Submitted to arXiv in August 2021 Physics Education paper The development of
an educational software for aircraft flight mechanics calculations Claudio C.
Pellegrini1*, Mateus S. Rodrigues2, Erika O. Moreira2 Educational software,
aeronautical engineering, aircraft performance, SAE Brasil AeroDesign
competition. Due to its versatility and low cost, the use of unmanned aerial
vehicles has been rapidly spreading in recent years, in applications ranging
form military operations, to land mapping, rescuing of lost people, aiding of
natural disaster victims and many others. To properly design and operate such
a vehicle, it is necessary to know its flight mechanics in the various stages
of the flight. Despite the fact that the physic behind the analysis of an
aircraft’s flight mechanics is well known and purely based on Classical
Mechanics, the large quantity of input and output data involved favors the use
of a computational toll. This work presents the development of a toolbox
called APT (Aircraft Performance Toolbox), able to make preliminary aircraft
flight mechanics calculations regarding its performance. To achieve this goal,
a number of Matlab scripts were created to perform the calculations, and a
graphical interface was created to unify them and to allow the end-user to
perform the analysis in a clear and intuitive way. To illustrate the potential
of the toolbox, we use APT to in the analysis of an UAV meant to participate
in the SAE Brasil AeroDesign competition of the year of 2014.

###### Contents

  1. 1 Introduction
  2. 2 Computational tools as a teaching resource
  3. 3 Aircraft performance
    1. 3.1 Input variables
    2. 3.2 Take-off
    3. 3.3 Climb
    4. 3.4 Steady, level flight
    5. 3.5 Level turn
    6. 3.6 Gliding
    7. 3.7 Landing
    8. 3.8 Payload variation
    9. 3.9 Flight envelope
  4. 4 Software Design
  5. 5 A case study
  6. 6 Didactic Experience
  7. 7 Conclusions
  8. Acknowledgments

##  1 Introduction

The use of unmanned aerial vehicles (UAVs) has been widely disseminated over
the past few years. This fact surely stems from its versatility: they are able
to operate in different environments, weather conditions and in situations
where human activity is considered risky or even impossible [1]. Also, UAVs
have extremely low acquisition and operating costs compared to conventional
aircraft.

Among the activities in which UAVs have been intensely used, we may cite
surveillance operations (military or not), the mapping of land use/land cover
for geographical and meteorological purposes, the control of clandestine
activities as agricultural, mineral extraction, hunting and fishing in
protected areas, the control of damage caused by natural disasters and the
search and rescue operations that follow, the search and rescue for people
lost in inhospitable places, as mountaineers and sailors, the sending of
supplies to isolated groups, and many others. In addiction, UAVs may represent
an important tool in the mapping of infectious diseases such as the new
Covid-19. UAVs can provide spatially and temporally accurate data to help
understand the links between disease transmission and environmental factors
[2].

Both the design and the operation of UAVs require analysis of the aircraft’s
flight mechanics, and in particular of its performance in various flight
situations. Such analyses determine whether the aircraft is capable of
fulfilling the objective for which it was designed. The idea is to be able to
answer questions such as: can the UAV take off and land on the stipulated
runway dimensions? Can it carry the necessary load to the prescribed distance
in the stipulated time? Can it reach the expected speed?

However, such an analysis requires input data related to the aerodynamic
characteristics of the aircraft, its propulsion system, its dimensions and
weight, the runway length available for takeoff and landing, the thermodynamic
properties of the air at the operating location, etc. In addition, it
inevitably ends up generating a large amount of output data.

Despite the fact that the physics used in the analysis of an aircraft’s
performance are well described both in the traditional and recent literature
[3, 4, 5, 6], the large quantity of input and output data involved favors the
use of a computational toll. For the authors, the need for such a tool
appeared during their participation in the SAE Brazil AeroDesign competition
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 diffusion and exchange of Aeronautical Engineering techniques
> and knowledge among students and future mobility engineering professionals,
> through practical applications and competition between teams

For the competition, students must form teams composed by engineering
undergraduates and a faculty advisor. The team must design and build a cargo
UAV to complete a mission that change every year. The competition aim is to
disseminate knowledge in aeronautical engineering, challenging students from
all over Brazil to face each other of in an event occurring every October.

