Aerospace Design Project
Light Business Jet Family Design Challenge
Authors: N. BOUVIER Faculty representatives: N. ESTEVES DE SOUSA G. DIMITRIADIS G. GOFFARD L. NOELS P. LAFONTAINE A. CROVATO K. MASROUR T. DOSSOGNE B. MOCKEL B. ROULETTE
May 11, 2017 1
Name First Name AIAA Membership Signature
BOUVIER Nicolas 775892
ESTEVESDE SOUSA Nicolas 775894
GOFFARD Gilles 775769
LAFONTAINE Pierre 775803
MASROUR Khadija 775893
MOCKEL Brice 775891
ROULETTE Benjamin 775889
Signature of the project members CONTENTS 2
Contents
1 Introduction 5
2 Market, Mission & Design 5
2.1 Market drivers ...... 5
2.2 The light segment ...... 5
2.3 Market research and competition analysis ...... 6
2.4 Design Methodology ...... 7
2.5 Mission requirements ...... 8
3 Aircraft design choices 10
3.1 Fuselage ...... 10
3.1.1 Cabin design ...... 10
3.1.2 Fuselage Length ...... 11
3.2 The wing ...... 12
3.2.1 Super-critical airfoil: NACA SC(2)-0714 ...... 13
3.2.2 Wing geometry ...... 14
3.2.3 Flaps design ...... 15
3.2.4 Results ...... 18
3.3 Propulsion ...... 18
3.3.1 Engine selection ...... 18
3.3.2 Placement and installation ...... 20
3.3.3 6 seats engine choice ...... 21
3.4 Empennage ...... 21
3.4.1 V tail principle ...... 21
3.4.2 Statistically prescribed horizontal and vertical surfaces ...... 22
3.4.3 Design choices and geometry determination ...... 23
3.5 Undercarriage ...... 23
3.5.1 Rotation clearance angle ...... 24
3.5.2 Wheel track and wheel base ...... 24
3.5.3 Tire size ...... 26
3.5.4 Results ...... 26
3.6 Weight considerations ...... 27
3.6.1 Empty weight definition ...... 27 CONTENTS 3
3.6.2 Payload weight ...... 29
3.6.3 Weight of fuel ...... 29
3.6.4 Results ...... 30
3.6.5 Center of gravity ...... 30
3.7 Catia model ...... 31
4 Trade-off study 33
4.1 Aspect ratio of the wing ...... 34
4.2 3D lift coefficient (CL,w) ...... 35 4.3 Fuselage length ...... 36
5 Optimization 37
5.1 Stability ...... 37
5.1.1 Enforcing equilibrium ...... 38
5.1.2 Longitudinal stability ...... 39
5.1.3 Lateral stability ...... 40
5.1.4 6 seats consideration ...... 41
5.2 Aerodynamics ...... 41
5.2.1 Computation of CL,plane and CD,plane ...... 41 5.2.2 TRANAIR ...... 42
5.2.3 Drag analysis ...... 47
5.3 Performance ...... 49
5.3.1 Take-off ...... 49
5.3.2 Climb ...... 52
5.3.3 Cruise ...... 54
5.3.4 Turning rate ...... 54
5.3.5 Landing ...... 57
5.3.6 Payload-Range Diagram ...... 60
5.4 Aircraft structure ...... 61
5.4.1 Flight envelope ...... 61
5.4.2 Aerodynamic loading ...... 64
5.4.3 Structural loading ...... 66
5.4.4 Materials selection ...... 68
5.4.5 Structure preliminary design ...... 69
5.4.6 FEM Analysis - Preliminary results ...... 75 CONTENTS 4
5.4.7 Further improvements ...... 76
6 Costs Analysis 77
6.1 RAND DAPCA-IV Method - Eastlake Model ...... 77
6.2 Effect of inflation on the costs ...... 78
6.3 Selling Price Definition ...... 78
6.4 Production rate ...... 79
6.5 Certification Cost ...... 79
6.6 Production Cost ...... 81
6.7 Break-even analysis ...... 83
6.8 Operating Costs ...... 84
6.9 Conclusion ...... 85
7 Conclusion 86
7.1 Contextualization ...... 86
7.2 Methods ...... 86
7.3 Results ...... 86
7.4 Critical assessment of the conceptual and preliminary design stages ...... 87
A Empirical gross Take-Off Weight estimation 90
A.1 Empiric estimates of each terms ...... 90
B Uninstalled maximum cruise rating 91
C Wing geometry parameters: statistics 92
D Experimental data of the NACA SC(2)-0714 93
E Summary of the wing geometrical parameters 94
F Material Selection: properties of aluminium 95
G Aircraft views 96 1 INTRODUCTION 5
1 Introduction
This report is produced in the context of the American Institute of Aeronautics and Astronautics (AIAA) 2017 Light Business
Jet Family Graduate Team Aircraft Design Competition.
In parallel, the contest is the object of a dedicated school subject throughout the 1st year of master in Aerospace Engineer- ing of the University of Liege. The students participating to this project have access to lectures, coaching and advises from the faculty representatives, Professors L. Noels, G. Dimitriadis, A. Crovato and T. Dossogne.
2 Market, Mission & Design
Nearly ten years have passed since the sub-prime crisis dealt a significant blow to the business jet market. As this latter is strongly correlated to GDP growth, the path towards recovery has been steady, but slow, although we can say most of the manufacturers have already healed up nicely. The future of the trade looks bright, as every aspect of the world’s current development inescapably tends to increase the demand for this particular mean of transportation.
2.1 Market drivers
Global trade treaties are in vogue, and lead to the conclusion that the world economy is progressively shifting to a global and homogeneous system. With boundaries expanding, one understands that the reach of people must also follow the trend, both in time, and space. Let us also add that the emerging economies show pockets of high growth, which leads to sudden creation of a market in the business jet sector where the demand was scarce up until then.
Secondly, the concept of full ownership might just become one of many alternatives, if not a completely outdated concept, to enjoy the benefits of business jet travels. Game changers are arising in the form of service companies. One can already book a flight in private jet with Private Fly, Victor and Wijet, or to obtain a jet card which will allow him access to a private jet for a set number of hours. The huge player Uber might also cause a lot of movement in the market when his UberJET initiative comes to a point where it is fully operational. Fractional ownership is also now possible with the help of NetJets for example. All these new accessibility models are highly disruptive for the market and will tend to make it more transparent, value-oriented, and comparative[1].
Finally, the demand for replacements has grown exponentially the recent years as we start to retire more and more planes.
A hefty market share of 1,825 units are expected to be retired during the next 10 years[2], which will drive the increase of demand.
2.2 The light segment
The light segment does not display the highest gross revenue, but represents the highest units sold. It is especially true in North
America, where it holds 53% of market share. Not only is this segment the most interesting in term of return on investment 2 MARKET, MISSION & DESIGN 6 perspective due to a higher number of transactions, but the market trending to accessibility and low-cost solutions also drives the cost efficiency criteria up. With that in mind, the development of a "cheap" business jet is paramount to be granted a wide opening in the market, which ineluctably go through optimisation of fuel efficiency, as it make up for up to 25% of the operational costs[1].
The other crucial aspect of such development is innovation. Standing out from the competition makes you more likely to be attractive. With that in mind, daring to be experimental may be what gives a project a step further than another one, especially when the public has little regard for technicals but craves novelty and aesthetic evolutions, even if this means being relieved of a couple more figures. For all these reasons, we decided to conceptualise an unconventional V-tail light jet.
2.3 Market research and competition analysis
In this section, the light business-jets market will be assessed, and the competition evaluated. These data will give us valuable insights on the potential ways of improvements of existing solutions. It will also be a key element to assess the feasibility and the attractiveness of the business jets, as a costs analysis will be performed in the second part of this report.
As the medium business jets market has been hit hard by the 2008’s financial crisis, and the market is not back on tracks yet (and will not be in the next 5 years), crushing the competition in terms of performance, and costs (maintenance, operation costs,...) are crucial to generate some profit.
Finally, this competition study will give us a way to validate rough approximations made regarding the weight of the aircraft.
In Tab. 2.1, the most popular aircraft from the light-business jet category are presented.
Brand/Model MTOW [lbs] Reference Area [ft2] Pax Wingspan [ft] Length [ft] Cessna - CJ3+ 13 873 294 9 53.34 51.11 Cessna - CJ4 17 110 330 10 50.82 53.34 Cessna - Citation SII 15,100 342.6 8 52.16 47.21 Cessna - Citation V 16,300 342.6 8 52.16 48.91 Cessna - Citation XLS+ 20,200 9 56.33 52.5 SyberJet - SJ30 13 500 190.7 6 42.32 46.78 Embraer Phenom 300 17 968 7 53.14 52.16 Bombardier-Learjet 25D 15,000 231.7 8 35.56 47.57 Bombardier-Learjet 24F 13,500 231.7 6 35.56 43.24 Bombardier-Learjet 40 21 000 311.6 7 47.76 55.54 Bombardier-Learjet 45 21 500 311.6 9 47.83 58 Pilatus PC-24 17 650 10 55.77 55.11 Bombardier-Learjet 75 21 500 311.6 9 45.93 57.74 Hawker 200 13 800 lb 6 45.57 46 Sabreliner T3J-1/T-39D 17 760 342.1 7 44.48 44
Table 2.1: Main competitors identified.
From this analysis, we have a better idea of the common parameters for the category of operation: 2 MARKET, MISSION & DESIGN 7
1. MTOW usually between 14,000 and 20,000 [lbs].
2. Wingspan up of about 49 − 52 [ft].
In addition, some price researches indicates that on average, the last (least price) is 9.331 US$ Million. Nevertheless, let us notice that the standard deviation is 1.93 mainly due to the passengers capability. Ranging from 6 to 8 passengers, it falls at
0.53. In the same manner, for 8 to 10 passengers, standard deviation is no more than 1.
Since the low price variation in the category assigned by the project, ingenuity and savings will be necessary.
2.4 Design Methodology
The methodology that we follow throughout the project is deeply inspired by the books of P. Raymer (see ref. [3]) and Ilan
Kroo (see ref. [4]), as the introductory class was mainly based on these references, and as it provided meaningful answers to the questionings throughout the project. Moreover, those books follow AIAA’s methodology for both conceptual and preliminary design phases, which are required in the competition as well, it then makes even more sense to follow steps and methods of the book.
In section 2.3, an extensive market research of the Light Business Jets class has been performed. This market research aims to give directions in the basic geometry, the typically involved weights and the type of engines that were used to accomplish the job. This section is also important, as it will be the basis to judge, at the end of the whole project, if the plane is, or is not competitive in terms of costs, as well as to assess its life cycle (sustainability, environmental impact, energy,...). In Section
INPUTS Fuselage design MISSION Mission Cruise velocity Payload-range Mission ZFW/MTOW Payload Estimation Range Cruise alt. Cruise speed Wing design no Configuration Choice of engine Performance ? Wing Tail yes Engines Equilibrium Technology Weights and CG location of each group. Wing position (stability). OUTPUTS Airfoils Evolution of CG with payload. Engines Tail sizing Undercarriage Materials Evolution of CG in terms of fuel consumed Plane drawing Static margin evol. Polar
no ZFW & MTOW yes correct?
Figure 2.1: Design process. Source: L. Noels, Aerospace Design Project - ULg 2 MARKET, MISSION & DESIGN 8
2.5, the request for proposal (RFP) will be analyzed to define the typical mission phases of the aircrafts under design, the typical velocity of operation, and the necessities in terms of payload and performances.
Then, empirical expressions will be used to assess the gross take-off weight, and derive the requested lift, and therefore to begin the wings sizing (a detailed methodology will be presented in Section 3.2.2 regarding the geometry of the wings).
The aircrafts geometries will be precised. Some parts, like the section of the fuselage, and the dimensions of the nose, the body, and the aft, will be defined directly by the RFP, as well as the number of passengers that need to be considered. Some other parts, such as the wings, the tail, and the propulsion, will have to be refined iteratively to reach optimal geometries and performances. Once the whole geometry will be defined, a study of the location and the respective weights of all the parts, using mostly empiric expressions, and estimates will be performed. This study will be the object of the Section 3.6.Using those weights and locations, we will then be able, in Section 3.6.5 to compute the position of the centre of gravity of the plane, and its evolution throughout the flight, as the fuel is being burnt by the engines. With the centre of gravity positioned, all the tools are available to determine the stability conditions in the Section 5.1. Using conditions on the relative position of the position xCG of the centre of gravity of the plane and the neutral point of the wing, we will optimize the wing position along the fuselage, and size the tail in order to ensure both good manoeuvrability and stability throughout the flight.
As a first step, all computations will be made with respect to the a maximum takeoff mass configuration. Then, a trade-off study will be performed by varying some principal parameters of 5% around their chosen value to confirm the suitability of the chosen parameters. Afterward, additional details of design settings will be provided and performances assessments will be carried out to finally end with an exhaustive cost analysis in the Section 6.
2.5 Mission requirements
The aim of the project is to develop the technology used for the category of light business jets to offer higher cruise speed, larger cabins, and updated technology compatibility. The research work asks for a two-member aircraft family envisioned to have a high level of part commonality between two family members to minimise the development and production costs.
The light business jets are designed to carry up to 6 and 8 passengers and the entry into service is 2020 for the first model and 2022 for the second model.
The general requirements for both the families are:
• Maximum Cruise Speed of Mach 0.85 at 35,000 [ft];
• Rate of Climb of 3,500 [fpm];
• Service Ceiling of 45,000 [ft];
• Maximum Sea Level Takeoff Balanced Field Length of 4,000 [ft] at Maximum Gross Weight with dry pavement;
• Maximum Landing Field Length of 3,600 [ft] at Typical Landing Weight.
In the particular case of six seat family member the requirements to accomplish are: 2 MARKET, MISSION & DESIGN 9
2500 nm - M.85
Cruise/Approach
W3 W2 Climb (Loiter) 35,000 ft.
W4
W W 1 to W5 Take-Off Landing
Figure 2.2: Schematic of a typical mission.
• Must meet FAA Federal Aviation Regulations Part 23 Airworthiness Standards for certification;
• Minimum range of 2,500 [nmi] at Long Range Cruise (LRC) assuming NBAA IFR Range with 100 [nm!] Alternate (1
pilot + 2 passengers);
• passenger/pilot at 200 [lbs] each;
• Baggage capacity of 500 pounds/30 cubic feet;
• 1 or 2 flight crew;
• 6 passengers, including 1 in the cockpit if there is no copilot.
The additional requirements to accomplish in the case of eight seat family member are instead:
• Must meet FAA Federal Aviation Regulations Part 25 Airworthiness;
• Standards for certification;
• Minimum range of 2,500 [nmi] at Long Range Cruise (LRC) assuming NBAA IFR Range with 100 nm Alternate (4
passengers; passenger/pilot at 200 [lbs] each);
• Baggage capacity of 1,000 pounds/60 cubic feet;
• 8 passengers.
A family of 2 aircraft is thus asked to be designed with a maximum of common components. The approach followed is therefore to design the the 8 seats and then see what it can by done for the 6 seats. Basically, the major change between the
2 configurations is a certain length of the fuselage that will be removed (this approach will be validated in the trade-off study section 4). Further details of the 6 seats settings will be specified in parallel with the 8 one. 3 AIRCRAFT DESIGN CHOICES 10
3 Aircraft design choices
3.1 Fuselage
To define the fuselage dimensions, the cabin section will first of all be established to ensure the optimum level of comfort, and then the seats and compartments will be organized to derive the respective lengths of the fuselage body, nose, and aft.
3.1.1 Cabin design
The cabin section’s design is a key factor of the overall design process. Indeed, the goal is to balance the level of comfort and equipment, and the induced drag of a higher section. In the proposed design, the cabin section has been optimized to the highest, and the result is presented on the Fig. 3.1.
15.35 in. 0.39 m.
5 ft. 1.52 m.
5 ft. 3 in. 1.60 m
Figure 3.1: Cabin section.
This design is compliant with the FAR23 and FAR25 specifications (in terms of width of aisle). Moreover, the seats and spaces are designed to ensure an optimal comfort to a statistical sample of American men up to the 95th percentile1. On Fig.
3.2, the cabin section is compared with competitors.
The Fig. 3.3 shows the seating diagonal space (24.8 in). The fuselage body has been designed to optimize the space within the cabin. As the fuselage section is relatively small, the seat pitch has been increased to 47 inches to give a sensation of space and freedom to the customers during the flight.
1The section design has been performed using CATIA v5’s ergonomic module, ensuring that the seating dimensions fit to the best the expected customers. 3 AIRCRAFT DESIGN CHOICES 11
Citation CJ4 Embraer Phenom 300 Learjet 40XR
(a) vs. Citation CJ4. (b) vs. Phenom 300. (c) vs. Learjet 40XR.
Figure 3.2: Comparison of our section with some competitors.
3.1.2 Fuselage Length
The fuselage contains a large galley/minibar area, a wide lavatory/bathroom area, and two big luggage compartments (on both parts of the rear-body/aft beginning). Both are pressurized. The Aft factor (AF ) has been set to 2, while the nose factor (NF ), is 1.8. This gives us the definitive fuselage dimensions of the Fig. 3.3.
Most of the fuselage remains identical to the 8 seats version, excepted that the cabin’s length is reduced by about 2,13 feet.
The 6 seats version has therefore more space per occupant, as shown on the Fig. 3.4.
Nose Fuselage Body Aft 10.20 ft. 24.9 ft. 11.33 feet 3.11 m 7.59 m 3.45 m
Seat Pitch 47 in. 1.20m
Galley Lugg. Minibar Comp.
Fuselage Length: 46.43 ft. / 14.15 m
63 cms 24.8 in.
Figure 3.3: Fuselage dimensions and cabin. 3 AIRCRAFT DESIGN CHOICES 12
49,9 feet
47,7 feet
Figure 3.4: 8 seats vs. 6 seats fuselage lengths.
3.2 The wing
The conceptual design of the wing is a very complex and important part since it is strongly linked to every studied component.
First, design choices must be made in order to guarantee consistency throughout the wing design procedure:
1. CL = 0.3: here, low wing loading is preferred over optimal aerodynamics performance. Indeed, optimal lift coefficient associated with maximum Lift-to-Drag ratio lies in the typical range of [0.4,0.6], as illustrated in Fig.3.5. This can be
shown using the statistical drag polar relationship,
2 CL CD = C + , (3.1) D,0 π ·e·AR
in which CD,0 and e are respectively the zero-lift drag coefficient and Oswald’s factor. However, such values lead to an increase in wing loading, and so compel the structure designer to allow for higher structural strength. This is generally
performed using other more appropriate materials and is eventually more costly in terms of computation time. However,
a sufficient lift coefficient must be selected so that the jet is still able to fly. Thus, the typical interval which CL must lie in is [0.2,0.4].
2. M = 0.85: The specification states that the plane has to be able to fly at maximum Mach of 0.85, which means that a
super-critical airfoil is required.
3. Another important choice that directly affects the selection of the airfoil is about the takeoff and landing field lengths.
Indeed, the family business jet should ensure really quick takeoffs and landings, which is suitable for a regular use and
enables to offer all the flexibility that customers of business jet may require.
In order to address the design of the wing, a preliminary weight assessment must be performed, using bold assumptions 3 AIRCRAFT DESIGN CHOICES 13
22 20 18 16 14
D 12 /C L
C 10 8 AR = 5 AR = 6 6 AR = 7 AR = 8 4 AR = 9 2 AR = 10 0 0 0.2 0.4 0.6 0.8 1 C L
Figure 3.5: Optimal range for CL. and empirical estimates (see section A for a description of the gross weight estimation method). As a result, the weight at mid-cruise (i.e. when the fuel is half-consumed) is determined. This leads to
1 L = W = ·ρ ·V 2 ·C ·S, (3.2) 2 ∞ l where S is the gross wing area, CL is the 3D design lift coefficient of the wing, ρ is the air density, V∞ is the airspeed, L is the total lift of the plane for mid-cruise conditions. On the other hand, an important quantity to be determined as soon as possible is the aspect ratio, AR, the expression of which is given as follows
AR = b2/S, (3.3)
where b is the span of the wing. As the gross area is fixed by the choice of CL, the aspect ratio gives directly the span. Its value is to be as high as possible since it is inversely proportional to the Lift to Drag ratio. Typically, it lies in [6,10], as it can be seen in Tab. C.1 and C.2, in App. C. Here, to be consistent with what is found in actual business jets, namely in terms of wing geometry, AR is imposed to 9.
3.2.1 Super-critical airfoil: NACA SC(2)-0714
Assessing several airfoils and considering the availability of the data, it has been decided that the NACA SC(2)-0714 is able to fully satisfy the imposed requirements. Indeed, this airfoil belongs to the super-critical family, a category of transonic airfoils that have the particularity to reduce dramatically the drag by increasing its drag divergence mach number. Although the design 3 AIRCRAFT DESIGN CHOICES 14 point of this airfoil is at 2D lift coefficient of 0.7, we can use this airfoil for a lower range of 2D lift coefficient, since we stay in the low drag bucket region at cruise conditions. Furthermore, the high slope, cl,α , of the curve ensures a very quick rise of lift during cruise if a change of altitude is needed.
This airfoil has a thickness to chord ratio (t/c) of 0.14 and is designed for a normal component of airspeed associated with
Mach 0.725. It means that the wing must have a sufficient sweep angle so that the airspeed in the normal direction is reduced up to maximum Mach 0.72. The drag-rise Mach number associated with this airfoil is 0.79. Thus, going slightly beyond the design Mach number do not cause a dramatic increase in drag provided this ultimate limit is not overcome.
