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. Vertical tail volume coefficient 𝑉𝑣 19. Vertical tail arm 𝑙𝑣 20. Vertical tail planform area 𝑆𝑉 21. Vertical tail aspect ratio 𝐴𝑅𝑣 22. Vertical tail span 𝑏𝑣 23. Vertical tail sweep angle Λ𝑣

24. Vertical tail minimum lift curve slope 𝐶𝐿𝛼 𝑣

25. Vertical tail airfoil 26. Overall Vertical Stabilizer Geometry 27. Overall Aircraft Static Direction Stability 28. 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 𝑙𝑎𝑐𝑙𝑓

= 0.6

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 =𝑙𝑡𝑆𝑡𝑆𝑐

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 𝑓𝑡2

𝑉𝐻 =𝑙𝑡𝑆𝑡𝑆𝑐

= 1

𝑆𝑡 = 3.6352 𝑓𝑡2 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 𝐶𝐿𝑚𝑎𝑥 𝑡

d) Maximize 𝐶𝐿𝛼 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:

𝐴𝑅𝑡 =2

3𝐴𝑅𝑤

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 𝐶𝑚𝑐𝑔

= 𝐶𝑚𝑐𝑔 𝑤+ 𝐶𝑚𝑐𝑔 𝑡

+ 𝐶𝑚𝑐𝑔 𝑓= 0

𝐶𝑚0𝑤

+ 𝐶𝑚𝛼𝑤𝛼𝑤𝑐𝑟𝑢𝑖𝑠𝑒

+ 𝐶𝑚0𝑡+ 𝐶𝑚𝛼 𝑡

𝛼𝑤𝑐𝑟𝑢𝑖𝑠𝑒+ 𝐶𝑚0𝑓

+ 𝐶𝑚𝛼𝑓𝛼𝑤𝑐𝑟𝑢𝑖𝑠𝑒

= 0

𝐶𝑚𝑎𝑐 𝑤+ 𝐶𝐿0𝑤

𝑥𝑐𝑔

𝑐 −

𝑥𝑎𝑐

𝑐 + 𝐶𝐿𝛼𝑤

𝑥𝑐𝑔

𝑐 −

𝑥𝑎𝑐

𝑐 𝛼𝑤𝑐𝑟𝑢𝑖𝑠𝑒 + 𝜂𝑉𝐻𝐶𝐿𝛼 𝑡

𝜀0 + 𝑖𝑤 − 𝑖𝑡 − 𝜂𝑉𝐻𝐶𝐿𝛼 𝑡 1 −

𝑑𝜀

𝑑𝛼 𝛼𝑤𝑐𝑟𝑢𝑖𝑠𝑒 +

𝑘2−𝑘1

36.5𝑆𝑐 𝑤𝑓

2 𝛼0𝑤+ 𝑖𝑓 Δ𝑥

𝑥=𝑙𝑓𝑥=0 +

1

36.5𝑆𝑐 𝑤𝑓

2 𝜕𝜀𝑢

𝜕𝛼Δ𝑥

𝑥=𝑙𝑓𝑥=0 𝛼𝑤𝑐𝑟𝑢𝑖𝑠𝑒

= 0

𝑖𝑡 = 4.2300 𝑑𝑒𝑔 13. Neutral Point 𝒙𝑵𝑷 :0.7067𝑐

𝑥𝑛𝑝

𝑐 =

𝑥𝑎𝑐𝑐

−𝐶𝑚𝛼 𝑓

𝐶𝐿𝛼𝑤

+ 𝜂𝑉𝐻𝐶𝐿𝛼 𝑡

𝐶𝐿𝛼𝑤

1 −𝑑𝜀

𝑑𝛼

𝑥𝑁𝑃 = 0.7067𝑐 14. Static Margin: 0.3548 𝑆𝑡𝑎𝑡𝑖𝑐 𝑀𝑎𝑟𝑔𝑖𝑛 = 𝑥𝑁𝑃 − 𝑥𝑐𝑔

𝑆𝑡𝑎𝑡𝑖𝑐 𝑀𝑎𝑟𝑔𝑖𝑛 = 0.3548

