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