On the year of 2009 the team Trem Ki Voa Micro was formed at the University
Federal of São João del Rei, with one of the authors as its academic adviser.
It made its debut at the 2010 competition and took part of every subsequent
issue of the competition, until the present year. In a decade of existence the
team involved more than 200 students from undergraduate courses of Mechanical,
Electrical and Production Engineering, many of whom now work in the
aeronautical industry. The educational benefits and risks of this kind of
competition is generally recognized by teachers and students. It is reviewed
in a number of studies ([8], [9] and [10]for example).

With the AeroDesign competition in mind, in the year of 2014 the authors set
out to obtain a computational tool to aid the team in the calculations. Their
experience showed that, in general, the computational packages developed for
performance analysis must meet the following characteristics:

  * •

They must be supported by mathematical models that allow a complete assessment
of the aircraft’s performance characteristics;

  * •

They must have a graphical interface that allows the creation or loading of a
database containing the information of the aircraft to be analyzed, and
allowing the selection 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 can be easily adapted to the user’s needs;

  * •

They must be able to be implemented in a computational environment
disseminated in the academic environment and be freely distributed.

Unfortunately, a large part of aeronautical science is developed by private
companies, that keep confidentiality about the tools developed by them or
charge their use. Thus, just a few computational packages developed for
aircraft performance analysis are presently available and the ones available
do not meet some — or even all — of the requirements mentioned. Therefore, the
authors realized that the only way to assure that the team would have access
to an efficient and complete computation tool, within an affordable cost,
would be to develop their own.

This situation is not exclusive of the aircraft performance area. Thus, [11]
for example, developed an interactive multimedia package on aircraft stability
and control to fulfill the gap and [12] did the same with an statistical
calculations package.

To further illustrate the problem, the license for the ULTRA-NAV® Performance
Software111Available at http://www.ultranav.com costs US$ 595–795, but the
package only performs take-off and landing analyses. The Atlas® and
iPreFlight® packages222Available at https://www.flyapg.com/atlas.html and
https://www.flyapg.com/ipreflight.html, respectively, developed by the APG
(Aircraft Performance Group), perform only take-off and payload analyses
respectively, and charge a “price on request” for their licenses. The Aircraft
Performance Program®333Available at http://www.darcorp.com/aircraft-
performance-program-software developed by DARcorporation fulfills most of the
requirements mentioned above, but its license costs from US$ 5,000
(intermediate version) to almost US$ 8,000 (full version).

The objective of the present article is to present the design and development
of the Aircraft Performance Toolbox (APT), created using the Matlab® (student
version) environment. It was designed to be a complete and free-to-use tool to
help students calculate the most relevant aircraft performance parameters and
help them explore the numerous possibilities associated with different
aircraft configurations. A brief account of the didactic experience with the
APT in the Trem Ki Voa Micro team is also presented.

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

##  2 Computational tools as a teaching resource

This is a topic that has been extensively covered in many studies in the last
decades, since the popularization of personal computers started. Nevertheless,
some general characteristics previously noted by some selected authors are
worth mentioning.

Experience has shown that active visualization-based learning using
interactive tolls can greatly enhance the student’s understanding and
retention. A direct interactive visual approach may help to remove many of the
conventional barriers that hinder the effective learning of abstract
engineering concepts [13]. However, computational packages and visualisation
programs are not the only possibility. Teachers have been using non-
computational tools for a long time, in a number of creative ways, including
experimental set-ups, scale models and toolkits. Extensive use has also been
made in the past of video presentations by lecturers (usually in Super 8
format) and slideshows, using the physical photographic slides.

Nevertheless, soon after the popularization of personal computers, they begin
to substitute the ”physical” apparatuses whenever possible, due to their
flexibility and low cost (a basic didactic wind tunnel can easily cost as much
as 100,000 Euros). For example, a study by [14] describes a numerical
simulation experiments being used in the late 70’s to teach basic physics at
the New York Institute of Technology. In the following decades, this tendency
accelerated noticeably and at the moment, educational software is used in
almost every area of knowledge, including human sciences.

In a study on the use of computers as a tool in the teaching of physical
sciences, [15]444In Portuguese conclude that the pedagogical potential of
computers can only be fully realized if there are enough educational programs
of sufficient quality. In this respect, many papers highlight the
characteristics that an educational software must have ([13] and [16], for
example). The most frequent adjectives are: interactive programs,
visualization-based learning, motivating, user-friendly interface, robustness,
easy of use, and, of course, free to download and free to use.

The didactic value of computational packages, in connection to the teaching of
basic physics, is summarized by [17]555Translated from Portuguese. as:

> The development of educational software allows teachers and students to
> perform simulations and create animations related to challenging problems in
> Newtonian kinematics and dynamics, both in high school and in undergraduate
> introductory Mechanics courses. Problems that do not require the student to
> be familiar with advanced mathematical methods but that allow for a wide
> variation of the relevant parameters, can be studied with great advantage.

