Additional Special Forms of the Equations of Motion

Trajectory or point-mass equations of airplane motion, lacking the torque or mo­ment equations, have been found useful for flight performance studies. In these applications, angles of attack and sideslip are assumed functions of time or are found in simple closed loops, instead of being the result of attitude adjustments influenced by control surface an­gles. Trajectory equations of motion have only 6 nonlinear state equations, as compared with 12 for the complete rigid-body equations. The savings in computer time are unimpor­tant with modern digital computers, but there is a conceptual advantage for performance studies in needing to specify only lift, drag, and thrust parameters.

Another special form of the equations of airplane motion puts the origin of body axes at an arbitrary location, not necessarily the center of gravity. The first use of such equations seems to have been for fully submerged marine vehicles, such as torpedoes and submarines. With the center of body axes at the center of buoyancy, there are no buoyancy moment changes due to changes in attitude (Strumpf, 1979). An equivalent set for airplanes came later (Abzug and Rodden, 1993).

Apparent mass and buoyancy terms in the equations of airplane motion are discussed in Chapter 13, “Ultralight and Human-Powered Airplanes.” The various special forms of the equations of airplane motion for representing aeroelastic effects are discussed in the next chapter, “The Elastic Airplane.”

Equationsof motion for an airplane with an internal moving load that isthen dropped were developed by Bernstein (1998). The motivation is the parachute extraction and dropping of loads from military transport airplanes. A control strategy using feedback from disturbance variables to the elevator was able to minimize perturbations in airplane path and airspeed during the extraction and dropping process.