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ASEN 2004Aerodynamic Performance of the Tempest/TTwister UAS

Experimental LabGroup 9, Section 015 - March 8, 2018

Nathan Castile, ∗ Jarrod Puseman, † Jeffrey Jones, ‡ Eric Vanderwolf § Bao Tran,¶ Ashkan Bafti, ‖ Matthew McCallum ∗∗

University of Colorado - Boulder

Our group analyzed the performance of the battery powered Tempest and TTwistor Un-manned Aircraft System (UAS) based on their physical properties and synthetic flight data.Computational fluid dynamics (CFD) produced the flight data used in the experiment. Weconstructed the drag polar and used it to find the aircrafts aerodynamic properties. Withthis data our group computed the drone’s maximum speed of 55 m/s and maximum rate ofclimb (R/C) of 20 m/s for an average storm-intercept mission. The range and endurancewere explored for varied velocity values and battery discharge rates. We found that rangeand endurance are extended when using the discharge rate of lithium-polymer batteriesinstead of ideal batteries.

Nomenclature

α = Angle of Attack [degrees]η = Battery Efficiencyηtot = Overall Propulsive Efficiencyρ∞ = Airstream Density [ kgm3 ]AR = Aspect RatioC = Capacity [Ah]CD = Coefficient of DragCL = Coefficient of LiftCD,0 = Coefficient of Drag at zero liftCD,min = Minimum Coefficient of DragCL,min = Minimum Coefficient of LiftD = Drag [N]E = Endurance [hr]

e = Oswald EfficiencyI = Current [A]L = Lift [N]n = Battery Discharge RateP∞ = Airstream Static Pressure [Pa]PA = Power available [W]PR = Power required [W]R = Range [km]R/C = Rate of Climb[ms ]Rt = Battery Hour Rating [h]t = Battery Discharge Rate [h]TR = Thrust required [N]V = Voltage [V]V∞ = Airstream Velocity [ms ]W = Weight [kg]

∗106240158†104003252‡106267737§104383652¶103370381‖107028894

∗∗106279461

ASEN 2004 Section 015 1 of 18 Spring 2018

Contents

I Introduction 3

II Background and Theory 3

III Experimental Apparatus and Procedure 5

IV Summary of Results 7

V Discussion 10V.A Effect of Varying Angle of Attack on Coefficient of Lift . . . . . . . . . . . . . . . . . . . . 10V.B Effect of Varying Angle of Attack on Coefficient of Drag . . . . . . . . . . . . . . . . . . . . 10V.C Effect of Varying Angle of Attack on Lift-to-Drag Ratio . . . . . . . . . . . . . . . . . . . . 10V.D The Drag Polar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10V.E Maximum Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11V.F Effect of Varying Velocity on Aircraft Endurance . . . . . . . . . . . . . . . . . . . . . . . . 11V.G Effect of Varying Velocity on Aircraft Range . . . . . . . . . . . . . . . . . . . . . . . . . . 11V.H Effect of Varying Velocity on Aircraft Rate of Climb . . . . . . . . . . . . . . . . . . . . . . 11V.I Effect of Varying Discharge Parameter on Battery Discharge Rate . . . . . . . . . . . . . . 11

VI Conclusions 12

2

I. Introduction

The purpose of this experiment was to estimate the performance (speed, rate of climb R/C, range,endurance) of a battery powered aircraft. The Tempest/TTwister UAS were the focus of this study, whosemain missions are to record atmospheric data in and around large storm veins. The performance of batterypowered aircraft varies from traditional aircraft because both power available and weight are constant.Atmospheric density does not affect the powerplant and therefore power available is independent of altitude,and because no fuel is being burned, weight does not change. To understand the aerodynamic properties ofthe Tempest/TTwister, we needed to form the drag polar for the airframe. This was accomplished by usingsynthetic flight data generated via CFD as well as directly modeling it through the drag polar equation.Bringing together the physical, aerodynamic, and power plant properties of the Tempest/TTwister UAS,we were able to find its range and endurance over a range of cruise speeds and for either ideal or realisticbattery discharge rates.

II. Background and Theory

The flight performance of the Tempest/TTwister UAS is primarily dependent of the aerodynamic prop-erties of the airplane. The first critical aerodynamic property is the drag polar, which is given by equation(1). The drag polar relates the total drag, equation (4), to the lift through the drag coefficient as a functionof the lift coefficient, equation (3) for the given plane in steady level flight. The drag is a crucial element ofairplane design because it is able to effect the max velocity, rate of climb, range, and endurance of a givenplane, as well as many of important aspects of aircraft performance.

