−7 =−q7−8. Taking the shear flow q1−2 as reference, it therefore be expressed as:
qs,0 = q1−2 =Tz
Iterm/D2fus
(∑6j=2 z j +∑
7j=2 z j)
2(5.46)
From Eq. 5.45 the average shear flow on each panel is therefore determined. The stress induced by this shear flow is
τi =qi
t(5.47)
where t is the skin thickness. Considering that t is uniform, it is will be designed accordingly to the maximum shear stress.
The corresponding shear flow shall be named qmax.
Another effect inducing shear stress in the skin is the pressure differential due to the pressurization of the fuselage. On
this point the FAR (part 25, Section 841 - Pressurized cabins) stipulates that "[...] the cabin must be equipped to provide a
cabin pressure altitude of not more than 8,000 f eet at the maximum operating altitude of the airplane under normal operating
conditions." Hence since the maximal altitude that the aircraft is required to reach is 45,000 [ft], the pressure differential
acting on the panel of the fuselage is ∆P = P45,000−P8,000. Since the skin is expected to be very thin in comparison with the
fuselage cross-section, the thin-walled pressure vessels formula can be used to quantify the additional stress induced by the
pressure:
σθ =∆PDfus
2t. (5.48)
Therefore, expressing the maximum distortion energy criterion:
σvon Mises =
√σ2
θ+ 3
(qmax
t
)2(5.49)
This failure stress must be smaller than τmax. Thus Eq. 5.47 and 5.48) provide an expression of the skin thickness:
t =1
τmax
√(∆Dfuselage
2
)2+ 3q2
max ; (5.50)
Consequently, the minimum skin thickness is t = 0.01181 [in] = 11.81 [th]. But since the skin is fastened to the frames
and stringers, it must support the the rivets. Moreover a thicker skin helps avoid deformation during maintenance operation
for example, as concentrated loads might accidentally be applied on the skin. Therefore a standard thickness is chosen:
t = 0.0393 [in] = 39.37 [th].
The simpleness of the model used to design the stringers area can be refined by taking account of the direct stresses actually
carried by the skin. Each booms area is therefore increased by an area equivalent to the direct stress carrying capability of
the adjacent skin panels. As T.H.G Megson states in ref. [18], providing that the stringers are spaced closely enough the skin
between them can be approximated as flat. The direct stresses on the panel can thus be assumed to vary linearly between the
values σi and σi+1 of the constraints in two consecutive booms.
5 OPTIMIZATION 73
b
z
y
x
t
σ1
xx
σ2
xx
z
y
x
σ1
xx
σ2
xx
A1
A2
b
Figure 5.25: Idealization of the direct stress carrying panel.
As presented in Fig. 5.25 the panels can be idealized by a skin without thickness and two booms carrying direct stresses
of area A1 and A2:
A1 =tb6
(2+
σ2xx
σ1xx
)(5.51)
A2 =tb6
(2+
σ1xx
σ2xx
)(5.52)
Each boom is therefore affected by the contribution of the booms of the two contiguous panels. The formula for the
stringer area Si can therefore be written:
Si = B− tb6
(2+
σ i+1xx
σ ixx
)− tb
6
(2+
σ i−1xx
σ ixx
)
= B− tb3
(2+
σ i+1xx
σ ixx
) (5.53)
In practice it is not convenient that all the stringers have different sizes: there are more different pieces to manufacture and
there is a risk that the assembly workers make mistakes between two stringers of slightly different cross section. Therefore
each stringer should have the same cross-sectional area, determined using the boom that bear the most important stress.
Applying Eq. 5.53 generate an odd result: the required stringer area is very small, with Sstringer = 0.0077 [in2]. This
is partly due to the fact that the skin thickness is oversized; the panels are therefore able to carry an important part of the
direct stresses. Considering the simplifying hypothesis on the loading and particularly the fact that some loads are not applied
since there is no way other than numerical to estimate them quantitatively, it is safer to choose the most conservative option.
Therefore the stringers area is determined from the solution of Eq. 5.43. Practically a standard size of stringers is chosen:
S = 0.0775 [in2].
5 OPTIMIZATION 74
5.4.5.2 Wing root
The wing root structure design is similar to the fuselage section design, although the cross section is not symmetrical and is
composed of several cells delimited by spars. The chosen configuration is typical for a wing design with 3 spars defining 4
cells, even though the last one in not considered in the structural design as its purpose is only aerodynamic. It therefore does
not bear the structural loading.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
Figure 5.26: Representation of the wing cross-section modelling.
