[PDF] 6. Aerodynamic Moments - 2018 pitching moment coefficient is invariant

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Aerodynamic Moments (i.e., Torques)Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018Copyright 2018 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.htmlhttp://www.princeton.edu/~stengel/FlightDynamics.htmlLearning Objectives•Aerodynamic balance and moment•Aerodynamic center, center of pressure, neutral point, and static margin•Configuration and angle-of-attack effects on pitching moment and stability•Configuration and sideslip-angle effects on lateral-directional (i.e., rolling and yawing) aerodynamic moments•Tail design effects on airplane aerodynamicsReading:Flight DynamicsAerodynamic Coefficients, 96-1181Review Questions§Why is induced drag proportional to angle of attack squared?§What spanwiselift distribution gives minimum induced drag?§Why can lift and drag coefficients be approximated by the Newtonian-flow assumption at very high angle of attack?§How does profile drag vary with Mach number?§What are some functions of secondary wing structures?§What is the primary function of leading edge extensions?§What is the "Area Rule"?2

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Handbook Approach to Aerodynamic Estimation•Build estimates from component effects-Technical reports, textbooks, ...-USAF Stability and Control DATCOM (download at http://www.pdas.com/datcomb.html)-USAF Digital DATCOM (see Wikipediapage)-ESDU Data Sheets (see Wikipediapage)

Interference

Effects

Interference

Effects

Wing

Aerodynamics

Fuselage

Aerodynamics

Tail

Aerodynamics

Interference

Effects

3Moments of the Airplane 4

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Airplane Balance•Conventional aft-tail configuration-c.m. near wing's aerodynamic center [point at which wing's pitching moment coefficient is invariant with angle of attack ~25% mean aerodynamic chord (mac)]•Tailless airplane: c.m. ahead of the neutral pointDouglas DC-3Northrop N-9M5Airplane Balance•Canard configuration: -Neutral point moved forward by canard surfaces-Center of mass may be behind the neutral point, requiring closed-loop stabilization•Fly-by-wire feedback controlcan expand envelope of allowable center-of-mass locationsGrumman X-29McDonnell-Douglas X-366

4 r×f= ijk xyz f x f y f z =yf z -zf y i+zf x -xf z j+xf y -yf x k

Moment Produced By Force on a Particle

m= m x m y m z yf z -zf y zf x -xf z xf y -yf x =r×f! rf= 0-zy z0-x -yx0 f x f y f z r: Cross-product-equivalent matrix Cross Product of Vectors7Forces and Moments Acting on Entire Airplane f B X B Y B Z B m B L B M B N B

Force VectorMoment Vector8

5 Aerodynamic Force and Moment Vectors of the Airplane m B yf z -zf y zf x -xf z xf y -yf x dxdydz=

Surface

L B M B N B f B f x f y f z dxdydz

Surface

X B Y B Z B

9Force VectorMoment VectorPitching Moment of the Airplane 10

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Body-Axis Reference Frames•Reference frame originis arbitrary; it is a fiducialpoint-xaxis along centerline-Tip of nose:All values of xon airframe are negative, but nose shape could change-Forward-most bulkhead:Fixed for all manufacturing measurements-Center of mass:Rotational center, but changes with fuel use, payload, etc.11F-86 Nose Variations12F-86AF-86DF-86DF-86E

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Pitching Moment(moment about the yaxis)Pressureand shear stressdifferentials x moment arms Integrate over the airplane surfaceto produce a net pitching moment

Body-AxisPitchingMoment=M

B =-Δp z x,y +Δs z x,y x-x cm dxdy surface +Δp x y,z +Δs x y,z Δp x z-z cm dydz surface

13Pitching Moment(moment about the yaxis)

M B ≈-Z i x i -x cm i=1 I +X i z i -z cm i=1 I +InterferenceEffects+PureCouples

Distributed effects can be aggregated to local centers of pressureindexed by iNet effect expressed as

M B =C m qSc 14 8

Pure Couple•Net force =0RocketsCambered Lifting SurfaceFuselage•Cross-sectional area, S(x)•xpositive to the right•At small α-Positive lift slope with dS(x)/dx> 0-Negative lift slope with dS(x)/dx< 0•Fuselage typically produces a destabilizing (positive) pitching moment [ "Apparent mass" effect]•Net moment ≠0Rockets15Lift Coefficient of a ConeMunk'sairship theory (potential flow)16

C L cone !±2fcn

Length

Diameter

S base S !±2

0.4,LenDia=2

0.84,LenDia=5

0.94,LenDia=10

S base S Munk, NACA-TR-184, 1924Sbase: cross-sectional area where flow separates 9

Net Center of Pressure Local centers of pressure can be aggregated at a net center of pressure (or neutral point) along the body xaxis

x cp net x cp C N wing +x cp C N fuselage +x cp C N tail C N total

Body AxesWind Axes(w.r.t, velocity vector)

C N =-C Z C A =-C X

S = reference area17Static Margin

StaticMargin!SM=

100x
cm -x cp net B c ≡100h cm -h cp net

•Static margin(SM)reflects the distance between the center of mass (cm)and the net center of pressure (cp)•Body axes•Normalized by mean aerodynamic chord•Does not reflect zposition of center of pressure•PositiveSMifcpis behindcm18

h cm x cm c 10

Static Margin

StaticMargin=SM

100x
cm -x cp net c ≡100h cm -h cp net

19Effect of Static Margin on Pitching Coefficient •Zero crossing determinestrim angle of attack, i.e., sum of moments = 0•Negative slope required for static stability•Slope, ∂Cm/∂α, varies with static margin