CD = CDmin + k(CL − CLmin)2 (1)

k =1

πeAR(2)

L =1

2ρ∞V

2∞SCL (3)

D =1

2ρ∞V

2∞SCD (4)

The Oswald efficiency factor, e in equation (2), is another critical aspect to airplane design yet to bediscussed. The Oswald efficiency factor is a scalar to help account for deviation of the lift distribution fromthat of an elliptical lift distribution. The elliptical lift distribution produces the minimum induced drag,hence e = 1 for an elliptical wing. The Oswald efficiency factor e decreases as the wing lift distributiondeviates from that of an elliptical one, which increases k, thus increasing the lift induced drag from equation(1).

The power available is for the aircraft is given by the total efficiency of the battery and propeller multipliedby the power of the battery. This is the total power given from the provided battery.

PA = ηIV (5)

The power required is the amount of power the aircraft will need in order to fly at a certain velocity.In leveled, unaccelerated flight, the power required can be defined in equation (6). Also under the sameconditions, the thrust is equal to drag and lift is equal to the weight from equations (4) and (3). From thegiven relationship, an equation of power required as a function of velocity can be derived from Equations(1), (4), and (6), defined in equation (7)[2].

PR = TRV∞ (6)

PR =1

2ρ∞V

3∞SCD,0 +

W 2

12ρ∞V∞S

πeAR(7)

From obtaining power available constant and power required curve, the maximum velocity is the intersec-tion of the two. It is the intersection of the two powers because power available is the maximum power the


aircraft can operate at, and the power required is the minimum power needed to fly at a certain operatingpoint.

The rate of climb for a given aircraft is given by the excess power over the gross weight of the vehicle.The excess power at any given free stream velocity is the difference between the power available and thepower required. The rate of climb is given in equation (8).

R/C =PA − PR

W=ηIV − TRV∞

W(8)

Since the Tempest/TTwister UAS is a battery powered propeller driven aircraft, the range and enduranceis determined using battery power variations of the range and endurance equations for propeller aircraft.The understanding for battery powered range and endurance came from the document provided on D2L”Range and Endurance Estimates for Battery Powered Aircraft” [4]. The range and endurance for the Tem-pest/TTwister from the estimates document is given in Equations (9) and (10) respectively with maximumrange and endurance given by Equations (11) and (12). Where Rt is the discharge time determined fromthe capacity, also known as the battery rating[4].

R = Rt1−n

[ηtotV C

12ρV

3∞SCD,0 + (2W 2k/ρV∞S)

]n√√√√2W

ρS

√k

CD,0

)[km] (9)

E = Rt1−n

[ηtotV C

12ρV

3∞SCD,0 + (2W 2k/ρV∞S)

]n[h] (10)

Rmax = Rt1−n

[ηtotV C

(1/√ρS)C

1/4D,0(2W

√k)3/2

]n√√√√2W

ρS

√k

CD,0∗ 3.6[km] (11)

Emax = Rt1−n

[ηtotV C

(2/√ρS)C

1/4D,0(2W

√k)

13

]n[h] (12)

The range and endurance of a battery powered aircraft is effected by the battery discharge parameter, n.The discharge parameter is determined by the type and temperature of a given battery, and it determinesthe time to completely drain the battery at a given current, battery hour rating and battery capacity. Thisrelationship is given in Equation (13).

t =(Rtin)( CRt

)n[h] (13)


III. Experimental Apparatus and Procedure

The Tempest/TTwister UAS is a battery-powered, propeller aircraft. The geometry, performance char-acteristics, and battery data of the Tempest/TTwister UAS [1] are in Figures 1, 2a, and 2b respectively.

Figure 1: Geometry of the Tempest UAS [1]Dimensions of the aircraft are in meters.

(a) Performance Characteristics of Tempest UAS Thisdata is for a typical storm-intercept mission for the Tempestin 2010 for the VORTEX2 field campaign.

(b) Battery specifications on the Tempest/TTwisterUAS Battery packs comes in units of two, resulting in twicethe total power available per aircraft.

Figure 2: Specifications of Tempest/TTwister UAS [1]

In order to investigate the aerodynamics of the design, a good place to start is to determine the aerody-namic coefficients of the aircraft. This can be obtained using a few methods, such as using a wind tunnel,test flights, etc. In this experiment, the drag and lift coefficients were calculated using computational fluiddynamics (CFD). This method was used to generate synthetic flight data shown below in Figure 3. This datawas given to the team from Professor Argrow to use to analyze further flight characteristics of the aircraft.

The table in Figure 3 includes CL and CD in respect to angle of attack. From this data, the team beganwith creating the following plots:

1. CL vs. α shown in Figure 4a

2. CD vs. α shown in Figure 4b

3. LD vs. α shown in Figure 5

4. CD vs. CL (Drag Polar) shown in Figure 6

From combining Equation (3) and (4) [2]:L

D=CLCD

(14)


Figure 3: Synthetic Flight Data [1] This data was obtained from using CFD to calculate the lift and dragcoefficient with respect to angle of attack.

