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The Cormorán project:
a new concept in commercial
aircraft design
Gómez González, Vı́ctor
Izquierdo Collado, Emilio José
Stockholm - 30th of January 2013
Abstract
This paper presents a new revolutionary design in commercial aircraft: the conventional vertical and horizontal tails are not present as generally known, and their contribution to the manoeuvering of the aircraft, namely the presence of the rudder and the
elevators, is achieved by locating them at the wingtips and in the canard, respectively.
Substituting the horizontal tail with the canard, the possibility of dividing the fuel between the wing (where it is located conventionally) and the canard allows the pilot to
change the center of gravity during the flight with more freedom, while the effect of the
elevators continues present. Locating the vertical stabilizers at the wingtips combines
the effect of the vertical stabilizer and the winglet all in one, with the corresponding lost
of weight. In this sense, the aerodynamic, stability and aeroelastic characteristics of an
aircraft such as the one described have been analyzed using different modules belonging
to CEASIOM program, and the results are very encouraging, showing that it is really
feasible to change the current concept of the commercial aircraft without penalizing the
performance.
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Acknowledgments
We would like to address special thanks to Professor Arthur Rizzi for giving us the
great opportunity to carry out our master’s thesis project in his department and also for
give us the chance to participate in the AIAA-Pegasus students conference representing
the Royal Institute of Technology (KTH). Moreover we would like to thank Professor
Sergio Ricci from Politecnico di Milano his help to carry out this project as well as José
Pedro Magraner Rullán from Universitat Politècnica de València for the useful information
provided.
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Contents
1 Introduction
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2 What is CEASIOM?
2.1 AcBuilder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Weights and Balance . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Aerodynamic module builder (AMB) . . . . . . . . . . . . . . . . . .
2.4 Simulation and Dynamic Stability Analyzer (SDSA) . . . . . . . . . .
2.5 Next generation Conceptual Aero-Structural Sizing Suite (NeoCASS)
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3 Geometry.
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4 Implementation in AcBuilder.
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4.1 Weights & balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Aerodynamic Model Builder (AMB).
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5.1 Code modifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6 Analysis of stability and control.
6.1 Previous procedure. Propulsion module
6.2 Stability and control . . . . . . . . . .
6.3 Phugoid mode. . . . . . . . . . . . . .
6.4 Short Period mode. . . . . . . . . . . .
6.5 Dutch Roll mode. . . . . . . . . . . . .
6.6 Roll mode. . . . . . . . . . . . . . . . .
6.7 Spiral mode. . . . . . . . . . . . . . . .
6.8 Conclusion of the stability analysis. . .
6.9 Manoeuver test. . . . . . . . . . . . . .
6.10 Performances. . . . . . . . . . . . . . .
6.10.1 Drag polar. . . . . . . . . . . .
6.10.2 Flight envelope. . . . . . . . . .
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7 Structural and aeroelastic analysis.
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7.1 Code modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8 Payload-Range diagram
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9 Cost estimation
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10 Conclusion
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4
A Aerodynamic derivatives
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5
List of Figures
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Beechcraft Starship . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Beechcraft Starship 3 views . . . . . . . . . . . . . . . . . . . . . . . .
Curtiss-Wright XP-55 Ascender . . . . . . . . . . . . . . . . . . . . . .
Curtiss-Wright XP-55 Ascender 3 views . . . . . . . . . . . . . . . . . .
Miles M39B Libellula . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Miles M39B Libellula 3 views . . . . . . . . . . . . . . . . . . . . . . .
Rutan Long-EZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rutan Long-EZ 3 views . . . . . . . . . . . . . . . . . . . . . . . . . .
Kyushu J7W1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kyushu J7W1 3 views . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conceptual design process . . . . . . . . . . . . . . . . . . . . . . . . .
3D model of Cormorán . . . . . . . . . . . . . . . . . . . . . . . . . . .
CEASIOM package . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cormorán’s planform view. Configuration 1. . . . . . . . . . . . . . . .
Cormorán’s front view. Configuration 1. . . . . . . . . . . . . . . . . .
Cormorán’s side view. Configuration 1. . . . . . . . . . . . . . . . . . .
Cormorán’s planform view. Configuration 2. . . . . . . . . . . . . . . .
Cormorán’s front view. Configuration 2. . . . . . . . . . . . . . . . . .
Cormorán’s side view. Configuration 2. . . . . . . . . . . . . . . . . . .
Cormorán’s AcBuilder model. Configuration 1. . . . . . . . . . . . . . .
Cormorán’s AcBuilder model. Configuration 2. . . . . . . . . . . . . . .
New options needed and implemented in AcBuilder. . . . . . . . . . . .
Fuel tanks in both wings. . . . . . . . . . . . . . . . . . . . . . . . . . .
New options needed and implemented in AcBuilder. . . . . . . . . . . .
Centers of gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
‘tornado geo.m’code. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3D panels, collocation points and normals for the first configuration. . .
3D panels, collocation points and normals for the second configuration.
‘run tornado’code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Three views for both configurations. . . . . . . . . . . . . . . . . . . . .
Showtable.m code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cl - α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cd - α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Typical Cl - α plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parameters of Propulsion module . . . . . . . . . . . . . . . . . . . . .
Results of Propulsion module . . . . . . . . . . . . . . . . . . . . . . .
Static margin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Recommended Phugoid Characteristics. Configuration 1. . . . . . . . .
Recommended Phugoid Characteristics. Configuration 2. . . . . . . . .
Recommended Short Period Characteristics. Configuration 1. . . . . . .
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Recommended Short Period Characteristics. Configuration 2. .
Recommended Dutch Roll Characteristics. Configuration 1. .
Recommended Dutch Roll Characteristics. Configuration 2. .
Recommended Roll Characteristics. Configuration 1. . . . . .
Recommended Roll Characteristics. Configuration 2. . . . . .
Recommended Spiral Characteristics. Configuration 1. . . . .
Recommended Spiral Characteristics. Configuration 2. . . . .
Control deflections due to the manoeuver. . . . . . . . . . . .
Polar with SDSA. . . . . . . . . . . . . . . . . . . . . . . . . .
Polar with flat-plate theory. . . . . . . . . . . . . . . . . . . .
Flight envelope. . . . . . . . . . . . . . . . . . . . . . . . . . .
Aerodynamic model . . . . . . . . . . . . . . . . . . . . . . . .
Structural model . . . . . . . . . . . . . . . . . . . . . . . . .
Aeroelastic model . . . . . . . . . . . . . . . . . . . . . . . . .
‘SymmXZ.m’code. . . . . . . . . . . . . . . . . . . . . . . . .
‘link structs.m’code. . . . . . . . . . . . . . . . . . . . . . . .
‘plot beam defo.m’code. . . . . . . . . . . . . . . . . . . . . .
Deformed shape for mode 3 . . . . . . . . . . . . . . . . . . .
V-g at h = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Payload-range diagram. . . . . . . . . . . . . . . . . . . . . . .
Range with maximum payload. . . . . . . . . . . . . . . . . .
Range with maximum fuel weight. . . . . . . . . . . . . . . . .
Example of the .txt that the showtable.m function gives. . . .
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List of Tables
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Geometry data for the fuselage. . . . . . . .
Geometry data for the wing. . . . . . . . . .
Geometry data for the canard. . . . . . . . .
Aerodynamic derivatives. . . . . . . . . . . .
Geometry data for vertical stabilizers. . . . .
Centers of gravity location. . . . . . . . . . .
Inertia matrix. . . . . . . . . . . . . . . . . .
Contribution to the CD0 of each component.
Cormorán weights. . . . . . . . . . . . . . .
Vibration modes. . . . . . . . . . . . . . . .
Development and procurement costs. . . . .
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8
Nomenclature
c1
c2
croot
ctip
b
l
α
λLE
Λ
Y
L
N
ρ
S
V0
φ
δR
δA
δE
T
aEO
Inboard chord
Outboard chord
Vertical Stabilizer root chord
Vertical Stabilizer tip chord
Span
Total fuselage length
Angle of attack
Leading edge sweep angle
Dihedral angle
Lateral force
Rolling moment
Yaw moment
Density
Wing area
Aircraft velocity
Roll angle
Rudder deflection
Aileron deflection
Elevator deflection
Thrust
Arm of the dead engine
β
CYδR
CYδA
CYβ
ClδR
ClδA
Clβ
CNδR
CNδA
CNβ
CNEO
dvert.Af t
dhoriz.Af t
dvert.F ore
dhoriz.F ore
u
v
w
p
q
r
Sideslip angle
Derivative for lateral force due to rudder deflection
Derivative for lateral force due to aileron deflection
Derivative for lateral force due to sideslip angle
Derivative for rolling moment due to rudder deflection
Derivative for rolling moment due to aileron deflection
Derivative for rolling moment due to sideslip angle
Derivative for yawing moment due to rudder deflection
Derivative for yawing moment due to aileron deflection
Derivative for yawing moment due to sideslip angle
Derivative for yawing moment due to engine failure
Vertical diameter aft fuselage
Horizontal diameter aft fuselage
Vertical diameter fore fuselage
Horizontal diameter fore fuselage
x-axis velocity
y-axis velocity
z-axis velocity
x-axis angular velocity
y-axis angular velocity
z-axis angular velocity
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1
Introduction
The aim of the work is to explain the development of a new concept of aircraft. This revolutionary design consists of a tailless aircraft whose vertical stabilizers are located at the
wingtip. This design has several advantages in comparison with the typical configurations
but, logically, also disadvantages exist which will be commented. Before this the most
representative characteristics and the aircraft design requirements must be explained.
