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7/28/2019 Project Progress Nov 2012
1/21
CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
Reduction of Wind Turbine Blade Model
M. Khorsand Vakilzadeh
Department of Applied Mechanics,
Chalmers University of Technology,
Gothenburg, Sweden.

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 2
Outline of the ThesisModel validation of wind turbine rotor blade will be considered in two levels:
1 A detailed structural dynamics model which has good correlation with
experimental of wind turbine testing.
A FEM model motivated by its connection to the observed physical
phenomena, such as wing twisting, nonlinear effects, spatial ortemporal load variations.
2 A loworder model needs to be validated by a good modeltomodel
correlation.
It gives a correct representation of thestimulitoresponse
characteristics of the system in an efficient
simulation environment.

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 3
Outline of the ThesisHighorder blade model basedon 5MW reference wind
turbine defined by NationalRenewable Energy Laboratory.
Apply developed modalreduction method based on
inputoutput relationpreservation
Loworder model
Loworder beam model basedon 5MW reference windturbine defined by National
Renewable Energy Laboratory.

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 4
Highorder model of NREL bladeA master thesis has been defined to build a highorder blade model based on 5MW reference windturbine defined by National Renewable Energy
Laboratory.

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 5
Highorder model of NREL blade

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 6
An Improved Modal Approach For Model Reduction Based On Inputoutput RelationPresented in ISMA2012 and would be submitted as journal paper
Structural dynamicsmodel
Model reduction Modal truncation
Complex structures
Large FE models
Time consuming
and computationally
intensive
Simple loworder
models
Keep important
features
Approximation
error
A tradeoff between
accuracy and
simplicity
Can handle very
large models with
lightly damped or
unstable modes
Keep the dominant
eigenvalue subset ofthe full system
Lack of a general
modal dominancy
analysis

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 7
An Improved Modal Approach For Model Reduction Based On Inputoutput RelationPresented in ISMA2012 and would be submitted as journal paperAn improved modal based model reduction containing dominant eigensolutions of thefull order model
Dominancy metric: the squared norm contribution of each eigensolution to the system
output.
Numerical example: FE based model of an aluminum plate
Advantages
Computationally efficient
Detection and elimination of unobservable and uncontrollable modes

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 8
Modal TruncationConsider a continuous, linear, time invariant, asymptotically stable dynamic system
wherex(t) , u(t) , y(t) .
Lets assume that system is diagonalizable as
( ) ( ) ( )
( ) ( ) ( )
x t Ax t Bu t
y t Cx t Du t
( ) ( ) ( )
( ) ( ) ( )
z t z t Bu t
y t Cz t Du t
1
1 2
1
1 2
1 2
( , ,..., ) ,
,
.
n
n
n
diag A
b
b
B B
b
C C c c c
xn un yn
where
A
( ) ( )x t z t

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 9
Modal Truncation( ) ( ) ( )
( ) ( ) ( )
z t z t Bu t
y t Cz t Du t
1 1 1 1
2 2 2 2
1
1 2
2
00
z z B uz z B
zy C C Du
z
1 1 1 ( , , , )r B C D
Truncated system
Properties
.
Hnorm of error system is upper bounded
1
1
( )
( ) ( ) sup ( ( ) ( ))Re( )
n
i i
r rs i i k i
c bG s G s G s G s
where
z1 k
Challenge
What are the dominant eigenvalues!?!

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 10
Modal Dominancy Analysis
i i i i
i i i
z z b u
y c z
1 ( )i i i iY c s bU
0( )Hi i iM y y dt
0( ( ) ( ))Hi i iM Y Y d
0
1 1
H H
i i i i i
ii
M b c c b djj
( )
1
arctan( )Re( ) Re( )
i
H H
i i i i i
i i Im
M b c c b
( ) ( ) ( )( ) ( ) ( )
z t z t Bu t
y t Cz t Du t
Define dominancy metric as
Extract ith modal
coordinate
Laplace
transform
Parsevals
Theorem u(t) is a zero mean white noise withunit intensity, U(s)=1

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 11
Modal Dominancy Analysis
( )
1
arctan( )Re( ) Re( )
i
H H
i i i i i
i i Im
M b c c b
Properties
Computationally efficient,
Able to detect the unobservable and uncontrollable modes,
Applicable for both real and complex eigenvalues,
Independent from the other eigensolutions.

