Horizontal Tail Model

At a conceptual design level, the purpose of the horizontal tail is threefold: to trim the aircraft such that it can fly in steady level flight, to provide longitudinal stability, and to give the pilot pitch control authority over a range of flight conditions.

Model Assumptions

The horizontal tail model assumes that the horizontal stabilizer is mounted to the fuselage and nominally produces downforce in cruise.

Model Description

Variable tables are available for download below:

• Free variables
• Fixed variables

Horizontal Tail Geometry and Structure

The horizontal tail model employs many of the same geometric constraints as the wing and vertical tail. More specifically, analogous versions of Constraints [eq:planformarea,eq:meanaerochord,eq:spanwisemac,eq:taperratio,eq:mintaperratio] and Constraints [eq:vtmomentarm,eq:vtleading,eq:vttrailing] enforce planform relationships and constrain the horizontal tail moment arm, respectively. As with the vertical tail, Constraint  needs to be implemented as a signomial equality constraint. The horizontal tail also reuses the same structural model from [Hoburg, 2013].

Trim Condition

The first sizing requirement is that the aircraft must satisfy the trim condition [Burton, 2017], which implicitly requires that the full aircraft moment coefficient be zero.

\[\frac{x_w}{\bar{c}_w} \leq \frac{x_{CG}}{\bar{c}_w} + \frac{C_{m_{ac}}}{C_{L_w}} + \frac{V_{ht} C_{L_{ht}}}{C_{L_w}}\]

Thin airfoil theory is used to constrain the horizontal tail’s isolated lift curve slope [Anderson, 2001].

\[\begin{aligned} C_{L_{ht}} &= C_{L_{\alpha,ht}} \alpha\end{aligned}\]

However, the horizontal tail’s lift curve slope is reduced by downwash, \(\epsilon\), from the wing and fuselage [Kroo, 2001]. Note \(\eta_{h_{lift}}\) is the horizontal tail sectional lift efficiency.

\[C_{L_{\alpha,ht}} = C_{L_{\alpha,ht_0}} \left(1 - \frac{\partial \epsilon} {\partial \alpha}\right) \eta_{h_{lift}}\]

The downwash can be approximated as the downwash far behind an elliptically loaded wing.

\[\begin{split}\begin{aligned} \epsilon &\approx \frac{2 C_{L_w}}{\pi AR_w} \\ \implies \frac{\partial \epsilon}{\partial \alpha} &\approx \frac{2 C_{L_{\alpha,w}}}{\pi AR_w}\end{aligned}\end{split}\]

Thus, an additional posynomial constraint is introduced to constrain the corrected lift curve slope.

\[C_{L_{\alpha,ht}} + \frac{2 C_{L_{\alpha,w}} }{\pi AR_w} \eta_{ht} C_{L_{\alpha,ht_0}} \leq C_{L_{\alpha,ht_0}} \eta_{ht}\]

Minimum Stability Margin

The second condition is that the aircraft must maintain a minimum stability margin at both the forward and aft limits[Burton, 2017].

\[\begin{aligned} \label{e:SM_CG} S.M._{min} + \frac{\Delta x_{CG}}{\bar{c}_w} + \frac{C_{m_{ac}}}{C_{L_{w,max}}} &\leq V_{ht} m_{ratio} + \frac{V_{ht} C_{L_{h,max}}}{C_{L_{w,max}}}\end{aligned}\]

The ratio of the horizontal tail and wing lift curve slopes, \(m_{ratio}\), appears in Equation and is constrained using the relationship in [Burton, 2017]. The constraint is a signomial equality because it is not possible to know a priori whether there will be upward or downward pressure on \(m_{ratio}\).

\[m_{ratio} = \left(1+\frac{2}{AR_w}\right) 1 + \frac{2}{AR_{ht}}\]

Stability Margin

The third condition is that the stability margin must be greater than a minimum specified value for all intermediate locations.

\[\begin{split}\begin{aligned} S.M. &\leq \frac{x_w - x_{CG}}{\bar{c}_w}\\ S.M. &\geq S.M._{min}\end{aligned}\end{split}\]

Horizontal Tail Drag

The horizontal tail employs the same drag model as the wing (Constraints [eq:wingdrag,eq:wingdragcoeff,eq:wingpdragcoeff,eq:wingRe,eq:induceddrag]), with the exception of the parasitic drag coefficient fit. The wing’s parasitic drag fit  is replaced by a fit to XFOIL [Drela, 1989] data for the TASOPT[Drela, 2011] T-series airfoils. The TASOPT T-series airfoils are horizontal tail airfoils intended for transonic use. The fit considers airfoil thickness, Reynolds number, and Mach number. The softmax affine function fit is developed with GPfit and has an RMS error of 1.14%.

\[\begin{split}\begin{aligned} \label{e:HT_drag} {C_{D_{0_{ht}}}}^{6.49} & \geq 5.288\times10^{-20} (Re_{h})^{0.901} (\tau_{h})^{0.912} (M)^{8.645}\\ &+ 1.676\times10^{-28} (Re_{h})^{0.351} (\tau_{h})^{6.292} (M)^{10.256} \nonumber \\ &+ 7.098\times10^{-25} (Re_{h})^{1.395} (\tau_{h})^{1.962} (M)^{0.567} \nonumber \\ &+ 3.731\times10^{-14} (Re_{h})^{-2.574} (\tau_{h})^{3.128} (M)^{0.448} \nonumber \\ &+ 1.443\times10^{-12} (Re_{h})^{-3.910} (\tau_{h})^{4.663} (M)^{7.689} \nonumber \end{aligned}\end{split}\]