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Optimal Vibration Control of Guyed Masts
B.Błachowski, W.Gutkowski
Institute of Fundamental Technological Research
Polish Academy of Sciences
Warsaw, Poland
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Motivations
Ice and windpressure are themost commonreasons of mastfailures.
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Mechanical model
Wind forcesVelocity
measurement
Control forces
Performance output
Feedback control
Small displacements
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Guyed mast dynamics
Finite element model
)()()()()( ttfttt uf uBBKqqDqM
Discrete time state-space model
nt)(Measureme
output) ce(Performan
)(
)(
2
1
211
kkk
kk
kc
kc
kkkkkt
tf
vxCy
xCz
η
ηxuBBAxx
Modal truncation
)()()()( tttt ccrrcc ηφηφηφq
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Wind fluctuations
Davenport filter
)()( fSjH
tcoefficien drag density,air where
)1(4800)(
spectrum and forces Wind
2/322
2
2
1
1
0
sin
cos
0
sin
cos
da
f
da
da
da
da
f
Cρ
S
CA
CA
CA
CA
B
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Controllability
modes ofnumber where
2rank 212
22
22
N
NN BABAABB
Classical criterion of controllability
Modal controllability index
NjjTuu
Tjj ,,2,1 φBBφ
Modal transformation
scoordinate modal)( where
)()()(2)( 2
t
tttt
c
uT
ccc
η
uBφηΩηΞΩη
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Mode 1 Mode 4
Mode 7 Mode 23
Modal analysis shown displacements are out of scale
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Control system scheme
Hydraulic actuator
Digital controller
Velocity measurement
Cable
2P
1P
Force
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Control strategy
Quadratic Programming
State estimator
Controller
Wind
Control input Measured output
Model updating
Performance output
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State estimator concept
2B
1B
Delay 2C
kv
ku
2B Delay
A
2C
ky
A
eK
kf
kx̂
1kx kx
1ˆ kx kx̂
Guyed mast
Guyed mast model
Estimator
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Optimal estimator
klTlk
klTlk
T
E
EE
Rvv
QffMxx
)(
errors tmeasuremen and
)()(
esdisturbanc condition, Initial
000
TTkk
kT
kT
kkk
Tke
kkekkk
m
mk
111
21
222
12
0221
)(
,,1,0
givenˆ)ˆ(ˆˆ
state of estimate likelihood Maximum
QBBAAPM
MCRCMCCMMP
RCAPK
xxCyKuBxAx
x
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Optimal control
Performance index
output eperformanc where
1
0
k
kTk
n
kk
Tknn
Tn
z
ΨuuΦzzzΦzJ
Quadratic programming with constraints
maxmin
2
UUU
VUWUU
T0JJ
1
1
0
nu
u
u
U
VUWUUUbLU
2minarg
TOPT
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Numerical simulations
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Conclusions
• A 3D FEM model of guyed mast under control forces wascreated
• External loading was included in the form of a stochasticwind model of Davenport spectrum
• Controllability of individual mode shapes of the mast wasdetermined
• A model based state estimator for the mast was constructed
• Numerical simulation of a control process was presented,displaying significant reduction in vibration amplitudes ofthe top of the mast
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Acknowledgements
The research was carried out with the financial supportprovided by the State Committee for Scientific Research ofPoland (KBN) under grant No. 5T07A 001 23 and this isgratefully acknowledged.