AAE 451 Aircraft Design
Final Report
Team 4
Sean Bhise
Kyle Brite
Philip Catania
Thomas Horan
Timothy Ma
Jason Wirth
Code of Ethics
Taken from the Purdue University Handbook, Student Code of Honor:
“The purpose of the Purdue University academic community is to search for truth and to
endeavor to communicate with each other. Self-discipline and a sense of social obligation within
each individual are necessary for the fulfillment of these goals. It is the responsibility of all
Purdue students to live by this code, not out of fear of the consequences of its violation, but out
of personal self-respect. As human beings we are obliged to conduct ourselves with high
integrity. As members of the civil community we have to conduct ourselves as responsible
citizens in accordance with the rules and regulations governing all residents of the state of
Indiana and of the local community. As members of the Purdue University community, we have
the responsibility to observe all University regulations.
To foster a climate of trust and high standards of academic achievement, Purdue University
is committed to cultivating academic integrity and expects students to exhibit the highest
standards of honor in their scholastic endeavors. Academic integrity is essential to the success of
Purdue University’s mission. As members of the academic community, our foremost interest is
toward achieving noble educational goals and our foremost responsibility is to ensure that
academic honesty prevails.”
The members of Team 4 agree with and uphold to the above Code of Ethics and maintain
that the information contained within this report is original unless otherwise referenced.
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Table of Contents
1. Executive Summary .............................................................................................................. 1
2. Aircraft Introduction ............................................................................................................ 2
3. Mission Requirements, Concept Selection, Initial Sizing .................................................. 3
- Jason Wirth; Sean Bhise; Philip Catania
3.1. Mission Requirements ..................................................................................................... 3
3.2. Concept Selection ............................................................................................................ 3
3.3. Initial Weight Estimation ................................................................................................. 4
3.4. Constraint Diagrams ........................................................................................................ 5
4. Structures and Weights ........................................................................................................ 6
- Timothy Ma
4.1. Introduction...................................................................................................................... 6
4.2. Material Properties........................................................................................................... 6
4.3. Weight Determination ..................................................................................................... 7
4.4. Geometric Layout of Wing Structure .............................................................................. 8
4.5. Analysis of Wing Loads .................................................................................................. 8
4.6. Landing Gear Configuration ............................................................................................ 9
5. Aerodynamics ........................................................................................................................ 10
- Thomas Horan; Sean Bhise
5.1. Introduction...................................................................................................................... 10
5.2. Lift Production ................................................................................................................. 11
5.3. Drag Minimization........................................................................................................... 12
5.4. Wing Design .................................................................................................................... 13
5.5. Stability ............................................................................................................................ 13
6. Propulsion .............................................................................................................................. 14
- Philip Catania
6.1. Introduction...................................................................................................................... 14
6.2. Propeller Selection ........................................................................................................... 14
6.3. Motor/Gearbox Selection................................................................................................. 16
6.4. Speed Controller Selection .............................................................................................. 17
6.5. Battery Selection .............................................................................................................. 17
7. Dynamics and Controls ........................................................................................................ 18
- Jason Wirth
7.1. Introduction...................................................................................................................... 18
7.2. Tail Surface Sizing .......................................................................................................... 18
7.3. Control Surface Sizing ..................................................................................................... 21
7.4. Roll Mode Approximation ............................................................................................... 21
7.5. Gain Selection .................................................................................................................. 21
7.6. Flight Characteristics ....................................................................................................... 22
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8. Economics .............................................................................................................................. 23
- Kyle Brite
9. References .............................................................................................................................. 25
APPENDIX A – Initial Sizing and Concept Selection ............................................................. 26
APPENDIX B – Structures and Weights .................................................................................. 33
APPENDIX C – Aerodynamics ................................................................................................. 37
APPENDIX D – Propulsion ....................................................................................................... 48
APPENDIX E – Dynamics and Controls .................................................................................. 58
APPENDIX F – Economics / Project Management ................................................................. 81
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1. Executive Summary
The purpose for this project is to design, build, and successfully fly an aircraft within the
confines of the Purdue Armory. This project is intended to demonstrate to Dr. Andrisani that an
interdisciplinary team of senior-level aeronautical engineering students can successfully design
an aircraft that meets all mission requirements. There are several mission requirements that must
be met for this aircraft to be considered successful. The aircraft must navigate through several
phases of flight before ultimately pitching up into a hover maneuver.
A roll rate gyro will be employed for this project to successfully maintain the hover. This
gyro will assist the pilot in maintaining a steady roll angle while allowing him to keep the
aircraft in a nose up configuration within the confines of the Purdue Armory. Team 4’s aircraft
uses an approximation of the roll mode to determine a nominal gain for the rate gyro..
The philosophy behind Team 4’s aircraft is a simple to build, easy to repair aircraft. The
aircraft that has been designed uses a large rectangular wing with a NACA 6412 airfoil in order
to minimize stall and cruise speed. The optimum cruise speed for our aircraft is 19.5 ft/s, which
allows an experienced pilot more than enough time to turn and avoid any obstacles inside of the
Armory. Since this aircraft is being flown indoors, a considerable amount of the design reflects
those considerations. Team 4 feels that our large, slow flying aircraft will be more controllable
in an indoor environment than a smaller and faster aircraft.
Another mission requirement for this aircraft is to be built on an operating budget of
$150. After careful evaluation and consideration, Team 4 found this to be insufficient for the
mission that the aircraft is required to perform. With an eye on costs, Team 4’s entire parts
budget was kept under $200. As this aircraft is potentially being marketed to a teenage audience,
the marketability of the aircraft is another concern. With this in mind, Team 4’s aircraft will use
a bright and bold paint scheme to increase possible sales.
The aircraft that Team 4 designed is anticipated to meet all mission requirements. The
construction will be simple and easily repairable to prevent extended periods of downtime in the
event of a crash. The aircraft is designed to be sufficiently maneuverable to be able to perform
aerobatic maneuvers without significant effort by the pilot.
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2. Aircraft Description
Figure 1 - Aircraft 3-View
Wing Span
Wing Area
Wing Chord
Aspect Ratio
Dihedral
Fuselage Length
Aircraft Weight
Horizontal Tail Area
Vertical Tail Area
4.8
4.8
1
4.8
3
3.75
1.94
1.75
0.35
ft
ft2
ft
o
ft
lb
ft2
ft2
Propellor Size
Motor Power
Battery Cell Type
Battery Cell Number
Battery Power
Battery Voltage
Gearbox Ratio
Feedback Controller Axis
Gear Type
13 x 6.5
150 W
Lithium Polymer
1x3
1320 mAh
11.1 V
2.5:1
roll
taildragger
Table 1 - Aircraft Specifications
2
3. Mission Requirements, Concept Selection, and Initial Sizing
3.1 Mission Requirements
There are several mission requirements that the aircraft must meet. Per the Request For
Proposal (RFP), the aircraft must be able to be flown within the confines of the Purdue Armory.
It must also be robust to crashes and be easy to fly. It must meet all Level IA military flying
qualities. The aircraft must be built for less than $150. During takeoff, the total ground roll
must not exceed 10 feet. During the climb stage, the aircraft must perform a 45° banked turn. It
must then loiter for a period of three minutes while maintaining a stall speed of no greater than
15 ft/s. The aircraft must then perform a pitch up maneuver and maintain a ±20° roll angle while
in a hover for a period of two minutes. To assist the pilot during this maneuver, a roll rate
feedback gyro will be used to assist in aileron control. Finally the aircraft must be demonstrated
to land safely.
3.2 Concept Selection
After considering the requirements set forth in the RFP, all team members were asked to
come up with an initial conceptual design for an aircraft capable of completing all aspects of the
mission. Three designs were then chosen by the team as finalists. These three designs included
a conventional single-engine aircraft, a twin-engine aircraft, and a single-engine design with a
dual-boom fuselage with integrated vertical tails that resembled a “double extruded plus-sign”.
Three-view drawings of these designs can be seen in Appendix A.
The next step in determining the final concept was using Pugh’s Method to compare
pertinent design variables and qualities. Concept 1, which was the conventional single-engine
aircraft, was chosen as the baseline because it is the most common design of the three. Concepts
2 and 3 were then compared with Concept 1 in each of the areas shown in Table , and even
though Concept 3, the “double plus-sign”, shows a final value of +2, a decision was made that
the complexity was not weighted heavily enough and would not be worth the benefits gained in
other areas. This ended up ruling out both Concepts 2 and 3, leaving Concept 1, with some
appropriate modifications, as the design of choice for Team 4.
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Design 1 Design 2 Design 3
Wing Area
Roll Control
Aspect Ratio
Cost
Weight
Cruise Speed
Stall Speed
Durability
Complexity
Repairability
+
S
S
+
S
+
S
S
2
4
4
+
S
S
S
+
+
+
S
4
2
4
Table 2 - Pugh's Method Comparison of Three Final Conceptual Designs
3.3 Initial Weight Estimation
To initially estimate the weight of this aircraft, a historical approach was employed.
Team 4 used a database of 10 commercial radio controlled (R/C) aircraft and found a linear
relationship between required battery weight and total weight for the aircraft. The table and
resulting plot are located in Appendix A. However this trend line only gives data for historical
r/c aircraft which may not have the same mission requirements as those given in the RFP. To
determine an accurate estimation of the battery weight required for the mission given for this
project, MATLAB code was used to analyze each phase of the flight. Detailed analysis for the
estimation of battery fraction necessary for each flight phase is located in Appendix A.
4
Weight estimation using historical weight data
0.8
Historical data
Estimated weight
0.7
W-We and Wb+Wp~lbf
0.6
0.5
0.4
0.3
0.2
Estimated aircraft weight is 1.8623 pounds.
Estimated battery weight is 0.20854 pounds.
0.1
0
Payload weight is 0.25 pounds.
0
0.5
1
1.5
Weight~lbf
2
2.5
3
Figure 2 - Aircraft Weight Estimation
3.4 Constraint Diagram
The constraint diagram shown below allowed us to pick a feasible design point for the
aircraft. The condition that constrained the aircraft the most was the hover condition which can
be seen on the graph as the horizontal lines. The loiter and cruise phases of the mission had no
effect on picking a feasible design point. As our simulations showed the CLmax = 1.66 for the
NACA 6412 airfoil, the design point was chosen to be at a CLmax = 1.61 in order to have a small
buffer area. The stall constraints can be seen as the vertical lines in the graph. The results from
choosing this design point yielded a wing loading of 0.405 lb/ft2 and power loading of 15 lbf/hp
at Vstall = 15 ft/s.
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Constraint Diagram
30
25
Power loading (lbf/hp)
20
Hover
Constraint
15
Design
Point
CLmax : 1.60 1.61 1.62
10
5
0
0.39
0.395
0.4
0.405
0.41
0.415
Wing loading lbf/ft 2
Figure 3. Plot of Constraint Diagram
4. Structures and Weights
4.1. Introduction
Within the mission specifications, the requirements that set parameters for the structural
aspect of the aircraft are that it must be robust to crashes, lightweight, easy to construct, and
inexpensive. Preliminary research on various model R/C aircraft, specifically in the Park Flyer
aircraft category, was done to enhance the understanding of the weights and typical structures
that should be expected. After collecting the data, the aircraft could be designed with an
approximate weight and model in mind. Considering the requirements for the mission, extruded
polystyrene (EPS) foam will be a major material used to compose the main structural member of
our aircraft. Compiling the historical knowledge of model aircraft, researching different design
philosophies, and performing different structural analyses, the structure of the aircraft can be
designed.
4.2. Material Properties
The material properties chosen for the aircraft needed to be lightweight, durable, easily
workable in construction, and relatively inexpensive. The main materials of the aircraft are
extruded polystyrene (EPS) foam, aluminum, and film covering. The properties of the main
materials used in the aircraft are summarized below in Table 3.
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Material Properties
EPS Foam
1.8
Aluminum
0.098
Corregated Plastic
14.02
MonoKote
0.0021
lb/ft^3
lb/ft^3
lb/ft^3
oz/in^2
Table 3 - Moments and Products of Inertia
Most aircraft in the Park Flyer category of R/C aircraft are constructed out of foam,
heavily influencing the design to be composed of foam. Since the wing is the primary structure
of our aircraft and is approximately a third of the weight, using EPS foam will help assist with
weight concerns. The qualities that EPS foam possess that it is rigid, lightweight, easily shaped
into the needed airfoil contour, and inexpensive. A film covering will be used as a wing skin to
assist in keeping the rigidity and shape of the wing and also to help prevent punctures, dents, and
scrapes in the EPS foam. The aluminum will be mainly used in the aircraft for the fuselage and
ribs for the fairing. An aluminum square tube for the fuselage will provide a strong foundation
for all the other aircraft structures to attach to. The thin sheet aluminum will be used to construct
the ribs for the fairing and wing connectors (images of these can be seen in Appendix B). The
aluminum fairing ribs, which holds all the necessary and vital propulsion and control
components, and EPS foam covering will provide a sturdy structure to house and protect the
parts during flight conditions and during a rough landing or crash. The horizontal and vertical
tails will be constructed from corrugated plastic. Images of the horizontal and vertical tail can be
seen in Figure 24 and Figure 25 in Appendix B.
4.3. Weight Determination
An analysis was done to approximate the weight of the aircraft for use in preliminary
calculations and design. The approximation was based on historical data of other commercially
sold R/C aircraft in production today. In determining the center of gravity (CG) location, a
tabular listing of weights and location of all parts of the aircraft was used and is provided below
inTable 4 . The weights are calculated using volume and material density properties or from
research done on commercial websites. To calculate the actual CG location, equation 1 in
Appendix B was used.
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Location from
Propeller (in)
0.00
0.50
1.13
8.00
8.00
12.00
12.00
13.00
11.00
12.00
22.50
42.50
40.00
Propeller
Motor
Gear box
Speed Controller
Batteries
Wing
Fairing
Servos (4)
Receiver
Rate Gyro
Fuselage
Vertical Stablizer
Horizontal Stablizer
Weight (lb)
0.04
0.14
0.13
0.04
0.13
0.60
0.17
0.08
0.06
0.06
0.37
0.02
0.14
Table 4 - Tabular Listing of Weights and Locations of Aircraft Parts
The aircraft weight is 1.94 pounds and the CG location of the aircraft is 14.17 inches
from the nose. The CATIA computer drafting program was used in calculating the moments and
products of inertias of the aircraft, which are listed in Table .
Moments of Inertia
Products of Inertia
2
Ixy
0 slug*ft2
3
Ixz
0 slug*ft
4
Iyz
0 slug*ft
Ixx
0.038 slug*ft
Iyy
0.044 slug*ft
Izz
0.081 slug*ft
3
4
Table 5 - Products and Moments of Inertias
4.4. Geometric Layout of Wing Structure
The primary geometry for the wing will be solid EPS foam using a film covering as a
wing skin. No other wing structures such as spars, stringers, or ribs are needed. As the aircraft is
lightweight and rigid, it will not see excessive loads that will warrant extra wing structures. The
wing will have a span of 4.8 feet and a chord length of 1 foot. It will be shaped into a NACA
6412 airfoil using a hot-wire CNC machine. In order to help with stability and control, the wing
will also have dihedral of 3 degrees.
4.5. Analysis of Wing Loads
From calculations done by other sections of the team and using equations 7-9, a V-n
diagram was created (Figure 4). The V-n diagram represents load factor versus velocity. The
FAA has defined the range of load factor for aerobatic aircraft from -3 to 6 g’s. From the
maximum lift curve, at the load of 6 g’s the velocity is approximately 35 ft/s. The wing load is
modeled using an elliptical load distribution approximation (Equation 8) and can also be seen in
Figure 8, both in appendix B. From that, to calculate the maximum root bending moment, a
8
maximum load of 6 g’s was used to determine lift (Equation 8 in appendix B) and calculating the
distance to the centroid of a quarter ellipse (Equation 7 in appendix B) where the load is
assumed. This gives a moment of 5.91 ft-lbs (Appendix B, equation 4). Using Equation 9 in
appendix B, the bending stress is 19.4 psi. Although this number seems high, loads of this
magnitude are not expected to be experienced by the aircraft. Furthermore, the calculations did
not take into account the film covering, which will help with the bending moments and stresses
of the entire wing. From these calculations, it directed the structural development of the aircraft.
V-n Diagram
8
6
Load Factor (g)
4
2
0
-2
-4
0
5
10
15
20
25
Velocity (ft/s)
30
35
40
Figure 4 - V-n Diagram
4.6. Landing Gear Configuration
The design strategy for the landing gear is to survive multiple landings, especially ones
which are rough and unplanned. The landing gear design is referenced from Raymer as seen
from in Figure 5.
9
Figure 5 - Taildragger Landing Gear Diagram from Raymer
The tail-down angle should be between 10-15˚ with the gear in the static position and
aircraft is modeled with a 12.5˚ angle. The CG should fall between 16˚-25˚ back from the
vertically measured main wheel location and the aircraft is designed with a 21˚ placement. If the
CG is too far forward, the aircraft will nose over and if it is too far back it will groundloop. The
main wheels must be laterally separated beyond 25˚ off the CG to prevent the aircraft from
overturning and the aircraft has a 35˚ separation. Also, a one inch clearance from the ground to
the propeller is added in to prevent a prop-strike when the tail begins to lift during takeoff. The
tail wheel will also be steerable as it is directly connected to the rudder.
5. Aerodynamics
5.1 Introduction
The mission requirements for our aircraft dictated the final design every step of the way.
