Solution of Eigenvalue Problems for Non-Proportionally Damped Systems with Multiple Frequencies
-
Upload
sage-finch -
Category
Documents
-
view
36 -
download
3
description
Transcript of Solution of Eigenvalue Problems for Non-Proportionally Damped Systems with Multiple Frequencies
Solution of Eigenvalue Problems for Non-Proportionally Damped Systems with Multiple Frequencies
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 Korea
Fourth World Congress on Computational MechanicsFourth World Congress on Computational MechanicsBuenos Aires, ArgentinaBuenos Aires, ArgentinaJune 30, 1998June 30, 1998 Session V-C : Structural Dynamics ISession V-C : Structural Dynamics I
Structural Dynamics & Vibration Control Lab., KAIST, Korea 2
OUTLINE Introduction
Objectives and scope Current methods
Proposed method Newton-Raphson technique Modified Newton-Raphson technique
Numerical examples Grid structure with lumped dampers Three-dimensional framed structure with lumped dampers
Conclusions
Structural Dynamics & Vibration Control Lab., KAIST, Korea 3
INTRODUCTION
Free vibration of proportional damping system
(1)
where : Mass matrix
: Damping matrix
: Stiffness matrix
: Displacement vector
0)()()( tuKtuCtuM
M
C
K)(tu
Objectives and Scope
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 analysis of non-proportional damping system
(4)
M
KA
0
0
where
0M
MCB
tu
tuz
0 tAztzB (3)
(5) tetu Let
tt eetz
(6)
, then
Structural Dynamics & Vibration Control Lab., KAIST, Korea 6
Therefore, an efficient eigensolution technique fornon-proportional damping system is required.
nmiBA iii 2,,2,1 (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.
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 METHOD Find p smallest multiple eigenpairs
p 21
iii BA Solve
ijjTi B Subject to
For i iand pi ,,2,1 : multiple
BA
pT IB
where
p ,,, 21
pdiag ,,, 21
Structural Dynamics & Vibration Control Lab., KAIST, Korea 9
Relations between and vectors in the subspace of
BA
p ,,, 21
pdiag ,,, 21 where
X
(7)
(8)
(9)
XZ
pT IBXX
Let be the vectors in the subspace of , and be orthonormal with respect to , then
X pxxxX ,,, 21
(10)
(11)
B
Structural Dynamics & Vibration Control Lab., KAIST, Korea 10
ZDZ
where pdddD ,,, 21 : Symmetric
Let (13)
Introducing Eq.(10) into Eq.(7)
BXZAXZ (12)
BXDZAXZ
BXDAX
or piBXdAx ii ,,2,1
Then
or
(14)
(15)
(16)
Note : If , from Eq.(13)p 21
D
X
(17)(18)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 11
pidXBxA ki
kki ,,2,10)1()1()1(
pkTk IXMX )1()1( )(
(19)
(20)
)()()1( ki
ki
ki ddd
)()()1( ki
ki
ki xxx
)1()1(2
)1(1
)1( ,...,, kp
kkk xxxX
,)(kid )(k
ix
where
: unknown incremental values
(21)
(22)
(23)
Newton-Raphson Technique
Structural Dynamics & Vibration Control Lab., KAIST, Korea 12
)()()()( ki
kki
ki dXBAxr where : residual vector
)()()()()()( ki
ki
kki
kii
ki rdBXxBdxA
0)( )()( ki
Tk xBX
(23)
(24)
Introducing Eqs.(21) and (22) into Eqs.(19) and (20)
and neglecting nonlinear terms
Matrix form of Eqs.(23) and (24)
pi
r
d
x
X
BXBdA ki
ki
ki
Tk
kkii
,,2,1
00B)(
)(
)(
)(
)(
)()(
(25)
Coefficient matrix : • Symmetric• Nonsingular
Structural Dynamics & Vibration Control Lab., KAIST, Korea 13
pi
r
d
x
X
BXBdA ki
ki
ki
Tk
kkii
,,2,1
00B)(
