THE UNIVERSITY OF QUEENSLAND
ENGINEERING THESIS
Experimental Analysis of Paper Plane Flight Characteristics
Author: Natalia COOK, 43181737
Course Code: MECH4501
Supervisor: Dr Alexander Klimenko
Submission date: 2nd June 2017
A thesis submitted in partial fulfilment of the requirements of the Bachelor of Engineering degree in Mechanical Engineering
UQ Engineering
Faculty of Engineering, Architecture and Information Technology
ABSTRACT Although the concept of paper planes appears at first straight forward, understanding how the best planes can be made is a difficult and dynamic problem. They possess unstable aerodynamics, and beyond the point of launch they experience entirely uncontrolled flight. These factors compound to create a problem without a simple formulaic answer. The purpose of this project was to reach a scientific understanding of how paper aeroplanes behave, and the design characteristics required to attain predictable flight trajectories. A series of rigorous scientific experiments was established in order to gain insight into the behaviour of paper aeroplanes.
After assessing the literature available that pertained to both the flight mechanisms of real aircraft and paper aircraft, it was determined that the best approach to the problem would be to iteratively test the design characteristics linked to the stability modes – longitudinal, directional and lateral – associated with flight. A total of twenty plane designs were tested with measurements taken of their distance and air time, which were used to characterise each design’s overall performance. Of these designs, the best performing were analysed to determine which of their design characteristics contributed to their superior performance.
The findings of this analysis suggested a strong correlation between the fuselage depth and the flight performance of the tested paper planes. The fuselage depth is linked to both longitudinal and directional stability modes, however observations of the flight performance suggest the longitudinal mode contributes the greatest to the flight characteristics.
A simulation of the flight of the best performing paper plane was developed using a series of equations of motion established by Stengel (2004). These equations were calibrated in order to better match the experimental results.
It was recommended that further study expound upon the relationship found between the fuselage depth and the flight performance of the planes, in particular in scenarios outside the scope of this report. It was also recommended that other plane designs be modelled by the simulation code in order to determine if the calibration used could be linked to specific design characteristics.
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CONTENTS Abstract ...... i
Contents ...... ii
List of Figures and Tables ...... v
Chapter 1 : Introduction to Paper Planes ...... 2-1
1.1 Context ...... 2-1
1.1.1 Background ...... 2-1
1.1.2 Goals of Project ...... 2-1
1.1.3 Project Scope ...... 2-2
1.2 Literature Review ...... 2-2
1.2.1 Characteristics of Flight ...... 2-2
1.2.2 Characteristics of Paper Planes...... 2-6
1.2.3 Aeroelasticity and Aerodynamic Flutter...... 2-10
1.2.4 Previous Experiments ...... 2-10
1.3 Chapter Summary ...... 2-11
Chapter 2 : Design and Construction ...... 2-12
2.1 Paper Plane Designs ...... 2-12
2.2 Paper Plane Design Characteristics ...... 2-13
2.2.1 Characteristics of Base Paper Plane Designs ...... 2-15
2.3 Paper Plane Launcher ...... 2-16
2.3.1 Component Specification ...... 2-16
2.3.2 Construction ...... 2-16
2.4 Chapter Summary ...... 2-18
Chapter 3 : Experimentation ...... 3-19
3.1 Experimental Method ...... 3-19
3.1.1 Apparatus ...... 3-19
3.1.2 Experimental Set Up ...... 3-19
3.1.3 Method ...... 3-20
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3.2 Results ...... 3-21
3.2.1 Base Designs ...... 3-21
3.2.2 Iterations of Design 3 ...... 3-27
3.2.3 Iterations of Design 4 ...... 3-29
3.2.4 Iterations of Design 5 ...... 3-31
3.2.5 Compilation of Paper Plane Designs ...... 3-31
3.3 Chapter Summary ...... 3-33
Chapter 4 : Analysis and Discussion of Results ...... 4-34
4.1 Design 3 Characteristics ...... 4-34
4.1.1 Design 3 Fuselage ...... 4-36
4.1.2 Design 3 Centre of Gravity and Static Margin ...... 4-36
4.2 Design 4 Characteristics ...... 4-36
4.3 Design 5 Characteristics ...... 4-39
4.4 Design 1 and 2 Characteristics ...... 4-39
4.5 Design Comparisons ...... 4-39
4.6 Error Analysis ...... 4-45
4.7 Chapter Summary ...... 4-46
Chapter 5 : Simulation Comparison ...... 5-47
5.1 Equations of Motion ...... 5-47
5.1.1 Modelling a Point Mass...... 5-47
5.1.2 Rates of Change ...... 5-48
5.2 Code Inputs ...... 5-50
5.3 Code Outputs ...... 5-51
5.4 Chapter Summary ...... 5-53
Chapter 6 : Summary, Conclusions & Recommendations ...... 6-54
6.1 Summary ...... 6-54
6.2 Conclusions ...... 6-54
6.2 Recommendations ...... 6-55
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Chapter 7 References ...... 7-57
Chapter 8 Appendices ...... 8-59
A. Paper Plane Design Instructions ...... 8-59
Design 1: ...... 8-59
Design 2: ...... 8-60
Design 3: ...... 8-60
Design 4: ...... 8-61
Design 5: ...... 8-61
B. Raw Results of Experiments ...... 8-62
C. Flight Simulations Code ...... 8-66
Equations of Motion: ...... 8-66
Equations of Motion – Maximum Lift-Drag Ratio ...... 8-66
Paper Plane Flight ...... 8-66
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LIST OF FIGURES AND TABLES Figure 1.1: The Popular Explanation of Lift (Kroon, 2015) ...... 2-3 Figure 1.2: Principle of Equal Transit Time (Anderson & Eberhardt, 1999) ...... 2-3 Figure 1.3: Free Stream Velocity of Air (Anderson AND Eberhardt, 1999) ...... 2-4 Figure 1.4: True Airflow over a Wing ...... 2-5 Figure 1.5: Static Stability (Administration, 2012) ...... 2-6 Figure 1.6: Dynamic & Static Stability (Administration, 2012) ...... 2-7 Figure 1.7: Mechanisms of Stability (Model Aircraft, 2016) ...... 2-8 Figure 1.8: Dihedral Wing Shape (Blackburn, 2006) ...... 2-8 Figure 2.1: Paper Plane Designs, 1-5 from left to right ...... 2-12 Figure 2.2: Wing Area ...... 2-13 Figure 2.3: Wing Span ...... 2-13 Figure 2.4: Fuselage Depth ...... 2-14 Figure 2.5: Root and Tip Chords ...... 2-14 Figure 2.6: Mean Aerodynamic Chord (Johnson, 2015) ...... 2-15 Figure 2.7: : Launcher Disks ...... 2-16 Figure 2.8: Launcher Base ...... 2-16 Figure 2.9: Paper Plane Launcher ...... 2-17 Figure 3.1: Experimental Setup ...... 3-20 Figure 3.2: Base Design Results - Distance ...... 3-26 Figure 3.3: Base Design Results - Time ...... 3-26 Figure 3.4: Design 3 Results - Distance ...... 3-28 Figure 3.5: Design 3 Results - Time ...... 3-28 Figure 3.6: Design 4 Results - Distance ...... 3-30 Figure 3.7: Design 4 Results - Time ...... 3-30 Figure 3.8: Design 5 Results - Distance ...... 3-31 Figure 3.9: Design 5 Results - Time ...... 3-31 Figure 3.10: All Designs Results - Distance ...... 3-32 Figure 3.11: All Designs Results - Time ...... 3-32 Figure 4.1: Best Comparison - Distance ...... 4-40 Figure 4.2: Best Comparison - Time ...... 4-40 Figure 4.3: Best Comparison - Fuselage/Root Chord Ratio ...... 4-43 Figure 4.4: All Designs - Fuselage/Root Chord Ratio ...... 4-44 Figure 5.1: Paper Plane Coordinate System ...... 5-47 Figure 5.2: Flight Trajectory of Design 3 Alt 2 ...... 5-52 v
Figure 5.3: Flight Trajectory with Calibrations ...... 5-53
Table 2.1: Base Design Characteristics ...... 2-15 Table 2.2: Launcher Components ...... 2-16 Table 3.1: Base Designs - Distance ...... 3-22 Table 3.2: Base Designs - Time...... 3-23 Table 3.3: Base Designs - Processed Data (Distance) ...... 3-24 Table 3.4: Base Designs – Processed Data (Time) ...... 3-25 Table 3.5: Design 3 Alterations ...... 3-27 Table 3.6: Design 4 Alterations ...... 3-29 Table 4.1: Design 3 Alterations - Design Characteristics ...... 4-35 Table 4.2: Design 4 Alterations - Design Characteristics ...... 4-38 Table 4.3: Best Designs - Design Characteristics ...... 4-41 Table 4.4: Best Designs - Fuselage/Root Chord Ratio ...... 4-42 Table 4.5: Standard Deviation ...... 4-45 Table 5.1: Code Inputs ...... 5-50 Table 5.2: Code Outputs ...... 5-51 Table 6.1: Best Performing Plane - Design Characteristics ...... 6-55 Table 8.1: Raw Results - Design 1 ...... 8-62 Table 8.2: Raw Results - Design 2 ...... 8-62 Table 8.3: Raw Results - Design 3 ...... 8-63 Table 8.4: Raw Results - Design 4 ...... 8-64 Table 8.5: Raw Results - Design 5 ...... 8-65
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CHAPTER 1 : INTRODUCTION TO PAPER PLANES Chapter 1 of this report explores the concepts surrounding paper planes, including the history of their application, the variability of the problem surrounding them, and the measures taken to reduce the problem to an approachable level. This chapter also examines the concepts of flight in real aircraft, and how these concepts may or may not be applied in the context of paper planes. This knowledge could then be used to guide the direction of experimentation.
1.1 CONTEXT This section of the report provides the context surrounding paper planes and outlines the expectations of the project.
1.1.1 BACKGROUND Paper has been used as a method of flight simulation for a number of centuries. China is accredited with the invention of the first flying paper craft in the form of paper kites, recorded as early as 400 BC. Paper was used in the design of a number of Leonardo di Vinci’s flying vehicles, such as his ornithopter. Hot air balloons, and the floating lanterns that inspired them, also used paper as an initial design material.
In modern incarnations, paper has been used to construct simulations of aeroplanes. The origin of this methodology is still in dispute (Paper Plane Mafia, 2016), however evidences suggests that the Wright Brothers used paper planes in their initial developments of the first flying machines, which from then on inspired further iterations of the paper craft.
Although the concept of paper planes appears straight forward, understanding how the best planes can be made is a difficult and dynamic problem. They possess unsteady aerodynamics, and beyond the point of launch they experience entirely uncontrolled flight. These factors compounded to create a problem without a simple answer. In order to gain insight into the behaviour of paper aeroplanes, a series of rigorous scientific experiments were required.
1.1.2 GOALS OF PROJECT The main goal of the project was to reach a scientific understanding of how paper aeroplanes behave. This overall goal accompanied a number of sub goals, including:
• To design a system and facility that was both reliable and replicable for the testing of flight characteristics of paper aeroplanes. It was desirable that this mechanism may be able to be altered, but not essential. • To determine, in a replicable manner, the requisite design characteristics of paper planes to achieve specified criteria, and whether this design criterion was different in order to maximise the distance versus maximising the air time achieved.
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• To simulate the flight trajectories in two-dimensions of the best performing paper plane designs using mathematical models.
1.1.3 PROJECT SCOPE This project had the potential to follow a number of diverging pathways. It was therefore important to define the scope of the project in order to assist in determining the project goals. The aspects inside the project scope include:
• Experimentation upon the effects of different paper plane designs upon their flight characteristics; • Experimentation upon the effects of altering various design characteristics of the paper planes upon their flight characteristics; • Evaluation of the stability control measures; • Evaluation of effects of paper as a construction material; • Construction of a reliable launching device; • Basic mathematical simulations of flight.
Outside of the project scope includes aspects such as:
• The effects of construction materials other than paper; • Use of wind tunnel in the evaluation of lift and drag coefficients; • Comprehensive CFD model of paper plane flight.
