milestone 1: aircraft designm2i.aere.iastate.edu/cardinalflight/files/2017/12/aerostruct...milestone...

Iowa State University

AerE 294X/AerE 494X

Make to Innovate

Milestone 1: Aircraft Design

Project: Cardinal FlightTeam: Aerodynamics and Structure Team

Author(s):Ryan StoryJoshua BuettnerJoseph CairoMichael LonderganJose MontesinosTheodore PermulaSamuel Ruhlin

Role:Team Leader

Team MemberTeam MemberTeam MemberTeam MemberTeam MemberTeam Member

Faculty Adviser: Cdr. Daniel Buhr

November 10, 2017

Milestone 1: Aircraft Design Ryan Story

Contents

Abstract 2

1 Introduction 4

2 Background 42.1 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Deliverable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Problem Identification 6

4 Problem Solution 64.1 Tilting Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Fuselage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Design Theory 125.1 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6 Design Verification 136.1 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Discussion 15

8 Conclusion 15

A Assembly Photos 16

B Linkage Calculator 17

C Aerodynamic and Wing Force Calculator 18

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Abstract

Cardinal Flight has set out to design, build, and fly a tilt-wing aircraft leveragingvertical launching capabilities and efficient forward flight. The Aerodynamics andStructures team is tasked with the aerodynamic design of the aircraft and the designof all the structural components of the aircraft. The design started with relativelyfew constraints and analysis was done to incorporate all of the desired features forthe aircraft. Design changes were made throughout the process as more componentswere modeled and were analyzed in SolidWorks. The deliverable for this milestoneincludes the report that consists of the design considerations/decisions, a descriptionof the design, and a complete SolidWorks model. This report also details problemsthat arose during the process and makes recommendations for improvements tofuture design work.

Figure 1: Aircraft in forward flight configuration

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Figure 2: Aircraft in VTOL flight configuration

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1 Introduction

Milestone 1 of Cardinal Flight, Aerodynamics and Structures team, is to createa complete design of the project aircraft that will be constructed in the Springsemester. This includes designing and modeling the entire aircraft structure, includ-ing the tilting mechanism for the wing/tail. These components were then insertedinto a larger assembly to model the entire aircraft. In addition, structural andaerodynamic analysis was performed using classical methods to ensure that flightcritical parts were properly sized while reducing the weight as much as possible. Bymodeling the assembly, the team was also able to provided weight estimates to theElectrical team for motor and electrical system sizing. All CAD modeling was donein SolidWorks.

2 Background

For the 2017-2018 academic year Cardinal Flight has set out to design and buildan aircraft similar to NASA’s Greased Lightning aircraft. The aircraft will use atilting wing and stabilator to transition between vertical and forward flight. Theproject seeks to develop a design that can capitalize on the vertical takeoff or landing(VTOL) capability while adding in the efficiency of fixed-wing forward flight. Thiswill allow the aircraft to carry a wide range of payloads and support them throughall flight regimes.

2.1 Tasks

This milestone was comprised of the following tasks:

1.1 11 Sept 2017 - Preliminary Tilting Mechanism Design

1.2 25 Sept 2017 - Final Tilting Mechanism Design

1.3 23 Oct 2017 - Model Wing in SolidWorks

1.3.1 Model wing tilt mechanism

1.3.2 Model internal components

1.4 23 Oct 2017 - Model fuselage in SolidWorks

1.4.1 Model internal components

1.4.2 Model tail

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1.5 30 Oct 2017 - Complete SolidWorks Assembly

1.6 10 Nov 2017 - Complete Milestone Report

2.2 Deliverable

Milestone Report

– Design Considerations and Decisions

– Design Description

– Full SolidWorks Model

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3 Problem Identification

The Aerodynamics and Structures team’s goal for this milestone was to take aconceptual idea that was developed the previous academic year and turn it into afunctional and safe design. There are a few technical issues that had to be overcometo accomplish the aircraft design. First the team needed to determine if the projectwas feasible, followed by designing the aircraft, and then designing all the internalcomponents for the aircraft.

The aircraft design process began with a constraint of a wingspan of six feet. Thiswas due to transport concerns if the wing was larger six feet. A root chord ofeighteen inches, and a tip chord of ten inches for the wing was decided on afterpreliminary lift calculations. The desired wing design would include a wing sweepto be determined in the design phase to manipulate the center of lift and the centerof gravity. The team needed to evaluate the aerodynamic properties of the chosenClark Y airfoil to determine if the wing will produce sufficient lift for the designedstructure.

