A New Mathematical Blueprint for Flight: Closed-Form Equations Bring Aerobatic Manoeuvres Into the Classroom

For more than a century, the science of aircraft flight has lived a strange double life. In university lecture halls, students learn the subject through George Bryan’s 1911 framework of stability modes—the short period, the phugoid, the Dutch roll—while aspiring pilots in ground schools absorb an almost entirely qualitative tradition rooted in Wolfgang Langewiesche’s 1944 classic Stick and Rudder. Now a team of researchers from the University of Maryland, the Tata Institute of Fundamental Research and IISER Bhopal has proposed a way to bridge that gulf. In a study published in Frontiers in Aerospace Engineering, they present a closed-form system of nonlinear equations describing the pitch-plane, or longitudinal, motion of a fixed-wing aircraft—an explicit model free of the data tables and interpolated functions that have long kept full manoeuvre analysis out of textbooks and training syllabi.

The core obstacle the team confronted is a familiar one to aerodynamicists. Since Bryan’s era, virtually every equation of aircraft motion has relied on coefficients of lift, drag and pitching moment—C_L, C_D and C_m—which are unspecified functions of speed, angle of attack and other variables. These functions exist only as tables built from wind tunnel experiments or computational fluid dynamics, and working with them forces engineers into either linearized approximations valid only near a trim point, or laborious polynomial interpolation of the tabulated data. The only truly explicit flight equation in the literature, derived independently by Frederick Lanchester and Nikolai Zhukovsky more than a century ago, describes a glider and ignores pitch altogether—too crude, the authors argue, to capture how real pilots actually fly. Complete data tables, meanwhile, exist in the public domain for only a handful of aircraft, such as the F-4, F-16 and FA-18, making generic modelling nearly impossible.

The researchers’ way around this bottleneck is a compact empirical formula for aerodynamic force that agrees with experiment while remaining analytically tractable. When an airfoil sits in an airflow at an angle of attack, the formula gives a force whose component perpendicular to the flow—the lift—scales with the square of the airspeed and, for the small angles relevant to unstalled flight, linearly with the effective angle of attack, exactly as experiments demand. The same expression automatically produces an induced drag proportional to the lift, a hallmark of every real wing, and yields a lift-to-drag ratio that decreases as angle of attack grows, matching wind tunnel behaviour beyond the regime where parasitic drag dominates. Because the formula is a quadratic polynomial in the velocity components, everything built on it stays algebraically simple. A standard parasitic drag law for the fuselage completes the force picture, and the team treats the horizontal tail as a single pivoted airfoil—an elevator—whose lift force, rather than the pilot’s stick torque itself, serves as the fundamental control input.

Assembling these ingredients with the help of the computer algebra package WXMaxima, the authors arrived at a sixth-order nonlinear system governing speed, flight path elevation, pitch, angular velocity and position, driven by just two external functions: thrust and elevator force. The model contains nine constant parameters—mass, wing lift constant, tail lift constant, drag constant, moment of inertia, rotational damping, and geometric distances—plus the optional camber and scale factors. Because the unknowns are constants rather than multivariate functions, a modeller can tune them to represent a passenger airliner, a piston-engine trainer or a stealth fighter without hunting for proprietary data. For their demonstrations, the team chose parameters mimicking a 200-seat airliner in the mould of the Airbus A321, with a maximum takeoff weight of 100,000 kilograms and takeoff thrust of 300,000 newtons.

Validation came first through the classics. Solving for equilibrium flight conditions and examining the eigenvalues of the system’s Jacobian, the researchers recovered the two canonical longitudinal modes known to every aerospace student. One mode, with strongly negative real parts, corresponds to the short period: a rapid, heavily damped oscillation in which pitch and angle of attack settle back to equilibrium within a couple of seconds. The other, whose stability depends on airspeed, is the phugoid—a slow, long-period exchange between altitude and speed that the simulations reproduce in full. The team also recovered the well-known regions of normal and reversed command. At a fixed 40 percent thrust, their model produces two equilibrium speeds for each climb rate; on the right side of the thrust curve, higher pitch means higher climb rate, while on the left the relationship inverts, so that the same power yields lower speed coupled to lower climb rate—a counterintuitive behaviour familiar to every instrument-rated pilot approaching a slow, high-drag regime.

With the model validated, the authors turned to its central purpose: simulating full aerobatic manoeuvres, something the standard curriculum almost never attempts. Using an

Subject of Research: Closed-form nonlinear modelling of longitudinal aircraft flight dynamics and manoeuvre simulation

Article Title: Model-based manoeuvre analysis: a path to a new paradigm in aircraft flight dynamics

Article References: Shayak, B., Girdhar, S., & Malviya, S. (2024). Model-based manoeuvre analysis: a path to a new paradigm in aircraft flight dynamics. Frontiers in Aerospace Engineering, 3, Article 1308872. https://doi.org/10.3389/fpace.2024.1308872

Image Credits: AI Generated

DOI: 10.3389/fpace.2024.1308872

Keywords: flight dynamics, aircraft manoeuvres, nonlinear equations, aerodynamics, lift and drag, phugoid mode, short period mode, Immelmann turn, vertical dive, pilot training, flight simulation, aerospace engineering