1.2 LITERATURE REVIEW This section of the report examines the mechanisms behind flight in real aircraft, and how this can be applied to paper planes and their design. It first examines the physics of lift, and which design characteristics are the most relevant to paper plane design. This section then examines a number of stability modes associated with flight, and how these stability modes connect with the features of paper plane design.
1.2.1 CHARACTERISTICS OF FLIGHT Understanding the characteristics of flight provides a vital foundation upon which to build the theory of paper plane flight trajectories and expectations. The fundamentals of flight and how they are applied in the design of real-world aircraft will be used in the smaller-scale design of the paper plane aircraft.
During flight, all aircraft experience four forces that determine the speed and flight characteristics of the aircraft. These forces are lift, thrust, drag and gravity. Of these forces, drag – the effect of friction as a result of air flow – and gravity occur irrespective of the design of the aircraft, and typically act to negate the motion of the aircraft (Airplane 2-2
Discovery Box, 2001). Lift and thrust act positively on the motion of the aircraft. Thrust is a result of a directly applied force, such as an engine, or in the case of a paper plane, other instruments of launch such as throwing. However the mechanisms behind lift are complex, and require further explanation to understand.
1.2.1.1 THE POPULAR EXPLATATION According to D. Anderson of the Fermi National Accelerator Laboratory and S. Eberhardt, formerly of the Department of Aeronautics and Astronautics, University of Washington, (Anderson & Eberhardt, 1999) textbooks and training manuals alike tend to use a description of lift that simplifies and overlooks a number of significant phenomena of flight. This description is referred to as the ‘Popular Explanation’ of lift. It is based upon an inaccurate application of Bernoulli’s theorem. A basic deconstruction of Bernoulli’s theorem states that when air is accelerated its pressure is lowered. The popular explanation suggests that the wing of an aircraft is shaped such that air flows faster over the top and slower underneath, as shown in Error! Reference source not found..1.
FIGURE 1.1: THE POPULAR EXPLANATION OF LIFT (KROON, 2015)
The region of air above the wing therefore has a lower pressure than the region of air below the wing, and this results in the phenomenon known as lift. It is suggested that the shape of the aerofoil gives rise to this change in velocity of the air. When air separates at the leading edge, it is assumed that the section of air that travels over the top of the wing must converge with the same section of air that travels along the underside of the wing. This assumption is known as the ‘principle of equal transit’, as shown in Error! Reference source not found..
FIGURE 1.2: PRINCIPLE OF EQUAL TRANSIT TIME (ANDERSON & EBERHARDT, 1999)
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Anderson and Eberhadt are supported in their denouncement of the popular explanation by C. Woodford, an author who specialises in physics and mathematics (Woodford, 2016). Woodford further adds that, in practice, it is possible to have symmetrical aerofoils that produce lift, and this is not supported by the popular explanation.
Under further scrutiny, the popular explanation continues to break down. Such an explanation for lift cannot account for flying phenomena as inverted flight, the ground effect, or the importance of the angle of attack. The reason the theory fails is due to the ‘principle of equal transit times’ assumption. In reality, the air that flows under the wing is slowed down from the ‘free stream’ velocity of the air, meaning the air that flows over the top of the wing reaches the trailing edge first, as shown in Fig 1.3.
FIGURE 1.3: FREE STREAM VELOCITY OF AIR (ANDERSON AND EBERHARDT, 1999)
This shows that, although the Bernoulli principle is correct, information is missing in its application to flight and lift.
This explanation suggests that aerofoils should not be used as the basis of paper plane design as only the shape of the aerofoil of the wing would be taken into account, and not the numerous other features of an aircraft which will impact upon its flight characteristics.
THE PHYSICAL DESCRIPTION Instead of the popular explanation of lift, Anderson and Eberhardt use what they call the ‘Physical Description’ to explain the phenomena of lift. The physical description is based on Newton’s Laws, as opposed to the Bernoulli principle.
The first of Newton’s laws states: a body at rest will stay at rest, and a body in motion will stay in motion unless acted upon by a force. This means that if the flow of air is changed, or if air that begins at rest is accelerated into motion, there must be a force acting upon it.
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Newton’s third law states: for every action there is an equal and opposite reaction. This suggests that in order to generate lift, a wing must do something to the air. In this instance, what the wing does to the air is the action, whilst the lift that results is the reaction. Returning to Fig 1.2, it can be seen that there is no net action occurring on the air, and as such no lift can be produced. Figure 1.4 shows how air is diverted around the wing, showing upwash and downwash.
FIGURE 1.4: TRUE AIRFLOW OVER A WING
From this it can be determined that the lift of a wing is equal to the rate of change in momentum of the air it is diverting downward. By using a variation of Newton’s second law, acceleration is produced when a force acts upon a mass, lift is therefore proportional to the amount of air diverted down per second times the downward velocity of that air. For more lift, the wing can either divert more air or increase its downward velocity.
The angle of attack also plays a significant role in how the air is diverted. With an increase in the angle of attack comes a greater vertical velocity. Likewise, for the same angle of attack, should the speed of the wing increase the vertical velocity shall also increase.
Whilst the relationship between angle of attack and lift is of great significance in real aircraft, it is not a reasonable basis upon which to begin the design of paper planes. However, understanding the mechanisms of flight as they are applied to real-world aircraft provides a foundation upon which to build theories that relate to paper planes and their flight characteristics. It is important to highlight the significance of the popular explanation versus the physical description of lift in regards to further research into paper plane design. The popular explanation relies solely upon the shape of an aerofoil to generate lift, which is a difficult design feature to alter in most typical paper plane designs. However, by following the physical description of lift, it can be seen that a number of other design features of the paper plane will come into effect during the design process. The following section refers to the mechanisms of flight that directly relate to paper planes. 2-5
1.2.2 CHARACTERISTICS OF PAPER PLANES Using the foundation of the physical theories behind flight in real aircraft, these theories can then be applied to paper planes. However, not all mechanisms of flight in real aircraft can be applied to paper planes, and this is where the problem starts to diverge and become more complicated.
1.2.2.1 UNPILOTED VERSUS PILOTED FLIGHT Modern aircraft are equipped with a number of design features that enable command over their flight mechanisms. Aside from the ability to control the direction of the aircraft, pilots are also equipped with the ability to correct the aircraft’s trajectory should any disturbances occur that may result in instability.
Static and dynamic stability are terms used to describe an aircraft’s tendency to return to its prescribed flight condition after a disturbance has been introduced (Administration, 2012). Figure 1.5 shows an aircraft that has undergone some form of static imbalance, and the various reactions that may result.
FIGURE 1.5: STATIC STABILITY (ADMINISTRATION, 2012)
For a statically unstable aircraft, the flight path will tend in the direction of the disturbance, increasing the change in trajectory. For a statically neutral aircraft, there exists no further changes in the aircraft’s trajectory after the disturbance. An aircraft is considered to be statically stable if it is able to correct itself to the equilibrium flight position.
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Dynamic stability refers to the oscillatory motion of the aircraft after it has encountered a disturbance. If the oscillatory motions decrease toward the equilibrium condition, it is said to be dynamically stable. If the magnitude of the oscillatory motion increases and the aircraft orientation starts to change with increasing frequency, the aircraft is considered dynamically unstable. Figure 1.6 shows the effects of dynamic stability versus instability, in conjunction with static stability or instability. A statically and dynamically stable aircraft will reduce its oscillations until it returns to its equilibrium flight condition. A statically and dynamically unstable aircraft will increase its oscillatory motion.
FIGURE 1.6: DYNAMIC & STATIC STABILITY (ADMINISTRATION, 2012)
To ensure statically and dynamically stable flight, pilots are able to counter the effects of disturbances. The significance of unpiloted flight in terms of paper planes is that all mechanisms for stability must be inherent in the design at the point of launch. Once in the air, the plane cannot be intentionally altered in the way that a pilot may alter an aircraft’s flaps or elevators. This leads to a complex problem of design in terms of achieving stability, which will be further explained in the following section.
1.2.2.2 STABILITY The traditional paper plane differs from typical real-world aircraft in a number of significant ways beyond scale and building material. They are simplistic in design and lack a number of components which are used in the correction of destabilisation. The stability of an aircraft is defined as ‘the aircraft’s ability to sustain a specific, prescribed flight condition’ (Administration, 2012). As seen previously, the stability of an aircraft’s flight trajectory may described in terms of static and dynamic stability. However, there also exist stability issues in relation to the aircraft itself, not only its flight path. Stability Concepts by Model Aircraft 2-7
Association states that, during flight, there are three types of stability that must be taken into consideration: lateral stability, which relates to an aircraft’s tendency to roll; directional stability, which relates to an aircraft’s tendency to yaw; and longitudinal stability, which relates to the aircraft’s ability to pitch (Model Aircraft, 2016). These stability mechanisms are also mentioned by Doherty (2000). They are shown in Fig 1.7.
FIGURE 1.7: MECHANISMS OF STABILITY (MODEL AIRCRAFT, 2016)
K. Blackburn makes note of the variations between paper planes and real aircraft in reference to stability (Blackburn, 2006). Blackburn is a past world-record holder for the longest air-time in paper plane throwing. He uses predominately experimental methods when performing his analysis of paper planes, however he has conducted research into the flight characteristics of small aircraft and has attempted to reconcile this research with his experiences with real paper planes. He describes that, in real aircraft, lateral stability is achieved through a dihedral wing shape, shown in Fig 1.8.
FIGURE 1.8: DIHEDRAL WING SHAPE (BLACKBURN, 2006) The Model Aircraft Association adds that lateral stability is achieved through a sweptback shape of wings, the keel effect and a proper distribution of weight (Model Aircraft, 2016). The dihedral effect is simple to replicate in paper planes by correcting the wing shape before launch. A sweptback wing shape is a feature that may be included in the paper plane design. The weight distribution for paper planes can be neglected.
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Directional stability is addressed in real aircraft by the tail fin and to some degree the vertical side areas of the fuselage. Paper planes do not typically possess a tail fin, and thus their directional stability must be a result of the fuselage alone.
Longitudinal stability depends upon the location of the centre of gravity. Blackburn notes that many real aircraft possess a vertical tail, whereas a paper plane does not. He states that the vertical tail allows for a greater degree of freedom in relation to weight distribution, or more specifically the aircraft’s centre of gravity. In real aircraft, the centre of gravity may shift depending upon passenger and cargo loadings. However, all aircraft become unstable if the centre of gravity moves aft of a point called the neutral point. For increased stability, the location of the centre of gravity in relation to the neutral point must be placed further forward. However this results in an increase in the required up elevator. Elevators on the tails of planes can provide this, and thus these aircraft may have an increased range for their centre of gravity. For paper planes, the centre of gravity is a fixed point, and therefore does not require the additional stability provided by the vertical tail. Whilst this assertion is not incorrect, it does simplify the issue and ignore other functions of the tail. N. Brown notes that the vertical tail also assists in counteracting adverse yaw (Brown, 2015). Blackburn does suggest that a central fuselage of a paper plane (the folded section at the centre typically gripped during a throw) performs a similar function to a vertical tail in this regard, and for that reason a tail is not required.
In addition to this traditional explanation of longitudinal stability, an observable phenomenon linked to this stability mode is when a paper plane is able to regain lift by pitching upward during its flight trajectory after it has gained the required speed to do so during its downward descent. This was not observed in all paper plane designs, however, and is further examined later in this report.
An issue also raised by Blackburn is that of low-Reynolds number aerofoils, and suggests this may have an impact upon the stability of a paper plane. After further research, it was found that the issue to which he refers is the problem of laminar separation bubbles, as described by P. Lissman in their report on the effects of low-Reynolds-number aerofoils (Lissaman, 1983). Laminar separation bubbles occur due to the inability of the flow to make a transition to turbulent flow in the attached boundary layer on the surface of the aerofoil. Instead, the laminar flow separates before transition. When this happens, separation occurs in the free shear layer of flow, and the separation bubble is formed when the turbulent flow reattaches to the aerofoil surface further downstream. This bubble can result in high drag. The phenomena can affect small model aircraft, as discussed by M. Hepperle, and can be reduced through the 2-9 use of turbulators (Hepperle, 2003). However it was determined that, despite this phenomena’s relevance to model airplanes and other small aircraft, its significance to the flight characteristics of paper planes could be considered negligible.