The structure of the aircraft also needed analysis to ensure a capable design. Forcecalculations will be completed for major structural components such as the wingspar and tail boom. In addition, an analysis will be performed on the wing tiltmechanism to ensure the servo is capable of the handling the load from the wing.

4 Problem Solution

4.1 Tilting Mechanism

To determine whether or not the project was feasible, the team first examined thetilting mechanism for the wing and tail to determine how it could be accomplished.Several solutions were explored. The first and most desirable solution was to usea linear actuator for the wing. This was the most desirable solution due to theload bearing capability of the linear actuator and once power was removed from theactuator, it would remain at it’s position and would be able to handle very highloads. Weight and space constraints lead the team to move away from this planin addition to the need for it’s own power supply. The linear actuator also movedslower than desired for the transition from vertical to forward flight

The next idea was to use a servo gearbox. This would allow us to place a normalservo into a gearing mechanism to greatly increase the torque produced by the servo.This solution was better because it allowed the wing to transition quicker than the

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linear actuator and it was lighter than the linear actuator. As the team continuedthe aircraft design, it became apparent that the aircraft structure weight needed tobe reduced. This meant that the servo gearbox was too heavy and another solutionwas needed.

Figure 3: Initial wing tilt design with servo

The next solution was a high quality, metal geared servo. This is the most lightweight solution that the team was able to find. It gives the aircraft the ability totransition quickly and with a more compact footprint for the mechanism inside theaircraft. The downside to this is that the required torque had to more carefullycalculated to determine if the servo was sufficient.

To determine the feasibility of different linkage combinations for the wing and tail,team members developed a simple MATLAB program that would take geometricalinputs and output the available degrees of rotation for the system.

A feasible linkage was designed to be placed into the SolidWorks assembly. Afterinstalling the system into the full assembly, it was determined that the linkage wouldnot work in its current position. The servo had been mounted at the forward edgeof the payload plate which is 1” aft of the leading edge (LE) of the wing. This ledto an evaluation of other possible linkage combinations. If no other suitable solutioncould be found, the fuselage would need to be modified to accommodate the tiltingmechanism. This preliminary design can be seen in Figure 3.

Fortunately, a suitable solution was found that did not necessitate the need to changethe design of the fuselage. The servo was relocated to 7” aft of the wing LE. It wasalso placed onto the tail boom that extends into the fuselage. This will provide asolid mounting platform and moves the servo 1” up. These two changes combined

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Figure 4: Final wing tilt design with servo moved

allow the mechanism to use a shorter linkage that is capable of fitting in the fuselageand providing a full 90◦of rotation. This satisfies the requirement for the wing tiltingmechanism. The final design can be seen in Figure 4. If it is determined that theselected servo is not capable of sustaining the load from the wing, an additionalservo will be added to the system.

The tail tilting proved to be a simpler problem due to it being mounted on the tailboom with no space constraints. The MATLAB program Appendix B was utilizedto determine the best linkage size for the system. The servo is located approximately14” forward of the stabilator spar, which is the pivot point. The stabilator design waschosen to simplify the design and reduce the need for additional servos to manipulateelevators. The linkage between the stabilator and servo arm is 15”. This systemprovides rotation to 90◦vertical and -45◦. This is more than adequate for use as anelevator and tilting mechanism. The design is shown in Figure 5.

4.2 Fuselage

After several iterations for the fuselage design, the team decided to use standardaircraft construction methods. The fuselage is a semi-monocoque design using bulk-heads to define the shape and stringers/longerons to increase the strength of thefuselage. The bulkheads made from carbon fiber plate. The stringers/longerons arecarbon fiber pultruded rods that will fit into cutouts made in the bulkheads. Thefuselage will be covered in Monokote with a removable fiberglass nose cone.

An additional structural component of the fuselage is a carbon fiber and foam sand-

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Figure 5: Tail tilting design

wich plate. This is located near the center of the aircraft. It provides an area toattach the payload pod, but it also is used to mount the brackets to hold the wingand the attach point for the tail boom.

The fuselage is designed so that the electronic components can be located in theforward part of the fuselage to assist in center of gravity manipulation. This areawill be accessed through a removable fiberglass nose cone. In addition since thebrackets to hold the wing are attached to the payload plate, the brackets can beeasily removed which translates to easily removed wings. This capability is essentialfor transporting an aircraft of this size.

