Nomenclature
The following variables and symbols are used throughout this sizing report.
| Symbol | Variable Name | Definition |
|---|---|---|
| \(b\) | Main Wing Span | The total length of the main wing from tip to tip. |
| \(b_{cs}\) | Control Surface Span | The spanwise length of the movable surface (aileron, elevator, rudder). |
| \(b_{total}\) | Total Span | The full span of the lifting surface to which the control surface is attached. |
| \(b_R\) | Rudder Span | The span (height) of the rudder surface. |
| \(b_v\) | Vertical Tail Span | The total span (height) of the vertical stabilizer. |
| \(\bar{c}\) | Mean Aerodynamic Chord | The average chord of the main wing. |
| \(c_{cs}\) | Control Surface Chord | The width of the movable surface aft of the hinge line. |
| \(c_{total}\) | Total Chord | The width of the entire stabilizer assembly (fixed + moving). |
| \(C_{L\alpha v}\) | Vert. Tail Lift Slope | The lift curve slope of the vertical tail airfoil. |
| \(C_{n\delta R}\) | Control Power Derivative | The change in yawing moment coefficient due to rudder deflection. |
| \(\eta_v\) | Tail Efficiency | Dynamic pressure ratio at the tail (\(q_{tail}/q_{\infty}\)), typically 0.95–1.0. |
| \(L_h\) | Horiz. Tail Moment Arm | Distance from Aircraft CG to the Aerodynamic Center (AC) of the horizontal tail. |
| \(L_v\) | Vert. Tail Moment Arm | Distance from Aircraft CG to the Aerodynamic Center (AC) of the vertical tail. |
| \(S_{ref}\) | Wing Area | Planform area of the main wing. |
| \(S_h\) | Horizontal Tail Area | Planform area of the horizontal stabilizer. |
| \(S_v\) | Vertical Tail Area | Planform area of the vertical stabilizer. |
| \(\tau_r\) | Flap Effectiveness | Angle of attack effectiveness parameter (based on \(c_{cs}/c_{total}\) ratio). |
| \(V_h\) | Horiz. Tail Volume Coeff. | Non-dimensional coefficient representing pitch stability authority. |
| \(V_v\) | Vert. Tail Volume Coeff. | Non-dimensional coefficient representing yaw stability authority. |
Preliminary Control Surface Sizing
The preliminary sizing of the aircraft control surfaces follows the historical “Rules of Thumb” and volume coefficient methods outlined in Sadraey’s Aircraft Design (Chapter 12) and Raymer’s Aircraft Design: A Conceptual Approach. Due to the specific flight envelope of SAE Aero Design aircraft (low Reynolds number, high payload fraction, and short takeoff requirements), these historical ratios have been adjusted to ensure sufficient control authority.
Control Surface Geometric Ratios
The initial dimensions for the Ailerons, Elevators, and Rudder are determined using geometric ratios relative to their respective lifting surfaces. These values are derived from the “Strip Theory” approach, where the control surface is treated as a hinged flap that alters the camber of the airfoil.
| Surface | Span Ratio (\(b_{cs}/b_{total}\)) | Chord Ratio (\(c_{cs}/c_{total}\)) | Primary Function |
|---|---|---|---|
| Aileron | 0.30 – 0.50 | 0.20 – 0.25 | Roll Control |
| Elevator | 0.80 – 1.00 | 0.35 – 0.45 | Takeoff Rotation |
| Rudder | 0.70 – 1.00 | 0.30 – 0.40 | Yaw/Ground Steering |
| Flaps | 0.40 – 0.60 | 0.25 – 0.30 | Lift Augmentation |
Note on SAE particular use-case:
Elevators: Standard General Aviation aircraft typically use a chord ratio of 25-30%. For this design, a ratio of 35-45% is selected to generate sufficient pitching moment (\(C_m\)) to rotate the aircraft at low takeoff speeds while carrying a heavy payload.
Rudder: A higher chord ratio is selected to combat the high adverse yaw generated by high-lift wings and to provide steering authority during the takeoff roll.
Tail Volume Coefficients
Before the control surfaces can be detailed, the reference areas for the Horizontal (\(S_h\)) and Vertical (\(S_v\)) stabilizers are sized using the Tail Volume Coefficient method. This ensures the aircraft possesses inherent static stability.
Horizontal Tail Volume (\(V_h\))
The horizontal tail volume coefficient relates the tail moment arm and area to the main wing’s geometry to ensure longitudinal stability and pitch authority.
\[V_h = \frac{L_h S_h}{\bar{c} S_{ref}}\]
Vertical Tail Volume (\(V_v\))
The vertical tail volume coefficient ensures directional stability and provides the rudder with sufficient lever arm to control yaw.
\[V_v = \frac{L_v S_v}{b S_{ref}}\]
Target Values for Stability
Table 2 outlines the target coefficients used for this design compared to standard transport aircraft. SAE Aero aircraft typically require larger tail volumes due to the slow flight speeds and potential for shifting CG with payload.
| Parameter | Typical Transport Jet | Target SAE Aero |
|---|---|---|
| Horizontal \(V_h\) | 0.40 – 0.50 | 0.50 – 0.70 |
| Vertical \(V_v\) | 0.04 – 0.05 | 0.05 – 0.07 |
Control Derivative Estimation
Following the methodology in Sadraey (Section 12.6), the control power effectiveness for the rudder (and analogously for the elevator) is defined by the derivative \(C_{n\delta}\). This may be verified using XFLR5 analysis.
\[C_{n\delta R} = -C_{L\alpha v} V_v \eta_v \tau_r \frac{b_R}{b_v}\]
This equation relates the physical size of the rudder (\(b_R/b_v\)) and its chord effectiveness (\(\tau_r\)) to the total yawing moment the aircraft can generate.
In-depth Control Surface Sizing
Sources
Raymer, Daniel P. Aircraft Design: A Conceptual Approach. 5th ed., American Institute of Aeronautics and Astronautics, 2012.
Sadraey, Mohammad H. Aircraft Design: A Systems Engineering Approach. Wiley, 2012.