Update on FlexFlight G. Guanqun, N.D. Tantaroudas, and A . Da Ronch [email protected]
Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock
description
Transcript of Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock
![Page 1: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/1.jpg)
![Page 2: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/2.jpg)
Nonlinear Model Reduction of an Aeroelastic System
A. Da Ronch and K.J. BadcockUniversity of Liverpool, UK
Bristol, 20 October 2011
![Page 3: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/3.jpg)
Objective
• Framework for control of a flexible nonlinear aeroelastic
system
• Rigid body dynamics
• CFD for realistic predictions
• Model reduction to reduce size and cost of the FOM
• Design of a control law to close the loop
![Page 4: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/4.jpg)
Full-Order Model
Aeroelastic system in the form of ODEs
• Large dimension
• Expensive to solve in routine manner
• Independent parameter, U* (altitude, density)
nw
UwRdtdw
R
, *
![Page 5: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/5.jpg)
Taylor Series
Equilibrium point, w0
• Expand residual in a Taylor series
• Manipulable control, uc, and external disturbance, ud
dd
cc
uuRu
uRwwwC
wwBwwAwRwR
',','61
','21'''
0
0
0, *0 UwR
![Page 6: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/6.jpg)
Generalized Eigenvalue Problem
Calculate right/left eigensolutions of the Jacobian, A(w’)
• Biorthogonality conditions
• Small number of eigenvectors, m<n
m
m
,,,,
1
1
ijiij
ijij
ii
A
,,
1,
![Page 7: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/7.jpg)
Model Reduction
Project the FOM onto a small basis of aeroelastic
eigenmodes (slow modes)
• Transformation of coordinates FOM to ROM
where m<n
m
n
z
w
zzw
C
R'
'
![Page 8: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/8.jpg)
Linear Reduced-Order Model
Linear dynamics of FOM around w0
From the transformation of coordinates, a linear ROM
mi
uuRu
uRz
dtdz
dd
cc
Ti
ii
,,1
dd
cc
uuRu
uRwwAwR
'''
![Page 9: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/9.jpg)
Nonlinear Reduced-Order Model
Include higher order terms in the FOM residual
B involves ~m2 Jacobian evaluations, while C ~m3. Look at
the documentation
m
r
m
ssrsrsrsr
srsrsrsr
zzBzzB
zzBzzBwwB
1 1 ,,
,,','
',','
61','
21 wwwCwwBT
i
![Page 10: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/10.jpg)
Governing Equations
• FE matrices for structure
• Linear potential flow (Wagner, Küssner), convolution
• IDEs to ODEs by introducing aerodynamic states
gagcacaaasasa
ggccaasasasaa
assssss
uAuAwAwAw
uBuBwDwKwCwMFFwKwCwM
![Page 11: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/11.jpg)
ag
gg
ac
cc
ssas
a
a
s
s
A
BMBA
BMB
AADMCMKM
IA
www
w
11
111
~0
,~0
0
~~~~~00
,
ggcc uBuBAwdtdw
''
State Space Form
![Page 12: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/12.jpg)
Procedure
FOM
• Time-integration
ROM
• Right/left eigensolution of Jacobian, A(w’)
• Transformation of coordinates w’ to z
• Form ROM terms: matrix-free product for Jacobian
evaluations
• Time-integration of a small system
*,UwRdtdw
![Page 13: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/13.jpg)
Examples
Aerofoil section
• nonlinear struct + linear potential flow
Wing model
• nonlinear struct + linear potential flow
Wing model
• nonlinear struct + CFD
![Page 14: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/14.jpg)
Aerofoil Section
![Page 15: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/15.jpg)
Aerofoil Section
2 dof structural model
• Flap for control
• Gust perturbation
Nonlinear restoring forces
Gust as external disturbance (not part of the problem)
Taaa
Ts
Tass
www
hw
wwww
81 ,,
,
,,
5
53
55
3
ˆ
ˆ
KK
KK
![Page 16: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/16.jpg)
Aeroelastic Eigenvalues at 0.95 of linear flutter speed
![Page 17: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/17.jpg)
FOM/ROM gust response – linear structural model
![Page 18: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/18.jpg)
FOM gust response – linear/nonlinear structural model
![Page 19: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/19.jpg)
FOM/ROM gust response – nonlinear structural model
![Page 20: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/20.jpg)
Wing model
Slender wing with aileron
20 states per node
Tj
aj
aj
a
Tjz
jy
jx
jz
jy
jx
js
Tjy
jz
j
www
wwww
MFF
81 ,,
,,,,,
0,,0,,0,0
![Page 21: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/21.jpg)
Wing model
Linear flutter speed: 13m/s
First bending: 7.0Hz
Highest modes: 106Hz
For time integration of FOM: dt~10-7
ROM with few slow aeroelastic modes: dt~10-2
Span 1.2mWidth 0.3mThickness
0.003m
E 50GPa
![Page 22: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/22.jpg)
Gust disturbance at 0.01 of linear flutter speed
![Page 23: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/23.jpg)
Gust disturbance at 0.99 of linear flutter speed
![Page 24: Nonlinear Model Reduction of an Aeroelastic System A. Da Ronch and K.J. Badcock](https://reader036.fdocuments.net/reader036/viewer/2022062811/56816122550346895dd0797d/html5/thumbnails/24.jpg)
Ongoing work
• Control problem for linear aerofoil section: apply control
law to the FOM
• Same iteration for a wing model
• Include rigid body dynamics and test model reduction
• Extend aerodynamic models to CFD