1.2.3 AEROELASTICITY AND AERODYNAMIC FLUTTER Aeroelasticity, in application to aeronautics, is the study between the inertial, elastic and aerodynamic forces the occur when an elastic body is exposed to fluid flow. Similarly, flutter occurs as a result of interactions between aerodynamics, stiffness and inertial forces in an aircraft (Hebert & Cowan, 2008). In an aircraft, as the speed of the airflow increases, it may reach a point where the structural damping is insufficient to damp out the motions which are increasing due to aerodynamic energy being added to the structure.
There are various types of flutter that can occur in real aircraft, and each of them present their own set of problems and complications to the aircraft’s flight characteristics and structural stability. In the simplest terms, for paper planes, flutter introduces further unsteady aerodynamics. Solving these issues exist outside the scope of this project.
It is suggested by Stengel (2004) in his Princeton University presentations that, for paper planes, it is recommended that a the launch speed does not exceed 5 m/s in order to avoid the issue of flutter.
1.2.4 PREVIOUS EXPERIMENTS Two prior experiments have been analysed to determine their relevance to the project. The first of these experiments was conducted by K. Chen and W. Lai of the Xiamen Foreign Language School (Chen & Wenxin, 2011). The primary motivation behind the experiment was to determine the pneumatic parameters of several different designs of paper plane. They approached the experiment by choosing a number of initial designs and testing each to determine the simplest and most stable models. They measured the lift and drag force of the chosen designs at different angles of attack using a simplified wind tunnel. These values were then used to calculate the lift and drag coefficients using Equation 1.1 and Equation 1.2.
푳 푪 = 푳 ퟏ (1.1) 흆푽ퟐ푺 ퟐ 푫 푪푫 = (1.2) ퟏ ퟐ ퟐ 흆푽 푺 A secondary motivation behind this experiment was to determine the relationship between angle of elevation and distance. Planes were launched at different angles and the distance travelled for each test measured. This allowed for the optimum launch angle for each design 2-10 to be determined. This portion of the experiment in particular is relevant to the project, as it is a quantifiable way of measuring the effectiveness of each design.
This experiment goes on to analyse a number of other features of paper plane flight, including the influence of the position of ailerons and the influence of an empennage. These design features fall outside the scope of the project, and as such are of less relevance. It should also be noted that each of the designs that were chosen for this experiment were not subsequently altered, and thus the design itself was not a factor that was considered in the paper plane’s performance beyond the initial tests.
The second of these experiments was conducted by B. K. Ng et al, of the Nayang Technological University, School of Mechanical and Aerospace Engineering, Singapore (Ng, Kng, Pey, & Schluter, 2009). They seek to determine the aerodynamic performance of different paper planes in order to open new avenues into the design of Micro Air Vehicles (MAV). A parallel is drawn between the aerodynamic performances of paper planes and the MAVs due to their operation at relatively low Reynolds-numbers. The experiment analyses the performance of a typical dart paper plane. They approached the calculations of the pneumatic parameters in a similar fashions as the first experiment discussed. However, they used a method of launch that introduced a significant variation in the data gathered. Instead of using a replicable device, they used people to throw the plane. This means that the reliability of their data is questionable.
Both experiments offer some value in determining a direction of the project. However, each has their own flaws, particularly in reference to their method of execution and relevance to the overall goal of the project.
1.3 CHAPTER SUMMARY This chapter explored the context surrounding paper planes and their design and the most reasonable ways in which to approach the problem of designing the best paper planes to achieve a specified trajectory, whether that be measured in terms of flight distance, flight airtime or trajectory predictability. It examined the mechanisms behind flight in real aircraft and showed the significance of the link between stability modes and the design of paper planes. It gave a foundation upon which to decide which design characteristics of the paper planes to target in order to test how they affect the flight characteristics of the planes. It also examined a number of previous experiments, and noted the portions that were significant to the experiments.
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CHAPTER 2 : DESIGN AND CONSTRUCTION This chapter details the design process of both the initial paper planes and the paper plane launcher. Section 2.1 outlines the criteria used in determining the initial base designs of five paper planes. Section 2.2 outlines the design characteristics that will be altered in an attempt to improve each plane’s flight characteristics over the course of the experimentation. Section 2.3 details the design and construction of the paper plane launcher.
2.1 PAPER PLANE DESIGNS Paper planes can, in theory, have an infinite number of variations in their design. For this reason a criteria was established to provide a limit on the variations possible for the purposes of experimentation. The criteria was based upon the official International Paper Aircraft Association rules (Chipling, 2006), which are used in paper plane throwing competitions around the world. The criterion is outlined below:
• The planes must be constructed from a single piece of A4 paper, with no cuts, tears, or adhesives used in their design; • The same type and stock of paper must be used for every new design, and; • All features of the planes must be a result of folding, only.
This criterion reduced the number of potential paper plane designs, however this still left a large scope within which to work. It was for this reason that an iterative approach to the design of the planes was chosen. Several base designs were chosen, with reference to previous works (Chen & Wenxin, 2011) and recommendations found at (Fold'N'Fly, 2015). It should also be noted that paper is, in comparison to various alternatives, a poor building material in terms of aeronautical applications. These factors also contributed to the chosen designs. Each design was tested and altered during experimentation based upon the research gathered. An explanation of the chosen base designs can be found in Appendix A. Figure 2.1 shows Designs 1 – 5, from left to right.
FIGURE 2.1: PAPER PLANE DESIGNS, 1-5 FROM LEFT TO RIGHT
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2.2 PAPER PLANE DESIGN CHARACTERISTICS Prior to the execution of each experiment, the designs were required to undergo an analysis in order to determine their characteristics. This analysis assisted in the adaption of each design in order to optimise its flight performance. Each characteristic that was measured is outlined below:
Wing area, S: The area of the wing. Calculated through geometry, depending upon the wing design. This is shown in Fig 2.2.
FIGURE 2.2: WING AREA Wingspan, b: The distance between the tips of each wing at their widest point. This is shown in Fig 2.3.
FIGURE 2.3: WING SPAN Aspect ratio, AR: The ratio of the wing’s span to its mean chord. Calculated using Equation 2.1.
풃ퟐ 푨푹 = (2.1) 푺
Fuselage depth: The length of the folded centre section of the plane, from the wings to the base. This is shown in Fig 2.4. 2-13
FIGURE 2.4: FUSELAGE DEPTH Centre of Gravity (CG): The location over which the plane will naturally balance. This was manually determined by balancing the plane on a single point until it was stable.
Mean Aerodynamic chord, MAC: Used to determine the Neutral Point of an aircraft. The neutral point of an aircraft is calculated based upon the MAC of the wings and the tail. For a paper plane, which has no tail, the neutral point is equal to the MAC for the wings. The following information on calculating the MAC was taken from Airfield Models, a website dedicated to model aeroplanes (Johnson, 2015).
Figure 2.5 shows the location of the root chord and tip chord in terms of a paper plane, and Fig 2.6 shows how the MAC was calculated. It gives rise to the expression in Equation 2.2.
FIGURE 2.5: ROOT AND TIP CHORDS
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FIGURE 2.6: MEAN AERODYNAMIC CHORD (JOHNSON, 2015)
푻풊풑 푪풉풐풓풅 푻풊풑 푪풉풐풓풅 ퟐ (ퟏ + + ( ) ) ퟐ 푹풐풐풕 푪풉풐풓풅 푹풐풐풕 푪풉풐풓풅 푴푨푪 = 푹풐풐풕 푪풐풓풅 ∗ ∗ (2.2) 푻풊풑 푪풉풐풓풅 ퟑ ퟏ + 푹풐풐풕 푪풉풐풓풅 These characteristics relate to different modes of stability for the paper plane. Depending upon the results of the initial experiments for each design, these characteristics were changed as necessary to optimise the stability of the paper plane, and thereby theoretically improving its flight capabilities.
These design characteristics are given for each plane design in the following sections.
2.2.1 CHARACTERISTICS OF BASE PAPER PLANE DESIGNS Table 2.1 summarises the design characteristics for each plane.
TABLE 2.1: BASE DESIGN CHARACTERISTICS Fuselage Wing Wing Aspect Tip Root MAC MAC CG Static [mm] Area Span Ratio Chord Chord [mm] Location [mm] Margin [mm2] [mm] [mm] [mm] [mm] [mm] Design 1 28-45 8721 130 1.9378 45 169 118.97 141 62 79
Design 2 31-34 12724. 145 1.6523 118 204 164.82 131 80 51 5 Design 3 28-34 8524 150 2.6396 87 138 114.42 83 48 35
Design 4 15-37 10894. 142 1.8508 42 222 152.45 192 95 97 5 Design 5 0-52.5 8211.5 139 2.3529 74 186 138.04 140 81 59
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2.3 PAPER PLANE LAUNCHER The Paper Plane Launcher was purpose built to serve the parameters of the experiment. It was inspired by a design from SonicDad.com (2017), however was ultimately constructed without the aid of instructions or other similar material. The following sections outline the components of the launcher and the process of construction.
2.3.1 COMPONENT SPECIFICATION Table 2.2 documents the components used in the construction of the launcher along with their relevant specifications.
TABLE 2.2: LAUNCHER COMPONENTS Component Quantity Specifications 365 DC 6-12V Output Shaft Diameter: 2.3mm 5000RPM Hobby Motor Output Shaft Length: 12mm Working Voltage: DC 6-12V 2 No-load Current: 0.05A@6V 0.055A@12V No-load Speed: 2000RPM@6V 5000RPM@12V 6VDC 2.2A Regulated Input: 100-240VAC 50/60Hz 2 Plugpack Output: 6VDC 2.2A Female DC Connector 2 Size: 5.5mm OD x 2.5mm Compact Disk (CD) 2 Diameter: 120mm Plumbers Tape 1 Size: 20mm x 10m Wooden Board 1 Size: 300mm x 300mm x 3mm Wooden Dowel 1 Length: 300mm
These components, in addition to various tools and fixatives, constitute the composition of the launcher. Section 2.3.2 details the assembly of the launcher using these components.
2.3.2 CONSTRUCTION The launcher was constructed in three stages. The first stage involved the assembly of the launcher disks, as shown in Fig 2.7. The second stage involved the construction of the base, as shown in Fig 2.8. The final stage involved connecting all electrical components.
FIGURE 2.8: LAUNCHER BASE FIGURE 2.7: : LAUNCHER DISKS
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The launcher disks were constructed using the two CDs, two small (30 mm x 30 mm) off-cuts of the wooden board and two sections of the wooden dowel (15 mm length). First, a single hole was pressed into the centres of both the wooden dowel. The holes were of the same diameter as the motor output shaft and approximately 8 mm in depth. Then, using the first CD, one of the wooden board off-cuts was superglued to the top side of the CD around the hole in its centre. On the opposite side at the exact centre of the CD the wooden dowel was also glued, connected on the side without the tapped hole. This process was repeated with the other CD and wooden board and dowel. Once dried, the CDs had a strip of plumbers tape wrapped around their edges to increase the surface area and also to increase the friction of the material at that point.
The base was constructed using the wooden board and remaining dowel. The dowel was cut into six equal pieces approximately 25 mm in length. This dowel was used to provide a grove into which the motors would then be fixed once connected with the launcher disks.
Once the base was appropriately positioned, the launcher disks were attached to the motor shafts using the holes in the dowel. The motors were connected to each DC connector and fixed into place on the base using plumbers tape. The DC connecters were then plugged into each plugpack. These plugpacks could then be powered by a mains source when operation of the launcher was required. Figure 2.9 shows the fully constructed paper plane launcher.
FIGURE 2.9: PAPER PLANE LAUNCHER
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2.4 CHAPTER SUMMARY This chapter addressed the method used to measure each plane and relate those measurements back to the design characteristics. This ensured the experiments were being conducted under the direction of empirical results in addition to observational analysis of the performance of the paper plane designs. It also addressed the first of the main sub-goals outlined in Chapter 1; to design a system and facility that is both reliable and replicable for the testing of flight characteristics of paper aeroplanes. The system did not however meet the desirable goal of having adjustable launch velocities, as this requirement was removed in favour of ensuring the reliability of the build.