Of the given requirements in the RFP, those with aerodynamic consideration are:
-
Slow flight with Vstall ≤ 15 ft/s
Short takeoff ability with takeoff ground roll ≤ 10 ft
Climbing right-angle following takeoff with γmin ≥ 45˚
Aircraft must be easily controlled and able to fly within the Purdue Armory
From these considerations, several specific design areas must be addressed. The main areas of
concern are the lift production, drag minimization, and the stability & control characteristics.
These design areas, combined with the requirements as stated above, will drive the design and
subsequent performance of the aircraft. The overall number of constraints will be minimized as
best possible in order to leave a wide design area.
During the initial sizing analysis, it became very apparent that the stall speed requirement
was going to require a high CLmax in order to be able to perform at such low speeds and still have
control over the aircraft. So while the design needs to be able to produce a large amount of lift,
drag needs to be minimized in order to curtail weight and power expenditures with the
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propulsion system. This will, in turn, reduce the overall aircraft weight, another very important
aspect of the design.
As this aircraft is being designed to be marketed and sold to teenagers, it must exhibit
good flight characteristics as well. Several aspects of the final design were specifically included
to maximize aircraft handling, and as a result, they have a direct aerodynamic response.
5.2 Lift Production
The center of lift production lies in the airfoil selection. By selecting the proper airfoil
given the aircraft’s mission, lift will be maximized while drag is subsequently reduced. It will
also allow the design to meet the requirements of a 15 ft/s stall speed, in addition to being able to
achieve flight after a fairly short ground roll. As a result of those two requirements, airfoil
selection was driven mainly by the Cl max, which would as a result, dictate the size and weight of
our wing. Before any airfoil analysis could be performed, the aircraft’s operating conditions
were studied. A Reynolds number (Re) for the flight conditions of 119,000 was calculated using
atmospheric conditions, aircraft operating speed, and wing chord length (Equation 13). After
calculation and subsequent research, it was determined that our aircraft would be operating in the
low Reynolds number regime (generally described as Re < 800,000).
Operating in the low Reynolds number regime introduces its own set of design
challenges. Namely, the flow tends to want to resist the transition to turbulent flow and remain
laminar. As the aircraft will have a relatively low operating velocity, there tends to be a
relatively adverse pressure gradient at the boundary of the airfoil surface and the air. The
resulting interaction tends to lead to a less “full” velocity profile, as compared to one from a
turbulent boundary layer. This can cause flow separation, which will cause a loss of lift and a
sharp increase of drag.
In the pursuit to find the optimal airfoil, several avenues were investigated. The airfoils
of several aircraft with similar missions were considered, in addition to experimental high-lift
airfoils (some with a Cl max purported to be over 2.0). From the large field of potential airfoils,
two airfoils were selected for further analysis: the NACA 6412 and the Selig & Donovan 7062
(as shown in Figure 6). The driving factor for the airfoil selection came down to lift production
and ease of construction. Some of the higher-lift airfoils require manufacturing tolerances
greater then what our building abilities allow, and as a result, were removed from consideration.
After generating airfoil data in XFOIL, a panel-based flow simulation program, we then
analyzed the lift curve plot and the drag polar (Figure 7) for those two airfoils.
Figure 6 - Airfoil Candidates
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1.8000
0.3
1.6000
0.25
1.4000
1.2000
0.2
Cl
Cd
1.0000
0.15
0.8000
0.6000
0.1
0.4000
0.05
0.2000
0.0000
-10.000
-5.000
0
0.000
5.000
10.000
15.000
20.000
25.000
0.0000
0.5000
alpha (degrees)
NACA 6412
1.0000
1.5000
2.0000
Cl
SD 7062
NACA 6412
SD 7062
Figure 7 - Airfoil Candidates Lift Curve & Drag Polar
The NACA 6412 was selected for its good lift characteristics and low drag response at high
angles of attack. From the constraint diagram, a total wing area of 4.8 ft2 is required using this
airfoil.
From the beginning, the largest area of concern has been the aircraft’s ability to meet the
stall speed constraint. Computer models of this scenario are difficult to create and produce
questionable results at best, mainly due to the flow separation. Therefore, Raymer’s method for
predicting wing lift coefficient was used (Raymer, “Aircraft Design”).
5.3 Drag Minimization
The elimination of drag was a facet of the design from the beginning stages. Although
we are unable to do the fine-tuned drag optimization using CFD codes and wind tunnels as used
in industry, we still tried to model the drag characteristics of the aircraft as best as possible.
Every aspect of the flight profile is influenced in some way by the drag produced by the aircraft.
The aircraft drag was calculated from Nicolai:
𝐶𝐷 = 𝐶𝐷𝑚𝑖𝑛 + (𝐾 + 𝐾 ′ )(𝐶𝐿 − 𝐶𝐿𝑚𝑖𝑛 )2
Equation 1
Where the total drag is a sum of the parasite drag (𝐶𝐷𝑚𝑖𝑛 ) and the drag produced by lift.
The parasite drag is a result of the whole aircraft body, and is equal to the sum of all the wetted
surface areas, multiplied by the coefficient of skin friction and form factors for each piece of
geometry on the aircraft. The skin friction is based upon the Re, which is derived from the
aircraft flight conditions. Due to a majority of the aircraft sitting in the slipstream from the
propeller, all airflow over the aircraft surfaces was assumed to be turbulent. In order to decrease
its own parasite drag contribution, the cylindrical size and shape of the fuselage was optimized as
part of a trade study (see Appendix C.4). The wing tips will also be optimized accordingly as
well during the build.
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The wing was designed in such a way as to ensure elliptical span-wise lift distribution, in
order to minimize induced drag. In order to account for any non-elliptical results, an Oswald’s
efficiency factor was calculated as well (Equation 14). Using the equation, an e = .9073 was
calculated, higher than the historical average for small, high-wing aircraft, but in line with values
produced by other teams in the design class.
As the aircraft flies, its trim condition is always changing due to differing flight
characteristics. Therefore, the produced drag is always changing as well, from the downwash
produced by the horizontal tail countering the pitching moment of the wing. Taking this into
account during our pitch stability analysis, a MATLAB script was used to generate the whole
aircraft drag polar (Figure 28).
5.4 Wing Design
Once a required wing area was generated, the wing design was started. The aspect ratio
was calculated to find the best possible compromise between induced drag and the wing root
bending moment. An aspect ratio of 4.8 was chosen. As a result of our required wing area, this
sets the wing chord at a constant 1 foot, due to our wing being untapered. Although wing taper
does produce a more elliptical lift distribution, analysis indicated that it was unnecessary due to
our wing size.
5.5 Stability
The aircraft has been designed to be stable and capable of handling aerobatic-type
maneuvers. The most important task that was taken into account when designing the aircraft was
the hover condition that had to be met. In order to remain stable in the hover, the aircraft has
large ailerons which will allow for more control since they will be in the prop wash.
In order to meet a minimum static margin of 10-20% MAC for our aircraft, the
aerodynamic center must be behind the CG. In order to meet these conditions, our wing was
placed 6 inches aft of the nose of the aircraft in order to move the aerodynamic center towards
the back of the aircraft. The aircraft also features a larger horizontal tail area of 1.75ft2 for this
same reason. The size of the horizontal tail was determined by a Class II sizing analysis which
will be further discussed in the Dynamics and Controls section. Using Equation 30 located in the
appendix for aerodynamic center, our static margin was calculated to be 14% of the mean
aerodynamic chord (MAC).
The CG of the aircraft was determined to be at 32% MAC as described in the Structures
section of the report. Using this CG, the trim conditions for the aircraft were able to be
determined using the trim diagram shown in Figure 8. The elevator deflection angle at αstall = 11°
is -7° with a deflection range from -1° to -7° and a horizontal tail angle of incidence = 0°. The
aircraft will be able to trim at stall since we will be able to achieve a CL = 0.98 with the CLmax of
the aircraft equal to 0.92. The CLmax determination of the aircraft was explained earlier in the
section.
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Figure 8 - Aircraft Trim Diagram
6. Propulsion
6.1
Introduction
The mission specification set out several important parameters that had to be met while
determining the components of the propulsion system for the aircraft, primarily the restriction to
using a battery-powered electric motor. This limited the propulsion system to a propeller,
electric motor and gearbox, speed controller, and batteries. Another major constraint that had to
be taken into account was that the aircraft had to be able to transition out of straight and level
flight after 3 minutes into a hover and hold there for 2 minutes before landing. The hover phase
of the flight ended up being the major contributor in determining the power required for the
mission.
The first parameter to be decided upon before the next steps of the analysis was that of a
cruise velocity. This was determined from the provided constraint of a maximum stall speed of
15 ft/s. The assumption was made to use this limiting case of 15 ft/s when calculating a cruise
speed. Cruise speed was then taken to be 130% of the stall speed, resulting in a cruise speed of
19.5 ft/s.
6.2
Propeller Selection
All of the propeller analysis for this project was done using two provided MATLAB
scripts, both of which are provided in Appendix D. The first uses Goldstein’s method of
propeller design to determine advance ratio and the thrust and power coefficients for a given
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propeller geometry, which in this case was assumed to be that of a Clark Y airfoil because of its
common use in propellers for small model aircraft. The second script uses parameters such as
cruise speed, aircraft weight, and airfoil geometry to step through the entire propulsion system
and determine power and other requirements for each component. After the aerodynamics team
selected the airfoil shape of the main wing, the necessary parameters were input into the two
scripts. Over 50 different propeller diameter and pitch combinations, with diameters ranging
from 6 inches to 14 inches and pitches ranging from 3 inches to 14 inches, were analyzed to
determine efficiency, power required, RPM, and endurance. An equation calculating power
required for the hover phase had to be determined and then inserted into
Main_System_Design.m, since it was initially optimized for cruising flight and not hover. The
equation used to determine the power required during the hover phase was the disc loading
equation derived in helicopter theory. It is listed as Equation 2 below.
𝑊
𝐷 2
𝑃𝐻𝑃 =
𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝜋 ( )
2
2𝜌𝐴
550 ∗ 𝜂𝑃 √ 𝑊
Equation 2
Endurance (min)
Once this was done, the propellers all had to be run through the scripts again to get a
more accurate power requirement and therefore a more accurate sizing of all of the components
of the propulsion system. After compiling all of the data, it was seen that a larger diameter and a
smaller pitch both led to increased endurance, allowing for a smaller battery to be selected.
Hover Endurance vs. Propeller
Diameter for P/D = 0.5
8
7
6
5
4
3
2
1
0
7
9
11
13
Propeller Diameter (in)
15
Figure 9 - Hover Endurance vs. Propeller Pitch for 13 in. Propeller
15
Endurance (min)
Hover Endurance vs. Pitch for 13 in.
Propeller
6.35
6.30
6.25
6.20
6.15
6.10
6.05
6.00
5.95
5.90
5
7
9
11
13
15
Propeller Pitch (in)
Figure 10 - Hover Endurance vs. Propeller Pitch for P/D = 0.5
Two final factors had to be taken into account, the first being the fact that a larger
propeller would provide a larger area of prop wash flowing over the ailerons, which would assist
the pilot in keeping the roll angle of the aircraft steady during hover. The second factor was that
most motors in the power range required for the aircraft could not physically handle some of the
larger propellers considered for this mission. This second fact, coupled with a recommendation
from Sean Henady, one of the test pilots, to strictly follow the propeller size range limitations
listed for the motor, caused some of the larger propeller options to be discarded. The size
limitations also factored in to the motor selection, which will be discussed below.
All of these factors together led the propulsion team to choose a 13”x6” APC Sport
Propeller; however, after a discussion with the pilots, it was determined that the chosen propeller
was too heavy, since it had been made for gas-powered aircraft. After a search online, a
13”x6.5” APC Thin Electric propeller was chosen for Team 4’s aircraft since it was the closest
geometry to the original chosen propeller.
6.3
Motor and Gearbox Selection
The next step in the design process for the propulsion system was to step backwards from
the propeller through the gearbox and motor. Using the selected propeller motor data provided
in a subroutine of Main_System_Design.m, along with the calculated input power to the
propeller, the power, voltage, and current required to the motor were calculated, effectively
providing minimum values that must be met when choosing a motor and gearbox. These values
were 0.143 hp, 7.8V, and 17.7A. For the chosen propeller size of 13”x6.5”,
Main_System_Design.m calculated the optimal gear ratio to be 2.1:1; however, the most
common gear ratio in that range was 2.5:1, which yielded the same input power of .143 hp, while
decreasing the current to 15.3A and increasing voltage to 9V.
16
Because of the limited number of motors available and the propeller size range
limitations previously mentioned, Team 4 chose to utilize a 150W (0.201 hp) motor, even though
only 106W (0.143 hp) is necessary for hover according to the results of Main_System_Design.m.
This is due mostly to the fact that the 110W motors cannot physically handle a propeller large
enough to produce a satisfactory amount of prop wash over the ailerons, which would result in
an inability to damp out the roll rate created by the motor during hover. 110W motors would
also not produce enough extra thrust to accommodate additional construction weight or give the
pilots the extra maneuverability they had asked for.
The other pertinent parameter to be mentioned at this point is the thrust generated by this
setup. With the chosen 13”x6.5” propeller, this propulsion setup generates roughly 4.5 lbf of
thrust, which produces a thrust-to-weight ratio of approximately 2.3 at full power, which is well
above the pilot recommendation of a thrust-to-weight ratio of approximately 1.3. This was
calculated with Error! Reference source not found. aboveusing maximum output power of the
motor, propeller area and propeller efficiency. Solving for weight yielded the maximum weight
that this motor-propeller combination could hold in a hover, which is equivalent to the maximum
thrust produced. The weight estimate of the aircraft does not take into account the extra weight
that will be present in the form of adhesive, wiring, the motor mount, and other small elements of
the aircraft not considered as part of the weight thus far. By estimating that up to an additional
0.5 lbf of weight, per the test pilots’ suggestion, will be added, the thrust-to-weight ratio drops to
1.8.
The final choice for the motor is the Himax HC2812-0850 150W Electric Brushless
Outrunner Motor, and the final gearbox choice is the Electrifly GD-600 Electric Flight Gear
Drive 2.5:1.
6.4
Speed Controller Selection
One of the values in the output from Main_System_Design.m was the input current to the
motor, which in this case is 17.7A. The smallest electric speed controller that could be found
that could handle at least 17.7A was a 25A speed controller. The one chosen for this aircraft was
the Castle Phoenix 25 ESC because it was sold in a bundle with the motor that was chosen to
power the aircraft and therefore was guaranteed to be able to safely handle the current
requirements of this motor and was provided free of charge by the test pilots themselves.
6.5
Battery Selection
After running Main_System_Design.m for the chosen propeller size and cruise speed,
several battery properties were given in the output, including the voltage required and the
endurance. This endurance value assumed that the aircraft was hovering for the entire mission,
which was a limiting case for this analysis since the hover phase uses more energy than any other
phase of the mission. Main_System_Design.m was also run assuming that the aircraft would be
cruising the entire time instead of hovering in order to determine the energy necessary to cruise,
17
since that would be the phase of the mission in which the aircraft would be spending the most
time.
The given battery data included in one of the subroutines assumed that the plane would
be powered by an n x m array of lithium polymer battery packs of a given voltage and energy,
where n is determined by the energy required and m is determined by the voltage required. The
voltage and energy values were varied using different common sizes of battery pack. After
running Main_System_Design.m, the required voltage was output as 9V. The two battery pack
options given were a 1x1 array, yielding 6.3 minutes of endurance in hover or 32 minutes in
cruise, and a 2x1 array, yielding 12.6 minutes of endurance in hover or 64 minutes in cruise. It
was decided that the energy required for 6.3 minutes of endurance in hover would be more than
adequate to complete the entire mission since the energy required for hover was roughly a factor
of five higher than the energy required for the same time in cruising flight. Therefore, the 2
minutes of hover would still leave roughly 68% of the battery life for all other mission phases,
allowing for 18 extra minutes of cruise after the loiter and hover phases of the mission were
completed. After comparing multiple batteries online at www.towerhobbies.com and
www.rctoys.com, decision was made to purchase the Thunder Power RC Pro Lite 1320mAh
11.1V 3 Cell Li Poly 3S battery pack. This provided the necessary energy and voltage and
stayed under the maximum current allowed by the speed controller.
7. Dynamics and Controls
7.1 Introduction
7.1.1 Stability Requirements
There are several dynamics and controls requirements for this aircraft. Of primary
concern is to ensure that the aircraft is stable in all relevant flight modes while also meeting
control anticipation parameter requirements. This is done first by ensuring that the aircraft has a
large enough static margin to ensure stability. The derivation of the static margin is found in
section 5.5. In addition to this, the aircraft must meet all Level IA flying qualities for military
and civilian aircraft as defined in Roskam. Specific information regarding these flight modes is
located in section 7.6. Team 4’s aircraft is found to meet all requirements for Level IA flight.
7.1.2 Dynamic Approximation
Another requirement that must be fulfilled is to correctly model the dynamic properties of
the aircraft. Per the RFP, the aircraft must be able to hold itself in a hover while maintaining a
±20˚ roll angle. A pilot would have a difficult time of being able to control the roll while the
aircraft is in the hover configuration so a roll rate gyro is installed on the aircraft to assist the
pilot in maintaining roll angle. A nominal gain is chosen and set on the rate gyro before the
aircraft takes off. The gain is selected by using a roll mode approximation for the hover as found
in section 7.4.
