Final Report Fixed Wing 2

DEPARTMENT OF AEROSPACE ENGINEERING REPORT AIRCRAFT DESIGN II FIXED-WING NO.2 THAI QUANG THINH NGUYEN HIEP HUNG LE NGUYEN HOANG PHI LE TRONG NHAN NGUYEN QUANG VINH 6/2014 G1003223 G1001386 G1002369 G1002229 G1004006 CLASS GT10HK The main goals of subject The project Aircraft Design is the following step of Aircraft Design which is carried on by the resulting data of Aircraft Design In this project, the QFD as well as the pugh matrix of the project is precisely determined Those are the foundation of the project, based on the need of the customers, the design must be the perfect product where the customer’s need meet the engineer’s requirements Besides, the project illustrates the specific calculating steps and resulting data for Stability and Performance In addition to Aircraft Design 1, those results guild the engineers to the next phase of the design sequence, which is Detail Design Contents Contents PART I - ENGINEERING DESIGN 1 QUALITY FUNCTION DEPLOYMENT 1.1 Concept 1.2 The House of Quality Configurations (HOQ) 1.3 Steps for Building a House of Quality 1.4 Quality Function Development was applied to our project “Fixed Wing seats” Evaluation Method 2.1 Pugh concept selector method 2.2 Analytic Hierarchy Process (AHP) method 13 PART II - STABILITY 17 A brief of stability 17 Introduction 17 2.1 Geometry parameters 17 2.2 Layout 17 Static stability and control .19 3.1 Static stability and control 19 3.2 Longitudinal control 23 3.3 Sensitivity Analyses 24 3.4 Directional control 26 3.5 Aileron control 29 3.6 Table of comparison between the same class aircrafts 29 Equations of motion 30 4.1 Estimation of 𝐶𝐷𝑝 30 4.2 Longitudinal stability derivative coefficients 35 4.3 Lateral stability derivative coefficients 36 4.4 Prediction of inertia moments 37 4.5 Longitudinal derivatives 37 4.6 Lateral directional derivatives 38 Dynamic stability 39 5.1 Longitudinal dynamic stability 39 5.2 Directional dynamic stability 41 5.3 Lateral dynamic stability 42 Transfer Functions 44 6.1 Short-Period Dynamics 44 6.2 Long-period Dynamics 47 6.3 Roll Dynamic 50 6.4 Dutch-roll Approximation 52 PART III PERFORMANCE 59 Steady Flight 59 1.1 Thrust required (Drag) 59 ii iii AIRCRAFT DESIGN II 1.2 Ratio 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 The fundamental parameters: Thrust-To-Weight Ratio, Wing Loading, Drag Polar, and Lift-To-Drag 60 Power required and Power available 61 Calculation of Stalling Velocity: Role of (CL )max 62 Rate of climb 64 Gliding (Unpowered) Flight 65 Service and Absolute ceilings 66 Time to climb 67 Range 68 Endurance 69 Accelerated Flight 69 2.1 LEVEL TURN 69 2.2 THE V-n DIAGRAM 70 2.3 ENERGY CONCEPTS: ACCELERATED RATE OF CLIMB 73 TAKEOFF PERFORMANCE 76 3.1 Calculation of Ground Roll 78 3.2 Calculation of Distance While Airborne to Clear an Obstacle 79 LANDING PERFORMANCE 80 REFERENCES 83 APPENDIX 83 List of Figures List of Figures FIGURE I-1 DIAGRAM SHOWING THE FOUR HOUSES OF THE COMPLETE QFD PROCESS FIGURE I-2 THE HOUSE OF QUALITY TRANSLATES THE VOICE OF THE CUSTOMER, INPUT AS CRS IN ROOM 1, INTO TARGET VALUES FOR ECS IN ROOM FIGURE I-3 THE MINIMAL HOQ TEMPLATE INCLUDES ROOMS 1, 2, 4, AND FIGURE I-4 ROOF OF HOQ FIGURE I-5 CUSTOMER REQUIREMENTS OF HOQ LEVEL FIGURE I-6 ENGINEERING CHARACTERISTICS OF HOQ LEVEL FIGURE I-7 SHOWS DIRECT DECOMPOSITION OF A AEROBATIC TWO-SEAT LIGHTPLANE INTO SUBASSEMBLIES 10 FIGURE I-8 HIERARCHICAL STRUCTURE OF THE SCOOTER 14 FIGURE II-1 WING CONTRIBUTION TO THE PITCHING MOMENT 19 FIGURE II-2 THE LIFT CURVE OF NACA 4415 AIRFOIL 19 FIGURE II-3 AFT TAIL CONTRIBUTION TO THE PITCHING MOMENT 20 FIGURE II-4 THE LIFT CURVE OF NACA 0012 AIRFOIL 20 FIGURE II-5 FUSELAGE CONTRIBUTION – ESTIMATE 𝐶𝑚0𝑓 21 FIGURE II-6 FUSELAGE CONTRIBUTION – ESTIMATE 𝐶𝑚𝛼𝑓 22 FIGURE II-7 WING BODY INTERFERENCE FACTOR 27 FIGURE II-8 CUTOFF REYNOLDS NUMBER 31 FIGURE II-9 TURBULENT MEAN SKIN-FRICTION COEFFICIENT ON AN INSULATED FLAT PLATE – ESTIMATE 𝐶𝑓𝑤 32 FIGURE II-10 WING-BODY INTERFERENCE CORRELATION FACTOR 33 FIGURE II-11 TURBULENT MEAN SKIN-FRICTION COEFFICIENT ON AN INSULATED PLAT PLATE – ESTIMATE 𝐶𝑓𝐻 AND 𝐶𝑓𝑉 34 FIGURE III-1 THRUST REQUIRE CURVE AT 3000M 60 FIGURE III-2 THE POWER REQUIRE CURVE AND THE POWER AVAILABLE AT 3000M 62 FIGURE III-3 THE 𝑉 − 𝑛 DIAGRAM FOR A TYPICAL JET TRAINER AIRCRAFT FREE-STREAM VELOCITY 𝑉∞ IS GIVEN IN KNOTS 72 FIGURE III-4 THE 𝑉 − 𝑛 DIAGRAM 73 FIGURE III-5 OVERLAY OF 𝑃𝑠 CONTOURS AND SPECIFIC ENERGY STATES ON AN ALTITUDE-MACH NUMBER MAP THE 𝑃𝑠 VALUES SHOWN HERE APPROXIMATELY CORRESPOND TO A LOCKHEED F-104G SUPERSONIC FIGHTER LOAD FACTOR 𝑛 = AND 𝑊 = 18,000 𝑙𝑏 AIRPLANE IS AT MAXIMUM THRUST THE PATH GIVEN BY POINTS A THROUGH I IS THE FLIGHT PATH FOR MINIMUM TIME TO CLIMB 76 FIGURE III-6 ILLUSTRATION OF GROUND ROLL 𝑠𝑔 , AIRBORNE DISTANCE 𝑠𝑜 , AND TOTAL TAKEOFF DISTANCE 77 FIGURE III-7 SKETCH FOR THE CALCULATION OF DISTANCE WHILE AIRBORNE 79 FIGURE III-8 THE LANDING PATH AND LANDING DISTANCE 81 List of Tables TABLE I-1 RELATIVE WEIGHTING OF HOQ LEVEL TABLE I-2 RELATIVE WEIGHTING OF HOQ LEVEL TABLE I-3 RELATIVE WEIGHTING OF HOQ LEVEL TABLE I-4 PUGH MATRIX OF WING 10 TABLE I-5 PUGH MATRIX OF TAIL 11 TABLE I-6 DECISION MATRIX OF ENGINES 11 TABLE I-7 PUGH MATRIX OF LANDING GEAR 12 TABLE I-8 PUGH MATRIX FOR EACH CONCEPT 13 TABLE I-9 MATRIX C 14 TABLE I-10 MATRIX CONSISTENCY AND CR VALUE 14 TABLE I-11 DECISION MATRIX 15 TABLE I-12 EXPLAINING TABLE FOR DECISION MATRIX 15 TABLE II-1 COMPARISON BETWEEN THE SAME CLASS AIRCRAFTS 29 TABLE II-2 SHORT-PERIOD TRANSFER FUNCTION APPROXIMATIONS 44 TABLE II-3 LONG-PERIOD TRANSFER FUNCTION APPROXIMATIONS 48 TABLE II-4 DUTCH-ROLL TRANSFER FUNCTION APPROXIMATIONS 52 iv AIRCRAFT DESIGN II Part I - ENGINEERING DESIGN QUALITY FUNCTION DEPLOYMENT 1.1 Concept Quality function deployment (QFD) is a planning and team problem-solving tool that has been adopted by a wide variety of companies as the tool of choice for focusing a design team’s attention on satisfying customer needs throughout the product development process The QFD process is known as a methodology for infusing the voice of the customer into every aspect of the design process The House of Quality translates customer requirements into quantifiable design variables, called engineering characteristics This mapping of customer wants to engineering characteristics enables the remainder of the design process More information can be interpreted from the House of Quality It can also be used to determine which engineering characteristics should be treated as constraints for the design process and which should become decision criteria for selecting the best design concept Figure I-1 Diagram showing the four houses of the complete QFD process 1.2 The House of Quality Configurations (HOQ) The HOQ takes information developed by the design team and translates it into a format that is more useful for new product generation This text uses an eight-room version of the House of Quality as shown in Fig  Room 1: Customer Requirements “WHAT” from users and customer (CRs) ENGINEERING DESIGN        Room 2: Engineering Characteristics “HOW” from engineering (Ecs) Room 3: Correlation Matrix means relationship of engineering characteristics Room 4: Relationship Matrix “Whats related to Hows” Room 5: Importance Ranking for decision what is the most important in WHATs Room 6: Customer Assessment Of Competing Products Room 7: Technical Assessment Room 8: Target Values Figure I-2 The House of Quality translates the voice of the customer, input as CRs in Room 1, into target values for ECs in Room AIRCRAFT DESIGN II The end result of the HOQ is the set of target values for ECs that flow through the HOQ and exit at the bottom of the house in Room This set of target values guides the selection and evaluation of potential design concepts Note that the overall purpose of the HOQ process is broader than establishing target values The HOQ will become one of the most important reference documents created during the design process Like most design documents, the QFD should be updated as more information is developed about the design 1.3 Steps for Building a House of Quality Figure I-3 The Minimal HOQ Template includes Rooms 1, 2, 4, and Room 1: Customer Requirements Customer requirements are listed by rows in Room The CRs and their importance ratings are gathered by the team in design process Also included in this room is a column with an importance rating for each CR The ratings range from to 5, importance ratings of and Room 2: Engineering Characteristics Engineering characteristics are listed by columns in Room ECs are product performance measures and features that have been identified as the means to satisfy the ENGINEERING DESIGN CRs One basic way is to look at a particular CR and answer the question, “What can I control that allows me to meet my customer’s needs?” ECs are measurable values (unlike the CRs) and their units that are placed near the top of Room Symbols indicating the preferred improvement direction of each EC are placed at the top of Room Thus a  symbol indicates that a higher value of this EC is better, and a  symbol indicates that a lower value is better It is also possible that an EC will not have an improvement direction Room 4: Relationship Matrix The relationship matrices at the center of an HOQ Each cell in the matrix is marked with a symbol that indicates the strength of the causal association between the EC of its column and the CR of its row Rules include significantly (9), moderately (3), or slightly (1) The cell is left blank if the EC had no impact on the CR Rules: Significantly  Moderately  Slightly  Room Importance Ranking of Ecs Absolute importance is calculated in two steps First multiply the numerical value in each of the cells of the Relationship Matrix by the associated CR’s importance rating Then, sum the results for each column, placing the total in Room 5a Relative importance (Room 5b) is the absolute importance of each EC, normalized on a scale from to and expressed as a percentage of 100 Rank order of ECs (Room 5c) is a row that ranks the