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CHAPTER 3 : EXPERIMENTATION The main purpose of this project was to gain insight into the behaviour of paper planes through iterative experimentation. This chapter outlines the experimental method used and the results gained through this process. The process of how and why each design was changed in the way it was and what the results indicate will be discussed in Chapter 4.
3.1 EXPERIMENTAL METHOD This section outlines the method used to experimentally assess the flight characteristics of the various paper plane designs. As on overview, each plane was launched at a constant angle of 15º (0.2618 rads) and constant speed of 2 m/s. Each plane had the distance flown until initial touchdown, air time and general notes on flight trajectory recorded. Each plane underwent seven tests.
3.1.1 APPARATUS The experiment required the following apparatus:
• Paper Plane Launcher; • Paper Planes – 5 base designs and their iterations as the tests progress; • Desk atop which to place the Launcher, approximately 1 m high; • Video recording equipment, including camera and tripod; • Designated experimental area; • Measuring tape; • Timer, and; • At least two personnel, one to operate the launcher, and one to operate the timer and video camera.
3.1.2 EXPERIMENTAL SET UP The apparatus was placed in the appropriate locations before the experiment was commenced. First, the desk was placed at the far end of the designated experimental area. The launcher was placed atop the desk facing the direction of measurement. The tape measure was extended along the floor of the experimental area from the launcher to the end of the area. The video recorder was placed above the launcher facing the same direction of plane launch. The personnel operating the stopwatch stood at the approximate location of touchdown for each design.
Figure 3.1 shows the setup.
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FIGURE 3.1: EXPERIMENTAL SETUP
3.1.3 METHOD After the setup of all experimental apparatus, the following method was adhered to.
1. The apparatus was assembled in accordance with the Experimental Setup. 2. The launcher was angled at 15°. 3. The launcher was connected to the mains power supply and started. 4. Design 1 plane was readied for launch. 5. An initial test run of the first design was undertaken to give an approximation of the expected flight trajectory. The plane was loaded into the launcher and its flight path observed in order to better place personnel. 6. The plane was fetched and checked over to ensure it had not become damaged during its flight. 7. The official tests were then undertaken. The video recorder was switched on before the paper plane was launched. The timer was begun at the point of launch and stopped at the first point of touchdown and the time recorded. 8. The distance along the floor from the launcher to the site of touchdown was measured and recorded. 9. Any notable features of the flight path was also recorded. This final point may be expounded upon once the video footage was been reviewed. 10. Again, the plane was checked over to ensure it had not become damaged during flight or landing. The wings were returned to a dihedral position if needed. 11. Steps 1 – 10 were repeated for this plane design for a total of seven recorded tests. 3-20
12. For each base design, once all seven tests were undertaken, the next set of tests for the next design were begun until all five planes were tested.
Once this initial experiment had been undertaken, a preliminary analysis of the results could be carried out. After this analysis, each design was altered to theoretically optimise its flight path, with particular focus on increasing the stability of each design. With each new design, the method was repeated until the optimal design for both flight time and flight distance was found. This resulted in the testing of 20 separate plane designs.
3.2 RESULTS The results of the experiments outlined above in Section 3.1 can be found summarised in Tables 3.1 – 3.6.
3.2.1 BASE DESIGNS Table 3.1 shows the test results in terms of distance of Designs 1 – 5. Table 3.2 shows the results of the time of Designs 1 – 5. For each design, a total of 7 recordings were taken, as this was considered to give a reasonable spread of the data. The measurements for the distance were measured to the nearest 0.01 m and the measurements for the time were measured to the nearest 0.01 s.
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TABLE 3.1: BASE DESIGNS - DISTANCE Distance [m] Take 1 2.66 Take 2 1.75 Take 3 2.1 Design 1 Take 4 2.26 Take 5 2.47 Take 6 2.23 Take 7 2.46 Take 1 1.44 Take 2 1.84 Take 3 1.84 Design 2 Take 4 1.39 Take 5 1.58 Take 6 1.73 Take 7 1.72 Take 1 2.85 Take 2 3.82 Take 3 3.92 Design 3 Take 4 2.41 Take 5 4.06 Take 6 3.85 Take 7 3.56 Take 1 2.79 Take 2 3.26 Take 3 2.14 Design 4 Take 4 1.56 Take 5 2.62 Take 6 2.93 Take 7 2.52 Take 1 3.28 Take 2 3.31 Take 3 2.65 Design 5 Take 4 2.68 Take 5 2.6 Take 6 2.43 Take 7 2.49
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TABLE 3.2: BASE DESIGNS - TIME Time [s] Take 1 1.25 Take 2 0.98 Take 3 1.33 Design 1 Take 4 1.37 Take 5 1.08 Take 6 1.43 Take 7 1.35 Take 1 1.32 Take 2 1.3 Take 3 1.25 Design 2 Take 4 1.07 Take 5 1.14 Take 6 1.23 Take 7 1.15 Take 1 1.21 Take 2 1.51 Take 3 1.35 Design 3 Take 4 0.97 Take 5 1.49 Take 6 1.36 Take 7 1.33 Take 1 1.08 Take 2 1.33 Take 3 1.43 Design 4 Take 4 1.36 Take 5 1.32 Take 6 1.12 Take 7 0.85 Take 1 1.08 Take 2 1.35 Take 3 0.95 Design 5 Take 4 1.09 Take 5 0.97 Take 6 1.1 Take 7 1.04
The data in Tables 3.1 and 3.2 were processed to produce box plots. For each set of data the minimum, first quartile, median, third quartile and maximum data point was found. These results are summarised in Tables 3.3 and 3.4.
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TABLE 3.3: BASE DESIGNS - PROCESSED DATA (DISTANCE)
Distance Minimum Value 1.75 First Quartile 2.165 Design 1 Median Value 2.26 Third Quartile 2.465 Maximum Value 2.66 Minimum Value 1.39 First Quartile 1.51 Design 2 Median Value 1.72 Third Quartile 1.785 Maximum Value 1.84 Minimum Value 2.41 First Quartile 3.205 Design 3 Median Value 3.82 Third Quartile 3.885 Maximum Value 4.06 Minimum Value 1.56 First Quartile 2.33 Design 4 Median Value 2.62 Third Quartile 2.86 Maximum Value 3.26 Minimum Value 2.43 First Quartile 2.545 Design 5 Median Value 2.65 Third Quartile 2.98 Maximum Value 3.31
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TABLE 3.4: BASE DESIGNS – PROCESSED DATA (TIME)
Time Minimum Value 0.98 First Quartile 1.165 Design 1 Median Value 1.33 Third Quartile 1.36 Maximum Value 1.43 Minimum Value 1.07 First Quartile 1.145 Design 2 Median Value 1.23 Third Quartile 1.275 Maximum Value 1.32 Minimum Value 0.97 First Quartile 1.27 Design 3 Median Value 1.35 Third Quartile 1.425 Maximum Value 1.51 Minimum Value 0.85 First Quartile 1.1 Design 4 Median Value 1.32 Third Quartile 1.345 Maximum Value 1.43 Minimum Value 0.95 First Quartile 1.005 Design 5 Median Value 1.08 Third Quartile 1.095 Maximum Value 1.35
Using this processed data, two graphs depicting box plots were generated, one each for the distance and time measurements. The box plots give an indication of the spread of data. The whiskers of each plot denote the maximum and minimum measurements, whilst the box shows the median, first quartile and third quartile. Shorter boxes and whiskers suggest a more precise spread of data. This data is shown in Fig 3.2 and 3.3.
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Base Designs - Distance 4.5
4
3.5
3
2.5
2
Distance [m] Distance 1.5
1
0.5
0 Design 1 Design 2 Design 3 Design 4 Design 5 Design
FIGURE 3.2: BASE DESIGN RESULTS - DISTANCE
Base Designs - Time 1.6
1.4
1.2
1
0.8
0.6 Distance [m] Distance
0.4
0.2
0 Design 1 Design 2 Design 3 Design 4 Design 5 Design
FIGURE 3.3: BASE DESIGN RESULTS - TIME 3-26
3.2.2 ITERATIONS OF DESIGN 3 After analysing the results of the Base Designs tests, Design 3 was chosen to move forward with the analysis of the design characteristics of the paper planes, as it had the overall best flight performance in terms of both distance and airtime measured. A total of seven alternative designs, denoted ‘Alt’ designs, were generated by changing the various design characteristics outlined in Section 2.2. Table 3.5 summarises the differences in the design characteristics of each alternative design.
TABLE 3.5: DESIGN 3 ALTERATIONS Design Name Design Characteristic Explanation Alt 1 Fuselage (Shallow) As the fuselage depth of the Base Design 3 was chosen arbitrarily, a shallow fuselage depth was chosen to determine the effects of changing the fuselage. Alt 2 Fuselage (Deep) As with Alt 1, a deeper fuselage depth was chosen to determine the effects of changing this characteristic. Alt 3 Centre of Gravity The centre of gravity (CG) was brought closer to the tip of the plane by increasing the number of folds close to the tip. Alt 4 Static Margin (Increase) The static margin was increased by increasing the length of the plane, making a shallow fold at the tip of the plane. Alt 5 Static Margin (Decrease) The static margin was decreased by decreasing the length of the plane, making a deep fold at the tip of the plane. Alt 6 Fuselage (Deeper) After the comparative success of Alt 2 and its deep fuselage, the fuselage depth was increased again. Alt 7 Fuselage (Angled) The fuselage was angled to be shallow at the front of the plane and deep at the back.
For further explanation as to the specific design changes made for each alteration, see Section 4.1.
Following the process outlined in Section 3.2.1 above, Figs 3.4 and 3.5 show the test results of the alternative designs. To view the raw distance and time data used to generate these graphs, see Appendix B.
An initial analysis of these results suggested the fuselage of the paper plane designs was one of the key design features.
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Design 3 - Distance 6
5
4
3
2 Distance [m] Distance
1
0 Alt 1 Alt 2 Alt 3 Alt 4 Alt 5 Alt 6 Alt 7 Base Design
FIGURE 3.4: DESIGN 3 RESULTS - DISTANCE
Design 3 - Time 1.8
1.6
1.4
1.2
1
0.8 Time[s] 0.6
0.4
0.2
0 Alt 1 Alt 2 Alt 3 Alt 4 Alt 5 Alt 6 Alt 7 Base Design
FIGURE 3.5: DESIGN 3 RESULTS - TIME
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3.2.3 ITERATIONS OF DESIGN 4 Using the data gained from an analysis of the flight characteristics of the Design 3 alterations, Design 4 underwent a number of alterations to assess the effectiveness of these design changes considering the differences between the base designs. These design changes are summarised in Table 3.6.
TABLE 3.6: DESIGN 4 ALTERATIONS Design Name Design Characteristic Explanation Alt 1 Fuselage (Deep) As the fuselage depth of the Base Design 4 was chosen arbitrarily, and considering the results of the previous Design 3 tests, a deep fuselage depth was chosen to determine the effects of changing the fuselage. Alt 2 Static Margin (Increase) The static margin was increased by increasing the length of the plane, making a shallow fold at the tip of the plane. Alt 3 Static Margin (Decrease) The static margin was decreased by decreasing the length of the plane, making a deep fold at the tip of the plane. Alt 4 Static Margin (Decrease) Considering the results of Alt 1 and Alt 3, a plane and Fuselage (Deep) combining both a decreased static margin and a deeper fuselage was tests.
For further explanation as to the specific design changes made for each alteration, see Section 4.1.
The results of these tests can be seen in Fig 3.6 and 3.7. To view the raw distance and time data, see Appendix B.
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Design 4 - Distance 5
4.5
4
3.5
3
2.5
2
Distance [m] Distance 1.5
1
0.5
0 Alt 1 Alt 2 Alt 3 Alt 4 Base Design
FIGURE 3.6: DESIGN 4 RESULTS - DISTANCE
Design 4 - Time 1.8
1.6
1.4
1.2
1
0.8 Time[s] 0.6
0.4
0.2
0 Alt 1 Alt 2 Alt 3 Alt 4 Base Design
FIGURE 3.7: DESIGN 4 RESULTS - TIME
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3.2.4 ITERATIONS OF DESIGN 5 As with Design 4, Design 5 underwent a number of alterations. Alt 1 had a deep tip fold and Alt 2 had a deeper fuselage and tip fold. Figures 3.8 and 3.9 show the processed results of the tests.
