Aircraft Stability Analysis Including Unsteady Aerodynamic Effects

62Since all the effects due to the sideslip or to the roll angle are now included in the termCN , i( φ)∗Δ of Eq. (2-16), the parameters and σ related to the localization andα(•)x, (•)shape of the sigmoids given by Eqs. (2-19) and (2-20) are not let free to vary with the rollangle in this case. The values of αeffare determined through the equationαeff , i= αi−τ& 2, ααi(2-21)with τ 2 , α being a time-delay constant related to the vortex burst location. Equation (2-14)is similar in structure to (2-13), and its nomenclature is given as follows,τ1y= the transient time-constant related to the vortices core spanwise displacement.yiyi = = non-dimensional distance between the panel normal lifting force point ofbapplication and the longitudinal axis.Uφ( Δ )1+φ= Δ sign2( Δ )Δ1−sign φD φ=2(2-22)(2-23)withΔ φ = φ j +1−φor Δ φ = φ( t+1) −φ( t ) respectively in the quasi-static and in thejjjdynamic cases, where, in the quasi-static case, the given sequence of the static roll anglesisφ , j = 1,2,…,l, and where φ = φ( t ) −φ( )j{ φ ( ti) 0 ≤ ti≤ tnΔ+1for the given time histories of roll anglesit i, }. The time-delay effects on the vortex movements due to the roll angleare taken into account throughφeff = φ −τ2,φ, (2-24)φ &where τ 2 , φ is a time-delay constant related to the vortex core position, to be found fromwind tunnel data.Since the vortex strength and vertical core position effects are considered to be lumpedinto Eq. (2-15), we want the state variables y ito behave qualitatively like the wind