Submitted to arXiv in August 2021Physics Education paper

The development of an educational software foraircraft flight mechanics calculationsClaudio C. Pellegrini1*, Mateus S. Rodrigues2, Erika O. Moreira2

AbstractDue to its versatility and low cost, the use of unmanned aerial vehicles has been rapidly spreading in recentyears, in applications ranging form military operations, to land mapping, rescuing of lost people, aiding of naturaldisaster victims and many others. To properly design and operate such a vehicle, it is necessary to know its flightmechanics in the various stages of the flight. Despite the fact that the physic behind the analysis of an aircraft’sflight mechanics is well known and purely based on Classical Mechanics, the large quantity of input and outputdata involved favors the use of a computational toll. This work presents the development of a toolbox calledAPT (Aircraft Performance Toolbox), able to make preliminary aircraft flight mechanics calculations regarding itsperformance. To achieve this goal, a number of Matlab scripts were created to perform the calculations, anda graphical interface was created to unify them and to allow the end-user to perform the analysis in a clearand intuitive way. To illustrate the potential of the toolbox, we use APT to in the analysis of an UAV meant toparticipate in the SAE Brasil AeroDesign competition of the year of 2014.

KeywordsEducational software, aeronautical engineering, aircraft performance, SAE Brasil AeroDesign competition.

1Department of Thermal and Fluid Sciences, Federal University of Sao Joao del-Rei, Brazil2Department of Electrical Engineering, Federal University of Sao Joao del-Rei, Brazil*Corresponding author: pelle@ufsj.edu.br

Contents

1 Introduction 1

2 Computational tools as a teaching resource 3

3 Aircraft performance 3

3.1 Input variables . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Take-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Climb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 Steady, level flight . . . . . . . . . . . . . . . . . . . . . . 43.5 Level turn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.6 Gliding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.7 Landing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.8 Payload variation . . . . . . . . . . . . . . . . . . . . . . . 53.9 Flight envelope . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Software Design 5

5 A case study 6

6 Didactic Experience 8

7 Conclusions 10

Acknowledgments 10

References 11

1. IntroductionThe use of unmanned aerial vehicles (UAVs) has been widelydisseminated over the past few years. This fact surely stemsfrom its versatility: they are able to operate in different envi-ronments, weather conditions and in situations where humanactivity is considered risky or even impossible [1]. Also,UAVs have extremely low acquisition and operating costscompared to conventional aircraft.

Among the activities in which UAVs have been intenselyused, we may cite surveillance operations (military or not),the mapping of land use/land cover for geographical and me-teorological purposes, the control of clandestine activities asagricultural, mineral extraction, hunting and fishing in pro-tected areas, the control of damage caused by natural disastersand the search and rescue operations that follow, the searchand rescue for people lost in inhospitable places, as moun-taineers and sailors, the sending of supplies to isolated groups,and many others. In addiction, UAVs may represent an im-portant tool in the mapping of infectious diseases such as thenew Covid-19. UAVs can provide spatially and temporallyaccurate data to help understand the links between diseasetransmission and environmental factors [2].

Both the design and the operation of UAVs require anal-ysis of the aircraft’s flight mechanics, and in particular ofits performance in various flight situations. Such analysesdetermine whether the aircraft is capable of fulfilling the ob-

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The development of an educational software for aircraft flight mechanics calculations — 2/12

jective for which it was designed. The idea is to be able toanswer questions such as: can the UAV take off and land onthe stipulated runway dimensions? Can it carry the necessaryload to the prescribed distance in the stipulated time? Can itreach the expected speed?

However, such an analysis requires input data related tothe aerodynamic characteristics of the aircraft, its propulsionsystem, its dimensions and weight, the runway length avail-able for takeoff and landing, the thermodynamic properties ofthe air at the operating location, etc. In addition, it inevitablyends up generating a large amount of output data.

Despite the fact that the physics used in the analysis ofan aircraft’s performance are well described both in the tradi-tional and recent literature [3, 4, 5, 6], the large quantity ofinput and output data involved favors the use of a computa-tional toll. For the authors, the need for such a tool appearedduring their participation in the SAE Brazil AeroDesign com-petition of 2014. The SAE Brasil AeroDesign competition,according to the organizers [7],

Is a challenge launched to Engineering students,whose main objective is to promote the diffusionand exchange of Aeronautical Engineering tech-niques and knowledge among students and futuremobility engineering professionals, through prac-tical applications and competition between teams

For the competition, students must form teams composedby engineering undergraduates and a faculty advisor. Theteam must design and build a cargo UAV to complete a mis-sion that change every year. The competition aim is to dis-seminate knowledge in aeronautical engineering, challengingstudents from all over Brazil to face each other of in an eventoccurring every October.

On the year of 2009 the team Trem Ki Voa Micro wasformed at the University Federal of Sao Joao del Rei, withone of the authors as its academic adviser. It made its debutat the 2010 competition and took part of every subsequentissue of the competition, until the present year. In a decade ofexistence the team involved more than 200 students from un-dergraduate courses of Mechanical, Electrical and ProductionEngineering, many of whom now work in the aeronauticalindustry. The educational benefits and risks of this kind ofcompetition is generally recognized by teachers and students.It is reviewed in a number of studies ([8], [9] and [10]forexample).