4.3 Wing

The aircraft’s wing is a more complicated structure than previous Cardinal FlightDesigns. The main reason being due to the wing being tilted in flight. This requiredsome analysis for spar strength and also determining the best place to pivot thewing at. To do this, the neutral point (NP) of the aircraft was calculated using theMATLAB program which can be found in Appendix C.

The NP of the aircraft is the point on the aircraft where the change in moment dueto a change in angle of attack (AOA) is zero. If the center of gravity (CG) of theaircraft is aft of this point, the aircraft will be uncontrollable without a computerflight control system. Since the CG of the plane needs to be aft of the wing when

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Figure 6: Internal structure of wing

it is in VTOL mode, the pivot point on the wing needed to be positioned so that itwas forward of the NP, but also forward of the CG when in VTOL operations. Todo this, the team decided to pivot the wing nine inches aft of the leading edge ofthe wing. This would meet all of the conditions for CG and NP locations.

This aircraft will be using multiple motors on the wing for both VTOL and forwardflight modes. This required evaluation of the structure and a design that couldsupport the eight motors distributed across the wing. This included using two sparsin the wing. The main spar is located at the quarter chord location throughout theentire wing. The main spar in each wing is joined in the center section to makea one piece spar. This spar is also what the motor mounts will be attached to inaddition to a pultruded rod that spans the leading edge of the wing. There was alsoa need for structure at the pivot point for the wing. To do this, a second spar wasadded that extends out to the middle of the wing. This adds additional strength tothe wing in both VTOL and forward flight modes. The second spar is located 9”aft of the wing LE.

4.4 Tail

Another step taken in the design process involved the sizing and modeling of thestabilator, the vertical stabilizer, and the rudder. The stabilator design was chosenin order to simplify the horizontal tail. Since the entire control surface would berotating for the hovering configuration, it would be much more complex to add afunctioning elevator within the horizontal stabilizer and it would increase weight.

The stabilator was designed using carbon fiber ribs with constant chord. There is a0.5” carbon fiber tube running down its quarter chord. In order to have the controlsurface rotate, two bearing housings will be mounted to the sides of the rear fuselage.

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Figure 7: Stabilator

These will hold the stabilator spar in place while a servo can rotate one of the centerribs via a control horn thereby rotating the stabilator.

The vertical stabilizer is designed to be constructed of tapered carbon fiber ribsusing a NACA0012 airfoil with a carbon fiber tube spar located in the center of therib. The rudder will also be constructed using carbon fiber ribs.

Figure 8: Vertical Stabilizer

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5 Design Theory

5.1 Structures

To evaluate if the spars were of sufficient strength, bending stress calculations wereused. An ultimate stress of 500 ksi was used and the team worked backwards todetermine the load that would be required to reach that point. These equations areover simplified for composites, but a reasonable estimate of the load and factor ofsafety can be found.

σb =My

I(1)

M = Σ(F ∗ d) (2)

I =π(d4outer − d4inner)

64(3)

5.2 Aerodynamics

The airfoil used for the wing is a Clark Y and the tail uses a symmetrical NACA0012. The Clark Y airfoil is a common airfoil used for many purposes. The airfoilperforms well at the speed and altitude the project aircraft will be operating at.Analysis was done on the wing to ensure that sufficient lift will be created duringforward flight. Lift will be calculated using Equation 4.

L =1

2ρV 2CLS (4)

To determine the tail specifications, the following method was used.

Vh =ShlhSc

(5)

In Equation 5, Vh represents the horizontal tail volume coefficient, Sh = horizontaltail area, lh = the horizontal tail moment arm, S = wing area, and c = average wingchord. Vh for a stable aircraft should fall between the values of 0.30 and 0.60. The

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value of 0.60 was chosen while designing the stabilator to ensure ample pitch control.Next, the vertical tail volume coefficient, Vv, was determined using Equation 6.

Vv =SvlvSb

(6)

Vv, in Equation 6, represents the vertical tail volume coefficient, Sv = vertical tailarea, lv = the vertical tail moment arm, S = wing area, and b = wing span. Vvshould fall between the values of 0.02 and 0.05. Due to problems in the past withinsufficient size of control surfaces, the team chose the value of 0.05 for the verticaltail volume coefficient.

