Quasi-Rigid Equations
While the normal modes are by far the most common way to account for aeroelastic effects on airplane stability and control, the less abstract, approximate normal mode method called quasi-rigid analysis deserves mention. In quasi-rigid analysis, the flexible airplane structure is represented by a chain of linked rigid bodies, held in position by springs. An approximate normal mode is introduced for each link.
The earliest dynamic stability and control analysis in which an airplane bending mode was represented appears to have been the quasi-rigid analysis of the Boeing XB-47 (White, 1947). A single yaw pivot was assumed behind the wing trailing edge. This effectively represents the airplane’s first asymmetric bending mode.
Ground vibration tests had established the B-47’s first bending mode frequency at 2.3 cycles per second. Using the known weight of the airplane’s aft section, the effective spring was sized to produce the measured bending frequency. A closed-loop servo analysis agreed with flight results that showed some vibration at the first bending mode frequency when the yaw damper rate gyro was located near the vertical tail. Moving the gyro near the airplane’s center of gravity corrected the problem.
Just as quasi-rigid analysis has provided a simple, approximate alternative to normal mode analysis, it has done the same for the quasi-static aeroelastic problem. The longitudinal neutral and maneuver points of the Northrop YF-17 were found in this way (Abzug, 1974).