Team 2 Thiokol Draft - Purdue University



AAE 451

Thiokol Final Design Report:

Design of a Stylish RC Aircraft

First Draft

March 10, 2005

Team 2

Michael Caldwell

Jeff Haddin

Asif Hossain

James Kobyra

John McKinnis

Kathleen Mondino

Andrew Rodenbeck

Jason Tang

Joe Taylor

Tyler Wilhelm

Executive Summary

A Stylish Remotely Piloted Aerial Vehicle

In an effort to efficiently accomplish the mission of senior design, some aircraft of the past had seemingly lacked a certain degree of style. Hence, in addition to the basic mission requirements to be discussed later, a major aspect of the design process for this group focused on styling. In as much as style can be rather subjective, for this team the attempt was in creating a unique design, yet with a theme somewhat reminiscent of more archaic fliers. Additionally, the design was to be suitable for the confines of indoor flight, specifically Mollenkopf Athletic Center. These two things in mind led us to having a thematic airplane that would appeal to beginning RC pilots. The ability to fly indoors allows flexibility in where the plane can fly in general. This easy-to-fly airplane opens RC aircraft flying to customers previously restricted by space.

The “Classic Flyer” is electrically powered with a brushless AXI 22 12/20 motor and Kokam lithium-polymer battery allowing for a total flight endurance of just over 25 minutes. The motor/propeller combination used allows for enough excess power such that more advanced maneuvers are possible for the capable pilot. Weighing a total of just under 2 lbs, the aircraft is easily handled and portable. If a suitable area is not available for takeoff, it is comfortably hand launched. The 5.5 ft. wing blends into the center fuselage and has an aspect ratio of 6. Wing twist distribution and taper ratio (.6) help achieve a near elliptical span-wise lift distribution. The wing has a slight dihedral of about 5o for stability. Favorable lift to drag ratios are obtained by using the Wortman FX 63-137 high camber airfoil section, specifically designed for low Reynolds number flight. The canard configuration remains non-standard in the industry, although it was used on the first powered flier, and is a style point for the plane. Two outer fuselages act as supports for the canard and landing gear. An elevator control feedback has also been implemented so that inexperienced pilots will have less difficulty in pitch control. The overall look of the aircraft is that of a large but docile machine, functional in form as craft from the earliest era of powered aviation. To amplify this effect, landing skids, rather than traditional tires, have been attached for a more antiquated look. The attachments for the skids easily allow for replacement with tires if conditions dictate such.

Building and flight tests are still pending for this aircraft. Once completed, more detail will be given on the subject. As designed, the aircraft is minimal in material cost, sturdy, and easy to fly. Another goal of the design was to achieve relatively slow flight speeds. For an amateur flier to be successful flying indoors or other restrictive spaces, this is necessary for adequate reaction time. As stated before however, the excess power of the motor may be utilized to perform more advanced or aerobatic type maneuvers based on the experience of the pilot. The classic, stylish appeal and versatility of performance make this a great trainer, first airplane, or nice addition to a current collection.

Introduction

The desired product in this endeavor was a stylish and unique airplane that could be easily flown indoors by a minimally experienced pilot. More specific mission requirements set for the by the instructor and the team included the following: electric powered; aircraft fits on a 6’X6’ tabletop; a loiter speed of less than 25 ft/s; stall speed at or below 15 ft/s; takeoff roll of less than 60 ft; flight endurance of 15 minutes; a climb angle of 20o and descent angle greater than -5.5o; the aircraft should be capable of flying a figure 8 in the 150 ft. width of the indoor field; lastly, the airplane must implement at least one feedback controller. No additional payload is required for the flight.

Having these as guiding parameters, each team member provided several design concepts, from which a team concept design was formed. Sizing and analysis lead to a more detailed design. With the design frozen, design for manufacturing began. Naturally following is prototype fabrication, which will be discussed in the final draft when it has been completed. The aircraft will then be flown and tested and comparisons made between the actual airplane and the design. In the following report the details of each process are discussed to include: concept selection, component design such as aerodynamics, structures, etc., the building process, and finally the flight testing and analysis. If time is sufficient it is reasonable to expect that there may need to be some re-design after noting characteristics during the first flight. All such decisions will be noted in this report.

[pic]

Figure 1: Preliminary Team Concept Sketch

Concept Selection

Generating solution concepts is the fundamental aspect of design. Understanding the mission requirements as a basis for generating them will result in a quality product. Selecting a concept design for this project was done using two different techniques: weighted objectives and Pugh’s Method.

For the weighted objectives, twelve objectives were selected which affected the mission requirements. Rankings and percentages were then assigned to each objective. Each objective was ranked either 1 – Poor, 3 – Average, or 9 – Excellent for the nine design concepts. Each objective score was then multiplied by its corresponding weighted average and added together to give a total score for each design (Figure 2).