6 Design Verification

6.1 Structures

The main spar is a standard modulus carbon fiber tube with an outer diameter (OD)of 1” and an inner diameter (ID) of 0.875”. The second spar is a high modulus carbonfiber tube with an OD of 0.5” and an inner diameter of 0.25”.

For the main spar it was determined that the spar would fail when subjected to amoment of 33.6E3 lb-in. If the moment is taken about the wing tip, the requiredforce to get that moment would be 930 lbs. With the planned aircraft weight of15 lbs and adding 10 lbs for the thrust from the motors, the factor of safety forthe main spar is 37. Since the calculations have been simplified with respect tocomposite analysis, this factor of safety is acceptable and excessive for the needs ofthe structure.

The second spar was evaluated similarly and provided a factor of safety of 11. Again,these methods are simplified and the strength of the spars is likely to be much greaterdue to the layup of the carbon fiber plies in the spar.

6.2 Aerodynamics

The team determined that the airfoil will be capable of creating at 16 lbs of lift at anairspeed of 30 mph and an AOA of 5 degrees. As the speed is increased in forwardflight, the AOA could be reduced while maintaining sufficient lift. This calculatedwas the determining factor in sizing the entire aircraft and is how the other Cardinal

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Flight teams worked on sizing the electrical/propulsions systems and the availablepayloads.

Using the methods for sizing the tail, it was determined that the vertical stabilizerwould be 12” tall. The chord of the vertical stabilizer ribs starts at 12” and tapersto 9.5” at the top. The stabilator uses a constant chord of 10” with a 30” span.

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7 Discussion

The milestone was a success. The team was able to take the project concept andcreate a complete design that will be constructed in the Spring. There were a fewchallenges that necessitated changes to certain components or assemblies throughoutthe design process. All the changes led to a better design that will also be easierto manufacture. As is the case with most aerospace designs, the design continuallystruggled with weight. The design materials have been used in previous CardinalFlight projects, so the team was able to take that knowledge and focus more on thedesign of the structure instead of selecting materials.

As it stands, the aircraft structure weighs six pounds. This estimate is from Solid-Works with information from the commercial components that will be used in theaircraft. A future goal will be to explore other ways to reduce weight in the design tomaximize the payload capacity. This effort to reduce weight also created challenges,as stated previously, in the tilting mechanisms and the components that could beused to solve that problem.

A recommendation to future design teams is to incorporate a SolidWorks coursein the beginning of the semester to get all the team members up to speed andensure a smooth design phase. The team was productive, but tasks could have beenaccomplished faster if team members had been given some baseline knowledge.

8 Conclusion

Overall, the design phase went smoothly. The most problematic area was the designof the tilting mechanism, but this challenge was expected prior to beginning theproject. After this, the only big challenges were working through the many partsthat needed to be modeled, which took time. By using previous project knowledge,material selection was greatly simplified and the team was able to focus mainlyon the design of the aircraft. SolidWorks drawings and a material list can nowbe created in preparation for the build phase, which will begin in the Spring 2018semester.

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A Assembly Photos

Figure 9: Assembly in forward flight configuration without skins

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Figure 10: Aileron structure

Figure 11: Motor Mount

B Linkage Calculator

MATLAB program to calculate different linkage situations: LinkageCode.m

1 c l ea r , c l c2

3 R1 = input ( ’ Enter se rvo arm length : ’ ) ;4 r1 = input ( ’ Enter se rvo rad iu s : ’ ) ;5 R2 = input ( ’ Enter t a i l /wing l ength : ’ ) ;6 r2 = input ( ’ Enter t a i l /wing rad iu s : ’ ) ;7 H = input ( ’ Enter he ight d i f f e r e n c e : ’ ) ;8 D = input ( ’ Enter d i s t anc e between r o t a t i o n po in t s : ’ ) ;9 P = input ( ’ Enter l i n k ag e l ength : ’ ) ;

10

11 f o r theta1 = 0 : 1 8 0 ;

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12 theta = theta1 ∗ pi /180 ;13 phi = theta ;14

15 p s i = acos ( (D+R2∗ cos ( phi )−R1∗ cos ( theta ) ) /P) ;16 f p r i n t f ( ’\n At %f degree s p s i i s %f ’ , theta1 , p s i ) ;17 dx = D − R1∗ cos ( theta )+R2∗ cos ( phi ) ;18

