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\chapter{Aerodynamic Characteristics}  \label{cha:areo}  %Please do not forget to add new symbols to the nomenclature e.g.: \nomenclature{$t_{element}$}{Thickness coefficient of plate \nomunit{[$-$]}}  \section{Initial estimation of airplane drag}  \label{sec:airplanedrag}  With the first airplane characteristics like wing area, power and weight estimation (Class I) completed, it is now time to get an portrayal of the drag polar of the E-SPARC aircraft. This is needed to quantify the amount of drag acting on the aircraft, as function of the lift required at given flight conditions. It yields a unique relationship between $C_L$ and $C_D$ which is known as the drag polar.   A drag estimation in the early design phase of the aircraft can be based on different methods where each method can be refined. To determine the true aerodynamics in this stage is a very elaborate exercise and not even really useful in the early design phases of the aircraft, because a lot can still change in the design. For the E-SPARC aircraft a first drag estimation is done by using Torenbeek method~\cite{TB76}. The general drag formula that is used for most low-speed drag polars is given by Equation~\ref{eq:dragcoefficient}. The drag is a function of Reynolds number and the lift coefficient; $C_D(Re, C_L)$.   \begin{equation}  C_{D}= C_{D_0} + \frac{{C_L}^2}{\pi A e}  \label{eq:dragcoefficient}  \end{equation}  Two different methods will be given here for the calculation of the zero drag coefficient. The first method is based on Raymer and was used for the Class I method where they were used for the W/S - W/P plot as input. Equation~\ref{eq:raymerdrag} shows that the skin friction drag coefficient is multiplied with the ratio of wetted area over reference area. The skin-friction drag coefficient is taken 0,0050 for the category "Light aircraft - single engine" from Raymer \cite{Zerodragraymer}. Together with an estimated ratio of wetted area over reference area of 5 this gives an $C_{D0}$ of 0,025. The decision was made to use values roughly between those of a Cessna Skyplane and the F104, since that is probably a good estimate of where a subsonic racing aircraft would be~\cite{wettedarearatio}.   \begin{equation}  C_{D_0} = C_{fe} \frac{S_{wet}}{S_{ref}}  \label{eq:raymerdrag}  \end{equation}  The Torenbeek method is a more elaborate analytical approach especially useful in calculating the zero drag coefficient as a first estimation if more is known about the design already resulting from Class I. The method is also still based on statistical data, but is sufficient for the early design stage. For the final design it will be important to make a much more precise drag prediction, because this can determine the success or failure of the design in the racing. The method sums up the drag areas of the various components of the airplane. This is only applicable for a wing with thickness/chord ratios up to 20\% and slender fuselages (length/diameter ratio greater than 4)~\cite{TB76}, which will be the case for this aircraft.   \begin{equation}  C_{D_0} S = r_{RE} r_{uc} [r_t((C_DS)_w + (C_DS)_f)+(C_DS)_n]  \end{equation}  \begin{equation}  r_{RE} = 47 * Re_f^{-0.2}  \label{eq:reynoldsnumber}  \end{equation}  This formula make use of parameters that are both based on reference aircraft of other aerobatic racing aircraft and on the class I estimation (wing area and aspect ratio). If the drag of a wing and fuselage are analyzed separately and then together, in reality there is always a higher amount of drag in the latter case. Therefore a correction factor is taken into account in the formula to correct for this phenomenon. This is added in the Reynolds number correction factor given by Equation~\ref{eq:reynoldsnumber}. An overview of the parameters is given in Table~\ref{tab:zerodragparameters}.   The oswald efficiency factor of 0,794 is determined with formula~\ref{eq:oswaldfactor}. This is also a function of the aspect ratio which is taken 6 from other aerobatic racing reference aircraft.  \begin{equation}  e = \frac{e_v}{1 + k_L C_{D_0} \pi A e_v}  \label{eq:oswaldfactor}  \end{equation}  The drag is minimum when the aircraft is flying at $C_L = 0$. For cambered wings and for certain wing fuselage settings, the minimum drag is obtained at $C_L$ values slightly higher then 0. This is not taken into account in the method of Torenbeek as can be also seen in Figure~\ref{fig:dragpolarclcd}. But for preliminary drag polar prediction, this can be neglected without affecting the quality of the results. The difference between the two methods, the latter more elaborate than the other, which calculate zero drag coefficient is given in table~\ref{tab:dragtwomethods}. \\  \begin{table}[htbp]  \centering  \begin{tabular}{clclc}  \toprule  & \textbf{Raymer method (Class I)} &\textbf{ Torenbeek method (Class II)} \\  \toprule  $C_{D0}$ & 0.025 & 0.029 \\  \bottomrule  \end{tabular}  \caption{Zero drag coefficient calculated for two different methods}  \label{tab:dragtwomethods}  \end{table}  \begin{table}[htbp]  \centering  \caption{Parameter input values for drag polar}  \label{tab:zerodragparameters}  \begin{tabular}{l l l l}  \toprule  \textbf{Parameter} & \textbf{Value} & \textbf{Unit} & \textbf{Description} \\   \toprule  $\frac{t}{c}$ & 0.1 & $[-]$ & Mean thickness/chord ratio \\   $\Lambda_{0.25}$ & 3 & $[rad]$ & Sweep angle at the quarter-chord line \\   $S$ & 6.03 & $[m^2]$ & Wing area \\   $r_w$ & 1 & $[-]$ & Type of wing support \\   $r_f$ & 1.15 & $[-]$ & Fuselage shape factor \\   $l_f$ & 5 & $[m]$ & Fuselage length \\   $b_f$ & 0.95 & $[m]$ & Fuselage maximum width \\   $h_f$ & 1.05 & $[m]$ & Fuselage maximum height \\   $r_t$ & 24 & $[\%]$ & Tail drag as percentage of wing and fuselage drag \\   $r_{uc}$ & 1.25 & $[-]$ & Fixed gear, streamlined wheel fairings and struts \\  $Re_f$ & $1.26*10^7$ & $[-]$ & Reynolds number at cruise altitude of 5000m \\  $e_v$ & 0.95 & $[-]$ & Span efficiency factor \\  $k_L$ & 0.38 & $[-]$ & Sweep correction factor \\  $A$ & 6 & $[-]$ & Aspect ratio \\  \bottomrule  \end{tabular}  \end{table}  \nomenclature{$\Lambda_{0.25}$}{Sweep angle at the quarter-chord line \nomunit{[$[rad]$]}}  \nomenclature{$r_w$}{Type of wing support \nomunit{[$-$]}}  \nomenclature{$r_f$}{Fuselage shape factor \nomunit{[$-$]}}  \nomenclature{$l_f$}{Fuselage length \nomunit{[$m$]}}  \nomenclature{$b_f$}{Fuselage maximum width \nomunit{[$m$]}}  \nomenclature{$h_f$}{Fuselage maximum height \nomunit{[$m$]}}  \nomenclature{$r_t$}{Tail drag as percentage of wing and fuselage drag \nomunit{[$\%$]}}  \nomenclature{$e_v$}{Span efficiency factor \nomunit{[$-$]}}  \nomenclature{$k_L$}{Sweep correction factor \nomunit{[$-$]}}  \begin{figure}[htbp]  \centering   \includegraphics[width=0.8\textwidth]{Figures/dragpolarclcd.PNG}  \caption{Drag polar: $C_L$ versus $C_D$}  \label{fig:dragpolarclcd}  \end{figure}  \section{Initial estimation of required lift coefficient}  \label{sec:airplanelift}  In order to choose the right airfoils the required values for $C_l$ have to be known. The two most critical situations were identified from the Class I method to be during landing and high-G turns. In both situations the canard will produce positive lift, so that the main wing area only is required to have 80\% of the total required lift taken, where the other positive lift is generated by the 20\% canard area. From the stability analysis in Chapter \ref{cha:stability} it was deducted that $S_H/S = 0.25$, for which also the stability and control requirement for a positive c.g. range are fulfilled. Due to sweep and canard down wash however, the main wing will need a slightly larger total surface area. This does not increase the effective lift and is therefore not added in the calculations in the conceptual design as a first estimation.  The aircraft needs to to be able to turn or pull up under high G-loads during fast and tight turns in the RBAR that allow for the fastest lab time. In order to ensure the safety of the pilots, the rules limit the maximum load factor to 10g. From the known load factor, the necessary $C_L$ value can then easily be calculated using Equation \ref{eq:CL}  \begin{equation}  \label{eq:CL}  C_L = \frac{W\cdot n\cdot 2}{\rho V^2S}  \end{equation}  With an airspeed of 85\,m/s at the entry of a turn or pull-up, a wing loading W/S = 790 (Chapter \ref{cha:classI}) and air density of 1.225\,$kg/m^3$ this results in a $C_L$ of 1.78.  