The geometry of this airfoil is shown in Fig.3.6. Fig.D.1a and D.1b in App. D respectively shows the behaviour of the airfoil at cruise and takeoff/landing configurations.
0.2
0.1
0 t/c [−] −0.1
−0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/c [−]
Figure 3.6: Airfoil NACA SC(2)-0714.
Fig. D.1a and D.1b respectively shows the behaviour of the airfoil at cruise and takeoff or landing configuration.
It can directly be seen on those figures that the lift coefficient approachable by the airfoil is really high, which is also important for optimization of the takeoff and landing field lengths.
3.2.2 Wing geometry
Here, the wing design is performed for the 8 seats configuration, which is almost identical to the 6 seats configuration (but a fortiori less optimal). A discussion about the 6 seats configuration will be carried out later in Section 3.2.4.
Taper ratio lies statistically in the interval [0.2, 0.6] (see Tab. C.2 in App. C). It reduces the lift at the wing tips, so allows to reduce the structural weight or rise the fuel capacity. Strong taper ratio increases occurring of tip stall. Thus, considering those effects, a taper ratio of 0.3 is imposed. Stall is a serious issue that should be anticipated in the conceptual design phase.
In order to exhibit a good behaviour near the stall angle or velocity, the wing is twisted. Twisted wing permits to redistribute the lift along the span. The purpose of lift redistribution is to ensure that the wing tip is the last part of the wing surface to stall. Indeed, a twisted configuration allows the pilot to stabilize the aircraft in case of increasing stall, and so avoid losing
◦ ◦ control of it. To this purpose, the geometric twist angle (εg,tip) is a washout angle typically in the range [-5 , 0 ], as explained in page 58, chapter 4 of reference book Aircraft Design: A Conceptual Approach [5].
As mentioned in the paragraph dedicated to airfoil, the maximum Mach number’s requirement makes necessary the wing 3 AIRCRAFT DESIGN CHOICES 15 to be swept. For the sake of simplicity, one decided to sweep it backwards (to the tail) so as to be in compliance with usual aircraft designs. Seeing the design Mach number, the sweep angle at the quarter chord is determined as follows
0.72 Λ = arccos = 32◦. (3.4) 0.85
As for the dihedral angle, its value is fixed when studying lateral stability. It is worthwhile noticing that the maximum sweep for an aircraft generally lies about 35◦, above which the performance might be reduced (see Lecture 4 of Mr. Noels’ course notes, page 58 [6]). Further studies might be performed to determine the maximum acceptable sweep.
3.2.3 Flaps design
The last geometrical parameters to determine for the wing are about the flaps. Flaps design is strongly related to the perfor- mance. Indeed, these are important to give an additional lift during the takeoff or enough drag for the approach and landing, in order to reduce the takeoff field length and landing distance as much as possible. To determine the flaps area, the strategy consists in estimating the CL,max necessary at takeoff and landing to meet the requirements in terms of field lengths. These requirements are resumed here below:
• maximum takeoff field length: 4000 ft at maximum gross weight with dry pavement;
• maximum landing field length: 3600 ft at typical landing weight;
• rate of climb: 3500 fpm.
All thermodynamical parameters are given using ISA tables.
A simple way to estimate the 3D lift coefficient (CL) is to use the wing partition method, presented in the chapter 10, pages 437 to 441 from the reference book "The Anatomy of Lift Enhancement" [7]. The idea is to cut the wing into several sections and compute the 3D lift coefficient of all the parts considering their respective properties (i.e., if this part is flapped or not and other eventual aerodynamic properties of the considered part). References coming from the Ilan Kroo’s course notes, page
257 [4], state that "typical flaps extend over 65 % to 80 % of the exposed semi-span, with the outboard sections reserved for ailerons. The resultant flapped areas ratio are generally in the range of 55 % to 70 % of the reference area". Considering this observation, it is possible to dimension flaps such as presented in the figure 3.7.
In this figure Sflaps represents the total surface of the flaps and Sw f the flapped surface of the wing. In our case, the flapped surface represents approximately 67.5 % of the gross area. The wing partition method applied to a swept wing allows to assess the lift coefficient in 3D given the following formula (from the reference book "The Anatomy of Lift Enhancement", page 438
[7]):
( N N ) 0.9 f f CL,max ≈ cl,max,i ·Si ·cos(Λhingeline,i) + Cl,i ·Si · cos Λ 1 , (3.5) ∑ ∑ 4 S i=1 i=1 3 AIRCRAFT DESIGN CHOICES 16
AR = 9 O y
2 x Λ1/4 = 32.1° Swing = 232.5 ft x 2 AC=7.84 ft Swf= 156.8 ft I S = 47.4 ft2 flaps II croot = 7.81 ft Λ = 34.5°
y MAC = 5.58 ft AC= 9.38 ft III
AC
IV ctip = 2.33 ft
16.40 ft 2.83 ft
1.98 ft 1.64 ft
45.7 ft
Figure 3.7: Left part: Flaps sizing and determination of the maximum lift coefficient: wing partition method. Right part: Representation of the wing’s main parameters. where:
• Si is the reference surface of the considered part of the wing;
• Nf is the number of the flapped surfaces;
• Cl,max,i is the maximum lift coefficient of the airfoil at takeoff or landing conditions considering the additional lift provided by the flaps (flapped section);
• Si is the surface of the considered section of the wing;
• Λhingeline is the angle between the hinge line of the flap and the vector normal to the aircraft plane of symmetry;
• Cl,i is the lift coefficient of the considered unflapped section at takeoff or landing conditions considering a decrease of the stall angle due to the deflection of the flaps;
• Λ 1 is the sweep at quarter chord. 4
The additional 2D lift coefficient of the flaps is given by the following empirical formula (presented in Ilan Kroo’s course notes, page 258 [4]):
∆Cl,max,flaps = K1 ·K2 ·∆Cl,max,ref, (3.6) 3 AIRCRAFT DESIGN CHOICES 17
where ∆Cl,max,ref is the 2D increment in Cl,max for a reference configuration and its value is determined from an experimentally- determined curve depending on the thickness of the considered airfoil. The constants K1 and K2 take into account the cor- rection due to the variation between the reference configuration and the configuration of interest, respectively in terms of flap chord extension and flap angle deflection. It is worthwhile noticing that this formula applies only for double slotted flaps. The course notes Aircraft Design - Synthesis and Analysis at page 258 (see ref. [4]) states that it is possible to adapt the result for different kind of flaps by multiplying it by a factor 0.98 for single slotted flaps and 1.08 for triple slotted flaps. In this conceptual design, it has been decided to choose single slotted flaps for both 6-seats and 8-seats configurations. This choice is motivated by several reasons:
• complexity of double and triple slotted flaps will certainly require a higher level of maintenance;
• the single slotted flap is cheaper to design and implement than both others;
• the single slotted flap is also lighter;
• the chosen airfoil has a high slope of the curve cl vs. α so able to provide a rapid increase in lift coefficient for the takeoff. That is, a single slotted configuration is enough to meet the requirements.
Considering an airfoil with a thickness to chord ratio of 14%, a flap extension of about 30%, formula 3.6 gives the results presented in the table here below (Fig.3.1).
Takeoff Landing Angle of deflection 20◦ 40◦
∆Cl,max 1.06 1.87
Table 3.1: Increment of 2D lift coefficient (∆Cl,max) due to a section of flap.
If needed, in case of emergency for instance, flaps can be eventually deflected up to 60◦. In that case, assessing the increment in 2D lift coefficient becomes less precise but by extrapolation we can compute a value of approximately 2.12.
Using these values of ∆Cl,max,flaps in the formula 3.5 gives results presented in Tab. 3.2.
Phase of flight Flaps in Flaps out Takeoff 1.5 2.1 Cruise 0.8 - Landing 1.5 2.6
Table 3.2: Value of the 3D maximum lift coefficient (CL) for different configurations.
From these value, takeoff and landing field length, stall velocity and decision speed will be computed in the second part of the report (Preliminary Design, section 5.3). The determination of the flight envelope (see section 5.4.1) will also strongly depends on these values. 3 AIRCRAFT DESIGN CHOICES 18
All the chosen and computed parameters concerning the flaps are presented in Tab. 3.3. The dimensioning of the flaps is drawn in Fig. 3.7
2 Flaps surface, Sflaps 47.4 [ft ] Flaps extension (relative to the span) 67.5%
Table 3.3: Summary of the principal parameters linked to the flaps.
3.2.4 Results
A sketch of the wing geometry with important geometrical parameters is presented in Fig 3.7. All the geometrical parameters and results concerning the wing are listed in the Tab. E.1. The wing designed for the 8 seats configuration will be reused for the 6 seats configuration in its entirety. Since the weight of the plane vary due to the fuselage shortening, the lift required for the wing decreases. Therefore, since its geometry shall not change, the required wing lift coefficient reduces to CL6seats = 0.27 ◦ at cruise. This leads to a variation of the angle of the wing at root which decreases to αroot = −0.5 [ ].
3.3 Propulsion
3.3.1 Engine selection
3.3.1.1 Preliminary researches
In this section, a quick market analysis is performed. Indeed, a good indicator to the choice of the engine is the choices that already have been made for aircrafts from the same class. Table 3.4 gathers data about the main competitor’s engines.2
Overall Dry Thrust sls SFC sls Diameter Manufacturer BPR (-) Length Mass [lbf ] (lbs/(h lbf)) (inch) and model (inch) (lbs) Williams SyberJet SJ30 2,300 0.46 4.1 59.8 21.7 530 International FJ44-2A Williams Cesna Citation 2,820 0.46 2.2 62.4 22.9 535 International CJ3+ FJ44-3A Embraer Pratt & Whitney 3,360 0.44 2.55 66.1 42.6 699 Phenom 300 PW535-E Bombardier- Honeywell 3,650 0.45 3.1 60.9 39.4 895 Learjet 40 TFE731-20BR
Table 3.4: Comparison between reference business-jets.
2The data has been collected either directly on the manufacturer website or in [8]. 3 AIRCRAFT DESIGN CHOICES 19
3.3.1.2 Engine matching
In order to choose the most suitable engine, the thrust-to-weight ratio is computed using Eq. 3.7. The wing loading W/S may be estimated using the data provided in section 3.2.
2 T 1 2 CL S = ρV CD + · , (3.7) W 2 0 e·π ·AR W
Where e is the Oswald efficiency factor, taken for the worst case (i.e. e = 0.75). CD is the drag coefficient, also considered in a worst case condition (i.e. CD0 = 0.02). CL is the wing lift coefficient in cruise (i.e. CL = 0.3, from Tab. E.1). AR = 9 is the Aspect Ratio (see Tab. E.1), S = 232 [ft2] is the Wing area (see Tab. E.1). The maximum take-off weight, MTOW = 18,272 lbs (see Fig. 3.13), and T is the thrust in cruise, set to 2·0.92·0.308·TSLS = 1,904 lbf. The thrust in cruise is computed from the SLS, nominal value provided by the manufacturer. As a first approximation, a correction of 0.92 is set to the SLS value, to represent the loss of thrust undergone by a mounted engine (vs. a uninstalled engine). In addition to that, the effect of the altitude on the thrust is added to this first installed approximation, leading to a drastic loss in thrust. The coefficients lie on the interpretation of the graph in App. B.
The Tab. 3.5 is the final result of a large iteration process, as the engine presents a double impact on the weight, firstly by its self weight, but also, and more importantly, by its fuel consumption. The engine fulfilling at best the criteria is the
PW535-E, manufactured by Pratt & Whitney Canada.
Weight W/S Actual Required lbs lbs/ ft2 T/W (-) T/W (-) Initial cruise 17,944 77.3 0.11 0.08 Mid-cruise 15,391 66.3 0.12 0.09 End cruise 13,138 56.6 0.14 0.11
Table 3.5: Comparison between results computed from conceptual design and approximations, with PW535-E as chosen engine.
3.3.1.3 Further comments about the chosen engine
Firstly, the manufacturer is the internationally located PRATT&WHITNEY. Meaning that support can be provided worldwide very quickly. Moreover, the engines family of PW500 sums up 12 millions of flight hours and their reliability is such that their time between overhauls (TBO) can go beyond 10,000 hours, directly translating into lower operational costs. Secondly, even if the PW535-E suits the cruise conditions of the aircraft under study, additional information on other flying phases implying its use were gathered.
• The Embraer Phenom 300 embeds 2 PW535-E, while having a comparable MTOW, and is yet able to take-off with a
3,138 ft long runway, which is clearly matching the RFP requirements. 3 AIRCRAFT DESIGN CHOICES 20
• In case of trouble for the climb, the PW535-A engine (same family) can provide a slightly higher thrust without any
change relative to the dry weight mass, BPR and SFC.
• In case of trouble with the landing field length, the same PW535-A engine is also certified with thrust reverser (it would
however induce some structural changes in the nacelle’s integration).
3.3.2 Placement and installation
Nacelle-engine set placement and incidence is determined using CFD and wind tunnel experiments, and is therefore out of the scope of this study. However, the main characteristics are detailed in the following.
The chosen configuration for the engine placement is aft fuselage. Main reasons are aestheticism, and level of noise comfort in the cabin. Some other advantages include
• less pylon interference (better lift and lesser drag),
• less yaw induced by an engine failure,
• shorter landing gear and ease of disembarking once arrived.
Nevertheless, some drawbacks are also to be taken into account.
• noise and vibration on the fuselage are severer (heavy insulators required and engines placed far away from passengers
and thus from the centre of gravity),
• supersonic flow throughout the fuselage-pylon-nacelle system is possible,
• structural advantage in the wing bending moment point of view are lost.
In the case of fuselage mounted nacelles, a gap of approximately one half of the nacelle’s diameter is recommended with respect to the fuselage. In addition to that, the vertical position of the nacelle is an important (yet hard to quantify, at the early stages of design) matter. As hot ejection gas exits the engine at a high velocity, a low-pressure zone is generated in the lower part of the tail, creating a downward force, and generating a pitching moment.
Regarding the wings and V-tail implantation, the vertical position of the engine axis is set to 1.67 ft above the cabin’s axis.
Figure 3.8: Nacelles implantation. 3 AIRCRAFT DESIGN CHOICES 21
3.3.3 6 seats engine choice
As the 6 seats’ aircraft weight is lower than the 8 seats version, the suitability of the engine has to be re-assessed. As a result, it is found that the FJ44-3AP model (WILLIAMS INTERNATIONAL) is also suitable to the RFP and FAR 25 requirements3.
As shown in Tab. 3.6, a slight fuel saving is noticed. However, for production cost minimizing reasons, the PW535-E will be maintained from the 8 to the 6 seats versions, as demonstrated in the costs analysis.
PW535-E FJ44-3AP Dry mass (lbs) 699 516 BPR (-) 2.55 2.2 SFC sls (lbs/(h lbf)) 0.44 0.46 Thrust sls (lbf) 3,360 3,052 Resulting TO mass (lbs) 16,225 15,397 Fuel required (lbs) 4,702 4,474
Table 3.6: Engine smackdown for the 6 seat business jet (reaching 2,500 nmi at full payload).
3.4 Empennage
The empennage balance the moments applied on the plane by the different forces acting on it. Therefore the tail plays a major role in both static and dynamic equilibrium. Among the different existing tail configurations , the most popular one for business jets is clearly the "T" tail. This choice is partly aesthetic and allows place on the aft for the engines. But the
"T" configuration also have few drawback that it may be interesting to avoid : structural complexity (thus higher weight and manufacturing costs) and propensity to be blanketed at high angles of attack, leading the plane into a deep stall from which it is hard to recover.
A way to dodge these flaws is to choose a simpler, less common yet well known tail configuration : The "V" tail. These tails present the asset to be structurally uncomplex since they are basically small, untwisted, symmetrical wings. They are therefore expected to be lighter and easier to manufacture than other types of tails. Moreover the aerodynamic characteristics of the "V" tails are interesting : the reduced number of junctions between surfaces is a favourable factor of reduction of the induced and interference drag of the tail. Besides, originality and innovation are sure important aspects in the scope of putting a new product on the market, and a jet equipped with a "V" tail is undoubtedly fresh.
3.4.1 V tail principle
A good method for designing a "V" tail is to start from a standard cruciform tail configuration. This design is conducted through a statistical method stated in Aircraft Design : A conceptual Approach [3]. The V surfaces then should be sized so that they provide the same total surface area 4:
3A complete performance study, as in section 5.3, was performed. However, in the interests of brevity, it won’t be detailed in this report. 4As stated in Experimental verification of a simplified vee tail theory... [9]. 3 AIRCRAFT DESIGN CHOICES 22
SVtail,exp = SH + SV , (3.8)
where SH and SV are the horizontal and vertical surfaces previously provided. The tail dihedral then should be set as follows :
r ! −1 SV Vangle = tan (3.9) SH
leading to the reference area :
SVTail = SVtail,exp cos(Vangle). (3.10)
Note that requirement brought by Eq. 3.8 implies that once projected in the horizontal and vertical plane, the surfaces will not be equal to the statistically provided ones.
3.4.2 Statistically prescribed horizontal and vertical surfaces
As stated previously, the surfaces of a cruciform tail can be sized statistically with respect to some dimensions of the plane. A good approach is to express the effectiveness of the tail in a non dimensional way : through the tail volume coefficient. These are defined as follows : lVTSVT cvt = for the vertical tail (3.11) bwSw and lHTSHT cht = for the horizontal tail, (3.12) CwSw with
• li the arm moment, commonly approximated as the distance between the tail quarter chord and the wing quarter chord.
• Cw is the wing mean chord.
• bw is the wing span.
Then typical values of these coefficient for business jet can be found in databases or as conservative averages based upon data.
Such values can be found in Aiplane Design ([10]), and typical values for business jet-like aircraft are cht = 1 and cvt = 0.09. At this stage the moment arm for both surfaces can be estimated by a percentage of the fuselage length: for an aircraft with aft-mounted engines, the tail arm is typically about 45-50% of the fuselage length. Inverting Eq. 3.11 and 3.12 provide statistically prescribed horizontal and vertical surfaces. Note that these are quite raw approximations but since they are meant to be modified by the projection in the V plane and in the scope of a preliminary design, the accuracy achieved on the surfaces is estimated good enough. 3 AIRCRAFT DESIGN CHOICES 23
3.4.3 Design choices and geometry determination
Now that the surface of the "V" tail is determined, the other geometrical parameters are determined using empirical relation- ships. These make use of main parameters of the wings and statistical range of values.
The choice of the airfoil for the tail is basically motivated by the same considerations as in the section 3.2.1. Tails usually have symmetrical airfoils: since the lift to produce is small there is no need for a camber. Drag reduction is an everlasting goal for the design, and a good method to achieve it for the tail is to choose a thin profile. On the other hand, thin profiles are structurally weak and might need more structural reinforcement than thicker ones, resulting in an heavier tail. Therefore a compromise has to be made between aerodynamic and structural/safety properties. The search is conducted in the NACA 4 digits series: these provides well known, easy to manufacture airfoils with good performances. Among the different available symmetrical airfoils, the NACA 0010 is chosen: this thickness is a good trade-off between the drag the profile will generate and the structural strength it provide to the tailplane.
Statistic data on subsonic jet from Aiplane Design ([10]) and Aircraft Design : A conceptual Approach ([3]) provide ranges for the geometric parameters of horizontal and vertical surfaces. But since the "V" tail is neither horizontal nor vertical, these ranges should be handled with caution. Typically, values in the mid range are chosen for some virtual horizontal and vertical surfaces, then projected in the V plane. This way the horizontal and vertical projection of the parameters of the "V" tail remain in the statistic range.
To ensure that the tailplane will stall after the wing, it is recommended for the horizontal tail to present a greater sweep angle than the wing (commonly 5◦). The sweep angle of the vertical tail is typically between 35 and 55◦. Once projected back in the V plane, the resulting sweep angle of the "V" tail equals 48.5◦ as resumed in the Tab.3.7.
The aspect ratio of the "V" tail can reasonably be approximated as a fraction of the one of the wing. Since the aspect ratio of the horizontal tail is approximately one half of the aspect ratio of the wing, and since the aspect ratio of the fin is usually lower than 2, it is clear that the factor for the AR of the "V" tail will be lower than 0.5. Indeed, after projection :
ARVTail = 0.36ARWing = 3.24 [−]. (3.13)
Assuming the value of the taper ratio in the middle of the range prescribed for an horizontal tail: taperVTail = 0.4 [−]. The remaining part of the geometrical parameters: span, mean chord, chord at the root, chord at the tip and mean aero- dynamic chord, can be determined from these choices according to empirical relationships. The results are presented in Tab.
3.7.
3.5 Undercarriage
In this section will be discussed the process of selecting the undercarriage geometry such as height, wheel base, wheel track, and the distance between main gear and aircraft center of gravity. In this selection several constraints have to be accomplished to guarantee safe landing, takeoff and taxi operations. 3 AIRCRAFT DESIGN CHOICES 24
Span: spanVtail 13.9 [ft] Aspect Ratio: ARVtail 3.24 2 Surface: SVtail 60 [ft ] 2 2 Total exposed surface: Sexp,Vtail 80.7 [ft ] Taper ratio: taperVtail 0.4 Chord at root: Chordroot,Vtail 5.9 [ft] Chord at tip: Chordtip,Vtail 2.67 [ft] Mean chord: Meanchord,Vtail 4.3 [ft] Mean aerodynamic chord: MACVtail 4.51 [ft] ◦ Sweep angle: sweepVtail 48.5
Table 3.7: Summary of the main parameters of the tail.
3.5.1 Rotation clearance angle
A first constraint to discuss is related to a good separation of the plane from the ground. This function is performed by providing an adapt rotation clearance angle (γ) (Fig. 3.9).