In conclusion, the use of computer simulations and computer experiments in the
teaching of sciences is very promising. However, as noted by [13],
implementing the idea may mean modifying not only the way of teaching but also
reconsidering the educational model and philosophy.

##  3 Aircraft performance

The performance characteristics of an aircraft requires several sets of
calculations, hereafter denominated _performance analyses_ or analyses for
short. Each contains a number of computations based on Classical Mechanics,
requires a considerable amount of input data and returns an equally
considerable amount of output data (39 variables to be precise).

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

###  3.1 Input variables

The input variables of the APT can be divided into three categories: take-off
variables, geometric parameters and aerodynamic parameters.

The take-off variables category includes the air density at the take-off
location and the runway length. The first can be 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 the national weather service.

The geometric parameters category includes the wing area, the wingspan, the
height of the wing above the ground, the aircraft empty weight and the weight
of the cargo. The rolling friction between the tires and the runway must also
be known. It is modeled according to $F=\mu_{R}$ [19], where $\mu_{R}$ is the
rolling 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 between the propulsion system thrust and the velocity
of the airplane, i.e., the thrust curve. In general, it is modeled as a second
degree polynomial, whose coefficients can be obtained by curve fitting to wind
tunnel data or using the low cost method proposed in [6].

Finally, in the aerodynamic parameters category, the drag polar curve is the
most important input information. The curve is composed by values of the lift
coefficient and their associated drag coefficient values, for every possible
angle of attack, $\alpha$. The polar curve is obtained by methods widely
disseminated in the aeronautical area, such as [3] and [5]. The category also
includes the drag and lift coefficients for stall angle and take-off angle,
and the Oswald efficiency factor.

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

###  3.2 Take-off

Taking off an aircraft consists of accelerating it from rest to a speed where
the lift is greater than its weight. The required runway length is evaluated
in this phase. The lift force is obtained through the well-known relationship:

$F_{L}=\frac{1}{2}\rho C_{L}V^{2}A$ (1)

where $\rho$ is the air density, $C_{L}$ is the lift coefficient (which
depends on the angle of attack of the wing), $V$ is the speed of the aircraft
in relation to the air and $A$ is the projected wing area.

The acceleration is created by the thrust force of the engine, $T$, which must
be greater than the sum of the two dissipative forces that act on the aircraft
during take-off, namely the total drag:

$F_{D}=\frac{1}{2}\rho C_{D}V^{2}A$ (2)

and the rolling resistance,

$F_{R}=\mu_{r}(W-F_{L})$ (3)

where $C_{D}$ is the drag coefficient for the entire aircraft, $N$ is the
normal force, and $\mu_{R}$ is the combined rolling resistance coefficient of
the complete landing gear.

According to the classic aeronautical literature ([3], for example), the
dependence of the engine thrust on the aircraft’s velocity relative to the
air, for the kind of propulsion system used, can be modeled by:

$T=\alpha V^{2}+\beta V+T_{0}$ (4)

where $\alpha,$ $\beta$ and $T_{0}$ are constants specific for each propulsion
system, $T_{0}$ being known as the static thrust.

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

$V_{to}=1.2\sqrt{\frac{2(W/A)}{\rho C_{Lstall}}}$ (5)

where $(W/A)$ is the wing load, $C_{Lstall}$ is the maximum lift coefficient,
occurring at the associated $\alpha_{stall}$ angle of attack. It is to be
attained by the airplane near the end of the runaway when the pilot _rotates_
the airplane, pitching its nose up. Using this value, the runway length
required for take-off, $S_{to}$, may be obtained from the closed analytical
model proposed by [6]:

$S_{to}=\frac{W}{2gB}\Bigg{[}\ln\left|\frac{BV_{to}^{2}+CV_{to}+D}{D}\right|\\\
-\frac{2C}{\sqrt{\Delta_{1}}}\bigg{(}\\!\arctan\frac{2BV_{to}+C}{\sqrt{\Delta_{1}}}-\arctan\frac{C}{\sqrt{\Delta_{1}}}\bigg{)}\Bigg{]}$
(6)

