Engineering.purdue.edu
Thiokol Final Report (Updated)
AAE 451 Aircraft Design
Fall 2006
12/2/06
Team 1
Dane Batema dbatema@purdue.edu USA
Benoit Blier bblier@purdue.edu France
Drew Capps dcapps@purdue.edu USA
Patricia Roman promanas@purdue.edu Panama
Kyle Ryan kpryan@purdue.edu USA
Audrey Serra aserra@purdue.edu France
John Tapee jtapee@purdue.edu USA
Carlos Vergara jcvergar@purdue.edu Peru
Table of contents
Introduction 1
1 Aerodynamics 2
1.1 Introduction 2
1.2 Airfoil Selection: Wing and Tail 2
1.3 Geometry: Wing and Tail 3
1.4 Mathematical Model: CL, CD, CM 5
2 Propulsion System 7
2.1 Propeller Analysis 7
2.2 Battery System Analysis 8
2.3 Gearbox/Motor Selection 10
2.4 Speed Controller Selection 10
2.5 Other Mission Requirements 11
3 Structures and Weights 13
3.1 Introduction 13
3.2 Preliminary Weight Estimation 13
3.3 Geometric Layout of Wing Structure 13
3.4 Analysis of Wing Loads 14
3.5 Horizontal Tail 16
3.6 Vertical Tail Structure 16
3.7 Fuselage Structure 16
3.8 Wing-Fuselage Attachment 17
3.9 Landing Gear 17
3.10 CG Estimation 17
4 Dynamics and Controls 19
4.1 Introduction 19
4.2 Preliminary Design Considerations 19
4.3 Center of Gravity and Aerodynamic Center 19
4.4 Sizing of Horizontal and Vertical Tail 20
4.5 Dynamic Stability 22
Conclusion 25
Addendum: Lessons Learned and Vehicle Summary 26
Tables of Constants 26
Propulsion 26
Aerodynamics 27
Controls 27
Structures 27
Resources Used 27
Appendix A: List of Symbols (Variables) 27
Appendix B: Aerodynamic Figures and Data 27
Appendix C: Propulsion Figures and Explanations 27
Appendix D: Structures Figures and Tables 27
Appendix E: Dynamics and Controls Figures and Tables 27
Introduction
This project’s mission is to design a high-speed, electric powered, unmanned aerial vehicle to be piloted by an experienced RC pilot. Designing an aircraft for a high speed mission has distinct challenges. The aircraft must be able to complete the following mission profile: takeoff, climb to 20 ft, high–speed dash to a target ¼ mile away, loiter for 5 minutes, and return home at the most economical speed. In addition, the aircraft must be able to carry a one pound payload, making this mission very unique since all preexisting RC racing aircraft carry no payload. In addition to combining payload capacity with high-speed, the aircraft is limited to a budget of $250 (excluding radio-control gar, speed controller, and rate gyro). These three requirements have made the design and construction of the aircraft rather difficult. Required performance properties include the following: Take-off distance and landing distance ≤ 120 ft of ground roll, take-off with minimum climb angle of 35 degrees, and Vstall ≤ 30 ft/sec, where Vstall is the level flight speed at CLmax. The primary factor which drove the design was, obviously the high speed requirement, however the $250 budget essentially limited the size of chosen propulsion system, thus the dash speed of the airplane. Secondarily, the stall speed of 30 ft/sec created the need to have a wing capable of not only flying at high speed but to produce large amounts of lift at low speed. Lastly, the takeoff and climb angle constraints proved to be minor with a propulsion system powerful enough to fly at speeds approaching 115 ft/s. More “soft” requirements included producing an aircraft that is efficient under loading, crash resistant, controllable and stable at the desired speeds, and buildable within a three-week window; these largely dictated the structural design and materials selection for the aircraft. Overall, the final design sufficiently fulfills these requirements; however, minor changes will be made during the actual building process which cannot be anticipated. Given the team members’ experience building and flying model aircraft, and a sound “on paper” design, the team is confident that this design will be successful.
Aerodynamics
1 Introduction
The problems faced in this mission with regard to aerodynamics are challenging and encompass a wide range of flight conditions. First and foremost, the aircraft will have to operate at a wide range of speeds, from the low stall speed (≤ 30 ft/s) to the high speed dash. The aircraft will also have to operate from a short field, with a takeoff and landing distance of less than 120 feet. As with any aircraft, weight and cost must be minimized as much as possible. Ultimately, the most desirable wing design will strike a balance across the design requirements.
For takeoff and climb, our aircraft will need to get off the ground in 120 feet and climb at 35 degrees to an altitude of 20 feet. Additionally, the aircraft must have a loiter time of at least seven minutes. These requirements push the design towards low drag and high lift wings related to the weight of the aircraft. However, for the high speed dash, smaller wings are more desirable than larger ones. Drag is another significant factor in the speed of the aircraft. To produce the fastest aircraft and still meet all design requirements, this team has decided to design the aerodynamics to barely meet the other design requirements and then minimize drag and weight for top speed. For example, we will cover our landing gear in an aerodynamic fairing to reduce drag. Thinner airfoils also reduce drag but suffer penalties in maximum lift; to account for this, devices like flaps can be used, but these add weight and cost. Flaperons are the best alternative, as they simply use an existing control surface to produce sufficient lift at low speeds.
Given seemingly conflicting aerodynamic requirements (slow v. fast), the best aerodynamic performance will be produced by first meeting all requirements except high speed and then minimizing drag and weight.
2 Airfoil Selection: Wing and Tail
One of the first steps in the aerodynamic design of any aircraft is the selection of the airfoil for the wing and tail. There are a number of considerations to make when choosing the airfoil. For this aircraft, the high speed performance is an important factor to consider. While the mission focuses on high speed, the speeds the aircraft will operate at are still relatively low (in terms of Mach number); coupled with the small size of the aircraft, this gives a low Reynolds number. A low Reynolds number adds some complication because many airfoils with good experimental data are designed for higher Reynolds numbers.
We primarily used historical trends to select the airfoil that would meet the high speed dash mission. Electric Pylon racers were analyzed because they most closely reflect the high speed mission. The Martin Hepperle MH 43 (Figure 1.1 below) is a common airfoil used in these Pylon racers. The MH 43 is also a low Reynolds number airfoil. NACA airfoils were not chosen because they are not as well optimized for our flight conditions. NACA 1-series attempt to get low drag and high critical Mach number. NACA 4-series have a large pressure spike at the leading edge. NACA 6-series are useful for transonic flow and NACA 7-series attempt to get a large percentage of laminar flow.
[pic]
Figure 1.1 – MH 43 (Wing)
For the tail, lift is not a driving factor, so a symmetric airfoil can be used. This has the benefit of a lower induced drag than that of a cambered airfoil of the same thickness. Although the NACA 0009 is a common airfoil used in tail surfaces in numerous aircraft, the NACA 0006 airfoil was chosen because the thinner airfoil will reduce parasite drag (Figure 1.2 below). Increasingly thin airfoils become difficult to build structurally, and the NACA 0006 strikes a balance between low drag and ease of construction.
[pic]
Figure 1.2 – NACA 0006 (Horizontal Tail)
3 Geometry: Wing and Tail
The preliminary sizing of the wing and tail were carried out by observing historical trends. The target maximum velocity as set by the propulsion team is 118 ft/s. The wing area was fixed from the constraint diagram (Appendix Figure B-3) and a initial weight estimate of 5 lbs. The taper ratio was fixed at .45 to closely approximate an elliptic lift distribution, which minimizes the induced drag. The aspect ratio was determined from a trade study to minimize drag (Appendix B-1.2). From these values, the span, root chord, tip chord, and sweep angle of the wing were found. For the tail, a similar method is used – a trade study optimized the aspect ratio of the tail with respect to minimum drag. The horizontal tail area was fixed by the dynamics and control sizing, and the taper ratio was fixed from historical trends. From these values, the span, root chord, tip chord, and sweep angle of the tail were found. Table 1.1 shows the important wing and tail sizing values.
Table 1.1 – Important Wing and Tail Parameters
| |Wing |Horizontal tail |
|Airfoil |MH 43 |NACA 0006 |
|Aspect ratio |5.76 |5.44 |
|Taper ratio |0.45 |0.60 |
|Sweep angle (quarter chord) |11.18° |7.85° |
|Span |4.90ft |2.01ft |
|Area |4.16ft2 |0.74ft2 |
To calculate the total wetted area, the aircraft was first broken up into general shapes, and their areas calculated (Appendix Figure B-4). The method and equations used can be found in Raymer’s book, Aircraft Design: A Conceptual Approach, Fourth Edition. Once completed, the CATIA model provided a more accurate value for the wetted area of 13.47 ft2.
One of the most important aerodynamic properties of an aircraft is the aspect ratio of the wing. Since the design mission contains both high speed and low speed requirements, selecting the aspect ratio becomes an exercise in balancing the different requirements. When choosing the aspect ratio, it is important to minimize drag, (impacting all areas of performance) and to keep the aircraft geometry in mind so that wing design is feasible.
The trade study showed that the optimum AR increases as the angle of attack, α, increases. This result agrees with modern aircraft designs, as low AR wings are faster in level flight than high AR wings. This is because the CDo term dominates drag at lower angles of attack, and CDo should be minimized for high speed level flight. Figure 1.3 shows this trend.
[pic]
Figure 1.3 – CD v. AR
While the optimum AR at α = 0° is 4, the length of the required root chord becomes impractical. At AR 4, the root chord is a little over half the total length of the aircraft. This causes problems with the dynamics and control of the aircraft, as roll becomes less stable, and the moment arm for the elevator decreases. At α = 8°, the optimum AR increases to 5.76. This AR has a more reasonable root chord to length ratio of a little over one third. Since the mission encompasses both speed and loiter phases, an AR of 5.76 was selected – this strikes a balance between the lower AR needed for speed, and the higher AR needed for loiter.
Although the primary focus of the design mission is high speed flight, there is also a stall speed requirement that must be respected (Vstall ≤ 30 ft/s). During the high speed phase, drag must be minimized, while during low speed phases (take off and landing), lift must be maximized. Adding flaps to the wing/airfoil is a good way to meet these two requirements. Flaps allow a low-lift airfoil, designed for high speed, to generate greater lift at low speeds by changing the airfoil shape to increase its camber. However, adding flaps means adding drag (due to the mechanism), complexity of structure (difficult to build), and, of course, weight (due to the mechanism). If flaperons are used, these issues with weight and drag increases are reduced since the same control surface is used as a flap and as an aileron.
An analysis of the aircraft geometry shows that the CLmax needed to meet the stall speed requirements is 1.155, yet the CLmax attainable without flaps is 0.81. After finding the optimum flap configuration (flap hinge location: x/c = 0.825 and deflection angle: δf = 10° – see appendix for details), the new CLmax (with flaps) is 1.16. This new CLmax should provide the necessary lift at low speed to ensure Vstall ≤ 30 ft/s. Using flaps is the best way to meet the requirements of both the high speed dash and the stall speed.
4 Mathematical Model: CL, CD, CM
The mathematical model used for the aerodynamics underwent several stages. The first stage was an approximate transition from 2-D information from XFOIL to 3-D. The preliminary lift coefficient was found by considering the wing only. This model was updated by adding in the effect of the tail. The preliminary parasite drag coefficient was found using the equivalent skin friction method. This method was updated by using the component buildup method. Equation 1.1 shows this method.