Next, using the CL vs. α plot, a best-fit line was created for the linear portion of the graph, also shownin Figure 4a.

The team then plot Equation (1) along with the Drag Polar on the same plot, shown in Figure 6, usingthe Oswald efficiency factor estimated by the given equation from the lab document [1].

e = 1.78(1− 0.045AR0.68)− 0.64 (15)

To further understand the performance of the Tempest/TTwister in a more recent flight on a typical day,the team analyzed the performance of the aircraft under these given conditions [1]:

1. Cruise speed of 18 ms

2. Altitude of 6000 ft (1829 m)

With the given information, the free-stream density and pressure can be acquired through Anderson [2]:ρ∞ = 1.024 kg

m3 and P∞ = 94276 Nm2 . From these information, the operation point of the Tempest/TTwister

is obtained by using Equation (3) to calculate CL and Equation (1) to calculate CD. The operation pointof the Tempest/TTwister is then plotted on Figure 6 as a black star.

The maximum speed is the intersection of the power available and power required curve on a Power vs.Velocity plot in Figure 7a. The power available can be calculated from the battery data in Figure 2b andusing Equation (5), meanwhile the power required curved is plotted using Equation (7).

The rate of climb is defined as excess power divided by weight, as seen in Equation (8). To find theexcess power, the maximum difference between the power available and power require curve is determined,from the power vs. velocity plot. The team was instructed to calculate the rate of climb for velocities12ms ≤ V∞ ≤ 40ms , as shown in Figure 7b.

For the same given velocities and overall propulsion efficiency ηtot = 0.5, the maximum range andendurance is calculated by using Equations (11) and (12), with endurance plotted in Figure 8a and rangein Figure 8b. Lastly, the team then investigated how the battery discharge rate is affected through differentdischarge parameters shown on Figure 9.


IV. Summary of Results

Included here are all relevant graphs and figures generated during this lab. These graphs show thedifferent design aspects that were investigated in order to characterize the aerodynamic performance of theTempest/TTwistor.

(a) Coefficient of Lift vs Angle of Attack The experi-mentally determined coefficients of lift vs angle of attack areplotted here. Also, a line of best fit is plotted before the Tem-pest/TTwister wing airflow begins to separate.

(b) Coefficient of Drag vs Angle of Attack Similarly,the coefficient of drag is plotted here versus angle of attack.

Figure 4: Coefficients of Lift and Drag Plotted Against Angle of Attack

Figure 5: Lift/Drag Ratio with Angle of Attack This plot shows the how the lift-to-drag ratio changes withangle of attack. See that the maximum lift to drag ratio occurs at a small positive angle of attack.


Figure 6: Drag Polar with Approximating Function. This is the experimental drag polar, plotted in orange,along with the approximating curve, from Equation (1), plotted in blue. Note that the curve matches closely withcoefficients of lift near CD,min. The black star marks the location of operation on the drag polar for flight during themore recent flight of the Tempest/TTwister on a standard day [1].

(a) Estimating Maximum Speed This plot shows boththe power required and power available of the aircraft as afunction of flight speed. Where the two curves intersect cor-responds to the maximum flight velocity of the aircraft. Here,it is 54.68 m

s.

(b) Rate of Climb The rate of climb as a function of flightspeed is plotted here. Note that the maximum rate of climb isnot at the maximum flight speed, as emphasized in Anderson[2].

Figure 7: Analyzing Power Characteristics


(a) Endurance The endurance (maximum flight time) asa function of velocity is plotted here. See that no matter thedischarge parameter n, the endurance shortens with increasedflight speed.

(b) Range of the Aircraft The range of the Tempest/T-Twister is plotted here. See how similar it appears to theendurance graph. This is because of its Equation (11) whichis merely the endurance multiplied by a scalar.

Figure 8: Endurance and Range

Figure 9: Battery Capacity vs Discharge Rate This plot shows the effects of the battery discharge rate on theeffective battery capacitance. The current draw from the battery is varied to affect the time the battery lasts.


V. Discussion

V.A. Effect of Varying Angle of Attack on Coefficient of Lift

As can be seen from Figure 4a, there appears to be a strongly linear correlation between α and CL, upuntil an angle of approximately 7°, where the experimental data begins to diverge from the linear relationand begins to gradually decline, due to separation in the flow causing less effective generation of lift. Alsonoted is the offset in the line, specifically how it is shifted to the left from the origin.

Although this may at first appear a relatively low angle of attack for flow to separate at, it makes moresense when considering the α shown are referring to that of an axis through whole plane, not that of thewing. The plane in question most likely follows suit of most other planes that have the wing mounted at acertain angle of attack with respect to the geometry of the plane, causing the line to shift to the left, andhence cause lift to be generated at an α of zero. Another contributer to this effect could be the camberpresent in the airfoil chosen, which also causes further shifting of the graph to the left, meaning that at evensmall angles of attack, considerable amounts of lift are being generated.