β
Rotation clearance angle (γ)
Wheel base
Figure 3.9: Rotation clearance angle representation.
The range of variation of γ is normally between 12◦and 16◦. As first approximation is possible to locate the main strut undercarriage at around 50% MAC; from these two parameters is possible to get the height of the plane from the ground that should be for a commercial jet between 0.2m and 1.2m.
Hf = Lend · tan(γ), (3.14)
where Hf is the height of the plane from ground and Lend is the distance of the main strut wheel from the end of the fuselage.
3.5.2 Wheel track and wheel base
Now, once the separation between the fuselage and ground has been defined, other geometric parameters of the undercarriage to define are the wheel base (i.e., the distance between the front and rear wheel axes in the vertical plane of symmetry) and 3 AIRCRAFT DESIGN CHOICES 25
Forwardmost CG Wheel track
β
ϑ hCG
Wheel base
Figure 3.10: Undercarriage main parameters. the wheel track (i.e., the distance between the main wheels in the lateral).
The wheel base and wheel track determine the aircraft turning radius on the ground. For this reason they have to be chosen to not overcome a maximum θ otherwise the aircraft would turn over on its side. It is better to maintain a lower angle θ to avoid an aircraft turning over, most of the aircraft have a θ between 40 and 50 ◦. It is necessary to evaluate first the wheel base dimension and from this it is possible to get the wheel track. Normally the main strut undercarriage sustain about 90 − 95% of the MTOW, while the nose strut takes only 5 − 10%. Once the load distribution is defined, the wheel base is given by the following equilibrium relation around the CG (Fig. 3.11):
2·Rm ·Lβ Lnose,rear,CG = . (3.15) Rn
L L nose, rear CG
CG CG for aft H CG
Wheel base R R n m
Figure 3.11: Wheel load geometry.
The wheel track is possible to get just by geometrical relations (Fig. 3.12). 3 AIRCRAFT DESIGN CHOICES 26
L
Ltrack αtrack
y
Lbase
Figure 3.12: Undercarriage base.
Once established, the maximum turnover angle (θ), y is given by:
y = tan(θ)·HCG. (3.16)
The wheel track is then given by: −1 y αtrack = sin (3.17) Lbase − Lβ
Ltrack = 2(Lbasetan˙ (αtrack)), (3.18)
where Lβ is the horizontal distance between the main strut location and the rear CG location, while αtrack is the semi- aperture of the triangular undercarriage base.
In practice, the mechanism of wheels retraction of the main struts, can be stored in the belly tank, while the wheels and struts can be stored at the root of the wings. In another hand, the nose wheel can be stored in the nose, which would be designed as a horizontally flattened cone in order to free enough space to store the front wheel as wheel as its retraction mechanism.
3.5.3 Tire size
The number of the wheels needed and the tire dimensions are defined by the load on the wheels. Referring to tire datasheets available is possible to conclude that one wheel for each strut is sufficient to bear the whole plane load. In case of our plane load, the tire dimensions for the main landing gears should be 21×7.25−10 where 21 inches is the outside tire diameter, 7.25 inches is the section width and 10 inches is the rim diameter. A smaller tire should be used for the nose wheel, i.e. 5.00 − 4.
3.5.4 Results
All the geometrical parameters and results concerning the undercarriage are listed in the Tab. 3.8 All these given parameters 3 AIRCRAFT DESIGN CHOICES 27
Clearance angle: γ 16◦
Fuselage ground height: Hf 2.00 [ft] nose-rear CG distance: Lnose,rear,CG 16.83 [ft]
rear CG-main strut distance: Lβ 1.77 [ft] main strut-end fuselage distance: Lend 6.94 [ft] maximum overturn angle: θ 42 [◦]
wheel base: Lbase 18.60 [ft] wheel track: Ltrack 9.92 [ft]
Table 3.8: Summary of the principal parameters of the undercarriage.
are the same for both 6 and 8 seats configurations. Indeed, the landing gears will entirely be reused for the design of the
6-seats business jet. A few adjustment can eventually be performed including tire modification more suitable in view of the change of TOW.
3.6 Weight considerations
From the full geometrical description detailed in the previous sections, an estimate of the respective weights of each of the aircraft’s components may be derived using the empirical relations presented in ref. [4].
3.6.1 Empty weight definition
3.6.1.1 Wing weight
The wing’s weight is given by Eq. 3.19.
√ n ·b3 · W ·ZFW·(1 + 2·λ) W = 4.22·S + 1.642·10−6 · ultim to (3.19) w t 2 c avg · cos (Λ) ·S· (1 + λ)
2 With S the gross area of the wing [ft ], ZFW the Zero-Fuel Weight of the aicraft [lbs], nultim, the ultimate load factor [-], b the wing span [ft], Wto the take-off weight [lbs], λ the taper ratio [-], t/c|avg the average thickness over chord ratio and Λ the sweep angle.
3.6.1.2 Fuselage weight
The fuselage weight is given by Eq. 3.20.
Wfus = (1.051 + 0.102·Ifus) ·Sfus,wet (3.20)
Where Ifus is the fuselage index, and Sfus,wet is the fuselage’s wet surface. The fuselage index is computed using Eq. 3.21, while the wet surface is approached by Eq. 3.22. 3 AIRCRAFT DESIGN CHOICES 28
Ip if Ip > Ib Ifus = I2 + I2 (3.21) p b if Ip < Ib 2·Ib With
−3 Ip = 1.5·10 ∆pmax ·Dfus the pressure index, ∆p being the pressure differential between the inside (8000 feet equivalent pressure) and the outside of the cabin (45000 feet equivalent pressure).
−4 Lfus Ib = 1.91·10 nlimit · (ZFW −Ww) · 2 Dfus is the bending index, where nlimit is the limit structural load, Lfus the length of the fuselage [ft] and Dfus the width of the fuselage [ft].
1 1 + 2·N · sin−1 1 − 2 q 2 F 2 Dfus 2 Dfus 4·NF Sfus,wet = Lfus ·Dfus ·π + π · · 1 + 4·AF + π · · s (3.22) 4 4 1 1 − 2 4·NF
With Lfus [ft] and Dfus [ft] respectively the length and the width of the fuselage, AF [-] the aft factor, NF [-] the nose factor.
3.6.1.3 Tail weight
No specific empirical estimation have been found for the case of a "V-Tail". Therefore, the weight of the horizontal components of a traditional "T" tail will be considered, and the weight is computed using Eq. 3.23.
3 p −6 nultim ·bT ·Wto ·c· ST,exp WT = 5.25ST,exp + 0.8·10 · (3.23) tT 2 3/2 · cos (ΛT) ·armT ·ST cT avg
2 Where ST,exp [ft ] is the exposed empennage area, bT [ft] is the tail span, c [ft] is the average aerodynamic chord of the wing,
tT/cT|avg [-] is the average thickness to chord ratio of the tail, ΛT [rad] is the sweep angle of the tail, armT [ft] is the distance between the aircraft’s center of gravity and the tail aerodynamic center, and ST [ft] is the gross area of the tail.
3.6.1.4 Engine weight
One engine’s dry weight is 700 [lbs]5, and the weight of an engine in use is given by Eq. 3.24.
Weng = 1.6·Weng,dry (3.24) 5Source: EASA, https://www.easa.europa.eu/system/files/dfu/EASA-TCDS-E.048_%28IM%29_Pratt_and_Whitney_ Canada_PW530_series_engines-03-28042008.pdf 3 AIRCRAFT DESIGN CHOICES 29
3.6.1.5 Systems weight
The weights of the different subsystems are computed with some rough empirical estimations in Tab. 3.9. All weights are provided in [lbs].
Landing gear Wgear = 0.04Wto Control Surfaces WCS = 3.5· ST,exp APU WAPU = 120 Instruments Winstr = 100 Hydraulics Whydr = 200 Electrics Welec ' 13·Nseats Electronics Wtronics = 300 Furnishing Wfurn ' (43.7 − 0.037·Nseats) ·Nseats + 46Nseats
Table 3.9: Weights of the different subsystems of the aircraft, provided in [lbs]
3.6.2 Payload weight
The payload weight is provided in the Request for Proposal, and is estimated to 3000 [lbs] (2 crew members weighing 200
[lbs] each, 8 passengers weighing 200 [lbs] each, and an additional 1000 [lbs] of luggage.
3.6.3 Weight of fuel
The maximum fuel weight to reach a given range is an important data, for both structural and financial reasons. In the following, empirical relations will be used to estimate the fuel requirements of the 8 seats aircraft to perform a 2500 nmi trip with full payload (2 crew + 8 pax + luggages).
Phase by phase, the following formulae, provided mainly by ref. [4] are used:
1. Taxi and takeoff:
Wfuel,T&TO = 0.0035·WTO (empirical). (3.25)
With WTO, the take-off weight.
2. Climb:
The empirical expression 3.26 is provided by [4], and is used for a climb angle of ' 10 ◦.
W alt 1 W = TO cruise + M2 . (3.26) fuel,climb 100 31600 2 cruise
With Mcruise = 0.85 the design mach number in cruise, and altcruise = 35,000 ft the cruise altitude.
3. Cruise: 3 AIRCRAFT DESIGN CHOICES 30
The Bréguet range equation is used to assess the fuel consumption in cruise.
1 R·SFC C W = W 1 − where ξ = √ cruise · D (3.27) fuel,cruise i ξ exp a0 · θ ·Mcruise CL
Where Wi = WTO −Wfuel,T-TO −Wfuel,climb is the weight at the beginning of the cruise phase. R ft is the range, expressed
in feet. a0 is the speed of sound, and θ is the temperature ratio Talt,cruise/TSL, computed using the International Standard
Atmosphere as a reference. CD [-] and CL [-] are respectively the drag and lift coefficients, available in Tab. E.1.
6 SFCcruise is the specific fuel consumption, corrected with respect to the altitude.
4. Reserve:
NBAA IFR Range with 100 nmi Alternate was demanded in the statement. One can approximate it by 100 nmi in cruise
condition. So, all other things being equal, Eq. 3.27 is re-used.
5. Descent:
As a first approximation, the descent is considered as a continuation of the cruise phase.
6. Landing and taxi:
Wfuel,L&T = 0.0035 WTO (empirical). (3.29)
3.6.4 Results
The different component’s masses are detailed on Fig. 3.13.
3.6.5 Center of gravity
The components of the planes and their respective weights have been estimated in the previous sections. Fixing the reference frame at the nose of the plane, the longitudinal location of the center of gravity is computed as follow:
n −1 xCG = Wtotal ∑ xCGiWi, (3.30) i=1 the i subscript designating the different components of the plane.
4 different configurations are considered:
• full plane with full fuel tank (corresponding to the maximum takeoff weight),
• full plane at mid-cruise,
6 From [11], the following empirical formula (valid for engines with BPR > 2) is used, evaluating the SFC at cruise with respect to the TSLS: p SFCcruise(lb/(h·lb f )) = 0.8 − 0.00096 Tsls. (3.28) Computation relative to the SFC of example business-jets show an increase about 65% at cruise condition. In the specific case of the PW535-E, the SFC rises to 0.77. App. B also confirms such a number. 3 AIRCRAFT DESIGN CHOICES 31
Take-Off Weight - 18272 lbs
Empty Weight Payload Fuel 9977 lbs 3000 lbs 5295 lbs 55% 16% 29%
4% 25% 22% 20% 29% 13% 53% 33% 7% 90% 3% Tail - 427 lbs Tail Crew - 400 lbs Wings - 1989 lbs Wings Cruise - 4743 lbs Reserve - 160 lbs Engines - 2237 lbs Systems - 2872 lbs Fuselage - 2452 lbs Luggages - 1000 lbs Passengers - 1600 lbs Taxi/Take-Off/Landing - 392 lbs Taxi/Take-Off/Landing
Figure 3.13: Aircraft’s mass distribution for 8 seats business-jet at MTOW and covering a range in cruise of 2500 nmi.
• full plane on reserve,
• totally empty plane.
The shifting of the center of gravity during the cruise will be important in determining if the plane meet the stability requirements at all time. The positioning of the components of the plane should therefore aim to reduce as much as possible the variation of the center of gravity between the different flight phases. This is partly achieved by placing the centre of gravity of the main source of mass variation, the fuel tank, at the centre of gravity of the plane. Solving the Eq. 3.30 gives:
xCGMTOW = 27.70 [ ft]
xCGmid-cruise = 27.73 [ ft]
xCGempty = 27.81 [ ft]
xCGreserve = 27.76 [ ft]
At it can be noticed the variation is very low ! However, this depict an idealized variation of the fuel weight, which assume that the center of gravity of the fuel does not move and which does not take into account the actual repartition mechanisms in the fuel tank Components locations following the same reference frame can be visualized in Fig. 3.14 and are numerically defined.
3.7 Catia model
A 3D model of the plane has been realized by Catia software in order to have a realistic view but at the same time an other tool to evaluate the CG location of the plane. 3 AIRCRAFT DESIGN CHOICES 32
z
x
1 2 3 4 5 6 7 10 11 12 13 14 8 9
Reserve Mid-Cruise Empty MTOW
# Item xi [ft] # Item xi [ft] # Item xi [ft] 1 AC 0 6 Furnitures 24.03 11 Propulsion 38.45 2 Crew 1.90 7 Fuel 27.56 12 Luggage (aft) 37.46 3 Instruments 6.23 8 Wings 28.13 13 APU 44.74 4 Landing gears 9.07 8bis Wings hydr. 28.13 14 Tail 47.96 5 Fuselage 23.68 9 Control Surfaces 28.68 14bis Tail hydr. 47.96 5bis Electrical 22.54 10 Luggage (cabin) 34.59
Figure 3.14: Determination of the CG of each constituent of the aircraft.
By the same software it is possible to have also an evaluation of the inertia contributions.
A whole 3D aircraft visualization is given in Fig. 3.15.
To compute the CG of the whole plane concentrated masses of the different components have been inserted in the Catia model in correspondence of the CG location of each of them basing on the data given by Matlab. In such study two cases have been analyzed, one corresponding to the MTOW and one corresponding to the ZFW.
The concentrated masses used and the corresponding CG location are shown in Tab.3.10.
In Tab.3.11 are instead shown the results obtained in the two cases.
The results obtained in Catia and Matlab are approximately the same and so we can conclude stating that we have a good estimation of the CG location. In particular it is possible to notice that in both the cases the symmetry of distribution of the weight with respect to the y axis is verified. 4 TRADE-OFF STUDY 33
Figure 3.15: CATIA 3D model
Component Mass[lbs] xCG[ ft] zCG[ ft]
Fuselage, nose and aft 2452.454 23.68 0.00 Wing 1989.461 28.13 -1.31 Tail 426.711 47.96 4.47 Propellers 2236.800 38.45 3.10 Landing gear 730.889 9.07 -1.31 Fuel 5295.000 27.56 -1.31 APU 120.000 44.74 2.83 Furnishing 782.320 24.03 0.00 Wings hydraulic system 578.820 28.13 0.00 Electric system 104.000 22.54 0.00 Electronic system 300.000 6.23 0.00 Control system 500.890 28.68 0.00 Deicing system 120.000 0.00 -2.83 Crew 400.000 1.90 0.00 Passengers row 1 400.000 23.94 0.00 Passengers row 2 400.000 27.88 0.00 Passengers row 3 400.000 31.82 0.00 Passengers row 4 400.000 35.75 0.00
Table 3.10: Concentrated masses and CG location of the different aircraft components.
MTOW ZFW CATIA MATLAB CATIA MATLAB Mass [lbs] 1.956e + 04 1.827e + 04 9.996e + 03 9.977e + 03
xCG[ ft] 27.12 27.70 27.56 27.81 yCG[ ft] 0 0 0 0 zCG[ ft] -0.21 -0.36 -0.18 -0.22 2 Ix[lbs/ ft ] 4.606e + 05 1.115e + 04 2 Iy[lbs/ ft ] 2.198e + 06 8.586e + 05 2 Iz[lbs/ ft ] 1.813e + 06 8.476e + 05
Table 3.11: Masses, CG location and inertia
4 Trade-off study
The trade-off study is performed for the 8-seats configuration, which is the optimal one. 4 TRADE-OFF STUDY 34
4.1 Aspect ratio of the wing
In the section dedicated to the wing design, the conclusion was made that the optimal value of the Aspect ratio was 9. Here, it is changed by 10% about this value in order to evaluate if this choice was optimal or if it needs to be changed.
An important parameter that is closely related to the aspect ratio is the aerodynamic wing loading, defined by the ratio L/S.
The variation of this quantity with respect to the aspect ratio is presented in Fig. 4.1 (Left). Despite the fact that the variation of the magnitude seems small, the represented graph depicts a trend that might be surprising at first approach: the higher the aspect ratio, the lower the wing loading. The small variation of the aspect ratio makes the magnitude of the aerodynamic loading variations very small too. For bigger variation of the aspect ratio, significant variation in amplitude can be expected.
The unexpected trend depicted in this graph is explained by the fact that a small increase of the aspect ratio does not necessarily imply an increase of the wing loading. Indeed, an increase in the aspect ratio can result in an increase of the exposed surface if the span also increases as well as an increase in the aspect ratio can imply a decrease of the lift due to a smaller surface of the wing, thus a smaller weight of the wing. The effect of the wing weight would seem negligible, but the depicted trend here is also of negligible amplitudes. As a result, it can be conclude that, in our case, a small variation of the aspect ratio around its initial value does not change the aerodynamic loading of the wing.
The influence of the aspect ratio on stability is also of big interest. Fig. 4.1 (Right) illustrates at what extent the Aspect ratio has an effect on stability. This figure shows that the plane becomes more stable and less manoeuvrable as the stability margin,
Kn increases. For an aspect ratio of 10, the stability margin at takeoff is about 10 % which exhibit less manoeuvrability. In conclusion, an aspect ratio of 9 is the best and the most reasonable choice of design considering the observations of Fig.
4.1 and discussions in section 3.2.
75.6176 0.1
75.6176 0.095 0.09 75.6176 0.085 75.6176 0.08 75.6176 0.075
75.6176 0.07 Takeoff
75.6176 Kn [ % ] Stability margin Cruise
Aerodynamic loading [ lbf ] 0.065 Landing 75.6176 0.06 8 8.5 9 9.5 10 8 8.5 9 9.5 10 Aspect ratio [-] Aspect ratio [-]
Figure 4.1: Evolution of some important parameters with the aspect ratio AR for several flight configurations. Left: Evolution of the aerodynamic wing loading, L/S. Right: Evolution of the stability margin Kn. 4 TRADE-OFF STUDY 35
4.2 3D lift coefficient (CL,w)
Four important parameters on which the variation of the 3D wing lift coefficient, CLw , has an impact are the takeoff weight, the aerodynamic loading of the wing, the CL,plane/CD,plane and the stability margin, Kn. Their variations with respect to the 3D lift coefficient are represented in the figures 4.2 and 4.3.
It can be seen in the figure 4.2a that the takeoff weight decreases with respect to an increase of the CLw parameter. This is explained by the following formula:
1 L = C ρv2S, (4.1) 2 L
where L is the total lift of the plane expressed in lbf, ρ is the density expressed in lb/ft3, v is the free stream velocity seen by the plane expressed in ft/s and S is the reference (or gross) area of the wing expressed in ft2. If the 3D wing lift coefficient increases, the gross area of the wing has to decrease in order to keep the same value of the lift which is fixed. Indeed this surface is the only free variable in Eq. 4.1. As the surface decreases, the weight of the wing also decrease as well as the weight of the tailplane, even if this last has a negligible effect.
Another effect close to this result is the variation of the aerodynamic loading with respect to the CLw . Indeed, if the CLw increase, the wing surface decreases in accordance with Eq. 4.1, while the lift remains the same. As a result, the aerodynamic loading increases with the value of the lift coefficient. This trend is represented in the figure 4.2b.
20 84 ] 18 2 82 [lbf/ft 16 80 L/S 14 78 Takeoff weight Fuel weight 12 76 & L [1000 lbs] to 10 74 W
8 72 Aerodynamic wing loading 6 70
4 68 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.27 0.28 0.29 0.3 0.31 0.32 0.33 C Lw [-] CLw [-]
(a) Takeoff weight and lift. (b) Aerodynamic wing loading.
Figure 4.2: Evolution of the takeoff weight, lift and aerodynamic loading with the 3D lift coefficient, CL.
The figure 4.3a shows the increase of the ratio CL,plane/CD,plane with respect to the CLw . This behaviour can be explained by the fact that the lift coefficient of the wing constitutes approximately 90 % of the total lift generated by the whole plane.
What is more, the low drag bucket region of the wing extends from values of the lift coefficient of approximately 0.2 to 0.7. 4 TRADE-OFF STUDY 36
Thus, an increase of the CLw in this range does not change the value of the drag generated by the wing, CDw . As a result, the ratio CL,plane/CD,plane increases when the CLw increases. Needless to say, this is true only for values of the CLw ranging from 0.2 to 0.7.
The last parameter to be considered is the stability margin, Kn for each phase of flight. This is presented in figure 4.3. This last figure shows that for an increase of 10 % of the CLw value, the stability margin is contained between 5 to 7 % which is acceptable. In another hand, for a decrease of 10 % of the CLw value, the stability margin reaches values ranging from 12 to 15 % depending the phase of flight. These are still in the required range for both stability and maneuverability. Therefore since the requirement on this parameter are met regardless of the lift coefficient variation, it is not critical in determining an optimum value.
As a result the chosen value of 0.3 for the 3D lift coefficient of the wing seems optimal: although a higher value would decrease the weight of the plane, it would drastically increase the aerodynamic loading of the wing. The structure of the wing would need to be stronger, therefore heavier. It is complex at this point to measure at which point the decrease of the weight brought by smaller wings is compensated by a heavier internal structure, as the latter should be designed multiple times.