7.2 Tail Surface Sizing
7.2.1 Class I
Class I sizing for the horizontal and vertical tails of the aircraft is done by using a method
as described in Raymer. This method uses the tail volume coefficient in conjunction with the
moment arm from the wing to the horizontal or vertical tail as well as the surface size of the tail.
18
To determine a sizing estimate typical values for horizontal tail volume coefficient ch (and cv for
the vertical tail) were found in the table below:
Aircraft Type
Sailplane
Homebuilt
GA Single Engine
GA Twin Engine
Jet Trainer
Military Cargo
Jet Transport
Horizontal
Ch
0.50
0.50
0.70
0.80
0.70
1.00
1.00
Vertical
Cv
0.02
0.04
0.04
0.07
0.06
0.08
0.09
Table 6 – Historical Tail Volume Coefficients
Using the results from the tail volume coefficient method values were obtained for the
horizontal and vertical tail sizing. The horizontal tail size obtained is Sh = 1.2 ft2 and the vertical
tail size is Sv = 0.35 ft2. These values are used as initial starting points for the Class II X-Plot
sizing.
7.2.2 Class II
The method used to refine the horizontal and vertical tail sizes of the aircraft is done
through an analytical method described in Roskam Part II. This method provides a significantly
more accurate method to approximate the tail size of an aircraft because it uses parameters
directly from the aircraft being designed to ensure proper stability. The horizontal tail is resized
using an X-Plot that plots the aircraft’s aerodynamic center and center of gravity against the
horizontal tail size. The plot is found below:
19
X-Plot
1.5
Xcg
Xac
1.4
Xcg and Xac bar
1.3
1.2
1.1
1
0.9
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Horizontal Tail Size Sh (ft2)
Figure 11 - X-Plot for Horizontal Tail Sizing
The important aircraft parameter derived from this plot is that the aircraft’s static margin
is determined by taking the difference between the aircraft aerodynamic center and center of
gravity. This plot shows that it is necessary to have a tail of at least Sh = 1.18 ft2 to maintain a
positive static margin. Historical data from Raymer suggests that the static margin must be in
the range of 10-25% for sufficient aircraft stability. This plot shows that the original Class I
sizing estimate for tail size was insufficient due to instability. A revised tail size is determined to
be Sh = 1.75 ft2 which yields a static margin of 14% for the aircraft.
The vertical tail is also resized by using a separate X-Plot as also determined by Roskam.
The weathercock stability derivative, 𝐶𝑛𝛽 , is plotted against vertical tail size. According to
Roskam this derivative must be larger than 0.001 deg-1 to ensure weathercock stability. The
resulting plot is found below:
-3
1.8
X-Plot for Directional Stability
x 10
1.6
Cn,beta (deg -1)
1.4
1.2
1
0.8
0.6
0.4
0.2
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Sv (ft2)
Figure 12 - X-Plot for Directional Stability
20
Figure 12shows that a vertical tail size of at least Sv = 0.28 ft2 must be chosen to ensure
weathercock stability. From this it is determined that the Class I sizing data is accurate for the
vertical tail and thus a final size of Sv = 0.35 ft2 is chosen.
7.3 Control Surface Sizing
Control surfaces for this aircraft are sized primarily according to historical data. Using
data from twenty general aviation single engine and light trainer models control surface sizing
ratios were determined for the elevator and rudder. The ailerons were sized according to
Roskam who suggests at least a ratio of 13% for full-span ailerons. Due to the additional aileron
control required during the hover phase of the aircraft’s flight, a value of 15% has been chosen to
ensure the ailerons are of sufficient size. A table with the results from this Class I sizing
estimate is found below.
Control
Surface
Aileron
Elevator
Rudder
Ratio
(Scs/S)
0.15
0.48
0.42
Value
(ft2)
1.1
0.83
0.13
Table 7 – Historical Control Surface Ratios
7.4 Roll Mode Approximation
The primary flight mode that must be modeled for this aircraft is the roll mode. This
mode is typically associated with the bank angle of the aircraft and important so that if perturbed
the aircraft will counter the perturbation in a time period Tr and return to trim conditions. For
this project the roll mode is of special importance because of the unique hover condition imposed
onto the design by the RFP. Since a roll rate feedback gyro will be used to help control the roll
angle it is important to accurately model the roll mode so that a nominal gain can be selected.
Using Roskam, the transfer function for the roll mode is found to be:
𝑳𝜹𝒂
𝝓(𝒔)
𝟕𝟑. 𝟑
= 𝟐
= 𝟐
𝜹𝒂 (𝒔) 𝒔 + (𝑳𝜹𝒂 𝑲 − 𝑳𝒑 )𝒔 𝒔 + (𝟕𝟑. 𝟑𝑲 − 𝟐. 𝟒𝟔)𝒔
Equation 3
This transfer function shows that the input is an impulsive aileron deflection by the pilot
and the output is a roll angle φ. The values for 𝑳𝜹𝒂 and 𝑳𝒑 are determined by using the Flat Earth
code provided by Dr. Andrisani. Specific values and calculations for these rolling moments can
be found in Appendix E. The results are shown in the equation above. The idea behind using a
roll rate feedback gyro is to reduce the roll mode time constant to stop the aircraft from rolling
during the hover.
7.5 Gain Selection
Using the approximation for the roll mode found in section 7.4, a Simulink model was
built to demonstrate the simulated flying qualities of the aircraft. This model is detailed below:
21
Figure 13 – Simulink Model of the Roll Mode Approximation
This model shows how the feedback control system works on the aircraft. An initial
aileron deflection is given to the aircraft which translates that into a roll rate. From equation (1)
the rate gyro moves the closed loop pole further negative, making the system more stable. This
roll rate is then turned into a roll angle via an integrator and output to the user. A root locus of
both the open- and closed-loop transfer functions can be found in Appendix E of this report. The
nominal gain selected for the rate gyro is 1.3 deg/deg/sec. This gain is chosen because for the
hover condition a nominal gain of 5 deg/deg/sec is necessary to reduce the roll mode time
constant to under 0.01 seconds. However, historical calibration data for the rate gyro shows that
the maximum absolute gain that can be set is 1.3 deg/deg/sec. Thus the largest possible value for
the gain is chosen. It should be noted that even though the rate gyro cannot achieve the nominal
gain calculated, the aircraft should still perform within requirements during the hover.
7.6 Flight Characteristics
There are several military and civilian flight requirements that must be met for this
aircraft. Per the RFP the aircraft designed must meet all Level I flying qualities. This means that
all requirements for flight modes must be met with the utmost precision. The aircraft has its
flying qualities modeled for a category A flight phase. This is a non-terminal flight phase
typically involving aerobatic or high maneuverability. The aircraft is also modeled as a Class
IV aircraft. This is the typical class for aerobatic and fighter aircraft. The combination of all
three of these requirements ensures that the aircraft built will meet the strictest flight qualities
and be able to perform well indoors under the control of an experienced pilot. A table of flight
characteristics is found below. Relevant calculations for these numbers can be found in
Appendix E.
22
Flight Mode
Level IA
Level IIA
Team 4
Meets
Level IA?
Short Period
0.35 ≤ ζsp ≤ 1.30
0.25 ≤ ζsp ≤ 2.00
ζsp = 0.84
x
Phugoid
ζph ≥ 0.04
ζph ≥ 0
ζph = 0.125
x
Dutch Roll
ζdr ≥ 0.19
ζdrωn,dr ≥ 0.35
ωn,dr ≥ 1.0
ζdr ≥ 0.02
ζdrωn,dr ≥ 0.05
ωn,dr ≥ 0.4
ζdr = 0.44
ζdrωn,dr = 2.01
ωn,dr = 4.57
x
Roll
Tr ≤ 1.0s
Tr ≤ 1.4s
Tr = 0.01s
x
Table 8 – Required Flight Qualities
In the table above the short period and phugoid flight modes both represent longitudinal
stability modes for the aircraft. The dutch roll and roll modes represent lateral stability. While
all of these modes are important the two most important for this project are the roll and short
period modes. The phugoid mode has the longest frequency of all the modes and as the aircraft
will be flown indoors it will not ever fly in a straight line long enough for this mode to be
significant. The roll mode ensures that if the aircraft is banked at an angle φ that it will return to
a trim condition after a certain time period Tr. This is important for this design because of the
hover roll angle requirement. The short period mode is also important because the aircraft
should return to a steady-state value when the aircraft is perturbed in the pitch axis over a short
distance. From the table above it is clear that the designed aircraft meets all Level IA flying
qualities and is stable for flight.
The other flight characteristic requirement that must be met for this project is to meet all
control anticipation parameter, or CAP, requirements. Using the master’s thesis prepared by
Mark Jacobs, he postulates that remotely controlled aircraft must have a CAP of at least 5.92 to
ensure stability. To determine the CAP of the aircraft, a MATLAB script provided by Mr.
Jacobs is used. From the code, the CAP of the designed aircraft is 18.02, clearly sufficient for
stability.
8. Economics
Aircraft cost plays a major role in every design. If an aircraft design exceeds its budget,
the number of customers who purchase the aircraft may be reduced. The budget intended for this
semester was $150. Components to be included into the budget were materials for the airframe,
the propulsion system, and control servos. Specifically excluded from the budget were radio
equipment (transmitter, receiver, rate gyro, and speed controller), which is provided by the
course instructor.
To determine the full cost of our airplane, the number of man-hours and a list of
construction costs were made. Using timesheets for each individual team member, an estimation
of total man-hours was made for the entire semester. The total number of estimated man-hours
devoted to the design, build, and test of the aircraft came to 1353 hours. Each man-hour was
determined to have the value of $100.00, which translates into $135,300.00 for the cost of the
total man-hours. The breakdown and distribution of time, for the team is below and the team
members, is outlined in Appendix F.
23
Team Hours Per Week
200
Hours
150
100
50
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14
Week
Figure 14 - Team Hours per Week Plot
The parts acquisition list in Appendix F shows a total cost of $197.12 for all parts,
bringing the total cost to be $135,497.12 for the entire aircraft design.
24
9. References
Brandt, Steven, et al. Introduction to Aeronautics: A Design Perspective. Reston: American
Institute of Aeronautics and Astronautics, 2004.
Nicolai, Leland M. “Estimating R/C Model Aerodynamics and Performance,” June 2002.
Raymer, Daniel P. Aircraft Design: A Conceptual Approach. Reston: American Institute of
Aeronautics and Astronautics, 2006.
Roskam, Jan. Airplane Design. Ottawa: Roskam Aviation and Engineering Corporation, 1985.
Roskam, Jan. Airplane Flight Dynamics and Automated Flight Controls. Ottawa: Roskam
Aviation and Engineering Corporation, 1995.
U.S. Military Specification MIL-F-8785C. “Flying Qualities of Piloted Airplanes,” November
1980.
25
APPENDIX A – Initial Sizing and Concept Selection
A.1 – Concept Selections
Figure 15 - Concept 1
26
Figure 16 - Concept 2
27
Figure 17 - Concept 3
A.2 – Weight Estimation
28
The following plot shows the relationship between battery weight and total weight of a
given r/c aircraft. The list of aircraft was compiled using specifications freely available on the
internet. This relationship gives a linear relationship between aircraft weight and battery size
required. It was used to give a rough estimate for the approximate battery size required for Team
4’s aircraft.
Battery Weight Vs. Total Weight
(Historical)
Battery Weight (lbf)
0.500
y = 0.2433x + 0.0065
R² = 0.7999
0.400
0.300
0.200
0.100
0.000
0.000
0.500
1.000
1.500
2.000
Total Weight (lbf)
Figure 18 - Historical Data for Battery and Total Aircraft Weight
RC AIRCRAFT
Mustang P-51D
Cessna 182
Decathlon
Edge 540T v.2
Aerobat
Gee Bee 3D
Super Cub Speed 400
Soareasy
Extra 330L
Fundango
Cessna 172
WING SPAN
(in)
30.0
38.5
38.5
36.6
38.0
30.3
46.5
42.5
38.0
35.0
44.0
LENGTH
(in)
24.0
29.0
29.0
33.1
30.5
29.5
29.0
31.8
33.0
30.0
33.0
TOTAL WEIGHT
(lb)
0.833
1.313
1.375
1.250
1.063
0.825
1.313
1.625
1.375
1.000
1.625
BATTERY
li-poly 7.4 volt 2 cell with balanced port
7 cell 8.4v 1000 mah nihm
8 cell 9.6v 1000 mah nihm
3 cell li-poly 1800 mah
11.1v li poly 1250 mah
2 cell 7.4v 1800mah
6 Cell 1100 X-Cell Sport NiMH Pack
8 Cell 1100 mAh Folded NiMH Pack w/Connector
11.1v li-poly 3 cell 1500 mah
8C 700 AA NiCd Flat Pack (GPMP0200)
3 cell 11.1V 1200mah Li-Poly
BATTERY WEIGHT
(lb)
0.192
0.362
0.397
0.309
0.249
0.223
0.281
0.375
0.265
0.400
0.250
Figure 19 - Historical Data for R/C Aircraft
MATLAB script weight_3.m:
% FILE: Weight_3.m
% Preliminary weight estimator for electric powereed aircraft
% Revised 9/5/06
disp(' '); disp('>>>>>>>>>Start here <<<<<<<<<'); disp(' ')
LoverDmax=12
% for fixed gear GA aircraft (Skyhawk) (See Raymer p.
22)
LoverD=.866*LoverDmax % for loiter (See Raymer p. 22)
Vloiter=40
% ft/sec, Estimated loiter speed
ETAmotor=0.75
29
ETAprop= 0.60
%RHOb=72900
% battery energy density for NiCad joule per pound
%RHOb=9.25E+04 % battery energy density for NiMH joule per pound
RHOb=2.39E+05 % battery energy density for Lithium polymer joule per pound
disp('Battery energy density for NiCad batteries, joules per pound')
EnduranceMIN=7
Wpayload=1 % payload weight pounds
EnduranceSEC=EnduranceMIN*60
TimeLoiterStraight=EnduranceSEC/2
% Loiter time in straight flight (sec)
TimeLoiterTurn=EnduranceSEC/2
% Loiter time in turning flight (sec)
g=32.17 % acceleration of gravity ft/sec^2
% For loiter in straight flight
WlsperW=Vloiter*1.356*TimeLoiterStraight/(ETAmotor*ETAprop*RHOb*LoverD)
% For loiter in turning flight
R=50 % Turn radius at loiter from mission spec.
phi=atan(Vloiter*Vloiter/(R*g)) % bank angle in the turn (rad)
WltperW=Vloiter*1.356*TimeLoiterTurn/(ETAmotor*ETAprop*RHOb*LoverD*cos(phi))
% For climbing flight
gamma=20/57.3 % climb angle (rad)
TimeClimb=12/(Vloiter*sin(gamma)) % time to climb to 12 feet
WclimbperW=Vloiter*1.356*TimeClimb*(cos(gamma)/LoverD+sin(gamma))/(ETAmotor*E
TAprop*RHOb)
% For Takeoff
disp('From integration of eoms at takeoff, assume that the battery')
disp(' weight fraction is .002.')
WtoperW=.002
% For warm-up assume takeoff times aree about 3 sec and
% warm-up times are about 30 seconds.
disp('Assume that the warmup weight fraction is 10 times the ')
disp(' takeoff weight fraction.')
WwarmperW=10*WtoperW
% Assemble the complete battery weight fraction.
WbperW=WlsperW+WltperW+WclimbperW+WtoperW+WwarmperW
Weight=0:1:10; %weight in pounds
echo on
WminusWe=.2103*Weight+.1243; % formula for historical data (pounds)
echo off
disp('Your weight estimate will only be as good at that historical data
represented in the equation above')
Wbattery=WbperW*Weight;
WbplusWpay=Wbattery+Wpayload;
30
plot(Weight,WminusWe,Weight,WbplusWpay)
xlabel('Weight~lbf')
ylabel('W-We and Wb+Wp~lbf')
% Determination of aircraft weight
delta=WminusWe-WbplusWpay;
% YI = INTERP1(X,Y,XI)
Waircraft=interp1(delta,Weight,0)
y=.2103*Waircraft+.1243;
string1=['Estimated aircraft weight is ',num2str(Waircraft),' pounds.']
text2(.25,.2,['
',string1])
title('Weight estimation using historical weight data')
legend('Historical data','Estimated weight')
hold on; plot(Waircraft,y,'o'); hold off
Wb=WbperW*Waircraft
string2=['Estimated battery weight is ',num2str(Wb),' pounds.']
text2(.25,.15,['
',string2])
string2=['Payload weight is ',num2str(Wpayload),' pounds.']
text2(.25,.1,['
',string2])
This code was used to determine the battery weight fractions necessary for each section
of the flight. Below is a table of hand-calculated values which confirm what weight3.m outputs.
Flight
Section
%
Battery
Endurance
Required
(m:s)
Energy
Required (J)
ETO
1.0%
0:02
149
EWU
7.5%
0:30
1120
ECL
0.4%
0:02
64
ELF
7.1%
3:00
1005
EHV
84%
2:00
12349
EB
100
5:34
27580
Table 2 - Energy Required per Flight Phase
31
Constraint Diagram
300
Power loading (lbf/hp)
250
200
150
100
50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Wing loading lbf/ft2
Figure 20 - Constraint Diagram
This is a zoomed out version of Team 4’s constraint diagram as found in section 3.4 of
the main report. It provides an overall view of the constraints on the aircraft as per this project.