For landing an $C_L$ of 1.8 is needed, as was shown in the Class I weight estimation in Chapter \ref{cha:classI}. This can be achieved either with the same airfoil or the addition of simple high lift devices (HLD) such as plain flaps which can generate an increase in $C_{L,max}$ of 0.3. For the airfoil selection the $C_L$ of 1.78 needed during high G turns is therefore design driving when the HLD are retracted, because the turns should be performed with minimum drag to be fastest.  The required 2D $C_l$ is calculated from this $C_L$ value using the DATCOM method from Raymer\cite{zerodragcoefficient}, here presented in Equation \ref{eq:cl_datcom} with Figure \ref{fig:cl_datcom} and an assumed NACA 4-series airfoil. The sweep is assumed to lie between 0 and 15 degrees and the sharpness parameter $\Delta y = 0.26$ (typical for NACA 4-series\cite{zerodragcoefficient}). The last term on the right hand side of equation~\ref{eq:cl_datcom} can be neglected as it only accounts for corrections above M=0.2, which is not the case for the RBAR aircraft. Solving Equation \ref{eq:cl_datcom} with $\Delta C_{L_{max}}=0$ yields a required $C_{l_{max}}$ of 1.98.  \begin{equation}  \label{eq:cl_datcom}  C_{L_{max}} = \left[\frac{C_{L_{max}}}{C_{l_{max}}}\right] \cdot C_{l_{max}} + \Delta C_{L_{max}}  \end{equation}  \begin{figure}[htbp]  \centering   \includegraphics[width=0.6\textwidth]{Figures/cl_datcom}  \caption{3D correction below M=0.2 \cite{zerodragcoefficient}}  \label{fig:cl_datcom}  \end{figure}  Using the DATCOM method, the lift coefficient slope can be determined as well for a wing with known aspect ratio and sweep. A higher aspect ratio and less sweep contribute to a more ideal (higher) lift slope $C_{L_{\alpha}}$ closer to $C_{l_{\alpha}}$. The canard will use a higher aspect ratio and no sweep and therefore have a higher lift curve slope than the main wing sweep and a lower aspect ratio. This plays an important role in the airfoil selection in Section \ref{sec:airfoil_selection}.  \section{Initial estimation of required moment coefficient}  \label{sec:airplanemoment}  The balancing of the aerodynamic moments around the center of gravity works different for a canard configuration than for a conventional aircraft. While for a conventional configuration the tail needs to generate negative lift in most flight conditions to counteract the negative pitching moment from the wing, a canard usually can counteract the moment with positive lift.  When the canard stalls, the main wing needs to ensure a pitch-down movement to make sure that instead of stalling itself and entering a deep stall, the canard is possible to recover from its stalling condition. A negative moment coefficient of the main wing airfoil is therefore beneficial that stays negative under high angles of attack. For the canard the moment coefficient of the airfoil is of less importance.  \section{Airfoil selection}  \label{sec:airfoil_selection}  The selection of the airfoil is a very important design choice depending on the configuration that is chosen. For the chosen canard configuration a proper combination of main wing versus canard airfoils and their relative position, as well as incidence angles, sweep and size is essential in designing a safe and controllable aircraft. Only the right choice and a good combination of the design parameters leads to a design that uses the full potential of a canard configuration.   The airfoil affects several performance characteristics in cruise, takeoff and landing. For example the aerodynamic efficiency and stall speed are much dependent on the airfoil type, especially the stall characteristics are very important for aerobatic racing, whereas the efficiency plays a bigger role during the 30 minutes cruise. This section will discuss the influence of airfoil parameters on the airfoil characteristics and give a first overview over their aerodynamic characteristics.   