With the AeroDesign competition in mind, in the yearof 2014 the authors set out to obtain a computational tool toaid the team in the calculations. Their experience showedthat, in general, the computational packages developed forperformance analysis must meet the following characteristics:

• They must be supported by mathematical models that al-low a complete assessment of the aircraft’s performancecharacteristics;

• They must have a graphical interface that allows thecreation or loading of a database containing the infor-

mation of the aircraft to be analyzed, and allowing theselection of the types of analyzes to be performed;

• They must generate a clear and flexible data output,allowing for the comparison of two or more aircraft;also, the resulting graphics must have layouts that canbe easily adapted to the user’s needs;

• They must be able to be implemented in a computa-tional environment disseminated in the academic envi-ronment and be freely distributed.

Unfortunately, a large part of aeronautical science is devel-oped by private companies, that keep confidentiality about thetools developed by them or charge their use. Thus, just a fewcomputational packages developed for aircraft performanceanalysis are presently available and the ones available do notmeet some — or even all — of the requirements mentioned.Therefore, the authors realized that the only way to assure thatthe team would have access to an efficient and complete com-putation tool, within an affordable cost, would be to developtheir own.

This situation is not exclusive of the aircraft performancearea. Thus, [11] for example, developed an interactive mul-timedia package on aircraft stability and control to fulfill thegap and [12] did the same with an statistical calculationspackage.

To further illustrate the problem, the license for the ULTRA-NAV® Performance Software1 costs US$ 595–795, but thepackage only performs take-off and landing analyses. TheAtlas® and iPreFlight® packages2, developed by the APG(Aircraft Performance Group), perform only take-off and pay-load analyses respectively, and charge a “price on request”for their licenses. The Aircraft Performance Program®3 de-veloped by DARcorporation fulfills most of the requirementsmentioned above, but its license costs from US$ 5,000 (inter-mediate version) to almost US$ 8,000 (full version).

The objective of the present article is to present the designand development of the Aircraft Performance Toolbox (APT),created using the Matlab® (student version) environment. Itwas designed to be a complete and free-to-use tool to helpstudents calculate the most relevant aircraft performance pa-rameters and help them explore the numerous possibilitiesassociated with different aircraft configurations. A brief ac-count of the didactic experience with the APT in the Trem KiVoa Micro team is also presented.

This study is part of a larger project that includes thecomplete evaluation of an aircraft flight mechanics. It involvesthe analysis of performance presented here, but also of staticstability, dynamic stability, classical feedback control, stabilityand control augmentation, and is currently underway.

1Available at http://www.ultranav.com2Available at https://www.flyapg.com/atlas.html and

https://www.flyapg.com/ipreflight.html, respectively3Available at

http://www.darcorp.com/aircraft-performance-program-software

The development of an educational software for aircraft flight mechanics calculations — 3/12

2. Computational tools as a teachingresource

This is a topic that has been extensively covered in many stud-ies in the last decades, since the popularization of personalcomputers started. Nevertheless, some general characteristicspreviously noted by some selected authors are worth mention-ing.

Experience has shown that active visualization-based learn-ing using interactive tolls can greatly enhance the student’sunderstanding and retention. A direct interactive visual ap-proach may help to remove many of the conventional barriersthat hinder the effective learning of abstract engineering con-cepts [13]. However, computational packages and visualisa-tion programs are not the only possibility. Teachers have beenusing non-computational tools for a long time, in a number ofcreative ways, including experimental set-ups, scale modelsand toolkits. Extensive use has also been made in the past ofvideo presentations by lecturers (usually in Super 8 format)and slideshows, using the physical photographic slides.

Nevertheless, soon after the popularization of personalcomputers, they begin to substitute the ”physical” apparatuseswhenever possible, due to their flexibility and low cost (abasic didactic wind tunnel can easily cost as much as 100,000Euros). For example, a study by [14] describes a numericalsimulation experiments being used in the late 70’s to teachbasic physics at the New York Institute of Technology. In thefollowing decades, this tendency accelerated noticeably andat the moment, educational software is used in almost everyarea of knowledge, including human sciences.

In a study on the use of computers as a tool in the teach-ing of physical sciences, [15]4 conclude that the pedagogicalpotential of computers can only be fully realized if thereare enough educational programs of sufficient quality. Inthis respect, many papers highlight the characteristics thatan educational software must have ([13] and [16], for exam-ple). The most frequent adjectives are: interactive programs,visualization-based learning, motivating, user-friendly inter-face, robustness, easy of use, and, of course, free to downloadand free to use.

The didactic value of computational packages, in connec-tion to the teaching of basic physics, is summarized by [17]5

as:

The development of educational software allowsteachers and students to perform simulations andcreate animations related to challenging problemsin Newtonian kinematics and dynamics, both inhigh school and in undergraduate introductoryMechanics courses. Problems that do not requirethe student to be familiar with advanced mathe-matical methods but that allow for a wide vari-ation of the relevant parameters, can be studiedwith great advantage.

4In Portuguese5Translated from Portuguese.