ECs’ Relative Importance from (highest % in Room 5b) to 10 This ranking allows viewers of the HOQ to quickly focus on ECs in order from most to least relevant to satisfying the customer requirements The Correlation Matrix or Roof of the House of Quality The correlation matrix shows the degree of interdependence among the engineering characteristics in the “roof of the house.” Figure I-4 Roof of HOQ Assessment of Competitor’s Products in House of Quality AIRCRAFT DESIGN II In Room 6, Competitive Assessment, a table displays how the top competitive products rank This information comes from direct customer surveys, industry consultants, and marketing departments Room in the lower levels of the House of Quality provides another area for the comparison to competing products Room 7, Technical Assessment, is located under the Relationship Matrix Technical Assessment data can be located above or below the Importance Ranking sections of Room Setting Target Values for Engineering Characteristics Room 8, Setting Target Values, is the final step in constructing the HOQ By knowing which are the most important ECs (Room 5), understanding the technical competition (Room 7), and having a feel for the technical difficulty (Room 7), the team is in a good position to set the targets for each engineering characteristic 1.4 Quality Function Development was applied to our project “Fixed Wing seats” 1.4.1 House of Quality level Customer Requirement Nowadays, the aviation industry have developed more than ever The first and crucial step in building an aircraft is to determine the requirements of customers through surveys There are several requirements come from the customers in designing a 2-seat aerobatic aircraft  Endurance: Since aircraft is a high-speed transportation, this is the first quality of the aircraft that the customers concern about High-speed saves time hence endurance is one of the most vital specifications of one airplane  Control: In a light-weight aircraft, the customers are mostly the pilots In other to approach a wide range of customer, the supplier have to make their product as controllable as possible  Price: This is a classical criterion in every single industry, aviation is not an exception  Economics: The word “Economics” here is used to mention about the cost of operating, repair and the ability to make profit of the airplane  Aesthetics: One of the most important factors of an aircraft  Safe: Moving too fast at such a high attitude makes aircraft have to be absolutely safe, one small error can lead to catastrophic consequences  Comfort: This is the factor contributes considerably to the competitiveness of the product  Range: Long range is one of the advantages of aircraft, one of the vital factors that will be considered when customer want to buy some kind of aircraft  Reliability 73 AIRCRAFT DESIGN II xlabel('V (m/s)') ylabel('Load factor') Results * Figure III-4 The 𝑽 − 𝒏 Diagram 2.3 ENERGY CONCEPTS: ACCELERATED RATE OF CLIMB 2.3.1 Energy Height Consider an airplane of mass m in flight at some altitude h and with some velocity 𝑉∞ Due to its altitude, the airplane has potential energy equal to mgh Due to its velocity, the airplane has kinetic energy equal to 𝑉∞ The total energy of the airplane is the sum of these energies, that is, Total aircraft energy = mgh + 1/2mV∞ The specific energy, denoted by 𝐻𝑒 is defined as total energy per unit weight and is obtained by dividing by W = mg This yields 1 𝑚𝑔ℎ + 𝑚𝑉∞2 𝑚𝑔ℎ + 𝑚𝑉∞2 2 𝐻𝑒 = = 𝑊 𝑚𝑔 Or 𝑉∞2 (40) 𝐻𝑒 = ℎ + 2𝑔 The specific energy He has units of height and is therefore also called the energy height of the aircraft Thus, let us become accustomed to quoting the energy of an airplane in terms of its energy height He, always remembering that it is simply the sum of the potential and kinetic energies of the airplane per unit weight Contours of constant He are given in Fig below, which is an "altitude-Mach number map.” Here the ordinate and abscissa are PERFORMANCE altitude h and Mach number M, respectively, and the dashed curves are lines of constant energy height We can draw an analogy between energy height and money in the bank Say that you have a sum of money in the bank split between a checking account and a savings account Say that you transfer part of your money in the savings account into your checking account You still have the same total; the distribution of funds between the two accounts is just different Energy height is analogous to the total of money in the bank; the distribution between kinetic energy and potential energy can change, but the total will be the same For example, consider two airplanes, one flying at an altitude of 30,000 ft at Mach 0.81 (point A) and the other flying at an altitude of 10,000 ft at Mach 1.3 (point B) Both airplanes have the same energy height of 40,000 ft However, airplane A has more potential energy and less kinetic energy (per unit weight) than airplane B If both airplanes maintain their same states of total energy, then both are capable of “zooming” to an