19 f o r x = R1∗ cos ( theta ) : . 1 : D+(R2∗ cos ( phi ) ) ;20 y = R1∗ s i n ( theta )+(0−R1∗ cos ( theta ) ) ∗( s i n ( p s i ) ) /(P∗dx )

+ x∗ s i n ( p s i ) /(P∗dx ) ;21

22 i f s q r t ( ( xˆ2)+(yˆ2) ) <= r1 ;23 f p r i n t f ( ’\n %f ’ , theta1 ) ;24 break25

26 e l s e i f s q r t ( ( (D−x ) ˆ2) +((y−H) ˆ2) ) <= r2 ;27 f p r i n t f ( ’\n %f g ’ , phi ) ;28 break29 end30 end31 end

C Aerodynamic and Wing Force Calculator

MATLAB program to calculate neutral point and force on the wing for a given pivotpoint: wing.m

1 c l c , c l e a r a l l2

3 [ dens , a , v i s c ] = StandardAtmosphere ( ’ 1000 ’ ) ;4

5 V = 20 ∗ 0 . 514444 ; % kts to m/ s6

7 roo t chord = 18 ; %in8 f u s e w idth = 6 ; %in9 t i p cho rd = 10 ; %in

10 wing span = 72 ; %in11 sweep d i s t = 15 ; %in − This i s the d i s t ance from the root LE

to the t i p LE

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12 p i v o t d i s t = 9 ; % in13

14 % Find LE and TE l i n e formula f o r swept wing15 l e b = 0 ;16 l e a = ( sweep d i s t − l e b ) / ( ( wing span − f u s e w idth ) ∗

0 . 5 ) ;17 t e b = root chord ;18 t e a = ( ( sweep d i s t + t ip cho rd ) − t e b ) / ( ( wing span −

f u s e w idth ) ∗ 0 . 5 ) ;19

20 % Find he lpe r l i n e formula21 mac b0 = −t i p cho rd ;22 mac a0 = ( ( sweep d i s t + t ip cho rd + root chord ) − mac b0 ) /

( ( wing span − f u s e w idth ) ∗ 0 . 5 ) ;23 mac b1 = root chord + t ip cho rd ;24 mac a1 = ( ( sweep d i s t − roo t chord ) − mac b1 ) / ( ( wing span

− f u s e w idth ) ∗ 0 . 5 ) ;25

26 % Determine MAC us ing i n t e r s e c t i o n o f he lpe r l i n e s27 mac x = ( mac b1 − mac b0 ) / ( mac a0 − mac a1 ) ;28

29 % Compute MAC i n t e r s e c t i o n with LE and TE30 l e mac y = l e a ∗ mac x + l e b ;31 te mac y = t e a ∗ mac x + te b ;32

33 %Calcu la te mac34 mac = te mac y − l e mac y ;35

36 % Get the wing area and area a f t o f the p ivot po int37 [ t o t a r ea , a r e a a f t ] = wingArea ( fuse width , root chord ,

t ip chord , sweep di s t , p i v o t d i s t , wing span ) ;38

39 % Wind Load Ca l cu l a t i on s with h o r i z o n t a l wind − Mult ip ly ingwing area to move to mˆ2

40 F w = 0.5 ∗ dens ∗ Vˆ2 ∗ ( t o t a r e a ∗0 .00064516) ; % N41 p dyn = (F w / ( t o t a r e a ∗0 .00064516) ) ; % N/m42

43 % Calcu la te fwd area and convert to mˆ244 fwd area m2 = ( t o t a r e a − a r e a a f t ) ∗ 0 .00064516 ; % mˆ245

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46 % Convert area from in ˆ2 to mˆ247 area a f t m2 = a r e a a f t ∗ 0 .00064516 ; %mˆ248

49 % Calcu la te the t o t a l f o r c e a f t o f p ivot po int duringv e r t i c a l and convert

50 % f o r c e from Newtons to l b s51 a f t f o r c e = ( area a f t m2 ∗ p dyn ) / 4 .4482216282509 ;52

53 fwd fo r c e = ( fwd area m2 ∗p dyn ) / 4 .4482216282509 ;54

55 fwd mom = (−1) ∗ fwd fo r c e ∗ ( p i v o t d i s t /2) %in−l b s56

57 aft mom = a f t f o r c e ∗ ( ( ( sweep d i s t+t ip cho rd )−p i v o t d i s t )/2)

wingArea.m

1 f unc t i on [ area1 , area2 ] = wingArea ( fuseWidth , rootChord ,tipChord , sweepDist , p ivotDist , span )