In conclusion, the use of computer simulations and com-puter experiments in the teaching of sciences is very promis-ing. However, as noted by [13], implementing the idea maymean modifying not only the way of teaching but also recon-sidering the educational model and philosophy.

3. Aircraft performance

The performance characteristics of an aircraft requires severalsets of calculations, hereafter denominated performance anal-yses or analyses for short. Each contains a number of compu-tations based on Classical Mechanics, requires a considerableamount of input data and returns an equally considerableamount of output data (39 variables to be precise).

This section briefly introduces the input variables anddescribes each of the analyses that compose the APT toolbox.The aeronautical terms used are traditional in the literatureand may be found in aviation terms dictionaries or glossaries,as [18] for example.

3.1 Input variablesThe input variables of the APT can be divided into threecategories: take-off variables, geometric parameters and aero-dynamic parameters.

The take-off variables category includes the air densityat the take-off location and the runway length. The first canbe measured locally using a portable meteorological station,obtained from the nearest surface meteorological station or,as a last resource, from the local climatology provided by thenational weather service.

The geometric parameters category includes the wing area,the wingspan, the height of the wing above the ground, theaircraft empty weight and the weight of the cargo. The rollingfriction between the tires and the runway must also be known.It is modeled according to F = µR [19], where µR is therolling friction coefficient obtained from the literature ([3]and [5], for example) or experimentally, as shown in [20] and[21]. Another important input variable is the relation betweenthe propulsion system thrust and the velocity of the airplane,i.e., the thrust curve. In general, it is modeled as a seconddegree polynomial, whose coefficients can be obtained bycurve fitting to wind tunnel data or using the low cost methodproposed in [6].

Finally, in the aerodynamic parameters category, the dragpolar curve is the most important input information. Thecurve is composed by values of the lift coefficient and theirassociated drag coefficient values, for every possible angleof attack, α . The polar curve is obtained by methods widelydisseminated in the aeronautical area, such as [3] and [5]. Thecategory also includes the drag and lift coefficients for stallangle and take-off angle, and the Oswald efficiency factor.

The calculations for each phase are described below. Ineach case, only the main output variables are mentioned. Thecomplete list can be found in Appendix A.

The development of an educational software for aircraft flight mechanics calculations — 4/12

3.2 Take-offTaking off an aircraft consists of accelerating it from rest to aspeed where the lift is greater than its weight. The requiredrunway length is evaluated in this phase. The lift force isobtained through the well-known relationship:

FL =12

ρCLV 2A (1)

where ρ is the air density, CL is the lift coefficient (whichdepends on the angle of attack of the wing), V is the speed ofthe aircraft in relation to the air and A is the projected wingarea.

The acceleration is created by the thrust force of the en-gine, T , which must be greater than the sum of the two dissi-pative forces that act on the aircraft during take-off, namelythe total drag:

FD =12

ρCDV 2A (2)

and the rolling resistance,

FR = µr(W −FL) (3)

where CD is the drag coefficient for the entire aircraft, N isthe normal force, and µR is the combined rolling resistancecoefficient of the complete landing gear.

According to the classic aeronautical literature ([3], forexample), the dependence of the engine thrust on the aircraft’svelocity relative to the air, for the kind of propulsion systemused, can be modeled by:

T = αV 2 +βV +T0 (4)

where α, β and T0 are constants specific for each propulsionsystem, T0 being known as the static thrust.

For the analysis of take-off, it is necessary to calculate thetake-off velocity, Vto, equaling the total weight of the aircraft,W , to the lift force under takeoff conditions and inserting theusual 1.2 safety factor [22]. Therefore:

Vto = 1.2

√2(W/A)ρCLstall

(5)

where (W/A) is the wing load, CLstall is the maximum liftcoefficient, occurring at the associated αstall angle of attack.It is to be attained by the airplane near the end of the runawaywhen the pilot rotates the airplane, pitching its nose up. Usingthis value, the runway length required for take-off, Sto, maybe obtained from the closed analytical model proposed by [6]:

Sto =W

2gB

[ln∣∣∣∣BV 2

to +CVto +DD

∣∣∣∣− 2C√

∆1

(arctan

2BVto +C√∆1

− arctanC√∆1

)](6)

for ∆1 = 4BD−C2 > 0, and

Sto =W

2gB

[ln∣∣∣∣BV 2

to +CVto +DD

∣∣∣∣− C√

∆2

(ln

2BVto +C−√

∆2

2BVto +C+√

∆2· C+

√∆2

C−√

∆2

)](7)

for ∆2 =C2−4BD > 0, where

B = α− ρ

2(CD−µrCL)S (8)

C = β −µrCLS (9)

D = T0−µrW (10)

The last result is not found in textbooks. It was recentlyproposed in substitution of the classic averaged approach, pre-sented for example in [3], where the necessary integration issimplified using average values of the forces involved, calcu-lated at 70% of Vto. Such a simplification yields good practicalresults but obscures the understanding of the take-off processand is limited by several factors [23]. The most relevant isthat the approximated integration does not yields a S = S(V )relation. Nevertheless, the value of Sto according to [3] is alsocalculated by APT as a reference.