Most aircraft that are competing at the Red Bull Air Race are originally aerobatic airplanes. Those aircraft mostly use symmetrical airfoils to sustain in an inverted flight during maneuvers. The E-SPARC aircraft however, is built specifically for the Red Bull Air Race which does not require inverted flight. Airfoil camber is therefore considered an option for both the main wing and the canard.  \subsection{Canard}  For a canard configuration it is required to have the canard stall first before the wing stalls. This makes the aircraft pitch down and moves the canard out of the stall condition. If the wing stalls first the overall moment coefficient becomes increasingly positive and the aircraft enters a deep stall situation from which it can not recover. To have the canard stall first different options exist. One way is assuming the same type of airfoil for both the canard and wing putting the canard at a positive incidence angle. This will yield a higher effective angle of attack for the canard which then reaches its maximum angle of attack earlier than the main wing. The disadvantage of putting the canard at a incidence angle is that it also generates more induced drag. Another way is therefore to use an airfoil for the canard with more camber which leads to a shift in the lift polar. This also leads to earlier stalling of the canard as the cambered airfoil reaches it maximum lift coefficient at lower angles of attack. Finally the canard can use an entirely different airfoil with a lower maximum lift coefficient, also leading to earlier stall of the canard.   The canard of E-SPARC will have an incidence angle of $0\,\degree$ and use a cambered airfoil for high maximum lift coefficient. The higher aspect ratio of the canard and the $0\,degree$ sweep however, will lead to a steeper lift slope (Section \ref{sec:airplanelift}) and allow it therefore to stall before the wing. The airfoil needs to be thick enough to provide enough stiffness for the high aspect ratio canard during high G turns. The canard will use a similar profile as the main wing with higher camber, leading to earlier stall and a higher thickness to support its higher aspect ratio. A first airfoil choice is the modified NACA 4-series airfoil NACA 8218-82 (Figure \ref{fig:canard_airfoil}). This airfoil has 18\% t/c with the point of max. thickness at 20\% xt/c, 8\% camber (f/c) at 25\% xf/c and 6.7\% leading edge radius (R/c).  \begin{figure}[htbp]  \centering   \includegraphics[width=0.6\textwidth]{Figures/Canard_airfoil}  \caption{Canard airfoil: \textbf{NACA 8218-82} at 0\degree \,incidence angle}  \label{fig:canard_airfoil}  \end{figure}  \subsection{Main wing}  The airfoil for the main wing will also be positioned at an incidence angle of 0\degree. A large maximum lift coefficient is needed to allow for steep turns during the races and short landing distance (requirement from Red Bull of 500m landing distance). In order to allow for high aerodynamic efficiency and because inverted flight is not part of the Red Bull Air Race, a moderately cambered airfoil is used. The higher camber and lift slope of the canard will make sure that the canard stalls first. Additionally the down wash from the canard will decrease the effective angle of attack at the main wing and delay main wing stall even further. In order to be longitudinally stable in a canard stall, the main wing airfoil needs to provide a negative moment coefficient that stays negative at higher angles of attack. This can also be achieved by using a cambered airfoil.  The critical Mach number is of no importance for the E-SPARC, because only speeds of up to 200 knots are allowed during the Red Bull Air Race. The point of maximum thickness for both the main wing and the canard can therefore be close to the leading edge and a large nose radius can be used to allow for higher maximum lift coefficient. For the main wing this leads to an airfoil of moderate thickness and camber with the point of maximum thickness close to the front and a large nose radius. A first airfoil that will still be altered and fine-tuned is the modified NACA 4-series airfoil NACA 6215-92   (Figure \ref{fig:mainwing_airfoil}). This airfoil has 15\% t/c at the point of max. thickness at 20\% xt/c, 6\% camber (f/c) at 25\% xf/c and 6.7\% leading edge radius (R/c). %The high thickness and nose radius of both airfoils lead to trailing edge stall, which is beneficial for a more gradient stalling behavior.  \begin{figure}[htbp]  \centering   \includegraphics[width=0.6\textwidth]{Figures/MainWing_airfoil}  \caption{Main wing airfoil: \textbf{NACA 6215-92} at 0\degree \,incidence angle}  \label{fig:mainwing_airfoil}  \end{figure}  \subsection{Tip stall}  Tip stall can be dangerous for a canard configuration, due to sweep and the effect of the canard. Sweep causes a span-wise air flow component that leads to a higher wing loading at the tip than at the root. The down wash and disturbed airflow from the canard at the same time lead to a reduction in lift on the inboard wing section and an increase on the outboard section of the main wing, increasing the risk of tip stall. This can be prevented by applying twist, varying the airfoil for the outboard wing sections or using a leading edge droop. The exact approach will be determined during the detail design.  \subsection{Overview aerodynamic parameters}  \label{subs:aerodynamic_overview}  Table \ref{tab:airfoilparameters} gives an overview over the most important parameters that were chosen in the preceding sections.  \begin{table}[htbp]  \centering  \begin{tabular}{clclc}  \toprule  \rowcolor{grey} Parameter & Main wing airfoil & Canard airfoil\\  \toprule  $t/c_{max} (max thickness)$ & 0.15 & 0.18 \\  $xt/c_{max} (pos of max thickness)$ & 0.2 & 0.2 \\  $f/c (camber)$ & 0.06 & 0.08 \\  $xf/c (pos of camber)$ & 0.25 & 0.29 \\  $C_{D0}$ & 0.025 & 0.029 \\  \bottomrule  \end{tabular}  \caption{Airfoil parameters for canard and main wing}  \label{tab:airfoilparameters}  \end{table}  %The thicker NACA0012 and NACA0015 are popular wing airfoils for aerobatic and sport aircraft. Or modern airfoil types???   %Clmax   %Speed at 10 g  %High lift devices  %Type of configuration: Canard, Conventional  %Aerodynamic moment  %Drag polar  diff --git a/Chapters/15.Stability_Control.tex b/Chapters/15.Stability_Control.tex  new file mode 100644  index 0000000..94a4eec  --- /dev/null  +++ b/Chapters/15.Stability_Control.tex 
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\chapter{Stability and Control Characteristics}  \label{cha:stability}  %Please do not forget to add new symbols to the nomenclature e.g.: \nomenclature{$t_{element}$}{Thickness coefficient of plate \nomunit{[$-$]}}  The stability and control characteristics can be analysed for the chosen design configuration by investigating the range of centre of gravity in which it is both controllable and longitudinally stable for a respective tail area.  The load during a flight does not change substantially. Fuel is not burned and expelled and the power units do not gain or lose weight. Also, the pilots weight does not change. The smoke that is expelled for visibility does not introduce an important weight drop. Thus, the centre of gravity is assumed to remain in the same position throughout the course of the flight.  Furthermore, there is no payload added before the flight, apart from the pilots weight and the power units. Thus, on ground stability only depends on the stability of the empty aircraft, the aircraft with internal systems and finally with the pilot in the cockpit.  The aircraft shall perform in aerobatic racing competitions and has to be very maneuverable. Therefore, to avoid being too stable which would limit the fast maneuverability, the aircraft is designed to be close to neutrally stable. As a result, no stability margin (SM) is assumed between the centre of gravity and the neutral point of the entire aircraft, meaning that they coincide (SM$=$0). This can also offer better controllability characteristics than a conventional configuration, since the canard can be used up to higher values of $C_{L}$~\cite{SE_stability}.  