* Hong-Ki Jo 1) , Kyu-Sik Park 2) , Hye-Rin Shin 3) and In-Won Lee 4) 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 Seminar Taipei, Taiwan, Dec. 7-8, 2000
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* Hong-Ki Jo 1) , Kyu-Sik Park 2) , Hye-Rin Shin 3) and In-Won Lee 4)
KKNN Seminar Taipei, Taiwan, Dec. 7-8, 2000. SIMPLIFIED ALGEBRAIC METHOD FOR COMPUTING EIGENPAIR SENSITIVITIES OF DAMPED SYSTEM. * Hong-Ki Jo 1) , Kyu-Sik Park 2) , Hye-Rin Shin 3) and In-Won Lee 4) 1) ~ 3) Graduate Student, Department of Civil Engineering, KAIST - PowerPoint PPT Presentation
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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
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
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