3.3 ClimbThe climb phase consists of raising the aircraft’s altitude. Theprocess takes place after the take-off so that the aircraft canreach the desired altitude. In this analysis, the maximumclimb rate, CRmax, and the maximum climb angle, θmax, arecalculated using the following equations:

CRmax =

(T −FD

W

)V (11)

θmax = arcsin(

T −FD

W

)(12)

The toolbox also plots the curve CR =CR(V ). The valueof CRmax is obtained through the plot and does not occurs atθmax.

3.4 Steady, level flightIn this phase the aircraft is in a situation of horizontal flight,at a constant speed. Thus, the lift force must equal the weight,and the thrust generated by the propulsion system must equalthe drag. The maximum speed that the aircraft can reach,the cruising speed, the maximum range and maximum rangespeed can be calculated by equating the values of availableand required power, PA and PR, i.e, by putting

PA = TV (13)

PR = FDV (14)

The resulting dependencies PA = PA(V ) and PR = PR(V ) arepresented graphically to the user to allow a better analysis ofthe aircraft’s behavior under different conditions.

The development of an educational software for aircraft flight mechanics calculations — 5/12

3.5 Level turnIt is a simple curve maneuver, in which the aircraft performs arolling movement around its longitudinal axis, tilting its wingsin relation to the ground, while moving in a horizontal plane.The vertical component of the lift is equal to the weight of theaircraft and the horizontal component creates the centripetalacceleration necessary to travel along a circular path. For thissituation the maximum angle of roll and the minimum radiusof curvature are calculated, respectively, by

φmax = cos−1(

2(W/A)ρ CLstallV 2

max

)(15)

and

Rmin =2(W/A)ρgCLstall

(16)

where CLmax is the maximum lift coefficient and Vmax is maxi-mum aircraft speed.

3.6 GlidingIt is a situation of descent with the propulsion system in idle,that is, with zero thrust. It is used in the landing phase ofsmall aircraft or in case of engine failure. The angle and therate of descent are calculated for maximum autonomy andmaximum reach, respectively,

θmin = tan−1(

1(CL/CD)max

)(17)

where (CL/CD)max is the maximum aerodynamic efficiency.

3.7 LandingIt consists in decelerating the aircraft from the speed withwhich it touched the runway, to the rest. In general, thrustis considered null. Other forces related to the use of brakes,lift spoilers or the reverse6 to shorten the length necessaryfor landing, are not considered here and, thus, the result isconservative. Here, APT uses the same formulation employedfor the take-off with T = 0, yielding

Sl =W

2gB

[ln(

BV 2l +DD

)](18)

where

Vl = 1,3

√2(W/A)ρCLstall

(19)

according to ([22]), and

B =ρ

2(CD−µCL)A (20)

D = µW (21)

6A configuration in which the propulsion system generates negative thrust

3.8 Payload variationThe payload is the weight that the aircraft is capable of carry-ing at different altitudes at which take-off can occur, since thevariation in altitude leads to a variation in air density, conse-quently altering the thrust. the lift and the drag. This load isevaluated through an iterative process, in which the take-offdistance is repeatedly calculated for increasing masses, until itexceeds the runway length (with a prescribed factor of safety).The whole process is then repeated for various altitudes, withtheir respective values of density. The result, given by thetoolbox in graphic form, shoes the dependence of the payloadwith altitude.

3.9 Flight envelopeThe flight envelope is the region of the speed vs. altitudegraph in which the aircraft can sustain straight and level flight.As altitude increases, stall, maneuver and minimum speedsincrease, however, maximum speed decreases. The enve-lope traced by taking these speeds into account, delimits theaircraft’s operating range. In this analysis, this envelope ispresented graphically.

4. Software Design

The Matlab® platform was the chosen development environ-ment due to its renowned ability to deal with large numbersof variables and to perform calculations with large indexedvariables. The free student version was choosen.

Matlab® is an interactive environment specialized in nu-merical and algebraic manipulation that uses as its basicvariables matrices that do not require dimensioning. Thelanguage is rather similar to standard mathematical writing,which makes coding easier than in languages as FORTRAN, Cand Java. Matlab® possesses a tool to aid developing graphicinterfaces called GUIDE (Graphical User Interface DesignEnvironment) that enables the construction of the layout andautomatically generates the associated code, that can be fur-ther edited to add other desired functionalities.

The numerical data output can be easily written in Matlab®

by the omission of the semicolon at the end of an equationor using the command disp. The graphic data output can bedone via the plot command, which has a very intuitive syntax.Even after one graph has been made, the user still has widepossibilities to change its details, altering titles, legends, axes,fitting curves, colors, etc.

Finally, in spite of the fact that Matlab® is a commercialsoftware, due to its huge numerical problem solving capabilityand simplicity of coding, it is widely spread in the academiccommunity, being available at most universities in its studentedition and higher. To ensure that the didactical objectives ofthe tool were fulfilled, the user interface was designed in asimple and intuitive way. Thus, students with little knowledgeof airplane performance and even of aeronautics are ableto carry out the analyses quickly and correctly. It is onlynecessary that the student enter the required data and select”perform analysis”, and all graphs and values are calculated

The development of an educational software for aircraft flight mechanics calculations — 6/12

and presented. Also, a large number of explanatory remarkswhere included in the source code, for those with access to it.For users with only access to the executable, the program wasexternally documented in an organized manner.

