*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
2Structural Dynamics and Vibration Control Lab., KAIST, Korea
OUTLINE
INTRODUCTION
PREVIOUS STUDIES
PROPOSED METHOD NUMERICAL EXAMPLE CONCLUSIONS
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.
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)
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
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.)
)(
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.
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)
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.
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.
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)
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
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.
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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• Analysis Methods
• Zeng’s method (1995)
• Lee’s method (1999)
• Proposed method
• Comparisons
• Solution time (CPU)
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
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
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
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
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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CONCLUSIONS
• Proposed method- is composed of simple algorithm- guarantees numerical stability - reduces the CPU time.
An efficient eigen-sensitivity technique !
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Thank you for your attention.
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• 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
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
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)
26Structural Dynamics and Vibration Control Lab., KAIST, Korea
0A)(det
Therefore
Numerical Stability is Guaranteed.Numerical Stability is Guaranteed.
(18)
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)
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)