This is simply for reference as it does not provide a detailed view of the design space. That plot
can be found in section 3.4 of the main report.
32
APPENDIX B – Structures and Weights
𝑥̅𝑐𝑔𝑎 =
𝛴𝑚𝑖 𝑟𝑖
𝛴𝑚𝑖
Equation 4
𝑞(𝑦) =
4𝑆
2𝑦 2
√1 − ( )
𝑏𝜋
𝑏
Equation 5
𝐿 =𝑛 ×𝑊
Equation 6
𝑥̅ =
4𝑎
3𝜋
Equation 7
𝑀 =𝑑 ×𝐿
Equation 8
𝜎=
𝑀𝑦
𝐼𝑥
Equation 9
𝑉𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 = √𝜌⁄𝜌𝑠𝑙 𝑉𝑎𝑐𝑡𝑢𝑎𝑙
Equation 10
𝐿=
1
𝜌𝐶𝐿𝑚𝑎𝑥 𝑠𝑉𝑒2
2
𝑛=
𝐿
𝑊
Equation 11
Equation 12
33
Elliptical Load Distribution
1.2
Wing Load (lb/ft)
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
Distance from Centerline of Aircraft (ft)
2.5
Figure 21 – Elliptical Load Distribution
Figure 22 – Wing Connector Rib
34
Figure 23 – Fairing Rib
Figure 24 – Vertical Tail
35
Figure 25 – Horizontal Tail
36
APPENDIX C – Aerodynamics
C.1 - Reynolds Number
𝜌 = .00226 𝑠𝑙𝑢𝑔/𝑓𝑡 2
𝑈 = 19.5
𝑓𝑡
𝑠
𝑐 = 1 𝑓𝑡
𝜇 = 3.72 ∙ 10−7 𝑠𝑙𝑢𝑔/(𝑓𝑡 ∙ 𝑠)
𝑅𝑒 =
𝜌𝑈𝑐
≈ 119,000
𝜇
Equation 13
C.2 - Drag Computation
𝐴𝑅 = 4.8
𝐶𝑓𝑒 = 0.0055
𝑆𝑤𝑒𝑡 = 10.59 𝑓𝑡 2
𝑆 = 4.8 𝑓𝑡 2
𝑒 = 1.78(1 − 0.045𝐴𝑅 0.68 ) − 0.64 = 0.9073
Equation 14
𝐶𝐷0 = 𝐶𝑓𝑒
𝑆𝑤𝑒𝑡
= 0.0144
𝑆
Equation 15
𝐶𝐿𝑚𝑖𝑛 ≈ 𝐶𝑙𝑚𝑖𝑛 = 0.2149
Equation 16
𝐾=
1
= 0.0731
𝜋𝐴𝑅𝑒
Equation 17
𝐾 ′ = 0.0137
Individual Component Minimum Aircraft Drag Calculation
37
-
Fuselage
Fuselage Fineness Ratio : 𝐹𝑅 =
𝑓𝑢𝑠𝑒𝑙𝑎𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ
1 𝑓𝑡
=
=4
𝑓𝑢𝑠𝑒𝑙𝑎𝑔𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 . 25 𝑓𝑡
Equation 18
Fuselage 𝐶𝐷𝑚𝑖𝑛 = (1 +
60
+ .0025 ∙ 𝐹𝑅) 𝐶𝐷0 = 0.0280
(𝐹𝑅)3
Equation 19
-
Wing
𝑡
𝐿 = 1.2 as maximum location is ≥ 0.3𝑐
𝑐
𝑡
= 0.12
𝑐
𝑅 = 1.05 for low-speed, unswept wing
𝑡
𝑡 4
Wing 𝐶𝐷𝑚𝑖𝑛 = [1 + 𝐿 ( ) + 100 ( ) ] 𝑅 ∙ 𝐶𝐷0 = 0.0176
𝑐
𝑐
Equation 20
-
Horizontal Tail
𝑡
= .01, 𝑅 = 1.05, 𝐿 = 1.2
𝑐
𝑡
𝑡 4
Horizontal Tail 𝐶𝐷𝑚𝑖𝑛 = [1 + 𝐿 ( ) + 100 ( ) ] 𝑅 ∙ 𝐶𝐷0 = 0.0153
𝑐
𝑐
Equation 21
-
Vertical Tail
𝑡
= .01, 𝑅 = 1.05, 𝐿 = 1.2
𝑐
𝑡
𝑡 4
Vertical Tail 𝐶𝐷𝑚𝑖𝑛 = [1 + 𝐿 ( ) + 100 ( ) ] 𝑅 ∙ 𝐶𝐷0 = 0.0153
𝑐
𝑐
Equation 22
-
Tail boom
Tail Boom 𝐶𝐷𝑚𝑖𝑛 = 1.05 ∙ 𝐶𝐷0 = 0.0151
Equation 23
-
Whole Aircraft
𝐶𝐷𝑚𝑖𝑛 = 𝐶𝐷𝑚𝑖𝑛
+ 𝐶𝐷𝑚𝑖𝑛
𝑓𝑢𝑠𝑒𝑙𝑎𝑔𝑒
𝑤𝑖𝑛𝑔
+ 𝐶𝐷𝑚𝑖𝑛
𝐻−𝑇𝑎𝑖𝑙
+ 𝐶𝐷𝑚𝑖𝑛
𝑉−𝑇𝑎𝑖𝑙
+ 𝐶𝐷𝑚𝑖𝑛
𝑇𝑎𝑖𝑙 𝐵𝑜𝑜𝑚
= 0.0913
Equation 24
Whole Aircraft Drag Equation
38
2
𝐶𝐷 = 𝐶𝐷𝑚𝑖𝑛 + (𝐾 + 𝐾 ′ )(𝐶𝐿 − 𝐶𝐿𝑚𝑖𝑛 ) = 0.0913 + (0.0731 + 0.0137)(𝐶𝐿 − 0.2149)2
Equation 25
C.3 - Lift & Moment Calculations
Lift and moment values were calculated using the FlatEarth.m code (inputs and outputs
cited in the Dynamics & Controls appendix). From the inputs, the FlatEarth program produced
the following mathematical models:
𝐶𝐿 = 𝐶𝐿0 + 𝐶𝐿𝛼 𝛼 + 𝐶𝐿𝛿𝑒 𝛿𝑒 = 0.2471 + 3.525𝛼 + 0.9613𝛿𝑒
Equation 26
𝐶𝑀 = 𝐶𝑀0 + 𝐶𝑀𝛼 𝛼 + 𝐶𝑀𝛿𝑒 𝛿𝑒 = −0.07567 − 0.489𝛼 − 1.667𝛿𝑒
Equation 27
The above lift and drag models were used to generate the lift and drag polars in sections 5 and 6
of this appendix.
C.4 - Fuselage Sizing Trade Study
Objective
The primary objective of this trade study is to analyze the relationship between the aspect
ratio, fuselage size, and the minimum drag of our aircraft design. Aspect ratio has been a bit of
an issue for everyone in the class, as essentially every aspect of the aircraft is driven by its
selection. However, due to practicality, as not all aircraft are as sleek as one may like, fuselage
size is a significant issue as well. Necessary flight electronics need to be stored, easily accessible
to certain areas of the aircraft, as well as protected. But bulky fuselages “tarnish” the
aerodynamic beauty of an aircraft. That is the tradeoff that has to be made. The proper design of
fuselages is something that will follow an aerospace engineer for the rest of their career. Real
world examples are everywhere: case in point, the Lockheed Martin F-22 Raptor. The F-22 is
the king of the sky, and it requires some extremely specialized equipment in order to hold its
throne there. However, there is a major maintenance issue whenever an internal component
breaks, as in some cases whole sections of the aircraft have to be completely disassembled in
order to replace a tiny part (I experienced this problem firsthand when I had the opportunity to
work with an Air Force maintenance unit on the F-22. Needless to say, there was some animosity
between the on-site engineers and the maintainers). Early on in our design process, we wanted
the aircraft to be easily repairable, and in order to accomplish that, every component of the
aircraft has to be accessed when the user wants to access it; there is no point in marketing a r/c
aircraft that requires complete disassembly in order to flip a switch.
39
Procedure
The code that was developed by our group for our aerodynamic lift and drag calculations
was modified to suit this trade study. Given some specifications of our aircraft, such as
component sizes, airfoil data, and operating conditions, the script outputs several values and
plots. The important data that was utilized is the CDmin values for the fuselage and the entire
aircraft, the CD vs. α plot, and the CD vs. CL plot. Earlier in our design process, we chose a half
cylinder to be our fairing cover for the fuselage. It is slung underneath the wing. A range of
aspect ratios, from 2 to 8, were inputted, and the radius of the fuselage was varied from 0.5
inches up to 2.0 inches. The outputted data was analyzed, and from this, our group will select
the proper fuselage radius in order to minimize drag as based upon our selected aspect ratio.
Results
An initial design point was selected for the total CDmin of the entire aircraft. We wanted
to keep our total drag estimate to be as close to 500 counts as possible, ± 10 counts. As of when
this trade study was performed, our working aspect ratio is 3.88. Before we actually went and
selected a fuselage radius, we first analyzed the data to look for trends. One large trend that was
discovered is that there is the change in the CDmin is not linear as aspect ratio and fuselage radius
is varied; it is closer to a quadratic approximation. This is illustrated in the CDmin vs alpha plot
below (figure 1). As a result, we immediately discovered that the radius had to be large, no
matter what aspect ratio was chosen. The aircraft drag builds quickly as the aspect ratio is
minimized, and the CDmin of the fuselage becomes a larger percentage of the total aircraft CDmin
as a result.
40
Radius of
AR
Fuselage
2
3
3.88
4
5
6
7
8
2
3
3.88
4
5
6
7
8
2
3
3.88
4
5
6
7
8
0.5
1
1.5
CDmin
CDmin
Fuselage
0.0297
0.0809
0.0198
0.0591
0.0153
0.0492
0.0148
0.0482
0.0119
0.0417
0.0099
0.0373
0.0085
0.0342
0.0074
0.0319
0.031
0.0822
0.0207
0.06
0.016
0.0499
0.0155
0.0489
0.0124
0.0422
0.0103
0.0378
0.0089
0.0346
0.0078
0.0322
0.0331
0.0843
0.0221
0.0614
0.0171
0.051
0.0166
0.0499
0.0132
0.0431
0.011
0.0385
0.0095
0.0352
0.0083
0.0328
Radius of
AR
Fuselage
2
2.5
3
2
3
3.88
4
5
6
7
8
2
3
3.88
4
5
6
7
8
2
3
3.88
4
5
6
7
8
CDmin
CDmin
Fuselage
0.0376
0.0888
0.0251
0.0644
0.0194
0.0533
0.0188
0.0522
0.0151
0.0449
0.0125
0.04
0.0108
0.0365
0.0083
0.0328
0.0453
0.0965
0.0302
0.0695
0.0233
0.0573
0.0226
0.056
0.0181
0.0479
0.0151
0.0425
0.0129
0.0387
0.0113
0.0358
0.0567
0.1079
0.0378
0.0771
0.0292
0.0632
0.0284
0.0617
0.0227
0.0525
0.0189
0.0464
0.0162
0.042
0.0142
0.0387
Table 3 - CDmin Results
From all this, for our current aspect ratio of 3.88, the optimized fuselage radius is 1.5
inches. Though the CDmin is at the edge of our limits for aircraft drag, any smaller and there are
going to be serious storage issues based upon the purported size of our electronic components.
The fuselage CDmin may be a bit large, but it is at the lower end of the CDmin values for all of the
fuselages radii, no matter the aspect ratio. As a result of this trade study, the new fuselage radius
was added to our final model.
CDmin
CDmin vs AR
0.105
0.095
0.085
0.075
0.065
0.055
0.045
0.035
0.025
2
3
4
5
6
7
8
AR
r = 0.5
r = 1.0
r = 1.5
r = 2.0
r = 2.5
r = 3.0
Figure 26 - CDmin vs. AR for entire aircraft
41
CDmin Fuselage vs AR
0.065
0.055
CDmin
0.045
0.035
0.025
0.015
0.005
2
3
4
5
6
7
8
AR
r = 0.5
r = 1.0
r = 1.5
r = 2.0
r = 2.5
r = 3.0
Figure 27 - CDmin vs. AR for fuselage
42
C.5 - Aircraft drag polar
CD vs CL - Team 4
0.24
0.22
0.2
0.18
C
D
0.16
0.14
e =-20o
0.12
e =-10o
0.1
e =0o
0.08
e =5o
0.06
e =10o
0.04
-0.5
0
0.5
1
1.5
2
CL
Figure 28 - Aircraft Drag Polar
C.6 - Aircraft lift versus α
CL vs  - Team 4
2
1.5
L
1
C
e =-20o
e =-10o
0.5
e =0o
e =5o
0
e =10o
-0.5
-5
0
5
10
15
20
25
 (degrees)
Figure 29 - Airfoil Lift Curve
43
C.7 - Aircraft Center of Gravity Computation
Weight
(lb)
Location from
Propeller (in)
0.00
0.50
1.13
8.00
8.00
12.00
12.00
13.00
11.00
12.00
22.50
42.50
40.00
Propeller
Motor
Gear box
Speed Controller
Batteries
Wing
Fairing
Servos (4)
Receiver
Rate Gyro
Fuselage
Vertical Stablizer
Horizontal Stablizer
0.04
0.14
0.13
0.04
0.13
0.60
0.17
0.08
0.06
0.06
0.37
0.02
0.14
Table 4 - Tabular Listing of Aircraft Equipment
Total Weight (lb)
Center of Gravity (in)
Aerodynamic Center (in)
Static Margin
1.94
14.17
15.92
0.14
Table 5 - Total Aircraft Weight and Static Margin
Static Margin Computation
Equation 28
𝐶𝐺 =
∑ 𝑟𝑖 𝑚𝑖
∑ 𝑚𝑖
Equation 29
𝑆𝑀 = 𝑥̅𝐴𝐶 − 𝑥̅𝐶𝐺
Equation 30
44
C.8 - Aircraft Trim Diagram code
% Aircraft trim diagram
clear all
%close all
disp(' '); disp('Start Here <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<')
disp(' First we will duplicate figure 4.5 on page 201')
disp(' Data Input section')
% Data is selected to duplicate Roskam figures 4.5
CL0=0.24711
CLalpha=3.525*(pi/180)
% Per degree
CLdeltaE=0.96133*(pi/180)
% Per degree
CLiH=3*CLdeltaE % Per degree
CM0=-0.075671
CMalpha=-0.48903*(pi/180)
% Per degree
CMdeltaE=-1.6671*(pi/180) %Per degree
CMiH=3*CMdeltaE %Per degree
deltaE=[-20 -10 0 10]
iH=0
% Degree
mom_ref_pt=.25
% Moment reference point in % chord
forward_cg=.2 % forward cg limit in % chord
aft_cg=.4
% aft cg limit in % chord
alpha_stall=11
% angle of attack (deg) for stall
Cl_PlotMax=1.1
% maximum Cl for plots in figure 2 (the aircraft trim
diagram)
alpha_PlotMax=11 % maximum angle of attack (deg) for figure 1
% End of data input section
% Plotting information
color=['-bo-gx-r+-c*-md-yv-k^'];
s1=['De=',num2str(deltaE(1)),' deg.'];
s2=['De=',num2str(deltaE(2)),' deg.'];
s3=['De=',num2str(deltaE(3)),' deg.'];
s4=['De=',num2str(deltaE(4)),' deg.'];
% s5=['De=',num2str(deltaE(5)),' deg.'];
% End plotting information
alpha=0:1:alpha_PlotMax;
dCMdCL=CMalpha/CLalpha;
CM0bar=CM0-dCMdCL*CL0;
CMiHbar=CMiH-dCMdCL*CLiH;
CMdeltaEbar=CMdeltaE-dCMdCL*CLdeltaE;
% Plot aircraft trim diagram
figure(2) % <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
clf
for i=1:length(deltaE)
dE=deltaE(i);
CL=CL0+CLalpha*alpha+CLiH*iH+CLdeltaE*dE;
CM=CM0bar+dCMdCL*CL+CMiHbar*iH+CMdeltaEbar*dE;
subplot(122);plot(CM,CL); hold on
45
subplot(121);plot(alpha,CL); hold on
end
% End temporary plot
% Extend plots
subplot(122); z1=axis;
clf
CLexpended=z1(3):.1:z1(4);
for i=1:length(deltaE)
dE=deltaE(i);
CL=CL0+CLalpha*alpha+CLiH*iH+CLdeltaE*dE;
subplot(121);plot(alpha,CL,color(3*(i-1)+1:3*(i-1)+3)); hold on
CMexpanded=CM0bar+dCMdCL*CLexpended+CMiHbar*iH+CMdeltaEbar*dE;
subplot(122); hold on; plot(CMexpanded,CLexpended,color(3*(i-1)+1:3*(i1)+3));
end
LOC='SouthEast';
subplot(122); legend(s1,s2,s3,s4,'Location',LOC)
subplot(121); legend(s1,s2,s3,s4,'Location',LOC)
% end of expanded plotting
subplot(122);plot([0 0],[0 z1(4)],'k') %plot zero line on CM plot
hold off
subplot(122);z1=axis;axis([z1(1) z1(2) 0 Cl_PlotMax])
% plot the forward cg line of the trim triangle
delta_cg1=mom_ref_pt-forward_cg;;
Cm_forward=+Cl_PlotMax*delta_cg1;
subplot(122);hold on; plot([0 Cm_forward],[0 Cl_PlotMax],'k') %plot zero line
on CM plot
% plot the aft cg line of the trim triangle
delta_cg2=mom_ref_pt-aft_cg;
Cm_aft=+Cl_PlotMax*delta_cg2;
subplot(122);hold on; plot([0 Cm_aft],[0 Cl_PlotMax],'k') %plot zero line on
CM plot
str1=['CM about ',num2str(mom_ref_pt),' c'];
xlabel(str1)
ylabel('CL')
title('CM about .25c vs. CL')
grid on
text(.18,.75,['iH= ',num2str(iH),' deg.'])
strXF=['fwd cg xbar=',num2str(forward_cg)];
strXR=['aft cg xbar=',num2str(aft_cg)];
text(.16,1.18,strXF)
text(-.02,1.18,strXR)
stralpha=['alpha stall=',num2str(alpha_stall), ' deg.'];
text(.09,.95,stralpha)
subplot(121);z2=axis; axis([z2(1) z2(2) 0 Cl_PlotMax]);
ylabel('CL')
xlabel('alpha (deg)')
title('angle of attack vs. CL')
grid on
text(.1,.95,['iH= ',num2str(iH),' deg.'])
hold off
46
% Plot alpha_stall line in figure 2
CL=CL0+CLalpha*alpha_stall+CLiH*iH+CLdeltaE*deltaE;
CM=CM0+CMalpha*alpha_stall+CMiH*iH+CMdeltaE*deltaE;
subplot(122); hold on; plot(CM,CL,'k');
axis([-.3 .3 0 1.2])
set(gca, 'XDir', 'reverse'); % reverse the plotting direction on the x axis
47
APPENDIX D – Propulsion
D.1 - Provided MATLAB Scripts for Propulsion Design
Main_System_Design.m
% Team 4 Main_System_Design
% modified with gold.m as a called function
clear
clear functions
close all
ifig=0;
hold off
disp(' '); disp('>>> Start of script. <<<'); disp(' ')
echo on
% Script to design an end-to-end propulsion system for an
% electric-powered propeller-driven aircraft.