For the development of the toolbox, several scripts wereinitially implemented to carry out the necessary performanceanalyzes. Each script contains the equations necessary toperform one of the analyzes, as well as the reading commandsfor the input data and writing or plotting commands for theoutput data. The set of scripts works interactively to allow theoutput data from one analysis to be used as input data fromanother.

A graphical interface was also created to allow the userto create a database designed to feed the scripts and to maketheir execution easy and intuitive, as shown in Figure 1. Newdatabases can be created simply by filling in the blanks withthe value of the variables. A previously created bank can beloaded via the ‘Load Database’ button. After inserting a dataset, the user has the option of saving them to a text file usingthe ‘Save Database” button.

Once the database is available, the user must select thetype of analysis he wants to perform by clicking on the corre-sponding option. By clicking on the button ‘Perform analysis’the data entered in the interface at that moment will becomeavailable to the system.

Before the code executes the script corresponding to theanalysis, it will verify that all fields have been filled in, andif not, a message will appear in the field for observations.The code will also check whether the indexed variables, dragcoefficient, and total aircraft lift, have the same number ofelements, a crucial detail for the calculations. If not, the userwill also be alerted in the manner previously mentioned.

After the execution of the script the numeric output data ispresented in the Matlab command window and the graphs aredisplayed each in its window. The user can save the graphicin one of several supported formats: png, jpg, bmp, pdf, etc.

At this point, the user can request to perform other typesof analysis, make changes to the database or perform thesame analysis again. He can also load another database orsimply clear the output data using the ‘Close graphs’ and‘Clear Command Window’ buttons.

If the user chooses to perform more than one analysiswithout closing the existing graphs, the graphs that relate tothe same quantities will be displayed in the same window.This feature will allow the user to graphically compare twodifferent aircraft or to analyze the variation of the outputparameters given the change in an input parameter. This willbe exemplified in the case study presented in the followingsection.

For better visualization of the structure of the APT, a blockdiagram was built, which can be seen in Figure 2.

5. A case studyAPT allows the user to quickly assess the influence of changesin design data on the performance of an aircraft by making

changes to the databases. This means that it can be used notonly for the evaluation of an existing aircraft but also duringthe conceptual design phase.

To illustrate this type of use, this section will show theprocess of selecting a battery fot the UAV designed by theTrem Ki Voa Micro team (Figure 3) for the 16th SAE BrasilAero design Competition. Initially, a brief description of thecompetition challenge will be presented and in the end, themost significant graphics and results will be shown.

The UAV propulsion system consists of a brushless elec-tric engine driven by a frequency inverter, commercially calledESC (Electronic Speed Control), and powered by a Lithium-Iron battery. Because of the weight restrictions imposed bythe competition regulations, the least possible weight batterythat allows the necessary flight performance must be chosen.

The thrust generated by the kind of engine in questionincreases as larger electrical voltage is applied to its terminals.For this, the voltage supplied by the battery must be increased.This is possible by increasing the number of cells in it, asthey are connected in series and have, by default, 3.3 V each.However, the greater the number of cells, the greater theirweight.

The APT was used to compare the performance of theUAV built by the team, using two 1300 mAh batteries, onewith 3 cells in series (called the 3S configuration) and the otherwith 4 cells in series (the 4S configuration), each weighing,respectively, 148 and 195 g. For comparison, two databaseswere used, differing only in the thrust coefficients and theempty weight of the aircraft. The values were 0.833 kgf forthe 3S battery and 0.880 kgf for the 4S.

The coefficients for the propulsion system thrust curve,given in Eq. 4, were obtained by the low cost methodology as

Configuration 3S:

α =−0.00922 N/(m/s)2

β =−0.355 N/(m/s)T0 = 14.387 N

(22)

Configuration 4S:

α =−0.0110 N/(m/s)2

β =−0.427 N/(m/s)T0 = 17.904 N

(23)

The other input variables, equal for both configurations,are shown in Table 1, where MP represents the mass of thepayload, Cstall

L and CstallD are the lift and drag coefficient at

stall angle, CtoL and Cto

D are the lift and drag coefficients attake-off angle, b is the wingspan, Hg is the height of the wingin relation to the ground, eo is Oswvald’s efficiency factor andSrun is available runway length.

The polar curve used in the calculations is shown in Fig-ure 4.

Some output data for the tested configurations are shownbelow. Figure 5, from the analysis of straight and level flight,shows the dependence of available and required thrust withvelocity for both configurations. It is evident that the availablethrust shows considerably higher values for the 4S configura-tion, but required thrust values are only slightly larger. The

The development of an educational software for aircraft flight mechanics calculations — 7/12

Aircraft Performance Toolbox

Performance AnalysisDatabase Comments

Designed by Mateus Rodrigues

Choose a language

English Português

Open Database Save Database Perform Analysis Close Graphs Clear Command Window:

Choose the type of analysis you want to performregininha 3S.txt

Steady, level flight and Speeds Evaluation:Steady, level flight and Speeds Evaluation:

Climb

Gliding

Landing

Level Turn

Take off

Best Cl for the take off ground roll

Take off - Anderson’s simplified method

Flight envelope

Payload evaluation

The numerical output will be written on the

Command Window and the graphs will be

plotted on separated windows.