The stability is highly dependent on the stabilizer size with respect to the wing area. The relation of the two is plotted over the location of the centre of gravity with respect to the leading edge of the mean aerodynamic chord. The derivation of the corresponding requirements for stability and controllability for the canard can be found in \cite{SE_stability} and~\cite{SE_control}. The results are presented in Equation~\ref{eq:stability_conventional}~and~\ref{eq:control_conventional}. Equation~\ref{eq:stability_conventional} shows the relation of the centre of gravity and the tail size within the limits of stability, resulting from the fact that in the neutral point the aerodynamic moment should not change with a disturbance in angle of attack. Equation~\ref{eq:control_conventional} depicts the relation of the centre of gravity and the tail size in controllability, with the underlying idea that, for a controllable, trimable aircraft, the moment around the neutral point in trimmed condition should be zero.  Due to the fact that the canard configuration has a negative additional lifting surface (or canard) arm and that the canard produces positive (upwards) lift, the allowable centre of gravity range for stability is drastically decreased and shifted forward as compared to a conventional configuration. This limits the design flexibility, but since the centre of gravity does not change during the flight, as was explained before, will not cause problems when this is considered during the distribution of the weights during design.  \begin{equation}  \frac{S_{h}}{S}=\frac{1}{\frac{C_{L\alpha_{h}}}{C_{L\alpha}}(1-\frac{d\epsilon}{d\alpha})\frac{l_h}{c}(\frac{V_{h}}{V})^2}x_{cg}-\frac{x_{ac}}{\frac{C_{L\alpha_{h}}}{C_{L\alpha}}(1-\frac{d\epsilon}{d\alpha})\frac{l_h}{c}(\frac{V_{h}}{V})^2}  \label{eq:stability_conventional}  \end{equation}  \begin{equation}  \frac{S_h}{S}=\frac{1}{\frac{C_{L_{h}}}{C_{L_{A-h}}}\frac{l_h}{c}(\frac{V_{h}}{V})^2}x_{cg}+\frac{\frac{C_{m_{ac}}}{C_{L_{A-h}}}-x_{ac}}{\frac{C_{L_{h}}}{C_{L_{A-h}}}\frac{l_h}{c}(\frac{V_{h}}{V})^2}  \label{eq:control_conventional}  \end{equation}  In the canard configuration under consideration, several aspect have to be taken into account when using these equations. %They are presented in table ??.  \begin{itemize}  \item \textbf{$\frac{d\epsilon}{d\alpha} = 1$} - The downwash a tail experiences normally due to being positioned in the wake of the wing is not considered for a canard~\cite{SE_stability}. Therefore, both wings experience the same angle of attack. However, if the trailing edge of the canard is too close to the leading edge of the main wing, closer than 2 times the root chord length, the design becomes close coupled and this assumption cannot be made any more~\cite{SE_stability}. The canard experiences an upwash and stall characteristics have to be considered.  \item \textbf{$\frac{V_{h}}{V} = 1$} - Both wings experience the same velocity, since the canard does not slow down the freestream substantially.  \item The effect of the canard on the main wing can be neglected. Therefore, no up- or downwash is taken into account on a general level. In detail, the inboard section of the main wing experiences a downwash, while the outboard section experiences an upwash. These detailed characteristics have to be considered in the final, detailed design. Equation~\ref{eq:canard-wing_wash} is used to determine the altered $C_{L_{\alpha}}$ of the main wing when the airfoil is determined~\cite{SE_stability}.  \end{itemize}  \begin{equation}  C_{L_{\alpha-due to canard}}=C_{L_{\alpha}}(1-\frac{2C_{L_{\alpha_{h}}}S_{h}/S}{\pi A_{w}k}  \label{eq:canard-wing_wash}  \end{equation}  The relation of the canard and the main wing can be determined when the dimensions of the aircraft are finalised. Then, it will be seen whether the configuration has to be considered as close coupled or whether aerodynamic interrelations can be neglected.  A first stability analysis was performed and the results shown in figure \ref{fig:stab-contr_graph} are found.  \begin{figure}[htbp]  \centering  \includegraphics[width = \textwidth]{Figures/stab-contr_graph.png}  \caption{The possible range of centre of gravity with a certain tail size in terms of stability and controllability.