Design 5 - Distance 3.5 3 2.5 2 1.5 1
Distacne [m] Distacne 0.5 0 Alt 1 Alt 2 Base Design
FIGURE 3.8: DESIGN 5 RESULTS - DISTANCE
Design 5 - Time 1.6 1.4 1.2 1 0.8 0.6
Time[s] 0.4 0.2 0 Alt 1 Alt 2 Base Design
FIGURE 3.9: DESIGN 5 RESULTS - TIME
3.2.5 COMPILATION OF PAPER PLANE DESIGNS Using the data gained from an analysis of the iterations of Design 3, 4 and 5, final improvements were made to Designs 1 and 2. The results of all base designs and their alterations can be seen in Fig 3.10 and 3.11.
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All Designs - Distance 6 5 4 3 2 1
Distance [m] Distance 0 Base Alt 1 Base Alt 1 Base Alt 1 Alt 2 Alt 3 Alt 4 Alt 5 Alt 6 Alt 7 Base Alt 1 Alt 2 Alt 3 Alt 4 Base Alt 1 Alt 2 Design 1 Design 2 Design 3 Design 4 Design 5 Design
FIGURE 3.10: ALL DESIGNS RESULTS - DISTANCE
All Designs - Time 7 6 5 4 3 2 Time[s] 1 0 Base Alt 1 Base Alt 1 Base Alt 1 Alt 2 Alt 3 Alt 4 Alt 5 Alt 6 Alt 7 Base Alt 1 Alt 2 Alt 3 Alt 4 Base Alt 1 Alt 2 Design 1 Design 2 Design 3 Design 4 Design 5 Design
FIGURE 3.11: ALL DESIGNS RESULTS - TIME
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3.3 CHAPTER SUMMARY This chapter explained the methods used to assess the flight characteristics of the various paper plane designs. It showed the results of the experiments in a format in which they could then be later analysed at a later point in the report. Preliminary analysis shows the best performing plane of the twenty tested to be Design 3 Alt 2, a design which focused on the effects of changing the fuselage depth of the plane. Further analysis of this result will be undertaken in Chapter 4.
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CHAPTER 4 : ANALYSIS AND DISCUSSION OF RESULTS A major goal of this project was to determine the specific design characteristics associated with certain paper plane performances. This goal is expanded upon in this chapter, as the results of the experiments are analysed against specific design changes.
An inherent aspect of the problem of paper plane design comes in the relationship between the various design characteristics. Altering one design characteristic may impact on one or more of the other characteristics, such as how increasing the fuselage depth of the design will directly decrease the wing span. This analysis will examine the relationships between the design characteristics, and determine which characteristics or combination of characteristics have the greatest impact upon flight performance.
4.1 DESIGN 3 CHARACTERISTICS Each base design underwent a number of iterative changes. To understand the decision making processes behind which design changes were considered an improvement upon the base designs, this section will examine the results of each design in the order they were created, rather than following the numerical naming system of the base designs. In this way, it can be understood why certain design characteristics persist throughout the testing process, and why others may feature only once.
As previously stated, Design 3 was chosen as an initial starting point after preliminary tests showed its improved performance in both distance and time over the other base designs. Objectively, it also experienced the most stable flight trajectory, despite the comparatively larger spread of data.
Table 4.1 shows the design characteristics of the base design and all of its alterations, with all measurements recorded to the nearest millimetre. The fuselage depth was measured at two points along the plane, at the tip and at the tail. A fuselage depth given as 20-40 suggests a depth of 20 mm at the tip of the plane and 40 mm at the tail.
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TABLE 4.1: DESIGN 3 ALTERATIONS - DESIGN CHARACTERISTICS Fuselage Average Wing Wing Aspect Tip Root MAC MAC CG Static [mm] Fuselage Area Span Ratio Chord Chord [mm] Location [mm] Margin [mm] [mm2] [mm] [mm] [mm] [mm] [mm] Base 28-34 31 8524 150 2.639 87 138 114.426 83 48 35 Alt 1 16-13 14.5 10908 183 3.070 87 138 114.426 84 45 39
Alt 2 32-42 37 8693 130 1.944 87 138 114.426 84 49 35 Alt 3 22-30 26 8815.5 161 2.940 87 120 104.376 72 32 40 Alt 4 28-34 31 10863 153 2.154 87 153 123.025 100 57 43 Design3 Alt 5 28-34 31 8310 150 2.707 87 126 107.690 75 39 36 Alt 6 45-50 47.5 6105 115 2.166 88 137 114.278 88 49 39 Alt 7 16-42 29 9278 163 2.863 87 140 115.562 100 49 51
Each of these values was calculated using the methods outlined in Section 2.2. The column indicating the MAC is the calculated length of the Mean Aerodynamic Chord. The column indicating the MAC location is the measured location of the MAC from the tip of the plane. CG is the centre of gravity, measured from the tip of the plane.
Each of these alterations can be grouped into the general characteristic each design was aiming to test. Alt 1, 2, 6 and 7 all tested changes made to the fuselage, Alt 3 tested changes to the centre of gravity, and Alt 4 and 5 tested changes to the static margin. The following sections will use this breakdown to individually assess the effectiveness of each of these design changes.
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4.1.1 DESIGN 3 FUSELAGE From the results seen in Section 3.2.2, Alt 1 – the shallow fuselage design – performs poorly in comparison to the other designs, at a median distance of only 0.47 m and median time of 0.66 seconds. Alt 2 – deep fuselage – performs the best of the fuselage designs, with a median distance of 4.44 m and median time of 1.43 seconds. It does however have one of the larger spreads of data. Alt 6 – deeper fuselage – also performs comparatively poorly, with a median distance of 1.9 m and time of 0.43 seconds. Alt 7 – angled fuselage – measures around the middle of the spectrum with a median distance of 2.52 m and time of 1.07 seconds. This was compared to the base design results of a median distance of 3.82 m and time of 1.35 seconds.
These results coupled with the other design characteristics measured suggest there is, for at least this base design, a correlation between the fuselage depth and the flight performance of the planes. Early analysis of these results suggested that the decline of the performance of the planes after a certain depth – i.e. the difference in performance between Alt 2 and Alt 6 – is due to the relationship between the fuselage depth and the wing area of the plane. From the results of the Alt 6 design, both empirical and observational, it is suggested that the wings no longer produce lift. As lift is related to the wing area of the plane, this in turn suggests that the wing area of this Alt 6 design has decreased past the point that lift can reasonably be produced.
This knowledge was used to ascertain a lower boundary condition for the wing area of future designs. All designs from that point forth had wing areas greater than 6105 mm2.
4.1.2 DESIGN 3 CENTRE OF GRAVITY AND STATIC MARGIN From the results seen in Section 3.2.2, Alt 3, 4 and 5 all performed poorer than the base design. Alt 3 – the double tip fold – had a median distance of 3.32 m and time of 1.17 seconds. It should be noted that this time measurement had the largest spread of data. Alt 4 – the shallow tip fold – had a median distance of 2.63 m and time of 1.03 seconds. Alt 5 – the deep tip fold – had a median distance of 3.33 m and time of 1.02 seconds. No correlations were drawn from these results in isolation.
4.2 DESIGN 4 CHARACTERISTICS The design changes made to Design 4 were based around the results of the Design 3 tests. For the designs that significantly lowered the performance of the plane, such as the shallow fuselage of Alt 1 or the comparatively deeper fuselage of Alt 6 were removed from the testing
4-36 procedures. This left four alternative designs to be tested. These designs were the deep fuselage, increase in static margin, decrease in static margin, and a combination design of a decreased static margin and deep fuselage. The design characteristics of these planes can be seen in Table 4.2.
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TABLE 4.2: DESIGN 4 ALTERATIONS - DESIGN CHARACTERISTICS Fuselage Average Wing Wing Aspect Tip Root Chord MAC MAC CG Static [mm] Fuselage Area Span Ratio Chord [mm] [mm] Location [mm] Margin [mm] [mm2] [mm] [mm] [mm] [mm] Base 15-37 26 10743 142 1.876 42 222 152.454 192 95 97 Alt 1 20-55 37.5 8177 111 1.506 43 224 153.950 192 97 95 Alt 2 15-37 26 11284 140 1.736 46 243 166.881 209 113 96
Design4 Alt 3 15-37 26 11061 141 1.797 45 204 141.421 169 83 86 Alt 4 20-40 30 9226 124 1.666 46 202 140.354 170 82 88
As with Table 4.1, each of the values of Table 4.2 was calculated using the methods outlined in Section 2.2. The column indicating the MAC is the calculated length of the Mean Aerodynamic Chord. The column indicating the MAC Location is the measured location of the MAC from the tip of the plane.
Each of these alterations tests a different design characteristic based upon the results of the Design 3 tests. Alt 1 tests the effects of a deep fuselage across different designs. Alt 2 and 3 test the effects of changing the static margin. Alt 4 is a combination of a deeper fuselage and a shorter static margin (deeper tip fold) to test the effects of changing more than a single design characteristic.
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From the results seen in Section 3.2.3, Alt 1 – deep fuselage - performs marginally better than the base design, with a median distance of 2.85 m and time of 1.02 seconds. Alt 2 – shallow tip fold – performs marginally poorer than the base design, with a median distance of 2.79 m and time of 0.93 seconds. Alt 3 – deep tip fold – is the best performing of the Design 4 alterations, with a median distance of 4.13 m and time of 1.36 seconds. Alt 4 – the combination of deep fuselage and deep tip fold – is the poorest performing the this set of designs, with a median distance of 2.62 m and time of 1.32 seconds.
For this design, it was apparent that reducing the static margin and bringing the centre of gravity closer to the tip of the plane improved its flight characteristics the most.
4.3 DESIGN 5 CHARACTERISTICS Furthering the results of the tests performed on Design 4 and Design 3, the Design 5 alterations were made to test the previous successful improvements across other designs. This left only two design characteristics to be tested: a deep fuselage and a deep tip fold to shorten the static margin. However, as can be seen from Fig 3.8 in Section 3.2.4, this design could not be improved upon using the methods outlined above.
4.4 DESIGN 1 AND 2 CHARACTERISTICS Considering the relative performances of the base Design 1 and Design 2, improving these designs to perform better than previous iterations of other designs was not feasible. However, to determine the overall effectiveness of the design improvements made to the other designs, Design 1 and Design 2 each underwent a single design change in order to improve their flight characteristics compared to their base design, rather than the other designs. These results are shown in the following Section 4.5.
4.5 DESIGN COMPARISONS After all experiments were conducted, it could be seen that the flight characteristics of each base design had been improved upon using some combination of changes in the design characteristics. Figures 4.1 and 4.2 offer a direct comparison between the original base designs and their best performing alternative design. Table 4.3 shows the design characteristics that correspond to these designs.
