* Hong-Ki Jo 1) , Kyu-Sik Park 2) , Hye-Rin Shin 3) and In-Won Lee 4)
description
Transcript of * Hong-Ki Jo 1) , Kyu-Sik Park 2) , Hye-Rin Shin 3) and In-Won Lee 4)
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*Hong-Ki Jo1), Kyu-Sik Park2), Hye-Rin Shin3) and In-Won Lee4)
1) ~ 3) Graduate Student, Department of Civil Engineering, KAIST 4) Professor, Department of Civil Engineering, KAIST
SIMPLIFIED ALGEBRAIC METHOD FOR COMPUTING EIGENPAIR SENSITIVITIES OF DAMPED SYSTEM
KKNN SeminarTaipei, Taiwan, Dec. 7-8, 2000
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2Structural Dynamics and Vibration Control Lab., KAIST, Korea
OUTLINE
INTRODUCTION
PREVIOUS STUDIES
PROPOSED METHOD NUMERICAL EXAMPLE CONCLUSIONS
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3Structural Dynamics and Vibration Control Lab., KAIST, Korea
INTODUCTION• Objective of Study
• Applications of Sensitivity Analysis
- Determination of the sensitivity of dynamic responses
- Optimization of natural frequencies and mode shapes
- Optimization of structures subject to natural frequencies.
- To find efficient sensitivity method of eigenvalues and eigenvectors of damped systems.
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4Structural Dynamics and Vibration Control Lab., KAIST, Korea
)( 2 0KCM jjj
• Problem Definition
(1)
shape) (moder eigenvectocomplex th :
frequency) (natural eigenvaluecomplex th : definite-semi positive matrix, Stiffness :
damping classical-non matrix, Damping : definite positive matrix, Mass :
11
j
j
j
j
K
CKMKCMCM
n) ,2 1,( j
- Eigenvalue problem of damped system (N-space)
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5Structural Dynamics and Vibration Control Lab., KAIST, Korea
(2)
- Normalization condition
- State space equation (2N-space)
jj
jj
jj
j
00
0M
MCM
K
(3)1)2( 0
jiT
ijj
jT
ii
i
CM
MMC
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6Structural Dynamics and Vibration Control Lab., KAIST, Korea
jj , ,K ,C ,M K, C, M,
jj ,
Given:
Find:
- Objective
* indicates derivatives with respect to design variables (length, area, moment of inertia, etc.)
)(
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7Structural Dynamics and Vibration Control Lab., KAIST, Korea
PREVIOUS STUDIES
- many eigenpairs are required to calculate eigenvector derivatives. (2N-space)
,)( jjTjjλ BA
2/)(
)()(
])()([ )(
*
*
*
*
*
*
11
jTjjjjj
jj
Tjj
M
j
j
kj
Tkk
M
k
j
kj
Tkk
M
k
j
mjj
a
aa
ABAA
ABAAB
1M
0m
a
N
jk,1k
(4)
(5)
• Q. H. Zeng, “Highly Accurate Modal Method for Calculating Eigenvector Derivatives in Viscous Damping System,” AIAA Journal, Vol. 33, No. 4, pp. 746-751, 1995.
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8Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Sondipon Adhikari, “Calculation of Derivative of Complex Modes Using Classical Normal Modes,” Computer & Structures, Vol. 77, No. 6, pp. 625-633, 2000.
- many eigenpairs are required to calculate eigenvector derivatives. (N-space) - applicable only when the elements of C are small.
N
k kjkj
kjTkj
jj
jTjjj
ji
kkkj
jiT
kkj
kj
jiTkkj
k
jjTjj
CiiC
where
FF
M
1*
)(*
*
))(()(
5.0
~)1(~)1(2
1
)(5.0
N
jk (6)
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9Structural Dynamics and Vibration Control Lab., KAIST, Korea
• I. W. Lee, D. O. Kim and G. H. Jung, “Natural Frequency and Mode Shape Sensitivities of Damped Systems: part I, Distinct Natural Frequencies,” Journal of Sound and Vibration, Vol. 223, No. 3, pp. 399-412, 1999.
