Small-Perturbation Analysis of Airplane Flight Dynamics { A Reappraisal. I: Longitudinal Modes

The theory of small-perturbation airplane igh t dynamics has been known to suer from several deciencies. Concerted eort in recent years has succeeded in removing some of these shortcomings. For example, the existing poor approximations to the slow mode dynamics were shown to be the result of ignoring the fast-mode static residual while evaluating the slow mode dynamics. Another example of such work is the recent derivation, using correctly dened fast and slow timescales, of consistent literal approximations to the longitudinal modes; that is, when the second-order polynomial expressions for the short period and phugoid approximations were multiplied, the fourth-order characteristic polynomial was correctly recovered. In the light of these developments, the present paper attempts to carry out a reappraisal of the small-perturbation theory with the aim of removing some of the anomalies that persist. An example of such an anomaly is the appearance of additional terms in the short period frequency approximation over and above the longitudinal static stability parameter Cm ; the additional terms cannot be explained in any fundamental manner. On tracing the evolution of the small-perturbation analysis from the early work by Bryan, we discover that a signican t reason for the anomalies in the theory is due to the manner in which the aerodynamic stability derivatives have been dened in the past. On redening the stability derivatives in a physically meaningful manner, we are able to correctly derive the small-perturbation equations in a suitably nondimensionalized form, where the fast and slow timescales become apparent. Using these equations, our analysis to obtain literal approximations to the longitudinal dynamic modes gives a physically and mathematically appealing theory which is rid of the anomalies that presently exist in the small-perturbation analysis. The discussion in this paper is limited to the airplane longitudinal dynamics; the small-perturbation lateral dynamics will be treated in another paper.