However,
13.5 0.16 Takeoff Cruise 0.14 Landing 13 [%]
n 0.12 K 12.5 D,plane
/C 0.1
L,plane 12 C 0.08 Stability margin
11.5 0.06
11 0.04 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.27 0.28 0.29 0.3 0.31 0.32 0.33 C CLw [-] Lw [-]
(a) CL,plane/CD,plane. (b) Stability margin, Kn.
Figure 4.3: Evolution of the CL,plane/CD,plane with the 3D lift coefficient of the wing, CLw , for several flight configurations.
4.3 Fuselage length
A last important parameter will complete this trade-off: the fuselage length.The result of this last trade off is shown in Fig.
4.4. It can be observed that any variation of the fuselage length about its length of design, tend to decrease the surface of the wing. That increases the aerodynamic loading and has a negligible effect on the weight as it can be seen in Fig. 4.4b. What is more, the stability margin diverges between the different phases of fly and reach unacceptable values about 50 ft. On the contrary, the stability margin is quite good below the length of design. However, the jet has been chosen to offer enough space 5 OPTIMIZATION 37
232.5 20
232 ] 2
[ft 15 S 231.5 Takeoff weight Fuel weight & L [1000 lbs]
231 to W 10 Wing gross surface,
230.5
230 5 45 46 47 48 49 50 51 45 46 47 48 49 50 51 Fuselage length [ft] Fuselage length [ft]
(a) Wing gross area, S. (b) Fuselage length.
0.1
0.095
0.09 [%] n K 0.085
0.08
Stability margin, 0.075 Takeoff Cruise 0.07 Landing
0.065 45 46 47 48 49 50 51 Fuselage length [ft]
(c) Stability margin, Kn.
Figure 4.4: Evolution of the wing gross surface, takeoff weight, fuel weight and stability margin Kn with respect to the fuselage length. to meet expectations of passengers in terms of comfort. Changing the fuselage length of the 8 seats configuration is thus to exclude. On the contrary, from the 6 seats point of view, such a results tend to enforce the fact that a shortage of the fuselage length is the most suitable approach for its design.
5 Optimization
5.1 Stability
The stability of an aircraft is defined as its ability to remain or return in its equilibrium position after a perturbation. Indeed although in flight the plane is in equilibrium, its flight dynamics must be such that unexpected phenomena do not lead to 5 OPTIMIZATION 38 durable nor violent perturbations of the flight. On the other hand a "too much" stable aircraft is hardly manoeuvrable; the stability is therefore a compromise between safety and manoeuvrability.
5.1.1 Enforcing equilibrium
According to the methodology most of the geometry of the plane is now fixed: the location of the center of gravity and therefore the torques caused by both the camber of the wing and the lift are known for different flight configurations.
Therefore defining ML as the force moment resulting from the aerodynamic moment of the wing and of the torque of the lift from the wing in mid-cruise:
ML = Mw − L(xL,app − xCG05 ), (5.1) the equilibrium of the plane is reached if the tail produce a lift counteracting this moment. The lever arm of the tail lift being known, the longitudinal equilibrium is enforced with:
ML Ltail = − . (5.2) armtail
To produce this lift, the V surfaces must have the incidence angle leading to the required lift coefficient. As a matter of fact this incidence angle should not be high since an adverse effect is that the tail would produce a lot of drag. Typical range of value is between 0 and 3◦. Note that since the chosen plane configuration is "CG forward", the tail has to produce a downlift.
Therefore this angle will be negative.
Aircraft Design - Synthesis and Analysis [4] provides an estimation of the lift coefficient taking account of the effect of the wings wake on the flow perceived by the tail.
CL,wing dε CL,tail = CL,α,tail · 1 − − αL,0,root − ηT . (5.3) CL,α,wing dα
where
dε • is the downwash gradient produced by the wing wake on the flow perceived by the tail. According to the definition dα provided in ([4]), this gradient is equal to 0.29.
• ηT = iT − iw is the angle between the wing chord and the tail chord at the root.
Note that the V tail disposition may change the way the flow is affected by the downwash gradient since it is not located at the same place in the wake as a conventional tail. At this stage of the design, without any clues to what extent the "V" tail is differently affected, the full effect is taken into account. Further studies may have to be conducted to quantify the effective effect of the wake on the tail.
A sweep of different values of the tail incidence angle allows to find the value for which the tail produces the requested lift. This angle equals -2.9 ◦which as required is lower than 3◦. 5 OPTIMIZATION 39
5.1.2 Longitudinal stability
As stated previously, the longitudinal equilibrium is enforced by Eq. 5.2. This equilibrium can be rewritten in the form of a residual momentum which has to be null:
Mres = ML + Ltail ·armtail
(5.4) = Mw − L·(xL,app − xCG05 ) + LTail ·armTail
= 0 (at equilibrium).
A good way to characterize the stability of the longitudinal equilibrium is by its derivative with respect to the angle of dM attack, as set out in Aircraft Design - Synthesis and Analysis [4]. A stable equilibrium is then characterized by res < 0. dα ∂Cm,plane Expressed in a more general way, this stability condition can be written as < 0, and since the lift coefficient CL is ∂α proportional to α the stability limit can be approximated as
∂C m,plane < 0. (5.5) ∂CL,wing
It is now possible to express the total momentum coefficient Cm as the sum of all its contributions, as proposed in [4] :
xCG05 − xL,app Stail ·armtail Cm,plane = Cm,0 +CL,wing +Cm,tail −CL,tail +Cm,fuselage. (5.6) c c·Sw
As it can be seen the second term depends on the position of the center of gravity of the plane. Therefore the variation of the center of gravity also impact the stability and is a major factor. The neutral point xn = hn ·c is defined as the position of the center of gravity for which the momentum derivative from Eq. 5.5 equals zero. Deriving Eq. 5.6 with respect to the lift coefficient provides the expression of hn :
xL,app STail dCL,tail dCm,fus hn = + armtail · · − ; (5.7) c c·Sw dCL,w dCL,w
Where
dC dε C • L,tail = 1 − · L,α,tail , dCL,w dα CL,α,w dC k ·fus2 ·fus • m,fus = fus width length , dCL,w Sw ·c·CL,α,w
• kfus is an empirical parameter, defined with respect to the fuselage geometry and the wing location.
A convenient way to quantify the stability of an aircraft is to express the distance between the center of gravity and the neutral point, normalized by the mean aerodynamic chord. This define the stability margin: 5 OPTIMIZATION 40
x − x K = n CG (5.8) n c
According to the Federal Aviation Administration requirements, the stability margin of an aircraft must be greater than 5% in order to ensure sufficient stability. Requirements on the upper limit vary depending on the tail configuration of the plane since some tails provide more stability than others. Without specific requirements concerning the "V" tail, the acceptable range of stability margin is assumed to be similar to the standard stated in the FAR, with a higher upper ceiling to favor stability: from 5% to 15%.
As the center of gravity vary during the flight, the stability margin fluctuate as well. Computed for the different flight configurations stated in the section 3.6.5, the stability margins are:
• Maximum takeoff weight : Kn = 9.24%.
• Mid cruise weight : Kn = 8.75 %.
• End cruise (on the fuel reserve) : Kn = 8.22 %.
• Empty weight : Kn = 7.3 %.
5.1.3 Lateral stability
The variation of the moment coefficient in yaw can be computed using the following formula presented in Aircraft Design -
Synthesis and Analysis [4])
dCN, C = plane ·β, (5.9) N,plane dβ
with,
dC dC dC dC N,plane = N + N,fus + N,w , (5.10) dβ dβ dβ dβ
where,
• CN,plane is the total 3D moment coefficient of the whole plane in yaw;
dCN,plane • dβ is the variation of the total 3D moment coefficient of the whole plane in yaw with respect to the yaw angle.
Eq. 5.10 is nothing else than the derivative of the moment coefficient in yaw with respect to the yaw angle to which we
dCN,w add respectively the effect of the fuselage and the wing. dβ is equal to 0.024 for low-mounted wing for low-mounted wing according to lecture 4, page 57 from Pr. Noels’ course notes [6].
The results of the moment coefficient derivatives are presented in Fig. 5.1. 5 OPTIMIZATION 41
x10-3 0 8 6 -0.04 4 2 -0.08 0 -2 -0.12 of the plane [-] of the plane [-] -4 N M C C -0.16 -6 -8 -0.2 -10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -3 -2 -1 0 1 2 3 4
CL,w of the wing [-] Yaw angle β [°]
Figure 5.1: Left: Variation of the moment coefficient of the overal plane in pitch with respect to the 3D lift coefficient of the wing. Right: Variation of the moment coefficient of the overall plane in yaw with respect to the yaw angle.
5.1.4 6 seats consideration
The "V" tail remain geometrically the same as for the 8 seats version. It still have to balance out the moments around the center of gravity, it therefore imply a variation of the tailplane angle with respect of the fuselage axis, compared to the other member of the plane family. To produce the adequate lift, the "V" tail must have an incidence angle of −3 [◦].
The stability of the 6 seats version is computed the same way as for the 8 seats: taking account of the diverse variations, notably of lift coefficients of the tail and wing, the stability margins are respectively:
• Maximum takeoff weight : Kn = 8.84%.
• Mid cruise weight : Kn = 11.33 %.
• End cruise (on the fuel reserve) : Kn = 14 %.
• Empty weight : Kn = 8.5 %.
These are well in the stability range defined previously in section.
5.2 Aerodynamics
5.2.1 Computation of CL,plane and CD,plane
In the conceptual design, a first approximation was made to assess the lift coefficient of the plane by assuming that all the lift was generated by the wing. Nevertheless, in the interest of rigour, lift of the tail had to be considered for stability computation.
A more accurate assessment of the lift coefficient is thus given by the following formula
S C = C +C T , (5.11) L,plane Lw LT S 5 OPTIMIZATION 42 where,
• CL,plane is the total lift coefficient of the plane, neglecting the effect of the fuselage;
• CLw is the lift coefficient of the wing, set in section 3.2;
• CLT is the lift coefficient of the tail;
• ST is the reference surface of the tail;
• S is the reference (gross) area of the wing.
Like the lift coefficient of the plane, the drag coefficient of the plane was first estimated from the drag generated by the wing on itself. However, to pursue the study more in details, we have to consider the interference drag due to interaction between components. This induce a drag component, necessary to compute at the end the total drag of the whole plane. To get a better idea of the drag coefficient, we can use the formula of the polar drag presented in the course notes in the slides from Mr. Noels [6]:
C2 C = C + L,plane , (5.12) D,plane D0 eπAR
where,
• CD,plane is the total drag coefficient of the plane, neglecting the effect of the fuselage;
• CD0 is the independent drag that takes the compressibility and 3D effects independent of the 3D lift into account;
• CL,plane is the total lift coefficient of the plane computed with the formula 5.11;
• e is the Oswald efficiency factor;
• AR is the Aspect ratio of the wing.
As well as the total lift coefficient of the whole plane, here the effect of the fuselage has been neglected. Although the approximation for the total lift coefficient of the plane is not so important, it is no longer true when it comes to the drag. A more accurate study has to be performed for the drag. This is the subject of section 5.2.3.
5.2.2 TRANAIR
TRANAIR is a software analogous to a typical panel method program developed to analyze compressible flow over complex configurations at subsonic, transonic or supersonic freestream Mach numbers. The numerical method solves the full potential equation subject to a set of boundary conditions. The solution is obtained on a sequence of successively refined grids which are constructed adaptively based on estimate solution errors.[12] 5 OPTIMIZATION 43
0.7 0.04 0.7
0.6 0.6 0.035 0.5 0.5
0.4 0.03 0.4
0.3 0.025 0.3 of the plane [-] of the plane [-] L D of the plane [-] C L 0.2 C 0.2 C 0.02 0.1 0.1
0 0.015 0 -3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 4 0.015 0.02 0.025 0.03 0.035 0.04
Angle of attack α [°] Angle of attack α [°] CD of the plane [-]
Figure 5.2: Left: Variation of the lift coefficient of the plane with respect to the angle of attack. Center: Variation of the drag coefficient of the plane with respect to the angle of attack. Right: Polar drag of the plane.
5.2.2.1 Structure mesh
The wing and tail elements being fully defined mathematically, it is quite easy to model them into a mesh. On the other hand, the fuselage has a more "free hand" aspect to it, and has been meshed at a 100% true to what has been drawn in our CAD models. This way, the defects and geometrical flaws will be easily tracked, not unlike what would be studied in a wind tunnel.
The full mesh is visible in Fig. 5.3 in a rendered version. Notice that only half the plane is there, Tranair resorting to symmetry to spare computation time.
Figure 5.3: Rendered plane mesh.
Winglet contribution This device helps reducing induced drag caused by wingtip vortices. The design of those winglets has been based on Whitcomb’s winglet design[13] and adapted to our geometry. As can be seen in Fig. 5.4, an optimum in this drag reduction is reachable. The chosen parameters are labelled; notice that the ultimate minimum has not been selected for the twist because it would have produced too steep of an angle at the junction with the wing. 5 OPTIMIZATION 44
0.0429 0.0438
0.0428 0.0436
0.0427 0.0434 [-] [-] 0.0432 0.0427 D D C C
0.0427 0.043
0.0426 0.0428 α : 14
λ: 1.37 CD: 0.04267
CD: 0.0426 0.0426 0.0426 1.32 1.34 1.36 1.38 1.4 1.42 1.44 0 2 4 6 8 10 12 14 16 18 λ [rad] α [degree] (a) Sweep angle. (b) Longitudinal geometrical twist.
Figure 5.4: Drag coefficient reduction with winglet parameters (wing element alone).
5.2.2.2 Cruise condition calculations
Launching a computation to simulate cruise condition, results are finally coming to life. As seen in Fig. 5.5, there is a strong
depression (corresponding to a shockwave) at the point of junction between the cabin and the aft. Indeed, this point has a
sharp angle and we have seen that the software struggles a bit with those. Nevertheless, it shows that the angle may be too
sharp and that the structure needs to be smoother at that junction. Moreover, we can see another depression point on top of
the nose, at the beginning of the cabin. Again, smoothing that connection with more flat curve would probably improve the
aerodynamics of the fuselage. Finally, a strong pressure concentration can be seen on the tip of the nose and at the angle
starting the bulge corresponding to the cockpit. Those area being constrained for practical reason, improving them may be
difficult but we would need more precise constraints to decree.
0.7456
0.5237
0.3019
0.08
-0.1418
-0.3637
-0.5856
-0.8074 -1.0293
-1.2511
-1.4730
Figure 5.5: Pressure distribution along the structure.
Other features of interest are the lift and pressure coefficient distribution. The latter, visible in Fig. 5.6 (left), shows a
rather strong and abnormally oblique shock. Two reasons may be causing this phenomenon:
• The study is made in non-viscous conditions, which means that in reality, the shock would be less strong and moved
forward along the chord. 5 OPTIMIZATION 45
• As the profile is cambered, the normal vector to the surface at the leading edge is also strongly leant backward. The
shock is oblique with respect to the absolute axis but is normal to the surface towards the trailing edge.
The lift coefficient distribution is also visible in Fig. 5.6 (right). Notice that the curve breaks at the wingtip due to the connection with the winglet, the shape of this curve is otherwise expected as the 2D lift coefficient is inversely proportional to the chord length.
1 1 Intrados Extrados 0.9 0.9
0.8 0.8
0.7 0.7 [-] [-] L p C C 0.6 0.6
0.5 0.5
0.4 0.4
0.3 0.3 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/c [-] y/span [-]
Figure 5.6: Left: Pressure coefficient around the mean aerodynamic chord. Right: Sectional lift coefficient distribution along the normalized span.
Finally, the moment distribution along the span can be seen in Fig. 5.7. The moment being extremely important at the trailing edge and of negative sign, it shows that the plane undergo a nose-down moment, in this case compensated by the downforce exerted by the tail.
5.2.2.3 Performances comparison with conceptual design
Graphs presented in Fig. 5.8 make the comparison between theoretical model and results obtained from the simulation with
Tranair. It is worthwhile noticing that the simulation was performed under the assumption of an inviscid flow of same free stream velocity than considered in theory. Differences can thus be expected, not only due to variation from the theoretical theory to the simulation of the flow around the plane, but also due to the modification of conditions.
It can be observed in left graph of the figure 5.8 that the theory predicts the same trend of the CL behaviour than the simulation. Indeed, the red curve is simply an offset of the blue one, such that the linear relation is nearly conserved. However, the differences in magnitude implies some changes in terms of stability. Taking into account that the lift generated by the wing constitutes almost 90 % of the total lift generated by the plane, it can be stated that the variation of Kn with respect to CL,plane follows the same trend than the variation of Kn with respect to CLw . As a result, it can be assumed that the graph presented in 5 OPTIMIZATION 46
QMY
1.8192
0.9534
0.0877
-0.7781
-1.6438
-2.5095
-3.3753
-4.2410
-5.1067 -5.9725
-6.8382
Figure 5.7: Pitching moment distribution along the span (extrados up, intrados down).
Fig. 4.3b depicts the the general behaviour of the stability with respect to the CL,plane. That is, an increase in the lift coefficient to 0.4 would decrease the stability margin to an unacceptable value well below the stability margin, such that the plane would not be stable.
Another issue concerns the wing loading. Such values of CL,plane would require a CLw bigger than 0.3 and considering Eq. 4.1, the wing surface would be too small implying a too high wing loading.
Fig. 5.8 shows that the value of the overall drag coefficient of the plane is the closest to the the value obtained theoretically for an angle of attack of about -0.5 ◦. The more we increase the angle of attack the more both solutions diverge.
◦ The right graph of Fig. 5.8 shows that for an angle of attack of −0.5 , CL,plane/CD,plane is worth 16.53 which is much more optimal than what theory predicts for an angle of attack of -0.17 ◦ (value of design). However as explained earlier in this discussion, such a value of the CL,plane does not lead to optimal results in terms of stability and wing loading.
The new derivative a for the whole plane corresponding to the slope of the CL with regard to the angle of attack is obtained by linear regression, which gives:
a = 0.098224 [deg−1], r = 0.99996.
The theoretical line gave a slope of a = 0.0898 [deg−1] which is a difference of 8.6%.
5.2.2.4 Further improvements
A rather important offset has been seen between the theoretical lift coefficient curve and the one computed in Tranair, corresponding more or less to a difference of 2◦of angle of attack. This last value is perfectly valid when a viscous modeled is compared with an inviscid one, and even though the design point is not met as shown by the inviscid output, building a viscous CFD model would undoubtedly lead to a result very close to the theoretical result. 5 OPTIMIZATION 47
0.7 0.05 0.7
0.6 0.045 0.6
0.5 0.04 0.5
0.4 0.035 0.4
0.3 0.03 0.3 of the plane [-] of the plane [-] L D of the plane [-] C L 0.2 C 0.025 0.2 C
0.1 0.02 0.1
0 0.015 0 -1 -0.5 0 0.5 1 1.5 2 2.5 -3 -2 -1 0 1 2 3 4 0.015 0.025 0.035 0.045
Angle of attack α [°] Angle of attack α [°] CD of the plane [-]
Conceptual design Tranair
Figure 5.8: Left: Lift coefficient with regards to the angle of attack. Center: Drag coefficient with regards to the angle of attack. Right: Performance comparison with the theory.
5.2.3 Drag analysis
In the following section, the total drag is evaluated in detail by using the collection of formulas presented in ref. [14]. Those methods only hold in cruise configuration for subcritical speeds (M < 0.85). Thus, transonic effects such as wave drag are not accounted for in this drag breakdown. Here, the drag study is performed for the 8-seat configuration.
When usually analyzing performance of aircrafts, the equation 3.1, introduced in a previous section is used. Instead, a further breakdown is preferred.
1. Vortex-induced drag: associated with the system of trailing vortices and downwash which a wing generates. This
particular drag is computed for each of aircraft’s main components regarded in isolation.
2. Profile drag: skin friction and pressure (form) drag result respectively from boundary layers and the form of the bodies
(so the separation of flows). Here, bodies are considered as streamlined, and isolated. As the aircraft is not designed
in detail, it is often convenient to use the relation between a well-streamlined body and a flat plate providing accurate
results. A skin friction coefficient is determined for the flat plate associated with the aircraft’s part studied. Then
correction factors are applied to allow for the body’s thickness and difference in development of the boundary layer.
3. Interference corrections: The interaction of flow around main aircraft components lead to consider corrections. Indeed,
flows of the wing and fuselage interact about the root for instance. This drag component is not usually more than 5-10%
of the total drag, when the aircraft is well-designed.
4. Protuberances, surfaces imperfections, etc: Windshield, external fuel tank and other excrescences are accounted for
drag study to be exhaustive. Despite the fact that the design is not yet entirely defined, statistical formulas and correc-
tions must in some cases be considered.
The Catia model is used to provide volumes, wetted areas of main parts of the airplane. Furthermore, the 3D lift coef- 5 OPTIMIZATION 48
ficient used to determine the CD aforementioned is that of the aircraft which combines CL,w and CL,Vtail. Fig.5.9 depicts the distribution of the total drag in terms of its components aforementioned .