)(
)(
)(
)(
)()(
Coefficient matrix : • Symmetric• Nonsingular
(25)
Introducing modified Newton-Raphson technique
00B)(
)(
)(
)(
)(
)()0( ki
ki
ki
Tk
kii r
d
x
X
BXBdA
)()()1( ki
ki
ki ddd
)()()1( ki
ki
ki xxx
(26)
(22)
(21)
Modified Newton-Raphson Technique
Structural Dynamics & Vibration Control Lab., KAIST, Korea 14
Step
Step 2: Solve for and )(kix )(k
id
00B)(
)(
)(
)(
)(
)()0( ki
ki
ki
Tk
kii r
d
x
X
BXBdA
Step 3: Compute)()()1( k
ik
ik
i ddd
)()()1( ki
ki
ki xxx
Step 1: Start with approximate eigenpairs ,)0(X )0(D
,)()0( kiix pid k
iii ,,2,1)()0(
Structural Dynamics & Vibration Control Lab., KAIST, Korea 15
Step 4: Check the error norm
Error norm =
If the error norm is more than the tolerance, then go to Step 2 and if not, go to Step 5
Step 5: Multiple case
)(lim k
kX
)(lim k
kDi
kii
kd
)(lim
ik
ik
x
)(lim
or
or
2
)(2
)()(
ki
ki
kii
xA
xBdA
Structural Dynamics & Vibration Control Lab., KAIST, Korea 16
NUMERICAL EXAMPLES
Structures Grid structure with lumped dampers Three-dimensional framed structure with lumped d
ampers Analysis methods
Proposed method Subspace iteration method (Leung 1988) Lanczos method (Chen 1993)
Structural Dynamics & Vibration Control Lab., KAIST, Korea 17
Comparisons Solution time(CPU) Convergence
Error norm =
Convex with 100 MIPS, 200 MFLOPS
2
)(2
)()(
ki
ki
kii
xA
xBdA
Structural Dynamics & Vibration Control Lab., KAIST, Korea 18
Grid Structure with Lumped Dampers Material Properties
Tangential Damper :c = 0.3
Rayleigh Damping : = = 0.001
Young’s Modulus :1,000
Mass Density :1
Cross-section Inertia :1
Cross-section Area :1
System Data
Number of Equations :590
Number of Matrix Elements :8,115
Maximum Half Bandwidths :15
Mean Half Bandwidths :14
Structural Dynamics & Vibration Control Lab., KAIST, Korea 19
Proposed MethodMode
Number
Error Norm ofStarting Eigenpair(Lanczos method)
Number ofIterations
Eigenvalue Error Norm
123456789101112
0.467621E-070.325701E-050.467621E-070.325701E-050.170965E-060.823644E-060.170965E-060.823644E-060.250670E-040.786392E-000.250670E-040.786392E-00
111100001111
-0.09590 + j 8.66792-0.09590 + j 8.66792-0.09590 - j 8.66792-0.09590 - j 8.66792-0.60556 + j 15.5371-0.60556 +j 15.5371-0.60556 - j 15.5371-0.60556 - j 15.5371-0.57725 + j 20.7299-0.57725 + j 20.7299-0.57725 - j 20.7299-0.57725 - j 20.7299
0.824521E-100.805278E-100.824521E-100.805278E-100.170965E-060.823644E-060.170965E-060.823644E-060.344167E-100.432693E-090.344167E-100.432693E-09
Table 1. Results of proposed method for grid structure
Structural Dynamics & Vibration Control Lab., KAIST, Korea 20
Method CPU time in seconds Ratio
Proposed method(Lanczos method + Iteration scheme)
Subspace Iteration Method
872.67(214.28 + 658.39)
3,096.62
1.00
3.55
Table 2. CPU time for twelve lowest eigenpairs of grid structure
Structural Dynamics & Vibration Control Lab., KAIST, Korea 21
0 1 2 3
Iteration Number
1.0E-11
1.0E-10
1.0E-9
1.0E-8
1.0E-7
1.0E-6
1.0E-5
1.0E-4
1.0E-3
1.0E-2
1.0E-1
1.0E+0
Err
or
No
rm
Error L im it
Lanczos method(48 Lanczos vectors)
Fig.2 Error norms of grid model by proposed method
0 5 10 15 20 25 30 35 40 45 50
Iteration Number
1.0E-8
1.0E-7
1.0E-6
1.0E-5
1.0E-4
1.0E-3
1.0E-2
1.0E-1
1.0E+0
Err