( High-Subsonic aircraft with a capacity of 164 passengers.
( 4000 km range with maximum take off weight.
( One cabin configuration with 3 + 3 seats.
( Similar qualities to the Boeing 737 − 800 and the Airbus 320.
( Low fuel consumption.
The idea of carrying out this configuration arises after seeing that there were some
advantages that made the study interesting. This is not the first time that something
similar is done, however, it has never been carried out on a plane of this size. Examples
are the Beechcraft Starship, Curtiss-Wright XP-55 Ascender, Miles M39B Libellula, Rutan Long-EZ or the Kyushu J7W1 Shinden.
In the following pictures are shown the named aircrafts with the 3 views in order to
compare it and note the differences that they have with the Cormorán. It is easy to see
that they are from very different ages, which indicates that tailless aircraft have been
studied for about 80 years. Initially they were studied as military aircrafts but recently
also for jets. Nowadays there are projects which are evaluating the possibilities of a big
tailless aircraft like the Cormorán [], but all this ideas are in a early phase so in the next
years we will continue seeing the typical configuration of the Boeing 737 or the Airbus 320.
Figure 1: Beechcraft Starship
Figure 2: Beechcraft Starship 3 views
10
Figure 3: Curtiss-Wright XP-55 Ascender
Figure 4: Curtiss-Wright XP-55 Ascender 3 views
Figure 5: Miles M39B Libellula
Figure 6: Miles M39B Libellula 3 views
Figure 7: Rutan Long-EZ
Figure 8: Rutan Long-EZ 3 views
Figure 9: Kyushu J7W1
Figure 10: Kyushu J7W1 3 views
11
The starting point comes from a previous work[1] in Valencia where a preliminary
study showed that the design was feasible. This step corresponds with the first sizing and
the initial layout of the figure 11. The diagram shows the followed steps in the design
process, starting with the design requirements and finishing with the refined sizing. In the
initial layout, which was done in the previous work, empirical methods from Raymer[2]
and Roskam[3] literature were used. Some of these methods are included in this report
in order to compare the empirical results with the ones obtained with CEASIOM. It has
been during the Master Thesis Project[4], performed at the Royal Institute of technology
(KTH), where we have continued the development of the Cormorán. In this Master Thesis
the study has been more accurate, that is the reason why the original design has suffered
several changes in order to achieve the expected flying qualities.
Figure 11: Conceptual design process
The fact of having the vertical stabilizers at the wingtips implies that the structure of
the wing must be more robust than the usual configurations. However, there are significative advantages as for example:
( The reduction of the aircraft total height.
( The increase of lift due to the sum of the canard and the wing.
( Low wing, which means that the landing gear is shorter and the ground effect is
higher. There is no discontinuity in the cabin or in the baggages deck.
( The position of the engines reduces the ingestion of objects or particles from the
runway. Furthermore, in case of engine failure the asymmetry is lower.
12
( The vertical stabilizers have two functions: rudder and winglet.
( The decrease of the drag due to the fact that there is no tail. Really the drag of
the tail is similar (a bit higher) to the drag of the winglets but as we have a tailless
aircraft there is no addition of the tail and the total drag is lower.
Different aspects as geometry, weights, aerodynamics, stability, control and aeroelasticity have been studied and are explained in the next sections. In these calculations
different programmes have been used, all of them belonging to CEASIOM[5] , [6] (Computerised Environment for Aircraft Synthesis and Integrated Optimization Methods). Those
programmes are AMB[7] , [8] (Aerodynamic model builder) which predicts the aerodynamic forces around the aircraft using a vortex-lattice method, SDSA (Simulation and
Dynamic Stability Analyser) which predicts the stability and the controllability through
flight simulations and the design of a flying control system based on a LQR, and NeoCASS
(Next generation Conceptual Aero-Structural Sizing Suite) which allows to dimension the
structure of the aircraft under development and to investigate its aeroelastic behavior.
CEASIOM and its modules will be explained in the next section.
In picture 12 there is a 3D view which shows the final chosen configuration.
Figure 12: 3D model of Cormorán
13
2
What is CEASIOM?
CEASIOM is developed in the scope of the SimSAC project. SimSAC stands for Simulating Stability And Control Characteristics for Use in Conceptual Design, and it is a
project approved for funding by the European Commission 6th Framework Program on
Research, Technological Development and Demonstration.
CEASIOM is a tool used for the preliminary design and analysis of aircraft. It allows
a user to, using a set of parameters, define the aircraft for various analysis tasks such as
stability analysis and aeroelastics. A brief description of the CEASIOM package is shown
in the Figure 13.
Figure 13: CEASIOM package
CEASIOM is meant to support engineers in the conceptual design process of the
aircraft, with emphasis on the improved prediction of stability and control properties
achieved by higher-fidelity methods than found in contemporary aircraft design tools.
Moreover, CEASIOM integrates into one application the main design disciplines, aerodynamics, structures, and flight dynamics, impacting on the aircraft’s performance. It
is thus a tri-disciplinary analysis brought to bear on the design of the aero-servoelastic
aircraft.
CEASIOM does not however carry out the entire conceptual design process. It requires as input an initial layout as the baseline configuration that it then refines and
outputs as the revised layout. In doing this, CEASIOM, through its simulation modules,
generates significant knowledge about the design in the performance, loads, and stability
and control databases. The information contained in these databases is sufficient input
to a six Degree of Freedom engineering flight simulator.
Below we describe the different modules CEASIOM.
14
2.1
AcBuilder
AcBuilder is a CAD (Computerized Aided-Design) tool permitting to carry out the conceptual aircraft design. It is composed of different parts: the geometric definition, the
fuel tanks definitions, the cabin and luggage definitions, the centers of gravity computation and the technology definition. In each part, the user can modify several parameters
defining the aircraft geometry or properties (the total fuselage length for example). From
those parameters, the program calculates the coordinates of the sections for creating each
element (fuselage, wing, fuel tanks...).
The AcBuilder module is essentially JAVA (Mixed with Matlab). Indeed, the coordinates calculation (which permit to create the elements surfaces) from the input parameters
is made in java as well as the whole graphic interface.
2.2
Weights and Balance
The weight and balance module allows the user to calculate the weights of the aircraft
by different methods. These methods are Raymer, Howe, Torenbeek, USAF and Cessna,
which are based in empirical data and correlations of the most typical aircrafts.
Hence, the module gives the weights for each element and then the user has to select
which of them matches better with his design. Once one method is chosen the programme
calculates the inertia moments and the center of gravity of each element and the whole
aircraft.
2.3
Aerodynamic module builder (AMB)
Traditionally, wind-tunnel measurements are used to fill look-up tables of forces and moments over the ?ight envelope but wind-tunnel models become available only late in the
design cycle. To date, most engineering tools for aircraft design rely on handbook methods or linear ?uid mechanics assumptions. The latter methods provide low cost reliable
aerodata as long as the aircraft remains well within the limits of the flight envelope. However, current trends in aircraft design towards augmented-stability and expanded flight
envelopes require an accurate description of the non-linear ?ight-dynamic behavior of the
aircraft. The obvious option is to use Computational Fluid Dynamics (CFD) early in the
design cycle. It has the predictive capability to generate data but the computational cost
is problematic, particularly if done by brute force: a calculation for every entry in the
table. Fortunately methods are available that can reduce the computational cost.
There are essentially three issues. The computational models considered here range
15
from handbook methods (USAF Digital DATCOM), through linear singularity methods (Vortex Lattice Method) to full non-linear Euler and RANS (Reynolds-averaged
Navier–Stokes method) compressible flow CFD packages.
In this work we have only used TORNADO. TORNADO, a vortex-lattice method
(VLM) for conceptual aircraft design and education has been integrated into CEASIOM
as the main tier I tool. TORNADO allows a user to define most types of contemporary aircraft designs with multiple wings, both cranked and twisted with multiple control
surfaces located at the trailing edge. Each wing is permitted to have unique definitions
of both camber and chord. The TORNADO solver computes forces, moments, and the
associated aerodynamic coefficients. The aerodynamic derivatives can be calculated with
respect to: angle of attack, angle of sideslip, roll-pitch-yaw rotations, and control surface
deflections.
To account for viscous effects, CEASIOM provides a correction to the steady vortex
lattice method by the strip theory that combines the linear potential results with the 2D
viscous airfoil code XFOIL.
2.4
Simulation and Dynamic Stability Analyzer (SDSA)
Once the aerodynamic coefficients have been obtained for the flexible aircraft using the
‘eta’ values (i.e. the S & C aerodata database is in hand) along with the mass and inertia
properties, the S & C analysis can begin with SDSA. SDSA covers the following functionalities: stability analysis (eigenvalues analysis of linearized model in open and closed
loop case), six Degree of Freedom flight simulation (test flights, including trim response),
flight Control System based on Linear Quadratic Regulator (LQR) theory, performance
prediction and other issues.
SDSA uses the same Six DoF mathematical nonlinear model of the aircraft motion
for all functions. For the eigenvalue analysis, the model is linearized by computing the
Jacobian matrix of the state derivatives around the equilibrium (trim) point numerically.