Cl, stall and Cd, stall are the lift and drag

coefficient at stall angle.

Cs,to e Ca,to are the lift and drag coefficient

for the aircraft incidence angle during the

takeoff ground roll.

Cl,tot e Cd,tot at the values for the total

lift and drag coefficient that build the entire

aircraft polar. You must separate the values

with a space and use a dot as a decimal

mark. Coefficients which are placed on the

same position must correspond to the same

angle of incidence.

Air density: Wing

Weight

Cl and Cd (Entire aircraft) Thrust (ax +bx+c)2

Available RunwayRolling resistance coefficient

Polar

Cl,tot:

Cd,tot:

0.0936 0.1867 0.2716 0.3593 0.4447 0.5343 0.6217 0.7073 0.8008 0.8826 0.8666 1.0473 1.1284 1.2075 1.2857 1.3634 1.4406 1.5183 1.5588 1.6428 1.7225

0.0551 0.0537 0.0563 0.0598 0.0646 0.0708 0.0787 0.0866 0.0941 0.1041 0.115 0.1271 0.1409 0.1575 0.1784 0.2017 0.2263 0.2509 0.2431 0.3041 0.3359

1.089

0.0833

2.28

1.418

0.3147

0.447

0.0646

0.11

0.32

0.9727

-0.00922

-0.355

14.6387

61

1.32

0.034Rho: Area (m ):2

Wingspan:

Hg:

eo:Payload (Kg):

Aircraft (Kg):

Cl,stall:

Cd,stall:

Cl,tot:

Cd,tot:

Crr: Sa:

a:

b:

c:

Figure 1. Graphic interface.

Table 1. Input variables.

Variable Unit Value

ρ kg/m3 1.089MP kg 2.280Cstall

L – 1.418Cstall

D – 0.314Cto

L – 0.447Cto

D – 0.064µR – 0.110A m2 0.340b m 1.320Hg m 0.320eo – 0.972Srun m 61.0

increase in available thrust is due to the increase in the bat-tery’s output voltage and presents a considerable gain in thechange from the 3S to the 4S configuration. The requiredthrust, however, does not show such a strong dependence onthe weight of the set, as shown in the figure. Thus, it is clearthat, from the aircraft’s performance point of view, the 4Sconfiguration is advantageous, increasing the surplus powerfor maneuvers and the maximum flight speed, as seen in thetable 2.

Figure 6 comes from the climb analysis and shows thecurves of the aircraft’s maximum climb rate. Once again, itis noticed that despite the slight increase in weight, the 4Sconfiguration gives the aircraft a higher climbing capacity,due to the increase in thrust surplus.

Figure 7 comes from the take-off analysis and shows thespeed along the runway, from rest to take-off speed. Althoughthe 4S configuration demands a slightly larger take-off speed,the aircraft can reach this speed in a distance 23% smallerthan with the 3S configuration.

Figure 8 shows the dependence between the minimum ra-dius of the level curve and the speed for the two configurations.It is clear that the lower power configuration allows for moretight turns for a given speed, due to its lower weight. This isa very important fact for the aircraft’s maneuverability andone the only negative points in the 4S configuration. However,the 4S configuration allows for an overall smaller minimumradius, as it allows the aircraft to reach a higher maximumspeed.

Finally, Figure 9 shows the load capacity for both config-urations. It is evident that, due to the increase in thrust, theaircraft powered by the 4S battery has a considerably largerload capacity than the aircraft with the 3S battery, an increaseof 11%.

Table 2 shows the main aircraft output data for variousanalyses. Here, Vstall is the stall speed, Vmax is the maximumspeed for steady, level flight, Vcru is the cruise speed, ϕmax isthe maximum climb angle and φ

rngmax is the maximum range an-

gle during gliding, RrngDmax is the descent reason for maximum

range during gliding, nCmax is the maximum load factor incurve, Rmin is the minimum radius in curve, θmax is the maxi-mum roll angle, Sto is the runway length needed to take-offand Sl is and the runway length needed to land.

It can be seen that almost all the performance parametersindicate the 4S configuration to be advantageous, except forthe aforementioned minimum radius of the level curve andfor the parameters resulting from glide and landing analysis,

The development of an educational software for aircraft flight mechanics calculations — 8/12

Figure 2. Block diagram.

Figure 3. UAV used at 16th SAE Brazil Aero Designcompetition by the Trem Ki Voa team.

where the engine thrust is set to zero. As the 3S configurationwould receive, according to the competition rules, only a fewmore points due to its lower weight, the team opted to use the4S battery. The decision proved to be correct and the teamwon the 2014 competition.

In subsequent years, the team also won the 2016, 2017and 2019 competitions using the APT toolbox, and is nowfour times Brazilian champion. It was also awarded a secondoverall place in the SAE East AeroDesign world competitionof 2015 and another second overall place in the SAE BrazilAeroDesign competition of 2020. It is now, the most awardedteam in Brazil, with four championships, two second places(2012 and 2020) and a world second place.