• I. W. Lee, D. O. Kim and G. H. Jung, “Natural Frequency and Mode Shape Sensitivities of Damped Systems: part II, Multiple Natural Frequencies,” Journal of Sound and Vibration, Vol. 223, No. 3, pp. 413-424, 1999.
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10Structural Dynamics and Vibration Control Lab., KAIST, Korea
Lee’s method (1999)
jjjT
jj KCM 2
jjjT
j
jjjjjj
j
jT
j
jjjj
CMMKCMCM
CMCMKCM
25.0)()2(
00)2()2(
2
2
(7)
(8)
- the corresponding eigenpairs only are required. (N-space)- the coefficient matrix is symmetric and non-singular. - eigenvalue and eigenvector derivatives are obtained separately.
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11Structural Dynamics and Vibration Control Lab., KAIST, Korea
PROPOSED METHOD
)( 2 0KCM jjj n) ,2 1,( j
• Rewriting basic equations
1)2( jjTj CM
- Eigenvalue problem
- Normalization condition
(9)
(10)
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12Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Differentiating eq.(9) with respect to design variable
jjj
jjjj
)(
)2( )(2
2
KCM
CMKCM
• Differentiating eq.(10) with respect to design variable
jjTj
jTjj
Tj
)2(5.0
)2(
CM
MCM
(11)
(12)
jj
jj
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13Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Combining eq.(11) and eq.(12) into a single matrix
jjT
j
jjj
j
j
jT
jjT
j
jjjj
)2(5.0)(
)2()2(
2
2
CMKCM
MCMCMKCM
(13)
- the corresponding eigenpairs only are required. (N-space)- the coefficient matrix is symmetric and non-singular.- eigenpair derivatives are obtained simultaneously.eigenpair derivatives are obtained simultaneously.
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14Structural Dynamics and Vibration Control Lab., KAIST, Korea
NUMERICAL EXAMPLE• Cantilever beam with lumped dampers
1 : (A) areasection -Cross1 : (I) inertiasection -Cross
1 : )(density Mass1000 :(E) Modulus sYoung'0.3 :(c)damper Tangential
Design parameter : depth of beam
Material Properties System Data
Number of elements : 20
Number of nodes : 21
Number of DOF : 40
v1
v2
1 2 3 4 2119 20
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15Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Analysis Methods
• Zeng’s method (1995)
• Lee’s method (1999)
• Proposed method
• Comparisons
• Solution time (CPU)
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16Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Results of Analysis (Eigenvalue)
Modenumber
Eigenvalue Eigenvalue derivative
1 -0.0035 - 1.0868i 0.0010 - 0.2997i2 -0.0035 + 1.0868i 0.0010 + 0.2997i3 -0.0203 - 6.0514i 0.0072 - 1.3173i4 -0.0203 + 6.0514i 0.0072 + 1.3173i5 -0.0422 - 14.7027i 0.0140 - 2.4536i6 -0.0422 + 14.7027i 0.0140 + 2.4536i7 -0.0719 - 24.7343i 0.0189 - 3.1194i
8 -0.0719 + 24.7343i 0.0189 + 3.1194i
9 -0.1106 - 35.3632i 0.0213 - 3.4203i
10 -0.1106 + 35.3632i 0.0213 + 3.4203i
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17Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Results of Analysis (First eigenvector)
DOFnumber Eigenvector Eigenvector derivative
1 0.0013 + 0.0013i -0.0004 - 0.0004i2 0.0050 + 0.0050i -0.0015 - 0.0015i
3 0.0049 + 0.0049i -0.0015 - 0.0015i
4 0.0096 + 0.0096i -0.0029 - 0.0029i
5 0.0108 + 0.0108i -0.0033 - 0.0032i6 0.0139 + 0.0139i -0.0042 - 0.0042i7 0.0188 + 0.0188i -0.0056 - 0.0056i8 0.0179 + 0.0178i -0.0054 - 0.0053i9 0.0287 + 0.0286i -0.0086 - 0.0085i
