Modern aeronautics has radically transformed the way we live and connect with the rest of the world. Mathematics has been the silent engine that drives aeronautics from its origins to the unimaginable heights of modernity. A thorough understanding of aerodynamics, drag, stability and control based on fundamental mathematical principles is required for efficient and safe aircraft design.
For example, the famous Euler’s theorem for polyhedrons –proposed by the Swiss mathematician in 1750–, is currently used in the design of aeronautical structures, such as frames and cells, to optimize their rigidity and strength. The theorem establishes a fundamental relationship between the vertices, edges and faces of any convex polyhedron –specifically, that the number of faces plus the number of edges equals the number of vertices minus two–. Well, in the construction of light aircraft and drones, this formula is used to calculate the minimum number of structural elements –such as beams and panels– necessary to maintain the stability and integrity of the aircraft, taking into account the forces and stresses acting on it. It is also useful in the design of composite materials used in aircraft construction, such as so-called panels honeycombsince it allows determining the optimal number of hexagonal cells (faces) and the junction points (vertices) necessary to balance the strength and lightness of the material.
On the other hand, in the design of aircraft it is also essential to analyze the air flow around the structure, especially the calculation of the aerodynamic forces; the four main are drag, lift, weight and thrust. To study in detail the interaction of these forces over the entire surface of the aircraft, the divergence theorem (or Gauss’s theorem). This relates the flow of a vector field –which is the speed of the air at each point around the plane– through a closed surface with the divergence of the field –which indicates how the speed of the air is changing at each point, if it is positive the air is entering at that point, if it is negative it is leaving–.
Furthermore, in order to be able to fly aircraft, it is necessary to study the flight controls and the response of the aircraft to different forces and disturbances. To do this, integral transformations are used, among other things – which allow a function to be expressed as the sum of other, more manageable functions – such as Laplace transform or Fourier. The first is used to analyze the dynamics of complex systems, such as aircraft and rockets, subject to time-varying forces, in order to understand their behavior. It is also used to model control systems that regulate the movement and attitude of a ship efficiently and accurately. It is also applied in the design of radio communication and navigation systems, using filters and signal processing systems to eliminate noise and improve the quality of communication.
The Fourier transform is used to break down a signal into its frequency components, which, in the aeronautical industry, is applied in the processing of signals generated by navigation systems, communication systems and on-board sensors, facilitating the detection of noise and interference, and improving signal quality. On the other hand, it is used to analyse vibrations – caused by engines, turbulence and changes in flight conditions – and break them down into their frequency components, which is key in the design of damping systems that guarantee structural integrity.
It is also important to analyze risks and evaluate flight safety systems, for which the Bayes’ theorema fundamental result of probability theory proposed more than 250 years ago, which establishes how update the probability of an event, after learning new data relevant to the phenomenon studied. For example, this theorem is applied in the analysis of aircraft accident data and in the assessment of contributing factors, such as weather, maintenance and human error, to improve the safety of future flights. It is also used to process alerts from real-time fault detection systems on modern aircraft, for example, to assess the likelihood that a system alert is a false positive or a genuine indication of a problem. This prevents unnecessary alarms that can distract pilots, while ensuring that genuine alerts are not overlooked. It is thus possible to estimate the probability of failure and assess the performance of electronic systems, improving aircraft safety and reliability.
Mathematics is also at the technological frontier of the aerospace industry. One of the fundamental concepts in the development of hypersonic commercial aircraft is the Prandtl-Glauert transformation. This establishes that, at speeds close to the speed of sound, the effects of air compressibility become significant and must be taken into account in the calculations of the drag and lift of the aircraft, which is not done in calculations for normal speeds. Without a doubt, in any future advances that aeronautics experiences, mathematics will be a fundamental tool.
Yoshua Diaz Interian He is a predoctoral researcher at the National Polytechnic Institute (Mexico).
Agatha Timon Garcia-Longoria She is the coordinator of the Mathematical Culture Unit of the Institute of Mathematical Sciences (ICMAT).
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
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