The next step was Pugh’s Method, where the concept which scored highest in the weighted objectives was set as the datum, a standard for which all other designs were compared. To compare each design, a score of “+” (better), “-“ (worse), or “s” (same) was assigned for each objective. The sum of each scoring criteria (“+”, “-“, or “s”) was taken and design strengths and weaknesses were determined. Design 6, a three fuselage design with canard and high aspect ratio wing, was selected from the weighted objectives and Pugh’s Method results.

Aerodynamics

Airfoil Selection

Knowing that the aircraft would be flying relatively slow, one of the initial and very critical design processes was airfoil selection. The basic criteria for selection consisted of the following: indicated performance at low Reynolds number; high lift-to-drag ratio; large minimum drag region and/or a small increase in drag with lift ; highest possible Cl max; finally, the geometry should be easily formed with the materials to be used. Actual wind tunnel data for airfoil sections could be obtained on the NASG online airfoil database: . For a selected airfoil, a drag polar with data values and a lift curve were provided for a specified Reynolds number. The section’s contour shape and coordinates were provided. This information was the basis for the airfoil study and subsequent selection.

Having a reasonably accurate range for Reynolds number was important in this selection process. This was clearer after the initial sizing where approximate weight and wing area were determined. The flight regime was already well set by the initial constraints; therefore, using the basic equation for lift, [pic][pic], and Reynolds number, [pic], a range of chord lengths could be determined. These corresponded to Re numbers where the airfoil was known to have good performance.

The Figure 1.1 in Appendix I show the comparison between the Wortmann FX 63-137 and the Selig 1210 airfoils. Though the Selig gives a greater max lift coefficient, it is actually more than necessary; furthermore, the Wortmann has a much smoother drag bucket and lower drag coefficient values for the range of lift coefficients in which the aircraft will be flying. Ultimately, the Wortmann airfoil was selected (Figure 1.2). The 3-D max lift coefficient ended up at approximately 1.4, where the 2-D indicates the max at around 1.6. The stall angle is also shifted up from 10 degrees to about 14 degrees for the 3-D wing.

3-D Wing Analysis

As an initial analysis for the 3-D effects of our wing, Prandtl’s lifting line theory was implemented. Mathematically, this theory models the wing as a series of horseshoe vortices in a spanwise distribution. The idea is to model the circulation at the wingtips, compute downwash, and therefore the induced drag and actual lifting capability. The method of implementation for this team concurrently analyzed those items above and developed a wing twist distribution for a specific design point flight speed.

The MATLAB scripts developed allowed the user to hold the wing area, taper ratio, and aspect ratio constant, as these were specified in other analyses. The design point was specified by flight velocity. Additionally, when total lift is specified (total weight) the distribution is assumed to be elliptical spanwise. From here the required spanwise Cl is determined. Using airfoil camber data, which directly relates to zero-lift angle of attack, and the lifting line model, a twist distribution is determined to achieve elliptical loading, which when integrated spanwise, equal the total weight of the aircraft. The design point was chosen as the loiter velocity (appx. 22 ft/s) since this is where the majority of the mission is spent. TWISTConstantSAR.m sets up the geometry and constants, lline.m (modified from Prof. Williams, AAE Purdue University) models the wing as the line of horseshoe vortices, and integ.m is the integrating function to verify total lift is as specified. Figures 1.3 and 1.4 show the required Cl and twist distribution. The drag polar of Figure 1.5 accounts only for induced drag as this is an inviscid model. Later this feeds into the aircraft drag model for a total aircraft drag polar. The twist of the wing also helps in stall characteristics. Since the root of the wing will always be at a higher angle of attack than the rest, it will stall first and aileron effectiveness will be prolonged in a stall.

Mathematical Drag Model

The drag coefficient of the aircraft was calculated using Equation 1.1. In Equation 1.2, the viscous drag coefficient is [pic] and the induced drag coefficient is[pic]. The viscous drag (also called parasite drag) was calculated using the component buildup method found in Raymer’s Aircraft Design: A Conceptual Approach. The method uses a flat-plate skin-friction coefficient (Cf), component wetted area (Swet), the pressure drag due to viscous separation estimated by a component “form factor” (FF), and a “Q” factor to estimate interference effects on the component drag. In addition, a miscellaneous drag coefficient ([pic]) is used to account for unretracted gear and is estimated to be approximately 10% of the component parasite drag. An additional estimate for leakages and protuberances ([pic] ) can also be added; however, for the design it was considered to be negligible. The parasite drag build up is shown in Equation 1.3. The equations for individual component parasite drag buildup are located in the Appendix. Table 1.1 lists the component wetted area and parasite drag along with the total component parasite drag, miscellaneous drag, and overall parasite drag.

The induced drag coefficient ([pic]) is a function of the lift coefficient (CL), as shown in Equation 1.3. In Equation 1.3, e is Oswald’s span efficiency factor and AR is the aspect ration. Equation 1.3 can also be written in the form of Equation 1.4 where e is related to δ by Equation 1.5. An Oswald’s span efficiency factor (e) of 0.85 was used for the design.