The flight simulation module can be used to perform test flights and record flight parameters in real-time. The recorded data can be used for identification of the typical modes
of motions and their parameters (period, damping coefficient, phase shift). The stability analysis results are presented as ”figures of merits” based on JAR/FAR, ICAO, and
MIL regulations. The SDSA embedded flight control system is based on a LQR approach.
16
2.5
Next generation Conceptual Aero-Structural Sizing Suite
(NeoCASS)
NeoCASS is a numerical analysis tool particularly suited to conceptual and preliminary
design of aircraft. Its main purpose is to enhance these early design phases with details
regarding bearing airframe which is usually poorly represented by the structural weight
coming from empirical formulas. Indeed, the software helps the designer to dimension
the structure of the aircraft under development and to investigate its aeroelastic behavior
by means of structural and aerodynamic numerical methods having physical basis. Thus,
statistical formulas, usually characterized by poorness of details and accuracy, are overcome so that the code can be used to design innovative and uncommon aircraft layouts.
The code is completely written in the MATLAB.
17
3
Geometry.
As explained in the introduction of the text, a commercial aircraft with an unusual configuration has been chosen. It is going to be analyzed in the several modules that CEASIOM
has, and the first module, as said, is the AcBuilder. The idea is to analyze the configuration explained in the introduction, with the vertical stabilizers located at the wingtips,
but we have also worked in an alternative configuration, with the vertical stabilizers located at the half of the wing, just in case the first configuration had any problems with
aerodynamics, control or aeroelasticity. The aim is to analyze both configurations and
decide if the first configuration (’Configuration 1’) is feasible, as thought in first therm,
or is it better to choose the second configuration (’Configuration 2’).
The geometry of the aircraft in its ’Configuration 1’ can be seen in the image of the
introduction section, with the 3D view. Also, the main characteristics has been described
in the introduction, so in this section the aim is to achieve a more accurate vision of
the geometry. To have a more detailed vision of the Cormorán, in the following three
figures the 3-view drawing of the aircraft is reported. Note that later in this section all
the dimensions will be described and detailed.
Figure 14: Cormorán’s planform view. Configuration 1.
18
Figure 15: Cormorán’s front view. Configuration 1.
Figure 16: Cormorán’s side view. Configuration 1.
Moreover, considering the second configuration of the aircraft, figures below show the
3-view drawing of the aircraft in its ’Configuration 2’.
Figure 17: Cormorán’s planform view. Configuration 2.
19
Figure 18: Cormorán’s front view. Configuration 2.
Figure 19: Cormorán’s side view. Configuration 2.
Moreover, in the tables reported in this section we have the most important geometrical parameters. Regarding to the fuselage, the basic data is reported in Table 1.
Table 1: Geometry data for
the fuselage.
Property
dvert.Af t (m)
dhoriz.Af t (m)
dvert.F ore (m)
dhoriz.F ore (m)
Nose length(m)
Tail length(m)
l(m)
3.60
3.32
2.91
3.32
2.11
5.4
36.2
As can be seen in Table 2, the wing is composed by three trapezoidal patches (in, mid
and out), and each of them is characterized by its chord inboard and outboard (c1 and c2
respectively), its span (b), its incidence angle (α), its leading edge sweep angle (λLE ) and
its dihedral angle (Λ, which is the same for the three patches).
It is also prominent the position of the wing in the aircraft. The wing is located at
25.82m since the nose of the aircraft, namely at the 71.3% of the total length of the
fuselage. This is very different from other commercial aircraft like the Boeing 737 − 800
or the Airbus A320, in which the wing is in a more forward position. In fact, the previous
calculus done in Valencia (described in the introduction) had as a result a more forward
position of the wing than the final position, which has been decided after the utilization
of CEASIOM and all its modules with an iterative process, paying particular attention to
the static margin, which is described in the aerodynamics section of the paper. However,
taking into account that this aircraft is a tailless one and the vertical stabilizers are at
20
Table 2: Geometry data for the wing.
Property In
c1 (m)
7
c2 (m)
4.37
b(m)
5
α(o )
0
λLE (o )
28
Λ(o )
3
Mid
4.37
2.70
5.50
1
28
3
Out
2.70
1.20
5.70
1
28
3
the wingtips, the position of the wing is reasonable. The fact of being reasonable can be
verified because in the group of aircraft with similar features that was named in the introduction section, all of them have the wing in a more rearward position,the only difference
is the size of the aircraft.
One basic fact is the presence of the canard. This is not common in commercial aviation, but has some advantages that we appreciate and were described in the introduction;
also, we want to give more importance to it. The idea is an aircraft without horizontal tail,
so that the horizontal stabilizers are placed in the forward horizontal surface, that means
in the canard. Moreover, the canard is essential for the location of the center of gravity
(which is fundamental for the longitudinal stability) because the fuel is divided between
the wing and the canard. This will require a more complex control system for the fuel, but
will allow more freedom in the location of the center of gravity and corresponding effect
on the stability during the flight. The geometry data for the canard is reported in Table 3.
Table 3: Geometry
data for the canard.
Property
c1 (m)
3.20
c2 (m)
1.50
b(m)
5.77
α(o )
1
λLE (o )
32
Λ(o )
0
Apart from the presence of the canard in a commercial aircraft, the other particularity of this aircraft is the vertical tail, which is divided into two vertical tails, as the twin
tail, but in this case they are located at the wingtip. Thus, they work as vertical tails
(with its corresponding rudder) and as winglets (with its benefits reducing the drag at
the wingtip). This location is the reason why it is necessary to use airfoils with a higher
thickness near the wingtip than the common aircrafts do: it is necessary to support the
weight of the vertical tail (it is bigger than a normal winglet) and to connect the cables
for the sensors and actuators of the rudder.
21
For the sizing of the vertical tail, it has been established that the the aircraft controls
have to be able to compensate aerodynamically the yaw moment caused by the failure of
an engine during the take off. In this sense, we have considered the well known in the
flight mechanics field ’Bryan Equations’[10]. The complete equations are not reported,
as that exceeds the purpose of this paper, but we can simplify them and focus in the
important ones for this case: we know that, with an engine failure, the rolling moment
(L) must be zero, and the yaw moment (N ) must compensate the moment generated by
the asymmetrical propulsion. Hence, we can solve the problem with a system with the
following three equations (1, 2 and 3) and four unknown parameters (the sideslip angle
β, the aileron’s deflection δA , the roll angle φ and the rudder’s deflection δR ):
Y = (CYδR δR + CYβ β + CYδA δA ) = CL · φ
(1)
L = (ClδR δR + Clβ β + ClδA δA ) = 0
(2)
N = (CNδR δR + CNβ β + CNδA δA ) = −CNEO
(3)
T ·aEO
Where CNEO = 0.5ρV
. Note that the aerodynamic derivatives[11] are given by the
2
0 Sb
software used, in this case, by AMB (Aerodynamic Model Builder). These values are
shown in the table 4.
Table 4: Aerodynamic derivatives.
Derivative
CYδR
-0.073
CYβ
-0.171
CYδA
-0.015
ClδR
0.012
ClδA
0.354
Derivative
Clβ
0.092
CNδR -0.017
CNδA 0.064
CNβ
-0.030
Hence, there are four unknown parameters and three equations, so it is necessary to
impose one parameter. The easiest option is that the steady condition with zero sideslip
may be accomplished by the use of rudder to balance the unsymmetrical engine-out condition. With these conditions the results are:
φ = −1.67 ◦
δA = −2.88 ◦
δR = 85.2 ◦
(4)
The solution δR = 85.2 ◦ is impossible, it is necessary to change the parameter that
we impose. The solution would be an algorithm that calculates which is the value of the
22
roll angle which minimize the values of δR , δA and β. With this method we would achieve
to compensate the yaw moment created with a feasible deflection value of the control
surfaces. In the most critical case (take off at sea level), the roll angle that minimize the
named values is φ = −1.62 ◦ , and these deflections are the following:
β = 23.73 ◦
δA = −7.08 ◦
δR = 27.82 ◦
(5)
Moreover, even more detail will be given in the aerodynamics section, a three meter’s
rudder has been selected, and this decision is not random because it is the minimum span
which gets proper results when it is tested on the SDSA module with the manoeuver that
will be detailed in the corresponding section.
As can be seen regarding to the results given by equation 5, the deflections that have
to be done for compensating the moment are not illogical, so the rudder’s sizing is proper.
For more details, in the Table 5 more data is reported regarding to the vertical stabilizers.
Table 5: Geometry data
for vertical stabilizers.
Property
croot (m)
ctip (m)
b(m)
α(o )
λLE (o )
Λ(o )
4
1.20
0.44
3
0
45
0
Implementation in AcBuilder.
Once the configuration is decided and plotted in Autocad, the next step is to ’build’
the model in AcBuilder, with all its particularities, as similar as possible to the original design, which has been shown in the previous figures. This task was carried out by
calculating the parameters needed for AcBuilder since the geometrical parameters that
the Cormorán has, and were reported in the tables reported in the previous subsection.
Moreover, it was necessary to do some changes in the code, in Matlab even in Java, to
achieve a proper model, which is represented below (as with the 3-view drawings, below
we have the configurations 1 and 2, respectively):
23
Figure 20: Cormorán’s AcBuilder model. Configuration 1.
Figure 21: Cormorán’s AcBuilder model. Configuration 2.