for ${\Delta_{1}=4BD-C^{2}}>0$, and

$S_{to}=\frac{W}{2gB}\Bigg{[}\ln\left|\frac{BV_{{to}}^{2}+CV_{to}+D}{D}\right|\\\
-\frac{C}{\sqrt{\Delta_{2}}}\bigg{(}\\!\ln\frac{2BV_{to}+C-\\!\sqrt{\Delta_{2}}}{2BV_{to}+C+\\!\sqrt{\Delta_{2}}}\cdot\frac{C+\\!\sqrt{\Delta_{2}}}{C-\\!\sqrt{\Delta_{2}}}\bigg{)}\Bigg{]}$
(7)

for ${\Delta_{2}=C^{2}-4BD}>0$, where

$B=\alpha-\frac{\rho}{2}(C_{D}-\mu_{r}C_{L})S$ (8) $C=\beta-\mu_{r}C_{L}S$ (9)
$D=T_{0}-\mu_{r}W$ (10)

The last result is not found in textbooks. It was recently proposed in
substitution of the classic averaged approach, presented for example in [3],
where the necessary integration is simplified using average values of the
forces involved, calculated at 70% of $V_{to}$. Such a simplification yields
good practical results but obscures the understanding of the take-off process
and is limited by several factors [23]. The most relevant is that the
approximated integration does not yields a $S=S(V)$ relation. Nevertheless,
the value of $S_{to}$ according to [3] is also calculated by APT as a
reference.

###  3.3 Climb

The climb phase consists of raising the aircraft’s altitude. The process takes
place after the take-off so that the aircraft can reach the desired altitude.
In this analysis, the maximum climb rate, ${C\\!R}_{max}$, and the maximum
climb angle, $\theta_{max}$, are calculated using the following equations:

$C\\!R_{max}=\left(\frac{T-F_{D}}{W}\right)V$ (11)
$\theta_{max}=\arcsin\left(\frac{T-F_{D}}{W}\right)$ (12)

The toolbox also plots the curve ${C\\!R}={C\\!R}(V)$. The value of
${C\\!R}_{max}$ is obtained through the plot and does not occurs at
$\theta_{max}$.

###  3.4 Steady, level flight

In 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 equal the drag. The maximum speed that
the aircraft can reach, the cruising speed, the maximum range and maximum
range speed can be calculated by equating the values of available and required
power, $P_{A}$ and $P_{R}$, i.e, by putting

$P_{A}={TV}$ (13) $P_{R}={F_{D}V}$ (14)

The resulting dependencies $P_{A}=P_{A}(V)$ and $P_{R}=P_{R}(V)$ are presented
graphically to the user to allow a better analysis of the aircraft’s behavior
under different conditions.

###  3.5 Level turn

It is a simple curve maneuver, in which the aircraft performs a rolling
movement around its longitudinal axis, tilting its wings in relation to the
ground, while moving in a horizontal plane. The vertical component of the lift
is equal to the weight of the aircraft and the horizontal component creates
the centripetal acceleration necessary to travel along a circular path. For
this situation the maximum angle of roll and the minimum radius of curvature
are calculated, respectively, by

$\phi_{max}=\cos^{-1}\left(\frac{2(W/A)}{\rho\,C_{Lstall}V_{max}^{2}}\right)$
(15)

and

$R_{min}=\frac{2(W/A)}{\rho gC_{Lstall}}$ (16)

where $C_{Lmax}$ is the maximum lift coefficient and $V_{max}$ is maximum
aircraft speed.

###  3.6 Gliding

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

$\theta_{min}=\tan^{-1}\left(\frac{1}{(C_{L}/C_{D})_{max}}\right)$ (17)

where $(C_{L}/C_{D})_{max}$ is the maximum aerodynamic efficiency.

###  3.7 Landing

It consists in decelerating the aircraft from the speed with which it touched
the runway, to the rest. In general, thrust is considered null. Other forces
related to the use of brakes, lift spoilers or the reverse666A configuration
in which the propulsion system generates negative thrust to shorten the length
necessary for landing, are not considered here and, thus, the result is
conservative. Here, APT uses the same formulation employed for the take-off
with $T=0$, yielding

$S_{l}=\frac{W}{2gB}\Bigg{[}\ln\left(\frac{BV_{l}^{2}+D}{D}\right)\Bigg{]}$
(18)

where

$V_{l}=1,3\sqrt{\frac{2(W/A)}{\rho C_{Lstall}}}$ (19)

according to ([22]), and

$B={\frac{\rho}{2}(C_{D}-\mu C_{L})A}$ (20) $D={\mu W}$ (21)

###  3.8 Payload variation

The payload is the weight that the aircraft is capable of carrying at
different altitudes at which take-off can occur, since the variation in
altitude leads to a variation in air density, consequently altering the
thrust. the lift and the drag. This load is evaluated through an iterative
process, in which the take-off distance is repeatedly calculated for
increasing masses, until it exceeds the runway length (with a prescribed
factor of safety). The whole process is then repeated for various altitudes,
with their respective values of density. The result, given by the toolbox in
graphic form, shoes the dependence of the payload with altitude.