}  \label{fig:stab-contr_graph}  \end{figure}  Figure \ref{fig:stab-contr_graph} shows the stability and controllability requirements for the centre of gravity location for a certain horizontal stabiliser area. The ratio of the canard area with respect to the main wing area main not lie under the red control line or above the blue stability line for a particular centre of gravity location. However, since the electric aircraft offers a great flexibility in moving the centre of gravity and keeping it at a certain location during the flight, the aircraft can be made stable and controllable by shifting the weight components. Therefore, it will be tried to achieve a small canard area by bringing the centre of gravity location close to the point where the two lines in figure \ref{fig:stab-contr_graph} cross. This is the case for a centre of gravity location of 0.14 $\frac{x_{cg}}{MAC}$ with a possible horizontal stabilizer to wing ratio of about $0.15$.  Since the $\frac{S_{h}}{S}$ ratio should not become too big the design is limited at $\frac{S_{h}}{S} = 0.25$ for this stage of the design. That would leave a centre of gravity range of $+0.14$ to $-0.13$ $\frac{x_{cg}}{MAC}$.  Updates to the aerodynamic characteristics and the dimensions of the aircraft have to be incorporated in the stability and control tool to investigate whether the design changes still allow for stable and controllable operation. If not, an iterative process has to be started to find the correct designs.  For figure \ref{fig:stab-contr_graph}, dimensional estimates that were also used in the Class 2 method are utilised. The aerodynamic characteristics that were used are shown in table~\ref{tab:stab-contr_input}.  \begin{table}[!htbp]  \centering  \caption{Factors used for gaining a first impression of the stability and control characteristics}  \begin{tabular}{lclc}  \toprule  \textbf{Factor} & \textbf{Symbol} & \textbf{Value} & \textbf{Unit} \\  \toprule  Lift coefficient, main wing & $C_{L}$, $C_{L_{A-h}}$ & 1.7 & - \\ %\midrule  Lift coefficient, horizontal canard & $C_{L_{h}}$ & 1.5 & - \\ %\midrule  Moment coefficient about aerodynamic centre & $C_{m_{ac}}$ & -0.2 & - \\ %\midrule  Lift gradient, main wing & $C_{L_{\alpha}}$ & $2\pi$ & $\frac{1}{rad}$ \\ %\midrule  Lift gradient, canard & $C_{L_{h_{\alpha}}}$ & $0.5\pi$ & $\frac{1}{rad}$ \\  \bottomrule  \end{tabular}%  \label{tab:stab-contr_input}%  \end{table}%  %There are several ways to determine the characteristics for the biplane. Firstly, one can assume that the biplane behaves like a conventional configuration and replace the two wings by one equivalent conventional wing. Whilst this yields an impression of the behaviour, it does introduce errors \cite{biplane_stab}. Following the approach presented in \cite{biplane_stab}, the stability characteristics of a biplane and the tail area with respect to the centre of gravity location can be analysed by considering the wings individually, using the subscripts s, i and t for the top wing, the bottom wing and the tail, respectively.  %According to \cite{biplane_stab}, the relation presented in \ref{eq:stab_bi} is found for the stability behaviour. It is assumed that the change of drag with change of angle of attack is small enough to be neglected.  %method 2; check with professionals whether approach is correct  %\begin{equation}  %\frac{S_t}{S_{eq}}= \frac{C_{Ls_{\alpha}}\frac{S_s}{S_{eq}}(\frac{s_s-x_s}{c_{eq}}+x_{cg})+C_{Li_{\alpha}}\frac{S_i}{S_{eq}}(x_{cg}-\frac{(x_i+s_i)}{c_{eq}})}{C_{Lt_{\alpha}}(1+\frac{d\epsilon}{d\alpha})(\frac{l}{c_{eq}}+\frac{x_{cg}-l_{LEMAC}}{c_{eq}}-x_{cg})}  %\label{eq:stab_bi}  %\end{equation}  %$S_{s,i,t}$ are the respective wing area, $s_{s,i,t}$ are the leading edge position on the respective wing mean aerodynamic chord, which was multiplied by the respective wing mean aerodynamic chord, which is represented by $c_{s,i,t}$, and divided by $c_{eq}$, the equivalent chord length of a representative single wing to standardise the relation between the different chords. $c_{eq}$ was found from Equation~\ref{eq:equiv_chord}. \cite{biplane_stab}  %\begin{equation}  %c_{eq}=\frac{c_{s}S_{s}+c_{i}S_{i}}{S_{s}+S_{i}}  %\label{eq:equiv_chord}  %\end{equation}  %The equivalent wing area, $S_{eq}$, is determined by the relation shown in Equation~\ref{eq:equiv_area}.  %\begin{equation}  %S_{eq} = S_{s}+S_{i}  %\label{eq:equiv_area}  %\end{equation}  %By setting the moment around the centre of gravity of the biplane to zero, the controlability characteristics can be analysed as well, based on the same distribution of lift over both main wings, as done in Equation~\ref{eq:stab_bi}. The relation is shown in Equation~\ref{eq:contr_bi}.  %\begin{equation}  %\frac{S_t}{S_{eq}}=\frac{C_{m_{0}}+C_{Ls}\frac{S_s}{S_{eq}}(\frac{s_s-x_s}{c_{eq}}+x_{cg})+C_{Li}\frac{S_i}{S_{eq}}(x_{cg}-\frac{(x_i+s_i)}{c_{eq}})-C_{D_{s}}\frac{S_{s}}{S_{eq}}(\frac{h-Y}{c_{eq}})-C_{D_{i}}\frac{S_{i}}{S_{eq}}\frac{Y}{c_{eq}}}{C_{L_{t}}(\frac{l_{t}}{c_{eq}}+x_{eq}-x_{cg})-C_{D_{t}(\frac{h_{EH}-Y}{c_{eq}})}}  %\label{eq:contr_bi}  %\end{equation}  %Here, $Y$ depicts the vertical distance form the bottom wing to the equivalent wing. Y can be determined using Equation~\ref{eq:equiv_location}. $h$ is the vertical distance between the two main wings, $h_{EH}$ shows the vertical distance from the horizontal tail wing to the bottom wing and $h_{t}$ is the vertical distance from the thrust line to the vertical tail.  %think about filling in figures showing the distances here  %\begin{equation}  %Y = \frac{C_{L_{s}}S_{s}}{C_{L_{s}}S_{s}+C_{L_{i}}S_{i}}  %\label{eq:equiv_location}  %\end{equation}  %The presented equations can be used to determine the X-plots, showing the centre of gravity locations and the respective tail areas needed for stability and controlability for the different configurations.  \nomenclature{$S_{h}$}{Horizontal tail area \nomunit{[$m^2$]}}  \nomenclature{$S$}{Wing area \nomunit{[$m^2$]}}  %\nomenclature{$S_{eq}$}{Equivalent wing area of biplane \nomunit{[$m^2$]}}  %\nomenclature{$S_{s}$}{Upper wing area \nomunit{[$m^2$]}}  %\nomenclature{$S_{i}$}{Lower wing area \nomunit{[$m^2$]}}  %\nomenclature{$S_{t}$}{Horizontal tail area of biplane \nomunit{[$m^2$]}}  %\nomenclature{$s_{s,i,t}$}{Leading edge position on wing mean aerodynamic chord of upper, lower, tail wing \nomunit{[$m$]}}  %\nomenclature{$x_{s,i,t}$}{Position of aerodynamic centre on chord line of upper, lower, tail wing \nomunit{[$m$]}}  \nomenclature{$x_{cg}$}{Longitudinal position of centre of gravity \nomunit{[$m$]}}  \nomenclature{$x_{ac}$}{Longitudinal position of aerodynamic centre \nomunit{[$m$]}}  \nomenclature{$C_{L_{\alpha_{h}}}$}{Lift gradient of tail wing \nomunit{[$1/deg$]}}  \nomenclature{$C_{L_{\alpha}}$}{Lift gradient \nomunit{[$1/deg$]}}  \nomenclature{$C_{L_{h}}$}{Lift coefficient of tail wing \nomunit{[$-$]}}  \nomenclature{$C_{L_{A-h}}$}{Lift coefficient of main wing without tail \nomunit{[$-$]}}  \nomenclature{$C_{m_{ac}}$}{Moment coefficient about aerodynamic centre \nomunit{[$-$]}}  %\nomenclature{$C_{L_{\alpha_{s}}}$}{Lift gradient of upper wing \nomunit{[$1/deg$]}}  %\nomenclature{$C_{L_{\alpha_{i}}}$}{Lift gradient of lower wing \nomunit{[$1/deg$]}}  %\nomenclature{$C_{L_{\alpha_{t}}}$}{Lift gradient of tail wing of biplane \nomunit{[$1/degree$]}}  \nomenclature{$\alpha$}{Angle of attack \nomunit{[$degree$]}}  \nomenclature{$\epsilon$}{Downwash angle \nomunit{[$degree$]}}  %\nomenclature{$C_{L_{s}}$}{Lift coefficient of upper wing \nomunit{[$-$]}}  %\nomenclature{$C_{L_{i}}$}{Lift coefficient of lower wing \nomunit{[$-$]}}  %\nomenclature{$C_{D_{s}}$}{Drag coefficient of upper wing \nomunit{[$-$]}}  %\nomenclature{$C_{D_{i}}$}{Drag coefficient of lower wing \nomunit{[$-$]}}  %\nomenclature{$C_{D_{t}}$}{Drag coefficient of tail wing of biplane \nomunit{[$-$]}}  \nomenclature{$C_{m_{0}}$}{Zero lift moment coefficient \nomunit{[$-$]}}  \nomenclature{$V$}{Free stream velocity the main wing is facing \nomunit{[$m/s$]}}  \nomenclature{$V_{h}$}{Velocity the tail wing is facing \nomunit{[$m/s$]}}  \nomenclature{$l_{h}$}{Tail arm; distance from $x_{cg}$ to the tail aerodynamic centre \nomunit{[$m$]}}  \nomenclature{$c$}{Chord length \nomunit{[$m$]}}  %\nomenclature{$l$}{Tail arm of biplane \nomunit{[$m$]}}  \nomenclature{$l_{LEMAC}$}{Longitudinal position of the LEMAC \nomunit{[$m$]}}  %\nomenclature{$h$}{Vertical distance from top wing CA to bottom wing CA \nomunit{[$m$]}}  %\nomenclature{$h_{EH}$}{Vertical distance from horizontal tail CA to bottom wing CA \nomunit{[$m$]}}  %\nomenclature{$Y$}{Vertical distance from bottom wing CA to equivalent wing CA \nomunit{[$m$]}}