The mentioned improvements added in AcBuilder in order to be able to get a proper
model are, basically, the following:
- In ’Components’, inside ’geometry’ module, the basic change was to make it possible
to put the vertical tail in the wing tips or in the half of the wing, depending on
which configuration we were working with. We couldn’t use the twin tail configuration because we wanted, with our changes in the code, that the vertical tail would
be always at the wing tips for the configuration 1 and at the half of the wing for
the configuration 2, and every change in the wing wouldn’t affect the position of
24
the vertical tail with respect to the wing. This was get with very robust equations
that depend on quite a lot parameters.
This can be set now in AcBuilder with the new options added, which appear in the
following image. There are two new options: you have to activate the box called
’Winglet rudder’ to allow the vertical tail to be at the wings, and then select the configuration, 1 or 2, depending on where the vertical tail is, as explained before. The
result of this changes in the code can be seen in the previous images of AcBuilder.
Figure 22: New options needed and implemented in AcBuilder.
- In ’Fuel’, inside ’geometry’ module also, we needed to do some changes, because we
wanted to use the horizontal tail, which is at the front of the aircraft, not only as
a typical horizontal tail but also as a ’second wing’ or a canard, in which we could
put fuel into it. In AcBuilder there is no option to put fuel in a second wing, so
we had to modify the code in Java. With our modifications we achieve to put fuel
in a second wing, not only in the main wing, and this allows us to get an accurate
center of gravity.
Here we had to do a procedure to achieve our aim: really we have an horizontal tail
at the front of the aircraft, but we can’t put fuel into it. The solution was to put
an horizontal tail to run the AMB, SDSA and NeoCASS modules, but put a second
wing instead of an horizontal tail to the AcBuilder module, in order to be allowed
to put fuel into it and to get the real center of gravity.
25
Figure 23: Fuel tanks in both wings.
Figure 24: New options needed and implemented in AcBuilder.
26
As can be seen in the previous images, we have fuel in both wings and also a few
more options were added to allow this.
- In the ’weights & balance’ module, the code has been changed in order to use the
correct equations for our aircraft. The previous code was proper for most aircraft,
what we had to change the code, mostly using equations that were so similar to the
ones that we had used to put the vertical tail in its new places. The difference is
that in this case the changes were in the Matlab code and before it had been in the
Java code.
With the corresponding changes, we were able to put each center of gravity of each
component in the proper place, as we can see in the following image. As can be
seen, the results seem to be reliable, because if we look at the fuel, which is one
of the most great changes we have done, we can see that the center of gravity of
the fuel tanks of the principal wing are in a logical place, and the respective of the
second wing (really the horizontal tail, but considered a second wing, as explained
before) also, but the most important result: the center of gravity of both fuel tanks
is a little bit ahead the center of gravity of the main wing, due to the effect of the
second wing, which is what was expected. The location of the different components
center of gravity can be seen in the following image:
Figure 25: Centers of gravity.
27
4.1
Weights & balance.
Mainly, the Weight & balance module of CEASIOM consist on obtaining the position of
the centers of gravity of each component of the aircraft, and also the matrix of inertia. It is
necessary to run the module before the AMB, in order to use the matrix of inertia created.
For further detail, in Table 6 and Table 7 the position of the different centers of
gravity (and the total center of gravity) and the inertia matrix are reported, respectively.
Moreover, the exact weight of the different parts of the aircraft and the total weight is
reported the section corresponding to ’Structural and Aeroelastic Analysis’.
Table 6: Centers of gravity location.
Component
Wing
Canard
Vertical Tail
Fuselage
Engines
Passengers
X(m)
29.69
10.07
35.8
19.48
31.31
17.11
Y (m)
0
0
0
0
0
0
Z(m)
−0.95
−0.54
0.95
−0.36
0.70
−0.58
Table 7: Inertia matrix.
Elements
Ixx
Iyy
Izz
Ixz
Ixy
Iyz
kg · m2
1174486.61
5916318.38
7005027.31
132976.21
0
0
28
5
Aerodynamic Model Builder (AMB).
The AMB module allows the user to create an aerodynamic database that will be used in
the SDSA module to study the stability. There are three possibilities to do that: DATCOM, Tornado and non-linear Euler and RANS compressible flow CFD packages.
In this case, the method that has been used is Tornado because gives proper results
for the subsonic range. DATCOM can not be used because it is difficult to adapt to this
unusual configuration, and CFD needs to much computational cost.
Tornado is a Vortex Lattice Method for linear aerodynamic wing design applications
in conceptual aircraft design. By modeling all lifting surfaces as thin plates, Tornado can
solve for most aerodynamic derivatives for a wide range of aircraft geometries. With a
very high computational speed, Tornado gives the user an immediate feedback on design
changes, making quantitative knowledge available earlier in the design process.
5.1
Code modifications.
Although Tornado allows the user to create a huge variety of geometries, it is not possible
to put the vertical stabilizers in the wing tip. For this reason the code has been changed.
In this section these changes will be explained showing some parts of the modified code.
The main change affects the file called ‘tornado geo.m’. In this file is where the geometry is defined and is, basically, where the equations of the position of the vertical stabilizers
are introduced. These equations are the same that have been used in the AcBuilder, the
only difference is that, in this case, they have been programmed in Matlab and not Java.
The procedure is to move the vertical stabilizer to the new position, and then create a
new one on the other wing. The next picture shows the part of the code that has been
described. Note that depending on the configuration the equations are different.
29
Figure 26: ‘tornado geo.m’code.
The result is a plot where the different panels in which the wings are divided are
shown. This is the geometry that Tornado uses in the calculation and for which we get
the results. In the pictures are shown the results for both configurations.
Figure 27: 3D panels, collocation points and normals for the first configuration.
30
Figure 28: 3D panels, collocation points and normals for the second configuration.
The previous file is the most important that has been modified, but there are another
files with changes. One of the ”problems” of Tornado is that the user does not know how
much time needs the process. To help the user in this task the time has been estimated
and now, when the program starts, a line with the total time needed is displayed. This
time depends on the options that are chosen. For example, in the most simple case each
point takes about three second but, if we choose viscous flow, each point takes about nine
seconds. In the next figure the lines of the code are shown.
Figure 29: ‘run tornado’code.
Know the time it is not essential but helps the user to chose the number of points
taking into account how much time they need. Another new feature is that also the
number of points that are going to be calculated are shown. This is useful to have an
idea of how good the database will be. Obviously a greater number of points means that
the database is better but the required time is also greater. The fact of knowing both
parameters allows the user to adjust better calculation. Another change is that the point
31
that is being calculated at each moment is displayed in the command window.
When the AMB module is started, it is shown a figure with the three views of the
aircraft. In the original version the vertical stabilizers were not shown in the correct position, for this reason the code has been changed in the same way as the ‘tornado geo file.m’.
Figure 30: Three views for both configurations.
Once the iteration has finished it is possible to see the results in the graphic interface
of the programme and save it in the xml file. However the values of the aerodynamic
derivatives are not shown, and it would be useful to see the values in order to understand
the behavior of the aircraft. For this reason, there is a new function called showtable.m
which stores all the aerodynamic derivatives in a .txt file called AMB results. A part of
the code is shown in the figure 31, and in the appendix A is shown the results of the .txt file.
Figure 31: Showtable.m code.
Finally, the last change is related with the viscous flow option. When the user chooses
32
this option, the loop goes into another part of the code. in this part there is a while loop
that requires that a condition is satisfied. This condition is obtained in the great majority
of the cases but sometimes not. And the problem is that if this condition is not true one
time, the program will continue trying to achieve it and there is no solution.
Fortunately, if the solution is forced for the points that do not get the condition, the
error is of order 10−4 , which is completely admissible. The procedure is the next, if after
ten times the condition is not achieved the solution is forced and the values are displayed.
Looking at the values the user can decide if the error is admissible.
5.2
Results.
Really, the most important are not the results, is the database. The AMB has a graphical utility where the user can see different graphics and check if the results are proper.
However, the SDSA has more options and is better to see in it the aerodynamic plots.
In this project all the databases have been done with viscous and compressible flow,
trying to obtain the most realistic solution. As an example, in the next picture are shown
two typical curves. However, the correction that AMB has for the viscous effects does
not give good results for the parasite drag. The parasite drag (Cd0 ) calculated is 0.00559
which is far from the empirical values (about 0.015-0.025), but without viscous effect the
value would be worse (about 0.0001).
Only the results for the first configuration are shown because, as we will see, this is
going to be the chosen configuration.
Figure 32: Cl - α.
33
Figure 33: Cd - α.
The results that we obtain are logical for the lift and a bit strange for the drag. In
the case of the lift the curve is a straight whose slope changes depending on the speed.
The Cl increases with the angle of attack but it is important to note that we are studying
only the linear region. For greater angles of attack would exist a region where the lift
coefficient falls, this is shown in the figure 34.
Figure 34: Typical Cl - α plot.
For the drag only remark that it increases with the speed. The maximum value that
has been calculated is M = 0.7 because greater numbers of mach would not give realistic
results.
34
The most important result that the AMB gives is the values of the aerodynamic derivatives which are used in the SDSA module to obtain the stability analysis. The summary
is shown in the following tables.
35
36
6
Analysis of stability and control.
6.1
Previous procedure. Propulsion module
In order to run the SDSA module, it is necessary to have run the ’AcBuilder’, ’Weights
& balance’ and ’Propulsion’ modules. This is because we need obviously a geometry implemented in AcBuilder, a matrix of inertia given by Weigths & Balance and propulsive
features, calculated with propulsion module.