From Equation 1.1 a Drag Polar can be found. The Drag Polar compares lift to drag for various angles of attack. It is important to note that there is no induced drag when the lift is zero; however, drag is nonzero when lift is equal to zero because the parasite drag is not dependent on the lift or angle of attack. Figure 1.6 show the Drag Polar for a Reynolds number (Re) of 147,820 which corresponds to the loiter phase of the mission.

CL Max

The maximum lift coefficient of the wing ([pic]) was calculated using Raymer’s approximation for high aspect ratio subsonic wings Equation 1.6. Equation 1.6 assumes moderate sweep and a large leading edge airfoil radius. The resulting maximum lift coefficient of the wing is [pic].

Endurance

One major constraint on the aircraft design was the endurance. This was a concern early in the design to verify that the aircraft can maintain loiter velocity for the given amount of time. To complete the analysis a few calculations were needed. First, the equations for CL and CD were used to create a function for L/D in terms of angle of attack, Equations 1.7 – 1.10. Next, L/D was calculated and since CLo, CLα, CDo, and k are all constants for a given design, L/D is only a function of angle of attack as shown in Equation 1.11. For a given set of geometry the velocity of the aircraft is a function of the lift required and the angle of attack. For specific flight conditions, i.e. straight and level flight or for constant altitude turning flight, the velocity becomes a function of only angle of attack. The velocity as a function of angle of attack is shown below in Equation 1.12.

For straight and level flight the lift needs to be equal to the weight of the aircraft to maintain a constant altitude. For turning flight the equation becomes a bit more complicated. The lift in constant altitude turning flight the wing must produce lift to offset the weight of the aircraft as well as produce the inward force to create a circular flight pattern. The radial acceleration required for a given radius turn is a function of velocity as shown in Equation 1.13. An illustration of the free body diagram is shown in Figure 1.7. The total lift required for constant altitude is a sum of the two perpendicular vectors and is shown in Equation 1.14. Combining Equation 1.14 with Equation 1.12 produces an equation for turning velocity that is only dependant on angle of attack.

Combining this equation with the Drag equation produces the thrust required by the propeller at a given angle of attack and velocity. With a set velocity and thrust, power can be calculated as shown in Equation 1.15. This is the output power of the propulsion system after the efficiencies of the motor, gearbox, and propeller have been taken into account. So the input power required to produce the output power is almost double, the efficiency of the entire engine is 0.50. Multiplying the input power and the time at the given power calculates the energy needed for the loiter flight. This number can then be used to size the amount of battery power needed for the entire flight. It has been assumed that half of the 15 minute flight time will be constant altitude turning flight and the other half in straight and level flight. This produces an energy requirement that is a function of the angle of attack of the straight flight and the angle of attack of the turning flight. The endurance of the aircraft is 16 minutes as shown in Figure 1.8 in the appendix.

Propulsion

Propeller Selection

Goldstein’s method was used to analyze the performance of the propellers. Goldstien’s Kappa factor was replaced by Prandlt’s tip loss factor to allow the computation of performance for propellers with an arbitrary number of blades. Prandlt’s tip loss factor is computed as a function of the ratio radial station along the blades to the blade radius, r/R, and the helix angle at the tip of the blade,

[pic].

Prandlt’s tip loss factor for n blades is[pic], and it is an excellent approximation of Goldstein’s kappa factor as shown in Figure 2.1. All figures, equations, and tables are located in Appendix II.

The MATLAB script nblade.m was used to implement Goldstein’s method with Prandtl’s tip loss factor for a wide variety of propellers. A two bladed 8 in diameter propeller with 5 inches of pitch was found to best fulfill the requirements. A three bladed 8 inch diameter propeller with 6 inches of pitch also does very well and will allow much more aerobatic aircraft performance if desired. Table 2.1 compares the two propellers for various mission phases.

Figure 2.2 compares the propeller efficiency of the propellers at loiter speed, 22 ft/s. The 2 bladed propeller has higher efficiency in the operating range where the required amount of thrust is produced. Figure 2.3 compares the thrust of the propellers at loiter speed, 22ft/s. The three bladed propeller produces more thrust and has a much higher maximum thrust. Figure 2.4 compares the power required by the propellers at loiter speed, 22 ft/s. The 2 bladed propeller requires significantly less power to operate.

Motor Selection

Engine sizing and battery selection began with a theoretical look at the thrust requirements of our aircraft. To do this flight was divided into 4 phases; take off, climb, loiter and cruise. The thrust required was determined by calculating the lift requirement for each phase of flight (Equations 2.1-2.4) and equating that to drag with the help of a drag polar Figure 2.5. As calculated the climb phase required the highest thrust, 4oz, thus becoming the baseline for engine selection. Thrust requirements for the various phases of flight are shown in Table 2.2.