V.B. Effect of Varying Angle of Attack on Coefficient of Drag

In the data shown in Figure 4b, a quadratic relation appears to be formed between the angle of attackand drag, with minimum values being generated in the range of angles between −3° and 2°, with variationand noise in the data making it hard to distinguish an exact minimum.

Although it would seem reasonable given the ambiguity in the data to assume that minimum dragoccurred at minimum lift, that can’t be said with certainty due to the fact that there are other devices onthe aircraft such as the tail, and push-rods used to actuate the control surfaces, antennae and so forth thatall contribute to the drag generated as well, and have their own relationship with α, and so we can’t makeconcrete answers on the location of CD,0 based on Figure 4a.

V.C. Effect of Varying Angle of Attack on Lift-to-Drag Ratio

The combination of the effects described for the previous two Figures is shown in Figure 5, where it canbe seen that a peak L/D ratio is reached, at approximately α = 4°, after which it drops off considerably dueto a combination in the diminishing of the CL after these values and the sharp increase in CD values. Thismeans that the most efficient flight conditions are in the range of α between 3° and 7°.

V.D. The Drag Polar

The experimental drag polar shown in Figure 6 demonstrates the relation that exists directly betweenCL and CD, resulting in a graph similar to most real planes. For moderate values of CL, an approximatelyquadratic curve is observed, with an offset both up and to the right. The vertical offset exists due to the factthat 0 drag can never be achieved, and so must have some positive minimum value. The offset to the rightsimply means that the minimal drag is not at minimal lift, due to the fact that it once again isn’t only a wingthat is being observed, but an entire plane, and so the fuselage, tail, and other components all affect thedrag produced in their own way which culminates in the minimum drag coefficient occurring with a positivelift coefficient. Furthermore, at the very right of the graph you can see the effects that happens when youreach higher values of α, where CL reaches its limit and CD increases more and more. This illustrates theeffects of flow separation, where the drawback of extremely high angles of attack is made clear.

Overlaid onto the graph is a calculated approximation of the drag polar, using Equations 1, 2, and 15 togenerate smooth curve that can be used to effectively approximate the experimental data to a good degree ofaccuracy for 0 ≤ CL ≤ 0.7. Shown with a star is the region of operation for the Tempest/TTwister aircrafton the recent test of the aircraft [1], and it appears to lie on both curves. It can also be noted that crossreferencing between Figures 4a and 5 reveals that said typical operation lies within the previously definedrange for maximum aerodynamic efficiency, which agrees with what would be expected of the operation ofthe craft.


V.E. Maximum Velocity

Shown in Figure 7a is the method used to calculate maximum velocity, where the curve for the powerrequired in level, unaccelerated flight at a constant altitude of 1829m is plotted, as well as the power available,assumed to be constant. For lower flight velocities, the available power exceeds that of the required power,and then as V∞ is increased, the difference between the 2 curves decreases until there is no excess powerleft, and power at which maximum velocity is achieved, is reached. To then acquire the velocity at whichthis happens, we can refer to Equation 7, and use what we know about ρ, S, CD,0, W, e and AR to acquirea V∞ of 54.68m/s.

V.F. Effect of Varying Velocity on Aircraft Endurance

As shown in Figure 8a, the endurance of the Tempest/TTwister decreases with velocity for flight velocitiesbetween 12 and 40 m/s. From Equation (10), the independent variable V∞ occurs as a factor of V 3

∞ in thedenominator of the fraction and only V∞ in the numerator. Therefore as flight speed increases, the fractionmostly decreases and thus endurance does as well. The endurance is largest for small flight velocities. Thisis because the battery lasts longer as a result of less current being taken out to fly.

The endurance chart also shows how the battery discharge parameter n affects endurance. When n =1.3, endurance is larger at smaller flight velocities, and it is only much later that a discharge parameter ofn = 1 performs slightly better. A more thorough discussion can be found section V.I below.

V.G. Effect of Varying Velocity on Aircraft Range

Displayed in Figure 8b, the range of the Tempest/TTwister also decreases with flight speed. In fact, therange graph is the endurance graph from Figure 8a scaled by a certain factor: UR [4]. See that range isgiven by Equation (9) and is the endurance equation only multiplied by this UR. As a result, it has many ofthe same features as the endurance so that maximum range occurs at slower flight speeds and the dischargeparameter of n = 1.3 has generally a better range than a battery of n = 1.

V.H. Effect of Varying Velocity on Aircraft Rate of Climb

Figure 7b shows how the maximum rate of climb depends on the flight velocity. As Anderson predicts[2], the maximum rate of climb does not occur at the maximum velocity, but rather somewhere before. Thisis because rate of climb is defined as the excess power divided by the weight. Since the power available forthis aircraft is constant, the most excess power occurs at the minimum power required flight velocity. Thiswould be the velocity that would also yield the greatest endurance.