Since it has no difficulties to run and nothing to be changed in the code, the propulsion
mode has been run without problems with the following parameters, which have been
chosen precisely in order to correspond with the range of altitudes and speeds that would
be usual for the aircraft during a supposed cruise flight. In our case, as we explained
for doing the database with AMB, our range in cruise flight is M = [0.2; 0.8] and h =
[9000; 12000]m. Taking into account what we have said, the parameters appear in the
following image:
Figure 35: Parameters of Propulsion module
And the results of the module once it has run:
Figure 36: Results of Propulsion module
37
6.2
Stability and control
An stability and aerodynamic analysis has been carried out with the module SDSA, in
order to get the main performances of the aircraft and to ensure that the aircraft is stable.
As it’s well known we have been working with 2 configurations, thus in the present section
the results of the SDSA module are shown for both configurations. Basically, the goal is
to analyze the stability results given by SDSA for each configuration, compare them and
extract some interesting initial conclusions.
However, firstly it is necessary to show that the static margin is acceptable, as reported in figure 37. As can be seen, the static margin with 0 degrees of incidence is about
10%, as should be. As explained in the geometry section of this paper, the location of the
wing has been changed since the previous calculus in Valencia until this calculus carried
out with CEASIOM and its modules. The fact of having the wing in a more rearward
position allows the center of gravity to be forward the neutral point, which makes the
aircraft longitudinally stable: as the wing is the main lift producer, the more rearward
the wing is, the more rearward the neutral point is, and this fact allows a proper static
margin for the stability.
Figure 37: Static margin.
Once the static margin is established, it is necessary to limit a proper interval for calculating the eigenvalues that would give us the necessary information about the stability,
in order to avoid the calculus of the stability in an illogical interval: a too high or too low
altitud, or a too high or too low speed. The analysis has been done for a cruise flight,
namely the altitude and the speed ranges are the same that we used in the Propulsion
module, explained in the previous subsection.
The named range of the most representative analysis is between 9 km and 12 km of
altitude and between 180 m/s and 240 m/s of speed, and the poles in this range show
that the aircraft is mostly stable, which is endorsed by the representation of the different
38
modes in some graphics that show for each type of mode the different regions of acceptable
or not acceptable stability: longitudinal modes (phugoid and short period) and lateraldirectional modes (dutch roll, roll and spiral, because no roll spiral was identified). This
is reported in figures 38 to 46, in which can be seen that the poles are in the region that
makes the aircraft stable.
6.3
Phugoid mode.
The phugoid mode is the first longitudinal mode. As can be seen in the images below,
the phugoid mode has the same characteristics in both configurations when there is no
regulator, namely the system by itself. All the points are in the acceptable area, so there
is no problem regarding to the stability due to the phugoid mode.
Thus, we have the following results for each configuration considering the phugoid
mode:
Figure 38: Recommended Phugoid Characteristics. Configuration 1.
39
Figure 39: Recommended Phugoid Characteristics. Configuration 2.
6.4
Short Period mode.
We have the following results for each configuration regard to the short period mode,
which is the second and last longitudinal mode. In the case of the short period mode,
there is not much difference between the two configurations of study, or even the first
configuration is the better one, because it has the points further from the unacceptable
area.
Figure 40: Recommended Short Period Characteristics. Configuration 1.
40
Figure 41: Recommended Short Period Characteristics. Configuration 2.
6.5
Dutch Roll mode.
We have the following results for each configuration regard to the dutch roll mode, in
which there are not significant variations between them. Some points are in the unacceptable area, but this has not much importance, because these points are not logical in
the analysis, they are combinations of velocities and altitudes that are not going to take
place in a normal cruise flight.
Figure 42: Recommended Dutch Roll Characteristics. Configuration 1.
41
Figure 43: Recommended Dutch Roll Characteristics. Configuration 2.
6.6
Roll mode.
We have the following results for each configuration regard to the roll mode. It seems to
be exactly the same results for both configurations, and the stability results are so proper,
as can be seen in the following figures: the area in which are all the points is the better one.
Figure 44: Recommended Roll Characteristics. Configuration 1.
42
Figure 45: Recommended Roll Characteristics. Configuration 2.
6.7
Spiral mode.
We have the following results for each configuration regard to the spiral mode. As occurs
with the dutch-roll mode, all the points are the acceptable area of stability, except a
few points that are combinations of altitudes and velocities that will not ever take place
normally. Once more, both configurations have the same behavior.
Figure 46: Recommended Spiral Characteristics. Configuration 1.
43
Figure 47: Recommended Spiral Characteristics. Configuration 2.
6.8
Conclusion of the stability analysis.
As a brief conclusion, it is clear since the results achieved that both configurations have
almost the same behavior considering the stability. In this sense, there is not enough
arguments to decline the first configuration, and it will be necessary to do an structural
analysis in the corresponding section of the paper, and this analysis will be the one that
decide weather the configuration 1 is definitely proper, or the configuration 2 is better.
6.9
Manoeuver test.
However the aircraft is stable for a normal cruise flight, when a modification in the trajectory is done, or it is managed to do an special manoeuver, it is not so stable and requires
a regulator that could be able to return the aircraft to its normal flight position. This
has been carried out with an LQR (Linear Quadratic Regulator): this LQR works in the
state-space, and the objective of the LQR control is to control the different states by feedback with the minimum cost; it consists on 2 matrices: Q and R. The Q matrix represents
the weight that we give to each state: the more weight we give to the state, the more
action is given by the regulator to control it. The R matrix represents the weight that we
give to each control action, and low values imply that we do a great control action in the
respective control. With SDSA, we have eight states and three control actions, detailed
in equations 6 and 7:
Q = [u v
w
p q
r
φ θ]
(6)
44
R = [pitch roll
yaw]
(7)
As explained before, the main aerodynamic characteristic of the aircraft is the position
of the rudder, at the wingtips. Hence, an special manoeuver has been carried out in SDSA
to see the behavior of the aircraft with the mentioned position of the vertical stabilizers,
and with and without the LQR. This manoeuver is an input that consists on a 20 seconds
step of 5 degrees in the rudder. Without the LQR, the aircraft is not instable, but it
turns infinitely due to the coupling of the lateral and directional movements, and when
the LQR is activated, the aircraft returns to its stable position at cruise flight. The values
given to the states and control actions are represented in equations 8 and 9, considering
the order given by equations 6 and 7.
Q = [1 1 1 10 10 1 10 10]
(8)
R = [30 0.001 0.1]
(9)
As can be seen, we have given more importance to the angular velocities in X and Y
axis, the path angle and the roll angle states, and to the roll and yaw control actuators.
This was decided after seeing the behavior of the aircraft during the named manoeuver,
when me decided the most critical states and control actuators to be controlled. The LQR
makes the aircraft stable, and the values of the deflections of the rudder, the ailerons and
the elevators are assumable: the initial stable position could be reestablished with the
deflections given in expression 10 and this could be carried out in 10 seconds, as reported
in figure 48.
Moreover, the effect of the LQR can be evaluated with the evolution of the sideslip
angle β, which is the one directly related to the deflection of the rudder, in figure 000000.
In this figure, it can be seen that when the manoeuver is taking place, namely the first 20
seconds, the evolution of the β angle is stable, and the trend is to stabilize in a determinate unknown point. However, when in the second 30 the LQR is activated, the β angle
goes to its stable position, 0 degrees.
δE = 0 δA = 0 δR = 0
(10)
45
Figure 48: Control deflections due to the manoeuver.
6.10
Performances.
6.10.1
Drag polar.
As performances of the Cormorán, the polar (figure 0000) and the flight envelope (figure
0000) must be reported. In the polar figure, the curve CD − CL is shown, and something
remarkable is that the CD0 has a very low value, below 0.01 (concretely 0.006), when a
normal value should be between 0.021 and 0.024, as in similar aircraft. This is due to
the fact of using ’Tornado’ in the AMB module, which does not take into account the
viscosity accurately, and the CD0 is mainly due to the viscosity effect.
Figure 49: Polar with SDSA.
46
As explained, the value of the CD0 was not proper, so that an alternative procedure
had to be done to estimate the correct polar of the aircraft. In this sense, the flat-plate
method has been carried out to obtain a more proper polar.
Since the geometry of the Cormorán is well known, the calculus can be carried out
more precisely. The aim is to obtain the contribution of the different elements of the aircraft to the parasite drag, which is the drag not directly associated with the production
of lift. Later, the induced drag will be calculated in order to obtain the complete drag
polar. Thus, a parabolic model will describe the drag coefficient versus lift coefficient
(equation11):
CL2
(11)
π · AR · e
For calculating the parasite drag, the flat-plate method has been followed, which
implies to consider the wing as a flat plate and to calculate the drag generated. However,
as the wing is not really a flat plate, it is necessary to scale the mentioned calculated
drag with a form factor, which is the way to include the concrete geometry of the wing,
because it contains parameters like the relative thickness of the airfoil, the leading edge
sweep angle, the position where the maximum thickness is located and the Mach number.