altitude of 40,000 ft at zero velocity (point C) simply by trading all their kinetic energy for potential energy Consider another airplane, flying at an altitude of 50,000 ft at Mach 1.85, denoted by point D This airplane will have an energy height of 100,000 ft and is indeed capable of zooming to an actual altitude of 100,000 ft by trading all its kinetic energy for potential energy Airplane D is in a much higher energy state (He = 100,000 ft) than airplanes A and B (which have He = 40,000 ft) Therefore airplane D has a much greater capability for speed and altitude performance than airplanes A and B In air combat, everything else being equal, it is advantageous to be in a higher energy state (have a higher He) than your adversary 2.3.2 Specific Excess Power How does an airplane change its energy state; for example, how could airplanes A and B increase their energy heights to equal that of D? The answer to this question has to with specific excess power, defined below 𝑇𝑉∞ − 𝐷𝑉∞ = excess power We define specific excess power, denoted by Ps 𝑒𝑥𝑐𝑒𝑠𝑠 𝑝𝑜𝑤𝑒𝑟 𝑇𝑉∞ − 𝐷𝑉∞ 𝑃𝑠 = = 𝑊 𝑊 𝑑ℎ 𝑉∞ 𝑑𝑉∞ 𝑃𝑠 = + (41) 𝑑𝑡 𝑔 𝑑𝑡 𝑑𝐻𝑒 (42) 𝑃𝑠 = 𝑑𝑡 That is, the time rate of change of energy height is equal to the specific excess power An airplane can increase its energy height simply by the application of excess power In Fig above, airplanes A and B can reach the energy height of airplane D if they have enough specific excess power to so 74 75 AIRCRAFT DESIGN II Question: How can we ascertain whether a given airplane has enough P, to reach a certain energy height? The answer has to with contours of constant Ps on an altitudeMach number map Let us see how such contours can be constructed 2.3.3 Rate of Climb and Time to Climb (Accelerated Performance) Accelerated rate of climb and time to climb can be treated by energy considerations Consider an airplane at a given altitude and Mach number The rate of climb for this specified accelerated condition is, 𝑑ℎ 𝑉∞ = 𝑃𝑠 − 𝐴 (43) 𝑑𝑡 𝑔 In Eq (43), all quantities on the right-hand side are known or specified; the equation gives the instantaneous maximum rate of climb that can be achieved at the instantaneous velocity 𝑉∞ , and the instantaneous acceleration A The time required for an airplane to change from one energy height 𝐻𝑒,1 to a larger energy height 𝐻𝑒,2 can be obtained as follows From Eq (42), 𝑑𝐻𝑒 𝑑𝑡 = (44) 𝑃𝑠 Integrating Eq (44) between time t1 where 𝐻𝑒 = 𝐻𝑒,1 and time t2 where 𝐻𝑒 = 𝐻𝑒,2 , we have 𝐻𝑒,2 𝑑𝐻𝑒 𝑡2 − 𝑡1 = ∫ (45) 𝐻𝑒,1 𝑃𝑠 For a given 𝑉∞,2 at energy height 𝐻𝑒,2 , and a given 𝑉∞,1 at energy height 𝐻𝑒,1 , equation (45) gives the time required to achieve this change in altitude; that is, it gives the time to climb from altitude h1 to altitude h2 when the airplane has accelerated (or decelerated) from velocity 𝑉∞,1 at altitude h1 to velocity 𝑉∞,2 at altitude h2 The time to climb t2 –t1 between 𝐻𝑒,1 and 𝐻𝑒,2 is not a unique value—it depends on the flight path taken in the altitude-Mach number map Examine again Figure III-5 In changing from 𝐻𝑒,1 to 𝐻𝑒,2 , there are an infinite number of variations of altitude and Mach number that will get you there In terms of Eq (45), there are an infinite number of different values of the integral because there are an infinite number of different possible variations of dHe/Ps between 𝐻𝑒,1 , 𝐻𝑒,2 However, once a specific path in Figure III-5 is chosen between 𝐻𝑒,1 and 𝐻𝑒,2 then dHe/Ps has a definite variation along this path, and a specific value of t2 – t1 is obtained This discussion has particular significance to the calculation of minimum time to climb to a given altitude, which is a unique value There is a unique path in the altitudeMach number map that corresponds to minimum time to climb We can see how to construct this path by examining Eq (45) The time to climb will be a minimum when Ps is a maximum value Looking at Figure III-5, for each He curve, we see there is a point where Ps is a maximum Indeed, at this point the Ps curve is tangent to the He curve Such points are illustrated by points A to I in Figure III-5 The heavy curve through these points illustrates the variation of altitude and Mach number along the flight path for minimum time to climb Along this path (the heavy curve), dHe/Ps varies in a definite way, and when these values of dHe/Ps are used in calculating the integral in Eq (45), the resulting value of t2 – t1 is the minimum time to climb between He,1 and He,2 In general, there is no analytical form of the integral in Eq (45); it is usually evaluated numerically PERFORMANCE We note in Figure III-5 that the segment of the flight path between D and D' represents a constant energy dive to accelerate through the drag-divergence region near Mach We also note that Eq (45) gives the time to climb between two energy heights, not necessarily that between two different altitudes However, at any given constant energy height, kinetic energy can be traded for potential energy, and the airplane can “zoom” to higher altitudes until all the kinetic energy is spent For example, in Figure