MSD : SAE Aero Aircraft Design & Build
Preliminary Horizontal and Vertical Stabilizer Design, Longitudinal and Directional Static Stability
Horizontal Stabilizer Parameters:
1. Ratio of horizontal tail-wing aerodynamic centers distance with respect to fuselage length
𝑙𝑎𝑐 /𝑙𝑓
2. Overall fuselage length 𝑙𝑓
3. Horizontal tail-wing aerodynamic centers distance 𝑙𝑎𝑐
4. Horizontal tail volume coefficient 𝑉𝐻
5. Center of gravity location 𝑥𝑐𝑔
6. Horizontal tail arm 𝑙𝑡
7. Horizontal tail planform area 𝑆𝑡
8. Horizontal tail airfoil
9. Horizontal tail aspect ratio 𝐴𝑅𝑡
10. Horizontal tail taper ratio 𝜆𝑡
11. Additional geometric parameters (Sweep Angle, Twist Angle, Dihedral)
12. Incidence Angle 𝑖𝑡
13. Neutral Point 𝑥𝑁𝑃
14. Static Margin
15. Overall Horizontal Stabilizer Geometry
16. Overall Aircraft Static Longitudinal Stability
17. Elevator (TBD in Control Surfaces Design)
Vertical Stabilizer Parameters:
18.
19.
20.
21.
22.
23.
Vertical tail volume coefficient 𝑉𝑣
Vertical tail arm 𝑙𝑣
Vertical tail planform area 𝑆𝑉
Vertical tail aspect ratio 𝐴𝑅𝑣
Vertical tail span 𝑏𝑣
Vertical tail sweep angle Λ 𝑣
24.
25.
26.
27.
28.
Vertical tail minimum lift curve slope 𝐶𝐿𝛼 𝑣
Vertical tail airfoil
Overall Vertical Stabilizer Geometry
Overall Aircraft Static Direction Stability
Rudder (TBD in Control Surfaces Design)
1. l/Lf Ratio: 0.6
The table below shows statistical ratios between the distance between the wing aerodynamic center
and the horizontal tail aerodynamic center 𝑙𝑎𝑐 with respect to the overall fuselage length (Lf).
2. Fuselage Length (Lf): 60.00 in
Choosing this value is an iterative process to meet longitudinal and vertical static stability, internal
storage, and center of gravity requirements, but preliminarily choose 𝑙𝑓 = 60.00 𝑖𝑛
3. Horizontal tail-wing aerodynamic centers distance 𝒍𝒂𝒄 : 36.00 in
𝑙𝑎𝑐
= 0.6
𝑙𝑓
𝑙𝑎𝑐 = 36.00 𝑖𝑛
4. Horizontal Tail Volume Coefficient (VH): 1.0
The table below shows the horizontal and vertical tail coefficients for various types of aircraft.
𝑉𝐻 =
𝑙𝑡 𝑆𝑡
𝑆𝑐
𝑉𝐻 = 1
5. Center of Gravity Location (xcg):𝟎. 𝟑𝟎𝑐
Justification:
a) Choose center of gravity aft of aerodynamic center to aid (give more room) with placing
components to meet specified center of gravity location.
b) If horizontal tail stabilizes aircraft pitching up, it generates a positive lift force, adding to the
wing lift.
c) 0.30𝑐 is the aft most recommended limit for center of gravity placement.
6. Horizontal Tail Arm 𝒍𝒕 : 35.2537 in
𝑙𝑡 = 𝑙𝑎𝑐 − 𝑥𝑐𝑔 − 𝑥𝑎𝑐
𝑙𝑡 = 35.2537 𝑖𝑛
7. Horizontal Tail Planform Area (St): 3.6352 𝑓𝑡 2
𝑉𝐻 =
𝑙𝑡 𝑆𝑡
=1
𝑆𝑐
𝑆𝑡 = 3.6352 𝑓𝑡 2
8. Airfoil Selection: NACA-0021
Justification:
a) Choose symmetric airfoil as the horizontal tail should behave in a similar manner when at a
positive or negative angle-of-attack
b) Horizontal tail should never stall, specifically it should at least stall later than the wing for
recovery
c) Maximize 𝐶𝐿 𝑚𝑎𝑥
𝑡
d) Maximize 𝐶𝐿 𝛼
𝑡
e) Minimize overall drag
f) Minimize overall size
NACA-0009:
NACA-0010:
NACA-0015:
NACA-0018
NACA-0021:
NACA-0024:
NACA-0018, NACA-0021, NACA-0024 Airfoil Comparison:
9. Aspect Ratio 𝑨𝑹𝒕 : 4
Justification:
a) It is recommended that the aspect ratio of the tail be such that the span is longer than the
propeller diameter to ensure that a portion of the tail is out of the wake or downwash of the
wing, increasing tail efficiency 𝜂 .
b) Horizontal tail aspect ratio should be lower than that of the wing to increase stall angle and
allow for recovery if needed
c) It is recommended that:
2
𝐴𝑅𝑡 = 𝐴𝑅𝑤
3
𝐴𝑅𝑡 = 4
10. Horizontal Tail Taper Ratio 𝝀𝒕 : 0.7
a) For transport aircraft, the horizontal tail taper ratio is usually between 0.4 and 0.7
b) To ensure a higher stall angle than the wing through a lower Oswald efficiency factor and a lift
distribution that is less elliptical, choose 𝜆𝑡 = 0.7
11. Additional geometric parameters (Sweep Angle, Twist Angle, Dihedral): N/A
a) For the benefits of applying the any of the above parameters to the horizontal geometry, refer
to the preliminary wing design parameter selection document
b) In the preliminary design phase, it is recommended to make these parameters have the same
values as those of the wing.