This procedure will also be carried out for the rest of surfaces of the aircraft. Hence, in
the case of the wing:
CD = CD0 + CDi = CD0 +
z The drag of the wing can be calculated with equation 12:
1
D = ρV02 Swet Cf
2
(12)
z From the equation above, the density and the velocity are known (considering M =
0.7 and an altitude of 10500 m), as well as the wetted area of the wing (125 m2 ).
z The Cf factor is the most complex one. This coefficient depends on the type of flow
along the wing: laminar or turbulent. A turbulent boundary layer over the wing
may be desirable, because even the laminar boundary layer generates less drag, the
flow can separate more easily. The Reynolds number is needed (equation 13) to
calculate the laminar and turbulent Cf (equations 14 and 15):
Re =
c · V0
= 198080
ν
(13)
Cfl aminar = 1.328 · Re−0.5 = 0.0029
(14)
Cft urbulent = 0.455 · (log Re)−2.58 · (1 + 0.144M 2 )−0.65 = 0.0057
(15)
47
The Cf chosen is the greater of both, namely Cft urbulent .
z The wing form factor is given by the equation 16:

F Fwing = 1 +
0.6
(x/c)
t
c
!
+ 100
t
c
!4 
 · 1.34M 0.18 (cos Λ)0.28 = 1.5246
(16)
For the rest of components the procedure is the same, and the form factors are the
following:
z For the fuselage, considering that f represents the slenderness (ratio between length
and diameter, with a value of 36.2 m and 4 m respectively), the form factor is
given by equation 17:
F Ff uselage = 1 +
60
f
+
= 1.1035
3
f
400
(17)
z For the engine nacelles, considering a length of 36.2 m and the diameter of 4 m,
the form factor is given by equation 18:
F Fnacelles = 1 +
0.35
= 1.2566
f
(18)
z Regarding the canard, the equation is the same as the wing (equation 16), but
with its particular geometrical parameters, which results the following value of the
canard’s form factor:
F Fcanard = 1.5075
(19)
z Regarding the vertical stabilizers, the equation is the same as the wing and the
canard (equation 16), but with its particular geometrical parameters, which results
the following value of the vertical stabilizers form factor:
F Fvert.stab. = 1.5787
(20)
Once the form factor of all of the 0 j 0 components is calculated, the CD 0 can be calculated by equation 21:
Cf,j Swet,j F Fj
Sw
of each component are in Table 8:
CD0,j =
So that the results of the CD0
(21)
48
Table 8: Contribution to the
CD0 of each component.
CD0,j
Wing
Fuselage
Nacelles
Canard
Vertical stabilizers
0.0096
0.0083
0.0014
0.0026
0.0006
And the total CD0 :
CD0 = 0.0226
(22)
As can be seen regarding to the result obtained, the value of the CD0 is more accurate
considering the typical range of values of this parameter. To have the complete polar
equation, the calculus of the induced drag is needed.
The induced drag is mainly caused by the vortex at the wing tips. The vortex induce
that the angle of attack seen by the airfoil is lower than the geometrical angle of attack,
which causes a negative component of the lift that adds to the drag. The induced drag is
given by equation 23:
CL2
(23)
π · AR · e
The aspect ratio (AR = 8.712) is a known value of the Cormorán, so the main aim
is to decide an Ostwald coefficient. In this sense, several statistical equations are used
commonly, and we have decide to use the proposal of Raymer (equation 24):
CDi =
e = 4.61 1 − 0.045AR0.68 (cos Λ)0.15 − 3.1 = 0.537
(24)
Substituting values, the complete polar equation is obtained (equation 25) and it is
plotted in figure 0000:
CD = CD0 + CDi = CD0 +
CL2
= 0.0226 + 0.068CL2
π · AR · e
(25)
49
Figure 50: Polar with flat-plate theory.
6.10.2
Flight envelope.
The flight envelope has been carried out considering two engines of 60kN each, and with
a bypass ratio of 5 : 1, which should be enough for this aircraft. As can be seen in figure,
there is a grey zone inside the whole flight envelope, which is the one where the aircraft
will be able to fly. The rest of the area is not a suitable area for flying with this particular aircraft considering the design parameters explained in the introduction section, but
appears because the SDSA module gives the envelope result based only in the maximum
thrust of the engines at each altitude.
Figure 51: Flight envelope.
50
7
Structural and aeroelastic analysis.
In this section flutter and divergence are studied[18] , [19] , [20] in order to prove if it is
structurally feasible to have such vertical stabilizers. To do that, the first step is to define
the aerodynamic and the structural models which have to be as realistic as possible. Once
both models are defined a mesh must be generated.
In the case of the aerodynamic mesh, it can be used for panel methods as Vortex
Lattice. This is the same method that was used for the stability and control results. The
reason is because panel methods are suitable for design phases instead of Finite Element
methods which require more time and more calculus power.
About the structural model, in the second section the weight analysis was explained,
however for the structural analysis it is necessary a more accurate study and the definition
of several variables as the stiffness and inertia of every component (skin, web, stiffeners...).
The way to do it is the following: according the EASA CS-25 rules ten maneuvers have
been defined which check the behavior of each component and control surface. Then the
algorithm determines all the unknown parameters by the minimum weight principle. This
means that the weight of the aircraft depends on how strong are the maneuvers.
In other words, after setting the maneuvers, the loads for each one are defined. These
loads are used to create a stick beam model and to calculate the mechanical properties.
And also to obtain a mass distribution over fuselage length and lifting surfaces structural
span. Obviously, all the elements have to be linked in order to have only one body. Figures 52 and 53 shows the aerodynamic and structural models, and the figure 54 shows
both models coupled.
Figure 52: Aerodynamic model
Figure 53: Structural model
51
Figure 54: Aeroelastic model
Note that not only the structural masses are taken into account for the estimations,
but also the lumped non structural densities per unit length are included to consider other
items as passengers, baggage, paint, furniture...All this process is performed by the Guess
module, which belongs to NeoCASS.
7.1
Code modifications
The original NeoCASS version allows the user to analyze typical aircraft configurations.
However, the Cormorán is not like these aircrafts, it is a new ”solution” that needs to
make some changes in the code. This modifications have required an important task to
understand how the program had been built, do not forget that all the code is compound
by hundreds of files. Obviously not all the files have been modified, only the ones that
are necessary to define the geometry that we are studying.
In the next paragraphs the main changes are going to be explained and also some
parts of the code will be displayed. Note that this is not the final report, so the details
are not going to be shown. The objective is to make in an easy way an explanation of the
improvements.
The first step is specify the exact position of the vertical stabilizers. To do that, the
equations of the AcBuilder have been used. These equations determine the position (In
the three coordinates x,y,z) using known parameters as the span, the sweep angle or the
dihedral angle. So at this time the position where the vertical stabilizer root has to be
set is clear. Depending on the chosen configuration the equations are different making
necessary a loop.
Once we know the position where the vertical stabilizer has to be set, it is necessary
52
to move it from the original position. What we have done is to estimate the difference between the original position and the new one, so that is the distance that the stabilizer has
to be moved. This process is the same that has been explained in detail in the AcBuilder
section.
All this steps have been done in the ‘Stick Points Vert NEW.m’whose final objective is
to create a matrix with the points of the stabilizer. The structure of the matrix is the next:
it has three rows and two columns (because this is the minimum size to describe the initial
and the last point), in the first column we have the coordinates of the vertical stabilizer
root and, in the second, the coordinates of the tip. For example, in the configuration 1
the matrix obtained is:

34.6952 38.3280
 16.4774 16.4774  .
−0.3154 3.6846

According to the configuration that we choose for the aircraft, the ‘duplication’of the
vertical stabilizer is needed, and it was not programmed, so we had to modify the file that
contains this part of the code, which is called ‘symmXZ.m’. In other words, originally the
program only is able to analyze one vertical stabilizer because this is the most common
configuration, but we need two. In the ‘Stick Points Vert NEW.m’file we have moved the
stabilizer to the correct position of one wing, and in ‘symmXZ.m’file we have duplicated it.
As can be seen in the following image, we got the objective creating the points, panels
and nodes of the stabilizer situated at the left wing firstly copying the points from the
right wing and then changing the sign of them, in order to get the symmetry (something
similar to the command lines referred to the twin-tail case). Note that only the second
row has his sign changed, and that is because the symmetry is referred to the X-Z plane,
so we have to change the sign of the ’y’ variable and keep the sign of the ’x’ and z’ variables.
53
Figure 55: ‘SymmXZ.m’code.
Hence, as in the other modifications of the code, the condition that has to be true to
get into the loop and do the programmed tasks is that a vertical stabilizer and the called
’winglet rudder’ must be present, obviously.
After the stabilizers are in the correct position the next step is attach it to the wing.
The process is like this, we have to verify if there is any point in common between both
structures. If this point exist the problem is solved, however the experience shows that
usually there is a small difference in the decimals. Hence, as we know that the stabilizer
and the wing have one point in common, the solution is to force this link.
The advantage is that the vectors have always the same structure, so the coordinates
of the wing tip are always in the same position inside the vector. This allow us to link
the stabilizer with this known point. Once the two structures have one point in common
(Because it has been forced) the next steps are easier, we only have to do that the point
of the stabilizer is a slave of the same point of the wing.
This process is similar to others such as connecting the wings to the fuselage or the
canard with the fuselage. Note that this step is very important, if the links between the
different surfaces are not done the modes that we will obtained are not true. The aircraft
is only one item, it is true that it is compound by different elements but the total behavior
is affected by each one.