CD,tot

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22

CL

,tot

0

0.5

1

1.5

Figure 4. drag polar curve.

6. Didactic Experience

To access the usefulness of the tollbox, a survey was con-ducted among students who work and have worked in theperformance area of the team, to learn their experiences in theanalysis before and after using the software. Unfortunately,the success of the tool meant that just a few students hadexperience before and after its creation. Indeed, from 2015onward, no student ever made the calculations by hand again.

The research was carried out through the Google formsplatform. Google forms allow users to ”collect and organizeinformation large and small for free”. The responses to asurvey are stored in spreadsheets (Google Sheets) and can beviewed in graphs or even raw in the spreadsheet. There aredifferent styles of questions and input methods for answers,as well as section breaks, the possibility of uploading files,displaying images or videos, and other features.

The development of an educational software for aircraft flight mechanics calculations — 9/12

Speed (m/s)10 12 14 16 18 20 22

Thru

st (

N)

1

2

3

4

5

6

Available thrust - 3S

Required thrust - 3S

Available thrust - 4S

Required thrust - 4S

Figure 5. Available and required thrust.

Speed (m/s)11 12 13 14 15 16 17 18 19 20 21

Rat

e of

Cli

mb (

m/s

)

0

0.5

1

1.5

2

2.5

3

Rate of climb - 3S

Rate of climb - 4S

Figure 6. Rate of climb.

According to the 8 forms received, students think that thetoolbox considerably speeds up the laborious calculations andhelps organize the output data, making it easy to test differentaircraft configurations in the initial phase of the project. Inaddition, it is believed to help in the preparation of the finalreport and presentations. The tool was also credited withproviding greater interaction between the areas, as it helps toorganize the input and output data.

Another interesting feature mentioned by the students isthat the toolbox can be used both in the forward and back-ward directions. In the forward direction, it is used for itsoriginal purpose, calculating performance characteristics ofthe UAV from the input data. In the reverse direction, it maybe used to calculate input data from measured performancecharacteristics.

An example of this, is the possibility of calculating the

Ground roll (m)0 5 10 15 20 25 30 35 40

Spee

d (

m/s

)

0

2

4

6

8

10

12

14

3S

4S

Figure 7. Aircraft speed along the take-off ground roll.

Speed (m/s)11 11.5 12 12.5 13 13.5

Turn

rad

ius

(m)

0

50

100

150

Turn radius - 3S

Turn radius - 4S

Figure 8. Minimum turning radius.

Density altitude (m)0 500 1000 1500 2000 2500 3000 3500

Mas

s (K

g)

2.5

3

3.5

4

4.5

Payload - 3S

Payload - 4S

Figure 9. Payload prediction chart.

value of the rolling resistance coefficient for a new set of tires,from the take-off distance measured experimentally .

The only disadvantage of APT detected in our diagnosiswas that, as students understand that the code is able makingthe calculations “on its own”, they tend not to study the theoryin depth as they should. This have lead to two problems in thepast years: (i) students do not acquire the proper understand-ing about the subject; (ii) students spend a long time usingthe toolbox in trail and error tests, just to arrive at conclusionsthat were obvious from the theory (that they did not studiedproperly!).

Unfortunately, this situation seems to be commonplacein modern engineering courses, whenever a software is avail-

Table 2. Main output variables.

Output variables Unit 3S 4S

Vstall m/s 10.78 10.86Vmax m/s 19.20 19.96Vcru m/s 17.28 17.97RCmax m/s 2.06 2.79ϕmax (0) 9.89 13.43φ

rngmax (0) 6.68 6.68

RrngDmax m/s -1.63 -1.65

nCmax - 1.30 1.48Rmin m 18.37 16.29θmax (0) 40.14 47.55Sto m 38.10 29.44Sl m 84.23 85.51

The development of an educational software for aircraft flight mechanics calculations — 10/12

able to help in the calculations, and is rather worrying. In aresearch conducted more than one decade ago, [24] describescharacteristics and study habits of first-year students of IowaState University’s electrical engineering course. Their find-ings, very consistent with our own observations, identifies15 common traits and behaviors of the 21st century students.One of then was particularly associated with the problems justmentioned:

Students do not seek to find understanding, onlyanswers. As a result, the students are resentfulof the confusion that arises naturally from, and isnecessary for, the learning process to be success-ful – that is, to foster understanding, and createlifelong learners. . . While the answer is important,the understanding of its discovery, or meaning, isof little value to them — that one has obtained ananswer is sufficient.

Another observation closely related with what we considera misuse of the computational tools is:

Students lack an understanding of the learningprocess. In effect, the students are attempting togain theoretical knowledge. . . merely by workingthrough numerous examples. Indeed, many stu-dents seem to believe that the process of learningconsists only of working through such examples.

And last but not least:

Students lack an understanding of the meaning ofhard work. Generally, unless they are captivatedby a subject, they will not undertake it in earnest.Consequently, they are unwilling to perform theroutine. . . necessary to develop their skills.