The same considerations than for the fuselage are to be taken into account concerning the spacing of the stringers and
frames; therefore the number of stringers is chosen to be 20, resulting in a spacing of 6.2 [in]. The frames are spaced 11.8
[in] apart as in the previous section. Considering the worst loading case among the different points of the flight enveloppe, the
bending moments considered in the following are Mx = -234788 [lbf · ft] and Mz = 6650 [lbf · ft].
The unsymmetrical section imply that the expressions for the stresses resulting from bending have to be adapted. This
way the centroid of the cross section has not to be determined and computation time is saved. From [18], the direct stresses
are written
σy =Mx(Izzz− Ixzx)
IxxIzz− I2xz
+Mz(Ixxx− Ixzz)
IxxIzz− I2xz
. (5.54)
Taking out the booms area from these equations, the forces resulting from the bending moments are therefore computed at the
different booms locations. It follows that due to the loading, the maximal force is reached at the boom 5 such as shown in Fig.
5.26. Therefore enforcing σy,max = σmax provides the required boom area for the wing: Bwing = 0.6745 [in2]. All the stringers
are considered to be of the same cross sectional area.
The wing skin is designed with the same methodology as for the fuselage: on each cell the average shear flow between
two booms is determined by ([18]):
qi+1−qi =−(SxIxx−SzIxz
IxxIzz− I2xz
)Bixi−
(SzIzz−SxIxz
IxxIzz− I2xz
)Bizi, (5.55)
the shear flow on each panel can therefore be expressed in terms of the closure shear flow qs,0,n of the nth cell. The unknowns
are therefore the values of the shear flow at each of the cuts, i.e qs,0,I ,qs,0,II and qs,0,III , and the twist rate. Under the assumption
of an undistorted cross-section, the twist rate is the same for each cell therefore the twist rate compatibility provide 3 equations.
A fourth equation is provided by balancing out the moments resulting from the shear flows from the individual cells and the
moments of the externally applied loads about the same point:
Txη0−Tzξ0 =3
∑R=1
Mq,R =3
∑R=1
∮R
qp0ds+3
∑R=1
2ARqs,0,R (5.56)
5 OPTIMIZATION 75
with
• η0 the distance between the line of action of Tx and the chosen moment center O,
• ξ0 the distance between the line of action of Tz and O,
• AR the area of the cell R.
Therefore the system of equations can be solved and the shear flow on each panel is determined accordingly. Enforcing the
linear elasticity everywhere on the skin of the wing leads to t = qmax,wingτmax
= 0.0315 [in] = 31.5 [th]. As this skin must support
the rivets, it takes the standard thickness of t = 39.37 [th].
As for the fuselage section, the fact that the skin actually carry direct stresses can be taken into account by Eq. 5.53 which
allow a reduction of the stringers cross sectional area: Swing = 0.474 [in2].
5.4.6 FEM Analysis - Preliminary results
In order to validate the proposed structural layout for both the fuselage and the wing, Finite Elements models have been
developed using CATIA v5. In this model are included all the airframe’s elements such as the skin, the frames/ribs and the
stringers/longerons to the detailed dimensions.
In order to save computation time, and as the element under study is exclusively the fuselage, we did not include the
engine’s mounts nor the V-Tail in this model. Nevertheless, all the weights and aerodynamic loads explicated previously have
been taken into account and integrated in the model.
Figure 5.27: Finite Elements Analysis of the Fuselage’s structure under nlimit. Deformation amplified 10 times.
As it can be seen in Fig. 5.27, the maximum Von-Mises stresses are located directly aft the wing. The max V.M. stress
in the structure is 7.68 [ksi], which complies with the plasticity criterion (< Re = 52.1[ksi]). Using this model, some further
refinements can be brought to remove matter (and weight) to the structure without altering its efficiency. In particular, the
model’s frames were very thick (and support a very low constraint - they appear in blue).
The same type of Finite Elements model has been performed to validate the results of the section 5.4.5.2. This model leads
to a max. Von Mises stress of 36.26 [ksi], which again, complies with the plasticity criterion (<Re). A wide deformation of
the wing tip is noticeable.
5 OPTIMIZATION 76
Figure 5.28: Finite Elements Analysis of the Wing’s structure under nlimit. Deformation not amplified.
5.4.7 Further improvements
Until now the design of the structure was narrowed to ensuring that it withstands the loads statically applied to it. This is no
longer adequate if the structure have to be planned further than the preliminary design as several aspects of what an actual
structure is are missed.