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Best Comparison - Distance
6
5
4
3
2 Distance [m] Distance
1
0 Base Alt 1 Base Alt 1 Base Alt 2 Base Alt 3 Base Design 1 Design 2 Design 3 Design 4 Design 5
Design
FIGURE 4.1: BEST COMPARISON - DISTANCE
Best Comparison - Time 1.8
1.6
1.4
1.2
1
0.8 Time[s] 0.6
0.4
0.2
0 Base Alt 1 Base Alt 1 Base Alt 2 Base Alt 3 Base Design 1 Design 2 Design 3 Design 4 Design 5 Design
FIGURE 4.2: BEST COMPARISON - TIME 4-40
TABLE 4.3: BEST DESIGNS - DESIGN CHARACTERISTICS Fuselage Wing Area Wing Span Aspect Ratio Tip Chord Root Chord MAC MAC CG Static Margin [mm] [mm2] [mm] [mm] [mm] [mm] Location [mm] [mm] [mm] Base 28-45 8721 130 1.937 45 169 118.975 141 62 79 Design 1 Alt 1 34-45 7797.5 130 2.167 43 157 110.83 129 64 65 Base 31-34 12724.5 145 1.652 118 204 164.828 131 80 51 Design 2 Alt 1 34-40 9790 136 1.889 66 181 132.423 135 70 65 Base 28-34 8924 150 2.521 87 138 114.426 83 48 35 Design 3 Alt 2 32-42 8693 130 1.944 87 138 114.426 84 49 35 Base 15-37 10894.5 142 1.850 42 222 152.454 192 95 97 Design 4 Alt 3 15-37 11061 141 1.797 45 204 141.421 169 83 86 Design 5 Base 0-52.5 8211.5 139 2.352 74 186 138.041 140 81 59
All paper plane designs, with the exception of Design 5, have undergone design changes that have improved their flight characteristics. Designs 1 and 2 are of note, as although the overall distance travelled by each plane increased with their design change, their air time conversely decreased. This is due to the change in their flight trajectory shape, observed during the tests. For the base designs of 1 and 2, their flight shape resembled a steep parabola, with a sharp incline close to launch followed by a nose dive to the ground. This explains their short distance results, but comparatively higher airtime results. With the changes made to their designs, this flight path shape was drastically changed to reflect closer to the other designs. Further on the shape of the flight trajectories, the best performing plane design, Design 3 Alt 2, had a trajectory more complex than a parabola. It experienced the initial parabolic pattern, but upon its descent reached a point of sufficient speed that lift could once again be produced by the wings and continued on an upward trajectory before finally coming to rest. This noteworthy flight pattern will be further examined in Chapter 5.
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Pinpointing a single design characteristic as responsible for the improved flight characteristics of the designs is not a realistic representation of the results of the experiment. However, by examining the relationship between certain design characteristics and how they affect one another, a general trend that reflects the results of the distance experiments can be shown.
The most significant design characteristics that have improved the flight trajectories of the paper planes have been the fuselage depth and the static margin. The first of these design characteristics was examined to determine its bearing upon the flight performance of the planes.
To give a better representation of the fuselage depth in terms of each individual design, an average fuselage depth was taken for each plane. Using this value, a ratio between the fuselage depth and the root chord measurement of each plane was calculated, shown in Table 4.4. The root chord was used as it gives a representation of the length of each plane. Upon plotting these results, it was seen that the trend of these values correlates with the general shape of the distance measurements taken for the designs. This is shown in Fig 4.3.
TABLE 4.4: BEST DESIGNS - FUSELAGE/ROOT CHORD RATIO Average Fuselage [mm] Root Chord [mm] Fuselage/Root Chord Design 1 Base 36.5 169 0.215976331 Alt 1 39.5 157 0.251592357 Design 2 Base 32.5 204 0.159313725 Alt 1 37 181 0.20441989 Design 3 Base 31 138 0.224637681 Alt 2 37 138 0.268115942 Design 4 Base 26 222 0.117117117 Alt 3 26 204 0.12745098 Design 5 Base 26.25 186 0.141129032
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Best Comparison - Distance 6
5
4
3 Distance [m] 2
1
0 Base Alt 1 Base Alt 1 Base Alt 2 Base Alt 3 Base Design 1 Design 2 Design 3 Design 4 Design 5 Design
FIGURE 4.3: BEST COMPARISON - FUSELAGE/ROOT CHORD RATIO This results suggests there is a strong relationship between the ratio of fuselage to the length of the plane and flight performance. Although Design 4 breaks away from the trend somewhat, it still follows that a higher fuselage-to-root chord ratio results in better flight performance.
Analysing the static margin in relation to the root chord in a similar way did not yield useful results. This suggests that for designs that showed improvement with the alteration of the static margin, such as was the case for Design 4, showed this improvement due to the change in root chord and therefore the fuselage-root chord ratio, rather than a direct link to the static margin itself.
However this is only observing these results over the base design compared to its best performing alteration. Applying this same analysis to all plane designs results in Fig 4.4.
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All Designs - Distance 6
5
4
3
2 Distance [m] Distance
1
0 Base Alt 1 Base Alt 1 Base Alt 1 Alt 2 Alt 3 Alt 4 Alt 5 Alt 6 Alt 7 Base Alt 1 Alt 2 Alt 3 Alt 4 Base Alt 1 Alt 2 Design 1 Design 2 Design 3 Design 4 Design 5 Design
FIGURE 4.4: ALL DESIGNS - FUSELAGE/ROOT CHORD RATIO As can be seen, the trend appears to hold for a predominate portion of the designs, with a few notable exceptions. The first major exception can be seen in Design 3 Alt 6, where the fuselage-root chord ratio far outstrips the performance of the design. This anomaly may be ignored, as it has been previously noted that this design lays outside of the lower boundary limit of the wing area required to produce lift. The other most notable exception to the trend can be seen in Design 4 Alt 3, which is the best performing design of that set but has a comparatively low fuselage-root chord ratio.
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4.6 ERROR ANALYSIS The spread of the data for each plane can be reasonably approximated by observing the graphs of the box plots. This spread of data is due predominately to human error in operating the measuring equipment, such as observing by eye the location of initial touchdown and the operation of the timer. This also applies to the operation of the launcher, which was constructed with the reduction of human error in mind, but still requires a human operator. The other main source of error and uncertainty is in the nature of the problem, as paper planes have unstable and unpredictable flight dynamics.
Table 4.5 summarises the standard deviation for each plane design for both their distance and time measurements.
TABLE 4.5: STANDARD DEVIATION Standard Deviation Design Distance [m] Time [s] Base 0.2969 0.1657 Design 1 Alt 1 0.6221 0.1499 Base 0.1826 0.0915 Design 2 Alt 1 0.2888 0.0891 Base 0.6230 0.1836 Alt 1 0.2229 0.1215 Alt 2 0.6935 0.1738 Alt 3 0.2883 0.3878 Design 3 Alt 4 0.6875 0.1390 Alt 5 0.6160 0.2628 Alt 6 0.2798 0.0783 Alt 7 0.7042 0.2671 Base 0.5568 0.2049 Alt 1 0.3446 0.1936 Design 4 Alt 2 0.3291 0.1000 Alt 3 0.5033 0.3054 Alt 4 0.3362 0.1224 Base 0.3644 0.1316 Design 5 Alt 1 0.5187 0.1951 Alt 2 0.5788 0.2027
A general trend can be observed that the poorer performing planes have a smaller spread and therefore a smaller deviation from the median measured result. This suggests that most of the uncertainty surrounding the better performing planes is due to the nature of their uncertain flight trajectories and less emphasis on the human error involved in the measurements. This in turn suggests that further study into improving the stability of these better performing designs may be required.
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4.7 CHAPTER SUMMARY This chapter analysed the results gained during the experiments undertaken in Chapter 3, and most predominately addressed the first sub goal of the project, which was determining the required design characteristics to achieve specific flight characteristics. The most significant outcome of this analysis was the correlation drawn between the fuselage-root chord ratio and the flight performance, which suggests that these two characteristics in combination have the greatest affect upon the flight performance and characteristics of paper planes. The limiting factor of this result was the link between increasing the fuselage depth and the corresponding decrease in wing span and wing area. There was shown to be a lower limit allowable for the wing area, before the plane could no longer produce lift.
Also shown in this chapter was an error analysis, which showed a trend of greater uncertainty associated with better performing planes versus the lower uncertainty of the poorer performing planes.
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CHAPTER 5 : SIMULATION COMPARISON In addition to determining the design characteristics related to predictable and preferable flight characteristics and trajectories, it is useful to simulate an expected flight path in two dimensions. This was achieved by examining the forces behind lift and drag.
This chapter explains the equations of motion behind flight and how they relate to paper planes. It also examines the various inputs required, and how these values may relate to certain plane designs and their respective design characteristics.
5.1 EQUATIONS OF MOTION This section explains the derivations of the equations used to model the flight trajectory.
5.1.1 MODELLING A POINT MASS To simplify the problem, the paper planes were modelled as a point mass. The location of the point mass was equal to the location of the centre of gravity. Figure 5.1 below shows the coordinate system used. This analysis only accounts for longitudinal motion.
α θ γ
FIGURE 5.1: PAPER PLANE COORDINATE SYSTEM
The yellow arrow in Fig 5.1 shows the direction the plane is facing. The red arrow shows the direction the plane is travelling, with its velocity denoted by the red V. As shown in Fig 5.1, the angle of elevation of the plane, here denoted θ, is not necessarily equal to the angle of direction, denoted γ. The difference between these angles is most commonly referred to as the angle of attack, here denoted α.
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The four forces acting upon the plane in flight are lift force, drag force, gravity and thrust. Lift force is given by Equation 5.1, supplied by Stengel (2004).
1 휕퐶 1 퐿푖푓푡 퐹표푟푐푒 = 퐶 휌푉2푆 = [퐶 + 퐿 훼] 휌푉2푆 (5.1) 퐿 2 퐿표 휕훼 2
Where CL is the lift coefficient, a parameter that characterises the wing of the plane, ρ is the density of air, V is the velocity, S is the wing area, and α is the angle of attack. In addition to lift force, drag force is given by Equation 5.2.
1 1 퐷푟푎푔 퐹표푟푐푒 = 퐶 휌푉2푆 = [ 퐶 + 휀퐶2] 휌푉2푆 (5.2) 퐷 2 퐷표 퐿 2
Where CD is the drag coefficient and ε is the Oswald efficiency factor.
In determining the equations of motion, four variables were identified by Stengel (2004), as shown by the matrices in Equations 5.3.
푉푥 푥̇ 푉 푧 푧̇ 퐹푥 [ ̇ ] = (5.3) 푉푥 푚 푉푧̇ 퐹푧 [푚]
Fx and Fz are functions of the forces of flight. They are given by Equation 5.4 and 5.5.
1 퐹 = (퐶 cos 휃 + 퐶 ) 휌(푉 2 + 푉 2)푆 (5.4) 푥 푇 푥 2 푥 푧 This equation is of the form of the lift force equation, however the lift coefficient has been replaced by a coefficient of thrust and a secondary coefficient.
1 퐹 = (−퐶 sin 휃 +퐶 ) 휌(푉 2 + 푉 2)푆 + 푚푔 (5.5) 푧 푇 푧 2 푥 푧 0
5.1.2 RATES OF CHANGE The rate of change of velocity and flight path is given by Equation 5.6.
푥̇ 푉 푉 cos 훾 [ ] = [ 푥] = [ ] (5.6) 푧̇ 푉푧 −푉 sin 훾
This implies Equation 5.7.
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√ 2 2 푉 푉푥 + 푉푧 [ ] = (5.7) 훾 푉푧 − sin−1 [ 푉 ] Differentiating this result gives Equation 5.8.
푑 √푉 2 + 푉 2 푉̇ 푥 푧 [ ] = 푑푡 (5.8) 훾̇ 푑 푉푧 − sin−1 [ 푑푡 푉 ]
Using the equations derived for Fx and Fz, equations for the differentials of V and γ are then given by Equations 5.9 and 5.10.
1 2 (퐶푇 cos 훼 −퐶퐷) 휌푉 푆 − 푚푔0 sin 훾 푉̇ (푡) = 2 (5.9) 푚
1 2 (퐶푇 sin 훼 +퐶퐿) 휌푉 푆 − 푚푔0 cos 훾 훾̇(푡) = 2 (5.10) 푚푉(푡) These equations can then be applied to the conditions of flight. For this simulation, it was assumed that the plane experiences steady, level (cruising) flight conditions. This implies that the thrust is equal to the drag force, the lift force is equal to the weight, and the angle of attack is infinitesimally small. This then gives Equations 5.11 – 5.14.
푉푥 = 푉(푡) cos 훾 = 푉푐푟푢푖푠푒 (5.11)
푉푥 = 푉(푡) sin 훾 = 0 (5.12)
1 2 (퐶푇 − 퐶퐷) 휌푉푐푟푢푖푠푒 푆 푉̇ (푡) = 0 = 2 (5.13) 푚
1 2 퐶퐿 휌푉푐푟푢푖푠푒 푆 − 푚푔0 훾̇(푡) = 0 = 2 (5.14) 푚푉푐푟푢푖푠푒
These equations were then used to model the flight trajectory of a chosen plane design, as shown in Section 5.2.