15. Horizontal Stabilizer Geometry

16. Overall Aircraft Longitudinal Stability Criteria for Longitudinal Static Stability

𝐶𝑚𝛼=

𝑑𝐶𝑚𝑑𝛼

< 0

𝐶𝑚0

> 0

𝐶𝑚0

= 𝐶𝑚0𝑤+ 𝐶𝑚0𝑡+𝐶𝑚0𝑓

𝐶𝑚0= 𝐶𝑚𝑎𝑐 𝑤

+ 𝐶𝐿0𝑤 𝑥𝑐𝑔

𝑐 −𝑥𝑎𝑐𝑐 + 𝜂𝑉𝐻𝐶𝐿𝛼 𝑡

𝜀0 + 𝑖𝑤 − 𝑖𝑡 +𝑘2 − 𝑘1

36.5𝑆𝑐 𝑤𝑓

2 𝛼0𝑤+ 𝑖𝑓 Δ𝑥

𝑥=𝑙𝑓

𝑥=0

𝐶𝑚𝛼

= 𝐶𝑚𝛼𝑤+ 𝐶𝑚𝛼 𝑡

+ 𝐶𝑚𝛼𝑓

𝐶𝑚𝛼= 𝐶𝐿𝛼𝑤

𝑥𝑐𝑔

𝑐 −𝑥𝑎𝑐𝑐 − 𝜂𝑉𝐻𝐶𝐿𝛼 𝑡

1 −𝑑𝜀

𝑑𝛼 +

1

36.5𝑆𝑐 𝑤𝑓

2𝜕𝜀𝑢𝜕𝛼

Δ𝑥

𝑥=𝑙𝑓

𝑥=0

𝑪𝒎𝜶 𝟏/𝒓𝒂𝒅 -1.4679

𝑪𝒎𝟎 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 =𝑙𝑣𝑆𝑣𝑆𝑏

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 𝑓𝑡2

𝑉𝑣 =𝑙𝑣𝑆𝑣𝑆𝑏

= 0.08

𝑆𝑣 = 1.1521 𝑓𝑡2

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). Λ𝑣 = 18.8 𝑑𝑒𝑔

24. Vertical tail minimum lift curve slope 𝑪𝑳𝜶𝒗 : 0.0011137 [1/deg]

- Determine minimum vertical tail lift curve slope so to meet the static directional stability

requirement 𝐶𝑛𝛽> 0

𝐶𝑛𝛽

= 𝐶𝑛𝛽𝑤𝑓+ 𝐶𝑛𝛽 𝑣

𝐶𝑛𝛽= −𝑘𝑛𝑘𝑅𝑙

𝑆𝑓𝑠 𝑙𝑓

𝑆𝑏+ 𝑉𝑣𝐶𝐿𝛼𝑣

𝜂𝑣 1 −𝑑𝜎

𝑑𝛽

𝐶𝐿𝛼𝑣𝑚𝑖𝑛

= 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 𝐶𝐿𝛼𝑣𝑚𝑖𝑛

c) Refer to symmetric airfoil plots when choosing the horizontal tail airfoil

NACA-0009 (As Stabilizer not Airfoil, from XFLR5)

𝛼 (deg) 𝐶𝐿𝑣

0 0

5.00 0.314

𝐶𝑙𝛼 =Δ𝐶𝐿𝑣Δ𝛼

𝐶𝐿𝛼𝑣= 0.1214 [1/deg]

26. Vertical Stabilizer Geometry

27. Overall Aircraft Directional Stability Criterion for Directional Static Stability

𝐶𝑛𝛽=

𝑑𝐶𝑛𝑑𝛽

> 0

𝐶𝑛𝛽

= 𝐶𝑛𝛽𝑤𝑓+ 𝐶𝑛𝛽 𝑣

𝐶𝑛𝛽= −𝑘𝑛𝑘𝑅𝑙

𝑆𝑓𝑠 𝑙𝑓

𝑆𝑏+ 𝑉𝑣𝐶𝐿𝛼𝑣

𝜂𝑣 1 −𝑑𝜎

𝑑𝛽

𝑪𝒏𝜷 𝟏/𝒓𝒂𝒅 0.2847