III-5 point I corresponds to Ps = The airplane cannot achieve any further increase in energy height However, after arriving at point I, the airplane can zoom to a minimum altitude equal to the value of He at point I—in Figure III-5, a maximum altitude well above 100,000 ft After the end of the zoom, 𝑉∞ = (by definition) and the corresponding value of h is the maximum obtainable altitude for accelerated flight conditions, achieved in a minimum amount of time Figure III-5 Overlay of 𝑃𝑠 contours and specific energy states on an altitude-Mach number map The 𝑃𝑠 values shown here approximately correspond to a Lockheed F-104G supersonic fighter Load factor 𝑛 = and 𝑊 = 18,000 𝑙𝑏 Airplane is at maximum thrust The path given by points A through I is the flight path for minimum time to climb TAKEOFF PERFORMANCE Consider an airplane standing motionless at the end of a runway This is denoted by location in Fig below The pilot releases the brakes and pushes the throttle to maximum takeoff power, and the airplane accelerates down the runway At some distance from its starting point, the airplane lifts into the air How much distance does the airplane cover along the runway before it lifts into the air? This is the central question in the analysis of takeoff performance Called the ground roll (or sometimes the ground run) and denoted by sg, it is a major focus of this section However, this is not the whole consideration The total takeoff distance also includes the extra distance covered over the ground after the airplane is airborne but before it clears an obstacle of a specified height This is denoted by sa in Fig below The height of the obstacle is generally specified to be 50 ft for military aircraft and 35 ft for commercial aircraft The sum of sg and sa is the total takeoff distance for the airplane 76 77 AIRCRAFT DESIGN II Figure III-6 Illustration of ground roll 𝒔𝒈 , airborne distance 𝒔𝒐 , and total takeoff distance As the airplane accelerates from zero velocity, at some point it will reach the stalling velocity Vstall, as noted in Figure III-6 The airplane continues to accelerate until it reaches the minimum control speed on the ground, denoted by Vmcg in Figure III-6 This is the minimum velocity at which enough aerodynamic force can be generated on the vertical fin with rudder deflection while the airplane is still rolling along the ground to produce a yawing moment sufficient to counteract that produced when there is an engine failure for a multiengine aircraft If the airplane were in the air (without the landing gear in contact with the ground), the minimum speed required for yaw control in case of engine failure is slightly greater than Vmcg This velocity is called the minimum control speed in the air, denoted by Vmca in Figure III-6 For the ground roll shown in Figure III-6, Vmca is essentially a reference speed - the airplane is still on the ground when this speed is reached The airplane continues to accelerate until it reaches the decision speed, denoted by V1 in Figure III-6 This is the speed at which the pilot can successfully continue the takeoff even though an engine failure (in a multiengine aircraft) would occur at that point This speed must be equal to or larger than Vmcg in order to maintain control of the airplane A more descriptive name for V1 is the critical engine failure speed If an engine fails before V1 is achieved, the takeoff must be stopped If an engine fails after V1 is reached, the takeoff can still be achieved The airplane continues to accelerate until the takeoff rotational speed, denoted by VR in Figure III-6, is achieved At this velocity, the pilot initiates by elevator deflection a rotation of the airplane in order to increase the angle of attack, hence to increase CL Clearly, the maximum angle of attack achieved during rotation should not exceed the stalling angle of attack Actually, all that is needed is an angle of attack high enough to produce a lift at the given velocity larger than the weight, so that the airplane will lift off the ground However, even this angle of attack may not be achievable because the tail may drag the ground (Ground clearance for the tail after rotation is an important design feature for the airplane, imposed by takeoff considerations.) PERFORMANCE If the rotation of the airplane is limited by ground clearance for the tail, the airplane must continue to accelerate while rolling along the ground after rotation is achieved, until a higher speed is reached where indeed the lift becomes larger than the weight This speed is called the minimum unstick speed, denoted by Vmu in Figure III-6 For the definition of Vmu, it is assumed that the angle of attack achieved during rotation is the maximum allowable by the tail clearance However, for increased safety, the angle of attack after rotation is slightly less than the maximum allowable by tail clearance, and the airplane continues to accelerate to a slightly higher velocity, called the liftoff speed, denoted by VLO in Figure III-6 This is the point at which