% Given:
%
drag polar,
%
aircraft weight, air density,
%
pitch to diameter ratio of the prop and prop data,
%
motor constants for a particular motor.
%
% Find:
%
speed for maximum endurance,
%
propeller diameter,
%
gear ratio,
%
voltage at which to operate the motor,
%
battery sizes to acheive the desited battery voltage,
%
endurance for single strand and dual strand batteries,
%
for an aircraft flying straight and level or turning with a
%
specified turn radius.
%
% PHASE 1: AIRCRAFT SUBSYSTEM
%
% This data is roughly for the Boiler Express Aircraft
CD0=.013; % drag coefficient when CL=0.
A=4.8; % aspect ratio span squared divided by reference area
e=.9; % Oswalds efficiency factor
V=10:1:40; % velocity in ft/sec
rho=.002264; % air density in slugs/ft^3
W= 1.94;
% aircraft weight in pounds (lbf)
S= 4.8;
% wing area
R= 40;
% Radius of steady turn (ft)
RPM=4100; %Propeller RPM
n=RPM/60; % propeller frequency (rev/sec) or (hz)
echo off
[Pop,Vop,EtaAircraft,Pre,Ve,ifig]=DesignAircraft2(W,rho,S,CD0,A,e,V,ifig,R);
%
% PHASE 2: PROPELLER SUBSYSTEM
%
Vmph = [0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5 35];
D = input('Propeller diameter in inches');
P = input('Propeller pitch in inches');
48
tau = P/D;
string1=['Data for hypothetical propeller with tau=p/D=',num2str(tau)];
disp(string1);
for i = 1:length(Vmph)
[J(i),CP(i),CT(i)] = gold(D,P,Vmph(i));
end
% J = [
0
0.1000
0.7000
0.8000];
% CT =[0.0933
0.0867
0.0160
0.0013];
% CP =[0.0400
0.0393
0.0147
0.0027];
0.2000
0.3000
0.4000
0.5000
0.6000
0.0773
0.0667
0.0553
0.0427
0.0293
0.0387
0.0367
0.0333
0.0293
0.0227
[Jstar,CTstar,CPstar,Etastar,ifig]=PlotProp(J,CT,CP,string1,ifig);
% Propeller power equals the operating power of the aircraft
% The propeller is operating at a forward speed given by Vop
% Find the required propeller diameter
[Dactual,Jactual,nactual,RPMactual,EtaPropactual,Poutactual,Ppinactual,ifig]=
...
DesignProp(rho,Pop,Vop,Jstar,CTstar,CPstar,Etastar,J,CT,CP,ifig); % <<< lots
of work here
% RPMprop=RPMactual;
% PinPropWatts=Ppinactual*1.356; % convert to watts from ft-lbf/sec
PlotProp2(Jactual,EtaPropactual,J,CT,CP,ifig);
Pitch=tau*Dactual*12;
string2=[' You have selected a ',num2str(Dactual*12),'x',num2str(Pitch),'
prop.'];
disp(string2); disp(' ');
%
% PHASE 3: GEAR BOX AND MOTOR SUBSYSTEMS
%
EtaGear=.96; % Efficiency of the gearbox
% %Uncomment below section for Hover Endurance
disp('ALERT: IF THIS MSG IS VISIBLE, YOU ARE RUNNING THE HOVER ENDURANCE
MISSION')
PropA = pi*(Dactual/2)^2;
Ppinactual = W/(550*EtaPropactual*sqrt(2*rho*PropA/W));
%hp
PinPropWatts=Ppinactual*745.7; % convert to watts from hp
RPMprop = (CT(1)/CP(1))*Ppinactual*550/(Dactual*W)*60;
% At this point in the code the gear ratio is unknown.
% Required motor output taking into account the inefficiency of the grarbox
PoutMotorWatts=PinPropWatts/EtaGear;
% Get array of motor data
motordata
[nmotor,col]=size(mdata);
% Select motor from the available choices
string20a=['Which of the ',num2str(nmotor),' available motors do you want to
use? Default=1 >>>>'];
disp(' ');
ii=input(string20a);
49
if isempty(ii)
ii=1; % Select motor #1
end
titstr=mname(ii,:);
string29=['You have selected the motor ',num2str(ii),', the ',titstr,'
motor.']; disp(' ');
disp(string29);
Kv=mdata(ii,1);
% RPM/volt
Kt=mdata(ii,2);
% inch-ounce per ampere
R= mdata(ii,3);
% Ohms
Io=mdata(ii,4);
% amperes
Vin=mdata(ii,6) ;
% nominal motor voltage, volts
HPnom=mdata(ii,7);
% nominal horsepower at the conditions for max
efficiency
%
%
[Vinstar,Iinstar,Pinwattsstar,RPMstar,PoutHP,EtaMotorMax,ifig]=MotorMaxEff(Po
utMotorWatts,Kv,Kt,R,Io,ifig); % <<<<<
disp(' '); disp('PRELIMINARY MOTOR AND GEARBOX ANALYSIS');
string20=[' The output power of this motor has to be
',num2str(PoutMotorWatts),' watts or ',num2str(PoutHP),' Hp.'];
string21=[' For maximum motor efficiency this motor must be provided with
',num2str(Vinstar),' input volts.'];
string22=[' Under these conditions, the input current will be
',num2str(Iinstar),' amperes'];
string23=[' and the input power will be ',num2str(Pinwattsstar),' watts.'];
string24=[' The motor shaft will be spinning at ',num2str(RPMstar),' RPM.'];
string25=[' The motor efficiency will be ',num2str(EtaMotorMax),'.'];
disp(string20); disp(string21); disp(string22); disp(string23);
disp(string24); disp(string25);
m=RPMstar/RPMprop;
string26=[' To match this motor to the propeller requires an optimal gear
box ratio (m*) of ',num2str(m),'.'];
disp(string26); disp(' ');
mactual=input('What gear box ratio (m) do you want to use? Default=m*
>>>>');
disp(' ');
if isempty(mactual)
mactual=m;
% use default gear ratio for running the
motor at maximum efficiency
string20d=['You have selected the default gear ratio
',num2str(mactual),'.'];
disp(string20d);
RPMactual=mactual*RPMprop;
else
RPMactual=mactual*RPMprop;
% Compute for specified gear ratio
string20d=['You have selected the gear ratio ',num2str(mactual),'.'];
disp(string20d);disp(' ');
string20b=['At this point the motor RPM and output power of the motor are
specified, so motor inputs can be found.'];
string20c=['
Motor RPM= ',num2str(RPMactual),' RPM and motor output
power= ',num2str(PoutMotorWatts),' watts'];
disp(string20b); disp(string20c);
% compute motor input properties
end
disp(' ');
50
%
[Vin,Iin,Pinwatts,PoutHP,EtaMotor,ifig]=MotorInputs(PoutWatts,RPM,Kv,Kt,R,Io,
ifig)
[Vinactual,Iinactual,Pinwattsactual,PoutHP,EtaMotoractual,ifig]=MotorInputs(P
outMotorWatts,RPMactual,Kv,Kt,R,Io,ifig);
string30a=['MOTOR AND GEAR BOX DESIGN SUMMARY'];
string30= [' The output power of this motor is ',num2str(PoutMotorWatts),'
watts or ',num2str(PoutHP),' Hp.'];
string31= [' The motor input voltage is
',num2str(Vinactual),'
volts.'];
string32= [' The motor input current is
',num2str(Iinactual),'
amperes.'];
string33= [' and the motor input power is
',num2str(Pinwattsactual),'
watts.'];
string34= [' The electric motor shaft is spinning at ',num2str(RPMactual),'
RPM.'];
string34a=[' The gear box output shaft is spinning at ',num2str(RPMprop),'
RPM.'];
string35= [' The gear box ratio (m) is
',num2str(mactual),'.'];
string36= [' The motor efficiency is
',num2str(EtaMotoractual),'.'];
string37= [' The gear box efficiency is
',num2str(EtaGear),'.'];
disp(string30a); disp(string30); disp(string31); disp(string32);
disp(string33); disp(string34); disp(string34a);
disp(string35); disp(string36); disp(string37)
% Comments on these motor operating conditions
string40e=['COMMENTS ON THE MOTOR OPERATING CONDITIONS'];
string40= [' You are asking a motor with nominal horsepower of
',num2str(HPnom),' to produce a horsepower of ',num2str(PoutHP),'.'];
string40a=[' Since the produced power is greater than the nominal power you
are overdriving the motor.'];
string40b=[' Consider a larger motor!'];
string40c=[' Since the produced power is less than the nominal power you are
underdriving the motor.'];
string40d=[' Consider a smaller motor!'];
disp(' '); disp(string40e); disp(string40);
if PoutHP>HPnom;
disp(string40a); disp(string40b);
else
disp(string40c); disp(string40d);
end
% Plot the motor operating properties for the given voltage.
Iintest=Io:30;
[Pouttest,Poutfppstest,PoutHptest,Pintest,Touttest,Toutfptest,RPMtest,etatest
]=motor(Vinactual,Iintest,Kv,Kt,R,Io);
[ifig]=MotorPlot2(Vinactual,Iinactual,Iintest,Pouttest,Touttest,RPMtest,etate
st,titstr,ifig);
%
% PHASE 4: BATTERY SUBSYSTEM
%
51
disp(' '); disp('BATTERY SUBSYSTEM'); disp(' The battery pack will be made
up of individual cells with the following properties:')
VoltsPerCell=11.1;
% volts per cell
mAmpsHoursPerCell=1320;
% milliamps hours per cell
gramspercell=57;
%grams per cell
slugspercell=(gramspercell/1000)/14.5939;
lbfpercell=32.17*slugspercell;
string54=[' ',num2str(VoltsPerCell),' volts per cell, and
',num2str(mAmpsHoursPerCell),' milliamp hours per cell, and
',num2str(gramspercell),' grams per cell.'];
disp(string54);
disp(' '); disp(' A single string battery pack designed for the above
conditions will have the following properties.')
nCells1=ceil(Vinactual/VoltsPerCell);
nVolts=nCells1*VoltsPerCell;
weight1=nCells1*lbfpercell;
BatteryEnergy1Joule=(mAmpsHoursPerCell*3600/1000)*nVolts; %
Joules=watt*sec=ampere*volt*sec
ActualEnduranceMin1=(1/60)*(BatteryEnergy1Joule/Pinwattsactual); % predicted
endurance for single strand battery, minutes
string50=[' ',num2str(nCells1),' total cells, arranged in a
1x',num2str(nCells1),' array.'];
string51=[' producing ',num2str(nVolts),' volts, and giving a predicted
endurance of ',num2str(ActualEnduranceMin1),' minutes,'];
string54=[' and weighing ',num2str(weight1),' lbf.'];
disp(string50); disp(string51); disp(string54);
disp(' '); disp(' A dual string battery pack designed for the above
conditions will have the following properties.')
nCells2=2*nCells1;
% Cells in both strands
weight2=nCells2*lbfpercell;
BatteryEnergy2Joule=2*BatteryEnergy1Joule; % Joules=watt*sec=ampere*volt*sec
ActualEnduranceMin2=2*ActualEnduranceMin1; % predicted endurance for dual
strand battery, minutes
string52=[' ',num2str(nCells2),' total cells, arranged in a
2x',num2str(nCells1),' array.'];
string53=[' producing ',num2str(nVolts),' volts, and giving a predicted
endurance of ',num2str(ActualEnduranceMin2),' minutes'];
string55=[' and weighing ',num2str(weight2),' lbf.'];
disp(string52); disp(string53); disp(string55);
disp(' '); disp(' These endurance numbers do not include energy spent in
other mission phases, (e.g., take off, climb, turning).')
%
% SUMMARY
%
disp(' '); disp('SYSTEM SUMMARY');
EtaOverall=EtaMotoractual*EtaGear*EtaPropactual*EtaAircraft;
string60=[' The overall efficiency of your design in the product of the
individual subsystem efficiencies'];
string61=[' The overall efficiency is ',num2str(EtaOverall),'.'];
disp(string60); disp(string61);
52
Gold.m
% Team 4 gold.m
% Called as a function of Main_System_Design.m
function [J,CP,CT,torque] = gold(Din,Pin,Vmph)
echo off
format compact
RPM=4100;
% RPM of propeller
rho=.002274; % air density (slug/ft**2) (sea level)
% Clark Y Data
aoldeg=-3.5;
%
beta0deg=.5; %
a=8.57;
%
Cd0=.006;
%
k=.00256;
%
B=2 ;
%
% angle of zero lift of the propeller (degrees)
measured from mean chord line (typically negative)
angle from flat part of the prop to mean chord line
lift curve slope of propeller
2-d minimum drag coefficient
Cd = Cd0+k*Cl*Cl
number of blades (2 for standard type propeller)
% input nondimensional properties at each radial location
% cR=c/R, x=r/R
x=[.3,.35,.4,.45,.5,.55,.6,.65,.7,.75,.8,.85,.9,.95,1.];
cR=.09*ones(size(x));
% END OF INPUTS
% derived constants
V=Vmph*88/60; % airspeed in ft/sec
D=Din/12;
% Diameter in feet
R=D/2;
% Radius in feet
n=RPM/60;
% propeller frequency (rev/sec) or (hz)
omega=2*pi*n; % frequency of revolution of the propeller (rad/sec)
lamda=V/(omega*R);
r2d=180/pi;
Vt=omega*R;
% tip velocity (ft/sec)
J=V/(n*D);
% advance ratio
%
%
%
%
%
%
%
%
%
%
%
Output scalar constants
disp(['Airspeed V= ',num2str(V),' ft/sec'])
disp(['Propeller Diameter D= ',num2str(D),' feet'])
disp(['Propeller Radius R= ',num2str(R),' feet'])
disp(['propeller RPS n= ',num2str(n),' hertz'])
disp(['omega= ',num2str(omega),' rad/sec'])
disp(['lamda= ',num2str(lamda)])
disp(['r2d= ',num2str(r2d),' deg/rad'])
disp(['Tip speed Vt= ',num2str(Vt),' ft/sec'])
disp(['Advance Ratio J= ',num2str(J)])
disp(' ')
%derived section constants
53
c=R*cR;
% chord in feet
cin=c*12; % chord in inches
%disp(' ')
%disp('x in r/R and is nondimensional, cR=c/R, cin in the chord in inches')
%echo on
%disp([x',
cR',
cin'])
%echo off
%disp(' ')
beta1=atan(((Pin/Din)/pi)./x);
beta=beta1+(beta0deg-aoldeg)/r2d;
sigma=B*c/(pi*R);
r=x*R;
Vr=Vt*sqrt(x.*x+lamda*lamda);
phi=atan(lamda./x);
WtVt=.02*ones(size(c)); %initial guess
nr=length(c);
nr1=nr-1;
aiold=zeros(size(c));
for ii=1:40
WaVt(1:nr1)=.5*(-lamda+sqrt(lamda*lamda+4*WtVt(1:nr1).*(x(1:nr1)WtVt(1:nr1))));
ai(1:nr1)=atan(WtVt(1:nr1)./WaVt(1:nr1))-phi(1:nr1);
%ai(1:nr1)=atan((V+WaVt(1:nr1)*Vt)./(omega*r(1:nr1)-WtVt(1:nr1)*Vt))phi(1:nr1);
e=sum(abs(ai(1:nr1)-aiold(1:nr1)));
iter=['Loop index= ',num2str(ii),' error= ',num2str(e)];
%disp(iter)
if e<.0001 ; break; end
aiold(1:nr1)=ai(1:nr1);
Cl(1:nr1)=a*(beta(1:nr1)-ai(1:nr1)-phi(1:nr1));
VeVt(1:nr1)=sqrt((lamda+WaVt(1:nr1)).^2+(x(1:nr1)-WtVt(1:nr1)).^2);
gamma(1:nr1)=.5*c(1:nr1).*Cl(1:nr1).*VeVt(1:nr1)*Vt;
sinphialp(1:nr1)=sin(phi(1:nr1)+ai(1:nr1));
kappa(1:nr1)=kappa2(x(1:nr1),sinphialp(1:nr1));
WtVt(1:nr1)=B*gamma(1:nr1)./(4*pi*Vt*r(1:nr1).*kappa(1:nr1));
end
Cl(nr)=0;
ai(nr)=beta(nr)-phi(nr);
VrVt=sqrt(lamda*lamda+1);
WaVt(nr)=VrVt*sin(ai(nr))*cos(ai(nr)+phi(nr));
WtVt(nr)=VrVt*sin(ai(nr))*sin(ai(nr)+phi(nr));
VeVt(nr)=sqrt((lamda+WaVt(nr))^2+(x(nr)-WtVt(nr))^2);
kappa(nr)=0;
Cd=Cd0+k*Cl.*Cl;
ZT=(pi/8)*(J*J+pi*pi*(x.*x)).*sigma;
ZP=pi*ZT.*x;
dCTdx=ZT.*(Cl.*cos(phi+ai)-Cd.*sin(phi+ai));
dCPdx=ZP.*(Cl.*sin(phi+ai)+Cd.*cos(phi+ai));
% Overall propeller performance
CT=trapi(dCTdx,x);
CP=trapi(dCPdx,x);
eta=CT*J/CP;
54
T=CT*rho*n^2*D^4;
P=CP*rho*n^3*D^5;
HP=P/550;
Pwatt=1.356*P;
torque=P/omega;
Clmax=max(Cl);
Toz=T*16;
PestWatts=1.31*D^4*(Pin/12)*(RPM/1000)^3;
% The above approximate formula works for
% Top Flite, Zinger and Master Airscrews reasonably well.