or
No
rm
Error L im it
Fig.3 Error norms of grid model by subspace iteration method
12 24 36 48 60 72 84 96 108
Num ber of Generated Lanczos Vectors
1.0E-10
1.0E-9
1.0E-8
1.0E-7
1.0E-6
1.0E-5
1.0E-4
1.0E-3
1.0E-2
1.0E-1
1.0E+0
Err
or
No
rm
E rror L im it
1st, 3rd eigenpairs
2nd, 4th eigenpairs
5th, 7th eigenpairs
6th, 8th eigenpairs
9th, 11th eigenpairs
10th, 12th eigenpairs
Fig.4 Error norms of grid model by Lanczos method
Structural Dynamics & Vibration Control Lab., KAIST, Korea 22
Three-Dimensional Framed Structure with Lumped Dampers
Structural Dynamics & Vibration Control Lab., KAIST, Korea 23
Material Properties
Lumped Damper :c = 12,000.0
Rayleigh Damping : =-0.1755 = 0.02005
Young’s Modulus :2.1E+11
Mass Density :7,850
Cross-section Inertia :8.3E-06
Cross-section Area :0.01
System Data
Number of Equations :1,128
Number of Matrix Elements :135,276
Maximum Half Bandwidths :300
Mean Half Bandwidths :120
Structural Dynamics & Vibration Control Lab., KAIST, Korea 24
Proposed MethodMode
Number
Error Norm ofStarting Eigenpair(Lanczos method)
Number ofIterations
Eigenvalue Error Norm
123456789101112
0.334872E-070.602910E-060.334872E-070.602910E-060.256380E-050.497414E-040.256380E-050.497414E-040.117746E-060.374128E-040.117746E-060.374128E-04
000022221111
-0.13811 + j 3.09308-0.13811 + j 3.09308-0.13811 - j 3.09308-0.13811 - j 3.09308-3.53017 + j 2.20867-3.53017 - j 2.20867-0.24297 + j 4.16980-0.24297 - j 4.16980-1.65509 + j 7.04244-1.65509 + j 7.04244-1.65509 - j 7.04244-1.65509 - j 7.04244
0.334872E-070.602910E-060.334872E-070.602910E-060.308387E-070.132855E-060.308387E-070.132855E-060.482385E-110.390433E-060.482385E-110.390433E-06
Table 3. Results of proposed method for three-dimensional framed structure
Structural Dynamics & Vibration Control Lab., KAIST, Korea 25
Method CPU time in seconds Ratio
Proposed method(Lanczos method + Iteration scheme)
Subspace Iteration Method
7,641.94(1,006.73 + 6,635.21)
8,337.60
1.00
1.09
Table 4. CPU time for twelve lowest eigenpairs of three-dimensional framed structure
Structural Dynamics & Vibration Control Lab., KAIST, Korea 26
0 1 2 3 4
Iteration Number
1.0E-12
1.0E-11
1.0E-10
1.0E-9
1.0E-8
1.0E-7
1.0E-6
1.0E-5
1.0E-4
1.0E-3
1.0E-2
1.0E-1
1.0E+0
Err
or
No
rm
Error L im it
Lanczos method(48 Lanczos vectors)
Fig.6 Error norms of 3-D. frame model by proposed method
0 4 8 12 16 20 24 28
Iteration Number
1.0E-8
1.0E-7
1.0E-6
1.0E-5
1.0E-4
1.0E-3
1.0E-2
1.0E-1
1.0E+0
Err
or
No
rm
Error L im it
Fig.7 Error norms of 3-D. frame model by subspace iteration method
12 24 36 48 60 72 84 96 108
Num ber of Generated Lanczos Vectors
1.0E-12
1.0E-11
1.0E-10
1.0E-9
1.0E-8
1.0E-7
1.0E-6
1.0E-5
1.0E-4
1.0E-3
1.0E-2
1.0E-1
1.0E+0
Err
or
No
rm
Error L im it
1st, 3rd eigenpairs
2nd, 4th eigenpairs
5th, 7th eigenpairs
6th, 8th eigenpairs
9th, 11th eigenpairs
10th, 12th eigenpairs
Fig.8 Error norms of 3-D. frame model by Lanczos method
Structural Dynamics & Vibration Control Lab., KAIST, Korea 27
CONCLUSIONS Proposed method
converges fast guarantees nonsingularity of coefficient matrix
Proposed method is efficient
Structural Dynamics & Vibration Control Lab., KAIST, Korea 28
Thank you for your attention.