the airplane actually lifts off the ground The total distance covered along the ground to this point is the ground roll sg 3.1 Calculation of Ground Roll 𝑑𝑉∞ (46) = 𝑇 − 𝐷 − 𝜇𝑟 (𝑊 − 𝐿) 𝑑𝑡 The drag D varies with velocity according to (47) 𝐷 = 𝜌∞ 𝑉∞2 𝑆𝐶𝐷 However, during the ground roll, CD in Eq (47) is not the same value as given by the conventional drag polar used for full flight in the atmosphere; the conventional drag polar: 𝐶𝐷 = 𝐶𝐷,0 + 𝐾𝐶𝐿2 This is for two primary reasons: (1) With the landing gear fully extended, 𝐶𝐷,0 is larger than when the landing gear is retracted; and (2) there is a reduction in the induced drag due to the close proximity of the wings to the ground - part of the “ground effect.” An approximate expression for the increase in CD,0 due to the extended landing gear is given in Ref 41 as 𝑊 (48) ∆𝐶𝐷,0 = 𝐾𝑢𝑐 𝑚−0.215 𝑆 where W/S is the wing loading, m is the maximum mass of the airplane, and the factor Kuc depends on the amount of flap deflection With flap deflection, the average airflow velocity over the bottom of the wing is lower than it would be with no flap deflection; that is, the deflected flap partially blocks the airflow over the bottom surface Hence, the landing gear drag is less with flap deflection than its value with no flap deflection In Eq (48), when W/S is in units of newtons per square meter and m is in units of kilograms, Kur = 5.81 x 10−5 for a zero flap deflection and 3.16 x 10−5 for maximum flap deflection These values are based on correlations for a number of civil transports, and are approximate only In regard to die induced drag during the ground roll, the downwash is somewhat inhibited by the proximity of the ground, and hence the induced drag contribution is less than for the ground roll, K must be reduced below that for the airplane in flight The reduction in the induced drag coefficient can be approximated by the relation from Ref 50 given here 𝐶𝐷𝑖 (𝑖𝑛 − 𝑔𝑟𝑜𝑢𝑛𝑑 𝑒𝑓𝑓𝑒𝑐𝑡) (16ℎ/𝑏)2 ≡𝐺= (49) 𝐶𝐷𝑖 (𝑜𝑢𝑡 − 𝑔𝑟𝑜𝑢𝑛𝑑 𝑒𝑓𝑓𝑒𝑐𝑡) + (16ℎ/𝑏)2 where h is the height of the wing above the ground and b is the wingspan 𝑚 78 79 AIRCRAFT DESIGN II 𝑆𝑔 = 3.2 𝐾𝐴 𝑙𝑛 (1 + 𝑉 ) + 𝑁𝑉𝐿𝑂 2𝑔𝐾𝐴 𝐾𝑇 𝐿𝑂 (50) Calculation of Distance While Airborne to Clear an Obstacle The flight path after liftoff is sketched in Fig below This is essentially the pull-up maneuver discussed in Section 6.3 In this Fig., R is the tum radius: 𝑉∞2 (51) 𝑅= 𝑔(𝑛 − 1) During the airborne phase, Federal Air Regulations (FAR) require that increase from 1.1 𝑉𝑠𝑡𝑎𝑙𝑙 at liftoff to 1.2𝑉𝑠𝑡𝑎𝑙𝑙 as it clears the obstacle of height h OB Therefore, we assume that in Eq (51) is an average value equal to 1.15 𝑉𝑠𝑡𝑎𝑙𝑙 The load factor n in Eq (51) is obtained as follows The average lift coefficient during this airborne phase is kept slightly less than (𝐶𝐿 )𝑚𝑎𝑥 for a margin of safety; we assume C L = 0.9(𝐶𝐿 )𝑚𝑎𝑥 Hence, 𝜌∞ (1.15𝑉𝑠𝑡𝑎𝑙𝑙 )2 𝑆(0.9)(𝐶𝐿 )𝑚𝑎𝑥 𝐿 (52) 𝑛= = 𝑊 𝑊 In Error! Reference source not found., 𝜃𝑂𝐵 is the included angle of the flight path between the point of takeoff and that for clearing the obstacle of height hOB From this figure, we see that 𝑅 − ℎ𝑂𝐵 ℎ𝑂𝐵 𝐶𝑜𝑠𝜃𝑂𝐵 = =1− 𝑅 𝑅 Or ℎ𝑂𝐵 (53) ) 𝜃𝑂𝐵 = 𝑐𝑜𝑠 −1 (1 − 𝑅 Also from the geometry of Fig., we have 𝑠𝑎 = 𝑅𝑠𝑖𝑛𝜃𝑂𝐵 (54) Figure III-7 Sketch for the calculation of distance while airborne Return to the design problems % TAKE-OFF CLmaxTO = 1.6; V_stallTO = sqrt(2/rho_SL*wingloading/CLmaxTO) V_LO = 1.1*V_stallTO PERFORMANCE T_TO = P_max/(0.7*V_LO) K_uc = 3.16e-5; Delta_CD0 = wingloading*K_uc*m^(-0.215) G = ((16*0.7/b_w)^2)/(1 + (16*0.7/b_w)^2) K_A = -rho_SL/(2*wingloading)*(CD0+Delta_CD0+(0.0221 + G/(pi*0.8*AR_w))*0.1^2 0.04*0.1) K_T = (T_TO/W - 0.04) s_groundTO = 1/(2*9.81*K_A)*log(1+K_A/K_T*V_LO^2)+3*V_LO R_TO = (1.15*V_stallTO)^2/(9.81*(1.15 - 1)) % n_TO = 1.2 theta_OB = acosd(1 - 10/R_TO) s_A = R_TO*sind(theta_OB) S_TO = s_groundTO + s_A Results 𝑉𝑠𝑡𝑎𝑙𝑙𝑇𝑂 = 24.0496 𝑚/𝑠 𝑉𝐿𝑂 = 26.4546 𝑚/𝑠 𝑇𝑇𝑂 = 9.2882e + 03 𝑁 Δ𝐶𝐷0 = 0.0046 𝐺 = 0.6836 𝐾𝐴 = −1.8492e − 05 𝐾𝑇 = 1.6537 𝑠𝑔𝑇𝑂 = 101.0176 𝑚 𝑅𝑇𝑂 = 519.8176 𝑚 𝜃𝑂𝐵 = 11.2567𝑜 𝑠𝐴 = 101.4709 𝑚 𝑠𝑇𝑂 = 202.4886 𝑚 LANDING PERFORMANCE The analysis of the landing performance of an airplane is somewhat analogous to that for takeoff, only in reverse Consider an airplane on a landing approach The landing distance, as sketched in Figure III-8, begins when the airplane clears an obstacle, which is taken to be 50 ft in height At that instant the airplane is following a straight approach path with angle Q a , as noted in Figure III-8 The velocity of the airplane at the instant it clears the obstacle, denoted by V0, is required to be equal to 1.3Vstall for commercial airplanes and 1.2Vstall for military airplanes At a distance h f above the ground, the airplane begins the flare, which is the transition from the straight approach path to the horizontal ground roll The flight path for the flare can be considered a circular arc with radius R, as shown in Figure III-8 The distance measured along the ground from the obstacle to the point of initiation of the flare is the approach distance sa Touchdown occurs when the wheels