###  3.9 Flight envelope

The flight envelope is the region of the speed vs. altitude graph in which the
aircraft can sustain straight and level flight. As altitude increases, stall,
maneuver and minimum speeds increase, however, maximum speed decreases. The
envelope traced by taking these speeds into account, delimits the aircraft’s
operating range. In this analysis, this envelope is presented graphically.

##  4 Software Design

The Matlab® platform was the chosen development environment due to its
renowned ability to deal with large numbers of variables and to perform
calculations with large indexed variables. The free student version was
choosen.

Matlab® is an interactive environment specialized in numerical and algebraic
manipulation that uses as its basic variables matrices that do not require
dimensioning. The language is rather similar to standard mathematical writing,
which makes coding easier than in languages as FORTRAN, C and Java. Matlab®
possesses a tool to aid developing graphic interfaces called GUIDE (Graphical
User Interface Design Environment) that enables the construction of the layout
and automatically generates the associated code, that can be further 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 equation or using the command _disp_. The
graphic data output can be done via the _plot_ command, which has a very
intuitive syntax. Even after one graph has been made, the user still has wide
possibilities to change its details, altering titles, legends, axes, fitting
curves, colors, etc.

Finally, in spite of the fact that Matlab® is a commercial software, due to
its huge numerical problem solving capability and simplicity of coding, it is
widely spread in the academic community, being available at most universities
in its student edition and higher. To ensure that the didactical objectives of
the tool were fulfilled, the user interface was designed in a simple and
intuitive way. Thus, students with little knowledge of airplane performance
and even of aeronautics are able to carry out the analyses quickly and
correctly. It is only necessary that the student enter the required data and
select ”perform analysis”, and all graphs and values are calculated and
presented. Also, a large number of explanatory remarks where included in the
source code, for those with access to it. For users with only access to the
executable, the program was externally documented in an organized manner.

For the development of the toolbox, several scripts were initially implemented
to carry out the necessary performance analyzes. Each script contains the
equations necessary to perform one of the analyzes, as well as the reading
commands for the input data and writing or plotting commands for the output
data. The set of scripts works interactively to allow the output data from one
analysis to be used as input data from another.

A graphical interface was also created to allow the user to create a database
designed to feed the scripts and to make their execution easy and intuitive,
as shown in Figure 1. New databases can be created simply by filling in the
blanks with the value of the variables. A previously created bank can be
loaded via the ‘Load Database’ button. After inserting a data set, the user
has the option of saving them to a text file using the ‘Save Database” button.

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

Before the code executes the script corresponding to the analysis, it will
verify that all fields have been filled in, and if not, a message will appear
in the field for observations. The code will also check whether the indexed
variables, drag coefficient, and total aircraft lift, have the same number of
elements, a crucial detail for the calculations. If not, the user will also be
alerted in the manner previously mentioned.

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

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

If the user chooses to perform more than one analysis without closing the
existing graphs, the graphs that relate to the same quantities will be
displayed in the same window. This feature will allow the user to graphically
compare two different aircraft or to analyze the variation of the output
parameters given the change in an input parameter. This will be exemplified in
the case study presented in the following section.

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

Figure 1: Graphic interface. Figure 2: Block diagram.

##  5 A case study

APT allows the user to quickly assess the influence of changes in design data
on the performance of an aircraft by making changes to the databases. This
means that it can be used not only for the evaluation of an existing aircraft
but also during the conceptual design phase.

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

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

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

The thrust generated by the kind of engine in question increases 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, as they are connected in series and have, by default, 3.3 V each.
However, the greater the number of cells, the greater their weight.

