*2.4. Handbook Methods for Vertical Tail Plane Sizing According to Roskam*

For the resizing of the vertical stabilizer, methods from Part II [24] and Part VI [25] of Roskam's book series on airplane design are used. This series covers the whole design process of an aircraft, from the preliminary design phase to the detailed construction of the different components and serves as standard literature in aircraft design. Most of the formulas from Roskam used in this paper are empirical correlations mainly derived from the DATCOM study of the United States Air Force [19], where an extensive number of experiments to gather data and to derive empirical equations was performed. As the methods used in this study are intended to be used within a preliminary aircraft design and iterated intensively during the mission data interpolation, the computation performance requirements should be kept at a minimum. According to Ciliberti et al. [4] this can be achieved using handbook methods. The error contained within the DATCOM method and hence their derivatives, Roskam [24,25] being one of them, lays around −3.0% for vertical tail aspect ratios of 1.0. As the investigated aspect ratios are around 0.7, the error can be neglected for the configuration in this paper.

According to Roskam [24], a first idea of the required VTP size for conventional configurations can be estimated using the so called volume coefficient, defined as

$$\mathcal{V}\_{\upsilon} = \mathbf{x}\_{\upsilon} \cdot \mathbf{S}\_{\upsilon} / \mathbf{S}\_{ref} \cdot b \tag{3}$$

where *V*¯ *<sup>v</sup>* is the vertical tail volume coefficient, *Sv* the VTP area, *Sref* the reference wing area, *b* the wing span and *xv* the longitudinal distance from the aircraft's centre of gravity (CG) to the VTP's aerodynamic centre. As seen in Equation (3) the formula lacks any significance concerning the controllability of the investigated design and is solely derived from the evaluation of mostly conventional configurations, already in service.

In Part II "Preliminary configuration design and integration of the propulsion system" [24], the rudder deflection required to keep the aeroplane stable in case of OEI is given as

$$\delta\_r = (N\_D + N\_{t\_{crit}}) / (q\_{mc} \cdot \text{S}\_{ref} \cdot b \cdot \text{C}\_{n\_{\delta\_r}}) \tag{4}$$

with the reference wing area *Sref* , the wing span *b* and the yawing moment of the remaining engine(s) *Ntcrit* . According to [24], the yawing moment resulting from the parasitic drag increase in the inoperative engine *ND* for a propeller-driven aircraft with variable pitch propellers is 0.1 times *Ntcrit* . The dynamic pressure *qmc* is calculated at the minimum control speed *vmc* = 1.2 · *vs* , with *vs* being the lowest stall speed. *Cn<sup>δ</sup><sup>r</sup>* is the so-called control power derivative. The value of *δ<sup>r</sup>* should not exceed 25◦ [24].

The control power derivative can be obtained via the following formula from Roskam's Part VI [25]:

$$\mathbb{C}\_{\mathfrak{n}\_{\delta\_{\mathbb{F}}}} = -\mathbb{C}\_{\mathfrak{y}\_{\delta\_{\mathbb{F}}}} \cdot \left( l\_{\upsilon} \cdot \cos \mathfrak{a} + z\_{\upsilon} \cdot \sin \mathfrak{a} \right) / b \tag{5}$$

where *lv* and *zv* are the horizontal and vertical distances, respectively, between the CG of the aircraft and the aerodynamic centre (*ACv*) of the vertical tail and *α* is the angle of attack of the aircraft. The "side-force-due-to-rudder" derivative *Cy<sup>δ</sup><sup>r</sup>* is calculated via:

$$\mathbb{C}\_{\mathcal{Y}\_{\delta}} = \mathbb{C}\_{L\_{\text{tr}}} \cdot k' \cdot K\_{\delta} \cdot ((\mathfrak{a}\_{\delta})\_{\mathbb{C}\_{L}} / (\mathfrak{a}\_{\delta})\_{\mathbb{C}\_{l}}) \cdot (\mathfrak{a}\_{\delta})\_{\mathbb{C}\_{l}} \cdot \mathbb{S}\_{\mathbb{D}} / \mathbb{S}\_{ref}. \tag{6}$$

where *Sv* is the surface area of the vertical tail and the coefficients *k* , *Kb*, ((*αδ*)*CL*/(*αδ*)*cl* ), (*αδ*)*cl* and the lift curve slope of the vertical tail *CL<sup>α</sup><sup>v</sup>* can be obtained from Roskam [pp. 228–261] [25].

$$\mathbb{C}\_{L\_{a\_{\mathcal{V}}}} = 2\pi \cdot A\_{v\_{eff}} / \left(2 + \sqrt{A\_{v\_{eff}}^2 \cdot \beta^2 / k^2 \cdot \left(1 + \tan^2(\Lambda\_{c/2}) / \beta^2\right) + 4}\right) \tag{7}$$

with the semi-chord sweep angle of the vertical tail Λ*c*/2 and

$$
\beta = \sqrt{1 - M\_{mc}^2} \tag{8}
$$

$$k = c\_{l\_{a\_M}} / 2\pi \tag{9}$$

where *Mmc* is the Mach number corresponding to the minimum control speed. *cl<sup>α</sup><sup>M</sup>* is the lift curve slope of the VTP at the same Mach number which can be calculated from

$$\mathfrak{c}\_{l\_{a\_M}} = \mathfrak{c}\_{l\_a} / \sqrt{\mathbf{1} - M\_{\rm mc}^2}. \tag{10}$$

The effective aspect ratio of the vertical tail *Aveff* in Equation (7) is obtained via

$$A\_{v\_{eff}} = \left(A\_{v(f)} / A\_{\upsilon} \right) \cdot A\_{\upsilon} \cdot \left(1 + K\_{vh} \cdot \left(A\_{v(hf)} / A\_{v(f)} - 1 \right) \right) \tag{11}$$

with the vertical tail aspect ratio

$$A\_v = b\_v^2 / S\_v \tag{12}$$

where *bv* and *Sv* are the span and the area of the vertical tail, respectively, whereas the coefficients (*Av*(*f*)/*Av*), *Av*(*h f*)/*Av*(*f*) and *Kvh* are obtained from Roskam [p.388–p.390] [25].