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5.2 CODE INPUTS To determine the reasonableness of using these equations of motion to model real paper planes, the best performing paper plane, Design 3 Alt 2, was chosen as the reference plane for the inputs of the code. These inputs are shown below in Table 5.1.
TABLE 5.1: CODE INPUTS Input Value Explanation Wing Area, S [m2] 0.0174 Given in Table 4.1 in Section 4.1 Wing Aspect Ratio, AR 0.972 Given in Table 4.1 in Section 4.1 Oswald Efficiency Factor, e 0.9 Correction factor that represents change in drag as compared with an ideal wing. Value given by Stengel (2004) Mass of Plane, m [kg] 0.0045 Mass of average piece of A4 printer paper Gravitational Acceleration, g [m/s2] 9.81 Gravity at surface of earth Air Density, ρ [kg/m3] 1.225 Air Density at Sea Level Zero Lift Drag Coefficient, CDo 0.02 Drag of falling plane, given by Stengel (2004) Launch Velocity, V [m/s] 2 Given in Section 3.1 Launch Angle, γ [radians] 0.2618 Given in Section 3.1
These design inputs were then used to calculate the coefficients of lift and drag required for the code. First, the induced drag factor was calculated, using Equation 5.15.
1 휀 = (5.15) 휋푒퐴푅 This could then be used to calculate the lift and drag coefficients, using the assumption that the lift of the plane was equal to its weight force. This gives Equation 5.16 and 5.17.
2푚푔 퐶 = (5.16) 퐿 휌푉2푆 2 퐶퐷 = 퐶퐷표 + 휀퐶퐿 (5.17) In addition to the actual conditions of the experiment, the code also calculates the conditions necessary for the maximum lift-drag ratio, and the corresponding launch angle and velocity required to meet these conditions. This is shown by Equations 5.18 and 5.19.
퐶 퐶 = √ 퐷표 (5.18) 퐿 휀
2 퐶퐷 = 퐶퐷표 + 휀퐶퐿 (5.10)
The maximum lift-drag ratio is given in Equation 5.20.
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퐶퐿 퐿퐷푚푎푥 = (5.20) 퐶퐷 The flight angle required to achieve this ratio is given in Equation 5.21.
1 훾 = − tan−1 (5.21) 퐿퐷푚푎푥 The velocity required to achieve this ratio is given by Equation 5.22.
2푚푔 푉 = √ (5.22) 휌푆(퐶퐿 cos 훾 − 퐶퐷 sin 훾)
Additionally, the lift coefficient slope of the wing was calculated in order to determine the angle of attack. The lift coefficient slope was given by Equation 5.23.
휋퐴푅 퐶 = 퐿푎 2 (5.23) √ 퐴푅 1 + 1 + ( 2 ) The angle of attack could then be calculated from Equation 5.24.
퐶 훼 = 퐿 (5.24) 퐶퐿푎 These relations were then compiled into a series of Matlab™ code to produce a graph of the trajectory of this plane configuration. For the full code, see Appendix C. The code used for this analysis was based upon a code created by Stengel (2004) and adjusted for the purposes of this report.
5.3 CODE OUTPUTS Using the code in Appendix C a number of outputs were given. Table 5.2 summarises the results of the calculations.
TABLE 5.2: CODE OUTPUTS Variable Value
Real Launch Speed, V [m/s] 2 Real Launch Angle, γ [rads] 0.2618
Induced Drag Factor, ε 0.3639 Real Real Lift Coefficient, CL 0.6633 Conditions Drag Coefficient, CD 0.1801
Best Lift Coefficient, CL 0.2344 -
Best Drag Coefficient, CD 0.04 Maximum Lift-Drag Ratio, LDmax 5.8612 Best Launch Angle, γ [rads] -0.1690 Best Launch Speed, V [m/s] 4.1749
Drag Ratio Drag Lift Coefficient Slope, CLa 1.446 Maximum Lift Maximum Angle of Attack, α [rads] 0.1621
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Coupling the results of Table 5.2 with the equations of motion derived in Section 5.1.1 gives the graph shown in Fig 5.2.
FIGURE 5.2: FLIGHT TRAJECTORY OF DESIGN 3 ALT 2
The blue line indicates the maximum lift-drag results, whilst the orange indicates the expected real results of this plane design. As can be seen, the landing point falls short of the experimental results, which have a median flight distance of 4.44 m. This suggests further calibration of the code is required.
Introducing two calibration coefficients, Ce and Cu allowed for the equations of motion model to be calibrated until the results better matched that observed in real life. These coefficients were applied in Equations 5.25 and 5.26.
1 2 퐶푈 [(−퐶퐷) 휌푉 푆 − 푚푔0 sin 훾] 푉̇ (푡) = 2 (5.25) 푚
1 2 퐶퐿 휌푉 푆 − 푚푔0 퐶푒cos 훾 훾̇(푡) = 2 (5.26) 푚푉(푡)
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Using these calibration coefficients with values of Cu = 0.35 and Ce = 0.6, the result can be seen to better match the shape of the plane’s flight trajectory and also aligns with the median measurement of distance. This is shown in Fig 5.3.
FIGURE 5.3: FLIGHT TRAJECTORY WITH CALIBRATIONS This result only pertains to Design 3 Alt 2. To determine whether these results would align with other plane designs, and further if these calibration coefficients could be linked in some way to any of the design characteristics, would require further experimentation and analysis.
5.4 CHAPTER SUMMARY This chapter examined the equations of motion behind the code used to model the flight trajectory of the best performing paper plane. Using only the basic equations of motion, it could be seen that the model required further calibration. With the addition of two calibration coefficients that effected both the velocity profile and the angular profile, a better match to the real experimental results could be drawn. However, relating these results to design characteristics, and whether these calibrations could apply to all paper planes and not just the design chosen would require further analysis.
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CHAPTER 6 : SUMMARY, CONCLUSIONS & RECOMMENDATIONS This chapter will explore the final results of the project and analyse the success or failure of the project in addressing its main goals as outlined in Chapter 1. It will address the results of the experiments and what conclusions could be drawn from the analysis of those results. It will also give further recommendations for future work to be conducted.
6.1 SUMMARY The changes made in the paper planes’ designs were chosen based upon the research conducted into the mechanisms behind the flight and real aircraft. After determining the most relevant conditions to monitor were the stability modes of the paper planes during flight – longitudinal, directional and lateral – a series of measureable design characteristics were chosen that directly linked to each of those stability modes. These design characteristics were the wing area, wing span, fuselage depth, centre of gravity, root chord and tip chord. These design characteristics could then be used to calculate further design details, such as the aspect ratio, the mean aerodynamic chord and the static margin.
From there, five base designs were chosen, and each of their design characteristics measured and recorded. Each design was tested and then according to the results of those tests, a series of iterative alterations were made to each design characteristic to in order to ascertain its effects upon the flight characteristics of that design.
Designs 1 and 2 gave a similar performance. Their flight trajectories followed the shape of a sharp incline, followed by a nosedive directly to the ground. These were not considered desirable flight characteristics, and so were not considered for immediate further testing. Design 3 performed the best of all design tested, followed by Design 5 and then Design 4.
Design 3 was chosen to continue the analysis, due to its comparatively better performance. Seven alternative planes were made using the Design 3 base, with changes made to various design characteristics. For Design 3, it was found that a deeper fuselage would result in a better flight performance.
Design 4 underwent a total of four design changes, and Design 5 underwent two. Designs 1 and 2 each had only one design change each, based upon the results of the previous designs, in an attempt to improve their relative performances.
6.2 CONCLUSIONS Of all of the total twenty designs tested, the design denoted Design 3 Alt 2 was the best performing in terms of distance – with a median measurement of 4.44 m – airtime – with a
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TABLE 6.1: BEST PERFORMING PLANE - DESIGN CHARACTERISTICS Fuselage Wing Wing Aspect Tip Root MAC MAC CG Static [mm] Area Span Ratio Chord Chord [mm] Location [mm] Margin [mm2] [mm] [mm] [mm] [mm] [mm] 32-42 8693 130 0.9720 87 138 114.42 84 49 35
The performance of this design was determined to correlate with the ratio between the fuselage depth measurement and the root chord measurement. The fuselage design characteristic was shown to link to the directional stability mode, preventing yaw during flight, but also to a lesser degree it affects the longitudinal stability. From the results of the experiment, the latter of these two modes is of the most significance. The improvement made to the flight distance was linked to the plane flying straighter – a correction made to its directional stability – but primarily it was due to a more stable pitch movement.
Using a series of equations of motion describing the forces experienced by a paper plane, a Matlab code was generated that was able to model the flight trajectory, in two-dimensions, of the best performing Design 3 Alt 2. However, in order to better match the experimental results, a number of calibration coefficients were required.
The main goal of this project was to reach a scientific understanding of how paper aeroplanes behave. This goal was achieved by designing a system that was reliable and replicable for the purposes of launching the paper planes. This system was the paper plane launcher, operating at a constant speed and constant launch angle. It was achieved by determining the correlation between the fuselage depth and the flight performance of the tested planes, suggesting the most relevant characteristic to paper plane design was the fuselage depth. It was determined here that the characteristics to achieve the greatest airtime also apply to achieving the greatest distance. The goal of the project was also achieved through the simulation of the paper plane flight trajectories.
6.2 RECOMMENDATIONS Should future work be conducted in this area of study following the methods outlined in this report, a number of recommendations should be made.
It is recommended that more tests should be performed specifically upon the relationship between the fuselage depth and the root chord of the plane, as this relationship appeared to
6-55 have the greatest impact upon the flight capabilities of the planes. By broadening the scope of the project to potentially include alterations outside of simple folding, this aspect of design could be further studied.
It is recommended that a greater number of repeat tests be performed upon the best performing designs in order to reduce the random error, since as N infinity, the calculated error true error. In general more tests would give a more reliable indication of each designs performance. Related to this, it is recommended to make observations about improving the predictability of the planes, as the error analysis suggested there was still large amounts of uncertainty associated with the better performing planes. This relates to the problem of instability and flight characteristics.
It is recommended to record qualitative video of the flight trajectories be taken from an angle that can observe the behaviour of the plane across an x-z coordinate system. This would allow for further calibration of the model to match up the results of the model with the real trajectories by plotting the flight path of the plane and using the calibration coefficients to match the simulation trajectory to the model trajectory.
Finally, it is recommended to use the code to model each of the tested designs to ascertain if there exists a single approximate calibration value that can be applied across all designs, and if that calibration can be related back to design characteristics such as the fuselage.
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CHAPTER 7 REFERENCES Administration. (2012, August). Difference Between Static Stability and Dynamic Stability. Retrieved from Difference Between: http://www.differencebetween.com/difference- between-static-stability-and-vs-dynamic-stability/
Airplane Discovery Box. (2001). Forces of Aerodynamics. Alabama: Department of Archives and History.
Anderson, D., & Eberhardt, S. (1999). How Airplanes Fly: A Physical Description of Lift. Seattle: Sport Aviation.
Blackburn, K. (2006, December). Paper Airplane Aerodynamics. Retrieved from PaperPlane: http://www.paperplane.org/Aerodynamics/paero.htm
Brown, N. (2015, May). Why do aircrafts require a sizeable vertical tail while birds do not? . Retrieved from Quora: https://www.quora.com/Why-do-aircrafts-require-a-sizeable- vertical-tail-while-birds-do-not
Chen, K., & Wenxin, L. (2011). Paper Plane Aerodynamics. Xiamen: Xiamen Foreign Language School.