One thing to note about this plot, however, is that it may not be entirely accurate. The maximum rateof climb Equation 8 from Anderson [2] assumes small angles of attack into the sky. Since the Tempest/T-Twister has fairly large excess power, it can climb at angles that make this approximation not so accurateanymore. For a more thorough study on the rate of climb, a force analysis could be performed to accountfor more complicated flight geometry of the craft and corresponding forces. For the purposes of this lab,this assumption still shows that the maximum rate of climb occurs not at maximum velocity, and that issufficient.

V.I. Effect of Varying Discharge Parameter on Battery Discharge Rate

Since the battery discharge parameter is dependent on the battery type and temperature, it is worthstudying its effect on aircraft performance. For the lithium batteries in the Tempest/TTwister, the dischargeparameter is taken to be about 1.3. Compare this to the so-called ideal battery that has a discharge parameterof 1. From the plots generated for range and endurance in Figure 8, however, the aircraft has larger rangeand endurance when the discharge parameter is 1.3. This is because the ”ideal” discharge parameter isideal in its behavior; it will deplete predictably and steadily. The Peukert effect [4] makes the batterywith discharge parameter 1.3 deplete in a slightly different way. As a result, the range and endurance ofthe Tempest/TTwister with n = 1.3 are greater than n = 1 for lower flight speeds. Past a certain point,however, the case for n = 1 will actually get better range and endurance.

To better understand this, the effect on time to discharge as a function of current draw from the batteryis plotted. This shows in another way how at small draw from the battery (a lower discharge rate), the


battery will last longer for n = 1.3 than the ”ideal” case of n = 1. In short, the lithium batteries on theTempest/TTwister with discharge parameter n = 1.3 function better than so-called ”ideal” batteries at someconfigurations of flight and power usage.

VI. Conclusions

Using the synthetic flight data from the CFD simulation and the calculation methods outlined in SectionII, we were able to estimate certain aspects of the Tempest/TTwister UAV’s in-flight performance. Insteady flight, the vehicle is limited by its stall angle, which occurs approximately at 7° angle of attack, andits theoretical maximum speed of 54.68 m

s . However, flying at these limits will not prove to be very efficient.For the best endurance and range, the Tempest/TTwister should be flown at the lowest possible speed itcan without stalling. Additionally, for the best flight performance, the aircraft should be flown near anangle of attack that creates a peak L

D ratio. For the Tempest/TTwister, this optimal angle is approximately4°. Making alterations to the aircraft’s power plant, a battery in this case, can have a significant effect onperformance as well. Using a battery with a discharge parameter of 1.3 will increase range and endurance forspeeds under 30.15 m

s . Above that threshold, however, the ”ideal” discharge parameter (n = 1) will improverange and endurance. In general, we feel that using a battery with a discharge rate of 1.3 would be best forthis aircraft, since we also recommend flying at low speeds for best performance.

Since this experiment was intended to be a first venture into using data to estimate aircraft performance,we were able to use CFD data instead of performing actual wind tunnel tests. Although modern CFDsimulations can produce useful synthetic data for an introductory experiment, they obviously cannot modelreality with total accuracy. Thus, the accuracy of this experiment could be improved by simply conductingactual wind tunnel tests on a scale model of the Tempest/TTwister. Additionally, our range and enduranceestimates relied entirely on theoretical calculations using properties of our battery. Although it is likelybeyond the scope of this experiment, the range and endurance estimates could be improved by having actualin-flight data from the aircraft, to see if our calculations are close to how power output changes over timeduring a real flight.


References

1Argrow, Brian. ASEN 2004 Experimental Lab 1: Aerodynamic Performance of the Tempest/TTwister UAV. CU, 2018.PDF.

2Anderson, John. Introduction to Flight, 8th Edition. McGraw Hill, 2016.

3Raymer, D. P., Aircraft Design: A Conceptual Approach, 5th Edition., AIAA Inc., Chap. 12 (2012).

4Roadman, J., Argrow, B., and Frew, E., Mission Performance of th Tempest Unmanned Aircraft System in SupercellStorms,” Journal of Aircraft, Vol.49, No.6, pp.1821-1830 (2012).

5Shevell, R. S., Fundamentals of Flight, 2nd Ed., Prentice Hall, Chap. 11 (1989).

6Traub, L. W., Range and Endurance Estimates for Battery-Powered Aircraft, Journal of Aircraft, Vol. 48, No. 2, pp.703-707 (2011).

Acknowledgments

We would like to thank Professor Argrow and the TA’s for their guidance and assistance developing thisanalysis and report.


