IB Mathematics Internal AssessmentWhat is the optimal wing configuration for subsonic flight?Topic: AerospaceName: Gerasimos KonidarisCourse: IB Mathematics HL 2I. IntroductionOver the years many designs of planes and wings have been introduced. The first plane that ever took flight was a biplane, meaning two wings. Humanity ditched this design in favor for a monoplane, or single wing.
Why did we do this, and why are there still biplanes being used and produced? We have discovered that two winged planes increase stability but also increase drag compared to the monoplanes which are faster but less stable. However, do the position and angle of the wings matter? How about the number or placement of the wings? What total configuration of wing(s) will lead to the most optimal mix of speed and stability for subsonic flight? What about each aspect adds together to form this peak configuration?For the purposes of this Internal Assessment, the definition of ‘optimal’ in the title will be fuel efficiency combined with speed potential.II. OutlineIn this IA, we must explore concepts like the lift:drag ratio and its accompanying equation, the effect of surface area on lift, and how this affects subsonic flight. We will also explore the main factors of wing configurations and the benefits and detriments of each factor.
These topics will require a basic understanding of lift and drag and what purpose airplane wings serve in a aerodynamic flight. We also must explore the differences between monoplanes, multiplanes, tandem wings, and canard wings in addition to angles and sizes of wings to be able to weigh all options and determine a clear winner based on experimentation and mathematical calculation.In order to explore these, we must derive and deconstruct existing equations to find what part of each equation actually affects flight, as well as graphing several equations to find the extrema which would allow us to discover the best and worst configurations of wings. We must also build off existing discoveries to sift through the multitude of configurations that time has proven do not work well in subsonic flight. We can also build models airplanes out of paper or other materials to physically test our findings on a small scale to be able to predict how a real aircraft would behave in a real world environment.
III. Mathematical FormulasII a. Lift-to-Drag Ratio 1This formula represents the maximum possible lift-to-drag ratio2.
In the equation, AR represents the aspect ratio3, epsilon represents the span efficiency factor4, a number less than but close to unity for long, straight edged wings, and CD,O is the zero-lift drag coefficient. This ratio is important because it is independent of the weight of the aircraft or the area of the wing.II b.
Lift L = CL((pv2)2)A 5L represents Lift, of course.p represents the density of the air that the plane is currently flying inv is the velocity of the aircraft in feet/ secondA represents the wing area of an airplane represented in square feetCL is the Coefficient of lift , which a variable given to represent all of the complex dependencies of shape, inclination,and some flow conditions on lift. This is difficult to calculate in complex situations so it can only be found with experimentation.II c.
Coefficient of LiftCl = L / (A * .5 * r * V^2)CL Represents the coefficient of liftLift is represented by LWing Area is represented by AVelocity is represented by VDensity is represented by rII c. Aspect RatioAR = a2sAR representing the aspect ratio of a plane’s wingA again represents the wing area of an airplane represented in square feetA representing the wing spanII d. Bernoulli’s PrincipleOh boyIV. Number of Wings per PlaneIII a. MonoplanesMonoplanes are planes with a single fixed wing, which is essentially every mass produced jet (with the exception of the PZL M-15).
They are considered very stable and extremely efficient airplanes due to the fact that they have lowest drag of any wing configuration. They are also the simplest to build, as only one wing would need to be accounted for. We are able to calculate their efficiency using the Lift-to-Drag equation and compare it to a biplane with similar dimensions.(calculations)III b.
BiplanesBiplanes are dual winged airplanes. These have been the staple of humanity up until jet planes took over because a biplane wing configuration has a structural advantage over a monoplane. However, it produces more drag than a similar monoplane wing, which we calculated above.
Improved structural techniques, better materials and the quest for greater speed made the biplane configuration obsolete for most purposes by the late 1930s. Biplanes offer several advantages over traditional monoplane designs because they allow for lighter wing structures, low wing loading and smaller span for a given wing area. These gains are offset because despite each wing being far smaller, there are two of them. Also, interference between the airflow over each wing in addition to the extra support required to brace two wings increases drag substantially.III c. TriplanesTriplanes, naturally, occupy the space between biplanes and multiplanes. They have three wings.
A triplane has a narrower wing chord than a biplane with similar wings. This gives each wing a slender appearance with higher aspect ratio, making it more efficient and giving more lift. This offers a faster rate of climb and smaller turning radius, both of which are important in a fighter. These advantages, however, are offset to a greater extent in any given design by the extra weight and drag of the extra bracing needed to support more wings and by the loss of lift resulting from aerodynamic interference between the wings in any stacked configuration. None were successful, and as biplane design advanced, it became clear that the disadvantages of the triplane outweighed its advantages.III d. MultiplanesMultiplanes have always been experimental, unconventional planes.
Not a single multiplane has ever proven successful. They are often comical in appearance, and the most wings ever installed on a real aircraft comes from one of Horatio Frederick Phillips’ famous multiplanes, a whopping 200 individual airfoils. The aerodynamic interference between the airfoils in any multiplane far outweighs and benefit of stability that could be offered. I believe there is no reason to calculate precisely why this aircraft failed because it should be simple enough. As for all the less extreme multiplanes, although they do provide far more stability, the drag between airfoils is far too high to make for an efficient airplane and thus the designs were cast away for a much better mono- or biplane configuration.
V. Swept WingsIV a. Right Angled WingsIV b. Swept ForwardIV c. Swept BackwardVI. Known Inferior DesignsVII.
ExperimentsFor this internal assessment, I constructed a model airplane made out of cardboard. It is modelled after traditional passenger jet airplanes but can only function as a glider due to the lack of resources and the absurd cost of fitting a simple device with an engine. The purpose of this is to be able to visualize the theoretical calculations done in this IA and actually implement them in reality to see if they would work as predicted. We fitted this plane with several different wing configurations and lengths which could be contenders for most optimal.
Below is a table of the findings of the experiment.VI a. Table(Insert Table)VI b. Graph(Insert Graph)VIII. ConclusionIn conclusion, IX. Works Citedhttps://www.grc.nasa.gov/www/k-12/airplane/size.htmlhttps://en.wikipedia.org/wiki/Canard_(aeronautics)https://en.wikipedia.org/wiki/Wing_configurationhttps://aviation.stackexchange.com/questions/19475/what-are-the-different-wing-planforms-what-are-eachs-advantages-where-are-thehttp://www.dept.aoe.vt.edu/~mason/Mason_f/ConfigAeroDesignOpt.pdfhttp://aero.stanford.edu/reports/vki_kroo_supersonics.pdfhttps://en.wikipedia.org/wiki/Supersonic_airfoilshttp://www.bbc.com/future/story/20140522-are-these-the-worlds-worst-planehttps://www.grc.nasa.gov/www/k-12/WindTunnel/Activities/lift_formula.htmlhttps://en.wikipedia.org/wiki/Monoplanehttps://io9.gizmodo.com/these-bizarre-multi-winged-planes-are-historys-most-a-1484799626https://en.wikipedia.org/wiki/Chord_(aeronautics)https://en.wikipedia.org/wiki/Triplanehttp://www.aviation-history.com/theory/aspect_ratio.htm