Chipling, A. J. (2006, June). International Rules of Paper Planes. Retrieved from Paper Aircraft Association: http://www.paperaircraft.com/
Doherty, P. (2000, October). Paper Airplane Science. Retrieved from Scientific Explorations: http://www.exo.net/~pauld/activities/flying/PaperAirplaneScience.html
Fold'N'Fly. (2015, February). Paper Airplane Designs. Retrieved from FoldNFly: http://www.foldnfly.com/#/1-1-1-1-1-1-1-1-2
Hebert, C., & Cowan, D. (2008, September). Aerodynamic Flutter. Retrieved from The American Institute of Aeronautics and Astronautics: http://dl.btc.pl/kamami_wa/hk_24474_2.pdf
Hepperle, M. (2003, September). Laminar Separation Bubbles. Retrieved from Aerodynamics of Model Aircraft: http://www.mh-aerotools.de/airfoils/bubbles.htm
Johnson, P. (2015, May). Finding the Mean Aerodynamic Chord (MAC). Retrieved from Airfield Models: http://www.airfieldmodels.com/about.htm
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Kroon, e. (2015, May). How Does and Aircraft Fly? Retrieved from KLM.com: https://blog.klm.com/how-does-an-aircraft-fly/
Lissaman, P. B. (1983). Low-Reynolds-Number Airfoils. Pasadena: AeroVironment Inc.
Model Aircraft. (2016, September). Aerodynamics: Stability Concepts. Retrieved from Welcome to Model Aircraft: http://adamone.rchomepage.com/index.html
Ng, B., Kng, Q., Pey, Y., & Schluter, J. U. (2009). On the Aerodynamics of Paper Airplanes. Singapore: Nanyang Technological University, School of Mechanical and Aerospace Engineering.
Paper Plane Mafia. (2016, March). History of Paper Airplanes. Retrieved from Paper Plane Mafia: https://paperplanemafia.com/history-of-paper-airplanes/
Sonic Paper Airplane Launcher. (2017, August). Retrieved from SonicDad.com: http://www.sonicdad.com/project-details/paper-airplane-launcher/
Stengel, R. F. (2004). Flight Dynamics. Princeton: Princeton University Press.
Woodford, C. (2016, May). How Do Planes Fly? Retrieved from ExplainThatStuff: http://www.explainthatstuff.com/howplaneswork.html
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CHAPTER 8 APPENDICES
A. PAPER PLANE DESIGN INSTRUCTIONS DESIGN 1:
Tip Fold
Fuselage Fold
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DESIGN 2:
DESIGN 3:
Tip Fold
Fuselage Fold
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DESIGN 4:
Tip Fold
Fuselage Fold
DESIGN 5:
Tip Fold
Fuselage Fold
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B. RAW RESULTS OF EXPERIMENTS
TABLE 8.1: RAW RESULTS - DESIGN 1
Design 1 Distance [m] Time [s] Take 1 2.66 1.25 Take 2 1.75 0.98 Take 3 2.1 1.33 Base Take 4 2.26 1.37 Take 5 2.47 1.08 Take 6 2.23 1.43 Take 7 2.46 1.35 Take 1 4.51 1.29 Take 2 3.31 1.08 Take 3 3.43 1.04 Alt 1 Take 4 3.47 1.13 Take 5 3.03 0.92 Take 6 2.84 0.9 Take 7 4.29 1.25
TABLE 8.2: RAW RESULTS - DESIGN 2
Design 2 Distance [m] Time [s] Take 1 1.44 1.32 Take 2 1.84 1.3 Take 3 1.84 1.25 Base Take 4 1.39 1.07 Take 5 1.58 1.14 Take 6 1.73 1.23 Take 7 1.72 1.15 Take 1 1.78 0.72 Take 2 2.53 0.65 Take 3 2.19 0.66 Alt 1 Take 4 2.57 0.86 Take 5 2.33 0.86 Take 6 2.61 0.82 Take 7 2.39 0.74
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TABLE 8.3: RAW RESULTS - DESIGN 3
Design 3 Distance [m] Time [s] Take 1 2.85 1.21 Take 2 3.82 1.51 Take 3 3.92 1.35 Base Take 4 2.41 0.97 Take 5 4.06 1.49 Take 6 3.85 1.36 Take 7 3.56 1.33 Take 1 0.6 0.7 Take 2 0.4 0.6 Take 3 1 0.66 Alt 1 Take 4 0.34 0.42 Take 5 0.41 0.81 Take 6 0.47 0.71 Take 7 0.58 0.61 Take 1 3.84 1.63 Take 2 3.15 1.33 Take 3 4.57 1.43 Alt 2 Take 4 5 1.7 Take 5 4.74 1.55 Take 6 3.47 1.21 Take 7 4.44 1.38 Take 1 3.07 0.83 Take 2 3.36 1.26 Take 3 3.61 1.17 Alt 3 Take 4 3.08 0.63 Take 5 2.87 0.55 Take 6 3.32 1.56 Take 7 3.64 1.38 Take 1 3.19 1.26 Take 2 2.63 1.03 Take 3 2.21 0.97 Alt 4 Take 4 3.58 1.31 Take 5 1.75 0.98 Take 6 1.81 1 Take 7 2.79 1.07 Take 1 3.57 1.4 Take 2 3.99 1.44 Take 3 3.33 1.02 Alt 5 Take 4 3.37 1.2 Take 5 2.82 0.85 Take 6 2.1 0.82 Take 7 2.83 0.87
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Design 3 Distance [m] Time [s] Take 1 2.04 0.55 Take 2 1.91 0.48 Take 3 1.58 0.43 Alt 6 Take 4 1.9 0.43 Take 5 1.43 0.32 Take 6 1.93 0.36 Take 7 1.33 0.37 Take 1 3.47 1.3 Take 2 2.24 0.83 Take 3 2.04 0.95 Alt 7 Take 4 2.43 0.82 Take 5 2.52 1.47 Take 6 3.43 1.07 Take 7 3.82 1.38
TABLE 8.4: RAW RESULTS - DESIGN 4
Design 4 Distance [m] Time [s] Take 1 2.79 1.08 Take 2 3.26 1.33 Take 3 2.14 1.43 Base Take 4 1.56 1.36 Take 5 2.62 1.32 Take 6 2.93 1.12 Take 7 2.52 0.85 Take 1 2.55 0.85 Take 2 2.65 1.17 Take 3 3.01 1.02 Alt 1 Take 4 3.11 1.33 Take 5 2.85 0.98 Take 6 2.43 0.92 Take 7 3.41 1.33 Take 1 2.82 1.01 Take 2 2.83 0.99 Take 3 2.79 1.08 Alt 2 Take 4 2.71 0.89 Take 5 2.62 0.78 Take 6 2.34 0.93 Take 7 3.43 0.87
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Design 4 Distance [m] Time [s] Take 1 4.37 1.63 Take 2 3.47 1.36 Take 3 4.33 1.57 Alt 3 Take 4 3.02 0.72 Take 5 4.22 1.44 Take 6 3.98 1.26 Take 7 4.13 1.17 Take 1 2.71 1.18 Take 2 2.08 1.02 Take 3 2.15 0.98 Alt 4 Take 4 2.08 1.21 Take 5 2.49 1.29 Take 6 1.65 1.02 Take 7 2.22 1.01
TABLE 8.5: RAW RESULTS - DESIGN 5
Design 5 Distance [m] Time [s] Take 1 3.28 1.08 Take 2 3.31 1.35 Take 3 2.65 0.95 Base Take 4 2.68 1.09 Take 5 2.6 0.97 Take 6 2.43 1.1 Take 7 2.49 1.04 Take 1 2.64 1.28 Take 2 2.84 1.35 Take 3 1.5 0.85 Alt 1 Take 4 1.67 1.12 Take 5 1.79 1.03 Take 6 2.08 1 Take 7 1.69 0.85 Take 1 1.49 0.68 Take 2 3.11 1.27 Take 3 1.62 0.84 Alt 2 Take 4 2.39 0.93 Take 5 2.17 1.05 Take 6 2.72 1.03 Take 7 2.01 0.74
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C. FLIGHT SIMULATIONS CODE
EQUATIONS OF MOTION: function xdot = EqMotion(t,x) % Fourth-Order Equations of Aircraft Motion
global CL CD S m g rho
V = x(1); Gam = x(2);
xdot = [(-CD * 0.5 * rho * V^2 * S - m * g * sin(Gam)) / m (CL * 0.5 * rho * V^2 * S - m * g * cos(Gam)) / (m * V) V * sin(Gam) V * cos(Gam)];
EQUATIONS OF MOTION – MAXIMUM LIFT-DRAG RATIO function xdot = EqMotion1(t,x) % Fourth-Order Equations of Aircraft Motion
global CL1 CD1 S m g rho
V1 = x(1); Gam1 = x(2);
xdot = [(-CD1 * 0.5 * rho * V1^2 * S - m * g * sin(Gam1)) / m (CL1 * 0.5 * rho * V1^2 * S - m * g * cos(Gam1)) / (m * V1) V1 * sin(Gam1)
PAPER PLANE FLIGHT
% Inputs from Design 3 Alt 2
global CL CD S m g rho CL1 CD1 S = 0.017386; % Wing Area, m^2 AR = 0.972; % Wing Aspect Ratio e = 0.9; % Oswald Efficiency Factor; m = 0.0045; % Mass, kg g = 9.81; % Gravitational acceleration, m/s^2 rho = 1.225; % Air density at Sea Level, kg/m^3 CDo = 0.02; % Zero-Lift Drag Coefficient V = 2.5; % Assumed Launch Speed, m/s Gam = 0.2618; % Launch Angle, rad
% Calculations for Actual Flight Path
epsilon = 1 / (3.141592 * e * AR)% Induced Drag Factor CL = 2 * m * g / (rho * V^2 * S) % CL for Maximum Lift/Drag Ratio CD = CDo + epsilon * CL^2 % Corresponding CD
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% Calculations for Maximised Flight Path
CL1 = sqrt(CDo / epsilon) % CL for Maximum Lift/Drag Ratio CD1 = CDo + epsilon * CL1^2 % Corresponding CD LDmax = CL1 / CD1 % Maximum Lift/Drag Ratio Gam1 = -atan(1 / LDmax) % Corresponding Flight Path Angle, rad V1 = sqrt(2 * m * g /(rho * S * (CL1 * cos(Gam1) - CD1 * sin(Gam1)))) % Corresponding Velocity, m/s CLa = 3.141592 * AR/(1 + sqrt(1 + (AR / 2)^2)) % Lift-Coefficient Slope, per rad Alpha = CL1 / CLa % Corresponding Angle of Attack, rad
% a) Equilibrium Glide at Maximum Lift/Drag Ratio H = 1.2; % Initial Height, m R = 0; % Initial Range, m to = 0; % Initial Time, sec tf = 6; % Final Time, sec tspan = [to tf]; xo = [V1;Gam1;H;R]; [ta,xa] = ode23('EqMotion1',tspan,xo);
% b) Actual Velocity and Launch Angles xo = [V;Gam;H;R]; [tb,xb] = ode23('EqMotion',tspan,xo);
% c) Oscillating Glide due to Zero Initial Flight Path Angle xo = [V;0;H;R]; [tc,xc] = ode23('EqMotion',tspan,xo);
% d) Effect of Increased Initial Velocity xo = [1.5*V;0;H;R]; [td,xd] = ode23('EqMotion',tspan,xo);
figure plot(xa(:,4),xa(:,3),xb(:,4),xb(:,3)) %,xc(:,4),xc(:,3),xd(:,4),xd(:,3)) axis([0 8 0 2]) xlabel('Range, m'), ylabel('Height, m'), grid title('Flight Trajectory of Design 3 Alt 2') legend('Equilibrium Glide at Maximum Lift/Drag Ratio','Actual Velocity and Launch Angles')
figure subplot(2,2,1) plot(ta,xa(:,1),tb,xb(:,1),tc,xc(:,1),td,xd(:,1)) xlabel('Time, s'), ylabel('Velocity, m/s'), grid subplot(2,2,2) plot(ta,xa(:,2),tb,xb(:,2),tc,xc(:,2),td,xd(:,2)) xlabel('Time, s'), ylabel('Flight Path Angle, rad'), grid subplot(2,2,3) plot(ta,xa(:,3),tb,xb(:,3),tc,xc(:,3),td,xd(:,3)) xlabel('Time, s'), ylabel('Altitude, m'), grid subplot(2,2,4) plot(ta,xa(:,4),tb,xb(:,4),tc,xc(:,4),td,xd(:,4)) xlabel('Time, s'), ylabel('Range, m'), grid
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