A theoretical model of the Power required of each phase of flight was necessary for battery selection. First, energy input required was calculated from Equation 2.5 where brushed motor efficiency was taken to be 60%, brushless 85%, and propeller efficiency was taken to be 75%. Power required was calculated from Equation 2.6, and used to solve directly for the necessary mAh a battery would need to fly this mission (Equations 2.7 & 2.8). Endurance required for brushed motor flight was near 420mAh, and 300mAh required for brushless motor flight (Table 2.3 & 2.4).

Area modeling experts urged us to revisit battery energy requirements when hearing of the calculated value of 300mAh. They claimed that the battery needed to be closer to 1300mAh of energy. The claim was made on the grounds that model aircraft missions are never ideal. Factors such as wind, pilot experience, and control systems can cause an aircraft to quickly stray from theoretical conditions. The prop2 code was used to validate the opinion of the modeling experts. The prop2 code calculated a 21 minute flight time for our aircraft configuration using a 1200mAh battery. Through iteration it was discovered that the best battery for our configuration according to the prop2 code was the Thunder Power 900mAh battery as it resulted in a flight time of 16 minutes. Prop2 however does not take into account the power requirements the servos, receiver or control system which will also be onboard. With these accounted for a battery of size 1044mAh is required according to prop2. This battery size is not available. Therefore the Kolkam 1200mAh battery was again chosen. The 1200mAh battery fits into the weight constraints set for the battery. As an added bonus, the Kolkam 1200 also has about 15% more energy than required which may be necessary to account for unknown variables during flight. Motocalc[1] was used to validate this information Figure 2.6 and 2.7. For various pre-programmed missions, Motocalc estimated between a 15 and 23 minute flight time with our chosen components.

A brushless Model Motors Axi 2212/20 motor was chosen to fly our aircraft. When compared to brushed motors the Axi motor wins in all respects. The smallest available brushed motor is overpowered by nearly 50% (it is designed to fly a 60oz aircraft). A limited number of available batteries results in the selection of a battery that will power flight for nearly 25 minutes (70% longer than constrained). This in addition to factors such as the high temperatures which brushed motors run, and the extreme weight difference, pointed to the Axi motor being the right choice for this mission. The pricing and weight differences can be seen in Tables 2.5 & 2.6. The weight of components necessary for brushed flight being double that of brushless flight, and the total cost of brushless components is only $45 more.

Structures

Structural Layout and Construction

The structural design consists of two outer fuselages, a main center fuselage, a canard, a rear main wing, and two vertical tails. The outer fuselages are 3 ft (36 in) long cylinders with a 0.083 ft (1 in) radius. At the tips they become conical. The distance between the outer fuselages is 1 ft (12 in). The main fuselage is a 1.75 ft (21 in) long cylinder with a 0.167 ft (2 in) radius also becoming conical at the tips. This length includes the space for the propeller and the propeller spinner. Both the canard and main wing are a blended body configuration. The canard has a span of 1.8 ft (21.6 in) with a 0.67 ft (8.04 in) chord with an aspect ratio of 2.69. Outside of the outer fuselages the canard has a taper ratio of 0.7 and a dihedral angle of -4°. The elevator has a span of 0.33 ft (4 in) and has a 1 ft (12 in) span running the entire length between the outer fuselages. The main wing is made from three sections. The middle section has a span of 1.67 ft (20 in) and a chord of 1 ft (12 in). A taper ratio of 0.7 and a dihedral angle of 4° begin at the outer sections. The total span of the wing is 5.24 ft (62.88 in) with an aspect ratio of 5.24. The ailerons have a 0.20 ft (2.4 in) chord and a 2.62 ft (31.44 in) span. They start 0.25 ft (3 in) from the wing tip. The vertical tails are mounted in the very back of the outer fuselages with the rudders hanging off the end. They are each swept back at a 45°, have a chord of 1 ft (12 in), and a span of 0.64 ft (7.68 in) with an aspect ratio of 0.64. The rudders have a span of 0.82 ft (9.84 in) and a chord of 0.50 ft (6 in) each.

EPP foam was milled using a CNC machine for the construction of all parts except the vertical tails and landing gear. Top and bottom halves were constructed for each fuselage with the airfoil sections cut out of the sides and the center hollowed for internal component storage on the main fuselage (Figure 3.1) and space for battery wires and a carbon fiber rod in the outer fuselages. The carbon fiber rod in the outer fuselages was added for reinforced rigidity. The canard was placed inside the outer fuselages’ airfoil cutout with the foam elevator attached to the canard using a tape hinge (Figure 3.2). A hole was cut out of the top of the canard just big enough to fit the battery inside. A hatch was created with the cutout piece. A carbon fiber rod was used as the spar at the quarter chord to provide additional strength. The middle section of the main wing was placed inside the outer fuselage as well (Figure 3.2). The outer fuselage halves were then glued together with epoxy. All of the internal components except the battery were placed inside the main fuselage. The two halves were kept together using small plastic taps evenly spaced on both sides of the hollow section that friction locked them in place. The main wing also had a carbon fiber rod placed inside as the spar at the quarter chord. The outer sections of the wing were connected to the center section at the spar and with plastic tabs near the trailing edge to keep the wing from rotating about the spar rod. The ailerons also had a tape hinge. UltraKote then covered the entire structure.

