Aircraft Forces and Moments
15
AEM 668 Lecture 8 Aircraft Forces and Moments Dr. Jinwei Shen University of Alabama Feburary 3, 2015
Transcript of Aircraft Forces and Moments
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
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