% For Rev Up props subract .5 in from the pitch.
% For APC props use constant 1.11 instead of 1.31.
% For thin carbon fiber folding props use 1.18 instead of 1.31.
% Ref: Electric Motor Handbook, by Robert J. Boucher,
%
AstroFlight, Inc.
disp(' ')
disp(['Advance Ratio J= ',num2str(J)])
disp(['Thrust Coefficient CT= ',num2str(CT)])
disp(['Power Coefficient CP= ',num2str(CP)])
disp(['Propeller efficiency eta= ',num2str(eta)])
disp(['Speed V= ',num2str(V),' ft/sec'])
disp(['RPM= ',num2str(RPM),' rpm'])
disp(['Thrust T= ',num2str(T),' pounds'])
disp(['Thrust Toz= ',num2str(Toz),' ounces'])
disp(['Power used P= ',num2str(P),' ft*lbf/sec'])
disp(['Horsepower used HP= ',num2str(HP),' HP'])
disp(['Power used Pwatt= ',num2str(P),' watts'])
disp(['Torque used Q= ',num2str(torque),' ft*lbf'])
disp(['Torque used Q= ',num2str(torque*192),' in-oz'])
disp(['Clmax= ',num2str(Clmax)])
disp(' ')
disp(['Estimated power used, PestWatts= ',num2str(PestWatts),' watts, Ref:
Boucher'])
disp(' ')
55
D.2 - Thrust Coefficient, Power Coefficient, and Propeller Efficiency vs. Advance Ratio for
13”x6.5” Propeller
0.1
CT*= 0.019694 for J*= 0.54484
Thrust Coef, CT
0.08
0.06
0.04
0.02
0
0
0.1
0.2
0.3
0.4
0.5
Advance ratio, J=V/(nD)
0.6
0.7
Figure 30 - Thrust Coefficient vs. Advance Ratio for 13"x6.5" Propeller
0.03
Power Coef, CP
0.025
0.02
0.015
0.01
CP*= 0.012761 for J*= 0.54484
0.005
0
0
0.1
0.2
0.3
0.4
0.5
Advance ratio, J=V/(nD)
0.6
0.7
Figure 31 - Power Coefficient vs. Advance Ratio for 13"x6.5" Propeller
56
1
Eta*= 0.84089 for J*= 0.54484
Efficiency, eta
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
Advance ratio, J=V/(nD)
0.6
0.7
Figure 32 - Propeller Efficiency vs. Advance Ratio for 13"x6.5" Propeller
57
APPENDIX E – Dynamics and Controls
E.1 – Class I Sizing Plot
This section of the appendix covers the rationale used to size the horizontal and vertical
tails of the aircraft. This Class I sizing was done as a trade study by Jason Wirth. Please note
that the team’s design has changed significantly since this point but the study is included to
demonstrate the reasoning behind tail size selection.
The purpose of this trade study is to evaluate the effect that horizontal tail size has on specific
flight characteristics of the aircraft. In this study the size of the horizontal tail will be iterated to show a
range of results for flight characteristics that are related to the size. Specifically the focus of this trade
study will be the effect that horizontal tail size has on the aircraft’s static margin. This is a very important
flight dynamics and control characteristic of the aircraft because it controls the stability of the aircraft. A
larger static margin will result in a plane that is easier to control and vice versa. Finding an optimum
horizontal tail size will ensure that the static margin is sufficient for the given mission of the aircraft.
Design Variables:
For this study the only value changed will be the horizontal tail size denoted as SH. The
following table of variables contains other relevant variables which for this trade study will be held as
constants so as to not confuse the results of the study:
Property
AR
W
LH
𝑑𝜀ℎ
𝑑𝛼
̅̅̅̅
𝐶𝑤
Value
3.88
1.5
2.25
Units
lbf
ft
.55
1
1/rad
-
S
𝑋𝐴𝐶𝑤𝑏
𝑋𝐴𝐶ℎ
𝐶𝐿𝛼
3.88
.25
0.1375
ft2
-
2π
1/rad
ℎ
Table 6 - Tail Sizing Design Variables
These variables are all directly related in some way to horizontal tail size or static margin. The
source for these values comes several places. Some geometric variables are derived from Team 4’s
aircraft concept. Other quantities were derived using Roskam and Raymer by making educated
assumptions. The goal is to determine a horizontal tail size that works well with Team 4’s concept by
evaluating a range of tail sizes and determining how they affect other important parameters such as static
margin.
Measures of Merit:
As stated previously the primary method of evaluation for this study is to show the effect that
propeller size has on the power loading of the aircraft. However there are still several other factors to
consider that could prove to be important in demonstrating their relationship with horizontal tail size.
Below is a table of measures of merit:
58
Property
SM
Units
-
cH
𝑋𝐴𝐶𝑎
-
Table 7 - Tail Sizing Measures of Merit
Of these three values, SM is the focus for this study. After consulting Raymer and other
teammates a safe range for stability results in a static margin of 0.05-0.15. This number is given as a
guideline in Raymer. It is important to note that aircraft with a SM of 0.05 are typically more aerobatic
and is typical for fighter aircraft. Larger static margins result in a more stable aircraft. This trade study
will attempt to size the horizontal tail by ensuring the SM is within that range. This trade study will also
investigate the affect that horizontal tail size has on cH, which is the tail volume coefficient. Raymer lists
typical values for cH and this trade study will use those values to confirm the selected tail size is
appropriate for the type of aircraft being built.
Description of Tools Used:
To complete this study several fundamental equations and tools were used. Historical data was
used to determine a rough estimate for the initial tail sizing. General equations for tail size and tail
volume coefficients were also used to determine the design space for SH. To determine the effect of the
tail size on the static margin of the aircraft an X-Plot was created using the relevant equations below:
The follow equations were important to this study and helped to influence the results:
̅𝒂𝒄𝒂
𝒙
𝑺
̅𝒂𝒄𝒘𝒃 + 𝑪𝑳 𝜶 (𝟏 − 𝒅𝜺⁄𝒅𝜶)( 𝒉⁄𝑺)𝒙
̅𝒂𝒄𝒉
𝒙
𝒉
=
𝑺
𝟏 + 𝑪𝑳 𝜶 (𝟏 − 𝒅𝜺⁄𝒅𝜶)( 𝒉⁄𝑺)
𝒉
Equation 31
∑ 𝒎𝒊 𝒓𝒊
∑ 𝒎𝒊
𝒄. 𝒈 =
Equation 32
𝒄𝑯 =
𝑳 𝑯 𝑺𝑯
𝒄̅𝑾 𝑺𝑾
Equation 33
𝑺𝑯 =
𝒄𝑯 𝒄̅𝑾 𝑺𝑾
𝑳𝑯
Equation 34
Results & Discussion:
The code was run for a range of horizontal tail sizes from 0.1 to 1.5 ft2. To establish a rough
estimate for the size of the tail a simple plot of SH ranges based off wing area is plotted below:
59
SH versus S for historical
approximation
2
SH (ft2)
1.5
1
20%
0.5
30%
0
0
1
2
3
4
5
6
S (ft2)
Figure 33 - Horizontal Tail Area Sizing Plot
This plot shows a range of horizontal tail values based off of a percentage of the wing area. Historical
data suggests that this range will provide an accurate estimate for a range of tail area. The estimated
range for SH for an aircraft with a wing area of 3.88 ft2 is . 76 ≤ 𝑆𝐻 ≤ 1.16 ft2.
Next an analysis of the horizontal tail volume coefficient cH was done to evaluate the range of tail
sizes appropriate for the aircraft. To do this equation 3 was used and tail size was again varied. The
results are below:
CH versus SH
1
0.8
CH
0.6
0.4
0.2
0
0
0.5
1
SH
1.5
2
(ft2)
Figure 34 - Horizontal Tail Volume and Area Sizing Plot
Using the initial guess of . 76 ≤ 𝑆𝐻 ≤ 1.16 ft2 the plot shows that ranges for cH in the estimated
area vary between 0.4 and 0.7. Included below is a table from Raymer which details common c H values
for different aircraft types:
60
Aircraft Type
Sailplane
Homebuilt
General Aviation – Single Engine
General Aviation – Twin Engine
Jet Trainer
Military Cargo
Jet Transport
Typical Horizontal
cH
0.50
0.50
0.70
0.80
0.70
1.00
1.00
Table 8 - Common Tail Volumes
From the table and graph one can conclude that the cH value for this aircraft should be on the
higher side of the range found in the plot above. This is because the aircraft will have to perform several
aerobatic maneuvers such as maintaining a hover for a specific period of time. From the above resources
it can be concluded that Team 4’s aircraft should have a cH value of approximately 0.7. While this value
does not serve as a design point for the horizontal tail area it does provide a good believability check for
when a tail size is chosen.
The next area that was considered as a measure of merit was how much the moment arm of the
tail factored into the calculation of tail size area. Since the purpose of the horizontal tail is to counter the
pitching moment created by the wing a larger distance between the wing and tail will decrease the size of
the horizontal tail needed. Below is a plot of this relationship using several different cH values:
SH versus LH for given CH values
2.5
SH required (ft2)
2
1.5
Ch = .5
1
Ch = .6
Ch = .7
0.5
Ch = .8
0
1.4
1.65
1.9
2.15
2.4
LH (ft)
Figure 35 - Horizontal Tail and Moment Arm Plot
This plot shows that there is a relationship between tail size and distance between the wing and
tail. For our purposes, LH is considered to be measured from the quarter chord of the wing to the quarter
chord of the horizontal tail. Using Team 4’s estimate for LH of 2.25 ft the plot shows that the required tail
area is at least 1.2 ft2 to maintain a cH value of 0.7. This is on the higher side of the initial estimate.
However if the moment arm distance is increased to 2.5 ft the required tail area is reduced to 1.08 ft 2, a
more reasonable value. Below is a magnification of the chart above detailing the potential design space
for the aircraft:
61
Selected Design Area
SH required (ft2)
1.45
1.35
1.25
1.15
1.05
Ch = .6
0.95
Ch = .7
0.85
0.75
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
LH (ft)
Figure 36 - Horizontal Tail Area Selected Design Area
This design space shows a range of feasible values for tail area and moment arm that stay within
general cH boundaries. Larger tail sizes with shorter moment arms have not been selected for this design
space because the weathercock stability is worse than that of those points selected above. From this
design space the potential tail area size can be reduced to 1.0 ≤ 𝑆𝐻 ≤ 1.15 ft2.
From this study several conclusions can be made. Tail size clearly plays a large role in the
stability of the aircraft and proper sizing is important to ensure dynamic stability. From the results given
above a new estimate for the horizontal tail size can be made. This new value is estimated to be SH = 1.1
ft2. This value was chosen because it falls well within historical parameters for tail area sizes and tail
volume coefficients. The static margin for this tail size is 0.07. While this value is on the lower side of
the range for stability it still allows for an experienced pilot to operate the aircraft. One parameter that
this trade study did not investigate is the effect that increasing the moment arm has on the static margin.
Theoretically moving the tail further aft should increase the static margin even further. If Team 4 feels
that a static margin of 0.07 is too low it is still possible to increase it by changing the moment arm LH.
Impact and Conclusions:
From this study several conclusions can be made. Tail size clearly plays a large role in the
stability of the aircraft and proper sizing is important to ensure dynamic stability. From the results given
above a new estimate for the horizontal tail size can be made. This new value is estimated to be SH = 1.1
ft2. This value was chosen because it falls well within historical parameters for tail area sizes and tail
volume coefficients. The static margin for this tail size is 0.07. While this value is on the lower side of
the range for stability it still allows for an experienced pilot to operate the aircraft. One parameter that
this trade study did not investigate is the effect that increasing the moment arm has on the static margin.
Theoretically moving the tail further aft should increase the static margin even further. If Team 4 feels
that a static margin of 0.07 is too low it is still possible to increase it by changing the moment arm LH.
E.2 – Class II Sizing Plots
𝑪𝒏𝜷 = 𝑪𝒏𝜷,𝒘𝒃 + 𝑪𝑳𝜶𝒗
𝒙𝒗 𝑺𝒗
𝒃 𝑺
Equation 35
62
Matlab Code sm_calc4.m:
clc
clear all
close all
% WEIGHTS
fprintf('\n Tail Size\t Fuselage Length
Area \t Req. Wing Area');
Static Margin\t
Weight\t
Est Wing
servo = .0025; % all weights in slugs
motor = .0044;
gearbox = .0039;
prop = .0012;
v_stab = .0006;
fusel = 45;
fuse = .75*.114*(fusel/12)/32.2;
batt = .0039;
s_cont = .0012;
gyro = .0019;
rec = .0019;
fairing = .0053;
S_h = 250;
%in^2
h_tail = .000252*S_h*.0787;
%slugs
%AC Derivatives
cl_alh =2*pi;
cl_alwf=5.80;
eta_h=1;
% S=558.72;
x_ach=fusel-4;
de_dal=.2; %?
x_acwf = 8;
%in^3
wingsize = 3.5:.01:5;
for i = 1:length(wingsize)
wing(i) = (1/12)*1.5*wingsize(i)/32.2;
S(i) = wingsize(i)*144;
tweight(i) = prop + gearbox + motor + wing(i) + fairing + servo + batt +
s_cont + gyro + rec + fuse + h_tail + v_stab;
CG(i) = (prop*0 + gearbox*1.125 + motor*.5 + wing(i)*12 + fairing*12 +
4*servo*13 + batt*8 + s_cont*8 + gyro*12 + rec*11 + fuse*(fusel/2) +
h_tail*(fusel-5) + v_stab*(fusel-2.5)) / ...
(prop + gearbox + motor + wing(i) + fairing + 4*servo + batt + s_cont +
gyro + rec + fuse + h_tail + v_stab)/12;
wload(i) = tweight(i)*32.2 / .405;
AC2(i) = ((x_acwf/12 + (cl_alh/cl_alwf)*eta_h*((S_h)/S(i))*x_ach/12*(1de_dal))/(1 + (cl_alh/cl_alwf)*eta_h*((S_h)/S(i))*(1-de_dal)));
end
63
SM2 = (AC2-CG);
for i = 1:length(wingsize)
fprintf('\n%6.2f \t
%6.2f
%6.2f\t
%6.4f \t
%6.2f\t
%6.2f', S_h/144, fusel/12, SM2(i), tweight(i)*32.2, wingsize(i),
wload(i));
end
%
%
%
%
%
%
fprintf('\n For a tail size of: ')
fprintf('%6.2f ft^2', S_h(167)/144)
fprintf('\n Your computed Static Margin is: ')
fprintf('%6.2f\n', SM2(167))
fprintf('\n Your aircraft total weight is: ')
fprintf('%6.4f lbf\n\n', tweight(167)*32.2)
%
%
%
%
%
%
%
%
plot(S_h/144,CG)
hold on
plot(S_h/144,AC2,'--r')
grid
xlabel('Horizontal Tail Size Sh (ft^2)');
ylabel('Xcg and Xac bar');
legend('Xcg','Xac')
title('X-Plot');
This MATLAB code was used to optimize the horizontal tail size of the aircraft. It runs
in an iterative loop that outputs the static margin and respective tail size.