[pic] (Equation 1.1)
The total CD was found using Equation 1.2, shown below.
[pic] (Equation 1.2)
The total CL was found using Equation 1.3, shown below.
[pic] (Equation 1.3)
Equation 1.4 was used to transform 2D coefficients to 3D.
[pic] (Equation 1.4)
In Equation 1.4, the subscript c represents either the wing or tail. The total moment coefficient was found using Equation 1.5.
[pic] (Equation 1.5)
Appendix Figure B-5 shows the lift curve and drag polar found from these models. Table 1.2 shows the results of the important aerodynamic coefficients.
Table 1.2 – Important Aerodynamic Coefficients
|Coefficient |Value |
|CLo |0.088 |
|CDo |0.025 |
|CMo |0.056 |
|CLα |4.821/rad |
|CLih |0.769/rad |
|CMih |-1.993/rad |
|CMα |-0.697/rad |
|CLδe |0.466/rad |
|CMδe |-1.206/rad |
Propulsion System
The propulsion system design was primarily focused on meeting the requirements of the high speed dash. To that extent, it was necessary to size the battery, motor, gearbox, and propeller for optimum high speed performance. After designing the propulsion system for dash performance, the system had to be thoroughly examined to ensure that the other mission requirements were also met.
1 Propeller Analysis
The first step in designing the propulsion system was determining how the different geometric properties of the propeller, i.e. its pitch and diameter, affected its performance. The primary program available for propeller analysis was the gold.m program. A discussion of gold.m program is given in Appendix C-1.1. Since there was an experience based factor that related previously measured power input to gold.m predicted power input, the default inputs to this code were used. For propeller speeds below 10,000 RPM, actual power required will be 1/0.75=33% higher than the power required based on gold.m analysis. In addition, for propeller speeds greater than 10,000 RPM, as much as twice the power predicted will be required for the actual propeller. This drastic change in propeller efficiency around 10,000 RPM imposed a maximum speed on the propeller. The results of the gold.m program were the non-dimensional thrust coefficients (CT) and power coefficients (CP) as a function of advance ratio (J) for several values of pitch/diameter ratio. These quantities were used to calculate the propeller efficiencies. Appendix Figure C-1 shows the propeller efficiency as a function of advance ratio for several different pitch/diameter ratios.
From this figure, it was seen that larger pitch/diameter ratios yielded larger peak efficiency and a wider range of advance ratios for a given efficiency level. Since the aircraft was required to fly efficiently at both high and low speeds, flexibility in flight advance ratio was a very important quality for the propeller to have. From this figure, it was also seen that as pitch/diameter ratio was increased the peak efficiency occurred at a higher advance ratio. For a fixed maximum airspeed, a larger diameter propeller was required to in order to fly at a higher advance ratio (see Appendix C-1.2). A study was conducted to determine the pitch/diameter ratio for optimum high speed dash propeller efficiency for a given propeller diameter.
The first step in this analysis was to determine the effect that freestream velocity had on propeller predicted efficiency. Appendix Figure C-2 shows the propeller efficiency vs. airspeed for a large variety of propeller diameters and pitch/diameter ratio of 1.0. This figure clearly shows that for high airspeeds, propeller efficiency is essentially constant for a given propeller diameter. This is because the needed coefficient of lift is extremely small at high dynamic pressures, so the coefficient of drag is very close to the constant CD0 value, leading to an essentially constant CD value. See Appendix C-1.3 for a complete discussion of this effect.
Next, a sampling of propeller diameters was examined to determine the required rotational speed for a given flight speed. Appendix Figure C-3 shows propeller rotational speed vs. airspeed. For all airspeeds that the aircraft can be expected to obtain, 11 inch propellers were the smallest that provided the power required at rotational speeds below 10,000 RPM. The smallest propeller will have the least frontal area, and require the shortest landing gear. Both of these will minimize the drag of the aircraft. Appendix Figure C-4 shows the propeller efficiency vs. aircraft speed for a selection of 1.0 pitch/diameter ratio propellers. From this figure it can be seen that at high speeds, the efficiency is essentially constant, while for lower speeds, the efficiency decreases rapidly. In the low speed regions, the efficiency of all of the propellers falls off very rapidly. It was seen in this figure, as expected, that propeller efficiency increased as diameter increased.
The final step in this study was to examine the pitch/diameter ratio for optimum efficiency for a given propeller diameter. Figure 2.1 below shows the propeller efficiency vs. pitch/diameter ratio for several different propeller diameters for the high speed phase of the flight. This figure shows that for propeller diameters larger than 10 inches, the peak efficiency occurs for pitch/diameter ratios of 1.0. For smaller diameter propellers, the peak efficiency would be obtained for smaller pitch/diameter ratios. Since propeller rotational speed limits required the use to 11 inch propellers, the optimum efficiency is seen to be for pitch/diameter ratios between about 0.9 and 1.0. These pitch/diameter ratios correspond to 11x10 and 11x11 propellers.
[pic]
Figure 2.1: Propeller Efficiency v. Pitch/Diameter Ratio
2 Battery System Analysis
Since the final battery system used in this aircraft was chosen from commercially available battery systems, the design of this component of the propulsion system became more of an analysis of the best possible solutions available rather than a design of the optimum theoretical setup. From initial analysis, lithium polymer (LiPo) batteries were identified as the battery type that would best meet the power and weight demands of the mission. The problem, however, was that the LiPo batteries were more expensive than the NiCad or NiMH alternatives. Given the $250 budget constraint and the main objective of the mission was high-speed flight, a limit was proposed for the total propulsion system to cost roughly $170, allotting $50 to the battery system. Obviously, this system needed to be as powerful as possible in order to fly the plane as fast as possible.
The first approach used to optimize the battery system involved plotting battery cost-per-cell against power-per-cell (see Figure C-6 and C-1.4 in Appendix). However, since the actual driving factors for the design are the total system cost and power output, this approach was deemed unsatisfactory. Instead, the total cost for the battery system required to produce a given power for each battery was calculated. Given a power required and a maximum current, the total voltage drop can be determined. Based on the voltage of a given battery pack which typically have two or three cells, the total number of cells required and total cost and weight of the system can be found. A spreadsheet was used to tabulate the total system cost for each battery and each value of power required. Appendix Table C-1 shows a section of this spreadsheet used to determine the optimal battery system. For example, this table shows that the minimum battery system cost for a required power of 0.7 hp is $50.00. This battery system lies at the self-imposed limit of $50, so it is a valid option. To arrive at the optimum system, the minimum system cost can be plotted against the power required (Figure 2.2 below).
[pic]
Figure 2.2: Minimum Battery Cost v. Battery Power Output
This plot shows that, for P ≤ 0.71 hp, a battery system can be designed that meets the cost constraint. Obviously, the maximum power attainable is desired; thus, the battery system that costs $50 and produces just over 0.7 hp is the ideal candidate. This system consists of two Apogee 3-cell, 1200 mAh, 20C battery packs, and is the same system as is distinguished in Table C-1. This battery system will operate at 22.2 V and a fairly low maximum current (24 A). Since the heat generated by circuit resistance is proportional to the square of current (P = I2R), a low current results in less power losses and better overall efficiency. Another notable battery characteristic is the "Watts/$" value, which provides a measure of cost efficiency for the entire battery system. Plotting this cost efficiency against battery power (Figure 2.2 above) shows a few distinguishable peaks at about 10.5 W/$ for different battery power output. Interestingly, each of these peaks corresponds to the battery type chosen for this mission (each peak represents a different number of packs), supporting the battery selection by this team.
3 Gearbox/Motor Selection
Once the optimum battery pack was selected, the supply voltage and maximum continuous current were known quantities. A motor needed to be selected that could handle both the input voltage and maximum battery output current in order to avoid wasting potential battery output power. When looking across many motor manufacturers, Mega Motors was found to offer the most cost effective motors. The motor needed to have a reasonably low Kv value as our high supply voltage would require gear ratios higher than those commercially available to spin our large propeller at its desired speed. The Mega Motor which met all of these requirements was the Mega ACn 16/25/3. This motor has a Kv of 1700 RPM/V, a no load current Io of 1.9 A, and a resistance of 17 mΩ. Both the no load current and the resistance are measures of the inefficiency for an electric motor. The low values for this motor were found to yield very high motor efficiencies.
Once the propeller, motor, and battery were selected, the propeller maximum power output could readily be calculated. The details of this calculation are shown in Appendix C-1.6. The power required by the aircraft was also calculated as a function of airspeed. Appendix Figure C-8 shows the propeller power available and power required curves vs. airspeed. The upper point where these two curves intersect is the theoretical top speed the aircraft can achieve. The maximum speed from this analysis was found to be 118.4 ft/s. This required a gear ratio of 4.015. The constraint on cost and the required power transmission limited the gear box options to essentially one distributor and gear box type, severely restricting the available gear ratios. The closest gear ratio to the ideal value that could be found was the MP Jet 4.1 gear ratio ball bearing gearbox. This gearbox caused the motor to run at a lower voltage than the ideal gear ratio. This decreased our actual maximum speed to 117.7 ft/s. Tables 2.1 and 2.2 below give the system operating point for the dash and loiter phases of the mission respectively.
4 Speed Controller Selection
The only considerations in choosing a speed controller are its maximum voltage and current capability. Some commercially available speed controllers have a Battery Eliminator Circuit (BEC), which eliminates the need for a separate battery to power the servos and receiver. As input voltage increases, the number of servos the BEC can power diminishes. Our high operating voltage exclude all but extremely expensive speed controllers. The Castle Creations Phoenix 60 was chosen as it was determined to be capable of handling the propulsion system’s maximum voltage and current output. Since we cannot use the BEC, the aircraft will be required to incorporate a separate battery pack to power the servos and receiver. This configuration allows the maximum battery output to be applied to the motor without also having to power the servos. Also, if the main battery system fails or exhausts its power, the auxiliary servo battery will enable controllability even without motor power.
5 Other Mission Requirements
The design mission has two distinct phases: high-speed dash and loiter/endurance. Since high-speed flight requires more power than the lower speeds at which an endurance mission will be flown, the design of the propulsion system was based on the high-speed requirement. Given a battery system that could provide enough voltage and current for high speed flight, additional batteries could be added in parallel to increase the energy available if the system could not meet the endurance requirement. Since the testing airfield will prohibit the aircraft from flying straight for the required loiter time (7 minutes), this team defined the endurance mission as a steady level turn at 40 ft/s with a 200 ft. radius. Analyzing the propulsion system for this mission showed that the battery designed for high-speed flight will provide more than sufficient loiter time (20.7 minutes).
A key concern of the propulsion group was that the high pitch of the selected propeller might not generate enough low-speed thrust to meet the takeoff constraint. To analyze this takeoff performance, the thrust and acceleration of the aircraft was integrated over time and distance to ensure that an acceptable takeoff speed could be reached by the end of the runway (see Appendix C-1.7). The maximum power available from the propulsion system for a given speed was calculated using the J, CP, and CT values from gold.m and the current and voltage limits of the battery, as shown in Appendix Figure C-9. The change in slope at higher speeds occurs when the system becomes voltage limited. The thrust available falls quickly after this point because the propeller has reached its maximum rotational speed. The 11x11 propeller was chosen because it can fly to higher speeds with before being voltage limited. This evaluation also revealed an issue that had previously been overlooked: at low airspeeds, the motor must produce more torque to turn a high pitch propeller at a given RPM than at high speeds, but the current limit for the battery cannot be exceeded for fear of catastrophic battery failure. Thus, the pilot will only be able to use approximately 80% throttle when taking off and accelerating from low speeds. Nevertheless, even with this throttle limit, the aircraft will reach takeoff speeds well before it traverses all 120 ft. of the runway (Figure 2.3 below).