touch the ground The distance over the ground covered during the flare is the flare distance s f The velocity at the touchdown VTD is 1.15Vstall for commercial airplanes and 1.1Vstall for military airplanes After touchdown, the airplane is in free roll for a few seconds before the pilot applies the brakes and/or thrust reverser The free-roll distance is short enough that the velocity over this length is assumed constant, equal to VTD The distance that the airplane rolls on the ground from touchdown to the point where the velocity goes to zero is called the ground roll s g 80 81 AIRCRAFT DESIGN II Figure III-8 The landing path and landing distance Calculation of Approach Distance 𝑠𝑎 = Calculation of Flare Distance Calculation of Ground Roll 50 − ℎ𝑓 𝑇𝑎𝑛𝜃𝑎 𝑠𝑓 = 𝑅𝑠𝑖𝑛𝜃𝑎 𝐽𝐴 ) 𝑙𝑛 (1 + 𝑉𝑇𝐷 2𝑔𝐽𝐴 𝐽𝑇 Specifically, s g depends on wing loading, maximum lift coefficient, and (if used) the reverse thrust-to-weight ratio We note that s g increases with an increase in W/S s g decreases with an increase in (C^)max Sg decreases with an increase in T m j W s g increases with a decrease in A»𝑠𝑔 = 𝑁𝑉𝑇𝐷 + Return to the design problems % LANDING CLmaxLD = 1.6; V_stallLD = sqrt(2/rho_SL*wingloading/CLmaxLD) V_f = 1.23*V_stallLD V_TD = 1.15*V_stallLD R_LD = V_f^2/(0.2*9.81) theta_A = h_f = R_LD*(1-cosd(theta_A)) s_approach = (10 - h_f)/tand(theta_A) s_f = R_LD*sind(theta_A) PERFORMANCE J_T = 0.7 Delta_CD0LD = 0.702*Delta_CD0 J_A = rho_SL/(2*wingloading)*(CD0 + Delta_CD0LD + (0.0221 + G/(pi*0.7*AR_w))*0.1^2 - 0.8*0.1) s_groundLD = 1*V_TD + 1/(2*9.81*J_A)*log(1 + J_A/J_T*V_TD^2) S_LD = s_approach + s_f + s_groundLD Results 𝑉𝑠𝑡𝑎𝑙𝑙𝐿𝐷 = 24.0496 𝑚/𝑠 𝑉𝑓 = 29.5810 𝑚/𝑠 𝑉𝑇𝐷 = 27.6570 𝑚/𝑠 𝑅𝐿𝐷 = 445.9917 𝑚 𝜃𝐴 = 7𝑜 ℎ𝑓 = 3.3244 𝑠𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ = 54.3688 𝑚 𝑠𝑓 = 54.3527 𝐽𝑇 = 0.7000 (Δ𝐶𝐷0 ) = 0.0032 𝐿𝐷 𝐽𝐴 = −6.5044e − 05 𝑠𝑔𝑟𝑜𝑢𝑛𝑑𝐿𝐷 = 85.4302 𝑚 𝑠𝐿𝐷 = 194.1516 𝑚 82 83 AIRCRAFT DESIGN II REFERENCES [1] Robert C Nelson, Flight Stability and Automatic Control, New York: McGraw-Hill, 1942 [2] John D Anderson, Aircraft Performance and Design, New York: McGraw-Hill, 1942 [3] George E Dieter, Linda C Schmidt, Engineering design, 4th ed., New York: McGraw-Hill, 2009 [4] Letures: Flight mechanics – Ph.D Ngô Khánh Hiếu, Department of Aerospace Engineering, HCM City- University of technology [5] John D Anderson, Jr., Fundamentals of Aerodynamics, 2nd ed., McGraw-Hill, New York, 1991 [6] Daniel P Raymer, Aircraft Design: A Conceptual Approach, 2nd ed., AIAA Education Series, American Institute of Aeronautics and Astronautics, Washington, 1992 [7] Jack D Mattingly, Aircraft Propulsion, McGraw-Hill, New York, 1995 [8] Nguyen X Vinh, Flight Mechanics of High-Performance Aircraft, Cambridge University Press, Cambridge, England, 1993 [9] E.L Houghton and P.W Carpenter, Aerodynamics for Engineering Students, 5th ed., Butterworth-Heinemann, 2003 APPENDIX MATLAB codes for Part II: Stability clear all clc V = 55; % van toc m/s m = 559; % khoi luong kg M = 0.1674; % so Mach % ON DINH TINH % thong so hinh hoc cua may bay % canh chinh S_w = 9.67; % dien tich canh m2 b_w = 7.62; % sai canh m AR_w = b_w^2/S_w; i_w = 2*pi/180; % goc dat canh c_w = 1.2587; % day cung canh (trung binh) % bien dang canh chinh NACA 4415, lay thong so tu profili Cl_alphaw = 5.867; CL_0w = 0.5196; Cm_acw = -0.09; CL_alphaw = Cl_alphaw/(1+Cl_alphaw/(pi*AR_w)); % duoi ngang NACA 0012, lay thong so tu profili b_t = 3.187; % 2.6 sai canh duoi ngang S_t = 1.6924; % dien tich duoi ngang AR_t = b_t^2/S_t; Cl_alphat = 6.486; CL_alphat = Cl_alphat/(1+Cl_alphat/(pi*AR_t)); l_t = 3.1289; % khoang canh giua tam ap suat cua duoi va tam may bay i_t = 2*pi/180; % goc dat duoi % anh huong cua duoi ngang V_H = (l_t*S_t)/(S_w*c_w); epsilon_0 = 2*CL_0w/(pi*AR_w); APPENDIX diff_epsilon_alpha = 2*CL_alphaw/(pi*AR_w); Cm_0t = V_H*CL_alphat*(epsilon_0 + i_w - i_t); Cm_alphat = -V_H*CL_alphat*(1-diff_epsilon_alpha); % anh huong cua than (fuselage) - uoc luong theo 2.3.6 Nelson Cm_0f = 0.00048; Cm_alphaf = 0.0136; % tong hop % o vi tri 25%c Cm_0 = Cm_acw + Cm_0t + Cm_0f Cm_alpha = Cm_alphat + Cm_alphaf alpha_trimpitch = -Cm_0/Cm_alpha*180/pi % stick fixed neutral point x_NP = 0.25 - Cm_alphaf/CL_alphaw + V_H*CL_alphat/CL_alphaw*(1diff_epsilon_alpha); % x_NP/c % dieu khien doc % tu S_e va S_t, theo thi 2.21 Nelson suy tau_e tau_e = 0.4; CL_deltae = S_t/S_w*CL_alphat*tau_e; Cm_deltae = -V_H*CL_alphat*tau_e; CL_level = m*9.81/(0.5*0.90925*V^2*9.67); CL_0 = CL_0w + S_t/S_w*CL_alphat*(i_t - i_w - epsilon_0); CL_alpha = CL_alphaw + S_t/S_w*CL_alphat*(1-diff_epsilon_alpha); CL_trim = CL_level - CL_0; % goc trim elevator delta_etrim = -(Cm_0*CL_alpha + Cm_alpha*CL_trim)/(Cm_deltae*CL_alpha Cm_alpha*CL_deltae); alpha_trim = (CL_trim - CL_deltae*delta_etrim)/CL_alpha*180/pi delta_etrim_degree = delta_etrim*180/pi % dieu khien huong % anh huong cua than va canh (dung pp sach Nelson) CL_alphav = CL_alphat l_v = 2.9554; S_v = 0.7317; S_r = 0.1785; z_v = 0; AR_v = 1.1747; V_v = l_v*S_v/(S_w*b_w); Cn_betawf = -0.0275; Cn_betav = 0.06945; Cn_beta = Cn_betawf + Cn_betav; % aileron control S_aileron = 0.5644; % dien tich aileron tau_a = 0.19; CL_deltaa = CL_alphaw*tau_a*S_aileron/(S_w*b_w); Cn_deltaa = -2*(-0.13)*CL_level*CL_deltaa; % rudder control dihedral_angle = 5; Cl_beta_dihedral = -0.00025*dihedral_angle*180/pi delta_Cl_beta = -0.0002; Cl_beta = Cl_beta_dihedral + delta_Cl_beta; % PHUONG TRINH LUC CAN % dung phuong phap roskam, tinh duoc CD = CD_0 + kCL^2 CD_0 = 0.01584; 84 85 AIRCRAFT DESIGN II k = 0.088; CD = CD_0 + k*CL_level^2; % CAC HE SO DAO HAM e = 0.6; K_w = 1/(pi*AR_w*e); K_t = *S_t/(S_w*pi*AR_t*e); K_v = 1*S_v/(S_w*pi*AR_v*e); I_y = 713.27; I_x = 538.5; I_z = 1080.88; V_level = V Q = 1/2*0.90925*V_level^2; u_o = V_level; Cl_0 = CL_level ; Cd_0 = CD; Cx_u = -3*Cd_0; Cx_alpha = Cl_0 - 2*Cl_0*CL_alpha/(pi*e*AR_w); Cz_u = -2*Cl_0; Cz_alpha = -(CL_alpha + Cd_0); Cz_alpha_cham = -2*CL_alphat*V_H*diff_epsilon_alpha; Cz_q = -2*CL_alphat*V_H; Cz_delta_e = - CL_deltae; Cm_u = 0; Cmalpha = Cm_alpha; Cmalpha_cham = -2*CL_alphat*V_H*l_t*diff_epsilon_alpha/c_w; Cmq = -2*CL_alphat*V_H*l_t/c_w; Cm_delta_e = -V_H*CL_alphat*tau_e; X_u = Cx_u*(1/u_o)*(Q*S_w/m); X_w = -((K_w+K_t+K_v)*CL_alpha-CL_0)*Q*S_w/(u_o*m); X_alpha = Cx_alpha*(Q*S_w/m); Z_u = Cz_u*(1/u_o)*(Q*S_w/m); Z_w = -(CL_alpha + Cd_0)*Q*S_w/(u_o*m); Z_alpha =Cz_alpha* (Q*S_w/m); Z_alpha_cham = Cz_alpha_cham*(c_w/(2*u_o))*(Q*S_w/m); Z_q = Cz_q*(c_w/(2*u_o))*(Q*S_w/m); Z_delta_e = Cz_delta_e*(Q*S_w/m); M_u = Cm_u*(1/u_o)*(Q*S_w*c_w/I_y); M_w = Cmalpha*Q*S_w*c_w/(u_o*I_y); M_w_cham = Cmalpha_cham*c_w*Q*S_w*c_w/(2*u_o^2*I_y); M_alpha = Cmalpha*(Q*S_w*c_w/I_y); M_alpha_cham = Cmalpha_cham*(c_w/(2*u_o))*(Q*S_w*c_w/I_y); M_q = Cmq*(c_w/(2*u_o))*(Q*S_w*c_w/I_y); M_delta_e = Cm_delta_e*(Q*S_w*c_w/I_y); Lamda = 1; S_r_S_v = S_r/S_v; To_r = 0.51; nu_v_xicma_beta = 1.0086; % tu phan on dinh tinh huong C_y_beta = -nu_v_xicma_beta*(S_v/S_w)*CL_alphav; C_y_p = 0;%unswept wing C_y_beta_tail = -(S_v/S_w)*CL_alphav; C_y_r = -2*(l_v/b_w)*C_y_beta_tail; C_y_delta_r = (S_v/S_w)*CL_alphav*To_r; C_n_p = -CL_level/8; C_n_r = -2*V_v*(l_v/b_w)*CL_alphav; C_n_delta_r = -V_v*To_r*CL_alphav; C_l_p = -(CL_alpha/12)*(1+3*Lamda)/(1+Lamda); C_l_r = CL_level/4 - 2*l_v*z_v*C_y_beta_tail/(b_w^2); APPENDIX C_l_delta_r = (S_v/S_w)*(z_v/b_w)*To_r*CL_alphaw; Y_beta = C_y_beta*(Q*S_w/m); Y_p = C_y_p*(b_w/(2*u_o))*(Q*S_w/m); Y_r = C_y_r*(b_w/(2*u_o))*(Q*S_w/m); Y_delta_r = C_y_delta_r*(Q*S_w/m); N_beta = Cn_beta*Q*S_w*b_w/I_z; N_p = C_n_p*Q*S_w*b_w^2/(2*I_x*u_o); N_r = C_n_r*Q*S_w*b_w^2/(2*I_x*u_o); N_delta_a = Cn_deltaa*Q*S_w*b_w/I_z; N_delta_r = C_n_delta_r*Q*S_w*b_w/I_z; L_beta = Cl_beta*Q*S_w*b_w/I_x; L_p = C_l_p*Q*S_w*b_w^2/(2*I_x*u_o); L_r = C_l_r*Q*S_w*b_w^2/(2*I_x*u_o); L_delta_a = CL_deltaa *Q*S_w*b_w/I_x; L_delta_r = C_l_delta_r*Q*S_w*b_w/I_x; % ON DINH DONG % on dinh dong doc A = [X_u X_w -9.81 Z_u Z_w u_o (M_u+M_w_cham*Z_u) (M_w+M_w_cham*Z_w) (M_q+M_w_cham*u_o) 0 0] tririeng1 = eig(A) % short-mode clear x y figure(1) x=(0:0.05:3); for i=1:length(x) y(i)=1exp(real(tririeng1(1))*x(i))*(cos(imag(tririeng1(1))*x(i)+abs(real(tririeng1(1))) /imag(tririeng1(1))*sin(imag(tririeng1(1))*x(i)))); end plot(x,y) grid on title('Short-period Mode'); xlabel('t'); % long-mode clear x y figure(2) x=(0:0.05:400); for i=1:length(x) y(i)=1exp(real(tririeng1(3))*x(i))*(cos(imag(tririeng1(3))*x(i)+abs(real(tririeng1(3))) /imag(tririeng1(3))*sin(imag(tririeng1(3))*x(i)))); end plot(x,y) grid on title('Long-period Mode'); xlabel('t'); % on dinh dong huong clear A A = [ -N_beta N_r] tririeng2 = eig(A) clear x y figure(3) x=(0:0.05:10); for i=1:length(x) 86 87 AIRCRAFT DESIGN II y(i)=1exp(real(tririeng2(1))*x(i))*(cos(imag(tririeng2(1))*x(i)+abs(real(tririeng2(1))) /imag(tririeng2(1))*sin(imag(tririeng2(1))*x(i)))); end plot(x,y) grid on title('Yawing Motion'); xlabel('t'); % on dinh dong ngang clear A A = [Y_beta/u_o Y_p/u_o -(1-Y_r/u_o) 9.81/u_o L_beta L_p L_r N_beta N_p N_r 0 0] tririeng3 = eig(A) % spiral mode clear x y figure(4) t=(0:0.05:200); delta_a=(0:1:4); for j=1:length(delta_a) % cot delta_a for i=1:length(t) % hang t y(i,j)=abs(L_delta_a/L_p)*(1-exp(t(i)*abs(real(tririeng3(4)))))*delta_a(j)*pi/180; end end plot(t,y(:,1),'r',t,y(:,2),'g',t,y(:,3),'b',t,y(:,4),'c',t,y(:,5),'k') grid on title('Spiral Mode'); xlabel('t'); legend('delta_a=0','delta_a=1','delta_a=2','delta_a=3','delta_a=4'); % roll mode clear x y figure(5) t=(0:0.05:1); delta_a=(0:1:4); for j=1:length(delta_a) % cot delta_a for i=1:length(t) % hang t y(i,j)=abs(L_delta_a/L_p)*(1-exp(t(i)*abs(real(tririeng3(1)))))*delta_a(j)*pi/180; end end plot(t,y(:,1),'r',t,y(:,2),'g',t,y(:,3),'b',t,y(:,4),'c',t,y(:,5),'k') grid on title('Roll Mode'); xlabel('t'); legend('delta_a=0','delta_a=1','delta_a=2','delta_a=3','delta_a=4'); % dutch-roll mode clear x y figure(6) x=(0:0.05:10); for i=1:length(x) y(i)=1exp(real(tririeng3(2))*x(i))*(cos(imag(tririeng3(2))*x(i)+abs(real(tririeng3(2))) /imag(tririeng3(2))*sin(imag(tririeng3(2))*x(i)))); end plot(x,y) grid on title('Dutch-roll Mode'); xlabel('t'); ... × 1 .25 87 2 12 2 6