The Cormoran project: a new concept in commercial aircraft design

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The Cormor´ an project:

a new concept in commercial aircraft design

G´ omez Gonz´ alez, V´ıctor Izquierdo Collado, Emilio Jos´ e

Stockholm - 30th of January 2013

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Abstract

This paper presents a new revolutionary design in commercial aircraft: the conven- tional 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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List of Figures

1 Beechcraft Starship . . . 10

2 Beechcraft Starship 3 views . . . 10

3 Curtiss-Wright XP-55 Ascender . . . 11

4 Curtiss-Wright XP-55 Ascender 3 views . . . 11

5 Miles M39B Libellula . . . 11

6 Miles M39B Libellula 3 views . . . 11

7 Rutan Long-EZ . . . 11

8 Rutan Long-EZ 3 views . . . 11

9 Kyushu J7W1 . . . 11

10 Kyushu J7W1 3 views . . . 11

11 Conceptual design process . . . 12

12 3D model of Cormor´an . . . 13

13 CEASIOM package . . . 14

14 Cormor´an’s planform view. Configuration 1. . . 18

15 Cormor´an’s front view. Configuration 1. . . 19

16 Cormor´an’s side view. Configuration 1. . . 19

17 Cormor´an’s planform view. Configuration 2. . . 19

18 Cormor´an’s front view. Configuration 2. . . 20

19 Cormor´an’s side view. Configuration 2. . . 20

20 Cormor´an’s AcBuilder model. Configuration 1. . . 24

21 Cormor´an’s AcBuilder model. Configuration 2. . . 24

22 New options needed and implemented in AcBuilder. . . 25

23 Fuel tanks in both wings. . . 26

24 New options needed and implemented in AcBuilder. . . 26

25 Centers of gravity. . . 27

26 ‘tornado geo.m’code. . . 30

27 3D panels, collocation points and normals for the first configuration. . . 30

28 3D panels, collocation points and normals for the second configuration. . . 31

29 ‘run tornado’code. . . 31

30 Three views for both configurations. . . 32

31 Showtable.m code. . . 32

32 C_l- α. . . 33

33 C_d - α. . . 34

34 Typical C_l - α plot. . . 34

35 Parameters of Propulsion module . . . 37

36 Results of Propulsion module . . . 37

37 Static margin. . . 38

38 Recommended Phugoid Characteristics. Configuration 1. . . 39

39 Recommended Phugoid Characteristics. Configuration 2. . . 40

40 Recommended Short Period Characteristics. Configuration 1. . . 40

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41 Recommended Short Period Characteristics. Configuration 2. . . 41

42 Recommended Dutch Roll Characteristics. Configuration 1. . . 41

43 Recommended Dutch Roll Characteristics. Configuration 2. . . 42

44 Recommended Roll Characteristics. Configuration 1. . . 42

45 Recommended Roll Characteristics. Configuration 2. . . 43

46 Recommended Spiral Characteristics. Configuration 1. . . 43

47 Recommended Spiral Characteristics. Configuration 2. . . 44

48 Control deflections due to the manoeuver. . . 46

49 Polar with SDSA. . . 46

50 Polar with flat-plate theory. . . 50

51 Flight envelope. . . 50

52 Aerodynamic model . . . 51

53 Structural model . . . 51

54 Aeroelastic model . . . 52

55 ‘SymmXZ.m’code. . . 54

56 ‘link structs.m’code. . . 55

57 ‘plot beam defo.m’code. . . 56

58 Deformed shape for mode 3 . . . 58

60 V-g at h = 0 . . . 59

63 Payload-range diagram. . . 61

64 Range with maximum payload. . . 62

65 Range with maximum fuel weight. . . 62

66 Example of the .txt that the showtable.m function gives. . . 70

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List of Tables

1 Geometry data for the fuselage. . . 20

2 Geometry data for the wing. . . 21

3 Geometry data for the canard. . . 21

4 Aerodynamic derivatives. . . 22

5 Geometry data for vertical stabilizers. . . 23

6 Centers of gravity location. . . 28

7 Inertia matrix. . . 28

8 Contribution to the C_D0 of each component. . . 49

9 Cormor´an weights. . . 57

10 Vibration modes. . . 58

11 Development and procurement costs. . . 66

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Nomenclature

c₁ c2

croot

ctip

b l α λ_LE Λ Y L N ρ S V0

φ δR

δA

δ_E 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

Cl_δR ClδA

Clβ

CN_δR CN_δA CNβ

CNEO

d_{vert.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, Ru- tan 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 Cormor´an. 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 Cormor´an [], 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

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Figure 3: Curtiss-Wright XP-55 Ascen- der

Figure 4: Curtiss-Wright XP-55 Ascen- der 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

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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 Cormor´an. 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 signi- ficative 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.

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( 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] (Comput- erised 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 Cormor´an

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2 What is CEASIOM?

CEASIOM is developed in the scope of the SimSAC project. SimSAC stands for Sim- ulating 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.

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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 computa- tion 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. How- ever, 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

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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 func- tionalities: 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.

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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 over- come so that the code can be used to design innovative and uncommon aircraft layouts.

The code is completely written in the MATLAB.

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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 Cormor´an, 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: Cormor´an’s planform view. Configuration 1.

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Figure 15: Cormor´an’s front view. Configuration 1.

