Aircraft Stability Analysis Including Unsteady Aerodynamic Effects

61.1.2 Indicial ApproachThe concept of indicial aerodynamic functions may be defined briefly as “theaerodynamic response of the airfoil as a function of time to an instantaneous change inone of the conditions determining the aerodynamic properties of the airfoil in a steadyflow” [13]. Recent developments and applications of the indicial approach to therepresentation of unsteady aerodynamics can be found in references [18], [19], and [22].A short summary based on these references follows.In this approach, the combined vector of the total aerodynamic coefficientsC r T= [ C C C C C C ]DYLlmnof an aircraft undergoing an arbitrary motion is theindicial response obtained in conjunction with the superposition principle [15], [16] orDuhamel’s integral theorem (Appendix A.1). This indicial response is given byr t= A ( t −τ ) hd&r τ(1-1)C∫0hiWhere = { }A is a matrix of indicial response functions for stepwise variations ofA C jkinematic parameters and control surfaces deflections combined in the vectorh r T= [ α β p q r δ δ δ ] . That means that, in this approach, the variationsaerwith time of the aircraft kinematic variables such as angle of attack and angular velocityare replaced by a large number of small instantaneous step changes. The transientaerodynamic reactions to these step changes are named “indicial functions”, and the totalresponse is obtained by their superposition.Even though this approach is considered a good representation of unsteady aerodynamics,it is difficult to combine it with the differential equations of motion. The indicial conceptwas originally conceived for linear time-invariant systems, but it has been generalized to