Integrated Aerodynamic/Structural/Control Modelingfor Flexible Aircraft (or Wind Turbines)

Mark Drela

MIT Department of Aeronautics and Astronautics

Motivation

Aerodynamics

Structures Controlactive dampingsensor placement

stability augmentationspanload optimizationcontrol surface stalltrim settings

static loads, deformationdivergencegust responseflutter

load alleviationflutter suppressionmode excitation

Modern air raft exhibit strong dis ipline oupling Design development an be bewildering What are the key intera tions? What are the key design drivers? What are potential failure s enarios?

Typi al Current Pra ti e

Independent Aero, Stru tural, Control modules Very general (e.g. NASTRAN, Vortex-Latti e, Simulink) Intera tions via in uen e matri es Data transfer usually via les Hampered preliminary design Numerous ases an require extensive setup and exe ution eort Nonlinear problems espe ially awkward Design hanges often outpa e analyses

ASWING Approach

• Simplest model which captures key interactions

• Compact, discretization-independent definition

• Interactive analysis/redesign interface

Fuselage beam

Wind−aligned vortex wake

Unloaded geometry

V

gravitySurface beam

Beam joint

Surface beam(lifting line)

(slender body)

propulsive force

Angular momentum

Point mass

Vgust

Envisioned Application to Wind Turbine

Unloaded geometry

gravity

Ω

V

wind field

(x,y,z,t)

Helicalvortex wake

Nonlinear beam, lifting line

blade control models

Solid ground mount,or floating platform with dynamic response

U

Generatorspeed/torque load model

Platform motion

Envisioned Application to Wind Turbine

Stopped unfeathered 100m turbine in 250 km/h wind

Model Denition General nonlinear bending/torsion beam properties: ~r0 #0 E(s) : : : predi ts large deformations predi ts shear stresses, extensional strains General lifting-line properties: o m d `max ` `Æ mÆ (s) : : : predi ts se tion loading, stall predi ts unsteady aero loads Slender-body properties: A df dp(s) : : : Point-obje t properties: m ~H CDA ~Fprop ~Mprop : : : represent on entrated masses, rotors, na elles represent propulsion units PID ontrol-law governs ap de e tion: ~Æ = F Z Udt; U; _U

State Des riptionState: U(t) = ~ri ~i ~Mi ~Fi ~ui ~!i i ~R ~ ~U ~ E ~Æ Governing equations: R U; _U;U = 0

Air raft Euler angles Inertial, Body, and Beam-se tion axes

x

y

z

c

n s

X

Z

Y

−x

Ψ

Φ

Θ

−X

Inertial Axes

V

Body Axes x y z

X Y Z

State Des riptionState: U(t) = ~ri ~i ~Mi ~Fi ~ui ~!i i ~R ~ ~U ~ E ~Æ Governing equations: R U; _U;U = 0Positions, velo ities, rotation rates Stress resultants(shown in sn axes)y

y

ω

u

c

sn

Ωx

Ωy

Ωz

z

y

x

z

z

ω

r

U

u

U

U

x

x

ω

u

R

n

s

c

Mn

s

Mc

n

s

c

Fs

Fn

Fc

M

Beam Se tion Properties~r0 geometry of unloaded beam#0 twist angle of unloaded beamE bending/torsion stiness tensorEA extensional stinessGK ;n shear stinesses1;2 se tion masses/length1;2 se tion inertia-tensors/length g, n g position of mass entroids ta, nta position of tension axis ea, nea position of elasti axisxo referen e axis lo ation hord (for lifting surfa e)R ylinder radius (for slender body) df , dp pressure,fri tion drag oeÆ ientsA, ` se tion lift properties `min, `max se tion stall properties `Æ , mÆ se tion ap derivativesÆF1;2::: se tion ap properties

c

n

cxo

c

ncsh sh

n

c

c

ncg

cg

mass centroid

, n

tension axis

elastic axis

cta

ntaea

ea

axial strain

Governing Equations | Stru turalU(t) = 8<: ~ri ~i ~Mi ~Fi ~ui ~!i| z i ~R ~ ~U ~ E ~Æ 9=;Rr T(~) d~r 8<: 01+ ~F ^s=EA0 9=; ds0 = 0R K(~) d~ K0 d~0 E1 T ~M ds = 0RM d ~M + ~mds + ~Mp d[1 + d~r ~F = 0RF d~F + ~f ds + ~Fp d[1 = 0Ru _~r ~u = 0R! K _~ T ~! = 0Applied loads: ~f = ~V ^s + ~g _~U _~u _~~r + : : :! + : : :~m = 12 V 2? 2 m ^s T T T _~ + _~!! + : : : m = mo + mÆ Æ ap

Governing Equations | Aerodynami U(t) = 8<: ~ri ~i ~Mi ~Fi ~ui ~!i i| z ~R ~ ~U ~ E ~Æ 9=;R ~V pi ^n pi V?2 Fstall( `) = 0 .p. relative velo ity: ~V p = ~U + ~u + ~~r + ~ + ~!~r p + ~Vind(~r;; _) + ~Vgust(~r; ~R; ~) .p. surfa e normal: ^n p = T Tf sinA 0 osA gTse tion zero-lift angle: A = o + `Æ ` Æ