The landing gear and vertical tails were constructed using balsa wood. The gear were attached to the fuselage by making 2 in blocks of balsa wood covered on top and bottom with ply wood cross grained (Figure 3.3). A hole was drilled through the block for a wire to go into. A bracket was then screwed onto the bottom with the wire bent into it. This prevents the wire from rotating in any direction. This wire created a strut for the landing gear. An axle was attached at the end which enabled the gear to be interchangeable between wheels, skids, and floats. The blocks were then placed inside four cutouts on the bottom of the outer fuselages at the middle of the canard and just behind the spar of the main wing. These cutouts were cut just barely big enough to fit the blocks creating a friction lock. Tape was then placed over the block for additional support. Doing this created a landing gear system that was designed to break gently so that the gear could easily be placed back into the fuselages after a rough landing.

For the vertical tails a skeleton structure was created and wrapped in UltraKote. At the bottom of the tail two extra pieces of balsa were glued on creating a tab (Figure 3.4). The tail was then slide into a slot cutout on the two upper halves of the outer fuselages and tape was placed over the slot to hold the tail in place from the back. The rudder was made from a solid sheet of balsa and attached using a plastic hinge.

Landing Gear

A major aspect of the aircraft design project is the selection of a launching and landing method. The design and positioning of a landing gear system is determined by the unique characteristics (mission requirements, weight, geometry) associated with each aircraft. Appropriate configurations can then be evaluated to determine how well they match with the aircraft structure and operational requirements.

One of the trade studies performed analyzed different launch and landing methods and determined which ones were best suited for the concept. Some design variables to consider were weight, cost, and performance. The gear should also be within the weight estimate and budget but the performance should not be hindered. It must also be structurally safe and rigid enough to withstand landing on the Astroturf inside Mollenkopf Athletic Center.

For launch techniques, there were several options to choose from: conventional launch, ski launch, hand launch, cart launch, or balloon launch. For landing techniques, there were several options to choose from: conventional landing, ski landing, belly landing, parachute, or hand catch.

After careful deliberation of each method’s advantages and disadvantages, the ski launching and landing method was selected. For the building process, the parts for the gear configuration should be very easy to find and assemble. For launching, the skis can be adjusted to factor in propeller strike. For landing, the wider stance of the skis will prevent tip-over and offer a smoother landing. The skis are also relatively cheap (around $10 - $15), won’t weight very much (about 1 oz.) and will add more style to the aircraft’s design.

Placement and technique for mounting the gear system on the aircraft was looked at. It was very important to take into account how much extra weight being adding to the aircraft and how that would affect the center of gravity depending on location. A wire strut will be attached to the bottom stringer of the outer fuselage for each gear. In order to take the load on impact, the bottom stringer would be increased from 1/8” to 1/2” in width. The wires would be held in place by a mounting bracket and screws and secured to the fuselage (Figures 3.5 and 3.6). They could be cut at a length that would give enough prop clearance (~6in.) and bent at any angle.

With the results of the analysis from the second trade study, the influence of positioning of the landing gear on the center of gravity is very nominal. Moving the front and back landing gears either forward or backward 2 inches will only change the CG by 0.006 feet or 0.072 inches. The important point is to secure the wire to a point of the bottom stringer of the outer fuselage that will be most stable. As seen in the top view of the aircraft, the most stable place to put the landing gears would be at the max thickness (where the I-beam will be placed) of the canard and wing. The load on landing will then be applied to the I-beam which will give more support than anywhere else along the length of either the canard or wing.

V-n Diagram

The purpose of a V-n diagram (Figure 3.7) is to describe the relationship between an aircraft’s speed, its longitudinal maneuverability, and its structural strength. The maximum lift line is the main focus of attention. It indicates how aggressively, at a given airspeed, the aircraft can pitch to change its flight path without stalling the wings or doing aerobatic maneuvers through excessive loads. The maximum lift line demonstrates how the load factor diminished as speed decreased, disappearing finally at the 1-g stall velocity. Table 12 in Raymer’s book listed typical limit load factors for different types of aircrafts. A general aviation aircraft was chosen with maximum and minimum load factors of 3-g and -1.5-g, respectively. When flying at a loiter speed of 25 ft/s, the max load factor needs to be at 2.7778-g or below.

Structural Design and Wing Loading

As the design process evolved, a new material, EPP or expanded polypropylene foam, became available. This foam had superior Young’s modulus and maximum yield stress to previous foams. Given this new material, our wing analysis was conducted again to determine if in fact a lighter wing could be constructed using EPP foam. A CAD model showed that each wing half, if constructed out of solid foam, would have an internal volume of 428.79 cubic inches, or .248 cubic feet. Given the density of EPP foam at 1.3 lbs/cu. ft., this gives a total wing weight of .6448 lbs, lower than our balsa construction wing, which was estimated to weigh .7 lbs. The use of this foam also simplifies construction techniques, as the entire structure can be machined using a 3 axis cnc mill from large foam billets.