Appendix A: MATLAB Code

1 %Vehic l e Design Aero Lab 12 %Group 93 %J e f f r e y Jones , Matthew McCallum , Er ic Vanderwolf , Nathan Cas t i l e ,4 %Jarrod Puseman , Bao Tran , Ash Ba f t i5 %Created 1/24/186 %Last Modif ied 3/6/1878 %P r o f e s s o r Argrow9 %ASEN 2003

1011 %Set Up12 c l c ; c l e a r ; c l o s e a l l13 %% TTwister Parameters14 b = 3 . 2 2 ; %[m]15 S = . 6 3 ; %[mˆ2 ]16 AR = 1 6 . 5 ;17 m = 4 . 5 ; %[ kg ]18 GTOW = 6 . 4 ; %[ kg ]19 g = 9 . 8 1 ; %[m/ s ]20 W = GTOW∗g ; %[N]2122 %Battery23 eta = . 5 ; %e f f i c i e n c y24 V = 2 6 . 1 ; %V25 I = 50 ; %A26 C = 18 ; %Capacitance − Ah27 Rt = 1 ; %Hour28 n = [ 1 1 . 3 ] ; %Discharge parameter dependent on batte ry p r o p e r t i e s2930 %At 6000 f t31 rho = 1.0269+(1.0167−1.0269) /(100) ∗29 ; %[ kg/mˆ3 ]32333435 %% Data Table from Experiment :36 a = (−5:16) ' ;37 a ( [ 2 1 1 9 ] ) = [ ] ;38 CL = [ − . 3 2 4 3 8 ; − . 2 1 5 0 3 ; − . 1 0 0 8 1 ; . 0 1 0 5 0 3 ; . 1 2 1 5 5 ; . 2 4 1 6 3 ; . 3 4 3 3 6 ; . 4 5 2 5 6 ; ...39 . 5 6 0 3 7 ; . 6 6 6 2 5 ; . 7 6 9 4 2 ; . 8 6 9 2 3 ; . 9 6 3 8 6 ; 1 . 0 4 4 1 ; 1 . 0 7 4 3 ; 1 . 0 8 0 7 ; 1 . 0 3 7 9 ; ...40 1 . 0 3 4 ; 1 . 0 1 5 6 ; . 9 7 9 4 6 ] ;41 CD = [ . 0 4 4 2 5 1 ; . 0 3 3 7 8 3 ; . 0 2 8 6 2 7 ; . 0 2 5 8 6 4 ; . 0 2 4 6 4 3 ; . 0 2 5 0 9 9 ; . 0 2 5 6 3 5 ; . 0 2 7 6 6 ; ...42 . 0 3 0 6 7 7 ; . 0 3 4 8 5 5 ; . 0 4 0 4 0 3 ; . 0 4 7 5 9 ; . 0 5 7 1 0 8 ; . 0 7 0 1 3 2 ; . 0 9 0 9 2 1 ; . 1 1 1 9 3 ; ...43 . 1 3 2 5 4 ; . 1 5 6 4 5 ; . 2 0 9 5 9 ; . 2 5 6 6 8 ] ;44 data = [ a CL CD] ;4546 %% Curve Fitting C L47 P = p o l y f i t ( a ( 1 : 1 2 ) ,CL( 1 : 1 2 ) ,1 ) ;48 Bes tF i tSt r = s p r i n t f ( ' Best−Fit Curve : C L = %.2 f ∗\x3B1 + %.2 f ' ,P(1 ) ,P(2 ) ) ;4950 %% Plotting51 %C o e f f i c e n t o f L i f t52 warning ( ' o f f ' , 'MATLAB: f p l o t : NotVector ized ' )53 p l o t ( a ,CL, 'k . ' ) ;


54 hold on55 p l o t ([−6 13] , [−6∗P(1)+P(2) , 13∗P(1)+P(2) ] )56 t i t l e ( ' C o e f f i c i e n t o f L i f t vs . \ alpha ' )57 y l a b e l ( 'C L ' )58 x l a b e l ( ' \ alpha [ degree s ] ' )59 legend ( ' Experimental Data ' , Bes tF i tSt r )60 g r id on61 p l o t ( [ 0 0 ] , ylim , 'k ' , ' l i n ew id th ' , . 4 ) ;62 p l o t ( xlim , [ 0 0 ] , 'k ' , ' l i n ew id th ' , . 4 ) ;63 p r i n t ( ' L i f t C o e f f ' , '−dpng ' )6465 %C o e f f i c i e n t o f Drag66 f i g u r e67 hold on68 p l o t ( a ,CD, 'k . ' ) ;69 t i t l e ( ' C o e f f i c i e n t o f Drag vs . \ alpha ' )70 y l a b e l ( 'C D ' )71 x l a b e l ( ' \ alpha [ degree s ] ' )72 g r id on73 p l o t ( [ 0 0 ] , ylim , 'k ' , ' l i n ew id th ' , . 4 ) ;74 p l o t ( xlim , [ 0 0 ] , 'k ' , ' l i n ew id th ' , . 4 ) ;75 p r i n t ( ' DragCoeff ' , '−dpng ' )7677 %Ratio o f L i f t and Drag vs ang le o f Attack78 f i g u r e79 hold on80 p l o t ( a , (CL. /CD) ) ;81 t i t l e ( 'L/D vs . \ alpha ' )82 y l a b e l ( 'C L/C D ' )83 x l a b e l ( ' \ alpha [ degree s ] ' )84 g r id on85 p l o t ( [ 0 0 ] , ylim , 'k ' , ' l i n ew id th ' , . 4 ) ;86 p l o t ( xlim , [ 0 0 ] , 'k ' , ' l i n ew id th ' , . 4 ) ;87 p r i n t ( ' Li f tDrag ' , '−dpng ' )8889 %Drag Polar90 f i g u r e91 hold on92 p l o t (CL,CD)93 t i t l e ( ' Drag Polar ' )94 y l a b e l ( 'C D ' )95 x l a b e l ( 'C L ' )96 % xL = xlim ;97 % yL = ylim ;98 g r id on99 p l o t ( [ 0 0 ] , ylim , 'k ' , ' l i n ew id th ' , . 4 ) ;