[pic]
Figure 2.3: Aircraft Velocity v. Ground Roll Distance for Takeoff
Structures and Weights
1 Introduction
The requirement to produce an aircraft that is efficient under loading, crash resistant, lightweight, and buildable within the three-week window will principally dictate the structural design and materials selection for the project. Inherently, the structures group must accommodate the needs of other systems first rather than dictate the design.
Due to high flight speeds, the aircraft must withstand the loading of high-speed, potentially tight turns. The size of wing spars and structural elements is directly related to predicted aerodynamic loading, estimated at 10gs, a decision driven mainly by expected turning radius, speed, and previous experience of the RC pilot. At the same time, the aircraft must be realistically crash resistant. Certain structural techniques such as breakaway structural elements and minor reinforcements to key structural areas like the wing attachment area were examined as possible ways to increase damage tolerance with minimal weight increase. The design must be highly weight efficient through proper member sizing and materials usage. Based on these concepts, a fuselage made of balsa wood and a wing core of lightweight polystyrene sheeted with balsa was chosen. The empennage will also be primarily made of balsa, with the horizontal stabilizer composed of a foam core/balsa skin similar to the wing. Based on trade studies and experience of the team members building and working with balsa, a fairly easy construction is expected.
2 Preliminary Weight Estimation
The maximum and loiter velocity performance was decided based on historical data and with Equation 3.1, those velocities were related to weight.
Equation 3.1:
With the battery weight ratio of every part of the flight mission, the battery weight and payload weight were plotted against the total weight for our aircraft. In addition, historical battery weight data was plotted and the intersection of the two lines gave the weight of 4.99 lbs for our aircraft (Appendix Figure D-1).
3 Geometric Layout of Wing Structure
The structural concept chosen for the wing is an expanded polystyrene core covered by a thin balsa skin (Figure 3.1). This construction requires fewer parts than an all-balsa internal wing structure, thus easier to build. A Balsa skin was chosen over fibreglass due to team member experience working with balsa sheeting and its superior surface quality and maintaining of an accurate airfoil shape.
[pic]
Figure 3.4: Wing Structural Concept
4 Analysis of Wing Loads
The loads provided by the aerodynamics group allowed the structure of the wing to be sized. The wing was discretized (Appendix Figure D-2) in several parts to calculate the loads on each part. The lift distribution was assumed to be linear and was calculated using Equation 3.2. The lift forces were located at the mean aerodynamic chord (MAC) of each part (Appendix Figure A-3 a-b) so that the bending moment (Figure 3.2 below) could be calculated for the wing with Equation 3.3.
Equation 3.2: [pic]
Equation 3.3 [pic]
[pic]
Figure 3.5: Bending Moment on the Wing
The wing was assumed to have a shape of an ellipse to calculate the moment of inertia which was used to calculate the stress (Equation 3.4) on the skin. Making the stress of the skin equal to the ultimate compression strength (UCS) of balsa, the optimal thickness for the minimal weight (Figure 3.3) was obtained.
Equation 3.4: [pic]
The deflection (Equation 3.5) for each section was calculated with the lift force then added to find the total deflection.
Equation 3.5: [pic]
From the polar of the airfoil, the maximum moment coefficient was obtained so the torque was calculated using Equation 3.6. Afterward the angle of twist was found using Equation 3.7.
Equation 3.6: [pic]
Equation 3.7: [pic]
Table 3.1 summarizes the properties found for the wing.
Table 3.1: Summary of Wing Properties
|Min thickness (in) |0.053 |
|Deflection (in) |0.11 |
|Max angle of twist (deg) |-0.4 |
5 Horizontal Tail
Calculations for the horizontal tail have been conducted in almost the same way as the wing. The difference is that in this case, only one discretization is considered for the half horizontal tail because of its small size. A load case of the high speed dash (115 ft/s) with a 20° deflection of the control surface was used to evaluate the structural properties needed. The total lift coefficient is given by the Equation 3.8. Then the lift force is obtained using the Equation 3.6. Afterward, the bending moment, the stress and the minimum thickness are calculated with the same method as the wing.
Equation 3.8:
All the values of this equation where calculated by the controls team. Using the values provided by the aerodynamics and controls teams, the minimum balsa skin thickness necessary is found to be 0.0138 in, producing a deflection of 0.41 in. The thinnest balsa sheet available is 1/32 in thick (0.065 in), and therefore will be the horizontal tail skin thickness.
6 Vertical Tail Structure
The same method utilized for the wing was used for the vertical tail. The load case used was the high speed dash with a 20° deflection of the rudder. The minimum thickness obtained was 8.09e-4 in. To have a more convenient construction, the vertical tail will be made out of a full sheet of balsa sanded to resemble the NACA 0006. The fuselage tail joint will consist of a notch extending off the bottom of the vertical tail which will rest in a slot cut in the top fuselage sheet, around which glue will be applied. The area around the slot will be padded up with an additional thickness of balsa to maximize bonding area between the tail and fuselage.
7 Fuselage Structure
The fuselage will be made mainly of balsa triangle rods and balsa sheets. It will have plywood reinforcement at the motor, landing gear, and possibly wing attachment locations. To shape the fuselage and to support the load applied by the landing gear and wing, balsa ribs will be used. The cross-section of the fuselage is shown in Figure 3.5 below. As it can be seen, it has four trianglular balsa rods will be glued to the corners produced by the balsa. To finish the shape of the fuselage, the exterior corners will be sanded.
[pic]
Figure 3.5: Fuselage Structure Cross section
8 Wing-Fuselage Attachment
To attach the wing to the fuselage, two nylon bolts and two carbon rods will be used as shown in Appendix Figure D-8. To size the attachment of the wing, we used the 10g load case. The maximum lift for 10g was divided by four so all the members carry the same amount of force.
For the nylon bolts, the cross-sectional area was calculated in order to get the stress on it (Appendix Figure D-9). After that, the stress obtained was compared to the ultimate tensile strength of nylon. The margin found was 34, meaning that the nylon bolts needs 34 times the maximum stress to fail—this suggests that the material which supports the bolts (balsa, nuts) will be the cause of failure, not the bolt. Therefore, extra care will be taken to ensure adequate structure to support the bolt.
On the carbon rod attachments, the balsa rib will fail before the carbon rod, similar to the nylon bolt. Consequently, the stress on the rib was the one calculated. The area for calculating the stress is the contact area between the carbon rod and the rib as shown on Appendix Figure D-10. After the maximum stress was obtained, it was compared to the ultimate compressive strength of balsa and a margin of 2.2 was found. The balsa ribs will be reinforced (or substituted) with plywood to avoid the elongation of the hole in the balsa due to vibrations.
9 Landing Gear
Roskam’s method on landing gear sizing and disposition was used along with some of Raymer’s recommendations. First, a fixed landing gear was chosen because of its cost and weight effectiveness. A taildragger configuration was chosen because if provides less drag due to smaller frontal area and less weight. A tip-over analysis was performed, and the required angle from the center of gravity to the wheel for such analysis is 20 degrees as shown in Appendix Figure D-4. The angle [pic] dictates the angle necessary for longitudinal tip-over stability (Appendix Figure D-5). The wing ground clearance criteria stated that the angle between the wheel and the tip of the wing should be [pic] 5 degrees. From online hobby catalogs, wheels were picked based on their lightness, strength and weight capacity (Appendix Figures D-6 and AD-7).
The main landing gear will be made out of aluminum to reduce cost and the tail landing gear will be composed of a lightweight tail wheel bracket kit. The main landing gear will be attached to the fuselage with four metal bolts and nuts, into a plywood reinforced section of the fuselage.
10 CG Estimation
The CG estimation for the aircraft is being calculated in two ways. A CATIA model was created to visually locate each component within the fuselage. These x-locations (distances from the nose in inches) were then used in the spreadsheet seen in Table A-1. Based on the volume each shape occupied in CATIA, a density was applied so each component’s virtual weight would accurately reflect its real world weight. The same weight was used in the spreadsheet to calculate the moment about the nose contributed by each component. This spreadsheet does not reflect the actual estimated CG, as the CATIA model will be used to estimated this; instead the spreadsheet was used simply as a tool along the design process. Ultimately, the actually CG of the aircraft will be adjusted by varying the payload position, an element of flexibility in the design which will prove useful. The goal is to continue adjusting the parts highlighted in green in the spreadsheet until the CG reported by the spreadsheet is close to the desired CG location for 15% static margin. This desired location, located at 33% of the Mean Aerodynamic Chord (MAC determination described in trade study #1), is set to be at 15.4 inches from the nose. The CG location reported by CATIA serves as a reliability check of the spreadsheet’s calculations and vice-versa. The current summed weight of components does not add up to the preliminary estimate of 5 lbs; this allows for just over 1 lb for glue, extra structural materials, wiring and fasteners.
Dynamics and Controls
1 Introduction
The stability of the aircraft will be defined by the characteristics of the aircraft such as control surfaces, center of gravity, and aerodynamic center. The budget constraint of $250 will not allow the use of complicated control systems such as autopilots. The design of the control system however has to comply with certain characteristics given on the mission statement such as: Dutch roll greater than 0.8, minimum takeoff climb angle greater or equal to 35 degrees, a descent angle less than -5.5 degrees, and a maximum stall velocity of 30 ft/sec. The focus of the dynamics and controls for the aircraft will be to design a stable aircraft which will use a control system to ensure the aircraft meets the design requirements.
2 Preliminary Design Considerations
The following are important design considerations: center of gravity position with respect to the aerodynamic center, tail position with respect to the center of gravity and tail area, rudder and elevator area, aileron size and position, dihedral angle, wing sweep angle, and tail height with respect to the thrust line of action. The position of the center of gravity has to be forward of the aerodynamic center to have longitudinal static stability. Since pitch is controlled by the elevators, adjusting the position of the payload to vary the location of the center of gravity will lessen the load on the servos.
Dutch roll is caused by a disturbance during flight that changes the yaw of the aircraft, which then affects the lift distribution on the wing and a rolling oscillation appears. To solve the Dutch roll problem, a yaw damping feedback controller will be used, which will be designed to achieve the required damping of 0.8. The vertical tail must be large enough to allow directional control and recover the airplane from rolling during stall.
Controllability of the aircraft also depends on the position of the wing. A high-wings experience less ground effects than lower wings, allowing the aircraft to land better. A high wing also offers a better stability and flight control than a mid or low wing because the center of gravity is located below the aerodynamic center. Velocity control is very important during the mission. To allow the best results, the propeller's thrust vector will be aligned with the centerline and the center of gravity to avoid decrease in speed due to high thrust.