Wing 0.0047 Fuselage and tail boom 0.0099 Jet engine nacelles 0.0007 Tailplane 0.0023 Total 0.0175
Table 5.1: Sources of interference drag ∆Cd
0.03
0.025
0.02 66% Vortex-induced drag Profile drag 0.015 Interference drag Protuberances drag Total drag 0.01 28.1%
0.005 11.4% 0 -5.5%
-0.005
Figure 5.9: Drag components
The reader may notice that the profile drag contributes to 66% of the total drag. Table 5.1 reports the contributions of the aircraft’s main components. It is clear that the fuselage accounts for more than 50% of the profile drag and the wing for at least 25%. The weaker contribution of the tailplane and engine nacelles results largely from their smaller size. Moreover, the lack of information about gas generator cowlings or the plug in the flow leads to regard this nacelle profile drag as minimum.
The second highest contribution is that associated with surface imperfections and protuberances, which the external fuel tank
(belly) accounts for 50%. Also, rough bodies are proven to experience more rapidly an increase in drag than smooth ones do.
This lead to allow for a certain amount of drag representing about 20% of the profile drag increment. Then, the vortex-induced drag component is for the most part due to the plane wing and horizontal tail, since the fuselage generates weak amount of lift. Eventually, the interference drag is negative due to the wing/tailplane interference. The downwash creates a tail upload and so reduces for a given airplane lift the vortex-induced drag of the wing.
The total drag coefficient and the drag force, using the plane lift coefficient obtained with Eq.5.11, are therefore
CD = 0.0266 (5.13) and, 1 D = ρv2SC = 1,348lbf. (5.14) 2 D 5 OPTIMIZATION 49
The lift-to-drag ratio considering the plane lift coefficient becomes,
C L,plane = 9.89 (5.15) CD
5.3 Performance
The aim of this section is to assess the flexibility of use of the aircraft, and the suitability of the aircraft’s component for each flying phase. In the last section, the drag study and the Tranair aerodynamic coefficients will be integrated to the assessment.
5.3.1 Take-off
As requested by the RFP, the aircraft must be compatible with a 4000 ft, FAR Part 25 compliant Take-Off Balanced Field
Length7. The upcoming section will assess this requirement.
A determining parameter is the decision speed V1, defined as the speed at which the full stop distance is equal to the obstacle clearance distance. Indeed, for a speed V > V1, the aircraft’s take-off is mandatory. In addition to that, the aircraft is required to clear an obstacle of 35 ft at a speed V2 ≥ Vstall at the end of the runway, as mentioned in the FAR Part 25.
5.3.1.1 Method
The following method, as well as some parameters are provided by ref. [15]. 5 distincs segments are identified as
A: all engines operating up to the decision speed V1,
B: one-engine inoperative acceleration from V1 to lift-off speed VLO,
C: one-engine inoperative acceleration from VLO to V2 by clearing the obstacle,
D: time interval separating the one-engine lost and the actual braking action,
E: deceleration from braking action speed VB to the stop.
The ground distances of each phase is then computed using Eq. 5.16.
Z Vi+1 V xi = dv. (5.16) Vi a
As an approximation, the acceleration a is considered constant and equal to V07 = 0.7· (Vi+1 − vi).
F = T − µR − D , with R = W − L. (5.17)
7The term "Balanced Field Length", as stated by ref. [15] refers to the one engine operative condition of take-off. 5 OPTIMIZATION 50
The average acceleration during ground run is therefore:
2 T 1/2 ρ CL V C a = g − µ − 07 D − µ . (5.18) W W CL
By translating the BFL definition in terms of those 5 segments,it follows that the decision speed corresponds to xB + xC = xD + xE .
5.3.1.2 Results
First of all, as the parameter provided the manufacturer is the uninstalled static sea-level thrust, some corrections must be taken into account. The loss due to the engine’s placement is chosen at its maximum statistical value (i.e. 8%), and the correction due to the aircraft’s motion at take-off is given by Eq. 5.19 (source: ref. [6]).
T 0.45M ·(1 + BPR) TO,available = 1 − √ + (0.6 + 0.11BPR)·M2 (5.19) TSLS 1 + 0.75BPR
• Segment A: In this segment, the aircraft speeds up from 0 velocity to the decision speed V1. The flaps settings are
◦ 2 assumed constant and equal to 20 , the take-off weight is WTO = 18272 lbs, and the wing loading W/S = 78.7 lbs/ft .
8 The stall velocity is Vstall = 166.5 ft/s, and the Thrust-to-Weight ratio is computed to T/W = 0.3 [-] . The tires on the dry pavement exert a friction coefficient on the aircraft of µ = 0.03 [-]. Finally, the lift coefficient and the drag to lift
9 ratio are respectively approximated by CL = 0.5[-] and CD/CL = 0.1
• Segment B: In segment B, an engine loss must be considered. Therefore, the Thrust-to-Weight ratio T/W = 0.15.
However, as the speed increases, the lift coefficient CL increases to 0.9 [-], and the drag to lift ratio slightly increases to 0.102 [-], due to the appearance of asymmetry (effect of the engine loss).
As the lift increases, the rolling friction coefficient lowers to 0.025.
Notice that the FAR imposes VLO ≥ 1.1Vstall. It is here chosen to respect the equality.
• Segment C: According to statistics, the time to flaring is set to 3 s. A uniform straight-line movement relation is thus
applied, using a velocity between of 1.2Vstall (between VLO and V2).
• Segment D: Estimating a 1s time for pilot recognition10, and 2s for the brakes to enter in service, the segment D is
assumed to last 3s overall from V1 to VB.
• Segment E: The segment E marks the deceleration from V1 to 0 [ft/s]. This deceleration is characterized by a zero thrust
as the remaining engine is shut down, and a breaking friction coefficient µbreak = 0.4 [-]. The lift coefficient to consider,
as stated by ref. [15] is CL = 0.5 [-].
8 As mentioned before, TSLS is corrected by a factor 0.8 9This value is advised by ref. [15]. It is hard to evaluate it without any wind-tunnel experiments taking the Reynolds and ground effects into account. 10A state of alert is assumed, as the take-off phase usually requires pilots to be focused. 5 OPTIMIZATION 51
The Fig. 5.10 represents the decision speed determination V1, and the Fig. 5.11 summarizes the main results of the take- off performance assessment. Specifically, it is shown that the take-off balanced field length is 3602 ft, which matches
the RFP’s requests, and gives flexibility to the jet’s owner to take-off from small runways.
180
170 [ f t/s ] 160 1
150
140
130 Decision speed V
120 2000 2500 3000 3500 4000 4500 5000 Balanced Field Length [ft]
V Take-off distance from 1 (segment A+B+C) V Stopping distance from 1 (segment A+D+E)
Figure 5.10: Balanced field length and related decision speed.
Balanced Take-Off Field Length - 3602 ft.
Flare (16%) Acceleration - 1 engine out (44%) Acceleration - all engines (40%)
Altitude [ft] 35 ft. H
V V V 2 LO stall V ft/s 1 = 1.2 Vstall = 1.1 Vstall 166 V 156 ft/s 0 V V SL 0 B RoC [ft/s]
Braking (47%) Recogniti- on (13%)
Figure 5.11: Left: Approached RoC evolution with the altitude. Right: Speed evolution and segment distances.
With the take-off parameters fully determined, the FAR 25’s second climb gradient must be assessed11. This design allows a climb angle of 3.5◦, and therefore a climb gradient of 6.1%, complying with the certification.
11The FAR states that the climb gradient must be ≥ 2.4% 5 OPTIMIZATION 52
5.3.1.3 6 seats take-off performance
The RFP requests does not differ from 8 seats one but it is important to notice that the FAR 23 imposes a 50 ft obstacle height to clear. As main results, it is obtained: V1 = 141 ft/s and BFL = 2936 ft.
5.3.2 Climb
The climb requirements are clearly stated in the request for proposal (Rate of climb (RoC) of 3,500 fpm and service ceiling of
45,000 ft). To assess the ability of the aircraft to reach the prescribed RoC, an approximate method for estimating the time to climb is used (source: ref [4]). This method considers bold assumptions:
1. The thrust is considered independent of the airspeed.
2. The RoC evolution is approached by a straight line from a 0 ft altitude to the service ceiling H.
This yields the RoC linear expression given by
h RoC = RoC 1 − (5.20) SL H
5.3.2.1 Details of the approximate method
If the unaccelerated rate of climb condition is assumed, it follows:
Z h2 1 Z h2 1 Z h2 1 H H − h t −t = dh = dh = dh = ln 1 (5.21) 2 1 RoC V sin h RoC H − h h1 h1 γ h1 RoCSL 1 − H SL 2
ROCSL is directly computed. Using the straight line properties, H can be found too. In fact, only the RoC of another point is necessary. The 35,000 ft (cruise altitude) one is chosen. Computations are carried out according to the subsequent set of equations:
1. The first assumption gives the following lift coefficient (associated to the maximum rate of climb):
q T T 2 −W + W + 12 CD0L K 1 CL = , with K = . (5.22) 2K e π AR
2. The corresponding airspeed and drag coefficient are thus equal to:
s W V = and CD = CD0L + KCL. (5.23) 0.5 ρ SCL
3. Hence, the flying path angle can be estimate via:
T − D T C sinγ = = − D . (5.24) W W CL 5 OPTIMIZATION 53
4. At last, the maximum rate of climb is given by:
R/C = V sinγ. (5.25)
5.3.2.2 Results
Using this method, it is possible to verify to the aircraft fulfils the RFP. As shown in Tab. 5.2, the time to climb up to the cruise altitude is lower than 10 minutes. Other time intervals are also computed for typical altitudes.
In addition, the approximation of the absolute aircraft ceiling leads to a value of 47,201 ft and a subsequent service ceiling
Altitude to reach in ft Time interval in min 10,000 1.74 20,000 4.03 30,000 7.39 35,000 9.9 40,000 13.76 45,000 22.43 47,000 40.00
Table 5.2: Summary time intervals to reach several altitudes.
altitude of 45,006 ft. Finally, mean climb angle is assessed to 6.5◦ (mean climb gradient equals 11.4%), all engines operative.
Such angle justifies therefore the employment of the empirical formula for the climb fuel consumption (see section 3.6.3). In case of one engine failure, mean climb angle is assessed to 1.9◦ (mean climb gradient equals 3.4%).
5.3.2.3 6 seats climb performance
Since both FAR and market demands does not change, it will be unthinkable to obtain worst results than for the 8 seat. Indeed, following main results are obtained: 8.3 minutes to reach 35,000 ft for a service ceiling at 45,478 ft.
5.3.2.4 Brief comparison
In order to assess the validity of the previous results, the comparison has been made with a business jet on Fig. 5.12. As stated by the author of the ref. [15], "the bizjet carries out the quasi-steady-state climb at a constant equivalent airspeed VEAS until it reaches Mach 0.7. Thereafter, it continues at a constant Mach number until it reaches the ceiling". Noting that the Tsls used by the author is 3,310 lbf by engine, which is thus very close to the PW535-E one, one can conclude by the reliability of the present aircraft results. 5 OPTIMIZATION 54
48,750 ft 47,201 ft
7,450 ft/min
6,451 ft/min
Figure 5.12: Comparison between computed results (orange for 8 seats and blue for 6 seats configuration) and a business jet example coming from [15].
5.3.3 Cruise
In the previous sections, the engines have been determined to meet the RFP requirements. It has been shown that the available thrust is 26% higher than necessary12.
5.3.3.1 6 seats cruise performance
It has been assessed that the available thrust is 28% higher than necessary. This slight change is largely due to a higher angle of attack for the 6 seats configuration.
5.3.4 Turning rate
In this section, it is assumed that the turn occurs in a horizontal plane according to a perfectly circular curve. This case is called by Raymer [3], a "sustained turn". Even if in most cases, a turn is accompanied by a change of altitude, this assumption provide a strong indicator of the actual turning performance.
12The method was based on the polar drag of the wing. As it will be shown in the further sections, a more detailed drag analysis may lead to some refinements. 5 OPTIMIZATION 55
Figure 5.13: Turning diagram.[6]
5.3.4.1 Method
Balancing the forces acting on the aircraft, as in figure 5.13, it appears a load factor n. Thinking in terms of load factor is convenient to assess structural limits but the human factor must be taken into account. Indeed, human comfort limits cannot range as far as structural one and for commercial transports, it is usually defined a nmax = 2.5. Due to this maximum value and with the lift coefficient which cannot be higher than CLmax, the upper limit of the turning rate is:
s dψ ρ SCLmax 1 = g nmax − . (5.26) dt max 2W nmax
As recommended by [3], a simple way to evaluate the maximum load factor in such a configuration is to think that:
n W n W = L ⇐⇒ CL = . (5.27) 1/2 ρ V 2 S
Since TL n = (5.28) WD must be verified at each time, this finally leads, expressing the drag by a drag polar, to:
s 1/2 ρ V 2 T 1/2 ρ V 2 C 1 n = − D0 , with K = . (5.29) K (W/S) W W/S e π AR
Expressing Eq. 5.29 in terms of wing loading and thrust to weight ratio emphasises the multiple computation options it offers.
5.3.4.2 Results
Only the cruise condition results are introduced here. The CL,max for the computation of maximum turning rate is therefore
6 equal to 0.8. This comes from the integration on the whole wing of the 2D cl,max in cruise condition (Re ' 9.6 ∗ 10 with respect to figure D.1), multiplied by a correction coefficient of 0.9 (recommended by [7]) to finally obtain the 3D CL,max. Moreover, a wing loading and thrust to weight ratio of respectively 65.7477 lbs/ ft2 and 0.1 are chosen, corresponding to the 5 OPTIMIZATION 56
mid-cruise aircraft mass and 80% of the available thrust.
6 180 Max turning rate 70 5 150 fmax
60
4 120 50
3 90 40
nmax [degree]
f 30
2 60 Turning radius [1000 ft]
20
Load factor [−] and turning rate [degree/s] 1 30
10
0 0 150 200 250 300 350 400 450 500 550 0 Airspeed V [knots] 150 200 250 300 350 400 450 500 550 Airspeed V [knots] (a) Load factor (solid line), turning rate (doted) and turning radius (green) evolution. (b) Aircraft bank angle φ evolution.
Figure 5.14: Turning aircraft performance with respect to airspeed V at cruise altitude, 80% of the thrust available and mid- cruise mass.
The first comment about figure 5.14 is that none of the upper limits defined previously are reached. In addition, the
airspeed cruise is about 490 knots. Considering the increase of the turning radius and the subsequent additional fuel mass, it
is clearly recommended to slow down.
5.3.4.3 6 seats turning performance
The following figure represents quite well the small variation occurring in such a configuration. 5 OPTIMIZATION 57
50
45 6 seats 8 seats 40
35
30
25
20
Turning radius [1000 ft] 15
10
5
0 150 200 250 300 350 400 450 500 550 Airspeed V [knots]
Figure 5.15: Zoom on the turning radius evolution for 8 and 6 seats configuration at cruise altitude, 80% of the thrust available and mid-cruise mass.
5.3.5 Landing
Following the RFP and FAR requirements, several conditions must be taken into account:
• a 50ft obstacle to clear at the approach,
• an approach speed Vappr ≥ 1.3 Vstall,
• an approach speed VTD ≥ 1.15 Vstall,
• a safety additional landing distance of 2/3 of the actual one,
• the ability to land under 3,600 ft at typical landing mass.
A basic assumption to consider is a zero thrust flying phase seeing that actually, it is performed at idling-engine rating.
5.3.5.1 Method
Instead of a traditional method in 3 segments (approach distance, flare distance and ground roll), it was decided, advised by ref. [16], to merge the 2 first segments. This initiative rests on the fact that typical civil aircraft descent rate at touchdown is between 12 and 22 ft/s. The second segment is a reminder of the segment E of the take-off section.
5.3.5.2 Results
• Segment A: 5 OPTIMIZATION 58
Parameters Values Comments
◦ Flaps setting 40 allowing a CLmax of 2.69 (further details in section 3.2.3)
Wland 12,888 lbs equal to 70.5% of MTOW, descent fuel consumption accounted W/S 55.5 lbs/ ft2 quite high since the wing surface is only 232 ft2 T/W 0 assumption
Vstall 135.5 ft/s
Vappr 176.2 ft/s taken as Vappr = 1.3 Vstall
VTD 155.8 ft/s VTD = 1.15 Vstall Elapsed time 2 s for actual brakes action and 4 s according to an advised descent rate between V and 6 s appr of 12.5 ft/s VTD
xA 913 ft
Table 5.3: Landing parameters for segment A.
• Segment B:
Parameters Values Comments Braking-friction 0.4 common value for business jets coefficient µB
CL 0.5 see section 5.3.1.2
CD/CL 0.1 see section 5.3.1.2
xB 1,043 ft braking speed VB ' VTD
Table 5.4: Landing parameters for segment B.
• Final computation of the LFL:
With the safety factor, the landing field length becomes, LFL = 5/3 (xA + xB) = 3260 ft. This value is well below the market requirements.
• Remark:
– Commonly, the LFL said as a similar length as BFL which is not really the case here.
– No that the touchdown speed is known, an advised (by[17]) thing to is to evaluate the minimum speed at which
hydroplaning begins as follow:
p Vmin hydro [knots] = 9 main tire pressure [psi] . (5.30)
The corresponding pressure is 352 psi, leading to Vmin hydro = 162 knots or 273 ft/s. The risk of hydroplaning is thus clearly eliminated. 5 OPTIMIZATION 59
• Graphical results
Landing field length - 3260 ft.
Approach/flare (28%) Braking distance (32%) Safety distance (40%) 40° flaps config.
50 ft.
VB V0
Vappr VTD
= 1.3 Vstall = 1.15 Vstall
Figure 5.16: Summary of results related to the landing field length computation.
Seeing that landing schedule, one can find a approach angle of 3.1◦, matching the statistical interval ranging from 2.5
and 3.5◦
• Emergency landing Although this was not in the statement, it would seem interesting to consider the landing in the
critical situation of an engine failure during take-off (after V1). Indeed, the landing analysis actually ranges from the take-off value to about 85% of the MTOW to address this case.
◦ In the present case, an emergency flaps deflection of 60 is allowed so much so that the CLmax increases to 2.79. Without considering the safety additional landing distance, the business jet is able to land under the prescribed distance until
88% of the MTOW.
Such a results lead to think that the pilots has to dump fuel but this solution is clearly not allowed. That is why, in further
steps of the aircraft design, use of device like slats or even spoilers,as part of solution has to be explored. Moreover,
a new engines could also help to reduce the LFL by the adoption of thrust reversers. As already discussed in section
3.3.1 PW535-A, very similar to the PW535-E, is certified for a thrust reverser employment. A careful study on the
asymmetric resulting drag will has to be carried out however.
5.3.5.3 6 seats landing performance
The RFP specifications does not differ from 8 seats one but it is important to notice that the FAR does not impose a safety additional landing distance of 2/3 of the actual one. As main results, it is obtained the LFL = 2,765 ft and that it is able to land on a 3,600 ft runway until 99.5% of the MTOW. 5 OPTIMIZATION 60
5.3.6 Payload-Range Diagram
5.3.6.1 Without correction
The RFP stated a minimum range of 2,500 nmi (for 4 pax/1 pilot), but it has been decided that exceed the expectations, and to be able to carry the full number of passengers throughout a typical east-to-west coast journey (∼ 2500 nm).
Accordingly, the fuel tank design depends on that choice. Therefore, it is impossible to add fuel in the tank past the Full payload & fuel point. Average payload-range diagram is composed of 3 distinct phases:
1. Full Payload, fuel is added to increase the range.
2. MTOW is reached, Fuel is added as payload is removed.
3. Fuel tank capacity is reached, removing payload is the only thing to do to increase the range.
Due to the design choice, Phases 2 will be thus absent from the figure. After computations, a range of up to 3,066 nmi could theoretically be reached with a payload composed of the pilot alone, and full fuel capacity. For the 6 seats, this theoretical maximum range is pretty much the same since the fuel thank is slightly reduced.
5.3.6.2 Tranair and drag study inclusion
Mentioned values and consecutive explanations are only related to the 8 seats configuration but, as already said, there is no apparent reason justifying that 6 seats goes worst. Besides, previous performance parts tend to confirm the opposite.
According to Tranair simulations (see section 5.2.2.2), the CL/CD ratio at 0deg (angle stated in the stability part for cruise conditions) of angle of attack is increasing from roughly 12 (theoretical) to 16. A piece of explanation about such an optimistic result lays on the fact that the simulation manage only inviscid flow. Accordingly, it is seems quite dangerous to trust it too much and re-compute the performance part with that input values.
According to the drag study (previous section 5.2.3), the cruise drag is evaluated to 1,348 lbf. In other words, 72% of the available thrust will be devoted to balance the drag. However, it is important to remind that the drag study does not explore the transonic case. Additional drag due to the aerodynamic shocks has thus to be taken into account. One can thus assume that the non-assessed additional drag is compensated by the non-fuel-savings review. A re-assessment of the payload-range diagram is performed in following figure. An important fact to emphasize is that the CL/CD in cruise drops from 12 to 10 as detailed by 5.15.
A large shifting of the maximum range is noticeable. Nonetheless, at 2,527 nmi is located the particular point where the payload is identified to 1 pilot, 4 passengers and corresponding luggage. In other words, the actual design point of the 5 OPTIMIZATION 61
×10 4 2 Full Payload & Fuel
Min Payload / Max Fuel 1.5
Design Point (1 pilot/4 pax) 1 Empty Weight Weight (lbs)
0.5
0 0 500 1000 1500 2000 2500 Range (nautical miles)
Additional fuel Fuel reserve Payload Max. empty weight
Figure 5.17: Payload range diagram for 8 seats configuration with corrected drag. requirements. In the first edited diagram, its associated range was ranging up to 3,127 nmi. The chosen approach of imposing the MTOW as design point in first time is thus validated.