δ

θ

n

∆rcp

r

cpAα

Uu

flow−tangencycontrol point

− axissω

Ω

z

xy

n n

c V

Vcp

Γ

ncp

cpV

V

Unsteady Indu ed Velo ity Exa t treatment sums over all shed vorti ity history Simplied model preferred Instantaneous part over bound vorti ity \exa t" History part approximated via urrent shedding rate Approximately reprodu es Theodorsen lag ee ts~Vind pi = Xj ~wijj bV? it ^n pib = 2= ( alibrated lag onstant)Γ(t)

indΓtV

−Vt

V 1Γγ = −

ignored

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

exact F(k)

implied F(k)

exact -G(k)

implied -G(k)

k

implied F (k) + iG(k) = 1 + 2ik1 + 2ikb

Se tion Stall Model Se tion ` based on airfoil-plane velo ity ~V?: ` = 2 V? ; ~V? = ~V (~V ^s) ^s \Leaky" ontrol point mimi s vis ous displa ement:~V p ^n p = V?2 Fstall( `) Apparent surfa e sour e adds to prole drag:~fdrag = 12 V ~V df + 12 V? ~V? dp + 2 ~V?V? ~V p ^n p2 c

s

VV

c minc∆stall

c

c max

>>1stall

Unstalled~V p ^n p = 0 Stalled~V p ^n p 6= 0 Resulting se tion properties

n

V

cp

cp Vcp

ncp

c max

c min

fd dpc +c

c dc

2 / (1+ )

αα2

stall

Governing Equations | Body Dynami sU(t) = 8<: ~ri ~i ~Mi ~Fi ~ui ~!i i ~R ~ ~U ~| z E ~Æ 9=;Unsteady An hored Free Stati RR _~R ~U = 0 ~R ~R = 0 ~R ~R = 0R T T K _~ ~ = 0 ~ ~ = 0 ~ ~ = 0RU Xi ~f s + ~Fp = 0 (V)(V) = 0 Xi ~f s + ~Fp = 0R Xi ~ms + ~Mp = 0 ~ ~ = 0 Xi ~ms + ~Mp = 0

Governing Equations | Control VariablesU(t) = 8<: ~ri ~i ~Mi ~Fi ~ui ~!i i ~R ~ ~U ~ E ~Æ| z 9=;error integrator: RE _E V V ; ; : : : T = 0 losed-loop ontroller: RÆ ~Æ CU; _U;U = 0open-loop ontroller: RÆ ~Æ ~Æ (t) = 0stati trim onstraint: RÆ ~ ~ = 0Typi al state data for physi al implementation of _E, C ...~R navigation data, altimeter ~U aero sensors (V1 , , )~ attitude gyros, ompass _~U a elerometers~ rate gyros

Solution Pro eduresResidual linearization about urrent U, _U, U . . .ÆR = RU U ÆU + R _U U Æ _U + RU U ÆU Options for al ulation of ÆU . . .Stati : Æ _U = 0 , ÆU = 0 , ÆR = RTime-mar h: Æ _U = 32t ÆU , ÆU = 0 , ÆR = RStati sensitivity: Æ _U = 0 , ÆU = f0 0::: 1 ::: 0g , ÆR = 0Freq. response: Æ _U = i! ÆU , ÆU = n0 0::: ei!t::: 0o , ÆR = 0Eigenmode: Æ _U = ÆU , ÆU = 0 , ÆR = 0Stati and time-mar h residuals R zeroed by Newton iteration . . .U U + ÆU

Numeri al Solutions Ja obians are 50005000 , moderately sparse Blo k-tridiagonal with numerous outliers Real 1212 blo ks for stati problems Real 1818 blo ks for time-mar h problems Complex 1818 blo ks for frequen y-response, eigenmode problems Dire t solution by disse tion with blo k-tridiagonal solvers One setup and solve in < 1 se on workstation Eigenmodes via inverse Arnoldi iteration (ARPACK) 20 roots for 10 operating points in 20 se Eigenmodes used for diagnosti s and ontrol-law design Modal oordinates not used for omputation Intera tive exe ution for all types of solutions

Predi tive Capabilities Stati and dynami deformations, strains, stresses ~r ~M Control-de e tion loads Gust loads Trim settings for spe ied ight onditions Æa Æe . . . Spanwise se tional loading, indu ed & total drag ` CDi CD Flexible-air raft stability and ontrol derivatives CL Cmq C`Æa . . . Stati divergen e, aileron reversal speeds Vdiv Vrev General eigenmodes | ight-dynami + stru tural ^U Open-loop or losed-loop air raft (in)stability Flutter Control-input frequen y response ^UÆa(!) ^UÆe(!) . . .

Application — Vertical Gust Encounter

Vertical-velocity contours Sailplane midway through encounter

Application — Vertical Gust Encounter

Aero loading, cℓ, ∆α snapshots Shear stress, normal-strain snapshots

Envisioned Applications for Wind Turbines

• Characterization of aeromechanical response of the entire

wind-turbine/tower/platform system via eigenmode and

Bode and Floquet analyses

• ROM construction for control law design

• Nonlinear simulation and evaluation of passive or active

load-alleviation control techniques

• Rapid and extensive nonlinear predictions of peak loads and

stresses

• Estimation of failure probability over turbine lifetime, via

Monte-Carlo simulations of atmospheric turbulence

response

• . . .