To express the importance of this file we are going to use a real example. Imagine that
the verticals stabilizers are not linked to the wings. In this case the movement of the wings
do not affect the stabilizers, so it is the same situation as if we solved the case of an aircraft without vertical stabilizers. NeoCASS does not show any error because is analyzing
the aircraft and the vertical stabilizers separately. In addition, we will obtain twice modes.
All this process has been the most difficult to implement because we have to link every
element to each other but in the right place. In other words, every wing has 20 nodes but
54
only one has to be linked with the fuselage (The root node). And, in the other hand, only
one node of the fuselage has to be linked with the wing. Repeating this process for every
element we finally obtain two slaves nodes for the wings, one for the canard and another
two for the vertical stabilizers.
Actually the process is more complex but is not the objective of this report to cover
all the details. The following picture shows the lines of the corresponding file which is in
this case ‘Link Structs.m’(Only the code for the right stabilizer and the first configuration
is shown).
Figure 56: ‘link structs.m’code.
Finally, as an improvement of the results, we have added an option to print in the
command window the deformed, in other words, shows the difference between the original
point and deformed point. The deformed model plot is very useful to see which type of
mode are we analyzing, however this extra option allows the user to know which is the
55
exact distance in meters.
Really only is shown the values for the wing tip and the vertical stabilizer tip because
are the most critical points for our aircraft, but any other point can be implemented. The
picture 57 shows the code.
Figure 57: ‘plot beam defo.m’code.
7.2
Results
The code with all the implementations that have been explained in the previous section
works and gives good results, but it is not tested. In other words, it is impossible to know
if the results are ok because it is necessary to do the same calculations with other programme like Nastran. For this reason the results that will be commented in the following
paragraphs have been obtained with a more recent version of NeoCASS that Sergio Ricci
has validated for the configuration of the Cormorán.
The original idea was to compare the results of both versions in order to validate the
version that we have implemented. However, during this task, we discovered that the new
version of NeoCASS has several differences in the algorithms and in the loads calculations.
For this reason it makes no sense to compare and contrast the results of both versions.
Really, the new NeoCASS version of Sergio Ricci is an adaptation of the twin tail that
already existed. The difference from the original code is that if the horizontal tail and
the vertical tail are not connected then the code links the vertical tail with the nearest
point of the wing. In the case of the Cormorán the horizontal tail does not exist, so the
programme detects this fact and links the vertical tail with the wingtip. This solution is
similar to the one that we did but is easier in the sense that less lines of code have been
changed.
56
Before starting the analysis of the results the most important is to notice that in this
point the second configuration is discarded. The reason is because the only point that
this configurations improves the configuration one is in the aeroelastic results, but the
results for the first configuration are good enough because flutter speed is outside the
flight envelope.
Namely, in all the aspects that have been studied before like aerodynamics or stability
the first configuration was better. The existence of the second configuration was because
the flutter could be a big problem if the vertical stabilizers were at the wingtip. However,
the flutter speed for the first configuration is so high that it is not necessary to continue
with the second configuration.
The main advantage of discard the second configuration is that we do not have the
problem of having a big vertical stabilizer and a winglet in each wing. This could create
aerodynamic problem because it is difficult to estimate the interaction between these elements and also the engines and the fuselage.
Starting with the results, it is very important the GUESS phase because if the model
does not seems to the original aircraft the results will be false. The results of the Guess
sizing are shown in the table 9. These values have been compared with the results of
other typical methods (In order to validate the model) for aircraft weight estimations like
Torenbeek[21] or Howe. There are differences between the methods, but all the values are
quite similar.
Table 9: Cormorán weights.
Component
Fuselage
Wing
Canard
Vertical stabilizers
Landing gear
Interior
Systems
Engines
Passengers
Baggage
Fuel
OEW
MZFW
MTOW
Guess
6801
9557
911
99
2409
6197
10400
7400
14877
1487
12317
44065
60431
72097
Torenbeek Howe
6215
7546
9075
3325
692
636
80
81
2737
2497
8200
6560
8000
9314
7510
12000
14786
14786
1450
1450
11666
11666
42083
43441
58319
59677
69885
71343
57
Table 10: Vibration modes.
Mode
1
2
3
4
5
6
7
8
9
10
Frequency
(Hz)
1.32427 · 10−6
1.87706 · 10−6
2.12776 · 10−6
3.57673 · 10−6
5.63686 · 10−6
6.17825 · 10−6
2.86468
3.8303
4.08739
5.26967
Mode
11
12
13
14
15
16
17
18
19
20
Frequency
(Hz)
6.14911
6.98
8.86266
8.97343
9.694
10.1608
11.1046
12.2225
12.9216
14.0079
Once the Guess sizing is finished and validated it is time to solve the modal analysis
to obtain the vibration modes. Only the first 20 frequencies have been solved because the
critical one will be one of these. This is expressed as ”It is a general rule that the modes
with the lowest frequencies are the ones which should be examined for evidence of flutter”
(Bisplinghoff 1983).The same author said that ”Experiences have shown that either first
bending or first torsion leads to the critical flutter mode”. This second statement it is
not true in this case because the flutter phenomena appears in a mode where the vertical
stabilizers are the critical element.
Table 10 shows the first 20 frequencies. The first 6 modes are rigid body modes so
there are not important for the flutter analysis. These modes are only translations and
rotations in the six degrees of freedom as is shown in figure 58 and 59 which corresponds
to the modes 3 (z-axis translation) and 4 (rolling). To understand the picture note that
the yellow nodes represent the deformed shape and the blue nodes the original model, and
the scale factors are 20 and 5 respectively in order to appreciate the details. However, the
rest of the modes are elastic body modes and are the ones that it is necessary to analyze
in order to obtain the flutter speed.
Figure 58: Deformed shape for mode 3
58
Flutter results at sea level show that the theoretical flutter speed is around 524m/s,
this means that flutter appears at supersonic speeds. For this reason the value of the
flutter speed is not reliable, because the aerodynamic method that has been used is for
subsonic aerodynamics. So, how could we interpret the results? It is impossible to predict
which is the real flutter speed but it is sure that this speed is outside the flight envelope
so the conclusion is that flutter never appears for this aircraft configuration.
Independently, it is interesting to see which is the flutter V-g plot and which is the
critical mode to understand how the structure works and the difficulties that the vertical
stabilizers produce. Figure 60 presents the V-g plot which shows that the critical mode,
with positive damping, is the number 12. This mode and its deformed shape is represented in the figure 62 whose scale factor is 20. The information that the deformed shape
gives is that the critical elements are the vertical stabilizers. This seems to be logical
because the wingtip is the point of the wing with the biggest deformation, and this is
added to the deformation of the vertical stabilizers.
Figure 60: V-g at h = 0
Moreover, the deformation of the fuselage and the canard is nearly null. This is due
to the fact that the critical mode excites the wings and the vertical stabilizers principally.
59
There are other modes that affects other elements such as the mode 10 in which the elements that are excited are the canard and the wings. This is shown in the figure 61 in
which it can be seen that the deformation at the wingtip is the most important but the
vertical stabilizers are not affected by this. Due to the configuration it would be expected
that the critical mode was related with the vertical stabilizers and this is exactly what
happens.
A flutter study of the vertical stabilizers showed that if the span of them is increased,
flutter appears in the flight envelope. For example, if the span of the vertical stabilizers
is 4 meters, flutter speed is around 210m/s and the critical mode is the same. For this
reason, a detailed study of the vertical stabilizers have been done in order to avoid flutter
and obtain a good response for manoeuvering, showing that three meters span is ok to
achieve both objectives.
60
8
Payload-Range diagram
An aircraft does not have a single number that represents its range. Even the maximum
range is subject to interpretation, since the maximum range is generally not very useful
as it is achieved with no payload. To represent the available trade-off between payload
and range, a range-payload diagram may be constructed. This diagram is shown in the
figure 63.
Figure 63: Payload-range diagram.
The first important point is the range with maximum payload, which is 4200 km. This
point is achieved with the maximum payload but not with the maximum fuel weight, for
this reason the range could increased. The following point is the range with the maximum
fuel weight which is 6019 km. This value is achieved when the fuel weight is the maximum
and the payload is the maximum amount that it is possible, but is lower than the payload
at the first point. Finally, the last point is the range without payload, which is 8037.85
km. This last point is a theoretical point but it is not realistic because no company uses
an aircraft without payload.
Considering that productivity is the product of the payload weight and the distance
that this payload is transported (range) we can approximate that the point of maximum
productivity is when the range is 5700 km.
In order to help the interpretation of the range results in the figures 64 and 65 it is
shown where can we arrive with the Cormorán for the point 1 and 2.
61
Figure 64: Range with maximum payload.
Figure 65: Range with maximum fuel weight.
62
9
Cost estimation
Once the aircraft is defined it is the turn to estimate the cost in order to know if it is
competitive respect the Boeing 737 − 800 and the Airbus 320. Logically, if the procurement cost of the Cormorán is higher, the possibilities of being sold are smaller. For this
reason the study of the cost is very important. To do that, the followed method is the
”Development And Procurement Cost Of Aircraft (DAPCA)”[22].
This method is based on the calculus of the development and manufacturing costs, in
other words, the method estimates all the costs that a new aircraft involves and gives the
minimum procurement cost to be profitable. The necessary inputs are the operational
empty weight, the cruise velocity and the number of units that are going to be produced.
The first two inputs are known and the number of units has been fixed in 100. Hence,
we are going to describe how to obtain each of the costs and then the result will be the
procurement cost. But before this it is important to note that all the results are in US
Dollars because the method is implemented in this currency. However in the table 11 the
results are shown in Euros to make it easy the comprehension.
