* * Man-Cheol Kim, Hyung-Jo Jung and In-W Man-Cheol Kim, Hyung-Jo Jung and In-W on Lee on Lee Structural Dynamics & Vibration Cont Structural Dynamics & Vibration Cont rol Lab. rol Lab. Korea Advanced Institute of Science & Technology Solution of Eigenvalue Problem for Non-Classically Damped System with Multiple Eigenvalues PSSC 1998 PSSC 1998 Fifth Pacific Structural Steel Conference Fifth Pacific Structural Steel Conference Seoul, Korea, 13-16 October 1998 Seoul, Korea, 13-16 October 1998
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*Man-Cheol Kim, Hyung-Jo Jung and In-Won Lee *Man-Cheol Kim, Hyung-Jo Jung and In-Won Lee Structural Dynamics & Vibration Control Lab. Structural Dynamics.
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**Man-Cheol Kim, Hyung-Jo Jung and In-Won LeeMan-Cheol Kim, Hyung-Jo Jung and In-Won Lee Structural Dynamics & Vibration Control Lab.Structural Dynamics & Vibration Control Lab. Korea Advanced Institute of Science & Technology
Solution of Eigenvalue Problem for Non-Classically Damped System with Multiple Eigenvalues
PSSC 1998 PSSC 1998
Fifth Pacific Structural Steel ConferenceFifth Pacific Structural Steel ConferenceSeoul, Korea, 13-16 October 1998Seoul, Korea, 13-16 October 1998
Structural Dynamics & Vibration Control Lab., KAIST, Korea
2
Problem Definition
Proposed Method
Numerical Examples
Conclusions
OUTLINE
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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PROBLEM DEFINITION
Dynamic Equation of Motion
)()()()( tftuKtuCtuM
M
C
K)(tu
)(tf
(1)
where : Mass matrix, Positive definite
: Damping matrix
: Stiffness matrix, Positive semi-definite
: Displacement vector
: Load vector
Structural Dynamics & Vibration Control Lab., KAIST, Korea
4
Methods of Dynamic Analysis Step by step integration method Mode superposition method
Mode Superposition Method Free vibration analysis should be first performed.
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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Condition of Classical Damping
Example : Rayleigh Damping
CMKKMC 11
KMC
(2)
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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Eigenproblem of classical damping systems
where : Real eigenvalue
: Natural frequency
: Real eigenvector(mode shape)
Low in cost Straightforward
niMK iii ,,2,1 (3)
2ii
ii
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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Quadratic eigenproblem of non-classically damped systems
niKCM iiiii ,,2,102 (4)
where : Complex eigenvalue
: Complex eigenvector(mode shape)i
i
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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niBA iii 2,,2,1 (5)
: Complex Eigenvector (6)
ii
ii
where
An efficient eigensolution technique of An efficient eigensolution technique of non-classically damped systems is required.non-classically damped systems is required.
: Complex Eigenvalue
Very expensive
M
KA
0
0
0M
MCB
i
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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Current Methods for Solving the Non-Classically Damped Eigenproblems
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
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PROPOSED METHOD Find p Smallest Eigenpairs
p 21
iii BA Solve
ijjTi B Subject to
For i iand pi ,,2,1
: multiple or close roots
BA
pT IB
where
p ,,, 21
pdiag ,,, 21
If p=1, then distinct root
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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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
pxxxX ,,, 21
(10)
(11)
BX
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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ZDZ
where AXXdddD Tp ,,, 21 : Symmetric
Let (13)
Introducing Eq.(10) into Eq.(7)
BXZAXZ (12)
BXDZAXZ
BXDAX
or piBXdAx ii ,,2,1
Then
or
(14)
(15)
(16)
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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Multiple or Close Eigenvalues
Multiple eigenvalues case : is a diagonal matrix.
Eigenvalues :
Eigenvectors :
Close eigenvalues case : is not a diagonal matrix. Solve the small standard eigenvalue problem.
Get the following eigenpairs.
Eigenvalues :
Eigenvectors :
ZDZ
D
D
XZ
DX
(13)
(10)
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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pidXBxA ki
kki ,,2,10)1()1()1(
pkTk IXMX )1()1( )(
(17)
(18)
)()()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
(19)
(20)
(21)
Newton-Raphson Technique
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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)()()()( ki
kki
ki dXBAxr where : residual vector
)()()()()()( ki
ki
kki
kii
ki rdBXxBdxA
0)( )()( ki
Tk xBX
(22)
(23)
Introducing Eqs.(19) and (20) into Eqs.(17) and (18) and neglecting nonlinear terms
Matrix form of Eqs.(22) and (23)
pi
r
d
x
X
BXBdA ki
ki
ki
Tk
kkii
,,2,1
00B)(
)(
)(
)(
)(
)()(
(24)
Coefficient matrix : • Symmetric• Nonsingular
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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pi
r
d
x
X
BXBdA ki
ki
ki
Tk
kkii
,,2,1
00B)(
)(
)(
)(
)(
)()(
Coefficient matrix : • Symmetric• Nonsingular
(24)
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
(25)
(20)
(19)
Modified Newton-Raphson Technique
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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Algorithm of Proposed Method
Step 2: Solve for and )(kid
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(
)(kix
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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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: Check if is a diagonal matrix, go to Step 6, if not, go to Step 7.
2
)(2
)()()(
ki
ki
kki
xA
dXBxA
D
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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Step 7: Close case Step 6: Multiple case
XD
Go to step 8. Go to step 8.
ZDZ
XZ
Step 8: Check the error norm.
piA
BA
i
iii,,1
2
2
Error norm =
Stop !
Structural Dynamics & Vibration Control Lab., KAIST, Korea
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Initial Values of the Proposed Method
Intermediate results of the iteration methods Vector iteration method Subspace iteration method
Results of the approximate methods Static Condensation method Lanczos method
Structural Dynamics & Vibration Control Lab., KAIST, Korea