3 Center of Gravity and Aerodynamic Center
The longitudinal controllability of any aircraft depends greatly on the position of two points: the total center of gravity (CG) and the total aerodynamic center (AC) of the airplane. The relative position between these to points is called static margin (SM); this value indicates how stable the airplane is at any attitude. An aircraft is stable if the static margin is positive – that is, the position of the aerodynamic center is aft the center of gravity – and the stability increases as the static margin becomes larger. The calculation of the CG comes from an accurate CAD model of the aircraft and distribution of the payload. In practice, this technique is not completely accurate because the exact material properties, fuselage interiors, and position of payload are not know during the initial states of the design; therefore, the calculation of the center of gravity using CAD model becomes inaccurate. The location of the aerodynamic center depends greatly on the size and geometry of the wing and horizontal tail; as well as the position of the horizontal tail with respect to the wing. A deeper explanation about the calculation of the aerodynamic center position is given on the following paragraphs.
Knowing that the exact position of the aircraft’s center of gravity is not well known during the initial stages of the design; it was assumed, in this project, that the position of the CG is located at distance equal to 33% of the length of the mean aerodynamic chord. This assumption simplifies the amount of excessive details needed during the initial part of the design. Assuming an initial position of the center of gravity also allows the designer more flexibility in terms of surface sizing since the position can be varied according to its needs. The center of gravity will then be positioned at the required location by arranging the payload and other components (such as batteries) inside the fuselage. Here, the value of static margin also becomes important because it limits the position of the center of gravity. If the center of gravity is too far forward of the aerodynamic center, the airplane will require larger elevators for pitch control, especially during take off and landing. Requiring large control surface deflection also increases the amount of drag on the airplane, thus reducing its performance.
It was mentioned above that the aerodynamic center depends on several geometric features of the wing and the tail. A more accurate description of the position of the aerodynamic center in terms of the wing mean aerodynamic chord is given by the following formula used by Roskam [1] (Note: a description of variables used in this report can be found in the appendix):
[pic] (Equation 4.1)
Equation 1 shows that the aerodynamic center depends on the position of the wing/fuselage aerodynamic center alone (first term in equation), the slope of the variation wing/fuselage, horizontal tail, total aircraft lift coefficient versus angle of attack, the surface of the wing and the horizontal tail, the ratio of the dynamic pressures of the horizontal tail to the wing tail, and the position of the tail aerodynamic center of the tail with respect to the leading edge of the MAC. The formulas used to calculate each of the components are shown in the appendix. Then, the tail size can be calculated by assuming an initial static margin of 15% as mentioned in [2]. The method used to size the horizontal and vertical tail is explained following this section.
As mentioned before, the variation of the aerodynamic center position also depends greatly on the geometry of the surface (wing / tail). Appendix Figures E-1 and E-2 show the variation of the chord-wise and span-wise positions of the aerodynamic center about the wing apex with variation of taper ratio and different leading edge sweep angles. From the figures it can be noticed that the chord-wise position of the aerodynamic center from the wing apex varies with the geometry of the surface; that is, as the taper ratio increases and/or the sweep angle increases, the aerodynamic center moves aft. Then, a wing/tail with high sweep angle and large taper ratio will move the chord-wise aerodynamic center position aft. The case of the span-wise position of the aerodynamic center is different (Figure E-2). It is shown that a variation of sweep angle does not have any effect on the span-wise position of the aerodynamic center; only the taper ratio of the surface has an effect on such position.
4 Sizing of Horizontal and Vertical Tail
It was mentioned that the stability of the aircraft depends on the relative position of the center of gravity and the aerodynamic center of the aircraft and the latter depends on the size of the horizontal tail. The methods used to calculate the size of the horizontal tail are explained as follows.
The first method used to sizing the airplane horizontal tail starts by assuming a value of static margin (in this case, %15 of the Wing MAC). From there, the required position of the aerodynamic center is calculated and finally the size of the horizontal tail is calculated by solving Equation 4.1. The main deficiency of this method is that the solution for the horizontal tail from Equation 4.1 does not take in consideration the variation of the center of gravity position with tail size. This method assumes that the center of gravity is fixed at a point and does not change with any tail size. It is a sufficient method for initial approximations, but a much better method is described as follows.
Jan Roskam³ describes how to successfully make a preliminary estimate of the horizontal tail size. The Class I method consists of the construction of a plot (called X-Plot) that shows the variation of the center of gravity and aerodynamic center with horizontal tail surface. Then, a static margin of 15% or above is suggested to calculate the required horizontal tail area; a Static Margin value of 17% was used to size the aircraft’s horizontal tail. Appendix Figure E-3 shows the X-Plot. The X-Plot indicates that the necessary horizontal tail area for good stability should be about[pic]. In fact, the solution given by X-Plot is close to the horizontal tail area calculated with the solution of Formula 1 which gives a required area of[pic]. However, the use of the Class I method is only accurate if the variation of the center of gravity with tail area is known, during the preliminary design of the aircraft, the exact material properties were not correctly known, therefore the calculated area of [pic] can be considered as the minimum horizontal tail area required for good stability.
The size of the vertical tail was also sized using the Class I method described by Roskam on [4]. It follows a procedure somewhat different than the one used to calculate the horizontal tail area. A figure is created by plotting the variation of the change of yawing moment coefficient with sideslip angle,[pic], as a function of vertical tail area. (The yawing moment coefficient is given by Equation 4.2). Roskam suggests that the overall level of directional stability should be assumed to be [pic]. Then, the minimum vertical tail area required is found by finding the corresponding area that meets the condition from the plot. Appendix Figure E-4 shows this plot.
[pic] (Equation 4.2)
The yawing moment coefficient with variation of sideslip angle ‘[pic]’ depends on the distance between the AC of the vertical tail and the CG of the airplane, the span and area of the wing, the variation of lift coefficient with angle of attack of the vertical tail and the variation of yawing moment with sideslip angle of the wing/fuselage alone. The appendix contains the formulas used to obtain such terms. Figure E-4 suggests that a vertical tail area of 0.35 ft2 is needed to meet the required[pic].
With the given wing and empennage areas, the control surfaces were sized using historical data with some geometry assumptions and by checking the derivative coefficients presented in the appendix. Appendix Figures E-5 and E-6 show control surface sizes of different homebuilt airplanes. The assumptions considered for the sizing of the control surfaces are as follow: The span of the rudder is equal to the span of the vertical tail; the span of the elevators is equal to the span of the horizontal stabilizer; the inboard position of the ailerons was assumed to have a value of 10% half wing span and outboard position of 85% half wing span (since flaperons will be used to increase the lift at take-off). These values approximate the historical data with the exception of the inboard position of the ailerons. The difference between the historical data and our airplane is the use of flaps to increase the lift of the airplane.
[pic]
Figure 4.1 – Trim diagram showing all the trimmable lift coefficients for different angles of attack and require elevator deflection angles required to trim
Figure 4.1 shows a trim diagram of the longitudinal pitch control system. This diagram allows the designer to see the range of trimmable lift coefficients at three different center of gravity positions. This plot also shows the amount of elevator deflection angles needed to trim the airplane at a given flight condition. The trim diagram shown in Figure 4.1 shows the position of the center of gravity at three positions: 18%, 25% and 48% of the wing mean aerodynamic chord measured from the leading edge. Notice that a required deflection angle of -2 degrees is required if the airplane is flying at maximum angle of attack and a close to maximum lift coefficient with the CG located at 18% wing chord. A larger deflection angle is required if the cg is located at 48% of the wing mean aerodynamic chord (this position coincides with the position of the aerodynamic center of the entire aircraft).
This concludes the section of Static Stability. Computer results for this part, including dimensions and aerodynamic coefficients are shown on in the Appendix E Part V. The next section describes the results of the Dynamic Stability simulation. It will be shown that a rate gyro will be used to compensate the yaw rate to rudder deflection to increase the damping of the Dutch roll. The results shown in the next section were obtained from the class Matlab code: FlatEarth_9.2.1.
5 Dynamic Stability
The dynamic stability and aircraft characteristics, including modes of motion, transfer functions and trim conditions, are calculated using the MATLAB code FlatEarth.m distributed by the instructor. This program has been very useful in calculating control coefficient derivatives and helped to determine necessary yaw rate to rudder deflection gain to damp the Dutch roll oscillation.
To analyze the dynamic stability of the aircraft, the linearized longitudinal and lateral-directional equations of motion of the aircraft were obtained using FlatEarth.m and represented in state space form (Equation 4.3). See Appendix-Part VI for values of the matrices A, B for the aircraft.
[pic] (Equation 4.3)
When an airplane is in flight, it has five inherent modes of motion: two longitudinal modes of motion known as Phugoid or Long Period motion and Short mode motion; and three lateral-directional modes of motion known as Spiral, Roll, and Dutch Roll. These modes of motion are further explained in the Appendix E Part VI. To calculate these five modes of motion, the eigenvalues of the matrix ‘A’ presented in Equation 4.3 were obtained since the matrix A describes the dynamics of the airplane due to its physical characteristics and cannot be modified unless such physical characteristics are modified (e.g. Change of wing area). The results of the modes of motion of the aircraft are presented in the following table:
[pic]
Table 4.1 – Aircraft Modes of Motion
The above table shows that the Phugoid mode has a period of 37.62 seconds and the Short mode has a period of 0.56 seconds. The Dutch Roll has a period of 6.28 seconds with a damping ratio of 0.11. Notice that the required damping ratio by the mission specifications is of 0.8. Then, to increase the low damping ratio, a yawing damping feedback system was designed as described below.
From the FlatEarth.m code, the transfer function corresponding to the yaw input to rudder deflection output was obtained and it is the only transfer function relevant to the analysis of yaw damping. The transfer function is shown in below:
[pic] (Equation 4.4)
Equation 4.4 is the transfer function obtained using FlatEarth.m. The actual transfer function of the yaw rate to rudder input is the negative of the above expression because a negative deflection of the rudder induces a positive response of the yaw rate. This is the open loop transfer function of the system of which the root locus is shown if Appendix Figure E-7. The figure shows that the system is unstable for gains smaller than 0.150 due to one positive pole. However, the closed loop of the system is stable for any gain. And the Dutch Roll damping is high; but the effects of the rudder servo are not yet included. The servo has the following transfer function:
[pic] (Equation 4.5)
The complete feedback block diagram for the system is as shown below:
Figure 4.2: Yaw rate to rudder input Block Diagram
Appendix Figure E-8 shows the root locus of the compensated system. To obtain a Dutch roll damping higher than 0.8, the necessary rate gyro gain has to be equal to -0.497. The gain value was found by using a SISO tool in MATLAB by imputing the Yaw to Rudder deflection transfer function, the servo transfer function and manual searching for the gain quantity that would increase the damping ratio of the closed loop poles to 0.8.
Conclusion
The design processes that were presented in this document are believed to meet the requirements of the mission specifications presented at the beginning of the semester. All the calculations made to obtain the aircraft characteristics were mostly relied on theoretical approximations rather than experimentation or numerical solution by computer. Several assumptions were made to simplify the complexity of the design but at the same time be able to meet the requirements for high speed flight.
The geometry and aerodynamics of the wing and tail were fixed following several trade of studied done during the course of the project; the use of flaperons will increase the lift of the airplane during take off and at the same time control the airplane during roll, therefore eliminating the use of ailerons. The electric motor and battery power were chosen to achieve take off between the required limits and a high flight speed during flight. The use of foam reduces the weight and complexity of the wing but gives a better structural strength than balsa wood.
