Integrated Aerodynamic/Structural/Control Modeling for...
Transcript of Integrated Aerodynamic/Structural/Control Modeling for...
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
• . . .