If a foam wing was to be used, wing loading analysis would need to be recalculated to determine if any internal stiffeners were required to reduce maximum bending stress at the wing root. Bending stress in a beam can be calculated using the Equation 3.1.

Equation 3.1

[pic]

Where M is the moment on the beam, y is the distance in the “z” direction from the beams neutral axis, and I is the moment of inertia of the wing. Using the elliptical load distribution given by the aerodynamics group, a lift generated by each wing half of 1.01 lbs, and the load factor due to maneuvering calculated to be 2.77, the bending moment was determined to be 52.87 in.*lbs. Given a maximum y distance of .95 inches, and a moment of inertia for our wing cross section of .1388 in^4, the maximum bending stress in the wing is 361.8 psi. Given a yield stress of EPP foam of 4000 psi, it was determined that no additional covering or internal support would be necessary to an EPP wing.

The twisting moment on the wing was determined using the Equation 3.2, based on aerodynamic loads.

Equation 3.2

[pic]

The maximum twisting moment on the wing section, given Cm for our airfoil and our flight velocities of no more than 30 ft/s, is 2.328 in.*lbs. The twist angle from this moment, is given by the Equation 3.3.

Equation 3.3

[pic]

EPP foam is quite stiff, featuring a Young’s modulus of 1000 ksi. This translates into a large shear modulus, G as well, and correspondingly the twist in the wing was found to be only .127 degrees.

Maximum wing tip deflection can be found using beam theory equations, if E and I are known and constant, as they are here. Integration of beam equations can be difficult for complex loading equations, such as our elliptical load distribution. A simple derivation exists for a cosine based load distribution, and manipulation of some constants allowed a cosine based loading distribution that very closely approximated an elliptical distribution (R^2 > .98). This produced a maximum wing tip deflection of .163 inches.

Dynamics & Controls

Canard and Vertical Tail Sizing

Multiple methods are available for sizing of horizontal and vertical surfaces. The class one sizing method, also known as the tail volume method, was the first method utilized. Historical information for this class of radio controlled aircraft was difficult to obtain. For that reason, typical values for a homebuilt aircraft were used. The volume coefficients, and the resulting tail sizes, can be seen in Table 4.1.

[pic]

Table 4.1: Class one tail sizing

The second method for sizing the canard and vertical tail is known as class two sizing. This method uses “x-plots” to determine the areas of the tail surfaces. The first line on the x-plot for the vertical tail sizing was made by plotting Cnβ versus the size of the vertical tail. A second line is plotted where Cnβ is constant at 0.0573 rad-1. This is the desired value of Cnβ given by both Roskam and Raymer. The intersection of the two lines sets the size of the vertical tail. The size of the canard was also set using an x-plot. Non-dimensional values of the location of the center of gravity and the location of the aerodynamic center are plotted versus canard size. The canard is sized by determining where the distance between the two lines is equal to the desired static margin. For this aircraft, the desired static margin is 15%. Table 4.2 summarizes the results of class 2 sizing.

[pic]

Table 4.2: Class two tail sizing

The size of the vertical tails was investigated further by varying the location of the tails, over a range of angles of attack. With the aerodynamic center of the vertical tails set 0.935 ft behind the aircraft center of gravity, and an angle of attack of -10 degrees, the maximum size each vertical tail becomes 0.847 ft2. The size of the canard is chosen from class one sizing because there is less estimation in the variables used for class one sizing as compared to class two sizing. The final design sizes of the tail surfaces are shown in Table 4.3.

[pic]

Table 4.3: Final design tail areas

With the areas of the canard and vertical tails established, the control surfaces were sized. Ailerons typically extend from about 50% to 90% of the wing span. Elevators and rudders are typical about 90% span of the horizontal and vertical tails respectively. The aileron chord is usually about 15-25% of the wing mean aerodynamic chord. The chord of the elevator and rudders are typically 25-50% of the horizontal and vertical tail chords respectively. For this design, the rudders extended the entire span of the vertical tails. The elevator had to fit between the two outer fuselages, so the span was restricted to 1 foot. Table 4.4 shows the dimensions of the control surfaces.

[pic]

Table 4.4: Control surface sizes

Center of Gravity and Aerodynamic Center

Determining the center of gravity is another very important part of aircraft design. The center of gravity is the mass-weighted average of the component locations. The aerodynamic center of the airfoil is where the aerodynamic moment remains constant. To have a stable aircraft, the center of gravity must be in front of the aerodynamic center. For the aircraft, the center of gravity is at 1.708 ft and the aerodynamic center or neutral point is at 1.856 ft (from the front of the aircraft), giving a static margin of 14.8% (Figure 4.1). This is a very good static margin for this type of aircraft design.

