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AEM 668 Lecture 8Aircraft Forces and Moments
Dr. Jinwei ShenUniversity of Alabama
Feburary 3, 2015
Aerodynamics
▶ Aero loads are dependent upon the vehicle’s velocityrelative to the air and the attitude of the body relativeto that relative velocity.
▶ The velocity of the body relative to the air is“relative velocity” = 𝐯𝑟𝑒𝑙 = “Airspeed vector”
AEM 668 Lecture 8 J. Shen 2/15
Axes and Angles
▶ Body axis▶ Stability axis
▶ 𝐶𝑠/𝑏 = 𝐶𝑏/𝑛(0, 𝛼𝑒, 0)▶ Wind axis
▶ 𝐶𝑤/𝑠 = 𝐶𝑏/𝑛(−𝛽, 0, 0)▶ 𝐶𝑤/𝑏 = 𝐶𝑏/𝑛(−𝛽, 𝛼𝑒, 0)
▶ 𝛼, 𝛽 undefined if 𝐯𝑟𝑒𝑙 = 0▶ 𝜓, 𝜃, 𝜙 always exist
AEM 668 Lecture 8 J. Shen 3/15
Velocities▶ Absolute Velocity
𝐯 = 𝐯𝑎𝑖𝑟 + 𝐯𝑟𝑒𝑙𝐯𝑟𝑒𝑙 = 𝐯 − 𝐯𝑎𝑖𝑟
𝐯𝑏𝐶𝑀/𝑒 = ⎡⎢⎢
⎣
𝑈𝑉𝑊
⎤⎥⎥⎦
𝐯𝑏𝑟𝑒𝑙 = ⎡⎢⎢
⎣
𝑈′
𝑉 ′
𝑊 ′
⎤⎥⎥⎦
𝐯𝑤𝑟𝑒𝑙 = ⎡⎢⎢
⎣
𝑉𝑇00
⎤⎥⎥⎦
tan(𝛼) = 𝑊′𝑈′
sin(𝛽) = 𝑉 ′𝑉𝑇
𝑉𝑇 = |𝐯𝑟𝑒𝑙|
▶ NED to 𝐹𝑏: 𝜓, 𝜃, 𝜙▶ NED to 𝐹𝑤: 𝜓𝑤, 𝜃𝑤, 𝜙𝑤
▶ 𝜓𝑤: trajectoryheading
▶ 𝜃𝑤: flight path angle 𝛾▶ 𝜙𝑤: wind frame bank
▶ Ex: 𝐯𝑏𝐶𝑀/𝑁 =
𝐶𝑏/𝑤𝐯𝑤𝑟𝑒𝑙 + 𝐶𝑏/𝑤𝐶𝑤/𝑁𝐯𝑁
𝑎𝑖𝑟
▶ Atmosphere is quiescent if 𝐯𝑎𝑖𝑟/𝑖 = 𝜔𝑒/𝑖 × 𝐩𝑎𝑖𝑟𝑝𝑜𝑖𝑛𝑡/𝑖
AEM 668 Lecture 8 J. Shen 4/15
Force and MomentAerodynamic Force
▶ Defined in wind frame
𝐅𝑤𝐴 = ⎡⎢⎢
⎣
−𝐷−𝐶−𝐿
⎤⎥⎥⎦
∥ 𝐯𝑟𝑒𝑙⟂ 𝐯𝑟𝑒𝑙⟂ 𝐯𝑟𝑒𝑙
▶ Lift, drag, cross windforce
Gravity Force𝐅𝑏
𝐺 = 𝐶𝑏/𝑁(𝜓, 𝜃, 𝜙)𝐖𝑁
𝐅𝑤𝐺 = 𝐶𝑤/𝑁(𝜓𝑤, 𝜃𝑤, 𝜙𝑤)𝐖𝑁
▶ In body frame
𝐅𝑏𝐴 = ⎡⎢⎢
⎣
𝑋𝑌𝑍
⎤⎥⎥⎦
X forceSide force
Z force▶ From 𝐅𝑤 to 𝐅𝑏:
𝐅𝑏 = 𝐶𝑏/𝑤(−𝛽, 𝛼)𝐅𝑤
Moments
𝐌𝑏𝐴 = ⎡⎢⎢
⎣
𝑙𝑚𝑛
⎤⎥⎥⎦
𝐌𝑤𝐴 = ⎡⎢⎢
⎣
𝑙𝑤𝑚𝑤𝑛𝑤
⎤⎥⎥⎦
AEM 668 Lecture 8 J. Shen 5/15
Aerodynamic Coefficients▶ For any aerodynamic force
▶ L, C, D, X, Y, Z▶ 𝐶𝐹𝐴 = 𝐹𝐴
1/2𝜌𝑉2𝑇 𝑆
▶ Dynamic pressure: 𝑞 = 1/2𝜌𝑉2𝑇
▶ For aerodynamic moments▶ Rolling: 𝐶𝑙 = 𝑙
𝑞𝑆𝑏▶ Pitching: 𝐶𝑚 = 𝑚
𝑞𝑆 𝑐▶ 𝑐: mean aerodynamic chord
▶ Yawing: 𝐶𝑛 = 𝑛𝑞𝑆𝑏
Wing-Planform Parameters𝑏 = wing span (tip to tip)𝑐 = wing chord (varies along span)
𝑐 = mean wing chord (mac)𝑆 = wing area (total)
AEM 668 Lecture 8 J. Shen 6/15