5.4 Aircraft structure
5.4.1 Flight envelope
The primary purpose of the structure is to transmit and resist loads, while providing an aerodynamic shapes to the aircraft and protect the passengers and payload. The loadings are numerous: aerodynamic loadings, thrust, weight and inertia loading, pressurization cycles, shocks (for example during the touchdown at landing). The fact that these loadings vary considerably depending on the flight configuration: parameters such as velocity, angle of attack, altitude, maneuver, gust, ... should be considered. Therefore there is a huge number of different flight configurations for which the structure loading should be characterized. A convenient way to avoid computing all these configurations is to represent the global loading of the structure using the load factor n defined as the resulting aerodynamic loads perpendicular to the longitudinal axis of the aircraft over the weight. When flying in steady flight, it then comes that n = 1. Other way, it is
L n = (5.31) W
Accounting on the placard diagram, two important design velocities are defined in Aircraft Structures for engineering 5 OPTIMIZATION 62 students ([18]):
• The design cruise Mach, defined as the Mach number obtained at maximum engine available thrust. It can be estimated
as : MC = 1.06·Mcruise.
• The design dive Mach, defined as the maximal Mach number that the plane can reach. The FAR define it as the minimum
◦ between 1.25·MC and the Mach number obtained after a 20 second dive at 7.5 and a 1.5g pullout. This Mach number
can be quantified as 1.07·MC.
Therefore the corresponding design velocities VC and VD can be determined, with VD being the minimum between the dive velocity (estimated as 1.15·VC and the velocity corresponding to MD. In the following developments, the velocities are expressed in equivalent velocities at sea level in order to alleviate the variation of density with the altitude: r ρ Ve = Vtrue , (5.32) ρ0
ρ0 being the density of the air at sea level. The envelopes computed afterwards are therefore valid for an altitude range around the design cruise altitude. For out of the range altitudes, the variation of density and speed of sound modify the compressibility effects, and the airspeed estimation is no longer accurate.
5.4.1.1 Manoeuver envelope
As stated in the Code of Federal Regulations concerning transport category airplanes, values for extreme load factors can be estimated depending on the weight and aircraft type. For a light (under 50,000 pounds) civilian aircraft, extreme loads factor are:
• nmin = −1.8 [−],
24,000 • n = minimum between 2.1 + and 3.8, the weight being expressed in pounds. This lead to n = max W + 10,000 max 3.04 [−].
These extreme load factors are defined for the two design velocities VC and VD. But they may be no longer relevant if the plane fly slower, since a pullout is limited by the lift the plane can withstand before stalling. Hence at lower speeds there is a dependency between the maximal load factor and the velocity. Since the load factor in flight is defined by Eq. 5.31, the maximum load factor that the plane can reach can be expressed in terms of equivalent velocity and lift coefficient:
ρ ·V 2 ·S·C n = 0 e L,flaps up , (5.33) stall 2W with CL,flaps up the maximum lift coefficient of the plane with the flaps up, calculated in the previous section. This equation define a stall line which intersections provide critical information: The intersection between the stall line and nmax gives the 5 OPTIMIZATION 63 maximal velocity at which a maximal deflection of the controls is authorized. On the other hand the intersection of the stall line with n = 1 yields the velocity under which the plane cannot achieve a loading such as L = W, i.e. the stall velocity in cruise.
Considering a negative load factor, a corresponding pulldown stall line can be constructed. Note that at high velocities, only a pullout is meaningfull as the aircraft considered is not a fighter nor an acrobatic plane. Other stall lines can be considered using the lift coefficients corresponding to the landing or takeoff flaps configurations. The deflection of the flaps is limited by the airspeed as they may reach their structural limit and be damaged. Therefore a limit velocity for the deflection of the flaps is defined: VF .
5
4
n 3 max
2
1 Load factor [-]
0
-1
n Vs1 VF min VC VD -2 0 50 100 150 200 250 300 350 Equivalent airspeed [knots]
Figure 5.18: Maneuver envelope of the plane.
There entire maneuver envelope is now constructed as shown in Fig. 5.18 defining the combinations of airspeed and loading that the pilot may reach during the flight by deflecting the controls.
5.4.1.2 Gust enveloppe
So far only intentional loadings have been considered: on one hand the structural loading is limited by the stall, on the other hand one may imagine that the board computer will prevent the pilot to exceed the extreme load factor defined for the plane.
But up to now the aircraft was assumed to fly in a perfectly steady flow, which is usually not the case. Actually wind speed fluctuations (gust) are very frequent and the structure must be able to withstand the increase of structural loading they produce. The gust velocities encountered vary depending on the altitude and plane velocity, and are therefore estimated from statistical data.
The analysis of the gust effect on the loading can be conducted through the calculation of the variation of aerodynamic 5 OPTIMIZATION 64 force it leads to. Considering a sudden vertical gust U, and assuming that the plane keeps temporary the same velocity V and
U attitude α0, the angle of attack becomes: α = α0 + ∆α ' α0 V . Therefore, according to Aircraft Structures for engineering students ([18]), the increase of lift due to the gust can be quantified as
2 ρ ·V ·S·CL, , ·∆α ρ ·V ·S·CL, , ·U ∆L = α plane ' α plane (5.34) 2 2
with CL,α,plane the variation of the lift coefficient of the whole plane with respect to the angle of attack, as computed in ... . Expressing the variation of load factor due to gust in terms of equivalent airspeed provides:
ρ ·Ve ·S·CL, , ·Ue ∆n ' 0 α plane (5.35) 2W
In practice the gust is not sudden and the aircraft may react to it before it reach its maximal value, therefore several effects participate to decrease the effect of the gust:
• The plane develops a vertical velocity due to the increase of lift, thus reducing the severity of the gust.
• The increase of lift will induce a pitching moment that will affect the loading.
• The structure may deform under loading therefore reducing the effect of the gust.
0.88µ 2W All these phenomena can be represented by a gust alleviation factor F = = 0.84 < 1, µ = = 5.3 + µ ρ ·CL,α,plane ·cmean ·g·S 127.8 being an evaluation of the airplane weight ratio. F is then introduced in Eq. 5.35.
Considering a linear approximation of the gust loading with respect to the velocity, the gust envelope is entirely determined as shown in Fig. 5.19.
The gust envelope provides nlimit = 3 [−] ,the maximum structural loading expected during service according to the gust statistics. Under this load factor the structure is expected not to undergo permanent deformations. From there the ultimate load factor is defined as nultimate = 1.5·nlimit = 4.5 [−] which is the design load factor of the structure.
5.4.2 Aerodynamic loading
The global loading of the structure is now determined for different airspeeds. To precise the loading analysis, the aerodynamic forces can be computed at the main points of the envelope B,C and D shown in Fig. 5.19. Note that since positive and negative loading are similar but in intensity, and in the scope of defining maximal aerodynamic loads to design the structure, the study of aerodynamic loading is limited to the positive loading part. For those extreme points, the plane should nevertheless be able to perform a maneuver without its structure being overloaded. To represent this potential additional loading, a pitch acceleration θ¨ = 60 [◦/s2] and a yaw angle ψ = 20 [◦] shall be considered for these points. Those maneuver angle have side effects, particularly the yaw angle: it induces an additional pitching moment, a torque due to an asymmetric slipstream on the tailplane, and a fin loading. These last two consequences are difficult to evaluate (especially for a "V" tail): the fuselage 5 OPTIMIZATION 65
5
nultimate
4
nlimit B C D 3
2
1 Load factor [-]
0
-1
V Vs1 B VC VD -2 0 50 100 150 200 250 300 350 Equivalent airspeed [knots]
Figure 5.19: Gust envelope of the plane. may partially blanket the tailplane and aggravate the lift variation due to the slip stream, therefore increasing the torque and
fin loading induced. But as the "V" tail partially stands out of the way of the yawing fuselage, it may not be fully affected.
These effects might be evaluated through a CFD model of the plane, but in the framework of a preliminary assessment of the aerodynamic loads they will not be considered.
The modification of pitching moment coefficient can be computed as stated in [18]:
CM,modified = CM − 0.0015ψ. (5.36)
The loads on the fuselage can now be calculated so that values of shear force, bending moment and torque acting on the structure can be determined. These loads are computed by balancing out the loading contributions and the induced torques, as they are represented in Fig.5.20.
Pi Ll
T Fuselage incidence CG Wing D + D incidence W body
ni W
Figure 5.20: Representation of the loading of the plane for the different points i of the flight envelope. 5 OPTIMIZATION 66
The bank angle of the plane α − iwing, where α stands for the angle of attack of the wing and iwing is the inclination angle of the wing with respect to the fuselage axis, is computed from the value of lift coefficient needed to achieve the load factor from each point of the flight envelope. An iterative process is therefore required for the points which are not located on a stall line.
The tailplane and wing aerodynamic loading, respectively noted Pi and Li, are computed for each main point of the flight envelope as shown in the Tab. 5.21.
Points of the en- ni α P [lbf] L [lbf] Mw [lbf · ft] velope B 2.99 8.4 3948.75 49873.15 -48906 C 3.04 9.3 4780.40 51419.85 -69575 D 3.04 8.3 5414.52 52081.67 -79657
Figure 5.21: Aerodynamic loads and bank angle corresponding to each main points of the flight envelope.
Note that for the following calculations a big approximation is made: the aerodynamic forces of drag and thrust are assumed not to vary from their value in cruise. This is indeed a raw approximation of the reality since the airspeed and angle of attack change (thus CL and CD, therefore the drag should vary !), but it is assumed that the increase of CD partly counterweight the lower airspeed. Indeed the difference should still be big, but since the arm of application of these forces is small, in the end these variations loose importance. Computed both with and without this approximation for point B of the envelope, the solutions are:
Case L [lbf ]P[lbf ] T and D constant 49873.15 3948.75 T and D recomputed 49696.76 4033.24
Figure 5.22: Comparison of the results with and without the simplifying assumptions.
Between these two approach, the value of P vary of 2% while the value of L vary of 0.3%. In the view of a preliminary design of the structure, these approximations are acceptable with regard to the simplifications they allows.
5.4.3 Structural loading
The different loadings applied on the structure are now available. In order to design the structure of the aircraft, the forces and moments that the loadings cause shall be computed at the points of interest. Considering a preliminary design of the structure, the structural loading are evaluated in critical locations: at the wing root and at the fuselage section directly aft the wing.
Using the mechanics of materials, the force and moments generated in these sections are put in equilibrium with
• the self weights multiplied by the load factor ni corresponding to the point i of the flight envelope,
• the aerodynamic loads computed in the previous section,
• the forces applied and the pressure differential (due to pressurization notably). 5 OPTIMIZATION 67
5.4.3.1 Fuselage section
The self weight of the fuselage part aft the section of interest is calculated using the weights estimations conducted in section
3.6, taking account of the contributions of all the components located in this part as shown in Fig. 5.23.
z
x y
Wtail
Wengine
Wluggage W fuel Part Wfuselage Part
Figure 5.23: Representation of the rear fuselage self weight components.
The weight and center of gravity of this fuselage part are thus:
Wtot,Part = 4946 [lb], (5.37) xCGtot,Part = 21.5 [ft].
Defining the shearing force S fi = ni ·Wtot,Part and bending moment BMi = ni · cos(αi − iwing)· (xCG,tot,Part − xsection) ·Wtot,Part in the section due to the self weight for the point i of the envelope, the equilib- rium with the loads is enforced by:
Tz,i = (S f ,i − Pi)cos(αi − iwing) Nx,i = Tcruise (5.38) My,i = BMi + Pi d cos(αi − iwing) − Tcruise · zCG,engine − zCG,mid cruise in the local axis of the fuselage section considered (as depicted in Fig. 5.23), with d the distance between the aerodynamic center of the tail and the section of fuselage considered.
5.4.3.2 Wing root
Concerning the wing, its self weight and center of gravity are easily defined from the weight estimations previously conducted, giving the location of the center of gravity along y:
y 1 = yAC = 9.37 [ft] and the weight of one wing: W1 = 1170 [lb]. (5.39) CG, 2 wing 2 wing 5 OPTIMIZATION 68
The pressure distribution on an airfoil can be modelled by a constant moment, a lift and a drag applied at the aerodynamic center of the airfoil. Concerning the 3D wing, the lift distribution is far from being uniform and depends on many geometrical parameters. However, the loading of the whole wing can be modeled by a total lift, drag and moment applied at the aerody- namic center of the wing as computed in ... . Since the structural loading considered results from one half of the wing, the aerodynamic forces are halved. The shear force and bending moment due to the self weight are written S f ,i = ni ·Wtot,wing and (yAC − ydist) BMi = ni · cos(Dihedral)· . Therefore the equilibrium equations of the stresses in the concerned cos(Dihedral)·Wtot,Wing wing section yields:
D Tx,i = · cos(αi) 2 Lwing,i Tz,i = (S f ,i − )· cos(Dihedral) 2 Lwing,i Mx,i = BMwing,i − (yAC − ydist)· · cos(Dihedral) (5.40) 2 My,i = Mwing,i Mz,i = (yAC − ydist)·D
The different structural loads at the wing root and at the fuselage section for all the points of the flight envelope are represented in respectively in Tab. 5.6 and Tab. 5.5.
BCD
S f ,i [lbf] 1,511 1,535 1,535 BMi [lbf· ft] 15,693 15,903 15,950 Tz,i [lbf] -2,411 -3,200 -3,836 My,i [lbf· ft] 80,452 94,389 105,305
Table 5.5: Structural loading at the fuselage section for the different points of the flight envelope.
BCD
S fi [lbf] 952 968 968 BMi [lbf· ft] 8,313 8,447 8,447 Tx,i [lbf] 701 699 701 Tz,i [lbf] -23,892 -24,647 -24,977 Mx,i [lbf· ft] -224,608 -231,697 -234,788 My,i [lbf· ft] -24,453 -34,787 -39,828 Mz,i [lbf· ft] 6,650 6,650 6,650
Table 5.6: Structural loading at the wing root for the different points of the flight envelope.
5.4.4 Materials selection
The materials selection can become a really tough task when it comes to aeronautics. Indeed many properties of the materials have to be taken into account to provide both a sufficient level of performance for the aircraft and safety for passengers. For 5 OPTIMIZATION 69 the design of an aircraft, many families ans sub-families of materials can be selected, depending on the aircraft’s specifications and material use.
Aircraft design and material selection require a deeper prospection in the following mechanical parameters:
• hardness;
• yield strength;
• density;
• ductility;
• toughness;
• fatigue strength at 107 cycles.
The strengths of non metallic materials are essentially the very low density and the possibility to produce the desire structure in one piece. Carbon composite, like carbon reinforced fibers for instance, may be attractive in the sense that this material exhibit a competitive yield strength to density ratio and Young modulus to density ratio. However, brittle fracture in this kind of material represents a challenge for cracks detection. This becomes even more problematic for riveting process.
On the other hand, the new processes used to manufacture these structures are still on research and actual ones can take a long time.
In the interest of safety and simplicity, it will be preferred to use metallic materials for which manufacturing processes are well-known and mechanical properties of an excellent level. To prevent from using too high density materials, it is common to use light alloy category, which consists mainly in the use of aluminum with alloying element, giving it mechanical properties as good as metallic materials. The aircraft will be designed using aluminum alloy as the main material. For the structure and the skin of the aircraft (the fuselage, the outer part of the belly and the wing), wrought aluminum alloy 7075 T6 has been selected. Wrought aluminium 5052-H32 is one of the most suitable material to design the inner part of the belly due to its excellent corrosion resistance and workability. The properties of these materials can be given using the software CES. These are shown in App. F.
5.4.5 Structure preliminary design
In order to perform a preliminary design of the structure, the complex structure is idealized into a simpler mechanical model under particular assumptions on the stress distribution, as defined in Aircraft Structures for engineering students ([18]).
Since the stringers are expected to have small cross sectional dimensions compared with the complete section of the wing or fuselage, the variation in stress over this cross-section is small. Assuming that the direct stresses are constant over the stringers cross-section, a good working hypothesis is therefore to idealize the layout of stringers and skin into a combination of direct stress bearing booms13 and shear stress bearing skin. This model actually loose the actual shear distribution as it only
13Concentrations of area. 5 OPTIMIZATION 70 depicts the average of the shear flow between two booms. Moreover, the expressions for direct stresses used in the following sections are based on the Euler-Bernoulli assumption that the plane sections remain plane after bending. This is actually not the case as the bending moments Mx,My and Mz are not constant along the fuselage nor the wing. In addition, these are considered to be of uniform cross-sections, which is false considering the taper of the wing and the variation of the fuselage cross section in the aft. Nevertheless, these assumptions provide a good first estimate of the structure design, although several aspects of stress distribution are missed. Further studies using more sophisticated models should be conducted in order to take account of these effects, eventually leading to a lighter and more effective structure design. However from the perspective of providing a preliminary design of the structure and after acknowledging the simplifications made throughout the loads computation, the following designs are limited to the basic model.
The sizing of the structure elements is based on a elastic design approach: as the material should deform in the elastic region the yield strength σyield and the tensile strength τstrength are taken as reference values. Then these are multiplied by a safety factor s = 1.5, providing the maximum value that the stresses shall reach:
σ 73 [ksi] σ = yield = = 48.5 [ksi] (5.41) max s 1.5 τ 48 [ksi] τ = strength = = 32 [ksi] (5.42) max s 1.5 defining the maximum stresses allowable in the elements.
5.4.5.1 Fuselage section
The fuselage section is circular, therefore a simple stringers configuration is to arrange them symmetrically and equally spaced.
For the sake of a quick preliminary design, all the stringers have the same cross-sectional area, which remains constant along the longitudinal axis (no sectional variation with respect to the x axis). As shown in Fig. 5.24, the stringers are replaced by booms located on mid-line of the skin. The number of stringers placed on the section is not to be chosen randomly as it is linked to the ability of the structure to avoid unstable failure such as buckling. Indeed, closely spaced stringers associated to the frames better maintain the fuselage shape. Moreover, ribs and stringers spacing play a major role in achieving a lighter and more efficient structure. As stated in Effect of Ribs and Stringer Spacings on the Weight of Aircraft Structure for Aluminum
Material ([19]), stringers spacing of 5.9 [in] is found to be stabilizing for the weight of the structure, i.e more smaller stringers would not make the structure weight less. This correspond to 36 stringers disposed around the fuselage section. The frames are mainly required to maintain the fuselage shape, therefore they are nominal in size and spaced 11.8 [in] apart.
As the structure must resist to the loading at all the points of the flight envelope, the worst case of loading have to be considered. The maximal value of bending moment is reached during dive, with My = 105,305 [lbf · ft].
The direct stresses in the fuselage section are induced by the moment My and the normal reaction force Nx. As the second 5 OPTIMIZATION 71
z z
y y D fus
Figure 5.24: Representation of the fuselage cross section modelling.
moment of area can be expressed in terms of the booms area B: Iyy = Iterm ·B·Dfus, the stresses are written:
My Mz Nx σxx = z − y + Iyy Izz B My Nx = z + (5.43) Iyy B My Nx = 2 z + . Iterm ·B·Dfus B
Considering booms of identical area, it can be deducted from Eq. 5.43 that the higher stresses are located on upper and lower
2 booms. Therefore, inverting the equation for z = Dfus and σxx = σmax provides the required booms area: B = 0.0736 [in ].
The skin must resist to the shear flow due to the shear load Tz. As for the direct stresses, the worst case is considered for the design loading: the design shear load is reached during the dive, with Tz = 3,836 [lbf]. As stated in Aircraft Structures for engineering students [18], the average shear flow between two booms i and i + 1 is:
i+1 i Tz Ty q − q = − Bizi − − Biyi Iyy Izz Tz = − Bizi (5.44) Iyy Tz = − 2 zi ItermDfus
From this equation it is possible to express by recurrence the shear flows on each panel with respect to the closure shear
flow value:
i+1 Tz q = qs,0 − 2 (5.45) ItermDfus
Since the loading Tz is applied through the shear center of the cross-section of the fuselage, no torque is induced by the shear force and the shear flow is symmetrical. Therefore the shear flows on the lower panels of the fuselage are opposite: 5 OPTIMIZATION 72 q6−7 = −q7−8. Taking the shear flow q1−2 as reference, it therefore be expressed as:
( 6 z + 7 z ) 1−2 Tz ∑ j=2 j ∑ j=2 j qs,0 = q = 2 (5.46) Iterm/Dfus 2
From Eq. 5.45 the average shear flow on each panel is therefore determined. The stress induced by this shear flow is
q τ = i (5.47) i t where t is the skin thickness. Considering that t is uniform, it is will be designed accordingly to the maximum shear stress.
The corresponding shear flow shall be named qmax. Another effect inducing shear stress in the skin is the pressure differential due to the pressurization of the fuselage. On this point the FAR (part 25, Section 841 - Pressurized cabins) stipulates that "[...] the cabin must be equipped to provide a cabin pressure altitude of not more than 8,000 f eet at the maximum operating altitude of the airplane under normal operating conditions." Hence since the maximal altitude that the aircraft is required to reach is 45,000 [ft], the pressure differential acting on the panel of the fuselage is ∆P = P45,000 − P8,000. Since the skin is expected to be very thin in comparison with the fuselage cross-section, the thin-walled pressure vessels formula can be used to quantify the additional stress induced by the pressure: ∆PD σ = fus . (5.48) θ 2t
Therefore, expressing the maximum distortion energy criterion:
r q 2 σ = σ 2 + 3 max (5.49) von Mises θ t
This failure stress must be smaller than τmax. Thus Eq. 5.47 and 5.48) provide an expression of the skin thickness:
r D 2 1 ∆ fuselage 2 t = + 3qmax ; (5.50) τmax 2
Consequently, the minimum skin thickness is t = 0.01181 [in] = 11.81 [th]. But since the skin is fastened to the frames and stringers, it must support the the rivets. Moreover a thicker skin helps avoid deformation during maintenance operation for example, as concentrated loads might accidentally be applied on the skin. Therefore a standard thickness is chosen: t = 0.0393 [in] = 39.37 [th].