These disregarded factors may be sorted in 3 domains: the modelling level, the structural complexity and the loading. As
stated in section 5.4.2, some aerodynamic loads are hardly estimable without using CFD simulation which implementation
time cost make that it is usually dedicated to more advanced design. Moreover, the structure should actually be designed to
withstands its dynamic loading, accounting notably the vibrations during flight and the shocks during landing. To proceed
further on this domain of the structure design, fatigue and crack propagation studies shall be conducted as the structure must
be able to fail-safe, i.e maintain the plane airworthy until emergency landing even in the case of a structural failure.
Another point is that the structure configuration considered in the preliminary design is simplistic and does not take account
of some practical aspects:
• The plane must include openings such as doors,
• The structural parts must be assembled one to each other,
• The mobile parts can not carry structural loads.
This results in the necessity to design cutouts in the structure. As they interfere with the good loading transmission, they
lead to a stress concentration. The structure at these locations should be designed to transfer the loads from skin, flanges and
shear webs around the cut-out. This inevitably lead to an increased cost and weight of the structure.
Even if the structure is considered in all its complexity, the modelling itself of the structural elements might be improved.
The assumptions followed throughout the preliminary design were that the direct stresses and shear stresses are decoupled,
therefore the structure could be modelled as an assembly of direct stress only carrying booms and shear only carrying skin. Ac-
tually much more complex mechanical phenomena such as twisting and wrapping shall be taken into account for an advanced
structure design. The easier way to do so is to create a Finite Element Model of the structure.
6 COSTS ANALYSIS 77
Finally thin skin structures are very prone to undergo unstable failure modes such as buckling. This critical phenomenon
in aircraft structures consists in a coupling of axial loading and bending mode, and shall be prevented through an adequate
design.
6 Costs Analysis
So far, a 2 aircrafts family of Business Jets has been designed and studied. As a design optimum has been reached, it is now
necessary to assess the competitiveness of this design on the current Business Jets’ market, and more specifically, to define
its price tag to the final customer, as well as the needs in terms of investment, and the expected returns on investments to the
shareholders.
In what follows, different categories of costs for the whole life-cycle of each aircraft will be assessed:
• The RDT & E (Research, Development, Test and Evaluation) costs include all the costs related to the design, the tooling
design, the prototyping, and the certification phases. In parallel, the management-related costs (H.R., Financials, Public
Relations) must also be taken into account, although it represents a smaller cost. These costs are mainly fixed costs to
the company.
• Production costs include labor and material resources that will be used to actually build the aircrafts. It also includes
the company running costs (management, sales teams,...). Those costs are obviously variable costs, as the overall costs
of the company will increase with respect to the number of units produced.
• Operations & Maintenance (O&M) costs cover all the indirect costs to run the aircraft. Fuel, Oil, crew members,
maintenance, and insurance must be assessed in this group. Even if O&M costs are to be charged directly to the
customer, they must be as low as possible to enforce the competitiveness of the Business Jet on the market. Indeed, low
Operations and Maintenance costs would justify a higher price tag for the jet at delivery.
Two types of Operations & Maintenance costs are to be taken into account:
– Estimated hourly variable costs: this estimate represents, all charges included, the cost of operation per hour of
1. Weights of the payload Wpayload and the crew Wcrew
The payload is composed of the passengers (8 passengers, 200 lbs eachs), and 1000 lbs worth of luggage. Which gives
Wpayload = 8 ·200+1000= 2600 lbs. A crew of 2 pilots will be considered (200 lbs each), and therefore, Wcrew = 400 lbs.
2. Empty weight fraction We/Wto
The empty weight fraction may be estimated using the empirical relationships provided by [3]. Having, for our category
of aircraft, with A = 1.02 and C =−0.06,We
Wto= AWC
to
3. Fuel weight fraction Wfuel/Wto
The fuel weight will obviously depend of the mission of the aircraft. In this case, we will stick to a typical transportation
mission, with the following phases:
(a) Taxi & takeoff: W1/Wto = 0.97
(b) Climb: W2/W1 = 0.985
(c) Cruise - Breguet Range equation
W3
W2= exp
−R ·SFCcruise
V · LD
' 0.77
With:
• R, the desired range (expressed in ft.)
• SFCcruise, the specific fuel consumption of the engine in cruise condition. As a first approximation, the
PW535-E is chosen, which has a SFC of about 0.77 lb/(h∆lb f )
B UNINSTALLED MAXIMUM CRUISE RATING 91
• V is the velocity of the plane (M = 0.85).
• L/D the lift to drag ratio. Estimated to ' 13 as a first approximation.
(d) Loiter (considered as slightly different conditions from cruise):W4
W3' 0.97.