E.3 – Control Surface Sizing
Control surfaces were sized from the tables below. Each of the S/Sref ratios were
averaged and then a final ratio was calculated. That ratio is found in section 7.4 of the main
report. This data was taken from Roskam Part II. General aviation and homebuilt aircraft were
used for historical data because the flight speed that the r/c aircraft will travel is closest to these
aircraft. High performance fighters or larger aircraft like tankers were not chosen because they
operate in the trans- to supersonic region of flight which has significantly different aerodynamics
than the flight condition the r/c aircraft will operate under.
64
Table 9 - Homebuilt Aircraft Horizontal Tail Data
Table 10 - Homebuilt Aircraft Vertical Tail Data
65
Table 11 - Single Engine Aircraft Horizontal Tail Data
Table 12 - Single Engine Aircraft Vertical Tail Data
66
E.4 – Roll Mode Approximation
The roll mode was approximated for the condition of cruise and hover conditions.
Values given below are for cruise conditions.
𝑳𝜹𝒂
𝝓(𝒔)
= 𝟐
𝜹𝒂 (𝒔) 𝒔 + (𝑳𝜹𝒂 𝑲 − 𝑳𝒑 )𝒔
Equation 36
𝑳𝜹𝒂 =
̅𝑺𝒃𝑪𝒍𝜹𝒂
𝒒
𝑰𝒙𝒙
Equation 37
𝑳𝒑 =
̅𝑺𝒃𝑪𝒍𝒑
𝒒
𝟐𝑰𝒙𝒙 𝑼𝟏
Equation 38
̅=
𝒒
𝟏 𝟐
𝝆𝑽
𝟐 𝒆𝒇𝒇
Equation 39
𝑪𝒍𝒑 =
𝝏𝑪𝒍
𝒑𝒃
𝝏(
𝟐𝑼𝟏 )
Equation 40
𝑪𝒍𝜹𝒂 =
𝝏𝑪𝒍
𝝏𝜹𝒂
Equation 41
Flat Earth Input File:
% BasicConstants_MPX5
Version 9.2 1/18/06
% This version requires Xcg and low_wing to be defined here.
%
% OBJECTIVE: Collect into one location all the vehicle specific constants
(a.k.a. basic constants).
%
From these basic constants all the stability and control
derivatives
%
can be determined.
% INPUTS: None
% OUTPUTS: Many basic constants defined in the Matlab workspace.
%
67
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
This version is the second one
Arbitrary reference point is the firewall
Moment reference point is the c.g.
Trim velocity assumed to be 90 ft/s
*********************************************
BasicConstants - Identifies, describes, and assigns all of the
the most basic variables for analyzing the control
and stability of a generic aircraft.
*********************************************
A&AE 565 Spring 2003 - Purdue University
Note: This code is provided for a first order approximation of the dynamic
analysis of an airplane and is not intended for final designs.
Equations/Figures can be found in :
(Ref.1) Roskam, Jan. "Airplane Flight Dynamics and Automatic Flight
Controls"
Published by DARcorporation
120 E. Ninth St., Suite 2
Lawrence, KS 66044
Third Printing, 2001.
(Ref.2) Roskam, Jan. "Methods for Estimating Stability and
Control Derivatives of Conventional Subsonic Airplanes"
Published by the Author
519 Boulder
Lawrence, Kansas 66044
Third Printing, 1997.
(Ref.3) Roskam, Jan. "Airplane Design: Part VI: Preliminary Calculation
of Aerodynamic, Thrust and Power Characteristics"
Published by Roskam Aviation and Engineering Corporation
Rt4, Box 274
Ottawa, Kansas 66067
Second Printing, 1990.
% x = our value
% x (maybe) = our value, might be wrong
% ? = need to figure it out
aircraft='Team 4';
adelf = 0;
%x Two dimensional lift effectiveness parameter
Ref.(2),Equ(8.7)
alpha = 0*pi/180;
%x Trim Ange of attack [rad]. This should be zero since
our
%
equations of motion are body axis system rather
then the stability axis system.
alpha_0 = -0.0829;
%x Airfoil zero-lift AOA [rad]
altitude= 627;
%x Trim altitude [ft]
AR_h = 2.92;
%x Aspect ratio of the horizontal tail
AR_w = 4.8;
%x Aspect ratio of the wing
b_f = 4.8;
%x Span of the flap [ft] (Alieron total span)****
68
b_h = 2.25;
%x Span of the horizontal tail [ft]
b_h_oe = b_h;
%x Elevator outboard position [ft]
b_h_ie = 0.08;
%x Elevator inboard position [ft]
b_w = 4.8;
%x Span of the wing [ft]
b_v = 0.8;
%x Vertical tail span measured from fuselage
centerline[ft]
b_v_or = b_v;
%x Outboard position of rudder [ft]
b_v_ir = 0.08;
%x Inboard position of rudder [ft]
c_a = .24;
%x Chord of aileron [ft]
C_bar_D_o = 0.0152;
%x Parasite drag
Cd_0 = 0.0152;
%x Drag coefficient at zero lift (parasite drag)
c_e = .4;
%x Elevator chord [ft]
cf = 0;
%x Chord of the wing flap [ft]
c_h = .771;
%x Mean aerodynamic chord of the horizontal tail [ft]
CL = 0.1472;
%x Lift coefficient (3-D) CL=W/(1/2*rho*U^2)
CL_hb = 0;
%x (maybe) Lift coefficient of the horzontal tail/body
CL_wb= 3.83;
%4.4066;
%x Lift coefficient of the wing/body assuming iw=0
Cl_alpha_h = 2*pi;
%x 2-D Lift curve slope of horizontal tail
Cl_alpha_v = 2*pi;
%x 2-D Lift curve slope of vertical tail
Cl_alpha = 3.83;
%x (maybe) Two-dimensional lift curve slope of whole
aircraft
Cl_alpha_w = 5.794;
%x Two-dimensional lift curve slope of wing
Cm_0_r = -0.1072;
%x Zero lift pitching moment coefficient of the wing
root
Cm_o_t = -0.1072;
%x Zero lift pitching moment coefficient of the wing
tip **Cm_0_r = Cm_o_t because wing has no twist
c_r = .15;
%x MEAN Chord of the rudder [ft]
c_w = 1;
%x Mean aerodynamic chord of the wing [ft]
c_v = .37;
%x Mean aerodynamic chord of the vertical tail [ft]
D_p = 13/12;
%x Diameter of propeller [ft]
d = 0.1667;
%x Average diameter of the fuselage [ft]
delf = 0;
%x Streamwise flap deflection [rad] NO FLAPS
delta_e = 0;
%x Elevator deflection [rad]
delta_r = 0;
%x Rudder deflection [rad]
dihedral = 0.0523;
%x Geometric dihedral angle of the wing [rad],
positive for
%
dihedral (wing tips up), negative for
%
anhedral(tips down) [rad] ***EST
dihedral_h = 0;
%x Geometric dihedral angle of the horizontal tail
[rad]
e = 0.9073;
%x Oswald's efficiency factor
epsilon_t = 0;
%x Horizontal tail twist angle [rad]
epsilon_0_h = 0;
%x Downwash angle at the horizontal tail (see Note in
%
Ref.(3) under section 8.1.5.2) [rad] ***EST
eta_h = 0.99;
%x Ratio of dynamic pressure at the horizontal tail to
that of the freestream ***EST
eta_ia = 0.1;
%x Percent semi-span position of inboard edge of
aileron
eta_oa = 0.95;
%x Percent semi-span position of outboard edge of
aileron
eta_p = 0.8;
%x Propeller Efficiency ***EST
eta_v = 1.0;
%x Ratio of the dynamic pressure at the vertical tail
%
to that of the freestream
h1_fuse =0.1667;
%x Height of the fuselage at 1/4 of the its length
h2_fuse = 0.021;
%x Height of the fuselage at 3/4 of the its length
h_h = 0;
%x Height from chord plane of wing to chord plane of
69
%
horizontal tail [ft] - Fig 3.7, Ref. 2
hmax_fuse = 0.1667;
%x Maximum height of the fuselage [ft]
Ixx = .038;
%x Airplane moment of inertia about x-axis [slug-ft^2]
*** With 4 lb load
Iyy = 0.044;
%x Airplane moment of inertia about y-axis [slug-ft^2]
Izz = 0.081;
%x Airplane moment of inertia about z-axis [slug-ft^2]
Ixz = 0;
%x Airplane product of inertia [slug-ft^2]
i_h = deg2rad(0);
%x Incidence angle of horizontal tail [rad]
i_w = deg2rad(0);
%x Incidence angle of wing [rad]
k = 0.0731;
%x k of the drag polar, generally= 1/(pi*AR*e)
Lambda = deg2rad(0);
%x Sweep angle of wing [rad]
Lambda_c2 = deg2rad(0);
%x Sweep angle at the c/2 of the wing [rad]
Lambda_c4 = deg2rad(0);
%x Sweep angle at the c/4 of the wing [rad]
Lambda_c2_v = 0;%deg2rad(14.7078); %x Sweep angle at the c/2 of the vertical
tail [rad]
Lambda_c4_v = 0;%deg2rad(20.8728); %x Sweep angle at the c/4 of the vertical
tail [rad]
Lambda_c2_h = 0;%deg2rad(5.1050); %x Sweep angle at the c/2 of the
horizontal tail [rad]
Lambda_c4_h = 0;%deg2rad(7.6833); %x Sweep angle at the c/4 of the
horizontal tail [rad]
lambda = 1;
%x Taper ratio of wing
lambda_h = 0.6;
%x Taper ratio of horizontal tail
lambda_v = 0.5;
%x Taper ratio of vertical tail
l_f = 3.75;
%x Horizontal length of fuselage [ft]
l_v = 3.25;
%x Horizontal distance from aircraft
arbitrary reference point to vertical tail AC [ft]
%Ref fig 2.1 in thesis for l_v, ref pt is c/4
low_wing=-1;
%x low_wing=-1 if the wing is high ; % low_wing=1
if the wing is low
% low_wing=0 if the wing is mid
%u = 115; % ft/sec
% Trim Airspeed <<<<<<<<<old
u = 19.5; % ft/sec
%x (maybe) Trim Airspeed <<<<<<<Frontside
%u = 20; % ft/sec
% Trim Airspeed <<<<<<<Backside
M = u/1125;
%x Mach number
S_b_s = 0.2725;
%x Body side area [ft^2]
S_h = 1.75;
%x Area of horizontal tail [ft^2]
S_h_slip = .9;
%x Area of horizontal tail that is covered in
%
prop-wash [ft^2] - See Fig.(8.64) - Ref.(3)
***EST
S_o = 0.02454;
%x Fuselage x-sectional area at Xo [ft^2] %
See Fig.(7.2) - Ref.(2)
%
Xo is determined by plugging X1/l_b into:
%
0.378 + 0.527 * (X1/l_b) = (Xo/l_b)
S_w = 4.8;
%x Surface area of wing [ft^2]
S_v = 0.5;
%x Surface area of vertical tail [ft^2]
tc_w = 0.12;
%x Thickness to chord ratio of wing
tc_h = 0.06;
%x Thickness to chord ratio of horizontal tail
theta = 0*pi/180;
%x Wing twist - negative for washout [rad]
theta_h = 0*pi/180; %x Horizontal tail twist between the root and tip
%
stations,negative for washout [rad]
two_r_one = 0.0417;%0.1837; %x Fuselage depth in region of vertical tail
[ft] Ref.(2),Figure 7.5
U = u/1.7; % knots
%x Free Stream Velocity (Trim velocity) [KNOTS true]
W = 1.94;
%x Weight of Airplane [lbf]
wingloc = 1;
%x If the aircraft is a highwing: (wingloc=1), lowwing:(wingloc=0)
70
wmax_fuse = 0.1667; %x Maximum fuselage width [ft]
X1 = 3.25;
% Distance from the front of the fuselage where the
%
x-sectional area decrease (dS_x/dx)
%
is greatest (most negative) [ft] Ref.(2),Fig. 7.2
x_m = .25;
% Distance from nose of aircraft to arbitrary reference
point [ft]
%
measured positive aftward.
x_over_c_v = 4/8;
% PARAMETER ACCOUNTING FOR THE RELATIVE POSITIONS OF THE
HORIZONTAL AND VERTICAL TAILS
%
defined as the fore-and-aft distance from
leading edge of vertical fin to the
%
aerodynamic center of the horizontal tail
divided by the chord of the vertical tail
%
[nondimensional] - See Fig 7.6 of Ref. 2
Xach = 1.667;
% Distance from the leading edge of the wing mean
aerodynamic chord
%
to the aerodynamic center of the horizontal
tail (positive aftward) [ft]
Xacwb = 0.25*c_w;
% Distance from the leading edge of the wing mean
aerodynamic chord
%
to the aerodynamic center of the wing and
body.
%
Measured as positive aft, starting from the
leading edge of the mean aero. chord. [ft]
Xacw = 0.2445*c_w;
% Distance from the leading edge of the wing mean
aerodynamic chord
%
to the aerodynamic center of the wing ALONE.
%
Measured as positive aft, starting from the
leading edge of the mean aero. chord. [ft]
Xref = 0.33*c_w;
% Distance from the leading edge of the wing mean
aerodynamic chord
%
to the arbitrary moment reference point. The
equivalent force system
%
for the aerodynamic force system is given
about this point.
%
Measured as positive aft, starting from the
leading edge of the mean aero. chord. [ft]
Xcg = 0.33*c_w;
% Distance from the leading edge of the wing mean
aerodynamic chord
%
to the center of gravity.
%
Measured as positive aft, starting from the
leading edge of the mean aero. chord. [ft]
%
% Xcg is ignored until Step 2. It an be changed later in
Step 2.
%
Z_h = -0.021;
% Negative of the VERTICAL distance from the fuselage
%
centerline to the horizontal tail aero center
%
(Z_h is a negative number FOR TAILS ABOVE THE
CENTERLINE)
%
- Ref.(2), Fig.7.6
%
***This produces a bunch of interpolation
errors because
%
Roskam doesn't have data for horizontal tails
below the
%
centerline of the fuselage
71
Z_v = 0.2;
% Vertical distance from the aircraft arbirary reference
point to the vertical
%
tail aero center (positive up) - Ref.(2), Fig.
7.18
Z_w = 0.141;
% This is the vertical distance from the wing root c/4 [ft]
%
to the fuselage centerline,
%
positive downward - Ref.(2), Equ(7.5)
Z_w1 = 0.141;
% Distance from body centerline to c/4 of wing root
%
chord,positive for c/4 point
%
below body centerline (ft) - Ref.(2), Fig. 7.1
Flat Earth Outputs:
Team 4
constant(1)= 1.94
W, Weight, pounds (lbf)
Always positive.
constant(2)= 32.1446
g, Acceleration of gravity, ft/(sec*sec)
Always 32.1741.
constant(3)= 0.060352
mass, slugs
constant(4)= 0.038
Ixx, slug*ft*ft
Always positive.
constant(5)= 0.044
Iyy, slug*ft*ft
Always positive.
constant(6)= 0.081
Izz, slug*ft*ft
Always positive.
constant(7)= 0
Ixz, slug*ft*ft
constant(8)= 0.8
propeller efficiency, eta, nondimensional
Typically .5 to .8
constant(9)= 0
unassigned
constant(10)= 0.003078
constant(4)*constant(6)-constant(7)*constant(7); %gamma
A computed constant.
constant(11)= -0.97368
=((constant(5)-constant(6))*constant(6)constant(7)*constant(7))/constant(10);% c1
A computed constant.
constant(12)= 0
=(constant(4)-constant(5)+constant(6))*constant(7)/constant(10);% c2
A computed constant.
constant(13)= 26.3158
=constant(6)/constant(10);% c3
A computed constant.
constant(14)= 0
=constant(7)/constant(10);% c4
A computed constant.
constant(15)= 0.97727
72
=(constant(6)-constant(4))/constant(5);% c5
A computed constant.
constant(16)= 0
=constant(7)/constant(5);% c6
A computed constant.
constant(17)= 22.7273
=1/constant(5);% c7
A computed constant.
constant(18)= -0.074074
=(constant(4)*(constant(4)constant(5))+constant(7)*constant(7))/constant(10);% c8
A computed constant.
constant(19)= 12.3457
=constant(4)/constant(10);% c9
A computed constant.
constant(20)= 4.8
S_w, wing area, ft^2
Always positive.
constant(21)= 1
c_w, mean geometric chord, ft
Always positive.
constant(22)= 4.8
b_w, wing span, ft
Always positive.
constant(23)= 0
phiT, thrust inclination angle, RADIANS
Typically <.09 radians
constant(24)= 0
dT, thrust offset distance, ft
Typically <5 ft
constant(25)= 0.0152
CDm, CD for minimum drag for drag polar CD=k(CLstatic-CLdm)^2 + CDm
Typically .0200 to .0300
constant(26)= 0.0731
k
Typically .04 to .07
constant(27)= 0
CLdm, CL at the minimum drag point
Typically 0
constant(28)= 0.24711
CL_0, For Lift Force Equation
Typically 0 to .5
constant(29)= 3.525
CL_alpha
Lift curve slope. Typically 3 to 6
constant(30)= 0.96133
CL_de
Typically .3 to .9
constant(31)= 2.2374
CL_alpha_dot
1 to 8
constant(32)= 4.6452
CL_q
Typically 4 to 10
constant(33)= 0
CY0, For Side Force Equation
Almost always 0
73
constant(34)= -0.25669
Cy_beta
Typically -.3 to -1
constant(35)= 0
Cy_da
Typically insignificant. <5% of Cy_dr
constant(36)= 0.12134
Cy_dr
Typically .1 to .2
constant(37)= -0.020829
Cy_p
Typically insignificant. 0 to -.3
constant(38)= 0.33506
Cy_r
Typically .2 to .5
constant(39)= 0
Cl0, For Rolling Moment Equation
Almost always 0.
constant(40)= -0.049392
Cl_beta
Dihedral effect. Typically -.09 to -.3
constant(41)= 0.27568
Cl_da
Aileron effectiveness. Typically .05 to .2
constant(42)= 0.0050558
Cl_dr
Typically 0 to .02 for high vertical tail aircraft. Negative for low vertical
tail (like the Predator)
constant(43)= -0.35325
Cl_p
Damping in roll. Typically -.3 to -.6
constant(44)= 0.27034
Cl_r
Typically .07 to .2
constant(45)= -0.075671
Cm_0, For Pitching Moment Equation
Important because it must be trimmed away for steady flight.
constant(46)= -0.48903
Cm_alpha
Static longitudinal stability parameter (pitch stiffness). Usually negative.
