Solution of Eigenproblem of Non-Proportional Damping Systems by Lanczos Method In-Won Lee, Professor, PE In-Won Lee, Professor, PE Structural Dynamics & Vibration Structural Dynamics & Vibration Control Lab. Control Lab. Korea Advanced Institute of Science & Technology The Fourth International Conference on Computational Structures The Fourth International Conference on Computational Structures Technology Technology Edinburgh, Scotland Edinburgh, Scotland 18th-20th August 1998 18th-20th August 1998
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Solution of Eigenproblem of Non-Proportional Damping Systems by Lanczos Method In-Won Lee, Professor, PE In-Won Lee, Professor, PE Structural Dynamics.
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Solution of Eigenproblem of Non-Proportional Damping Systems by Lanczos Method
In-Won Lee, Professor, PEIn-Won Lee, Professor, PE Structural Dynamics & Vibration Control Lab.Structural Dynamics & Vibration Control Lab. Korea Advanced Institute of Science & Technology
The Fourth International Conference on Computational Structures TechnologyThe Fourth International Conference on Computational Structures TechnologyEdinburgh, ScotlandEdinburgh, Scotland18th-20th August 199818th-20th August 1998
Structural Dynamics & Vibration Control Lab., KAIST, Korea 2
OUTLINE
Introduction
Method of analysis
Numerical examples
Conclusions
Structural Dynamics & Vibration Control Lab., KAIST, Korea 3
INTRODUCTION
Free vibration of proportional damping system
where : Mass matrix
: Proportional damping matrix
: Stiffness matrix
: Displacement vector
0)()()( tuKtuCtuM
M
C
K)(tu
(1)
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Eigenanalysis of proportional damping system
where : Real eigenvalue
: Natural frequency
: Real eigenvector(mode shape)
Low in cost Straightforward
niMK iii ,,2,1 (2)2ii
ii
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Free vibration of non-proportional damping system
(4),0
0
M
KA
where
0M
MCB
tu
tuz
0 tAztzB (3)
(5) tetu Let
tt eetz
(6)
, then
and
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Therefore, an efficient eigensolution technique is required.
iii BA (7)
(9): Orthogonality of eigenvector
mjiB ijjTi ,,2,1,
: Eigenvalue(complex conjugate)
: Eigenvector(complex conjugate)
i
(8)
ii
ii
where
Solution of Eq.(7) is very expensive.
nmi 2,,2,1
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Current Methods
Transformation method: Kaufman (1974)
Perturbation method: Meirovitch et al (1979)
Vector iteration method: Gupta (1974; 1981)
Subspace iteration method: Leung (1995)
Lanczos method: Chen (1993)
Efficient Methods
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Proposed Lanczos algorithm
retains the n order quadratic eigenproblems
is one-sided recursion scheme
extracts the Lanczos vectors in real domain
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METHOD OF ANALYSIS Free vibration of non-proportional damping system
where : Mass matrix
: Non-proportional damping matrix
: Stiffness matrix
: Displacement vector
0)()()( tuKtuCtuM
M
C
K)(tu
(11) tetu Let , then
(10)
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Quadratic eigenproblem
where : eigenvalue (complex conjugate)
: independent eigenvector (complex conjugate)
02 iiiii KCM
i
i
(12)
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where : dependent eigenvector
ji
ji
M
MC
ii
i
T
jj
j
1
0
0
iii *
(13)
ji
jiCMM i
T
ji
T
ji
T
j 1
0**