12. Horizontal Tail Incidence Angle 𝒊𝒕 : 4.2300 deg
-
Determine horizontal tail incidence angle to trim (longitudinal) aircraft at cruise
𝐶𝑚 𝑐𝑔 = 𝐶𝑚 𝑐𝑔
𝑤
+ 𝐶𝑚 𝑐𝑔 + 𝐶𝑚 𝑐𝑔 = 0
𝑡
𝑓
𝐶𝑚 0 𝑤 + 𝐶𝑚 𝛼 𝑤 𝛼𝑤 𝑐𝑟𝑢𝑖𝑠𝑒 + 𝐶𝑚 0 𝑡 + 𝐶𝑚 𝛼 𝑡 𝛼𝑤 𝑐𝑟𝑢𝑖𝑠𝑒 + 𝐶𝑚 0 𝑓 + 𝐶𝑚 𝛼 𝑓 𝛼𝑤 𝑐𝑟𝑢𝑖𝑠𝑒 = 0
𝐶𝑚 𝑎𝑐 𝑤 + 𝐶𝐿0 𝑤
𝑘 2 −𝑘 1
36.5𝑆𝑐
𝑥=𝑙 𝑓
𝑥=0
𝑥 𝑐𝑔
𝑐
−
𝑥 𝑎𝑐
𝑐
+ 𝐶𝐿𝛼 𝑤
𝑥 𝑐𝑔
𝑐
−
𝑥 𝑎𝑐
𝑐
𝑤𝑓 2 𝛼0 𝑤 + 𝑖𝑓 Δ𝑥 +
𝛼𝑤 𝑐𝑟𝑢𝑖𝑠𝑒 + 𝜂𝑉𝐻 𝐶𝐿 𝛼 𝜀0 + 𝑖𝑤 − 𝑖𝑡 − 𝜂𝑉𝐻 𝐶𝐿 𝛼
𝑡
1
36.5𝑆𝑐
𝑖𝑡 = 4.2300 𝑑𝑒𝑔
13. Neutral Point 𝒙𝑵𝑷 :0.7067𝑐
𝐶𝐿 𝛼
𝑥𝑛𝑝 𝑥𝑎𝑐 𝐶𝑚 𝛼 𝑓
𝑑𝜀
𝑡
=
−
+ 𝜂𝑉𝐻
1−
𝑐
𝑐
𝐶𝐿𝛼 𝑤
𝐶𝐿𝛼 𝑤
𝑑𝛼
𝑥𝑁𝑃 = 0.7067𝑐
14. Static Margin: 0.3548
𝑆𝑡𝑎𝑡𝑖𝑐 𝑀𝑎𝑟𝑔𝑖𝑛 = 𝑥𝑁𝑃 − 𝑥𝑐𝑔
𝑆𝑡𝑎𝑡𝑖𝑐 𝑀𝑎𝑟𝑔𝑖𝑛 = 0.3548
𝑥=𝑙 𝑓
𝑥=0
𝑤𝑓 2
𝜕𝜀 𝑢
𝜕𝛼
Δ𝑥
𝛼𝑤 𝑐𝑟𝑢𝑖𝑠𝑒 = 0
𝑡
1−
𝑑𝜀
𝑑𝛼
𝛼𝑤 𝑐𝑟𝑢𝑖𝑠𝑒 +
15. Horizontal Stabilizer Geometry
16. Overall Aircraft Longitudinal Stability
Criteria for Longitudinal Static Stability
𝐶𝑚 𝛼 =
𝑑𝐶𝑚
<0
𝑑𝛼
𝐶𝑚 0 > 0
𝐶𝑚 0 = 𝐶𝑚 0 𝑤 + 𝐶𝑚 0 𝑡 +𝐶𝑚 0 𝑓
𝐶𝑚 0 = 𝐶𝑚 𝑎𝑐 𝑤 + 𝐶𝐿0 𝑤
𝑥𝑐𝑔 𝑥𝑎𝑐
𝑘2 − 𝑘1
−
+ 𝜂𝑉𝐻 𝐶𝐿 𝛼 𝜀0 + 𝑖𝑤 − 𝑖𝑡 +
𝑡
𝑐
𝑐
36.5𝑆𝑐
𝑥=𝑙 𝑓
𝑤𝑓 2 𝛼0 𝑤 + 𝑖𝑓 Δ𝑥
𝑥=0
𝐶𝑚 𝛼 = 𝐶𝑚 𝛼 𝑤 + 𝐶𝑚 𝛼 𝑡 + 𝐶𝑚 𝛼 𝑓
𝐶𝑚 𝛼 = 𝐶𝐿𝛼 𝑤
𝑥𝑐𝑔 𝑥𝑎𝑐
𝑑𝜀
1
−
− 𝜂𝑉𝐻 𝐶𝐿 𝛼 1 −
+
𝑡
𝑐
𝑐
𝑑𝛼
36.5𝑆𝑐
𝑪𝒎 𝜶 𝟏/𝒓𝒂𝒅
𝑪𝒎 𝟎
𝑥=𝑙 𝑓
𝑤𝑓 2
𝑥=0
𝜕𝜀𝑢
Δ𝑥
𝜕𝛼
-1.4679
0.1281
18. Vertical tail volume coefficient 𝑽𝒗 : 0.06
The following table shows the vertical tail characteristics for various aircraft. Because our aircraft
configuration and mission requirements are very similar to the C-130, many vertical tail parameters are
chosen so that they match those of that aircraft.