The team expects the aircraft to fulfill all design requirements with a top speed of 118 ft/s and to have acceptable handling qualities. The construction phase will prove the soundness of the overall design and put all the theories and assumptions made to the test. The flight test of the aircraft will be the real test to the predicted performance qualities.
Addendum: Lessons Learned and Vehicle Summary
[pic]
Tables of Constants
|Aircraft Properties | |
|Length (in) |48.2 |
|Winspan (inch) |60.5 |
|Weight (lbf) (w/o payload) |4.4 |
| | |
|Propulsion Constants | |
|Propeller Dimensions |11x10 |
|Gear Ratio |4.44 |
|Motor Kv (RPM/Volt) |1700 |
|Motor R (Ohms) |0.017 |
|Motor I0 (Amps) |1.9 |
|Battery Nominal Voltage |22.2 V |
|Battery Peak Current |24.0 A |
|Aerodynamic Constants | |
|Wing Airfoil |MH 43 |
|Wing AR |5.87 |
|Wing Area (ft2) |4.33 |
|Horizontal Tail Airfoil |NACA 0006 |
|Horizontal Tail AR |4.92 |
|Horizontal Tail Area (ft2) |0.81 |
|Vertical Tail Airfoil |NACA 0006 |
|Vertical Tail Area (ft2) |0.31 |
|CD0 |.025 |
Propulsion
Due to improper dimensions specified by the gearbox manufacturer, the motor pinion was mounted in such a way that prevented complete meshing of this pinion with the gearbox gear teeth. This stripped the original gearbox (as well as an identical one found from previous semesters), rendering it useless. Without a replacement gearbox provided by the pilot, the aircraft was in jeopardy of not flying. This caused several aircraft-level changes, most significantly the mounting of the motor/gearbox. Luckily, the motor shaft could be extended so the original MEGA motor could be used with the new gearbox. All this occurred within days of flight testing, causing a great deal of last minute problem solving.
The only other change to the propulsion system was the use of a 11x10 prop instead of the planned 11x11, which was found to be only available for gas engines.
Aerodynamics
Fortunately, no significant changes were made to the aerodynamic design, aside from minor changes in wetted areas of components. An intake/outtake vent pair was added to cool the propulsion components, although this would have been impossible to predict the drag added, which was ignored.
Controls
No major changes were made to the control surfaces, aside from slightly increasing the elevator and aileron sizes. These decisions were governed simply on experience on our team and the designed sizes looking too small.
Structures
The wing structure was largely unchanged, with the exception of adding the carbon spar, a preliminary concept which was removed from the design. Though not necessary, according to the structural calculations, this spar (seen at left) was added for redundancy and for the reassurance of the team members. The result was a very stiff, durable, and still lightweight wing.
Since detailed designs of the fuselage main ribs were left to be determined during the building process, these were manufactured and included in the actual aircraft. The fuselage contained 2 main balsa/plywood/balsa ribs with lower sections cutout to reduce weight and allow component passthrough. The front rib had two holes into which the carbon spars for the wing attachment were inserted. On the wing, two brackets were epoxied into the underside of the balsa skin, through which these two rods were attached, as shown. The rear rib had a “shelf” into which the two nylon bolts were tightened. In retrospect, these ribs could probably have been thinner for an even lighter weight aircraft, however the extra structural rigidity provided some reassurance for the team members.
[pic] [pic]
A rear lightweight, balsa rib was included about halfway between the rear main rib and tail; holes were drilled to allow control wires to passthrough. Originally, a “firewall” rib was to be mounted in the nose of the aircraft to which attached the gearbox/motor assembly. However, this rib had to be removed in order to fit the replacement gearbox/motor assembly. This new gearbox was mounted to a plywood plate epoxied directly to the nose of the aircraft. With the original gearbox, access from the opening in the nose was sufficient, but an access hatch was needed for the new gearbox mounting and cut into the bottom of the nose. Also, an aluminum spacer plate was needed to use the original MEGA motor with the new gearbox
For the landing gear attachment, a plywood plate was used to carry the brunt of the impact load. It was to this plate that the aluminum landing gear were attached. Originally, three bolts were used to attach the gear, but it was found that the flexible gear allowed too much longitudinal oscillation of the aircraft when wings were attached. So, four bolts were used and spread out more, effectively widening the attachment area of the gear and solving the oscillation problem. A thin wire was also attached about midway up the height of the gear to reduce bending and effective spreading of the struts, also an easy fix.
The following pictures show the flight-ready internal component layout. This configuration placed the center of gravity at a stable location (static margin of 17%), as specified by the controls team. The final component locations closely reflected the planned layout in the CATIA model.
[pic] [pic]
Lessons Learned
The team learned that one cannot assume that manufacturers’ specifications are always correct. Had we measured the necessary position of the motor pinion for proper meshing with the gearbox, the original gearbox could have been used. Another interesting lesson was that not all balsa and plywood is the same, and they possess very different structural properties depending on their orientation. Had these been known during the design phase, lighter weight fuselage ribs could have been made. As a whole, the team realized the amount of sanding necessary for an all wood aircraft is enormous.
From a teaming perspective, the international composition of our team proved to be both an asset and an added difficulty. The backgrounds of the various team members and learning about each other’s cultures provided some enjoyment during the design and build process. Having three spoken languages on the team made technically communicating slightly more difficult but nonetheless manageable.
Resources Used
Roskam, Jan. Airplane Design Part I-VI. Ottawa, Kansas: Roskam Aviation and Engineering Corp., 1985.
Peters, Mark. Development of a Light Unmanned Aircraft for the Determination of Flying Qualities Requirements, May 1996.
Nelson, Robert. Flight Stability and Automatic Control, Second Edition, 1998.
Lennon, Andy. Basics of R/C Model Aircraft Design: Practical Techniques for Building Better Models. Air Age Publishing, September, 1996.
Raymer, Daniel. Aircraft Design: A Conceptual Approach. AIAA Education Series, January, 2006.
Section 2:
Appendices
Appendix A: List of Symbols (Variables)
Section 1: Aerodynamics
CLo : Lift coefficient at 0 a
CDo : Parasite drag coefficient
CMo : Moment coefficient at 0 a
CLα : 3D Lift coefficient slope
CMα : Moment coefficient slope
CLih : Lift coefficient slope due to tail incidence
CMih : Moment coefficient slope due to tail incidence
CLδe : Lift coefficient slope due to elevator deflection
CMδe : Moment coefficient slope due to elevator deflection
Cla : 2D Lift coefficient slope
α : Angle of attack
δe : Elevator deflection
ih : Tail incidence angle
iw : Wing incidence angle
Cf : Component skin friction coefficient
FF : Component form factor
Q : Component interference factor
Swet : Reference area, Wing area
CDmisc : Miscellaneous drag coefficient
Β : Mach constant
k : Cla constant
Λc/4 : Sweep at the quarter chord
Section 2: Propulsion
W : Weight
ρ : Air Density
S : Wing Area
CD0 : Zero Lift Drag Coefficient
AR : Aspect Ratio
e : Oswald’s Efficiency Factor
Kv : Motor Constant (RPM/Volt)
Kt : Motor Constant (in-oz/Amp)
Io : No Load Current
Rm : Resistance of Motor
Vmax : Maximum Battery Voltage
Imax : Maximum Continuous Current
D : Propeller Diameter
P/D : Pitch/Diameter Ratio
CT(J) : Coefficient of Thrust
CP(J) : Coefficient of Power
ηgear : Gearbox Efficiency
R : Gear Ratio
P : Power
τ : Torque
V : Velocity
μrf : Coefficient of Rolling Friction
Section 3: Structures
W : Total weight of the aircraft
WB : Total battery weight
WBTO : Take off battery weight
WBC : Climb battery weight
WBL : Loiter battery weight
WBR : Return battery weight
Flift : Lift force
Wload : The wing loading
S : The surface area of discretized section of wing
M : Bending moment of discretized section of wing
d : Distance from wing root to discretized section MAC (mean aerodynamic chord)
[pic] : UCS (ultimate compressive stress) of balsa
M : Bending moment
[pic] : Maximum thickness of wing skin for each discretized section
I : Area moment of inertia
y : Deflection of each descretized section
L : Distance from the root of each section to MAC
E : Modulus of elasticity
[pic] : Density of air
[pic] : Maximum speed of the aircraft
[pic] : Maximum moment coefficient
G : Shear modulus
J : Polar moment of inertia
[pic] : Total moment coefficient of the wing
[pic] : Lift coefficient due to the downwash of the wing
[pic] : Lift coefficient due to the angle of attack of the aircraft
[pic] : Lift due to the incidence angle
[pic]: Lift due to the deflection of the control surfaces
Section 4: Dynamics and Controls
[pic] : Aspect Ratio
[pic] : Wing Span
[pic] : Rate of Airfoil lift coefficient with angle of attack
[pic]: Rate of wing/fuselage lift coefficient with angle of attack
[pic] : Rate of horizontal tail lift coefficient with angle of attack
[pic] : Rate of vertical tail lift coefficient with angle of attack
[pic] : Total aircraft rate of lift coefficient with angle of attack
[pic] : Rate of horizontal tail lift coefficient with incidence angle
[pic] : Rate of change of lift coefficient with elevator deflection angle
[pic] : Rate of yawing moment of wing/fuselage coefficient with sideslip angle
[pic] : Rate of aircraft yawing moment coefficient with sideslip angle
[pic] : Equivalent fuselage diameter
[pic] : Rate of downwash angle with angle of attack
[pic] : Height from wing quarter chord plane to horizontal tail quarter chord plane
[pic] : Sweep angle at half chord
[pic] : Sweep angle at quarter chord
[pic] : Taper ratio
[pic] : Distance from wing quarter chord to horizontal tail quarter chord
[pic] : Flight Mach number
[pic] : Ratio of dynamic pressure between horizontal tail and wing
[pic] : Wing surface area
[pic] : Vertical tail surface area
[pic] : Horizontal tail surface area
[pic] : Flap effectiveness parameter
[pic] : Aerodynamic center position per unit wing MAC of whole aircraft
[pic] : Wing/Fuselage aerodynamic center position per unit chord of wing MAC
[pic] : Horizontal tail aerodynamic center position per unit chord of wing MAC from wing MAC leading edge.
[pic] : Distance between vertical tail aerodynamic center and aircraft center of gravity.
Appendix B: Aerodynamic Figures and Data
Figure B-1: MH 43 Airfoil Profile
[pic]
Figure B-2: NACA 0006 Airfoil Profile
[pic]
Figure B-3: Constraint Diagram
[pic]
Figure B-4: Aircraft Wetted Area Estimation
Figure B-5: Aircraft lift curve and drag polar
[pic]
Appendix B-1 Full Trade Studies
B-1.1 Flaps Trade Study by Audrey Serra
One of the big issues of our mission is due to the fact that we have to meet the requirements of a high speed dash and we also have to respect the stall speed condition (Vstall = 30 ft/s, which is a big constraint). During the high speed phase, the drag has to be minimized and during low speed phases (take off and landing), we need maximum lift.
Adding flaps on the airfoil is a good way to meet these two requirements. Indeed, we can keep our airfoil, designed for high speed, and take off the flaps for the low speed phases in order to increase the camber and consequently the lift.
However, adding flaps means adding drag (due to the mechanism), complexity on structure (difficult to build) and of course weight (due to extra servo and mechanism).