=-CL_alpha*(static margin)
constant(47)= -1.6671
Cm_de
Elevator effectiveness. Typically -1 to -2
constant(48)= -0.94777
Cm_a_dot
Important in damping the short period mode. Typically -3 to -15
constant(49)= -5.7383
Cm_q
Damping in pitch. Important in damping the short period mode. Typically -11
to -30
constant(50)= 0
CN0, For Yawing Moment Equation
Almost always 0.
constant(51)= 0.16781
Cn_beta
Weathercock stability derivative. Typically .06 to .2
74
constant(52)= -0.0094695
Cn_da
Might exhibit adverse (negative) or proverse (positive) aileron yaw.
Magnitude<10% of Cn_dr
constant(53)= -0.082157
Cn_dr
Rudder effectiveness. Typically -.06 to -.12
constant(54)= -0.0062002
Cn_p
Often insignificant. typically -.02 to -.2
constant(55)= -0.23206
Cn_r
Damping in yaw. Typically -.09 to -.4
constant(56)= 0.33
XbarRef, nondimensional
The equivalent force system for the input aerodynamic model uses this as its
reference point.
Arbitrary Ref. Point,
Xbarref= 0.33 (fraction of chord)
Static Margin
(Xbarac-Xbarref) = 0.13873 (fraction of chord)
Typically 0.05 to 0.50 of the reference chord.
NOTE: static margin above is relative the the arbitrary ref point, NOT the
c.g.
constant(57)= 0.33
XbarCG, nondimensional
The equations of motion will use this as a reference point. The correction
from XbarRef to XbarCG is handled in the Simulink simulation
C.G. location,
Xbarcg= 0.33 (fraction of chord)
Aerodynamic center location, Xbarac= 0.46873 (fraction of chord)
Static Margin
(Xbarac-Xbarcg) = 0.13873 (fraction of chord)
Typically 0.05 to 0.50 of the reference chord.
NOTE: static margin above is relative the the c.g. This is the static margin
that will be reflected in the subsequent simulations
constant(58)= 19.3602
Trim speed, Vt, ft/sec. These may or not be used by subsequent programs.
constant(59)= 627
Trim altitude, ft
constant(60)= 0
Trim alpha, >>>DEGREES<<<This is not used by LongSC
constant(61)= 0
CLu=0, These constants are used only by LongSC and LatSC
constant(62)= 0
CDu=0
constant(63)= 0
CTxu
constant(64)= 0
Cmu
constant(65)= 0
CmTu
constant(66)= 0
CmTalpha
constant(67)= 0
CDdeltae
The code above was used to determine many of the stability derivatives found in section
E.6. This software package was written for MATLAB and given to the class by Dr. Andrisani
75
for this project. The input values were calculated using the range of equations found throughout
this report. It is important to use correct values for this code as the stability and aerodynamic
values can be greatly affected by the code. This code also confirms Team 4’s static margin of
approximately 14%.
E.5 – Gain Selection
The gain selected for the roll rate gyro was made to be 1.3 deg/deg/sec because it is the
maximum gain allowable for the gyro based on historical data. Using the roll rate approximation
from section E.4 the nominal gain is 5 deg/deg/sec. Due to the fact that the rate gyro cannot set a
gain this high the largest possible value was chosen. The overall mission should still be able to
be accomplished but the roll mode time constant will be larger than what the team desired of
0.01s.
Roll Rate Vs Time
1
0.9
0.8
0.7
Roll Rate (deg/s)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
0.2
0.25
Time (s)
0.3
0.35
0.4
0.45
0.5
Figure 37 - Plot of roll rate versus time for a chosen nominal gain
76
Roll Angle Vs Time
0.5
0.45
0.4
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.05
0.1
0.15
0.2
0.25
Time (s)
0.3
0.35
0.4
0.45
0.5
Figure 38 - Plot of roll angle versus time for a chosen nominal gain
Root Locus
0.5
0.4
0.3
0.2
0.1
Imaginary Axis
Roll Angle (deg)
0.35
System: a
Gain: 1
Pole: -120
Damping: 1
Overshoot (%): 0
Frequency (rad/sec): 120
0
-0.1
-0.2
-0.3
-0.4
-0.5
-120
-100
-80
-60
-40
-20
0
Real Axis
Figure 39 - Root Locus of Open Loop TF
77
E.6 – Flight Characteristics
E.6.1 – Short Period Approximation
𝒁𝜶 𝑴𝒒
𝝎𝒏 𝒔𝒑 = √
− 𝑴𝜶
𝑼𝟏
Equation 42
𝜻𝒔𝒑 =
𝒁
− (𝑴𝒒 + 𝑼𝜶 + 𝑴𝜶̇ )
𝟏
𝟐𝝎𝒏 𝒔𝒑
Equation 43
E.6.2 – Phugoid Approximation
𝝎𝒏 𝒑 = √
−𝒈𝒁𝒖
𝒈
≅
√𝟐
𝑼𝟏
𝑼𝟏
Equation 44
𝜻𝒑 =
−𝑿𝒖
𝟐𝝎𝒏 𝒑
Equation 45
E.6.3 – Dutch Roll Approximation
𝝎𝒏 𝒅𝒓 = √𝑵𝜷
Equation 46
𝜻𝒅𝒓 =
𝒀𝜷
− (𝑵𝒓 + 𝑼 )
𝟏
𝟐𝝎𝒏 𝒅𝒓
Equation 47
E.6.4 – Roll Mode Approximation
𝑻𝒓 =
−𝟏
𝑳𝒑
Equation 48
78
E.6.5 – Longitudinal Stability Derivatives
𝒁𝜶 =
̅𝑺(𝑪𝑳𝜶 + 𝑪𝑫𝟏 )
−𝒒
𝒎
Equation 49
𝑴𝜶 =
̅𝑺𝒄̅𝑪𝒎𝜶
𝒒
𝑰𝒚𝒚
Equation 50
̅𝑺𝒄̅𝟐 𝑪𝒎𝜶̇
𝒒
𝑴𝜶̇ =
𝟐𝑰𝒚𝒚 𝑼𝟏
Equation 51
𝑴𝒒 =
̅𝑺𝒄̅𝟐 𝑪𝒎𝒒
𝒒
𝟐𝑰𝒚𝒚 𝑼𝟏
Equation 52
𝒁𝒖 =
̅𝑺(𝑪𝑳𝒖 + 𝑪𝑳𝟏 )
−𝒒
𝒎𝑼𝟏
Equation 53
𝑿𝒖 =
̅𝑺(𝑪𝑫𝒖 + 𝑪𝑫𝟏 )
−𝒒
𝒎𝑼𝟏
Equation 54
E.6.6 – Lateral Stability Derivatives
𝑵𝜷 =
̅𝑺𝒃𝑪𝒏𝜷
𝒒
𝑰𝒛𝒛
Equation 55
̅𝑺𝒃𝟐 𝑪𝒏𝒓
𝒒
𝑵𝒓 =
𝟐𝑰𝒛𝒛 𝑼𝟏
Equation 56
79
̅𝑺𝑪𝒚𝜷
𝒒
𝒀𝜷 =
𝒎
Equation 57
𝑳𝜷 =
̅𝑺𝒃𝑪𝒍𝜷
𝒒
𝑰𝒙𝒙
Equation 58
̅𝑺𝒃𝟐 𝑪𝒍𝒓
𝒒
𝑳𝒓 =
𝟐𝑰𝒙𝒙 𝑼𝟏
Equation 59
E.6.7 – Summary Of Values for Longitudinal and Lateral Stability Derivatives
Longitudinal Value
Units
Zα
-3.16
ft/sec2/rad
Mα
-23.37
rad/sec2/rad
Mαdot
-1.16
rad/sec2/(rad/sec)
Mq
-7.03
rad/sec2/(rad/sec)
Zu
n/a
ft/sec2/(ft/sec)
Xu
n/a
ft/sec2/(ft/sec)
Table 13 - Longitudinal Stability Derivative Values
Lateral
Value
Units
Nβ
20.91
rad/sec2/rad
Nr
-3.56
rad/sec2/(rad/sec)
Yβ
-8.98
ft/sec2/rad
Lβ
-13.12
rad/sec2/rad
Lδa
73.3
rad/sec2/rad
-2.46
1/sec
Lp
Table 14 - Lateral Stability Derivative Values
80
APPENDIX F – Economics / Project Management
Team Member Hours Per Week
35
30
Hours
25
20
15
10
5
0
1
2
Student 1
3
4
Student 2
5
6
Student 3
7
8
Week
9
Student 4
10
11
Student 5
12
13
14
Student 6
Figure 40 - Team Hours per Week
Table 15 - Total Parts List
81
APPENDIX G – Build Addendum
G.1 – Flight Test Results
•
•
•
•
Flight Test 1: Saturday 11/22/08
– Location : McAllister Park, Lafayette (outdoors)
– Taxi Test, Level Uniform Flight Test, Bank Test
– Results : 30s flight; static margin insufficient; landing gear not rolling smoothly
– Adjustments Made : Added 84g of lead weight to the nose of the aircraft; adjusted
landing gear axle to level the wheels with the ground; properly secured servo
arms with provided screws
Flight Test 2: Sunday 11/23/08
– Location : McAllister Park, Lafayette (outdoors)
– Taxi Test, Level Uniform Flight Test, Bank Test, Hover Test
– Results : Passed all tests other than hover; engine power insufficient; rudder too
small
– Adjustments Made : Added balsa extension to rudder, effectively doubling its size
Flight Test 3: Monday 11/24/08
– Location : Purdue Armory, West Lafayette (indoors)
– Indoor Flight Testing
– Results : Passed all tests; will not perform aerobatic maneuvers inside
– Adjustments Made : Shimmed horizontal tail so it would be level with the
aircraft; added aesthetic improvements
Official Flight 1: Tuesday 11/25/08
– Location : Purdue Armory, West Lafayette (indoors)
– Official First Flight
– Results : Passed all tests; fixing tail caused stability issue
– Adjustments Made : Attitudes were adjusted immediately post-flight at local
establishment
Figure 41 - Built Aircraft on First Flight Day at Purdue Armory
82
G.2 – Aircraft Comparison (Conceptual & Actual)
Attribute
Design Value
Final Value
Unit
Wing Span
4.8
4.8
ft
Wing Area
4.8
4.8
ft2
Wing Chord
1
1
ft
Aspect Ratio
4.8
4.8
-
Dihedral
3
0
°
Aircraft Length
45
48.5
in
Aircraft Weight
1.94
2.32
lbs
Horizontal Tail Area
1.75
1.75
ft2
Vertical Tail Area
0.35
0.418
ft2
Propeller Size
13 x 6.5
13 x 6.5
Max Motor Power
150
120
Battery Cell Type
Lithium Polymer Lithium Polymer
W
-
Battery Cell Number
1x3
1x3
-
Battery Power
1320
1320
mAh
Battery Voltage
11.1
11.1
V
Feedback Controller Axis
Roll
Roll
-
Stall Speed
15
<12
ft/s
Does it fly?
Lord we hope
YES!
-
Table 16 - Conceptual & Actual Aircraft Specifications
G.2.1 – Structures Issues & Lessons Learned
There were several structural aspects that were modified. This was mainly due to weight
considerations. A similar sized carbon fiber square rod was substituted for the square aluminum
rod, which was roughly one third of the weight, saving almost 0.4 lbs. The horizontal and
vertical tail fins were made out of the same EPS foam from which the wing was formed since it
was one ninth of the density of the corrugated plastic chosen initially. The center fairing rib was
not used, as the two wing connector ribs were deemed strong enough. Rubber bands were used
to secure the wing to the fuselage via the wing connector ribs instead of using metal rods due to
the extra complexity that would have arisen from having to create straight chord-length holes
through the wing. Issues arose with connecting the tail structures to the fuselage. Since nothing
was formally designed to attach the tail structure to the fuselage rod, one was machined out of a
83
piece of 2x4 and glued to the carbon fiber rod. The tail was then held on with nylon nuts and
bolts. While this provided a strong base for the tail structure, it adversely affected the center of
gravity of the aircraft, causing instability that was only corrected by adding 84g of lead weight to
the nose of the aircraft.
G.2.2 – Propulsion Issues & Lessons Learned
For the most part, there were not very many issues with the propulsion system of the aircraft, but
one of the issues was a major one. The chosen motor could not provide enough power to allow
the plane to hover. The 150W motor would only output 120W at full throttle. This, according to
Main_System_Design.m, still should have allowed the aircraft to achieve hover and possibly a
slow vertical climb. It is believed that the reason the plane would not hover, besides adding 84g
of weight to the nose, is that the propeller efficiency calculations were over-idealized, causing a
significant error in the calculations for required power. The projected efficiency was between
80% and 85%, which is believed to be approximately 130-140% of the actual value. This is
based on recalculation using the known output power of the motor, the actual build weight of the
aircraft, and the pilot’s estimation of the actual thrust-to-weight ratio of the aircraft. One other
small problem that occurred with the propulsion system was the incident in which the propeller
and its mount flew off the front of the aircraft when the pilots were running the motor up while
walking it to the flight line. This was a one-time incident; the lead propulsion team member
checked to make sure the propeller was securely tightened before every time the battery was
attached to the system, regardless of whether or not the motor was going to be used or not. This
was done to guarantee the safety everyone in the class, the pilots, and even the AIAA team when
testing was being done in the Armstrong lab.
G.2.3 – Dynamics & Control Issues & Lessons Learned
Once all of the individual components of the aircraft were built, an issue arose during the
assembly phase with the position of the vertical and horizontal tail and any possible interference.
The position of the vertical tail was moved forward two inches, which resolved the conflict of
the movement of the rudder and elevator. After the initial test flights, it was determined that our
rudder was not effective enough for the mission. Despite following the Class I sizing method,
the rudder size needed to be increased by approximately 50%. This was accomplished by adding
a thin piece of balsa wood into the rudder (as shown in Figure 41). The gyroscope was used for
the final day of flight testing and during the first official flight. The pilot did state after the first
indoor flight that it did allow for increased bank control during the indoor flying. However, as
our aircraft was unable to hover, we were unable to test the effectiveness of the gyro in the hover
mode.
G.2.4 – Aerodynamics Issues & Lessons Learned
During our first test flight, our aircraft had a static margin issue which seriously affected the
flying ability of the aircraft. Despite our best attempts, the center of gravity was too close to our
84
aerodynamic center. In order to fix this problem, 84g of lead weight, as well as the battery and
speed controller, were placed at the front of the aircraft in order to move the CG forward. The
wing for our aircraft also turned out to be more difficult to build than originally planned. Since
our wing span was 4.8 feet, we had to cut out two separate wings using the CNC foam cutter.
Two aluminum rods, placed at the ¼ and ¾ chord of the wing, were used to help connect the
wings. The wings were then glued together using high-strength glue to create the full span wing.
The wing was also held onto the airplane using about 4 rubber bands on each side, running the
length of the wing chord over the wings, connected to the fairing ribs. In order to keep the wing
stable about the fuselage, an H-shaped wooden insert was added right below the wing. The
horizontal tail needed to be shimmed for the final flight in order to have it level and in the same
plane as the wing. This, however, created instability in our aircraft and became very difficult to
control during our final flight. With this instability, the aircraft had to be flown with constant
elevator deflection added by the pilot.
G.2.5 – Building Issues & Lessons Learned
The actual building of an R/C aircraft proved to have a sharp learning curve to the novice. No
one on the team had any experience building aircraft of this type, and as a result, there were
several minor issues which arose that affected the positive outcome of the plane. Issues with
such items as the control horn, rod, and servo placement, and the other electronics onboard the
aircraft proved to be time-consuming and difficult to understand. Luckily the AIAA team that
graciously allowed us to intrude upon their lab provided very beneficial information and we will
forever be indebted to them for their service. Finally, the pilots gave us much advice through all
aspects of the project and without their efforts, this project would have been dead in the hangar
from day one. We also owe them many, many thanks.
G.3 – Aircraft Specifications & Constants
INSERT TABLE WITH ALL AERODYNAMIC, GEOMETRIC, MASS, AND PROPULSION
CONSTANTS
G.4 – Updated BasicConstants.m Input File
INSERT THIS FILE
85