With the sizes and weights of each component of the aircraft known, the aircraft center of gravity and aerodynamic center can be determined. The non-dimensional center of gravity is calculated using Equation 4.1.

Where M is the total mass of the aircraft, m is the mass of each component, and x is the distance from the leading edge of the mean aerodynamic chord of the wing to each component.

The non-dimensional total aircraft aerodynamic center, also known as the neutral point, is calculated using Equation 4.2.

Once the center of gravity and neutral point are known, the static margin can be calculated. The static margin is given by Equation 4.3.

Typical values of static margin are: fighter jets 0-5%, transport aircraft 5-10%, and model aircraft 10-15%. For this design, a 15% static margin is desired.

Trim Diagram

The trim diagram was found using the method outlined in Roskam’s Airplane Design Part IV. A condition of CM=0 is must be satisfied for trimmed flight. To achieve this condition canard incidence (ic) and the elevator deflection (δc) are used. The trim diagram for the aircraft is shown in Figure 4.2. The lift and moment coefficients were calculated using Equations 4.4 and 4.5, respectfully. The effect of elevator deflection on lift was calculated using Equations 4.6 and 4.7, where [pic] is the effect of the deflection. The effect of elevator deflection on moment was calculated using Equations 4.8 and 4.9, where [pic] is the effect of the deflection. For further explanation of the calculation of [pic] and [pic] refer to Appendix IV.

The deflection angle definition is shown in Figure 4.3. From the trim diagram, the elevator deflection can be found for trimmed flight at any desired lift condition. For our aircraft, the maximum trimmed lift coefficient ([pic]) is 1.54 at an elevator deflection of 5o and the loiter ([pic]) elevator deflection for trimmed flight is -2o.

Dynamics and Control Analysis

The dynamics and control analysis was performed on the design to help ensure longitudinal and lateral-directional stability of the chosen configuration. It turned out that the static stability criterion were sufficient to ensure stability. To ensure dynamic stability, a control system has been designed to incorporate rate feedback to the pitch axis of motion. Through feedback analysis, a gain was calculated that could be implemented in a feedback loop in order to augment the stability in the pitch axis.

Stability

Longitudinal Modes:

Aircraft starting from straight and level trimmed flight as small angle of attack experience only small perturbations experiences two independent natural motions acting about an aircraft’s pitch axis. It can be represented by a single fourth order differential equation. Taking Laplace transforms and converting to standard second order differential equations, the terms become:

[pic] and [pic]

One exhibits heavy damping and high frequency call short period mode while the other exhibits less damping and lower frequency call phugoid mode.

Lateral-Directional Modes:

Lateral directional equation of motion can also be expressed by a single fourth order differential equation. Factoring and taking Laplace transforms the equation can be converted to a standard second order differential equation and two first order differential equations:

[pic] [pic] and [pic]

These terms represents Dutch roll, roll and spiral modes respectively.

The flat earth predator software was used to compute the natural frequency and damping ratio of the short period, phugoid and dutch roll modes. They are,

Short Period:

Phugoid:

Dutch Roll:

Feedback Control

A feedback control system has been designed to incorporate rate feedback to one of the axes of motion using a mechanical rate gyro The dynamic model for the design is extensively analyzed in the pitch axis. A diagram of this loop closure is shown in Figure 1. Each component is analyzed individually and combined to obtain the overall model. Three major components are of interest for the analysis and gain determination, the rate gyro, the servo, and the aircraft model. The transfer function for the rate gyro turns out to be a single constant ‘k’.

[pic]

Figure 4.3: Block Diagram of Feedback Loop

From this diagram, it can be seen that the pilot enters an elevator command to the transmitter, which passes the signal through the air to the receiver located in the aircraft. This signal then travels to the servo, which deflects the elevator. The deflected elevator then acts to produce a pitch rate through the aircraft transfer function. This pitch rate is measured by the rate gyro, and then fed back to the control loop after being amplified by the gain. The sign of the gain could be adjusted to either stabilize or destabilize the dynamic pitch response of the aircraft. The models used for approximating the servo and pitch rate gyro transfer functions are given in Appendix.

The feedback gain of the design was calculated using SISO analysis. Flat earth predator software, code provided by Prof. Andrisani, was used to get the transfer function of the aircraft; pitch rate to the elevator deflection. That transfer function was then used to perform feedback analysis using Matlab SISO tool. A gain of 0.0857 was implemented to make the aircraft more stable in the pitch axis. To ensure that the closed loop poles of the system did not move into the right half plane, the region of instability, the implemented gain had to conform to gain and phase margin requirements.

The final transfer function calculated for the chosen design configuration was found to be:

q = pitch rate

(e = elevator deflection

[pic]

The mathematical model used for the rate gyro, also provided by Prof. Andrisani, was simply:

k = rate gyro feedback gain

qm = pitch rate measured by the rate gyro

[pic]

Conclusion

The conclusion is pending and will be prepared by the final report.