100 p l o t ( xlim , [ 0 0 ] , 'k ' , ' l i n ew id th ' , . 4 ) ;101 p r i n t ( ' DragPolar ' , '−dpng ' )102103104 %% Plot Eq. 1.1 using best guess of e105 %Get approximation o f e106 e = 1.78∗(1− .045∗ARˆ . 6 8 ) − .64;107 %Min drag108 CD min = min(CD) ;


109 CL minDrag = 0 .1 215 5 ; %Coef f l i f t at min drag110111 %C D approximation based on s h i f t e d drag po la r112 CD approx = @(t ) CD min + ( t−CL minDrag ) ˆ2/ p i /AR/e ;113 CD 0 = CD approx (0 ) ;% C D at 0 l i f t114115 %Plot the approximation116 f i g u r e117 hold on118 f p l o t ( CD approx , [ − . 3 , . 8 ] )119 g r id on120 p l o t (CL,CD)121 ylim ( [ 0 . 2 8 ] )122 t i t l e ( ' Drag Polar ' )123 y l a b e l ( 'C D ' )124 x l a b e l ( 'C L ' )125 p l o t ( [ 0 0 ] , ylim , 'k ' , ' l i n ew id th ' , . 4 ) ;126 p l o t ( xlim , [ 0 0 ] , 'k ' , ' l i n ew id th ' , . 4 ) ;127 legend ( ' Approximating Drag Polar ' , ' Experimental Polar ' )128129 %Where we are on the Drag Polar :130 Vinf = 18 ; %[m/ s ]131 CL cruise18 = W/(S ∗ ( . 5∗ rho∗Vinf ˆ2) ) ;132133 %Where we are134 pointCD = CD approx ( CL cruise18 ) ;135 p l o t ( CL cruise18 , pointCD , 'k∗ ' )136 p r i n t ( ' DragPolarPlus ' , '−dpng ' )137138139140 %% Performance Analysis141 % Max speed142 PA = eta ∗V∗ I ∗2 ; %Power Ava i l ab l e [W]143 PR = @(t ) s q r t ( (2∗Wˆ3∗CD approx ( (W/ ( . 5∗ rho∗S∗ t ˆ2) ) ) ˆ2) /...144 ( rho∗S∗(W/ ( . 5∗ rho∗S∗ t ˆ2) ) ˆ3) ) ; %Power as a func t i on o f V i n f i n i t y145 syms Q146 v max = r e a l ( double ( vpaso lve (PA==PR(Q) ,Q, 5 0 ) ) ) ; %V max [m/ s ]147148 %Plot i t149 f i g u r e150 hold on151 g r id on152 f p l o t (PR, [ 1 2 6 0 ] )153 hold on154 p l o t ( [ 1 2 6 0 ] , [PA PA] )155 p l o t ( v max ,PA, 'k∗ ' )156 p l o t ( [ v max v max ] , [ 0 ,PA] , 'k−− ' )157 t i t l e ( 'Max Speed ' )158 y l a b e l ( 'Power [W] ' )159 x l a b e l ( 'V \ i n f t y [m/ s ] ' )160 legend ( 'Power Required ' , 'Power Ava i l ab l e ' )161 p r i n t ( 'VMax ' , '−dpng ' )162163 %Rate o f Climb