Figure 16: Cormor´an’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: Cormor´an’s planform view. Configuration 2.

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Figure 18: Cormor´an’s front view. Configuration 2.

Figure 19: Cormor´an’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) 3.60 dhoriz.Af t(m) 3.32 dvert.F ore(m) 2.91 dhoriz.F ore(m) 3.32 Nose length(m) 2.11 Tail length(m) 5.4

l(m) 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 (λ_LE) 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

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Table 2: Geometry data for the wing.

Property In Mid Out

c1(m) 7 4.37 2.70

c₂(m) 4.37 2.70 1.20

b(m) 5 5.50 5.70

α(^o) 0 1 1

λ_LE(^o) 28 28 28

Λ(^o) 3 3 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 avia- tion, 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(ô) 1 λLE(ô) 32 Λ(ô) 0

Apart from the presence of the canard in a commercial aircraft, the other particular- ity 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.

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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 δ_A, the roll angle φ and the rudder’s deflection δ_R):

Y = (CY_δRδ_R+ CY_ββ + CY_δAδ_A) = CL · φ (1)

L = (Cl_δRδ_R+ Cl_ββ + Cl_δAδ_A) = 0 (2)

N = (CN_δRδ_R+ CN_ββ + CN_δAδ_A) = −CN_EO (3)

Where CN_EO = _0.5ρV^{T ·a}^EO2

0Sb. Note that the aerodynamic derivatives[11] are given by the software used, in this case, by AMB (Aerodynamic Model Builder). These values are shown in the table 4.

Table 4: Aerodynamic derivatives.

Derivative Derivative CYδR -0.073 Clβ 0.092 CYβ -0.171 CNδR -0.017 CYδA -0.015 CNδA 0.064 ClδR 0.012 CNβ -0.030 ClδA 0.354

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 δ_R = 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

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roll angle which minimize the values of δ_R, δ_Aand 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^◦ δ_A = −7.08^◦ δ_R= 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) 1.20 c_tip(m) 0.44 b(m) 3 α(ô) 0 λ_LE(ô) 45 Λ(ô) 0

4 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 Cormor´an 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):

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Figure 20: Cormor´an’s AcBuilder model. Configuration 1.

Figure 21: Cormor´an’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

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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.

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Figure 23: Fuel tanks in both wings.

Figure 24: New options needed and implemented in AcBuilder.

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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.

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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 X(m) Y (m) Z(m)

Wing 29.69 0 −0.95

Canard 10.07 0 −0.54

Vertical Tail 35.8 0 0.95

Fuselage 19.48 0 −0.36

Engines 31.31 0 0.70

Passengers 17.11 0 −0.58

Table 7: Inertia matrix.

Elements kg · m² Ixx 1174486.61 Iyy 5916318.38 Izz 7005027.31 Ixz 132976.21

Ixy 0

Iyz 0

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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: DAT- COM, 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.

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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.

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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

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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

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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 graph- ical 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: C_l- α.

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Figure 33: C_d - 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 C_lincreases 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 C_l - 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.

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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.

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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

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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

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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 lateral- directional 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.

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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.

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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.

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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.

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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.

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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 trajec- tory 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)

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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 determi- nate 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)

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Figure 48: Control deflections due to the manoeuver.

6.10 Performances.

6.10.1 Drag polar.

As performances of the Cormor´an, the polar (figure 0000) and the flight envelope (figure 0000) must be reported. In the polar figure, the curve C_D− C_L is shown, and something remarkable is that the C_D0 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 C_D0 is mainly due to the viscosity effect.

Figure 49: Polar with SDSA.

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As explained, the value of the C_D0 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 Cormor´an 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):

C_D = C_D0+ C_Di= C_D0+ C_L²

π · AR · e (11)

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:

z The drag of the wing can be calculated with equation 12:

D = 1

2ρV₀²S_wetC_f (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 C^f 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 C_f (equations 14 and 15):

Re = c · V₀

ν = 198080 (13)

C_f_l_aminar = 1.328 · Re^−0.5 = 0.0029 (14)

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The C_f chosen is the greater of both, namely C_f_t_urbulent. z The wing form factor is given by the equation 16:

F F_wing =



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 F_{f uselage} = 1 +60 f³+ f

400= 1.1035 (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 F_nacelles = 1 +0.35

f = 1.2566 (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 F_canard = 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 the⁰j⁰ components is calculated, the C_D0 can be calculated by equation 21:

C_D0,j = C_f,jS_wet,jF F_j

S_w (21)

So that the results of the C_D0 of each component are in Table 8:

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Table 8: Contribution to the CD0 of each component.

CD0,j

Wing 0.0096

Fuselage 0.0083

Nacelles 0.0014

Canard 0.0026

Vertical stabilizers 0.0006

And the total C_D0:

C_D0 = 0.0226 (22)

As can be seen regarding to the result obtained, the value of the C_D0 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:

C_Di = C_L²

π · AR · e (23)

The aspect ratio (AR = 8.712) is a known value of the Cormor´an, 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):

e = 4.61 1 − 0.045AR^0.68 (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:

C_D = C_D0+ C_Di= C_D0+ C_L²

π · AR · e= 0.0226 + 0.068C_L² (25)

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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.

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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. Fig- ures 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

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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 Cormor´an 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

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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.

The Cormoran project: a new concept in commercial aircraft design