Literal Approximations to the Modes
A literal approximation to a mode of airplane motion is defined as an approximate factor that is a combination of stability derivatives and flight parameters such as velocity or air density. This approximation is quite distinct from the factors that are obtained from the airplane’s fourth – or higher degree characteristic equations, factors that are necessarily in numerical form. Literal approximations to the modes have a long history, starting with Lanchester in 1908. A feedback systems analysis approach to developing and validating approximate modes was developed by Ashkenas and McRuer (1958).
A well-known and usually quite accurate literal approximation to the roll mode is for the roll mode time constant TR. The roll mode time constant is the time required for rolling velocity to rise to 63 percent of its steady value following an abrupt aileron displacement. The approximation is TR = -1 /Lp. The symbol Lp = Clpq Sb2/(2 VIx), where
Clp = dimensionless roll damping derivative, a function of wing planform parameters such as aspect ratio and sweep angle; q = flight dynamic pressure, (p/2) V2;
S = wing area;
b = wing span;
V = flight velocity;
p = air density;
Ix = roll moment of inertia.
Note that all of the individual parameters in the roll mode approximation would normally be known to an airplane designer. A large literature has been produced on literal approximations to the modes. McRuer (1973) lists four reasons for this interest, as follows:
1. Developing the insight required for the determination of airframe/automatic – control combinations that offer possible improvements on overall system complexity.
2. Assessing the effects of configuration changes on aircraft response and on air – frame/autopilot/pilot system characteristics.
3. Showing the detailed effects of particular stability derivatives (and their estimated accuracies) on the poles and zeros and hence on aircraft and air – frame/autopilot/pilot characteristics.
4. Obtaining stability derivatives from flight test data.
To this list one might add that mode approximations provide a reasonableness check on complete solutions generated within massive digital-computer programs, assuring that no input errors have been made. Literal approximations to the modes are obtainable only if the equations of motion themselves are simplified in some way, or if the factorization itself is approximated.
Mode approximations are useful in the ways McRuer lists as long as the approximations are simple ones, easy to grasp. One can improve the approximations, bringing the numerical values closer to the actual factors of the characteristic equation. This can provide additional insight into aircraft flight mechanics. However, if the literal expressions are lengthy, their utility suffers. The improvement to the classical Lanchester result for the phugoid mode period made by Regan (1993) and others (see Chapter 11, Sec. 13), which adds only one simple term but greatly improves accuracy at high airspeeds, is an example of a useful improved approximation, in the context of McRuer’s comments.
On the other hand, the improved modal approximations of Kamesh (1999) and Phillips (2000), while demonstrating considerable mathematical skills and adding to our understanding of flight dynamics, are probably too complex for the applications mentioned by McRuer.