2 %wingAreas − Given in fo rmat ion about the wing , c a l c u l a t e theareas

3 % S p e c i f i c a l l y , g iven a p ivot d i s t , c a l c u l a t e the areaa f t o f that

4 % point to be used in f o r c e c a l c u l a t i o n s .5

6 x = [ fuseWidth , ( span /2)+(fuseWidth /2) , ( span /2)+(fuseWidth /2), fuseWidth ,0 ,−( span /2)+(fuseWidth /2) ,−( span /2)+(fuseWidth/2) ,0 , fuseWidth ] ;

7 y = [0 ,− sweepDist ,−( sweepDist+tipChord ) ,−rootChord ,−rootChord ,−( sweepDist+tipChord ) ,−sweepDist , 0 , 0 ] ;

8

9 f i g u r e ( )10 p lo t (x , y ) ;11 xlim ([−45 4 5 ] )12 ylim ([−45 4 5 ] )13

14 area1 = polyarea (x , y ) ;15 x1 = [ ( p ivo tD i s t /( sweepDist / ( ( span /2)−(fuseWidth /2) ) ) )+

fuseWidth , ( span /2)+(fuseWidth /2) , ( span /2)+(fuseWidth /2) ,fuseWidth ,0 ,−( span /2)+(fuseWidth /2) ,−( span /2)+(fuseWidth/2) ,−( p ivo tD i s t /( sweepDist / ( ( span /2)−(fuseWidth /2) ) ) ) , (

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p ivo tD i s t /( sweepDist / ( ( span /2)−(fuseWidth /2) ) ) )+fuseWidth] ;

16 y1 = [−pivotDist ,− sweepDist ,−( sweepDist+tipChord ) ,−rootChord,−rootChord ,−( sweepDist+tipChord ) ,−sweepDist ,−pivotDist ,−p ivo tDi s t ] ;

17

18 f i g u r e ( )19 p lo t (x , y , x1 , y1 ) ;20 xlim ([−45 4 5 ] )21 ylim ([−45 4 5 ] )22

23 area2 = polyarea ( x1 , y1 ) ;24

25 area2 m2 = area2 ∗ 0 .00064516 ; %mˆ226

27 ( area2 m2 ∗ 141 .6693) / 4 .4482216282509 ;28

29 end

StandardAtmosphere.m

1 f unc t i on [ dens , a , v i s c ] = StandardAtmosphere ( a l t )2 %Read in a l t i t u d e and output v i s c o s i t y , dens i ty , and speed

o f sound3 %Units :4 %a l t i t u d e as a s t r i n g in f t5 %dens i ty in kg/mˆ36 %speed o f sound , a , in m/ s7

8 %Convert a l t i t u d e to number in meters9 a l t = str2num ( a l t ) ∗0 . 3048 ;

10

11 g = 9 .80665 ;12 R = 2 8 7 . 1 ;13 gamma = 1 . 4 ;14 i f a l t < 1100015 T1 = 2 8 8 . 1 6 ; %From Anderson , In t roduc t i on to Fl ight , 5

ed , pg 10916 l ap s = −6.5e−3;17 T = T1 + laps ∗( a l t ) ;18 dens1 = 1 . 2 2 5 0 ;

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19 dens = (T/T1)ˆ−(g /( l ap s ∗R)+1)∗dens1 ;20 e l s e i f a l t >= 11000 & a l t < 2500021 T1 = 2 1 6 . 6 6 ;22 T = T1 ;23 dens1 = 0 . 3 6 4 0 ;24 dens = exp(−g /(R∗T) ∗( a l t −11000) )∗dens1 ;25 e l s e26 T1 = 2 1 6 . 6 6 ;27 l ap s = 3e−3;28 T = T1 + laps ∗( a l t ) ;29 dens1 = 0 . 1 1 ;30 dens = (T/T1)ˆ−(g /( l ap s ∗R)+1)∗dens1 ;31 end32 a = s q r t (gamma∗R∗T) ;33 v i s c = 1.458 e−6∗Tˆ1 .5/ (T+110.1) ; %Bertin , Aerodynamics f o r

Engineers , 4 ed , pg 5

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