The APT was used to compare the performance of the UAV built by the team,
using two 1300 mAh batteries, one with 3 cells in series (called the 3S
configuration) and the other with 4 cells in series (the 4S configuration),
each weighing, respectively, 148 and 195 g. For comparison, two databases were
used, differing only in the thrust coefficients and the empty weight of the
aircraft. The values were 0.833 kgf for the 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

$\text{Configuration 3S:}\begin{cases}\alpha=-0.00922\text{ N/(m/s)}^{2}\\\
\beta=-0.355\text{ N/(m/s)}\\\ T_{0}=14.387\text{ N}\end{cases}$ (22)
$\text{Configuration 4S:}\begin{cases}\alpha=-0.0110\text{ N/(m/s)}^{2}\\\
\beta=-0.427\text{ N/(m/s)}\\\ T_{0}=17.904\text{ N}\end{cases}$ (23)

The other input variables, equal for both configurations, are shown in Table
1, where $M_{P}$ represents the mass of the payload, $C_{L}^{stall}$ and
$C_{D}^{stall}$ are the lift and drag coefficient at stall angle, $C_{L}^{to}$
and $C_{D}^{to}$ are the lift and drag coefficients at take-off angle, $b$ is
the wingspan, $H_{g}$ is the height of the wing in relation to the ground,
$e_{o}$ is Oswvald’s efficiency factor and $S_{run}$ is available runway
length.

Table 1: Input variables. Variable | Unit | Value
---|---|---
$\rho$ |  kg/m3 | 1.089
$M_{P}$ | kg | 2.280
$C_{L}^{stall}$ | – | 1.418
$C_{D}^{stall}$ | – | 0.314
$C_{L}^{to}$ | – | 0.447
$C_{D}^{to}$ | – | 0.064
$\mu_{R}$ | – | 0.110
$A$ |  m2 | 0.340
$b$ | m | 1.320
$H_{g}$ | m | 0.320
$e_{o}$ | – | 0.972
$S_{run}$ | m | 61.0

The polar curve used in the calculations is shown in Figure 4.

Figure 4: drag polar curve.

Some output data for the tested configurations are shown below. Figure 5, from
the analysis of straight and level flight, shows the dependence of available
and required thrust with velocity for both configurations. It is evident that
the available thrust shows considerably higher values for the 4S
configuration, but required thrust values are only slightly larger. The
increase in available thrust is due to the increase in the battery’s output
voltage and presents a considerable gain in the change from the 3S to the 4S
configuration. The required thrust, however, does not show such a strong
dependence on the weight of the set, as shown in the figure. Thus, it is clear
that, from the aircraft’s performance point of view, the 4S configuration is
advantageous, increasing the surplus power for maneuvers and the maximum
flight speed, as seen in the table 2.

Figure 5: Available and required thrust.

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

Figure 6: Rate of climb.

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

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

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

Figure 8: Minimum turning radius.

Finally, Figure 9 shows the load capacity for both configurations. It is
evident that, due to the increase in thrust, the aircraft powered by the 4S
battery has a considerably larger load capacity than the aircraft with the 3S
battery, an increase of 11%.

Figure 9: Payload prediction chart.

Table 2 shows the main aircraft output data for various analyses. Here,
$V_{stall}$ is the stall speed, $V_{max}$ is the maximum speed for steady,
level flight, $V_{cru}$ is the cruise speed, $\varphi_{max}$ is the maximum
climb angle and $\phi_{max}^{rng}$ is the maximum range angle during gliding,
$R_{Dmax}^{rng}$ is the descent reason for maximum range during gliding,
$n_{Cmax}$ is the maximum load factor in curve, $R_{min}$ is the minimum
radius in curve, $\theta_{max}$ is the maximum roll angle, $S_{to}$ is the
runway length needed to take-off and $S_{l}$ is and the runway length needed
to land.

Table 2: Main output variables. Output variables | Unit | 3S | 4S
---|---|---|---
$V_{stall}$ | m/s | 10.78 | 10.86
$V_{max}$ | m/s | 19.20 | 19.96
$V_{cru}$ | m/s | 17.28 | 17.97
$R_{Cmax}$ | m/s | 2.06 | 2.79
$\varphi_{max}$ | $({}^{0})$ | 9.89 | 13.43
$\phi_{max}^{rng}$ | $({}^{0})$ | 6.68 | 6.68
$R_{Dmax}^{rng}$ | m/s | -1.63 | -1.65
$n_{Cmax}$ | - | 1.30 | 1.48
$R_{min}$ | m | 18.37 | 16.29
$\theta_{max}$ | $({}^{0})$ | 40.14 | 47.55
$S_{to}$ | m | 38.10 | 29.44
$S_{l}$ | m | 84.23 | 85.51

It can be seen that almost all the performance parameters indicate the 4S
configuration to be advantageous, except for the aforementioned minimum radius
of the level curve and for the parameters resulting from glide and landing
analysis, where the engine thrust is set to zero. As the 3S configuration
would receive, according to the competition rules, only a few more points due
to its lower weight, the team opted to use the 4S battery. The decision proved
to be correct and the team won the 2014 competition.