Orthogonality of the eigenvectors
or
(14)
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Proposed Lanczos Algorithm
Assume that m independent
and dependent Lanczos vectors are found
Calculate preliminary vectors and
m ,,, 21
**2
*1 ,,, m
*11
ˆmmm MCK
mm *
1ˆ
1ˆ
m*
1ˆ
m
(15)
(16)
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Preliminary vectors can be expressed as
111
~ mmm
*11
*1
~ mmm
2111
~ˆmmmmmmmm
*
2*
1**
1*
1
~ˆmmmmmmmm
are the components of previous Lanczos vectors
(real values)
, and is the pseudo length of and
(17)
(18)
(19)
(20)
1
~m
*1
~m1m
,,, mmm real
where
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Orthogonality conditions of Lanczos vectors
miCMM m
T
im
T
im
T
i ,,1for011**
1
1
11111
*1
*11
orCMM mmmmmmm
TTT
111*
1*
111
~~~~~~ m
T
mm
T
mm
T
mm CMMsign
(21)
(22)
(23)
(19)
(20)
111
~ mmm
*11
*1
~ mmm
where
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Coefficient m
mm
T
mm
T
mm
T
mm CMM 11**
1ˆˆˆ
mm
T
mmm
T
m
T
mm MMCKCM *1*
the orthogonality conditions Eqs.(21) and (22)
CMT
m
T
m * MT
mEq.(17) + Eq.(18) and Applying
Using Eqs.(15) and (16)
(25)
(24)
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Coefficients and
1m 1m
1
2/1
11 mmm 2/1
11 mm
(26)
(27)
(28)
(29)
mm
T
mm
T
mm
T
m
mmm
CMM
111*
1*
11
111
~~~~~~
11 mm sign
Applying the orthogonality conditions Eqs.(21) and (22)
CMT
m
T
m 1*
1 MT
m 1Eq.(17) + Eq.(18) and
where
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Coefficients ,m
0
21*
1*
11
mm
T
mm
T
mm
T
mmm CMM (30)
Applying the orthogonality conditions Eqs.(21) and (22)
CMT
m
T
m 2*
2 MT
m 2Eq.(17) + Eq.(18) and
0
0
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(m+1)th Lanczos vectors and1m*
1m
1
1*1
1
11
~
m
mmmmmm
m
mm
MCK
1
*1
*
1
*1*
1
~
m
mmmmm
m
mm
mm sign
mmm 2/1
,2/1
11 mm
(31)
(32)
mm
T
mmm
T
m
T
mm MMCKCM *1*
mm
T
mm
T
mm
T
mm CMM 111*
1*
111
~~~~~~
where
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Reduction to Tri-Diagonal System
Rewriting quadratic eigenproblem
(33)
where (34)
(35)
(36)
where
*iiii MCK
iii *
ii
ii *~
m 21
**2
*1
*m
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iiiT (37)
Applying the orthogonality conditions Eqs.(21) and (22)
1* KMCTT
MT
Eq.(33) + Eq.(34) and
mm
m
TTTMMCKMCT
4
433
322
21
*1*
Unsymmetric
(38)
where
mmm ,, : Real values
nmi 2,,1for
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Eigenvalues and eigenvectors of the system
ii
1
ii
ii **
(39)
(40)
(41)
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Physical error norm(Bathe et al 1980)
and : Acceptable eigenpair
6
2
2
2
10][
i
iii
i K
KCMe
ii
Error Estimation
(42)
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Comparison of Operations
kk nmnm 23
21 2 nnmnmnm cmk 18482 Proposed method
kk nmnm 23
21 2 nnmnmnm cmk 276104 Rajakumar’s method
mm
kk
nmnm
nmnm
23
21
23
21
2
2
nnmnmnm cmk 216142 Chen’s method
MethodInitial operations(A)
Operations in each row of T(B)
Number of operations = A + p B
p : Number of Lanczos vectors
n : Number of equations
cmk mmm and, : Mean half bandwidths of K, M and C
where
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Example : Three-Dimensional Framed Structure
Proposed method
Rajakumar’s method
Chen’s method
Number of total operations Ratio
p = 30
n 1,008
cmk mmmm 81
Method
38.27e+6
53.23e+06
61.38e+06
1.00
1.39
1.60
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NUMERICAL EXAMPLES Structures
Cantilever beam with lumped dampers Three-dimensional framed structure with lumped d