Mathematics and aircraft design

Summary: Mathematics plays a pivotal role in designing, manufacturing, and enhancing aircraft components and launch platforms.

Achieving flight has been a dream of mankind since prehistory, one never abandoned. As early as Leonardo da Vinci, mathematics—the cornerstone of engineering and physics—was recognized as the key to realizing the dream. Da Vinci’s 1505 “Codex on the Flight of Birds,” for instance, is a brief illustration-heavy discussion attempting to discover the mechanics of birdflight in order to replicate those mechanics in manmade flying machines. Da Vinci considered not simply the wingspan and weight of birds but a fledgling notion of aerodynamics. He was the first to note that in a bird in flight, the center of gravity—the mean location of the gravitational forces acting on the bird—was located separately from its center of pressure where the total sum of the pressure field acts on the bird. This fact would be important in later centuries when aircraft were designed that are longitudinally stable. Today, mathematics is used in the study of all aspects of flight, from launch platform design to the physics of sonic booms.

Complex Analysis and the Joukowski Airfoil

Abstract mathematics can find its place in physical applications people experience quite often. For example, complex analysis and mappings play a vital role in aircraft. In layman’s terms, complex analysis essentially amounts to reformulating all the concepts of calculus using complex numbers as opposed to real numbers. This formulation leads to new concepts that cannot be achieved with only real numbers. In fact, the very notion of graphing complex functions, rather than real functions, is quite different—mathematicians often call the graphing of complex functions a “mapping.” Taking a simplistic geometric figure, like a circle, and then applying a complex function transforms the figure into a more complicated geometric structure. One figure that results from such a transformation looks like an airplane wing. Furthermore, one can consider the curves surrounding the circle as fluid flow, that is, air currents, and we obtain a rudimentary model of airflow around an airplane wing. This transformation is entitled the Joukowski Airfoil, which is named after the Russian mathematician and scientist Nikolai Joukowski (1847–1921), who is considered a pioneer in the field of aerodynamics. Variations of this transformation have been utilized in applications for the construction of airplane wings.

Nature-Inspired Algorithms

An example of how various fields of mathematics, science, and engineering coalesce is epitomized at the Morpheus Laboratory, where applications of methods and systems found in nature are applied to the study and design of various types of aircraft. For example, biologically inspired research is conducted by studying an assortment of details related to the mechanics of birds in flight.

Birds are an example of near perfection in flight, a fact that humans have long observed. Birds have been evolving for millions for years and have adapted to various environmental changes, thus altering their flight mechanics accordingly. By studying the mathematical properties related to their wing morphing, surface pressure sensing, lift, drag, and acceleration, among other aspects, the researchers at Morpheus Laboratory can use the knowledge they have gleaned and apply it to several different types of aircraft. In order to accomplish this feat, mechanical models of actual birds are constructed and analyzed. Morpheus researchers utilize an assortment of mathematics and physics, including fluid mechanics (the study of air flow in this case) and computer simulations, to analyze the data that result from studying the mechanical birds in flight. The analysis, in turn, results in novel perspectives in flight as well as the design of innovative types of planes.

In addition, many of the problems that arise regarding the machinery and components that comprise an aircraft carrier can also be potentially solved via Darwinian-inspired mathematical models. For example, the structural components of aircraft are constantly being optimized, as numerical performance is attempted to be maximized while cost is minimized.

The managing of cabin pressurization has made it possible for aircraft to fly safely under various weather conditions and landscape formations. This ability is due in large part to devices known as “pressure bulkheads,” which close the extremities of the pressurized cabins. Because of the wealth of physical phenomena that influence the stability of these bulkheads, such as varying pressures, it has been a challenge to optimize their design. In the early twenty-first century, it was proposed that the bulkheads should have a dome-like shape, as apposed to a flat one, which was suggested by both mathematical and biological evidence. Interestingly, these two structures demonstrate completely dissimilar mechanical behaviors, which lead researchers to consider different approaches to modeling the dome-like bulkheads.

The dome-like structured bulkheads are analogous to biological membranes and can be mathematically modeled in a similar fashion. In addition to the implementation of these membrane-like designs, the minimization of the cost of their construction and the assurance of their durability is mathematically modeled.

Simulating Sonic Booms

Every time an aircraft travels faster than the speed of sound, a very loud noise is produced called a “sonic boom.” The boom itself results when an aircraft travels faster than the speed of the corresponding sound waves. The boom is a continuous event, as opposed to an instantaneous sound, which is a result of the compression of the sound waves. Other fast-moving projectiles like bullets and missiles also produce sonic booms.