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 12
Improved Modal Truncation Algorithm
Given the stable system (A, B, C, D).
Solve eigenvalue problem ( ) for A.
Transfer the system to modal decomposed form.
Find multiple eigenvalues with dimension of multiplicity larger than the dimension of the
loading, nu.
Do QR decomposition of to make the eigenvectors found in step 4 orthogonal to load.
Compute the metric correspond to each modal coordinate.
Rearrange the modal coordinates as they appear in order:
Set the limit for error resulted from truncation and compute the loworder model order.
Do modal truncation along with residualization.
,
B
1 2 ... nM M M

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 13
Aluminum plate
Dimension: 500 500 1mm
Density: 2700kg/m3
Youngs modulus: 70 GPa
Poissons ratio: 0.33
4noded shell elements
16 16 elements mesh
Rayleigh viscous damping of
Numerical Example
3 610 , 10
V M K

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 14
Results
7 6 5 4 3 2 1 0
x 104
0
0.5
1
1.5
2
2.5
3
3.5
4x 10
5
radian/seconds
radian/seconds
N=3468 For the full state space model
Eigenvalue spectrum of full model; eigenvalues with positive imaginary part
N=1710 Guyan reduction is applied to remove massless rotational DOFs
CHALMERS

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 15
800 600 400 200 00
1
2
3
4x 10
4
radian/seconds
radian/seconds
Bal. Trunc.,k=288Prop. Mod. Trunc.,k=288
800 600 400 200 00
1
2
3
4x 10
4
radian/seconds
radian/seconds
Bal. Trunc., k=250Prop. Mod. Trunc., k=250
800 600 400 200 00
1
2
3
4x 10
4
radian/seconds
radian/seco
nds
Bal. Trunc., k=200
Prop. Mod. Trunc., k=200
1200 1000 800 600 400 200 00
1
2
3
4x 10
4
radian/seconds
radian/seco
nds
Bal. Trunc., k=150
Prop. Mod. Trunc., k=150
Eigenvalue spectrum of loworder models of plate;
eigenvalues with positive imaginary part
CHALMERS

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 16
Numerical statistics for reducedorder models with proposed modal truncation and balanced truncation using unitimpulse input signal; The Frobenius norm of the full system
output is 29.34 and the Hnorm of the full system is 28.42.
Proposed Modal Truncation Method Balanced TruncationReducedOrder
ModelRHN of the error
system (%)RFN of the error
system (%)Reduction time
(s)RHN of the error
system (%)RFN of the error
system (%)Reduction time
(s)k = 100 3.6e4 0.74 9.36 4.7e5 0.28 68.07k = 50 8.9e4 1.83 9.53 1.7e4 1.33 67.99k = 20 1.4e2 4.66 9.19 2.6e3 4.96 68.12
Numerical Statistics  Comparison with Balanced TruncationRelative Frobenius norm of error
Time domain analysis
Relative Hnorm of error system 
Frequency domain analysis
2
1 1
2
1 1
y
y
n m
ij
r i jF F
n m
F Fij
i j
ee y y
y yy
rG G
G
CHALMERS S C

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 17
Displacement as output, nr=20 Acceleration as output, nr=150
MethodRHN of the error
system (%)
RFN of the error
system (%)
RHN of the error
system (%)
RFN of the error
system (%)
Davison 0.19 16.01 3.08 96.40
Rommes 0.01 5.35 0.88 43.84
Aguirre 0.015 6.79 46.80 31.68
Proposed method 0.02 4.66 16.52 21.90
Numerical Statistics Comparison with other dominancy definitionsRelative Frobenius norm of error
Time domain analysis
Relative Hnorm of error system 
Frequency domain analysis
2
1 1
2
1 1
y
y
n m
ij
r i jF F
n m
F Fij
i j
ee y y
y yy
rG G
G
CHALMERS SWPTC

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 18
Conclusion
A new dominancy index is suggested to measure and rank the contribution of modes intoinputoutput relation:
 Squared norm of output deviation resulted from deflation of eigensolutions from
the system
The explicit formulation resulted from decomposed metric decreases computation time
needed for model reduction.
Comparison to balanced truncation showed that:
For a given model order, the balanced truncation is superior in approximation accuracy in
frequency domain analysis while the proposed modal truncation is superior in time domain
analysis.
Obtained loworder model includes a subset of the eigenvalues of the full model, same
physical interpretation, while balanced truncation alters the system eigenvalues.
The time required for reduction is decreased due to the explicit form of dominancy index as
compared with balancing reduction techniques.
CHALMERS SWPTC

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 19
Conclusion
Comparison to other dominancy definitions showed that: In a time domain analysis the proposed modal reduction technique yields less
approximation error.
This method can be efficiently applied to very largescale dynamics systems
while the application of the balanced truncation is restricted for very largescale
systems inasmuch as the solution of the Lyapunov equation is required in this
method.
CHALMERS SWPTC

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CHALMERSUNIVERSITY OF TECHNOLOGY
SWPTCNov. 2012
M. Khorsand Vakilzadeh, T. Abrahamsson 20
Future Works
Now it is time to apply the described inputoutput based modal reduction methodon the highorder model of NREL blade to obtain the reducedorder model.
On the other hand, the stateoftheart beam element based loworder model
would be developed to be compared with the reducedorder model in regard to
their inputoutput relation preservation.
CHALMERS SWPTC

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THANKS FOR YOUR ATTENTION