Normal Mode Analysis

Normal mode analysis, as applied to aeroelastic stability and control problems, is actually a form of the small oscillation theory about given states of motion. This goes back to the British teacher of applied mechanics E. J. Routh, in the nineteenth century. A body is supposed to be released from a set of initial restraints and allowed to vibrate freely. It will do so in a set of free vibrations about mean axes, whose linear and angular positions remain unchanged. The free vibrations occur at discrete frequencies (eigenfrequencies), in particular mode shapes (eigenvectors).

Of course, the airplane does not vibrate freely, but under the influence of aerodynamic forces and moments. These forces and moments are added to the vibration equations through a calculation of the work done during vibratory displacements. Likewise, the changes in aerodynamic forces and moments due to distortions must have an effect on the motion of mean axes, or what we would call the rigid-body motions.

According to Etkin’s criterion, if the separations in frequency are not large between the vibratory eigenfrequencies and the rigid-body motions such as the short-period longitudinal or Dutch roll oscillations, then normal mode equations should be added to the usual rigid – body equations. Each normal mode would add two states to the usual airframe state matrix (Figure 19.10). A useful example of adding flexible modes to a rigid-body simulation is provided by Schmidt and Raney (2001). Milne’s mean axes are used.

Normal mode aeroelastic controls-coupled analyses were made in recent times for the longitudinal motions of both the Northrop B-2 stealth bomber and the Grumman X-29A research airplane. In both cases, the system state matrix that combines rigid-body, nor­mal mode, low-order unsteady aerodynamic and pitch control system (including actuator dynamic) states was of order about 100 (Britt, 2000).