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

##  6 Didactic Experience

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

The research was carried out through the Google forms platform. Google forms
allow users to ”collect and organize information large and small for free”.
The responses to a survey are stored in spreadsheets (Google Sheets) and can
be viewed in graphs or even raw in the spreadsheet. There are different 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.

According to the 8 forms received, students think that the toolbox
considerably speeds up the laborious calculations and helps organize the
output data, making it easy to test different aircraft configurations in the
initial phase of the project. In addition, it is believed to help in the
preparation of the final report and presentations. The tool was also credited
with providing greater interaction between the areas, as it helps to organize
the input and output data.

Another interesting feature mentioned by the students is that the toolbox can
be used both in the forward and backward directions. In the forward direction,
it is used for its original purpose, calculating performance characteristics
of the UAV from the input data. In the reverse direction, it may be used to
calculate input data from measured performance characteristics.

An example of this, is the possibility of calculating the 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 diagnosis was that, as students
understand that the code is able making the calculations “on its own”, they
tend not to study the theory in depth as they should. This have lead to two
problems in the past years: (i) students do not acquire the proper
understanding about the subject; (ii) students spend a long time using the
toolbox in trail and error tests, just to arrive at conclusions that were
obvious from the theory (that they did not studied properly!).

Unfortunately, this situation seems to be commonplace in modern engineering
courses, whenever a software is available to help in the calculations, and is
rather worrying. In a research conducted more than one decade ago, [24]
describes characteristics and study habits of first-year students of Iowa
State University’s electrical engineering course. Their findings, very
consistent with our own observations, identifies 15 common traits and
behaviors of the 21st century students. One of then was particularly
associated with the problems just mentioned:

> Students do not seek to find understanding, only answers. As a result, the
> students are resentful of the confusion that arises naturally from, and is
> necessary for, the learning process to be successful – that is, to foster
> understanding, and create lifelong learners…While the answer is important,
> the understanding of its discovery, or meaning, is of little value to them —
> that one has obtained an answer is sufficient.

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

> Students lack an understanding of the learning process. In effect, the
> students are attempting to gain theoretical knowledge…merely by working
> through numerous examples. Indeed, many students seem to believe that the
> process of learning consists only of working through such examples.

And last but not least:

> Students lack an understanding of the meaning of hard work. Generally,
> unless they are captivated by a subject, they will not undertake it in
> earnest. Consequently, they are unwilling to perform the routine…necessary
> to develop their skills.

As for future versions of the toolbox, students showed interest in a module to
calculate manual launch take-offs. The development of toolboxes for other
areas of the project, with the capacity of interaction between them, was also
mentioned and is already underway. Priority was requested in the future
development of a module for a complete aerodynamic analysis toolbox, capable
of generating the polar drag curve, to remove the need for the user to enter
the data manually. An important, although obvious, detail about this request
is that it depends on the availability of dedicated members of the Team, with
adequate training in MatLab® and other necessary software. Of course, being a
member is not mandatory, but in our experience is highly recommended, as it
generally guarantee the necessary theoretical basis. Indeed, APT was developed
by one of the authors, a second year Electrical Engineering undergraduate
student, member of the performance sector.

##  7 Conclusions

We believe that APT was able to meet all the requirements stipulated
initially, including the didactic ones. The code was implemented in Matlab®,
available in the vast majority of Brazilian universities. The interface
created allows the user to create and modify input databases that can be used
to evaluate the performance characteristics of different aircraft designs, in
an easy and intuitive way. As shown in the case study, the data output is also
made in a simple and clear way, allowing for a quick comparison between the
different configurations studied. This is a much important quality, once time
is restricted in the kind of competition APT was designed to help with.

The evaluations of the tool returned by its users were considered to be
extremely positive. The only inconvenience, i.e., the tendency of the students
to treat the software as a black box, is being currently addressed but it
seems to be part of a much larger problem: the 21st century students study
habits. This is an issue that will certainly have to be dealt with, as
computational tools like the APT each day occupy a larger share of the
available tools for complex or laborious calculations. Our recommendation for
colleagues who are thinking of creating and implementing similar tools, is to
consider this question in as an early stage of the development of the tool as
possible.

Regarding the results obtained with the aid of APT, the titles conquered by
the Team in the competitions may be considered a good indicator of its
success. Of course, the group relied on many other skills to obtain those
titles, but the software was certainly one important factor in the process, as
pointed out by most students involved.