164 %R/C = exce s s /W165 %exce s s = PA − D∗V166 RC = @(t ) (PA − CD approx (W/ ( . 5∗ rho∗ t ˆ2∗S) ) ∗ . 5∗S∗ rho∗ t ˆ2) /W;167 f i g u r e168 hold on169 g r id on170 f p l o t (RC, [ 1 2 4 0 ] )171 t i t l e ( ' Rate o f Climb ' )172 y l a b e l ( 'R/C [m/ s ] ' )173 x l a b e l ( 'V \ i n f t y [m/ s ] ' )174 p r i n t ( ' RateClimb ' , '−dpng ' )175176 %% Endurance and Range177 k = 1/( p i ∗e∗AR) ;178 Endurance max =Rt.ˆ(1−n) . ∗ ( ( eta ∗V∗C) /((2/ s q r t ( rho∗S) ) ∗CD 0ˆ.25∗ ...179 (2∗W∗ s q r t ( k /3) ) ˆ 1 . 5 ) ) . ˆ n ;180 Range max = Rt.ˆ(1−n) . ∗ ( ( eta ∗V∗C) /((1/ s q r t ( rho∗S) ) ∗CD 0ˆ.25∗ ...181 (2∗W∗ s q r t ( k ) ) ˆ 1 . 5 ) ) . ˆ n∗( s q r t (2∗W/rho/S∗ s q r t ( k/CD 0) ) ∗3 . 6 ) ;182183 % Endurance Code184 % Endurance at n = 1185 E 1 = @(Vel ) Rtˆ(1−n (1) ) ∗ ( ( eta ∗V∗C) /(0 . 5∗ rho ∗( Vel ˆ3) ∗S∗CD 0 + ...186 (2∗k∗(Wˆ2) ) /( rho∗Vel∗S) ) ) ˆn (1 ) ; % Hours187 % Endurance at n = 1 .3188 E 2 = @(Vel ) Rtˆ(1−n (2) ) ∗ ( ( eta ∗V∗C) /(0 . 5∗ rho ∗( Vel ˆ3) ∗S∗CD 0 + ...189 (2∗k∗(Wˆ2) ) /( rho∗Vel∗S) ) ) ˆn (2 ) ;190191 U R = s q r t ( (2∗W) /( rho∗S) ∗ s q r t ( k/CD 0) ) ;192193 %Range at n=1194 R 1 = @(t ) E 1 ( t ) ∗ U R ;195 %Range at n=2196 R 2 = @(t ) E 2 ( t ) ∗ U R ;197198199 % Plot over a range o f v e l o c i t i e s200 Vels = l i n s p a c e (12 ,40 ,100) ;201202 E vec1 = ze ro s (1 ,100) ;203 E vec2 = ze ro s (1 ,100) ;204 R vec1 = ze ro s (1 ,100) ;205 R vec2 = ze ro s (1 ,100) ;206 f o r i = 1:100207 E vec1 ( i ) = E 1 ( Vels ( i ) ) ;208 E vec2 ( i )= E 2 ( Vels ( i ) ) ;209 R vec1 ( i ) = R 1 ( Vels ( i ) ) ;210 R vec2 ( i ) = R 2 ( Vels ( i ) ) ;211 end212213 %Plot i t214 f i g u r e215 hold on216 g r id on217 p l o t ( Vels , E vec1 )218 t i t l e ( ' Endurance ' ) ;


219 x l a b e l ( 'V \ i n f t y [m/ s ] ' )220 y l a b e l ( ' Endurance [ hrs ] ' )221 p l o t ( Vels , E vec2 )222 legend ( 'n = 1 ' , 'n = 1 .3 ' )223 p r i n t ( ' Endurance ' , '−dpng ' )224225 f i g u r e226 hold on227 g r id on228 p l o t ( Vels , R vec1 )229 t i t l e ( 'Range ' ) ;230 x l a b e l ( 'V \ i n f t y [m/ s ] ' )231 y l a b e l ( 'Range [km] ' )232 p l o t ( Vels , R vec2 )233 legend ( 'n = 1 ' , 'n = 1 .3 ' )234 p r i n t ( 'Range ' , '−dpng ' )235236 %% How discharge rate af fects battery capacity237 i 0 = l i n s p a c e (10 ,100 ,1000) ;238 t1 = (Rt . / ( 2∗ i 0 . ˆ n (1 ) ) ) ∗(C/Rt) ˆn (1 ) ; % time taken to d i s cha rge batte ry at239 %var i ous cur rent draw240 t2 = (Rt . / ( 2∗ i 0 . ˆ n (2 ) ) ) ∗(C/Rt) ˆn (2 ) ;241 %Plot i t242 f i g u r e243 hold on244 g r id on245 p l o t ( i0 , t1 )246 p l o t ( i0 , t2 )247 t i t l e ( ' Battery Discharge ' )248 x l a b e l ( ' Current Draw from Battery [A] ' )249 y l a b e l ( 'Time to Discharge [ hr ] ' )250 legend ( 'n = 1 .0 ' , 'n = 1 .3 ' )251 p r i n t ( ' BattDischarge ' , '−dpng ' )


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