Aerodynamic DerivativesDamping derivatives:
▶ Δ𝐶(𝑙,𝑚,𝑛) = 𝐶(𝑙,𝑚,𝑛) 𝑘2𝑉𝑇
(𝑝, 𝑞, 𝑟)▶ Example:
▶ Δ𝐶𝑙 = 𝐶𝑙𝑝 𝑏2𝑉𝑇
𝑝▶ Δ𝐶𝑚 = 𝐶𝑚𝑞 𝑐
2𝑉𝑇𝑞
▶ Δ𝐶𝑛 = 𝐶𝑛𝑟 𝑏2𝑉𝑇
𝑟
▶ 𝐶𝑙𝑝 = 𝜕𝐶𝑙𝜕𝑝
▶ 𝐶𝑚𝑞 = 𝜕𝐶𝑚𝜕𝑞
▶ 𝐶𝑛𝑟 = 𝜕𝐶𝑛𝜕𝑟
Acceleration derivatives▶ ��, 𝛽, 𝑉𝑇▶ Unsteady aerodynamics: 𝐶𝑙��, 𝐶𝑚��
Moment derivatives are important damping sources onthe natural modes of aircraft
AEM 668 Lecture 8 J. Shen 7/15
Aero-Coefficient Component Buildup 𝐶𝐷
▶ 𝐶( ) =𝐶( )(𝛼, 𝛽, 𝑀, ℎ, 𝛿𝑠, 𝑇𝐶)
▶ 𝐶𝐷 = 𝐶𝐷(𝛼, 𝛽, 𝑀, ℎ) +Δ𝐶𝐷(𝑀, 𝛿𝑒) +Δ𝐶𝐷(𝑀, 𝛿𝑟) + Δ𝐶𝐷(𝛿𝐹) +Δ𝐶𝐷(gear) + …
AEM 668 Lecture 8 J. Shen 8/15
Aero-Coefficient Component Buildup 𝐶𝐿
▶ 𝐶𝐿 = 𝐶𝐿(𝛼, 𝛽, 𝑀, 𝑇𝐶) +Δ𝐶𝐿(𝛿𝐹) + Δ𝐶𝐿(ℎ)
▶ Turboprop aircraft▶ Thrust Coefficient (TC)
effect on 𝐶𝐿
AEM 668 Lecture 8 J. Shen 9/15
Aero-Coefficient Component Buildup 𝐶𝑌
▶ 𝐶𝑌 = 𝐶𝑌(𝛼, 𝛽, 𝑀) +Δ𝐶𝑌(𝛼, 𝛽, 𝑀, 𝛿𝑟) +Δ𝐶𝑌𝛿𝑎
+ Δ𝐶𝑌𝑃 + Δ𝐶𝑌𝑅
▶ Linearized:▶ Δ𝐶𝑌𝛿𝑟
= 𝐶𝑌 𝛿𝑟 𝛿𝑟▶ Δ𝐶𝑌𝛿𝑎
= 𝐶𝑌 𝛿𝑎 𝛿𝑎
AEM 668 Lecture 8 J. Shen 10/15
Aero-Coefficient Component Buildup 𝐶𝑙
▶ 𝐶𝑙 =𝐶𝑙(𝛼, 𝛽, 𝑀) + Δ𝐶𝑙𝛿𝑎 +Δ𝐶𝑙𝛿𝑟 + Δ𝐶𝑙𝑝 + Δ𝐶𝑙𝑟
▶ Slide Slip▶ Stabilizing
▶ Dihedral▶ Wing backward
sweep▶ High wing
▶ Destablizing▶ Anhedral▶ Wing forward sweep▶ Low wing
AEM 668 Lecture 8 J. Shen 11/15
Aero-Coefficient Component Buildup 𝐶𝑚
▶ 𝐶𝑚 =𝐶𝑚(𝛼, 𝑀, ℎ, 𝛿𝐹 , 𝑇𝑐) +Δ𝐶𝑚𝛿𝑒 + Δ𝐶𝑚𝑞 + Δ𝐶𝑚�� +𝑋𝑅
𝑐 𝐶𝐿+Δ𝐶𝑚thrust+Δ𝐶𝑚gear▶ 𝐶𝑚𝛼 < 0: stabilizing
▶ tail contribution
AEM 668 Lecture 8 J. Shen 12/15
Aero-Coefficient Component Buildup 𝐶𝑛
▶ 𝐶𝑛 =𝐶𝑛(𝛼, 𝛽, 𝑀, 𝑇𝑐)+Δ𝐶𝑛𝛿𝑟 +Δ𝐶𝑛𝛿𝑎 + Δ𝐶𝑛𝑝 + Δ𝐶𝑛𝑟
▶ Slide Slip▶ Stabilizing
▶ Wing backwardsweep
▶ Stabilizer▶ Destablizing
▶ Wing forward sweep▶ Fuselage
AEM 668 Lecture 8 J. Shen 13/15
Aero-Coefficient Component Buildup 𝐶𝑌
▶ Simulation fidelity level▶ simplified tables▶ or large, fine tables
AEM 668 Lecture 8 J. Shen 14/15
Next lecture (SL 2.4)
▶ Static Analysis
0“Aircraft Control and Simulation, 2ed” by B.L. Stevens and F.L.Lewis, Wiley, 2003
AEM 668 Lecture 8 J. Shen 15/15