hong-ki jo 1), man-gi ko 2) and * in-won lee 3) 1) graduate student, dept. of civil engineering,...
TRANSCRIPT
Hong-Ki Jo1), Man-Gi Ko2) and *In-Won Lee3)
1) Graduate Student, Dept. of Civil Engineering, KAIST
2) Professor, Dept. of Civil Engineering, Kongju National University
3) Professor, Dept. of Civil Engineering, KAIST
Hong-Ki Jo1), Man-Gi Ko2) and *In-Won Lee3)
1) Graduate Student, Dept. of Civil Engineering, KAIST
2) Professor, Dept. of Civil Engineering, Kongju National University
3) Professor, Dept. of Civil Engineering, KAIST
SIMPLIFIED ALGEBRAIC METHOD FOR COMPUTING EIGENPAIR SENSITIVITIES OF DAMPED SYSTEMS
SIMPLIFIED ALGEBRAIC METHOD FOR COMPUTING EIGENPAIR SENSITIVITIES OF DAMPED SYSTEMS
ISEC-01Honolulu, HawaiiJanuary 24-26, 2001
2 2Structural Dynamics and Vibration Control Lab., KAIST, Korea
OUTLINE
INTRODUCTION
PREVIOUS STUDIES
PROPOSED METHOD
NUMERICAL EXAMPLE
CONCLUSIONS
3 3Structural Dynamics and Vibration Control Lab., KAIST, Korea
INTODUCTION
• Objective of Study• Objective of Study
• Applications of Sensitivity Analysis• 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.
4 4Structural Dynamics and Vibration Control Lab., KAIST, Korea
)( 2 0KCM jjj
• Problem Definition• 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
CKMKCM
C
M
n) ,2 1,( j
- Eigenvalue problem of damped system (N-space)
5 5Structural Dynamics and Vibration Control Lab., KAIST, Korea
(2)
- Normalization condition
- State space equation (2N-space)
jj
jj
jj
j
00
0
M
MC
M
K
(3)1)2( 0
jiT
ijj
jT
ii
i
CM
M
MC
6 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.)
)(
7 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.
Structural Dynamics and Vibration Control Lab., KAIST, Korea
8 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
Ci
iCwhere
FF
M
1*
)(*
*
))((
)(
5.0
~)1(
~)1(
2
1
)(5.0
N
jk (6)
Structural Dynamics and Vibration Control Lab., KAIST, Korea
9 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.
10 10Structural Dynamics and Vibration Control Lab., KAIST, Korea
Lee’s method (1999)Lee’s method (1999)
jjjT
jj KCM 2
jjjT
j
jjjjjj
j
jT
j
jjjj
CMM
KCMCM
CM
CMKCM
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.
11 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)
12 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
13 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
CM
KCM
MCM
CMKCM
(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.
- 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.
14 14Structural Dynamics and Vibration Control Lab., KAIST, Korea
NUMERICAL EXAMPLE• Cantilever beam with lumped dampers• Cantilever beam with lumped dampers
1 : (A) areasection -Cross
1 : (I) inertiasection -Cross
1 : )(density Mass
1000 :(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
2 3 4 2119 20
15 15Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Analysis Methods• Analysis Methods
• Zeng’s method (1995)
• Lee’s method (1999)
• Proposed method
• Comparisons• Comparisons
• Solution time (CPU)
16 16Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Results of Analysis (Eigenvalue)• Results of Analysis (Eigenvalue)
Mode
numberEigenvalue Eigenvalue derivative
1 -0.0035 - 1.0868i 0.0010 - 0.2997i
2 -0.0035 + 1.0868i 0.0010 + 0.2997i
3 -0.0203 - 6.0514i 0.0072 - 1.3173i
4 -0.0203 + 6.0514i 0.0072 + 1.3173i
5 -0.0422 - 14.7027i 0.0140 - 2.4536i
6 -0.0422 + 14.7027i 0.0140 + 2.4536i
7 -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
17 17Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Results of Analysis (First eigenvector)• Results of Analysis (First eigenvector)
DOFnumber Eigenvector Eigenvector derivative
1 0.0013 + 0.0013i -0.0004 - 0.0004i
2 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 - 0.0032i
6 0.0139 + 0.0139i -0.0042 - 0.0042i
7 0.0188 + 0.0188i -0.0056 - 0.0056i
8 0.0179 + 0.0178i -0.0054 - 0.0053i
9 0.0287 + 0.0286i -0.0086 - 0.0085i
10 0.0215 + 0.0215i -0.0064 - 0.0064i
18 18Structural Dynamics and Vibration Control Lab., KAIST, Korea
• CPU time for 40 Eigenpairs• 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
19 19Structural Dynamics and Vibration Control Lab., KAIST, Korea
Δ
: 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
CP
U t
ime
(sec
)
Δ Δ Δ ΔΔ Δ
184.05
61.47Improvement about 99%
2.211.59
Structural Dynamics and Vibration Control Lab., KAIST, Korea
20 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
CP
U t
ime
(sec
)
Δ
ΔΔ
ΔΔ
ΔΔ
Improvement about 25% 2.21
1.59
Structural Dynamics and Vibration Control Lab., KAIST, Korea
21 21Structural Dynamics and Vibration Control Lab., KAIST, Korea
CONCLUSIONS
• Proposed method
- is composed of simple algorithm
- guarantees numerical stability
- reduces the CPU time.
• Proposed method
- is composed of simple algorithm
- guarantees numerical stability
- reduces the CPU time.
An efficient eigen-sensitivity technique !
23 23Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Numerical Stability• 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
Ψ
0
0ΨY
A
(14)
APPENDIX
24 24Structural Dynamics and Vibration Control Lab., KAIST, Korea
rnonsingula , )1()1(:~
,0
~)( where 2
nn
jj
A
0
0AKCMT
1n : ~
,1~
)2(
,1
~
)2(
bbCM
bCM
T
T
jTj
jj
Then
(15)
jT
jjT
j
jjjj
jT
jjT
j
jjjj
MΨCM
CMΨΨKCMΨ
Ψ
MCM
CMKCMΨYAYT
)2(
)2()(
1)2(
)2(
1
T2T
2T
25 25Structural Dynamics and Vibration Control Lab., KAIST, Korea
Arranging eq.(15)
MT1
~10
~~
T
T
b
0
b0A
YAY
0
)A~
(det
~~~
1
10det)A
~(det
Y)A(Ydet
1
T
bA
bM T
T
(16)
Using the determinant property of partitioned matrix
(17)
26 26Structural Dynamics and Vibration Control Lab., KAIST, Korea
0A)(det
Therefore
Numerical Stability is Guaranteed.Numerical Stability is Guaranteed.
(18)
27 27Structural Dynamics and Vibration Control Lab., KAIST, Korea
• Lee’s method (1999)• 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)
28 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
CMM
KCMCM
CM
CMKCM
25.0
)()2(
00)2(
)2(
2
2
• Combining eq.(19) and eq.(21) into a matrix gives
eigenvector derivative.
(22)