(e) Landing: The Landing ratio is estimated, from [3] to be W5/W4 ' 0.995
In the end, the total mission weight fraction W5/Wto is obtained by multiplying all the terms,
W1
Wto·W2
W1·W3
W2·W4
W3·W5
W4=
W5
Wto' 0.7096
Adding the fuel reserve and trapped fuel (about 6% of the total fuel quantity),
Wf
Wto= 1.06 ·
(1− W5
Wto
)' 0.308
The value of Wto has been iterated multiple times, as the empty fraction weight is dependant from the TOW. A stable value
of ∼ 22500lbs is computed. However, the empirical relation used for the empty weight fraction is very rough, and is adapted
from a wide databases of jet aircrafts, most of which are much heavier than this light business jet. Therefore, the weight is
expected to be lower than this approximation. To make a compromise between this method and the market assessment results,
the 8 seats business jet is expected to reach a TOW of about 19000lbs as a first approximation.
B Uninstalled maximum cruise rating
Figure B.1: Unistalled maximum cruise rating (BPR ≤ 4).[16]
C WING GEOMETRY PARAMETERS: STATISTICS 92
C Wing geometry parameters: statistics
Aircarft Wing span [ft2] Aspect Ratio (AR)Airbus A 300-600B 147.11 7.7Airbus A 310-300 144.00 8.8Airbus A 320-200 111.25 9.4Airbus A 340-300 197.83 10Antonov AN 124 240.49 8.6BAe (Avro) RJ85 85.99 9.0Boeing B 737-600 112.57 9.4Boeing B 747-400 211.42 7.7
Boeing B 777 199.90 8.7Cessna 525 CitationJet 46.75 9.1
Learjet 60 43.77 7.2Embraer Phenom 300 52.20 8.9
McDonnell Douglas MD11 169.49 7.9Tupolev TU 204 137.80 9.7
Table C.1: Statistics about aspect ratio and wing span for different aircraft.
Aircarft Aspect Ratio (AR) Taper Ratio Sweep Angle [◦] Maximum MachVFW-Fokker 614 7.22 0.402 15 0.65Yakovlev Yak 40 9.00 0.396 0 0.70
Table C.2: Statistics about aspect ratio, taper ratio and sweep angle for different aircraft (source coming from the course notesof Mr. Noels, Lecture 4, page 59. [6])
D EXPERIMENTAL DATA OF THE NACA SC(2)-0714 93
D Experimental data of the NACA SC(2)-0714
(a) Lift, drag and moment coefficient at high Reynolds (Re = 10 ·106).
(b) Lift coefficient at low Reynolds (Re = 6 ·106).
Figure D.1: Experimental data for NACA SC(2)-0714 airfoil performed in wind tunnel.
E SUMMARY OF THE WING GEOMETRICAL PARAMETERS 94
E Summary of the wing geometrical parameters
Main wing parameters US/Imp
Span: b 45.7 [ft]Aspect Ratio: AR 9
Total (gross) area: S 232.2 [ft2]Total exposed surface: Sexp 191.3 [ft2]
Taper ratio: λ 0.30Chord at root: croot 7.81 [ft]Chord at tip: ctip 2.33 [ft]
Sweep angle at leading edge: Λ 34.5 [◦]Sweep angle at chord quarter: Λ1/4 32.1 [◦]Geometric twist (Washout): εgtip -1 [◦]Mean aerodynamic chord: MAC 5.58 [ft]
X coordinate of aerodynamic centre: Xac 7.84 [ft]Y coordinate of aerodynamic centre: Yac 9.38 [ft]
Compressibility parameter: β 0.3275Cruise Mach: M 0.85
Average airofoil thickness: t 0.72 [ft]Wing lift coefficient in cruise: CLw 0.3Wing lift coefficient derivative: a 4.77
Angle of attack at root (cruise): αroot -0.2 [◦]Zero-lift angle of attack of at root: αL0 -3.8 [◦]
Zero-lift angle of attack of the profile: αl0 -4 [◦]Aerodynamics twist coefficient: α01 -0.225
Aerodynamics twist: εatip -1 [◦]
Maximum wing lift coefficient (cruise): CLmax,cruise 0.8
Maximum wing lift coefficient (landing/takeoff, flaps in): CLmax,TO/Landing 1.5
Reynolds number (cruise): Re 9.6 ·106
Airofoil lift coefficient derivative: cla 2.9π
Maximum camber: cmax (expressed relative to the chord length) 2.5 %Lift generated by the wing (cruise): Lw 15397 [lbf ]
Table E.1: Summary of the principal parameters of the wing.