solution of eigenproblem of non-proportional damping systems by lanczos method in-won lee,...
TRANSCRIPT
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)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 4
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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 5
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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 6
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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 7
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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 8
Proposed Lanczos algorithm
retains the n order quadratic eigenproblems
is one-sided recursion scheme
extracts the Lanczos vectors in real domain
Structural Dynamics & Vibration Control Lab., KAIST, Korea 9
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)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 10
Quadratic eigenproblem
where : eigenvalue (complex conjugate)
: independent eigenvector (complex conjugate)
02 iiiii KCM
i
i
(12)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 11
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)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 12
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)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 13
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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 14
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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 15
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)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 16
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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 17
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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 18
(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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 19
Reduction to Tri-Diagonal System
Rewriting quadratic eigenproblem
(33)
where (34)
(35)
(36)
where
*iiii MCK
iii *
ii
ii *~
m 21
**2
*1
*m
Structural Dynamics & Vibration Control Lab., KAIST, Korea 20
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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 21
Eigenvalues and eigenvectors of the system
ii
1
ii
ii **
(39)
(40)
(41)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 22
Physical error norm(Bathe et al 1980)
and : Acceptable eigenpair
6
2
2
2
10][
i
iii
i K
KCMe
ii
Error Estimation
(42)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 23
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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 24
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
Structural Dynamics & Vibration Control Lab., KAIST, Korea 25
NUMERICAL EXAMPLES Structures
Cantilever beam with lumped dampers Three-dimensional framed structure with lumped d
ampers Analysis methods
Proposed method Rajakumar’s method (1993) Chen’s method (1988)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 26
Comparisons Solution time(CPU) Physical error norm
Convex with 100 MIPS, 200 MFLOPS
Structural Dynamics & Vibration Control Lab., KAIST, Korea 27
Cantilever Beam with Lumped Dampers
1 2 3 4 99 100 101
C
5
Material PropertiesTangential Damper :c = 0.3
Rayleigh Damping : = = 0.001
Young’s Modulus :1000
Mass Density :1
Cross-section Inertia :1
Cross-section Area :1
System DataNumber of Equations :200
Number of Matrix Elements :696
Maximum Half Bandwidths :4
Mean Half Bandwidths :4
Structural Dynamics & Vibration Control Lab., KAIST, Korea 28
Results of cantilever beam : Physical Error norm(number of Lanczos vectors=30)
ModeNumber
EigenvalueProposedMethod
Rajakumar’sMethod
Chen’sMethod
12345678910
-2.57457+j3.17201-2.57457-j3.17201-1.53800+j18.3566-1.53800-j18.3566-1.69581+j39.6477-1.69581-j39.6477-2.43492+j61.0104-2.43492-j61.0104-3.78360+j82.3222-3.78360-j82.3222
0.32e-070.32e-070.53e-080.53e-080.48e-080.48e-080.51e-070.51e-070.10e-060.10e-06
0.32e-070.32e-070.53e-080.53e-080.48e-080.48e-080.51e-070.51e-070.10e-060.10e-06
0.24e-030.24e-030.19e-030.19e-030.21e-040.21e-040.18e-040.18e-040.16e-040.16e-04
Solution time in second(ratio)
3.86(1.00)
5.21(1.35)
5.67(1.47)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 29
ModeNumber
EigenvalueProposedMethod
Rajakumar’sMethod
Chen’sMethod
12345678910
-2.57457+j3.17201-2.57457-j3.17201-1.53800+j18.3566-1.53800-j18.3566-1.69581+j39.6477-1.69581-j39.6477-2.43492+j61.0104-2.43492-j61.0104-3.78360+j82.3222-3.78360-j82.3222
0.25e-070.25e-070.53e-080.53e-080.48e-080.48e-080.51e-070.51e-070.36e-070.36e-07
0.25e-070.25e-070.53e-080.53e-080.48e-080.48e-080.51e-070.51e-070.36e-070.36e-07
0.24e-030.24e-030.19e-030.19e-030.21e-040.21e-040.18e-040.18e-040.13e-040.13e-04
Solution time in second(ratio)
11.63(1.00)
16.09(1.38)
17.10(1.47)
Results of cantilever beam : Physical Error norm(number of Lanczos vectors=60)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 30
Three-Dimensional Framed Structure with Lumped Dampers
Material Properties
Tangential Damper :c = 1,000
Rayleigh Damping : = -0.92
= 0.106
Young’s Modulus: 2.1E+11
Mass Density: 7,850
Cross-section Inertia: 8.3E-06
Cross-section Area: 001
System DataNumber of Equations: 1,008
Number of Matrix Elements :80,784
Maximum Half Bandwidths : 150
Mean Half Bandwidths : 81
Structural Dynamics & Vibration Control Lab., KAIST, Korea 31
Results of three-dimensional framed structure :
Physical Error norm (number of Lanczos vectors=30)
ModeNumber
EigenvalueProposedMethod
Rajakumar’sMethod
Chen’sMethod
12345678910
-0.015035+j3.03037-0.015035-j3.03037-0.024784+j3.09011-0.024784-j3.09011-0.243763+j3.65157-0.243763-j3.65157-3.83006+j7.78173-3.83006-j7.78173-3.42807+j8.04792-3.42807-j8.04792
0.13e-060.13e-060.17e-060.17e-060.25e-060.25e-060.93e-050.93e-050.21e-050.21e-05
0.13e-060.13e-060.17e-060.17e-060.25e-060.25e-060.93e-050.93e-050.21e-050.21e-05
0.46e-050.46e-050.46e-050.46e-050.35e-050.35e-050.33e-030.33e-030.14e-040.14e-04
Solution time in second(ratio)
100.27(1.00)
142.38(1.42)
164.44(1.64)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 32
ModeNumber
EigenvalueProposedMethod
Rajakumar’sMethod
Chen’sMethod
12345678910
-0.015035+j3.03037-0.015035-j3.03037-0.024784+j3.09011-0.024784-j3.09011-0.243763+j3.65157-0.243763-j3.65157-3.83006+j7.78173-3.83006-j7.78173-3.42807+j8.04792-3.42807-j8.04792
0.13e-060.13e-060.17e-060.17e-060.25e-060.25e-060.43e-050.43e-050.62e-060.62e-06
0.13e-060.13e-060.17e-060.17e-060.25e-060.25e-060.43e-050.43e-050.62e-060.62e-06
0.46e-050.46e-050.46e-050.46e-050.35e-050.35e-050.34e-060.34e-060.71e-050.71e-05
Solution time in second(ratio)
213.57(1.00)
323.56(1.51)
337.44(1.58)
Results of three-dimensional framed structure :
Physical Error norm (number of Lanczos vectors=60)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 33
An efficient solution technique!
CONCLUSIONS The proposed method
needs smaller storage space gives better solutions requires less solution time
than other methods.
Structural Dynamics & Vibration Control Lab., KAIST, Korea 34
Thank you for your attention.
Structural Dynamics & Vibration Control Lab., KAIST, Korea 35
111*
1*
111
~~~~~~ m
T
mm
T
mm
T
mm CMM (A-1)
(A-4)
(A-5)
(A-3)1
11
~
m
mm
(A-2)1
*1*
1
~
m
mm
where If 01 m , 2/1
11 mm
If 01 m , 2/1
11 mm j
To scale