Mathematically, this concept means that the velocity of an aircraft (v_a) exceeds the wave velocity of sound (v_s). The Mach Number (M), named after the Austrian physicist and philosopher Ernst Mach (1838–1916), is defined as the ratio of the velocity of an aircraft to the velocity of sound. This ratio is expressed mathematically as

When v_a < v_s , M < 1, the object is moving at what is often referred to as “subsonic speed.” If v_a = v_s , M = 1,and the object is moving at what is frequently called “sonic speed.” Whenever v_a > v_s , M > 1, and the object is moving at what is titled “supersonic speed.” Furthermore, whenever v_a > v_s , a shock wave is produced.

The shock waves from jet airplanes that travel at supersonic speeds carry a great amount of concentrated energy resulting in great pressure variations. In fact, two booms are often produced when jets fly at supersonic speeds. Usually, these two booms coalesce into an N-shaped sound wave that propagates in the atmosphere toward the ground. Although shock waves are exceedingly interesting, they can be unpleasant to the human ear and can also cause damage to buildings including the shattering of windows.

However, there is increasing economical interest in designing aircraft carriers that can travel at supersonic speeds with a low sonic boom. To demonstrate, the flight time for a trip from New York to Los Angeles can essentially be cut from 10% to 50% if the plane flies at a supersonic cruise speed instead of subsonic speed. Therefore, physicists are currently developing adaptive methods that model sonic booms in order to ultimately develop aircraft that can travel at supersonic speeds without causing structural damage—aircraft that create a low sonic boom. Aspects such as near-field airflow as well as pressure distribution have been analyzed in these models by utilizing techniques of mathematical analysis.

Aircraft Carriers

Airplanes were a major evolution in modern warfare. World War II aircraft carriers that moved airplanes closer to targets that would otherwise be well beyond their fuel ranges proved to be pivotal to many battles, especially in the Pacific. They continue to be a key component of many countries’ navies for rapid deployment of aircraft for surveillance, rescue, and other military uses. Launching from and landing airplanes on aircraft carriers is considered one of the most challenging pilot tasks because of the restricted length of the deck and the constant motion of the deck in three dimensions. A catapult launch system gives planes the added thrust they need to achieve liftoff and requires calculations that take into account mass, angles, force, and speed. Similar issues apply to the tailhook capture system that stops planes when they land.

There are also significant scheduling issues for multiple aircraft on a carrier, fuel use, weapons logistics, and radar systems used to monitor both friendly and enemy planes. Aircraft carriers are like large, self-contained floating cities. Mathematicians work in the nuclear or other power plants that provide electricity for the massive aircraft carriers of the twenty-first century and in many other logistics areas beyond direct flight launch and control. They also help design and improve aircraft carriers. For example, mathematician Nira Chamberlain modeled the lifetime running costs of aircraft carriers versus operating budgets to develop what are known as “cost capability trade-off models,” which were used to help make decisions about operations. He also worked on plans for efficiently equipping ships to optimize speedy access to spare components. Some of the mathematical methods he used include network theory, Monte Carlo simulation, and various mathematical optimization techniques.

Bibliography

Alauzet, Frederic, and Adrien Loseille. “Higher-Order Sonic Boom Modeling Based on Adaptive Methods.” Journal of Computational Physics 229 (2010).

Balogh, Andres. “Computational Analysis of a Boundary Controlled Aircraft Wing Model.” Sixth International Conference on Mathematical Problems in Engineering and Aerospace Sciences. Cambridge, England: Cambridge International Science Publishing, 2007.

Freiberger, Marianne. “Career Interview With Nira Chamberlain: Mathematical Modelling Consultant.” http://plus.maths.org/content/career-interview-mathematical-modelling-consultant.

Morpheus Laboratory. http://www.morpheus.umd.edu.

Niu, Michael Chun-Yung, and Mike Niu. Airframe Structural Design: Practical Design Information and Data on Aircraft Structures. Granada Hills, CA: Adaso Adastra Engineering Center, 1999.

Viana, Felippe, et al. “Optimization of Aircraft Structural Components by Using Nature-Inspired Algorithms and Multi-Fidelity Approximations.” Journal of Global Optimization 45 (2009).

Yong, Fan, et al. “Aeroservoelastic Model-Based Active Control for Large Civil Aircraft.” Science China: Technological Sciences 53 (2010).