10 0.0215 + 0.0215i -0.0064 - 0.0064i
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18Structural Dynamics and Vibration Control Lab., KAIST, Korea
• CPU time for 40 Eigenpairs
Method CPU time Ratio
Lee’s method 2.21 1.4
Proposed method 1.59 1.0
(sec)
Zeng’s method 184.05 115.8
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19Structural Dynamics and Vibration Control Lab., KAIST, Korea
: Zeng’s method (Using full modes(40), exact solution)
: Zeng’s method (Using two modes(2), 5% error)
� : Lee’s method (Exact solution) : Proposed method(Exact solution)
Fig 1. Comparison with previous method
Δ
5 10 15 20 25 30 35 400
50
100
150
200
Modes
CPU
tim
e (s
ec)
Δ Δ Δ ΔΔ Δ
184.05
61.47Improvement about 99%
Δ 2.21
1.59
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20Structural Dynamics and Vibration Control Lab., KAIST, Korea
� : Lee’s method (Exact solution) : Proposed method(Exact solution)
Fig 2. Comparison with Lee’s method
Δ
5 10 15 20 25 30 35 400
0.5
1
1.5
2
2.5
Modes
CPU
tim
e (s
ec)
Δ
ΔΔ
ΔΔ
ΔΔ
Improvement about 25% 2.21
1.59
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21Structural Dynamics and Vibration Control Lab., KAIST, Korea
CONCLUSIONS
• Proposed method- is composed of simple algorithm- guarantees numerical stability - reduces the CPU time.
An efficient eigen-sensitivity technique !
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22Structural Dynamics and Vibration Control Lab., KAIST, Korea
Thank you for your attention.
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23Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Numerical Stability
)det()det()det()det( YAYYAY TT
• The determinant property
), ..., n-, i( oft independen be chosen to t vectorsindependenArbitary :
nn: ]....[
singular-Non:
eq.(13) ofmatrix t coefficien The : where
j
i
jn
121
1
1321
Ψ0
0ΨY
A
(14)
APPENDIX
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24Structural Dynamics and Vibration Control Lab., KAIST, Korea
rnonsingula , )1()1(:~
,0
~)( where 2
nn
jj
A
00AKCMT
1n : ~ ,1~)2(
,1
~)2(
bbCM
bCM
T
T
jTj
jj
Then
(15)
jT
jjT
j
jjjj
jT
jjT
j
jjjj
MΨCM
CMΨΨKCMΨ
ΨMCM
CMKCMΨYAYT
)2(
)2()(
1)2()2(
1
T2T
2T
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25Structural Dynamics and Vibration Control Lab., KAIST, Korea
Arranging eq.(15)
MT1~
10
~~
T
T
b0
b0AYAY
0 )A~(det
~~~1
10det)A~(det
Y)A(Ydet
1
T
bA
bM T
T
(16)
Using the determinant property of partitionedmatrix
(17)
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26Structural Dynamics and Vibration Control Lab., KAIST, Korea
0A)(det
Therefore
Numerical Stability is Guaranteed.Numerical Stability is Guaranteed.
(18)
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27Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Lee’s method (1999)
• Differentiating eq.(1) with respect to design variable
(19)
• Pre-multiplying each side of eq.(19) by gives eigenvalue derivative.
jjjT
jj KCM 2
Tj
jjjjjj
jjj
)()2(
)(2
2
KCMCM
KCM
(20)
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28Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Differentiating eq.(3) with respect to design variable
jjjTj
jjTj
CMM
CM
)(25.0
)2((21)
jjjT
j
jjjjjj
j
jT
j
jjjj
CMMKCMCM
CMCMKCM
25.0)()2(
00)2()2(
2
2
• Combining eq.(19) and eq.(21) into a matrix gives eigenvector derivative.
(22)