Thus, the following table sums up the trade off:
|Advantages of Flaps |Drawbacks of flaps |
| |Weight, due to servo and mechanism |
|Lift increased at low speed |Difficult to build, structure complexity |
| |Drag, due to the mechanism |
The drag added by the mechanism is present even at 0° of deflection angle. This is not the drag due to the increase of camber but the drag due to the connection between the flap and the airfoil (vortices are created from this point). Thus, this is impossible to quantify this drag increase because the connection is modelled as a perfect liaison on XFOIL.
The MH 43 airfoil, well known for electrical powered model aircraft, has been chosen for our airplane. The XFOIL view is presented below:
[pic]
First, we are going to find the flap configuration that can achieve the optimum CLmax we need at low speed (flap hinge location and deflection angle).
First of all, we can calculate the CLmax needed thanks to this equation below:
Actually, we have to add the weight of the flaps to the actual weight of the aircraft (from the vehicle sizing study). We are going to assume that whatever the configuration of the flaps is, the weight is going be to quite the same. In reality, the mechanism stays the same but the servo gets bigger as the deflection angle gets higher, but this difference of weight is negligible. Thus, based on historical data, we chose the following weight:
Weight of the flap servo = 40 g
Weight of the flap mechanism = 20 g
Total weight added due to the flaps = 60 g or 0.13 lbs
Knowing:
W = 4.99+0.13 = 5.12 lbs (weight of the aircraft + weight of the flaps)
g = 9.81 m/s-²
Vstall = 30 ft/s
S = 4.16 ft² (wing area from the constraint diagram)
[pic]= 1.225 kg/m3 (Sea level condition)
We can figure out that:
Then, thanks to XFOIL, we can plot the lift coefficient (in 2D) versus the angle of attack, for the MH43 without flaps.
Knowing that: with and
And having:
[pic]= 6.18 /rad the 2D XFOIL curve.
[pic]= 0.0218 rad from the 2D XFOIL curve.
AR = 8.65
The Aspect ratio is 8.65 for the moment. Since another trade study is being made on this parameter, this value will certainly change.
We can also plot the lift coefficient in 3D. The following chart shows these two curves:
[pic]
We can graphically see that (blue curve): for
To find the CLmax in 3D, this equation can be used:
And, graphically (pink curve), we figure out that: for
We notice that the MH43 airfoil alone cannot achieve the .
There is a
Thus, the flaps have to correct this difference to finally reach the condition of
This value of the CLmax needed is in 3D and all the comparisons which are going to be made after are in 2D (XFOIL). Thus, we can use the following equation to find the lift coefficient needed in 2D:
This equation is reasonably valid for most subsonic aircraft of moderate sweep.
[pic] is the sweep angle between the horizontal and the quarter chord of the wing. Graphically, we found that: [pic]
So, we have:
To reach this Clmax, we are going to study different configurations of flaps making varying the flap hinge location (x/c) and the maximum flap deflection angle (δf).
First, the hinge location will be fixed to an average value while the deflection angle is going to vary and then the deflection angle will be fixed to an average value while the hinge location is going to change. Basically, the parameters are not coupled, they vary separately.
The sketch below shows an example of the MH43 airfoil with a flap (the deflection angle is 20° and the hinge location is 0.8):
[pic]
The Reynolds number we will use in XFOIL is given by this equation:
With:
[pic]=1.226 kg/m3
[pic]= 1.776 e-5 kg/m.s
Vmax = 115 ft/s (given by the propulsion group, will certainly be reduced)
c = 0.69 ft (average chord of the wing)
First, the hinge location is fixed at 0.8, value based on historical data, and the values tested for the deflection angle are: 6°, 8°, 9°, 10° and 20°. Using XFOIL, we obtain the following plot of lift coefficient versus the angle of attack:
[pic]
The table which follow sums up the values of Clmax obtained and the angle of attack associated for each configuration studied:
|Configurations |Alpha |Clmax |
|Without flaps |10 deg |1.082 |
|x/c = 0.8 δf = 6 deg |8.5 deg |1.233 |
|x/c = 0.8 δf = 8 deg |8 deg |1.281 |
|x/c = 0.8 δf = 9 deg |7.5 deg |1.296 |
|x/c = 0.8 δf = 10 deg |7.5 deg |1.327 |
|x/c = 0.8 δf = 20 deg |6 deg |1.509 |
Examining the results above, we can say that for a hinge location of 0.8, a deflection angle of 10 degrees is enough to reach the condition of . Indeed, we do not want a too high Clmax because this is going to increase the weight of the aircraft.
Then, fixing the deflection angle at 10 degrees, we will make vary the hinge position. The tested values for x/c are: 0.7, 0.75, 0.8, 0.825, 0.85 and 0.9. Using XFOIL, we obtain the following plot of lift coefficient versus the angle of attack:
[pic]
The table which follow sums up the values of Clmax and the angle of attack associated for each configuration studied:
|Configurations |Alpha |Clmax |
|x/c = 0.7 δf = 10 deg |7 deg |1.344 |
|x/c = 0.75 δf = 10 deg |7.5 deg |1.342 |
|x/c = 0.8 δf = 10 deg |7.5 deg |1.327 |
|x/c = 0.825 δf = 10 deg |7.5 deg |1.310 |
|x/c = 0.85 δf = 10 deg |8 deg |1.290 |
|x/c = 0.9 δf = 10 deg |8 deg |1.202 |
We notice that a hinge location of 0.825 satisfies the condition of .
This value (x/c=0.825) is quite close to the first value we picked at the beginning (x/c=0.8).
The hinge location is an important parameter because it is going to affect the aerodynamic behaviour of the airfoil. Indeed, a lot of vortices (disturbing flow) are created because of the connection between the flap and the airfoil (in reality there is like a hole between). Thus, the closer to the trailing edge is the hinge, the more efficient surface of the airfoil we get and the less drag due to the flap mechanism we have.
To conclude, the configuration which allows reaching just the Clmax needed is:
x/c = 0.825
δf = 10 deg
With this configuration, we have .
This value of the Clmax we have with the flaps is in 2D. Thus, we can again use the following equation to find the lift coefficient we have now in 3D:
So, we find: >
The value of CLmax we have with the flaps is greater than the value of CLmax needed to lift the aircraft at low speed.
To conclude, we can say that using flaps is a good way to meet both the requirements of a high speed dash and a low speed phase. The impact of the flaps on the design will be the more construction complexity and the more weight. We will need to study how we are going to link the servo to the two flaps (study more the connection).
B-1.2 Aspect Ratio Trade Study by Dane Batema
1. Purpose
Criterion: Select an aspect ratio.
Find: The relationship between aspect ratio and drag. Plot the drag coefficient verse aspect ratio. Examine the impact of aspect ratio on the current sizing of the aircraft.
Decision rule: Select the aspect ratio that minimizes drag and gives reasonable sizing. The root chord cannot be too long and the aircraft length too short.
2. Design Variables
- Wing area, fixed at 4.16 ft2 from constraint diagram
- Taper ratio of the wing, fixed at .45 for approximate elliptic lift distribution
- Wingspan, free to change with aspect ratio, cannot exceed 6ft
- Root and tip chord, free to change with aspect ratio, root chord cannot exceed half the length of the aircraft
- Aircraft length/height/width, free to change with aspect ratio
- Aspect ratio, parameter to be optimized
- Drag coefficient, parameter used to optimize aspect ratio
- Wetted area, free to change with aspect ratio
- Angle of attack, set to different flight regimes
3. Key Parameters
The key parameters for this study are aspect ratio, drag coefficient, root chord, and aircraft length. Drag will be used to optimize aspect ratio, and root chord and length will be used to further refine aspect ratio.
4. Tools and Assumptions
The main analysis tool used in this study was MATLAB. The code generates sizing for the aircraft based on the aspect ratio, which is used to calculate parasite and induced drag. Parasite drag is calculated using the component buildup method (note: due to preliminary sizing, both the vertical tail and landing gear were not modeled in the drag buildup). Induced drag is computed from the Oswald efficiency factor and the lift coefficient. The main assumption in the overall method is in regard to the sizing method. For the initial sizing, the team used data from other aircraft, which can be placed in terms of aspect ratio. Once all the values were calculated in MATLAB, several plots of CD, CDo, CDi verses AR at several alphas were made. From there, minimums were found.
5. Results
[pic]
Figure 1: Total CD vs AR for alphas 0° and 8°
[pic]
Figure 2: CD breakdown vs AR, for alpha = 0°
[pic]
Figure 3: CD breakdown vs AR, for alpha = 8°
[pic]
Figure 4: CDi vs AR for alphas 0°, 5°, and 8°
| |Min at alpha = 0°, for CD |Min at alpha = 8°, for CD |Min at alphas 0°, 5°, 8° for CDi |
|AR |4.00 |5.61 |6.21 |
|Span (ft) |4.08 |4.83 |5.08 |
|Aircraft Length (ft) |2.72 |3.22 |3.39 |
|Root Chord (ft) |1.41 |1.19 |1.13 |
6. Discussion
Figure 1 shows that the optimum AR decreases as alpha decreases, and increases when alpha increases. This is to be expected as low AR wings are faster in level flight than high AR wings. This is because the CDo term dominates at lower alphas, and you want to minimize CDo for high speed level flight. The inverse becomes true when you increase alpha. This can be seen in Figures 2 and 3. Interestingly, the minimum CDi seems independent of alpha, as can be seen in Figure 4. While the optimum AR at 0° alpha is 4, the root chord/length constraint is busted. At AR = 4, the root chord is a little over half the total length of the aircraft. This causes problems for the dynamics and control of the aircraft, as roll becomes less stable, and the moment arm for the elevator decreases. At 8° alpha, the optimum AR increases to 5.61. This AR has a more reasonable root chord to length ratio of a little over one third. Because loiter is an important part of the mission, the optimum AR in regards to just CDi was also found, which is 6.21. The root chord to length ratio for this AR was slightly smaller than that for an AR of 5.61. Since the mission encompasses both speed and loiter phases, I recommend an AR of 5.61, for a balance between the lower AR needed for speed, and the higher AR needed for loiter.
7. Impact
This study will have a large impact on our design. The AR will be reduced by 3, which will greatly impact the size of the aircraft, as well as the CDo. This change will also lead to further studies, particularly in regards to the sizing of the tail.
Appendix C: Propulsion Figures and Explanations
[pic]
Figure C-1: Propeller Efficiency v. Advance Ratio
[pic]
Figure C-2: Propeller Efficiency v. Airspeed for P/D = 1.0
[pic]
Figure C-3: Propeller Rotational Speed v. Airspeed for P/D = 1.0
[pic]
Figure C-4: Propeller Efficiency v. Airspeed for P/D = 1.0 (Full Velocity Spectrum)
[pic]
Figure C-5: Propeller Efficiency v. Pitch/Diameter Ratio
[pic]
Figure C-6: Cost/Cell v. Power/Cell for Many Different Batteries
[pic]
Figure C-7: Minimum Battery Cost v. Battery Power Output
[pic]
Figure C-8: Propeller Power Output Required and Available v. Flight Speed for P/D = 1.0
[pic]
Figure C-9: Maximum Achievable Thrust v. Aircraft Velocity
[pic]
Figure C-10: Aircraft Velocity v. Ground Roll Distance for Takeoff
Propulsion Tables
Table C-1 – Section of Battery System Selection Spreadsheet
[pic]
Note: The chosen battery system, the system with the lowest cost for a power of 0.7 hp, is boldfaced. Each row of this spreadsheet corresponds to a different motor, with the yellow background highlighting those batteries that are of limited supply (closeout and/or liquidation sales). This table does not show the complete set of data – there are roughly 70 batteries evaluated at powers of 0.2 hp to 1.0 hp at 0.1 hp increments. The complete battery specification chart is shown below.