As for future versions of the toolbox, students showedinterest in a module to calculate manual launch take-offs. Thedevelopment of toolboxes for other areas of the project, withthe capacity of interaction between them, was also mentionedand is already underway. Priority was requested in the futuredevelopment of a module for a complete aerodynamic anal-ysis toolbox, capable of generating the polar drag curve, toremove the need for the user to enter the data manually. Animportant, although obvious, detail about this request is that itdepends on the availability of dedicated members of the Team,with adequate training in MatLab® and other necessary soft-ware. Of course, being a member is not mandatory, but in ourexperience is highly recommended, as it generally guaranteethe necessary theoretical basis. Indeed, APT was developedby one of the authors, a second year Electrical Engineeringundergraduate student, member of the performance sector.

7. ConclusionsWe believe that APT was able to meet all the requirementsstipulated initially, including the didactic ones. The code was

implemented in Matlab®, available in the vast majority ofBrazilian universities. The interface created allows the user tocreate and modify input databases that can be used to evaluatethe performance characteristics of different aircraft designs, inan easy and intuitive way. As shown in the case study, the dataoutput is also made in a simple and clear way, allowing for aquick comparison between the different configurations studied.This is a much important quality, once time is restricted in thekind of competition APT was designed to help with.

The evaluations of the tool returned by its users wereconsidered to be extremely positive. The only inconvenience,i.e., the tendency of the students to treat the software as ablack box, is being currently addressed but it seems to be partof a much larger problem: the 21st century students studyhabits. This is an issue that will certainly have to be dealtwith, as computational tools like the APT each day occupya larger share of the available tools for complex or laboriouscalculations. Our recommendation for colleagues who arethinking of creating and implementing similar tools, is toconsider this question in as an early stage of the developmentof the tool as possible.

Regarding the results obtained with the aid of APT, thetitles conquered by the Team in the competitions may beconsidered a good indicator of its success. Of course, thegroup relied on many other skills to obtain those titles, but thesoftware was certainly one important factor in the process, aspointed out by most students involved.

AcknowledgmentsThe authors would like to thank the Department of Thermaland Fluid Sciences and the Departments of Electrical andMechanical Engineering at the Federal University of Sao Joaoda Rei for the infrastructure provided. They also thank theother members of the Trem Ki Voa Micro Team for their helpand support in the 2014—2020 c ompetitions.

This work was partially supported by FAPEMIG (MinasGerais State Agency for Research and Development) undergrant TEC-APQ-03108-13.

Appendix A - Output data by type ofanalysis

The following tables show all the output variables, listed bytype and, in the case of graphical variables, the parameteraccording to which they are plotted (Depend.). Symbols thathave not been defined throughout the text will be definedbefore the table where they will be inserted.

Table 3 presents the output data of the take-off analysis,in which the symbols have the following meaning. Vinst is theinstantaneous aircraft speed during take-off; Tp is the trackposition; Vv main wind speed; Wp weight of the transportedcargo.

Tables 4 and 5 show the output data of the analysis ofstraight and level flight and climb, respectively. In whichthe symbols have the following meaning: Vdiv diving speed,

The development of an educational software for aircraft flight mechanics calculations — 11/12

Table 3. Take-off.

Output variables Unit Type Dependency Unit

Sto m Numerical - -Vinst m/s Graphical Tp mSto m Graphical Vv m/sWp N Graphical Sto m

Vman maneuver speed, V maxaut maximum autonomy speed, V max

rchmaximum reach speed, Td thrust available, Tr required thrust,Tlo leftover thrust and PA available power, PR power required,θclimb climb angle.

Table 4. Steady, level flight.

Output variables Type Dependency

Vstall Numerical -Vto Numerical -Vmax Numerical -Vcru Numerical -Vdiv Numerical -Vman Numerical -V max

aut Numerical -V max

rch Numerical -Td Graphical Vel.Tr Graphical Vel.Tto Graphical Vel.

Table 5. Climb.

Output variables Type Dependency

RCmax Numerical -(R/C)max Numerical -ϕmax Numerical -ϕmax Numerical -PA Graphical Vel.PR Graphical Vel.θclimb Graphical Vel.

Table 6 shows the output data from the curve analysis.The symbols have the following meaning: n load factor; Rcurve radius; θ scroll angle.

Table 7 shows the output data from the glide analysis,where the symbols have the following meaning. φmax,aut angleof maximum autonomy in glide; RDmax,aut descent rate formaximum gliding autonomy, glide speed: Vgli, RD descentrate; φ gliding angle.

Tables 8, 9 and 10 show the output data for the landinganalysis, payload evaluation and flight envelope, respectively.

Table 6. Level turn.

Output variables Type Dependency

nCmax Numerical -Rmin Numerical -θmax Numerical -n Graphical Vel.R Graphical Vel.θ Graphical Vel.

Table 7. Gliding.

Output variables Type Dependency

φ autmax Numerical -

RDautmax Numerical -

Vgli forRDautmax Numerical -

φ rchmax Numerical -

RDrchmax Numerical -

Vgli for RDalcmax Numerical -

RD Numerical Vel.φ Graphical Vel

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Sl Numerical -

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Output variables Type Dependency

Payload prediction Graphical Altitude

Table 10. Flight envelope.

Output variables Type Dependency

Flight envelope Graphical Altitude

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