Researchers at NASA’s Armstrong Flight Research Center have developed a method that allows for the translation of frequency-domain aerodynamics from commercial code (e.g., ZAERO™ or NASTRAN®) to a time-domain formulation that can be easily understood by flight control engineers, and eliminates the complications inherent in previous methods. These previous methods were designed for structural control (independent of flight dynamics) and are therefore formulated in the inertial (stationary) modal reference frame, which cannot accurately capture phugoid mode dynamics without significant complexity. Thus, a non-inertial reference frame was required to correctly model flight dynamics — a goal previously achieved by applying a transformation to the final statespace model. However, the transformation method created numerical errors, leading to problems in model simulation, reduction of the order-for-control development, and decreased accuracy.

Armstrong addresses these shortcomings with a new method that:

  • Is familiar to flight control engineers
  • Eliminates the numerical errors seen in previous methods
  • Utilizes existing software to generate the magnitude of and delay in the aerodynamic forces of the aircraft
  • Avoids the generation of nonphysical errors in the conversion from frequency domain into the time domain

Armstrong’s method generates a more physically realistic least-squares estimate without introducing significant bias errors that would otherwise indicate some motions to be more important than they actually are.

This new method uses existing modeling software that ascertains magnitude and delay data as the aircraft moves in purely harmonic motion at particular frequencies relative to a stationary frame of reference on Earth. It transforms inputs to the aerodynamic forcing from the reference frame into a more physical frame of reference fixed to the aircraft. It then formulates a rational function that relates the aircraft’s harmonic motion in the non-stationary frame to the aerodynamic forces, expressing a linear differential equation that relates general (non-harmonic) motion to the time-varying aerodynamic forces on the aircraft. Engineers can use the linear differential equation directly in models of the flight dynamics of a flexible aircraft.

Building upon the well-established rational function approximation formulation by Kenneth L. Roger for a stationary reference frame, Armstrong’s new method can be used for a non-stationary frame of reference, deriving transformations from classical kinematic equations.

Armstrong’s new method uses mathematical libraries in MATLAB®, and currently takes its frequency domain aerodynamic models from ZAERO. However, the aerodynamic model generation could also use numerous other opensource, double-lattice methods (e.g., various versions of NASTRAN).

This work was done by Jeffrey Ouellette of Armstrong Flight Research Center. NASA is seeking partners to further develop this technology through joint cooperative research and development. For more information about this technology and to explore opportunities, please contact NASA Armstrong Technology Transfer Office at 661-276-3368 or by e-mail at This email address is being protected from spambots. You need JavaScript enabled to view it.. DRC-014-057

ZAERO is a trademark of Zona Technology Inc. NASTRAN is a registered trademark of MSC Software Corporation. MATLAB is a registered trademark of MathWorks, Inc.