Appendix C-1 – Calculations & Explanations
C-1.1 Discussion of gold.m program
The primary program available for analysis of propellers was the gold.m program. This program utilized momentum-blade element theory to calculate the quantities needed to perform a propeller efficiency study. The assumptions of this method were small angles of attack and high lift-drag ratios for the propeller airfoil sections. The program also utilized Goldstein’s classical vortex theory constant κ to account for vortex shedding of the propeller. There are many inputs to this program, some of which were straightforward and some which were very difficult to know without acquiring a propeller and performing extensive testing. The straightforward gold.m input quantities are propeller diameter, propeller pitch, rotational speed, airspeed, air density, and number of blades. The diameter, pitch, and speeds were used to establish the advance ratio at which the propeller is operating. These values were varied in this study to set up a range of advance ratios and pitch/diameter ratios for the analysis. Air density and number of blades are relatively easy quantities to define. Other quantities required by the program are the angle of zero lift, the lift curve slope of the propeller, the angle from the flat part of the prop to the mean chord line, the 2D minimum drag coefficient, and the induced drag constant k (CD=CD0+kCL2). Since there is no actual experimental data for the propeller, and there is some experience with the accuracy of the calculations for the default inputs, the default values were used in the analysis. This gave us an experience based factor to scale up the power required. For the angle of zero lift, this value is -6º. This value seems reasonable as the propulsion source book lists Master Airscrew propellers which all have a -6º angle of zero lift. The lift curve slope is set at 2π/rad. This value is very reasonable as most airfoil sections have a slope near this value. The angle between the flat part of the prop and the mean chord line is 0.5º, the 2D minimum drag coefficient is 0.00655, and the induced drag coefficient, k=0.01. The only one of these that is particularly concerning is the CD0 number which appears to be quite low. Since the only comparisons to actual data utilize these inputs, these numbers were kept in the analysis. Ideally, a wide range of model aircraft propellers would be wind tunnel tested to give reasonable numbers for these values, but this analysis was not realistic due to extremely tight design schedule.
C-1.2 Increase in J for a fixed airspeed
For a fixed airspeed, the thrust required is constant, and for fixed atmospheric conditions, the air density is constant. The equation for coefficient of thrust is: CT=T/(ρn2D4) where CT is the coefficient of thrust, T is the thrust, ρ is the air density, n is the propeller rotational speed, and D is the propeller diameter. The equation for advance ratio is: J=V/(nD) where J is the advance ratio and V is the freestream air velocity. For a given velocity and associated required thrust, increasing advance ratio can be done by decreasing nD. The only way this can be done is by increasing the diameter. For instance, if the diameter is increased to twice its value, to maintain the thrust level, the rotational speed must decrease to a quarter of its initial value.
[pic] (Equation 1)
The net result of the diameter increase is a decrease in nD or increase in advance ratio. The reverse is true for a decrease in diameter.
[pic] (Equation 2)
Therefore for a fixed airspeed, an increase in J can only be obtained through an increase in propeller diameter.
C-1.3 Constant efficiency for high speeds for a fixed propeller diameter
As the speed increases, the needed coefficient of lift decreases. At the range of high speeds examined, the needed lift coefficient is very close to zero, and doesn’t change much for the range of velocities examined. The total CD=CD0+k*CL2 equation for small values of CL yield essentially constant CD values. The total drag is then D=.5ρV2CD which must equal the thrust produced for steady level flight. T= ρn2D4CT If it is assumed that the J value remains constant for different airspeed, as is shown in the nearly constant efficiencies, the CT is also constant as CT is only a function of J. The thrust at different airspeeds then depends only on n2 as ρ, D, and CT are constant. Since the CD is essentially constant, the drag depends only on V2 since ρ and CD are constant. Since thrust must equal drag and these quantities are dependent on n2 and V2 respectively, an increase in velocity results in an equivalent increase in propeller speed, n. Since J=V/(nD), and by assuming J is constant, it was found that n and V change proportionally to each other, the assumption is verified that J will not change for a change in airspeed. The small changes in efficiency are due to the small changes in CD that are actually present in the aircraft.
C-1.4 Initial battery selection approach
Figure 6 shows the first approach used to select the most powerful battery system, which involved determining the power available per cell and plotting this value against the cost per cell for each battery. Using as many cells as needed to provide a given power to the motor was deemed a reasonable method of attaining the necessary power, and the two factors that would affect cost were the cost per cell and the power per cell since a greater power per cell would require fewer cells. This figure shows the expected trend: as power increases, cost increases. The ideal battery has a high power per cell and a low cost per cell. Thus, points that are furthest below and to the right of the trendline are the most desirable. Some of these batteries are cost prohibitive, even for a single-battery setup, while others provide a maximum current that exceeds the maximum current rating of affordable motors. The point circled in green shows that the selected battery has a very low cost-per-cell for its power-per-cell value. However, this figure shows that there are one or two batteries that have better ideal properties per cell than the selected battery. This is because of the discretization problem discussed below. The battery systems do not scale linearly, since you cannot design for a fractional amount of cells.
C-1.5 Battery, Motor, and Gearbox Discretization Problem
The design of the propulsion system encounters a problem of discretization – there is a discrete set of batteries, motors, and gearbox ratios available on the market. The final solution must be commercially available, which means that a given design providing a certain power, weight, etc. may not be feasible. For example, the analysis may show that the ideal motor will require an input current of 42.5A; however, there may not be any batteries on the market that can provide exactly 42.5A. None of the other design areas (dynamics & control, aerodynamics, and structures) encounter this problem to the same extent, since the dimensions, shape, and internal configuration of the aircraft are completely at the discretion of the designer.
C-1.6 Maximum propeller output power
The maximum power output of the propeller was calculated using the following method.
Given: ρ, Kv, Io, Rm, Vmax, Imax, D, P/D, CT(J), CP(J), ηgear, R
[pic] (Equation 1)
The first step is to guess a propeller rotational speed. Since the motor is connected to the propeller via fixed gear ratio, this enables the specification of the motor speed, and the calculation of the motor efficiency and operating voltage and current.
[pic]
The next step is to iterate on the propeller speed guess until the maximum propeller power output is obtained without exceeding neither the maximum battery supply voltage nor the maximum continuous current. Once these values have been obtained, the maximum thrust can be calculated as follows:
[pic] (Equation 3)
A similar method could be applied for normal operation if the propeller output power is set equal to the power required by the aircraft for a given flight speed. There would still be an iteration to guess the propeller speed, but the goal of this iteration would be to ensure that the predicted propeller output power was the same as the propeller output power required by the aircraft.
C-1.7 Numerical Takeoff Integration
Assuming that there is no effect from crosswind and the aircraft traverses the runway at α = 0˚, the equations of motion for takeoff reduce to a single, one-dimensional equation (Equation 1).
[pic] (Equation 1)
The forces acting on the aircraft are thrust, drag, and rolling friction. Since the normal force against the tires counteracts the weight of the plane and is perpendicular to the direction of motion, it can be ignored in the force summation. Thrust is simply a function of power and velocity, and drag depends only on velocity (Equations 2 and 3).
[pic] (Equation 2)
[pic] (Equation 3)
Rolling friction is determined from the normal force (in this case, the weight of the aircraft) and the coefficient of rolling friction, shown by Equation 4.
[pic] (Equation 4)
Rewriting acceleration in terms of velocity and substituting Equations 2, 3, and 4 into Equation 1 gives Equation 5.
[pic] (Equation 5)
Since the maximum power is most easily defined with an iterative process (converging to a maximum current and voltage for a given airspeed), integrating this equation of motion analytically would prove difficult. A numerical integration scheme begins with integrating the equation of motion as in Equation 6.
[pic] (Equation 6)
Using a small time step and assuming velocity is constant across this time step, an iterative equation for velocity can be found (Equation 7).
[pic] (Equation 7)
With a velocity value at each time step, the position of the aircraft can be updated as follows (Equation 8).
[pic] (Equation 8)
Thus, with initial conditions of t = 0 s, x = 0 ft, and V ≈ 0 ft/s (V cannot be zero since it lies in the denominator of the thrust expression), marching time forward until x ≥ 120 ft (runway constraint) will allow the designer to determine whether or not the aircraft will reach takeoff velocity before the end of the runway.
Appendix D: Structures Figures and Tables
Figure D-1: Preliminary Weight Estimation
[pic]
Figure D-2: Discretization of the Wing
Figure D-3 a: Mean Aerodynamic chord
[pic]
The mean aerodynamic chord was obtained by extending the length of the root chord on top of the tip and vice versa. Then the MAC is the intersection of the line connecting the extensions and the quarter chord.
Figure D-3b: Lift Distribution on wing
[pic]
Figure D-4: Lateral tip-over analysis
Figure D-5: Longitudinal tip-over analysis
[pic]
Figure D-6: Main wheels from Tower Hobbies
[pic]
Figure D-7: Tail wheel and bracket from Tower Hobbies
[pic] [pic]
Figure D-8: Wing-Fuselage Attachment
Figure D-9: Nylon screw loading
Figure D-10: Carbon Spar Attachment
Table D-1: CG Calculation spreadsheet
[pic]
Appendix D-1: Individual Trade Studies
Appendix D-1.1 Trade Study #1: Wing Materials
The purpose of this trade study is to see which materials can match the requirements of our wing. The loadings that the wing has to carry in bending and weight are the requirements taken into account in this trade study. As today, the group has been considering balsa for the skin of the wing, after this study the group should be able to justify the usage of this material.
Constants used:
UCS = Ultimate Compressive Strength (psi)
The ultimate compressive strength was taken from for aluminum and from the 1999 Forest Products Laboratory Wood Handbook. Since aluminum of a ductile material, the ultimate tensile strength is used as the ultimate compressive strength. Table # 1 summarizes the values used.
|Material |UCS (psi) |
|Aluminum 6061 T6 |45000 |
|Balsa |725.2 |
|Bass |4730 |
|Spruce |5180 |
Table #1 UCS for materials
Design Variables:
[pic] = Bending Stress (psi)
[pic] = Minimun thickness accepted (in)
The bending stress depends on the ultimate compressive strength of each material. Since the polar moment of inertia is on the formula for bending stress, the thickness of the skin depends on the bending stress.
Parameters controlled:
W/S = Wing Loading (lb/ft^2)
b = Wing Span (ft)
[pic] = Root chord (ft)
[pic] = Tip chord (ft)
t = Airfoil thickness (in)
W = Weight of aircraft (lb)
UL = Ultimate Load (g)
[pic] = Surface Area (ft^2)
Most of the geometry of the wing was taken from the results obtained from the aerodynamics group. Using these values, this study was made. The ultimate load was taken as 10 g because that was the number that the pilot suggested for the turning.
Measure of merit:
Weight of skin (lb)
After all the calculations are done, the weight will be use to compare the different materials. The one that makes the wing to be lighter will be choose as the best material.