Appendices

Appendix I: Aerodynamics

Appendix II: Propulsion

Appendix III: Structures & Weights

Appendix IV: Dynamics & Controls

Appendix I: Aero Dynamics

Equations:

Equation (1.1) [pic]

Equation (1.2) [pic]

Equation (1.3) [pic]

Equation (1.4) [pic]

Equation (1.5) [pic]

Equation (1.6) [pic]

Equations 1.17 – 1.28

[pic]

[pic]

[pic]

Figures:

[pic]Figure 1.1

[pic]

Figure 1.2

[pic]

Figure 1.3

[pic]

Figure 1.4

[pic]

Figure 1.6

[pic]

Figure 1.6

Tables:

[pic]

Table (1.1)

Appendix II: Propulsion

Equations:

Equation 2.1: Take off drag produced

[pic]

Equation 2.2: Climb phase lift requirement

[pic]

Equation 2.3: Cruise phase lift requirement

[pic]

Equation 2.4: Turning phase lift requirement

[pic]

Equation 2.5: Theoretical battery energy required

[pic]

Equation 2.6: Battery power input required

[pic]

Equation 2.7: Battery energy required for brushed motor flight.

[pic]

Equation 2.8: Battery energy required for brushed motor flight.

[pic]

Figures:

[pic]

Figure 2.1

[pic]

Figure 2.2

[pic]

Figure 2.3

[pic]

Figure 2.4

[pic]

Figure 2.5: Drag Polar produced by Wortman FX63-137 airfoil with twist.

[pic]

Figure 2.6: Airspeed and Amps vs. time for Motocalc 'Sedate' mission with brushless motor and components.

[pic]

Figure 2.7: Airspeed and Amps vs. time for Motocalc 'Trainer' mission with brushless motor and components.

.

[pic] Figure 2.8: Comparison between brushed & brushless motors running on nickel and Li-Poly batteries

Tables:

|2 Blades |8 in Diameter |5 in Pitch |  |  |  |

|Phase |Flight Speed [ft/s] |Thrust [oz] |% Throttle * |Propeller Efficiency |Power [W] ** |

|Take-off |18 |3.15 |50% |63% |10 |

|Climb |18 |4.00 |55% |59% |14 |

|Level Flight |22 |2.93 |51% |70% |10 |

|Turn |23 |3.24 |54% |70% |12 |

|Aerobatic |25 |8.00 |77% |59% |38 |

|3 Blades |8 in Diameter |6 in Pitch |  |  |  |

|Take-off |18 |3.15 |40% |65% |10 |

|Climb |18 |4.00 |44% |61% |13 |

|Level Flight |22 |2.93 |41% |72% |10 |

|Turn |23 |3.24 |43% |69% |12 |

|Aerobatic |25 |16.00 |83% |49% |92 |

|* max motor RPM is 9350, direct drive |  |  |  |

|** power required from the battery (assumes75% motor efficiency) |  |

Table 2.1

[pic]

Table 2.2: Thrust required for various flight phases

[pic]

Table 2.3: Energy required for brushed motors for various flight phases

[pic]

Table 2.4: Energy required for brushless motors for various flight phases

[pic]

Figure 2.5: Brushed motor & components weight and pricing

[pic]

Figure 2.6: Brushless motor & components weight and pricing

Appendix III: Structures

Figures:

[pic]

Figure 3.1: Main Fuselage / Wings Connections

[pic]

Figure 3.2: Outer Fuselages / Wings Connections

[pic]

Figure 3.3: Outer Fuselages / Landing Gear Connections

[pic]

Figure 3.4: Vertical Tail/Outer Fuselages Connections

[pic]

Figure 3.5: Landing Gear and Mounting Block

[pic]

Figure 3.6: Fuselage and Wheel/Ski/Float Attachments

[pic]

Figure 3.7: V-n Diagram

Appendix IV: Dynamics & Controls

Equations:

Equation (4.1) [pic]

Equation (4.2) [pic]

Equation (4.3) [pic]

Equation (4.4) [pic], where [pic] and [pic]

Equation (4.5) [pic], where [pic] and [pic]

Equation (4.6) [pic]

Equation (4.7) [pic]

Equation (4.8) [pic]

Equation (4.9) [pic]

Figures:

[pic]

Figure 4.1: Center of Gravity, Aerodynamic Centers, and Neutral Point

[pic]

Figure 4.2: Trim Diagram

[pic]

Figure 4.3: Elevator Deflection Definition

[pic]

[pic]

[pic]

[pic]

-----------------------

[1]

-----------------------

Wheel/Ski/Float Attachment

Fuselage Attachment

[pic]

[pic]

[pic]

Elevator

Servo

[pic]

[pic]

He(s)

q(s)/(e(s)

H (s)

K

(

Pilot

Input

Aircraft

(e(s)

q(s)

+/-

Pitch Rate

Gyro

Feedback

Gain

[pic]

[pic]

[pic]

[pic]

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