𝑉𝑣 =
𝑙𝑣 𝑆𝑣
𝑆𝑏
𝑉𝑣 = 0.06
19. Vertical tail arm 𝒍𝒗 : 36.1102 in
During the preliminary design phase, the vertical tail arm is selected to be equal to the horizontal tail
arm, then adjusted after further iterations if needed.
𝑙𝑣 = 36.1102 𝑖𝑛
20. Vertical tail planform area 𝑺𝑽 : 1.1521 𝑓𝑡 2
𝑉𝑣 =
𝑙𝑣 𝑆𝑣
= 0.08
𝑆𝑏
𝑆𝑣 = 1.1521 𝑓𝑡 2
21. Vertical tail aspect ratio 𝑨𝑹𝒗 : 1.84
Choose vertical tail aspect ratio such that it matches that of the C-130 (Table 6.6).
𝑨𝑹𝒗 = 𝟏. 𝟖𝟒
22. Vertical tail span 𝒃𝒗 : 17.4716 in
𝐴𝑅𝑣 =
𝑏𝑣 2
𝑆𝑣
𝑏𝑣 = 17.4716 𝑖𝑛
23. Vertical tail sweep angle 𝚲𝒗 : 18.8 deg
Choose vertical tail sweep angle such that it matches that of the C-130 (Table 6.6).
Λ 𝑣 = 18.8 𝑑𝑒𝑔
24. Vertical tail minimum lift curve slope 𝑪𝑳𝜶 𝒗 : 0.0011137 [1/deg]
-
Determine minimum vertical tail lift curve slope so to meet the static directional stability
requirement 𝐶𝑛 𝛽 > 0
𝐶𝑛 𝛽 = 𝐶𝑛 𝛽
𝑤𝑓
+ 𝐶𝑛 𝛽
𝐶𝑛 𝛽 = −𝑘𝑛 𝑘𝑅𝑙
𝐶𝐿 𝛼
𝑣 𝑚𝑖𝑛
𝑣
𝑆𝑓𝑠 𝑙𝑓
𝑑𝜎
+ 𝑉𝑣 𝐶𝐿 𝛼 𝜂𝑣 1 −
𝑣
𝑆𝑏
𝑑𝛽
= 0.0011137 1/𝑑𝑒𝑔
25. Vertical Tail Airfoil: NACA-0009
a) Choose symmetric airfoil as the vertical tail should behave in a similar manner when at a
positive or negative angle-of-attack
b) To minimize structure and weight, choose airfoil with smallest thickness that meets 𝐶𝐿 𝛼
𝑣 𝑚𝑖𝑛
c) Refer to symmetric airfoil plots when choosing the horizontal tail airfoil
𝐶𝑙 𝛼 =
Δ𝐶𝐿 𝑣
Δ𝛼
NACA-0009 (As Stabilizer not Airfoil, from XFLR5)
𝛼 (deg)
𝐶𝐿 𝑣
0
0
5.00
0.314
𝐶𝐿 𝛼 = 0.1214 [1/deg]
𝑣
26. Vertical Stabilizer Geometry
27. Overall Aircraft Directional Stability
Criterion for Directional Static Stability
𝐶𝑛 𝛽 =
𝑑𝐶𝑛
>0
𝑑𝛽
𝐶𝑛 𝛽 = 𝐶𝑛 𝛽
𝑤𝑓
+ 𝐶𝑛 𝛽
𝐶𝑛 𝛽 = −𝑘𝑛 𝑘𝑅𝑙
𝑣
𝑆𝑓𝑠 𝑙𝑓
𝑑𝜎
+ 𝑉𝑣 𝐶𝐿 𝛼 𝜂𝑣 1 −
𝑣
𝑆𝑏
𝑑𝛽
𝑪𝒏 𝜷 𝟏/𝒓𝒂𝒅
0.2847