The simpleness of the model used to design the stringers area can be refined by taking account of the direct stresses actually carried by the skin. Each booms area is therefore increased by an area equivalent to the direct stress carrying capability of the adjacent skin panels. As T.H.G Megson states in ref. [18], providing that the stringers are spaced closely enough the skin between them can be approximated as flat. The direct stresses on the panel can thus be assumed to vary linearly between the values σi and σi+1 of the constraints in two consecutive booms. 5 OPTIMIZATION 73
b b z t z
y y
x x
A1 A2
σ2 σ2 1 xx 1 xx σ xx σ xx
Figure 5.25: Idealization of the direct stress carrying panel.
As presented in Fig. 5.25 the panels can be idealized by a skin without thickness and two booms carrying direct stresses of area A1 and A2:
2 tb σxx A1 = 2 + 1 (5.51) 6 σxx
1 tb σxx A2 = 2 + 2 (5.52) 6 σxx
Each boom is therefore affected by the contribution of the booms of the two contiguous panels. The formula for the stringer area Si can therefore be written:
i+1 i−1 tb σxx tb σxx Si = B − 2 + i − 2 + i 6 σxx 6 σxx (5.53) i+1 tb σxx = B − 2 + i 3 σxx
In practice it is not convenient that all the stringers have different sizes: there are more different pieces to manufacture and there is a risk that the assembly workers make mistakes between two stringers of slightly different cross section. Therefore each stringer should have the same cross-sectional area, determined using the boom that bear the most important stress.
2 Applying Eq. 5.53 generate an odd result: the required stringer area is very small, with Sstringer = 0.0077 [in ]. This is partly due to the fact that the skin thickness is oversized; the panels are therefore able to carry an important part of the direct stresses. Considering the simplifying hypothesis on the loading and particularly the fact that some loads are not applied since there is no way other than numerical to estimate them quantitatively, it is safer to choose the most conservative option.
Therefore the stringers area is determined from the solution of Eq. 5.43. Practically a standard size of stringers is chosen:
S = 0.0775 [in2]. 5 OPTIMIZATION 74
5.4.5.2 Wing root
The wing root structure design is similar to the fuselage section design, although the cross section is not symmetrical and is composed of several cells delimited by spars. The chosen configuration is typical for a wing design with 3 spars defining 4 cells, even though the last one in not considered in the structural design as its purpose is only aerodynamic. It therefore does not bear the structural loading.
4 5 6 7 1 2 3 8 9 10
11 20 12 13 14 15 16 17 18 19
Figure 5.26: Representation of the wing cross-section modelling.
The same considerations than for the fuselage are to be taken into account concerning the spacing of the stringers and frames; therefore the number of stringers is chosen to be 20, resulting in a spacing of 6.2 [in]. The frames are spaced 11.8
[in] apart as in the previous section. Considering the worst loading case among the different points of the flight enveloppe, the bending moments considered in the following are Mx = -234788 [lbf·ft] and Mz = 6650 [lbf·ft]. The unsymmetrical section imply that the expressions for the stresses resulting from bending have to be adapted. This way the centroid of the cross section has not to be determined and computation time is saved. From [18], the direct stresses are written Mx(Izzz − Ixzx) Mz(Ixxx − Ixzz) σy = 2 + 2 . (5.54) IxxIzz − Ixz IxxIzz − Ixz
Taking out the booms area from these equations, the forces resulting from the bending moments are therefore computed at the different booms locations. It follows that due to the loading, the maximal force is reached at the boom 5 such as shown in Fig.
2 5.26. Therefore enforcing σy,max = σmax provides the required boom area for the wing: Bwing = 0.6745 [in ]. All the stringers are considered to be of the same cross sectional area.
The wing skin is designed with the same methodology as for the fuselage: on each cell the average shear flow between two booms is determined by ([18]):
i+1 i SxIxx − SzIxz SzIzz − SxIxz q − q = − 2 Bixi − 2 Bizi, (5.55) IxxIzz − Ixz IxxIzz − Ixz
th the shear flow on each panel can therefore be expressed in terms of the closure shear flow qs,0,n of the n cell. The unknowns are therefore the values of the shear flow at each of the cuts, i.e qs,0,I,qs,0,II and qs,0,III, and the twist rate. Under the assumption of an undistorted cross-section, the twist rate is the same for each cell therefore the twist rate compatibility provide 3 equations.
A fourth equation is provided by balancing out the moments resulting from the shear flows from the individual cells and the moments of the externally applied loads about the same point:
3 3 I 3 Txη0 − Tzξ0 = ∑ Mq,R = ∑ qp0ds + ∑ 2ARqs,0,R (5.56) R=1 R=1 R R=1 5 OPTIMIZATION 75 with
• η0 the distance between the line of action of Tx and the chosen moment center O,
• ξ0 the distance between the line of action of Tz and O,
• AR the area of the cell R.
Therefore the system of equations can be solved and the shear flow on each panel is determined accordingly. Enforcing the q linear elasticity everywhere on the skin of the wing leads to t = max,wing = 0.0315 [in] = 31.5 [th]. As this skin must support τmax the rivets, it takes the standard thickness of t = 39.37 [th].
As for the fuselage section, the fact that the skin actually carry direct stresses can be taken into account by Eq. 5.53 which
2 allow a reduction of the stringers cross sectional area: Swing = 0.474 [in ].
5.4.6 FEM Analysis - Preliminary results
In order to validate the proposed structural layout for both the fuselage and the wing, Finite Elements models have been developed using CATIA v5. In this model are included all the airframe’s elements such as the skin, the frames/ribs and the stringers/longerons to the detailed dimensions.
In order to save computation time, and as the element under study is exclusively the fuselage, we did not include the engine’s mounts nor the V-Tail in this model. Nevertheless, all the weights and aerodynamic loads explicated previously have been taken into account and integrated in the model.
Figure 5.27: Finite Elements Analysis of the Fuselage’s structure under nlimit. Deformation amplified 10 times.
As it can be seen in Fig. 5.27, the maximum Von-Mises stresses are located directly aft the wing. The max V.M. stress in the structure is 7.68 [ksi], which complies with the plasticity criterion (< Re = 52.1[ksi]). Using this model, some further refinements can be brought to remove matter (and weight) to the structure without altering its efficiency. In particular, the model’s frames were very thick (and support a very low constraint - they appear in blue).
The same type of Finite Elements model has been performed to validate the results of the section 5.4.5.2. This model leads to a max. Von Mises stress of 36.26 [ksi], which again, complies with the plasticity criterion ( Figure 5.28: Finite Elements Analysis of the Wing’s structure under nlimit. Deformation not amplified. 5.4.7 Further improvements Until now the design of the structure was narrowed to ensuring that it withstands the loads statically applied to it. This is no longer adequate if the structure have to be planned further than the preliminary design as several aspects of what an actual structure is are missed. These disregarded factors may be sorted in 3 domains: the modelling level, the structural complexity and the loading. As stated in section 5.4.2, some aerodynamic loads are hardly estimable without using CFD simulation which implementation time cost make that it is usually dedicated to more advanced design. Moreover, the structure should actually be designed to withstands its dynamic loading, accounting notably the vibrations during flight and the shocks during landing. To proceed further on this domain of the structure design, fatigue and crack propagation studies shall be conducted as the structure must be able to fail-safe, i.e maintain the plane airworthy until emergency landing even in the case of a structural failure. Another point is that the structure configuration considered in the preliminary design is simplistic and does not take account of some practical aspects: • The plane must include openings such as doors, • The structural parts must be assembled one to each other, • The mobile parts can not carry structural loads. This results in the necessity to design cutouts in the structure. As they interfere with the good loading transmission, they lead to a stress concentration. The structure at these locations should be designed to transfer the loads from skin, flanges and shear webs around the cut-out. This inevitably lead to an increased cost and weight of the structure. Even if the structure is considered in all its complexity, the modelling itself of the structural elements might be improved. The assumptions followed throughout the preliminary design were that the direct stresses and shear stresses are decoupled, therefore the structure could be modelled as an assembly of direct stress only carrying booms and shear only carrying skin. Ac- tually much more complex mechanical phenomena such as twisting and wrapping shall be taken into account for an advanced structure design. The easier way to do so is to create a Finite Element Model of the structure. 6 COSTS ANALYSIS 77 Finally thin skin structures are very prone to undergo unstable failure modes such as buckling. This critical phenomenon in aircraft structures consists in a coupling of axial loading and bending mode, and shall be prevented through an adequate design. 6 Costs Analysis So far, a 2 aircrafts family of Business Jets has been designed and studied. As a design optimum has been reached, it is now necessary to assess the competitiveness of this design on the current Business Jets’ market, and more specifically, to define its price tag to the final customer, as well as the needs in terms of investment, and the expected returns on investments to the shareholders. In what follows, different categories of costs for the whole life-cycle of each aircraft will be assessed: • The RDT & E (Research, Development, Test and Evaluation) costs include all the costs related to the design, the tooling design, the prototyping, and the certification phases. In parallel, the management-related costs (H.R., Financials, Public Relations) must also be taken into account, although it represents a smaller cost. These costs are mainly fixed costs to the company. • Production costs include labor and material resources that will be used to actually build the aircrafts. It also includes the company running costs (management, sales teams,...). Those costs are obviously variable costs, as the overall costs of the company will increase with respect to the number of units produced. • Operations & Maintenance (O&M) costs cover all the indirect costs to run the aircraft. Fuel, Oil, crew members, maintenance, and insurance must be assessed in this group. Even if O&M costs are to be charged directly to the customer, they must be as low as possible to enforce the competitiveness of the Business Jet on the market. Indeed, low Operations and Maintenance costs would justify a higher price tag for the jet at delivery. Two types of Operations & Maintenance costs are to be taken into account: – Estimated hourly variable costs: this estimate represents, all charges included, the cost of operation per hour of flight (fuel, maintenance, engine/APU allowance, crew member’s charges,...). – Estimated annual fixed costs: this estimate represents, all charges included, the fixed costs relative to the business jet (hangar rental, pilot’s salary, yearly insurance & liability costs). 6.1 RAND DAPCA-IV Method - Eastlake Model As stated in ref. [5], developing a full-scale cost analysis would be highly time consuming, and would happen only with a very detailed model of the aircrafts, knowing all the procedure needed for production. At this stage, though, a statistical method provided by the RAND Corporation, called DAPCA (Development and Procurement Costs of Aircraft) can be used. 6 COSTS ANALYSIS 78 This method was originally developed in the military context, but its cost estimating relationships (in the following, it will be refered as CER) have been adapted for General Aviation, and then more specifically to the Executive Aircrafts field (see ref. [20] and [21]). This model provides sets of equations to assess the different fixed and variable costs relative to the design, the certification and the production of the aircrafts under study. In addition to the financial cost, this method provides an insight of the workload needed for certification in terms of time resources, and therefore helps us to plan a release date given the size of the company, or, as presented here, to adjust the company’s resources in order to ensure a given release date. 6.2 Effect of inflation on the costs The inflation represents the fluctuations of the money’s value over the years. Inflation forecasts must be taken into account for the cost analysis. As the 8 seats aicraft’s entry into service is planned in 2020 (2022 for the 6 seats model), the prices must be adjusted to match those years’s consumer price index (CPI) forecasts, as presented in Fig. 6.1. In addition to that, as in ref. [21], the CER’s coefficients have been adjusted to match 2012’s CPI, the forecasted data for 2020 and 2022 must be given as a function of 2012’s basis. That yields to 114.1 for 2020 and 119.6 for 2022. Consumer Price Index evolution & forecasts 350 Past 300 Projected 250 200 150 CPI (100 = 1983) 100 50 0 1960 1970 1980 1990 2000 2010 2020 2030 Years Figure 6.1: Consumer Price Index (CPI) over the years, and 10 years forecast (1983 = 100). Source, US Congress Budget Office (ref. [22]). 6.3 Selling Price Definition In order to determine the selling price of the different aircrafts, the Eastlake’s adaptation of the DAPCA-IV method is used. In the following, we will use a CPI factor of 1.141 for the 8 seats aircraft (release planned for 2020), and a CPI factor of 1.196 for the 6 seats aircraft (release planned for 2022). 6 COSTS ANALYSIS 79 6.4 Production rate The production rate of each business jet must be defined in accordance with the typical sales rate for this class of aircraft. As a basis of comparison Embraer’s best seller Phenom 300 is the most delivered Business Jet in this category, with an average of 76 deliveries per year since 2013 (Source: Ref. [23]). Therefore, it is realistic that for this range of activity, the designed family of aircrafts reaches a production rate of 60/year (5/month), which represents 300 aircrafts produced over a 5 years duration. 6.5 Certification Cost The certification cost is a fixed cost. It gathers all expenses related to the work on the aircraft’s model prior to the certification phase. More into details, it includes a great amount of engineering resources, as well as tooling and manufacturing costs, as well as prototyping. • Engineering man-hours The engineering resources are given by the following equation: 0.777 0.894 0.163 Heng = 4.86·Wairframe ·VH ·N ·Fcert ·FCF ·Fcomp ·Fpress = 2,940,713 man-hours (6.1) Where Wairframe = 5599.51 lbs is the weight of the aircraft’s airframe (fuselage, wings, tail and landing gear’s summed weight). VH = 490 KTAS is the maximum airspeed. N = 300 is the number of aircrafts to be produced in a 5 years period after the EIS (entry into service). Fcert = 1.15 represents the additional cost due to the FAA Title 14 Part 25 certification (1 is for 14 Part 23). FCF = 1 as the flaps system is simple (a complex flaps system is 1.03). Fcomp = 1 as no composite is used on the structure (the level of use of composite makes this factor as high as 2). Fpress = 1.03 as the aircraft is pressured (1 otherwise). As in average, an engineer works 40 hours a week, for 48 weeks per year, and that the aircraft EIS is planned for 2020, this project alone would require close to 510 engineers. • Tooling man-hours This amount represents the number of man-hours required to design and build the tools necessary in the aircraft’s prototyping (fixtures, jigs, molds, and so on). 0.777 0 0.263 Htool = 5.99·Wairframe ·VH .696·N ·Fcert ·Ftaper ·FCF ·Fcomp ·Fpress = 1,710,070 man-hours (6.2) Where Fcert = 1.05 represents the additional cost due to the FAA Title 14 Part 25 certification (1 is for 14 Part 23). Ftaper = 1 as the wing is tapered. FCF = 1 as the flaps system is simple (a complex flaps system is 1.02). Fcomp = 1 as 6 COSTS ANALYSIS 80 no composite is used on the structure (the level of use of composite makes this factor as high as 2). Fpress = 1.01 as the aircraft is pressured (1 otherwise). Under the same assumptions than before, an EIS for 2020 would mean the hiring of 300 people in charge of the production and tooling processes. • Management costs (Development support) This section covers the expenses of all the supporting profiles, such as management board, marketing, H.R. that will be associated with the new aircraft. This cost is estimated by Eq. 6.3. 0.63 1.3 Cdev = 95.24·Wairframe ·VH ·CPI2020 ·Fcert ·FCF ·Fcomp ·Fpress = 91,140,973 USD (6.3) Where CPI2020 = 1.141 represents the effect of inflation until the EIS. Fcert = 1.10 represents the additional cost due to the FAA Title 14 Part 25 certification (1 is for 14 Part 23). FCF = 1 as the flaps system is simple (a complex flaps system is 1.01). Fcomp = 1 as no composite is used on the structure (the level of use of composite makes this factor as high as 2). Fpress = 1.03 as the aircraft is pressured (1 otherwise). • Flight Test Operations The flight test operation costs are all the expenses due to the completion of development and the due to the certification flight-test program. 0.325 0.822 1.21 CFT = 2606.51·Wairframe ·VH ·NP ·CPI2020 ·Fcert = 84,077,529 USD (6.4) Where – Fcert = 1.10 represents the additional cost due to the FAA Title 14 Part 25 certification (1 is for 14 Part 23). – NP = 5 represents the number of prototypes needed for certification. • Engineering/tooling costs The engineering and tooling costs are given by Eq. 6.5 Ci = 2.0969·Hi ·Ri ·CPIi (6.5) Where the 2.0969 factor is an adjustment due to the inflation rate between 1982 (often given as a reference for consumer price indexes in the literature), and 2012 (current basis for the CPI). Hi is the required man-hours, Ri is the rate/hour (USD) and the CPIi factor depends of the EIS year. 6 COSTS ANALYSIS 81 The certification (fixed) costs for the aircrafts certification are presented in Tab. 6.1 and 6.2 for respectively the 8 seats and the 6 seats version of the Business Jet. The reader may notice a significant difference between the certification costs respectively of the two jets, although there’s no significant weight change in the airframe. This is due to the fact that the certification requirements are not the same (as a reminder, the 6 seats aircraft must comply with FAA Title 14 Part 23, while the 8 seats aircraft must comply with the FAA Title 14 Part 25, which is more restrictive). Furthermore, these results are computed as if the two aircraft were independents. However, the re-use criterion of 70% of the components from one jet to the other should reduce significantly the costs (especially the tooling and engineering expenses). As an approximation, the engineering costs could be evaluated to 3/4 of the theoretical ones (pessimistic approx.), and the tooling costs to 1/2 of the theoretical ones. As a result, the certification cost of the 6 seats aircraft would drop to about 394,720,372 USD. The difference in the costs is slightly reduced due to the difference of release dates (respectively 2022 and 2020): the inflation makes the 6 seats-aircraft’s price rise. Man-hours Rate/hour (USD) Cost (USD) Engineering 2,940,714 92 308,692,594 Development Support 91,140,974 Tooling 1,717,070 61 119,509,803 Flight Test Operations 84,077,529 Overall Cost to Certify 603,420,900 Table 6.1: Details of the certification costs for the 8 seats aircraft. Man-hours Rate/hour (USD) Th. Cost (USD) Est. Cost (USD) Engineering (th.) 2,431.957 92 267,593,140 200,694,855 Development Support 79,760,258 79,760,258 Tooling 1,555,249 61 113,464,718 56,732,359 Flight Test Operations 57,532,899 57,532,899 Overall Cost to Certify 518,351,016 394,720,372 Table 6.2: Details of the certification costs for the 6 seats aircraft. 6.6 Production Cost • Manufacturing man-hours This amount represents the number of man-hours required to actually build the aircraft, during the production 6 COSTS ANALYSIS 82 phase (after certification). 0.82 0.484 0.641 Hmanufacturing = 7.37·Wairframe ·VH ·N ·Fcert ·FCF ·Fcomp = 6,341,898 man-hours (6.6) Where Fcert = 1,05 represents the additional cost due to the FAA Title 14 Part 25 certification (1 is for 14 Part 23). FCF = 1 as the flaps system is simple (a complex flaps system is 1.02). Fcomp = 1 as no composite is used on the structure (the level of use of composite makes this factor as high as 2). As a result, still under the same assumptions, we deduce it would require about 660 technicians to build the 300 aircrafts objective on a 5 years term. • Manufacturing costs With an estimate of the manufacturing workload, the costs Cmanufacturing can be computed using Eq. 6.5. • Quality Control Throughout the assembly line, quality standards must be met in order to guarantee the maximum possible level of safety, as well as traceability of the parts. The costs related to quality may be computed using Eq. 6.7. CQC = 0.133·Cmanufacturing ·Fcert ·Fcomp (6.7) Where – Fcert = 1.50 represents the additional cost due to the FAA Title 14 Part 25 certification (1 is for 14 Part 23). – Fcomp = 1 as no composite is used on the structure (the level of use of composite makes this factor as high as 1.5). • Materials cost Here are regrouped the costs linked to the different materials used for the aircraft. 