The first step is the calculation of the hours of engineering, assembly and manufacturing.
HE = 4.86OEW 0.777 · V 0.894 · Q0.163 = 9.94683 · 106
(26)
HA = 5.99OEW 0.777 · V 0.696 · Q0.263 = 5.62329 · 106
(27)
HM = 7.37OEW 0.82 · V 0.484 · Q0.641 = 1.64489 · 107
(28)
The costs of development, flight tests and materials are the followings.
CD = 45.42OEW 0.630 · V 1.3 = 1.186 · 108
(29)
CF = 1243.03OEW 0.325 · V 0.822 · F T A1.21 = 5.727 · 107
(30)
CM = 11OEW 0.921 · V 0.621 · Q0.799 = 3.47206 · 108
(31)
63
In the equation 30 FTA is the number of flight tests. The typical number of tests is
between 2 and 6 but, in order to be conservative, it has been taken 6 because the cost is
greater. The following cost to take into account is the cost of the engines. This value is
known once the engine is chosen but in this case we only know that it is a turbofan with a
maximum thrust of 60kN. For this reason the cost that has been used for the calculations
is the typical for an engine of this size.
Ceng = 7.5 · 106
(32)
To estimate the avionics cost, the method propose that the avionics weight is the 5%
of the OEW and the cost is 7000$/kg.
Cavionics = 7000 · 0.05 · OEW = 1.311 · 107
(33)
With all the previous results, the total costs of RDT & E (Research Development Test
& Evaluation) and manufacturing can be calculated as in the equation 34.
RDT &E = HE · RE + HA · RA + HM · RM + CD + CF + CM + 2Ceng + Cavionics
(34)
In the equation 34 RE , RA and RM are the engineering ratios. These ratios are used
to convert the hours that have been calculated into costs. To obtain these ratios it is
necessary to add the inflation from 1986 (Year when the method was created) until 2013
and a factor of 1.63 due to the use of composites.
RE = 1.63 · 59.1 · 1.957 = 188.52
(35)
RA = 1.63 · 60.7 · 1.957 = 193.6275
(36)
RM = 1.63 · 50.10 · 1.957 = 159.8144
(37)
Replacing these values in the equation 34 we get the following result.
64
RDT &E = 7.68 · 109
(38)
Really, the previous value is 25% greater than the real because the complexity of the
aircraft makes advisable to increase the cost. Note that the method has been developed
for the typical configurations, so it is conservative to guess that the cost must be greater
than the one that the method gives.
The costs of the interiors are difficult to calculate because it depends on what the
company demands. In this case the value is 2000$ per passenger, which is a typical value
for aircrafts with only tourists passages.
Cinteriors = 2000 · 164 = 32.8 · 104
(39)
Finally the total cost of the aircraft is
T otalcost = RDT &E + Q · Cinteriors = 7.7128 · 109
(40)
Hence, we know how much money it is necessary to produce 100 units of the Cormorán. With this data it is possible to fix the procurement cost or, in other words, the
price of the aircraft that makes the production profitable. If the manufacturer only wants
to recover the investment the price is the total cost divided by 100. Obviously, this price
does not have sense because the manufacturer must earn money. For this reason we are
going to calculate the price or the procurement cost for a benefit of the 18% and 12%. In
the first case we have the following result.
P rocurementCost(18%) = 1.18
RDT &E
+ Cinteriors
Q
= 91.01 · 106
(41)
The interpretation of the result is very simple, if the company sells the 100 the benefit
is the 18% but if not they need to sell 85 units to recover the investment. In other words,
it is necessary to sell 85 units because otherwise they will earn less money that they have
spent. The result for a benefit of the 12% is which is shown in the equation 42.
65
P rocurementCost(12%) = 1.12
RDT &E
+ Cinteriors
Q
= 86.38 · 106
(42)
In this case it is necessary to sell 89 units to recover the investment.
As we have said all the results are in US Dollars. In the table 11 are shown again
the results but in this case in Euros. Note that the values will change depending on the
difference between both currencies in each moment.
Table 11: Development and procurement costs.
Cost
Engineering
Assembly
Manufacturing
Development
Interiors
106 e
7.44
4.206
12.303
88.712
0.244
Cost
Flight tests
Materials
Engines
Avionics
106 e
42.83
259.71
5.6
9.806
Remembering that two cases have been calculated (18% and 12% of benefit), in the
first case the procurement cost is 67.613 · 106 e and for the second case is 64.18 · 106 e.
The method that we have been describing allows us to estimate a procurement cost
without taking into account the details of this aircraft. This means that the result is
approximated and a more rigorous study would be necessary to get a better estimation.
However, as a first approach, the method gives an starting value which gives an idea of
how expensive or not that the aircraft is.
The price of a 737 − 800 is about 72.5$, but it is difficult to know really how much
it cost because the price could change depending on the number of units purchased by
the company. The obtained price for the cormorán is bigger, but note that we have
increased this price the 25% to be conservative and, probably, the costs of assembly and
manufacturing are out of order because the processes have changed since then.
66
10
Conclusion
In the report the main characteristics of this new design have been explained and they
show the flying qualities of the Cormorán are reasonable. In other words, from the
aerodynamic viewpoint, the aerodynamic derivatives that have been calculated and the
subsequent stability analysis indicate that the aircraft with its geometry is stable and is
able to flight with the conditions of safety and manoeuvering expected in a commercial
aircraft.
And the structure, which is possibly the most critical aspect, is robust enough to resist
the aeroelastic phenomena because they appear outside the flight envelope, so the aircraft
safety is not compromised.
Obviously, this report shows the preliminary design of all the aircraft, so there are
elements and aspects which require a more detailed analysis to determine exactly which
is their behavior. However, this task will be part of an improvement of the design, and
do not affect the idea that the design works and must be taken into account for future
developments in the aerospace industry.
67
References
[1] Gómez,V.; Izquierdo,E.; Calatayud,A.; Sanmartı́n,J.,Cormorán project., Universitat
Politècnica de València, 2011.
[2] Raymer, D., Aircraft Design: A conceptual approach, AIAA Education Series, USA,
2006.
[3] Roskam, J., Airplane Design, University of Kansas Press, USA, 1985.
[4] Gómez,V.; Izquierdo,E.,The Cormorán project: a new concept in commercial aircraft
design., Report for the AIAA-Pegasus students conference 2013, Dept. of Aeronautical & Vehicle Engineering, Stockholm, Sweden, 2013.
[5] CEASIOM website. www.ceasiom.com.
[6] Puelles,A.,CEASIOM XML file Definition,Royal Institute of Technology,2010.
[7] Melin,T.,Tornado reference manual, 2001.
[8] Melin,T.,A Vortex Lattice MATLAB Implementation for Linear Aerodynamic Wing
Applications, Master Thesis.
[9] Pester,M.,Multi-disciplinary conceptual aircraft design using CEASIOM,Hamburg
university of applied sciences,2010.
[10] Magraner,J.P.,Flight Mechanics Notes., Universitat Politècnica de València, 2011.
[11] Hodgkinson, J., Aircraft Handling Qualities, AIAA Education Series, Oxford, 1999.
[12] SDSA user manual,Warsaw university of technology.
[13] Grabowski,T.; Mieszalski,D.; Marcinkiewicz,E., Stability analysis using SDSA tool,
Progress in Aerospace Sciences,Vol.47(8), pp.636-646, 2011.
[14] External performance module user manual,Warsaw university of technology.
[15] Raymer, D., Aircraft Design: A conceptual approach, AIAA Education Series, USA,
2012.
[16] Raymer,D.; Wilson,J.;Douglas,H.; Rizzi,A.; Zhang,M.; Ramı́rez,A.,Advanced technology subsonic transport study. N+3 technologies and design concepts.,NASA/TM2011-217130.
[17] Dorf,R.; Bishop,R.,Modern control systems., Pearson, 2011.
[18] Cavagna,L.; Ricci,S.;Da Ronch,A.,NeoCASS manual, Aerospace engineering department, Politecnico di milano,2009.
68
[19] Cavagna,L.; Ricci,S.;Riccobene,L.;De Gaspari,A.,NeoCASS Overview and GUIs Tutorial, Aerospace engineering department, Politecnico di milano,2009.
[20] Cavagna,L.; Ricci,S.; Travaglini,L., NeoCASS: An integrated tool for structural sizing,
aeroelastic analysis and MDO at conceptual design level , Progress in Aerospace
Sciences,Vol.47(8), pp.621-635, 2011.
[21] Torenbeek, E., Synthesis of Subsonic Airplane Design, Delft University Press, Netherlands, 1982.
[22] Hess,R.; Romanoff,H.,Aircraft Airframe Cost Estimating Relationships.,Rand Corporation, Santa Monica, CA, 1987.
[23] Abbott, I.H.; Von Doenhoff, A.E., Theory of wing sections, Courier Dover Publications, USA, 1959.
[24] Nickel,K., Tailless aircraft in theory and practice, AIAA education series , USA, 1994.
[25] izzi,A., Modeling and simulating aircraft stability and control-The SimSAC project,
Progress in Aerospace Sciences,Vol.47(8), pp.573-588, 2011.
69
A
Aerodynamic derivatives
Figure 66: Example of the .txt that the showtable.m function gives.
70

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