## Acknowledgments

The authors would like to thank the Department of Thermal and Fluid Sciences
and the Departments of Electrical and Mechanical Engineering at the Federal
University of São João da Rei for the infrastructure provided. They also thank
the other members of the Trem Ki Voa Micro Team for their help and support in
the 2014—2020 c ompetitions.

This work was partially supported by FAPEMIG (Minas Gerais State Agency for
Research and Development) under grant TEC-APQ-03108-13.

## Appendix A - Output data by type of analysis

The following tables show all the output variables, listed by type and, in the
case of graphical variables, the parameter according to which they are plotted
(Depend.). Symbols that have not been defined throughout the text will be
defined before 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. $V_{inst}$ is the instantaneous aircraft
speed during take-off; $T_{p}$ is the track position; $V_{v}$ main wind speed;
$W_{p}$ weight of the transported cargo.

Table 3: Take-off. Output variables | Unit | Type | Dependency | Unit
---|---|---|---|---
$S_{to}$ | m | Numerical | - | -
$V_{inst}$ | m/s | Graphical | $T_{p}$ | m
$S_{to}$ | m | Graphical | $V_{v}$ | m/s
$W_{p}$ | N | Graphical | $S_{to}$ | m

Tables 4 and 5 show the output data of the analysis of straight and level
flight and climb, respectively. In which the symbols have the following
meaning: $V_{div}$ diving speed, $V_{man}$ maneuver speed, $V_{aut}^{max}$
maximum autonomy speed, $V_{rch}^{max}$ maximum reach speed, $T_{d}$ thrust
available, $T_{r}$ required thrust, $T_{lo}$ leftover thrust and $P_{A}$
available power, $P_{R}$ power required, $\theta_{climb}$ climb angle.

Table 4: Steady, level flight. Output variables | Type | Dependency
---|---|---
$V_{stall}$ | Numerical | -
$V_{to}$ | Numerical | -
$V_{max}$ | Numerical | -
$V_{cru}$ | Numerical | -
$V_{div}$ | Numerical | -
$V_{man}$ | Numerical | -
$V_{aut}^{max}$ | Numerical | -
$V_{rch}^{max}$ | Numerical | -
$T_{d}$ | Graphical | Vel.
$T_{r}$ | Graphical | Vel.
$T_{to}$ | Graphical | Vel.

Table 5: Climb. Output variables | Type | Dependency
---|---|---
$R_{Cmax}$ | Numerical | -
$(R/C)_{max}$ | Numerical | -
$\varphi_{max}$ | Numerical | -
$\varphi_{max}$ | Numerical | -
$P_{A}$ | Graphical | Vel.
$P_{R}$ | Graphical | Vel.
$\theta_{climb}$ | Graphical | Vel.

Table 6 shows the output data from the curve analysis. The symbols have the
following meaning: $n$ load factor; $R$ curve radius; $\theta$ scroll angle.

Table 6: Level turn. Output variables | Type | Dependency
---|---|---
$n_{Cmax}$ | Numerical | -
$R_{min}$ | Numerical | -
$\theta_{max}$ | Numerical | -
$n$ | Graphical | Vel.
$R$ | Graphical | Vel.
$\theta$ | Graphical | Vel.

Table 7 shows the output data from the glide analysis, where the symbols have
the following meaning. $\phi_{max,aut}$ angle of maximum autonomy in glide;
$RD_{max,aut}$ descent rate for maximum gliding autonomy, glide speed:
$V_{gli}$, $RD$ descent rate; $\phi$ gliding angle.

Table 7: Gliding. Output variables | Type | Dependency
---|---|---
$\phi_{max}^{aut}$ | Numerical | -
$RD_{max}^{aut}$ | Numerical | -
$V_{gli}$ for$RD_{max}^{aut}$ | Numerical | -
$\phi_{max}^{rch}$ | Numerical | -
$RD_{max}^{rch}$ | Numerical | -
$V_{gli}$ for $RD_{max}^{alc}$ | Numerical | -
$RD$ | Numerical | Vel.
$\phi$ | Graphical | Vel

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

Table 8: Landing. Output variables | Type | Dependency
---|---|---
$S_{l}$ | Numerical | -

Table 9: Payload variation. Output variables | Type | Dependency
---|---|---
Payload prediction | Graphical | Altitude

Table 10: Flight envelope. Output variables | Type | Dependency
---|---|---
Flight envelope | Graphical | Altitude

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