Method used
The wing was divided in four parts (discretization of wing). The mean aerodynamic chord (MAC) was obtained extending the tip chord and root chord of the wing and intersecting the diagonals with a line that goes from 25% of the root to 25% of the tip (See Fig. #1).
[pic]
Fig. #1 Mechanism to find MAC
Each part was taken as a trapezoid to calculate the area. After that the lift for each part was calculated using Eq. #1.
[pic] Eq. #1
where W/S is the wing loading at ultimate load and A is the area of each part. Fig. # 2 shows the lift distribution. After that the bending moment due to lift was calculated for each part (Fig. #3).
[pic]
Fig. #2 Outline of the wing and Lift distribution
Fig. #3 Bending Moment in the wing
The bending stress (Eq. #2) was equal to the Ultimate compressive strength.
[pic] Eq. #2
Where [pic]is the bending moment previously calculated, [pic] is the maximum height of each part and [pic] is the polar moment of inertia. An assumption for this estimate is that the foam is not carrying any load, meaning that just the skin was used for this calculation. Since the polar moment of inertia is a function of the thickness, the “solver” function in Excel was used to calculate the thickness necessary to carry the bending stress. After the thickness for each part was obtained, a linearized thickness was obtained so the surface of the wing was uniform. Because it is hard to cut the sheet of material with different thicknesses, the highest resultant thickness of the discretized wing was taken as a constant thickness for the entire wing. Fig. #4, 5,6 and 7 shows the different thicknesses mentioned previously.
[pic]
Fig. # 4
[pic]
Fig. # 5
[pic]
Fig. # 6
[pic]
Fig. # 7
Table # 2 summarizes the thicknesses and total weight for the different materials.
|Material |Thickness of skin (in) |Total Weight (lbs) |
|Aluminum 6061 T6 |1.67E-03 |0.007 |
|Balsa |1.48E-01 |0.593 |
|Bass |1.64E-02 |0.066 |
|Spruce |1.50E-02 |0.06 |
Table #2
As it can be seen the best material for the skin of our wing is aluminum. Since it is very strong, the thickness is very small and the weight as well. The problem with this material is that it is very hard to use for construction. For this reason, it may not be considered for the skin. The material chosen by the group was balsa, as it can be seen from Table #2, Bass and Spruce seem to be better for our wing. They are around 88% lighter than balsa.
The group never considered other materials for the skin since one of the group members has experienced with balsa. After this trade study, the group should consider bass and spruce for the skin. It is very easy to find balsa in the market, but some research has to be done to find the availability and cost of bass and spruce.
Appendix D-1.2 Trade Study #2: Wing Skin Thickness
1. Introduction
The purpose of this trade study is to figure out the optimal thickness of the skin of the wing. The optimal thickness means the lower thickness to resist to the fixed loading case. Indeed, the main goal in designing any aircraft is to reduce the weight as much as possible and optimizing the thickness of the skin (thus the weight of the skin) is a good way to do. The first part of this study will only consider the skin as a structural element. In the second part, we will consider the skin and an added carbon spar as structural elements.
Note: The results of this study have already been presented in the first structural QDR presentation, but the method hasn’t been described in details due to the short time of the presentation. This document is going to present the study with more precision.
2. Method of the study
a. Structural Concept
The structural concept chosen for the wing is an expanded polystyrene core covered by a thin balsa skin. This is an easy to built concept (few pieces, few operations) and allows keeping a good airfoil shape and a good aerodynamic surface.
[pic]
Assumption for after: the foam is not a structural part of the wing.
b. Design Variables
The way to size the thickness of the skin is pretty simple: We know that for a given loading, the thinner the skin is, the higher the stresses inside are. Thus, if we size the skin for a given maximum loading case to resist to the ultimate stress of the balsa, we have the minimal thickness allowed for this loading case.
The method with the design variables and formulas can be summarized on the chart as follow:
c. Description of the method
➢ The first step is the discretization of the wing to be more accurate than a simple beam modelisation of the wing.
The wing is cut in 4 trapezoid parts. The location of the mean aerodynamic chord is figured out by a geometric method:
➢ The bending load (the lift) is figured out:
Flift=Wload x S
This is an approximation of the real parabolic distribution. This assumption gives a linear lift, for this wing geometry with a constant sweep angle.
➢ The bending moment is figured out with the relation M=Flift x d.
For the discretized wing, all the moment product by each parts are added to calculate the maximum moment at the root:
➢ The bending stress is given by the following relation:
For a given loading, the maximum stress occurs at a maximum y. In our case, it is the value of the maximum thickness of the airfoil.
By replacing the stress with the maximum stress until failure for balsa, we can figure out the optimal inertia resisting to the load.
[pic]
➢ The optimal inertia of the airfoil can be figured out from the previous relation. However it’s difficult to find the thickness of the skin from the inertia because the formula is too complicated. That’s why we assume the airfoil shape as an ellipse:
➢ The minimal thickness allowable is determined with the equation above. The excel solver is used to figure out this value.
➢ The weight is obtained by multiplying the volume of the skin with the density of balsa.
3. Results
a. Structural skin
➢ Data: The following data have been taken from the aerodynamic geometry and from the material properties.
|DATA |
|wing loading @ UL (lb/ft²) |14,4 |
|Sref (ft²) |3,8 |
|Mass (lb) |5,5 |
|Ultimate Load (.g) |10,0 |
|UL x Weight (lb) |55,1 |
|Ult. Comp. Stress (psi) |725,2 |
|Cm max |-0,0338 |
|Vmax (ft/s) |150,0 |
|young modulus (ksi) |185,64 |
|shear modulus (psi) |23061,0 |
|balsa density (lb/ft3) |6,24 |
|wingspan (m) |6,00 |
|root chord (m) |0,89 |
|wingtip chord (m) |0,39 |
|airfoil thickness (%) |8,45 |
➢ Calculations:
➢
|Surfaces |Trapezoid height |Area (ft²) |MAC x axis(ft) |Lift force (lbf) |Mbend (lbf.ft) |
|1) (root) |0,75 |0,62 |0,37 |8,88 |36,05 |
|2) |0,75 |0,53 |1,11 |7,55 |18,79 |
|3) |0,75 |0,43 |1,86 |6,23 |7,69 |
|4)(wingtip) |0,75 |0,34 |2,61 |4,90 |1,76 |
|Surfaces |Igx ellipse (ft4) |bending stress (Psi) |min. thickness accept. (in) |Mass with thick. Min |
| | | | |accept. (lb) |
|1) (root) |1,29E-05 |7,25E+02 |1,48E-01 |4,78E-02 |
|2) |5,80E-06 |7,25E+02 |9,47E-02 |2,59E-02 |
|3) |1,99E-06 |7,25E+02 |4,97E-02 |1,12E-02 |
|4)(wingtip) |3,67E-07 |7,25E+02 |1,57E-02 |2,80E-03 |
| | | |TOTAL 1 PANEL |0,088 |
|TOTAL WEIGHT 4 PANELS (lb) |0,351 |
|TOTAL WEIGHT, CONSTANT THICKNESS (lb) |0,593 |
[pic]
This chart represents the variation of thickness of the skin versus the distance from the root. Three types of thickness distribution are represented:
➢ The first one is the dicretized thickness which is the rough result of the discretization method. The front view of the sheet of balsa should looks like:
➢ The second one is a linearized distribution. This distribution is optimal.
➢ The problem of the optimal distribution is the difficulty to build. So that, a constant thickness is determined with the maximum thickness calculated with the discretization method (to be conservative).
The result of this study is a thickness of 1.48 in. By experience, this thickness is too much important: the bending of the balsa sheet over the foam core would too difficult, and the weight too important. That’s why a second study with another structural element is done.
b. Structural skin and spar
In this study, we consider a carbon tube spar to reinforce the skin of the wing (tube has a better rigidity/weight ratio than full fill circular beam). To simplify the calculations, a condition is made on the size of the carbon relative to the size of the skin:
We consider a bending iso-rigidity between the spar and the skin, which permit to say that 50% of the load is carried by each one.
(EI)skin=(EI)spar
Now we can redo the calculations with only 50% of the load in the skin. The conditions used with the solver to optimize the spar shape are:
_the weight is minimum.
_the outer diameter is thinner than half of the maximum thickness of the airfoil.
_the rigidity is the same as the balsa skin.
We obtain the following results:
|surfaces |trapezoid height |Area (ft²) |MAC x axis(ft) |Lift force (lbf) |lift force with carbon |Mbend (lbf.ft) |
| | | | | |spar (lbf) | |
|2) |0,75 |0,53 |1,11 |7,55 |3,78 |18,79 |
|3) |0,75 |0,43 |1,86 |6,23 |3,11 |7,69 |
|4)(wingtip) |0,75 |0,34 |2,61 |4,90 |2,45 |1,76 |
|surfaces |Igx ellipse (ft4) |bending stress (Psi) |min. thickness accept. (in) |Mass with thick. Min accept. (lb) |
|1) (root) |6,46E-06 |7,25E+02 |5,95E-02 |1,91E-02 |
|2) |2,90E-06 |7,25E+02 |4,06E-02 |1,11E-02 |
|3) |9,95E-07 |7,25E+02 |2,27E-02 |5,13E-03 |
|4)(wingtip) |1,84E-07 |7,25E+02 |7,61E-03 |1,35E-03 |
| | | |TOTAL 1 PANEL |0,037 |
| | | |TOTAL |0,147 |
| | |carbon tube |
|surfaces |E*I balsa (psi.ft4) |E*I carbon (psi.ft4) |spar inner dia. (in) |spar outer dia (in) |weight (lb) |
|1) (root) |1,20E+00 |1,20E+00 |3,28E-01 |4,49E-01 |3,85E-03 |
|2) |5,38E-01 |5,38E-01 |3,11E-01 |3,87E-01 |2,17E-03 |
|3) |1,85E-01 |1,85E-01 |2,85E-01 |3,24E-01 |9,83E-04 |
|4)(wingtip) |3,41E-02 |3,42E-02 |2,50E-01 |2,62E-01 |2,59E-04 |
| | | | | |7,26E-03 |
|TOTAL WEIGHT, 4 PANELS + ½ SPAR (lb) |0,150 |
|TOTAL WEIGHT, CONSTANT THICKNESS + ½ spar (lb) |0,241 |
The thickness needed for the skin with a carbon spar is thinner than the thickness for the skin structure without carbon spar, which seems logical. For the same reason as the previous study (more easy to build), the thickness is kept constant at the maximum value, 0.59 in (the value seem to be good according to personal experience).
The chart points out that at 1.5 foot from the root, the constant thickness is thicker than the thickness needed for the wing without spar. We can conclude that the carbon spar is not useful from 1.5 foot until the tip.
Weight results:
| |weight of skin alone (lb) |weight of skin +spar (lb) |
|discretized thickness |0,351 |0,15 |
|constant thickness |0,593 |0,241 |
The weight of the skin/spar solution is widely lower (60% lower for the constant thickness) than the other one.
4. Conclusion
As a result of this trade study the carbon spar with balsa skin is the best solution to use. The carbon spar permits to reduce the maximum value (at the root) of the minimum thickness needed without a lot of weight added. That’s why the weight is considerably reduced relative to the other solution. The values obtained are very close to those usually used in model aircraft (0.040 in< t ................
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