0.921 0.621 0.799 Cmat = 23.066·Wairframe ·VH ·N ·CPI2020 ·Fcert ·FCF ·Fpress (6.8) Where CPI2020 = 1.141 represents the effect of inflation until the EIS. Fcert = 1.15 represents the additional cost due to the FAA Title 14 Part 25 certification (1 is for 14 Part 23). FCF = 1 as the flaps system is simple (a complex flaps system is 1.02). Fpress = 1.01 as the aircraft is pressured (1 otherwise). • Cost of the landing gears The costs related to the retractable landing gears are already accounted for in the DAPCA-IV formulation. • Avionics It is hard to accurately predict the cost of the avionics. Following costs of Business Jets from the same class, using in-flight entertainment systems, the avionics will account for an approx. cost of 100,000 USD. As the designed aircrafts 6 COSTS ANALYSIS 83 must be to the edge of technology, a satellite based onboard WiFi system will be available, adding another 100,000 USD (see ref. [24] for detailed price estimate). The avionics costs vary with the level of certification. As the 6 seats aircraft is under the FAA’s Title 14 Part 23, a 60,000 USD avionics set is a reasonable approximation. The 100,000 USD in-flight WiFi system must also be added to the bill. • Engines Engines costs are well-guarded secrets from the manufacturers. Nevertheless, we can have a statistical estimate, using Eq. 6.9 0.8356 Cengines = 1035.99·NPP ·T ·CPI2020 = 2,090,463 (6.9) With NPP = 2 the number of engines and T = 3360 lbf the rated thrust. • Quantity Discount Factor The engines and the avionics are bought from external providers. Therefore, we must take into account that when signing a contract, the size of the order will impact the final pricetag. This effect is represented here by a "quantity discount factor" (QDF) as mentioned in Ref. [21]. The value of the QDF will depend on the ordered quantity (size of the deal), and on the "learning curve" (if a technician is "used" to some operation, he will do it more and more efficiently). It is already taken into account in the DAPCA-IV’s Eastlake CERs, but we must apply it to the engines and the avionics costs. This effect is another reason why it is particularly efficient to use the same engines, and to re-use most of the parts from one aircraft to the other. In figures, from a 300-units order over 5 years, we can expand to a 600-units order over 7 years (300 units in 2020-2025 for the 8 seats version and 300 units in 2022-2027 for the 6 seats version). Considering a 93% experience effectiveness (90% is often used, but it seems very optimistic), the cost for 300 pairs of engines would be 77.2% of the cost of one pair of engines, while the cost for 600 pairs of engines would be negotiated to 74.8% of the cost a one pair. This translate into a saving of close to 50,042$ per aircraft (considering it as a worst case). The same methodology is followed for avionics, although the avionics will be different from one aircraft to another (then, the max. order size is limited to 300). As a conclusion, the Tab. 6.3 displays the detailed costs analysis respectively for the 8 seats and the 6 seats aircraft. The profit margin is set to 15%, setting the final selling price of the aircraft to 9.1M$ (8 seats version) and 7.8M$ (6 seats version). 6.7 Break-even analysis In the last section, it has been demonstrated that to have the design certified would cost an initial investment of 600 M$. It is therefore important to know when the aircrafts will start generating profit to the company. This is the goal of the break-even analysis, which results are presented briefly on Tab. 6.4. 6 COSTS ANALYSIS 84 8 seats cost ($) 6 seats cost ($) Engineering 1,028,975 668,983 Development Support 303,803 265,868 Tooling 398,366 189,108 Flight Test Operations 168,155 115,066 Total Fixed Costs 2,011,403 1,315,735 Manufacturing 1,433,738 1,357,440 Quality Control 286,031 180,540 Materials 1,288,350 1,106,477 Avionics 228,200 191,360 Avionics Discount (QDF - 300) -52,131 -43,715 Engines 2,090,463 2,191,231 Engines Discount (QDF - 600) -527,600 -553,032 Total Variable Costs 4,747,049 4,430,299 Total Cost to Produce 6,758,452 5,746,034 Est. Liability Insurance (17%) 1,148,936 976,825 Minimum Selling Price 7,907,389 7,731,289 Table 6.3: Detailed costs estimates for the 8 seats & the 6 seats aircraft. 8 seats 6 seats Fixed Costs 603,420,900 $ 394,720,372$ Variable Costs 5,895,986 $ 5,407,125$ Selling Price 9,093,498 $ 7,731,289$ Break-even point 189 units 170 units Table 6.4: Break-even analysis 6.8 Operating Costs What really positions a Business Jet on the market is its operational costs. In the following, the operating cost will be approximated to the best under a series of approximations, and then compared with typical operational costs from other business jets of the same category using benchmarking data. 1. Fuel Costs: The fuel costs per hour can easily be determined using Eq. 6.10 Cfuel,h = 0.1497·SFCcruise ·Tcruise ·Cgal,fuel (6.10) Where the constant is issued from kg 1 m3 gal gal 0.4535 · ·264.17 = 0.1497 (6.11) lb 800 kg m3 lb 6 COSTS ANALYSIS 85 Averaging the fuel price per gallon to 4.70 $/gal (Source: [25]), the fuel cost per hours equals 743.78 $/hr. 2. Oil Costs: The oil costs are often estimated to 1% of the fuel costs. In our case, it represents Coil = 7.43 $/hr. 3. Maintenance Costs: • Labor costs: The labor cost must be differentiated from the parts cost. Assuming a 3 maintenance hours per flight hour (2.5 for the 6 seats, due to certifications differences), which is pessimistic for this range of aircraft, and assuming the rate of certified maintenance mechanics to 89$/hr (Source: Conklin & de Decker), a total hourly maintenance cost of 267 $/hr. • Parts costs: Typically, the prices involved into the parts and airframe replacements, are in a range slightly below the hourly maintenance cost. As an approximation, it will be set to 200.75 $/hr (75% of the engine operational costs). 4. Engine Restoration: The aircraft’s engines will require regular overhaul, as it is stipulated by the engine’s TBO (Time Between Overhauls). For the selected engines, the TBO is 5,000 hrs. The hourly engine restoration cost is approached by the Embraer’s Phenom 300’s engine restoration cost, since the same engines are used for this aircraft. The hourly cost of 267.67 $/hr has been determined (Source: Conklin & de Decker). All these direct operation costs are referenced in Tab. 6.5 8 seats 6 seats Fuel 773.64 773.64 Oil 7.74 7.74 Maintenance labor 267 222.50 Parts & Airframe 200.75 200.75 Engine restoration 267.67 267.67 Total operation cost / hour 1516.80 $ /hr 1472.30$/hr Table 6.5: Direct operation costs. The reader must not forget that other costs are also related to the acquisition of a Business Jet, such as the pilot/crew salaries, the hangar rental, the insurances (& single liability), but also the recurrents training and refurbishing expenses. One of the biggest indirect expense, though, is the aircraft’s value depreciation: due to the age, the aircraft is loosing some of its initial value, and this opportunity cost must be taken into account when buying an aircraft. 6.9 Conclusion In conclusion, as the designed aircraft is being benchmarked with its in-class competitors (see Tab. 6.6, it may be noticed that for a similar price tag and operation costs, the designed V-Tail business jet under study presents greater ranges and speeds, making it a serious competitor on the market. 7 CONCLUSION 86 Model Selling Price Direct Operation Costs 8 seats design 9.1 M$ 1516.80 $/h 6 seats design 7.8 M$ 1472.30 $/h Embraer Phenom 300 8.76 M$ 1328.74 $/h Bombardier Learjet 40XR 10.6 M$ 1836 $/h Cessna Citation CJ3+ 8.44 M$ 1353 $/h Table 6.6: Light Business Jets Benchmarking - Source: Conklin & de Decker. 7 Conclusion 7.1 Contextualization The study carried out in the scope of this report consists in the conceptual and preliminary design of two light business jets of the same family. These jets have to meet specific requirements detailed in Sec. 2.5 and are submitted to FAR 23 and FAR 25 for the 6 and 8 seats respectively. 7.2 Methods To achieve this objective all geometrical parameters have been studied first. This study encompasses the fuselage, the wing, the engines, the empennage and the undercarriage. All design choices have been selecting taking in consideration the potential cost it could generate. Innovation has also be put into practise when noticeable enhancement of the performance was intended. However, such geometrical innovation has to be carefully studied and further deeper studies can be complementary. After this conceptual design, a deeper preliminary study has been performed to assess the validity of the design parameters choice. To this point, a trade-off study has been carried out in Sec. 4 in order to evaluate the performance of the jet for a variation of 5% of some principal parameters around their design value. After that a model Catia as well as numerical simulation on the software Tranair have been carried out in order to extend the study. A more accurate evaluation of the whole plane’s drag has been performed in Sec.5.2.3. All these further improvements approve the design choices and enhance the first estimations on performances. 7.3 Results The final result consists of both very light aircraft, a 6 and 8 seats configurations that fully meet imposed requirements. Geometrical parameters have been studied to provide both high level of comfort and performance to the customer. That is, the cabin size is bigger than what typical business jets propose. The low-mounted wing configuration as well as the V-tail position allow a good stability and manoeuvrable qualities with stability margin comprised between 7.3 to 9.24 %. This V-tail, aside from being an aesthetic design choice, also allows to reduce the drag generate because of the simplicity of design and the 7 CONCLUSION 87 reduction of material used for production. Its particular position with respect to the engine turbulent flow ensure it to lower down the pressure below the inferior part of the V-tail and, as a result, to increase its efficiency. Both engines PW535 class E have been chosen for their relatively low specific fuel consumption and sufficient thrust able to fly at Mach 0.85 or able to takeoff with one engine in case of emergency. The comparison between other models has been indeed performed on Tab.3.4. A belly above the fuselage will contained all the fuel in order to avoid the center of gravity moving to much during different phase of cruise. Designing the plane this way allows to have a variation of the center of gravity of only 11 ft between the different phases of flight. What is more this choice allows to free space in the wing to store the undercarriage. 7.4 Critical assessment of the conceptual and preliminary design stages In the real life, actual aircraft engineers spent barely a few days to complete the conceptual and preliminary design stages. This rapidity rests on the employment of several assumption or semi-empirical correlations. Like those experts, some major results of this project rely on approximations, 3 of them especially. The first one is the mass estimations. The deduced structural mass percentage is too important relative to the payload one. A mass over-estimation is thus to dread. About the engines, the specific fuel consumption in cruise phase is data very difficult to quantify but can lead to sever variations of the consumption. Fuel savings considerations are therefore almost impossible to assess. At last, the drag study as well as the TRANAIR simulation accomplished does not include the transonic and viscous properties of the flight, leading to a drag under-estimation. REFERENCES 88 References [1] Emeric Delalandre. European business aviation market analysis 2016 - game changers. [2] Bombardier business aircraft. Market forecast 2015-2024. [3] Daniel P. Raymer. Aircraft Design: A Conceptual Approach. American Institute of Aeronautics & Astronautics, Reston, VA, 5 edition edition, August 2012. [4] Ilan Kroo. Aircraft Design - Synthesis and Analysis. Pages 252 – 264. Stanford, CA, January 2001. [5] Daniel P. Raymer. Aircraft Design: A Conceptual Approach. American Institute of Aeronautics & Astronautics, Reston, VA, 5 edition edition, August 2012. [6] Ludovic Noels. Aircraft Design Conceptual Design, 2016. [7] Snorri Gudmundsson. The Anatomy of Lift Enhancement. In General Aviation Aircraft Design, Applied Methods and Procedures, pages 437 – 441. Elsevier, Florida, United-States, 1st edition edition, September 2013. [8] Elodie Roux. Turbofan and turbojet engines: database handbook. Elodie Roux, 2007. [9] Paul E. Purser and John P. Campbell. Experimental verification of a simplified vee-tail theory and analysis of available data on complete models with vee tails. January 1945. [10] J. Roskam. Airplane Design. Roskam Aviation and Engineering Corp., Ottawa, 1983. [11] Charlie Svoboda. Turbofan engine database as a preliminary design tool. University of Kansas, 2004. [12] F. T. Johnson. Tranair: a full-potential, solution-adaptive, rectangular grid code for predicting subsonic, transonic, and supersonic lows about arbitrary configurations, December 1992. NASA Contractor Report 4348. [13] Richard T. Withcomb. A design approch and selected wind tunnel results at high subsonic speeds for wing-tip mounted winglets, July 1976. Nasa Technical Note D-8260. [14] Egbert Torenbeek. Synthesis of Subsonic Airplane Design: An Introduction to the Preliminary Design of Subsonic General Aviation and Transport Aircraft, With Emphasis on Layout, Aerodynamic Design, Propulsion. Kluwer Academic Publishers, Delft : The Hague : Hingham, MA, September 1982. [15] David Riordan Ajoy Kumar Kundu, Mark A. Price. Theory and Practice of Aircraft Performance. Wiley, 2016. [16] Ajoy Kumar Kundu. Aircraft design, volume 27. Cambridge University Press, 2010. [17] Federal Aviation Administration (FAA)/Aviation Supplies & Academics (Asa). Pilot’s Handbook of Aeronautical Knowledge. 2016. REFERENCES 89 [18] T.H.G Megson. Aircraft Structures for engineering students. Butterworth-Heinemann, 2003. [19] N. Lohith K.N. Arunkumar and B.B. Ganesha. Effect of Ribs and Stringer Spacings on the Weight of Aircraft Structure for Aluminum Material. Journal of Applied Sciences, 2012. [20] Charles N. Eastlake and Harry W. Blackwell. Cost Estimating Software for General Aviation Aircraft Design. age, 5:2, 2000. [21] Snorri Gudmundsson. General Aviation Aircraft Design: Applied Methods and Procedures. Butterworth-Heinemann, September 2013. [22] Budget and Economic Data. https://www.cbo.gov/about/products/budget-economic-data, Jan- uary 2016. [Online; accessed 20-April-2017]. [23] Embraer Phenom 300 is the world’s most delivered business jet for the fourth year in a row. http://www.embraer.com/en-US/ImprensaEventos/ Press-releases/noticias/Pages/Embraer-Phenom-300-is-the-world%E2%80% 99s-most-delivered-business-jet-for-the-fourth-year-in-a-row.aspx. [Online; accessed 20-April-2017]. [24] D. Rimmer. Tha Beginning of Airline Wifi: Broadband Takes Flight. https://www.thebalance.com/ airborne-wifi-faqs-broadband-takes-flight-282623. [Online; accessed 20-April-2017]. [25] 100ll - Aviation Fuel Prices. http://www.100ll.com/. [Online; accessed 20-April-2017]. A EMPIRICAL GROSS TAKE-OFF WEIGHT ESTIMATION 90 Appendix A Empirical gross Take-Off Weight estimation The estimate is based on the equation A.2. Wto = Wcrew +Wpayload +We +Wf (A.1) W +W = crew payload (A.2) W W 1 − e − f Wto Wto A.1 Empiric estimates of each terms 1. Weights of the payload Wpayload and the crew Wcrew The payload is composed of the passengers (8 passengers, 200 lbs eachs), and 1000 lbs worth of luggage. Which gives Wpayload = 8·200+1000 = 2600 lbs. A crew of 2 pilots will be considered (200 lbs each), and therefore, Wcrew = 400 lbs. 2. Empty weight fraction We/Wto The empty weight fraction may be estimated using the empirical relationships provided by [3]. Having, for our category of aircraft, with A = 1.02 and C = −0.06, We C = AWto Wto 3. Fuel weight fraction Wfuel/Wto The fuel weight will obviously depend of the mission of the aircraft. In this case, we will stick to a typical transportation mission, with the following phases: (a) Taxi & takeoff: W1/Wto = 0.97 (b) Climb: W2/W1 = 0.985 (c) Cruise - Breguet Range equation W R·SFC 3 = exp− cruise ' 0.77 W L 2 V · D With: • R, the desired range (expressed in ft.) • SFCcruise, the specific fuel consumption of the engine in cruise condition. As a first approximation, the PW535-E is chosen, which has a SFC of about 0.77 lb/(h∆lb f ) B UNINSTALLED MAXIMUM CRUISE RATING 91 • V is the velocity of the plane (M = 0.85). • L/D the lift to drag ratio. Estimated to ' 13 as a first approximation. W (d) Loiter (considered as slightly different conditions from cruise): 4 ' 0.97. W3 (e) Landing: The Landing ratio is estimated, from [3] to be W5/W4 ' 0.995 In the end, the total mission weight fraction W5/Wto is obtained by multiplying all the terms, W W W W W W 1 · 2 · 3 · 4 · 5 = 5 ' 0.7096 Wto W1 W2 W3 W4 Wto Adding the fuel reserve and trapped fuel (about 6% of the total fuel quantity), W W f = 1.06· 1 − 5 ' 0.308 Wto Wto The value of Wto has been iterated multiple times, as the empty fraction weight is dependant from the TOW. A stable value of ∼ 22500lbs is computed. However, the empirical relation used for the empty weight fraction is very rough, and is adapted from a wide databases of jet aircrafts, most of which are much heavier than this light business jet. Therefore, the weight is expected to be lower than this approximation. To make a compromise between this method and the market assessment results, the 8 seats business jet is expected to reach a TOW of about 19000lbs as a first approximation. B Uninstalled maximum cruise rating Figure B.1: Unistalled maximum cruise rating (BPR ≤ 4).[16] C WING GEOMETRY PARAMETERS: STATISTICS 92 C Wing geometry parameters: statistics Aircarft Wing span [ft2] Aspect Ratio (AR) Airbus A 300-600B 147.11 7.7 Airbus A 310-300 144.00 8.8 Airbus A 320-200 111.25 9.4 Airbus A 340-300 197.83 10 Antonov AN 124 240.49 8.6 BAe (Avro) RJ85 85.99 9.0 Boeing B 737-600 112.57 9.4 Boeing B 747-400 211.42 7.7 Boeing B 777 199.90 8.7 Cessna 525 CitationJet 46.75 9.1 Learjet 60 43.77 7.2 Embraer Phenom 300 52.20 8.9 McDonnell Douglas MD11 169.49 7.9 Tupolev TU 204 137.80 9.7 Table C.1: Statistics about aspect ratio and wing span for different aircraft. Aircarft Aspect Ratio (AR) Taper Ratio Sweep Angle [◦] Maximum Mach VFW-Fokker 614 7.22 0.402 15 0.65 Yakovlev Yak 40 9.00 0.396 0 0.70 Fokker-VFW F 28 7.27 0.355 16 0.75 BAC 1-11 200/400 8.00 0.321 20 0.78 Aerospatialle Caravelle 8.02 0.354 20 0.81 Boeing 737 100/200 8.83 0.251 25 0.84 Table C.2: Statistics about aspect ratio, taper ratio and sweep angle for different aircraft (source coming from the course notes of Mr. Noels, Lecture 4, page 59. [6]) D EXPERIMENTAL DATA OF THE NACA SC(2)-0714 93 D Experimental data of the NACA SC(2)-0714 (a) Lift, drag and moment coefficient at high Reynolds (Re = 10·106). (b) Lift coefficient at low Reynolds (Re = 6·106). Figure D.1: Experimental data for NACA SC(2)-0714 airfoil performed in wind tunnel. E SUMMARY OF THE WING GEOMETRICAL PARAMETERS 94 E Summary of the wing geometrical parameters Main wing parameters US/Imp Span: b 45.7 [ft] Aspect Ratio: AR 9 Total (gross) area: S 232.2 [ft2] 2 Total exposed surface: Sexp 191.3 [ft ] Taper ratio: λ 0.30 Chord at root: croot 7.81 [ft] Chord at tip: ctip 2.33 [ft] Sweep angle at leading edge: Λ 34.5 [◦] ◦ Sweep angle at chord quarter: Λ1/4 32.1 [ ] ◦ Geometric twist (Washout): εgtip -1 [ ] Mean aerodynamic chord: MAC 5.58 [ft] X coordinate of aerodynamic centre: Xac 7.84 [ft] Y coordinate of aerodynamic centre: Yac 9.38 [ft] Compressibility parameter: β 0.3275 Cruise Mach: M 0.85 Average airofoil thickness: t 0.72 [ft] Wing lift coefficient in cruise: CLw 0.3 Wing lift coefficient derivative: a 4.77 ◦ Angle of attack at root (cruise): αroot -0.2 [ ] ◦ Zero-lift angle of attack of at root: αL0 -3.8 [ ] ◦ Zero-lift angle of attack of the profile: αl0 -4 [ ] Aerodynamics twist coefficient: α01 -0.225 ◦ Aerodynamics twist: εatip -1 [ ] Maximum wing lift coefficient (cruise): CLmax,cruise 0.8 Maximum wing lift coefficient (landing/takeoff, flaps in): CLmax,TO/Landing 1.5 Reynolds number (cruise): Re 9.6·106 Airofoil lift coefficient derivative: cla 2.9π Maximum camber: cmax (expressed relative to the chord length) 2.5 % Lift generated by the wing (cruise): Lw 15397 [lbf ] Table E.1: Summary of the principal parameters of the wing. F MATERIAL SELECTION: PROPERTIES OF ALUMINIUM 95 F Material Selection: properties of aluminium Wrought aluminium alloy 7075-T6 5052-H32 Composition detail [%] Al (aluminium) 87.2 - 91.4 95.8 - 97.6 Cr (chromium) 0.18 - 0.28 0.15 - 0.35 Cu (copper) 1.2 - 2 0 - 0.1 Fe (iron) 0 - 0.5 0 - 0.4 Mg (magnesium) 2.1 - 2.9 2.2 - 2.8 Mn (manganese) 0 - 0.3 0 - 0.1 Si (silicon) 0 - 0.4 0 - 0.25 Ti (titanium) 0 - 0.2 0 Zn (zinc) 5.1 - 6.1 0 - 0.1 Other 0 - 0.15 0 - 0.15 Price 0.83 - 0.92 $/lb 0.78-0.86$/lb Mechanical properties Young’s modulus 10007.6 - 11022.9 ksi 10152.6 - 10674.8 ksi Yield strength (elastic limit) 52.1 - 76.9 ksi 22.0 - 24.9 ksi Tensile strength 62.9 - 84.1 ksi 31.9 - 34.4 ksi Elongation 2 - 10 % strain 4 - 12 % strain Compressive strength 57.0 - 76.9 ksi 22.0 - 24.4 ksi Flexural modulus 10007.6 - 11022.9 ksi 10152.6 - 10674.8 ksi Flexural strength (modulus of rupture) 52.1 - 76.9 ksi 22.0 - 24.9 ksi Shear modulus 3771.0 - 4061.1 ksi 3916.0 - 4119.1 ksi Bulk modulus 9717.5 - 10732.8 ksi 9906.1 - 10413.7 ksi Poisson’s ratio 0.325 - 0.335 0.33 - 0.343 Shape factor 16 35 Hardness - Vickers 152 - 168 HV 55 - 75 HV Fatigue strength at 107 cycles 22.0 - 24.4 ksi 16.0 - 17.4 ksi Impact & fracture properties √ √ Fracture toughness 7.00 - 7.04 ksi · ft 7.1 - 9.7 ksi · ft Durability Water (fresh) Excellent Excellent Water (salt) Acceptable Acceptable UV radiation (sunlight) Excellent Excellent Table F.1: Properties of wrought aluminium alloys 7075 T6 and 5052-H32. G AIRCRAFT VIEWS 96 G Aircraft views G AIRCRAFT VIEWS 97