Trim C m o C m M B =C m o +C m qSc 20 11

Pitch-Moment Coefficient Sensitivity to Angle of AttackFor small angle of attack and no control deflectionreferenced to wing area, S21

Cmα≈-CNαnethcm-hcpnet()≈-CLαnethcm-hcpnet()≈-CLαwingxcm-xcpwingc⎛⎝⎜⎞⎠⎟-CLαhtxcm-xcphtc⎛⎝⎜⎞⎠⎟Horizontal Tail Lift Sensitivity to Angle of Attack

C L ht horizontal tail ref=S =C L ht ref=S ht 1- elas S ht S V ht V N 2 •Downwasheffect on aft horizontal tail•Upwasheffect on a canard (i.e., forward) surface V ht : Airspeed at horizontal tail

ε: Downwash angle due to wing at tail

∂ε∂α: Sensitivity of downwash to angle of attack elas : Aeroelastic effect 22
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Aerodynamic Center and Center of Pressure of a WingNACA 0012NACA 2412NACA 4412Airfoil Toolshttp://airfoiltools.com23

xac=x for which ∂Cm∂α≡0=xcp for a symmetric airfoil≠xcp for an asymmetric airfoilFor small angle of attack and no control deflectionEffect of Static Margin on Pitching Moment24

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Typically, static margin is positiveand ∂Cm/∂αis negativefor static pitch stabilityEffect of Static Margin on Pitching MomentSum of moments is zero in trimmed condition25

MB=Cmo+Cmαα()qSc=0intrimmed(equilibrium)flightEffect of Elevator Deflection on Pitching Coefficient Control deflectionshifts curve up and down, affecting trim angle of attack

Trim 1 C m C m o +C m δE δE

Elevator deflection effectively changes C

m o M B =C m o +C m

α+C

m δE δE qSc 26
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Aviation in The Great War•1914-18:World War I changes the complexion of flying-Reconnaissance-Air superiority (dog fights) -Bombing-Personal transport•WrightsUS monopoly broken by licensing for war effort•Aircraft Design-Biplanes, a few mono-and triplanes-Design for practical functions-Multiple engines, larger aircraft-Aft tails-Increased maneuverability, speed, g-loads, altitude-Improved piston engines-Tractor propellersSPAD S.VII

Historical Factoids

27Maneuvering World War I Aircraft•Maneuverable aircraft with idiosyncrasies-Rotary engine-Small tail surfaces-Reliability issues•Maneuvering to stallsand spins•Snap roll: rudderand elevator•Barrel roll : aileron •Cross-control(e.g., right rudder, left stick)-glide path control during landing -good view of landing point•Unintended snap rollsled to spinsand accidents during takeoff or landinghttp://www.youtube.com/watch?v=OBH_Mb0Kj2shttp://www..com/watch?v=6ETc1mNNQg8youtube28DeHavillandDH-2SopwithTriplaneFokker E.III

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Stability OR Control?•Need for better understanding ofFlying (or Handling) Qualities -Stability and controllability characteristics as perceived by the pilot•Desired attributes:Stability of the S.E.-5 andcontrollability of the D.VII29Stability AND ControlLateral-Directional Effects of Sideslip Angle30

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Rolling and Yawing Moments of the AirplaneDistributed effects can be aggregated to local centers of pressure indexed by iRolling MomentYawing Moment31

NB≈Yixi-xcm()i=1I∑-Xiyi-ycm()i=1I∑+InterferenceEffects+PureCouplesSideslip Angle Produces Side Force, Yawing Moment, and Rolling Moment§Sideslip usually a small angle ( 5 deg)§Side force generally not a significant effect§Yawing and rolling moments are principal effects32

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Side Force due to Sideslip Angle

Y≈

∂C Y qS•β=C Y qS•β •Fuselage, vertical tail, nacelles, and wing are main contributorsS = reference area33

CYβ≈CYβ()Fuselage+CYβ()VerticalTail+CYβ()Nacelles+CYβ()WingSide Force due to Sideslip Angle

vt = Vertical tail efficiency (p. 96, Flight Dynamics) k=

πAR

1+1+AR

2

Γ= Wing dihedral angle, rad

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Yawing Moment dueto Sideslip Angle

N≈

∂C n ρV 2 2

Sb•β=C

n qSb•β §Side force contributions times respective moment arms-Non-dimensional stability derivative C n ≈C n

VerticalTail

+C n

Fuselage

+C n Wing +C n

Propeller

S = reference area35

C n

VerticalTail

≈-C Y vt vt S vt l vt Sb -C Y vt vt V VT

Vertical tail contribution

V VT S vt l vt Sb =VerticalTailVolumeRatio vt elas 1+ V vt 2 V N 2

Yawing Moment dueto Sideslip Angle

l vt  Vertical tail length (+) = distance from center of mass to tail center of pressure =x cm -x cpquotesdbs_dbs10.pdfusesText_16

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