SOLUTION OF EIGENVALUE PROBLEM FOR NON-CLASSICALLY DAMPED SYSTEM WITH MULTIPLE EIGENVALUES

* In-Won Lee: Professor, KAIST* In-Won Lee: Professor, KAIST Man-Cheol Kim: Senior Researcher, KRRIMan-Cheol Kim: Senior Researcher, KRRI Kyu-Hong Shim: Postdoctoral Researcher, KAIST Kyu-Hong Shim: Postdoctoral Researcher, KAIST

SOLUTION OF EIGENVALUE PROBLEM FOR NON-CLASSICALLY DAMPED SYSTEM WITH MULTIPLE EIGENVALUES

Nha Trang 2000Nha Trang 2000

Nha Trang, Vietnam, Aug. 14-18, 2000Nha Trang, Vietnam, Aug. 14-18, 2000

Structural Dynamics & Vibration Control Lab., KAIST, Korea

2

OUTLINE

PROBLEM DEFINITION

PROPOSED METHOD

NUMERICAL EXAMPLES

CONCLUSIONS


3

)()()()( tftuKtuCtuM

M

C

K)(tu

)(tf

(1)

where : Mass matrix, Positive definite

: Damping matrix

: Stiffness matrix, Positive semi-definite

: Displacement vector

: Load vector

PROBLEM DEFINITION• Dynamic Equation of Motion• Dynamic Equation of Motion


4

• Methods of Dynamic Analysis• Step by step integration method

• Mode superposition method

• Methods of Dynamic Analysis• Step by step integration method

• Mode superposition method

• Mode Superposition Method• Free vibration analysis should be first performed

• Mode Superposition Method• Free vibration analysis should be first performed


5

• Example : Rayleigh Damping

CMKKMC 11

KMC

(2)

• Condition of Classical Damping• Condition of Classical Damping


6

: Real eigenvalue

: Natural frequency

: Real eigenvector(mode shape)

niMK iii ,,2,1 (3)

2ii

ii

• Eigenproblem of classical damping systems• Eigenproblem of classical damping systems

- Low in cost

- Straightforward

where


7

niKCM iiiii ,,2,102 (4)

where : Complex eigenvalue

: Complex eigenvector(mode shape)i

i

• Quadratic eigenproblem of non-classically damped systems• Quadratic eigenproblem of non-classically damped systems


8

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


9

• 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

• Current Methods for Solving the Non-Classically Damped Eigenproblems• Current Methods for Solving the Non-Classically Damped Eigenproblems


10

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

PROPOSED METHOD

• Find p Smallest Eigenpairs• Find p Smallest Eigenpairs


11

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

• Relations between and Vectors in the Subspace of • Relations between and Vectors in the Subspace of


12

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)


13

ZDZ

D

D

XZ

DX

(13)

(10)

• 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 :

• 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 :


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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• Newton-Raphson Technique


15

)()()()( 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


16

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• Modified Newton-Raphson Technique


17

• 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

• Algorithm of Proposed Method• Algorithm of Proposed Method


18

• 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


19

• 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 !


20

• 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

• 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


21

NUMERICAL EXAMPLES

• Structures• Cantilever beam(distinct)

• Grid structure(multiple)

• Three-dimensional framed structure(close)

• Structures• Cantilever beam(distinct)

• Grid structure(multiple)

• Three-dimensional framed structure(close)

• Analysis Methods• Proposed method

• Subspace iteration method (Leung 1988)

• Lanczos method (Chen 1993)

• Analysis Methods• Proposed method

• Subspace iteration method (Leung 1988)

• Lanczos method (Chen 1993)


22

• Comparisons• Solution time(CPU)

• Convergence

• Comparisons• Solution time(CPU)

• Convergence

• Convex with 100 MIPS, 200 MFLOPS• Convex with 100 MIPS, 200 MFLOPS


23

Cantilever Beam with Lumped Dampers (Distinct Case)

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


24

Proposed MethodMode

Number

Error Norm ofStarting Eigenpair(Lanczos method)

Number ofIterations

Eigenvalue Error Norm

12345678910

0.234069E-030.234069E-030.192793E-030.192793E-030.148072E-040.148072E-040.619329E-020.619329E-020.377004E+000.377004E+00

1111111111

-2.57457 + j 3.17201-2.57457 - j 3.17201-1.53800 + j 18.3566-1.53800 - j 18.3566-1.69581 + j 39.6477-1.69581 - j 39.6477-2.43492 + j 61.0104-2.43492 - j 61.0104-3.78360 + j 82.3222-3.78360 - j 82.3222

0.232258E-070.232258E-070.428428E-090.428428E-090.727353E-100.727353E-100.204824E-100.204824E-100.997372E-070.997372E-07

• Results of Cantilever Beam Structure (Distinct)• Results of Cantilever Beam Structure (Distinct)


25

Method CPU time in seconds Ratio

Proposed method(Lanczos method + Iteration scheme)

Subspace Iteration Method

76.10(10.42 + 65.64)

100.94

1.00

1.33

• CPU Time for 10 Lowest Eigenpairs, Cantilever Beam• CPU Time for 10 Lowest Eigenpairs, Cantilever Beam


26

2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 11 0

N u m b er o f G en era ted L a n czos V ecto rs

1 .0 E -8

1 .0E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0E + 0E

rror

Nor

m

E rror L im it

Convergence by Lanczos method(Chen 1993)Cantilever beam (distinct)

Starting values of proposed method

: 1st, 2nd eigenpairs : 3rd, 4th eigenpairs : 5th, 6th eigenpairs : 7th, 8th eigenpairs : 9th, 10th eigenpairs


27

Convergence of the 1st eigenpairCantilever beam (distinct)

: Proposed Method : Subspace Iteration Method (q=2p)

0 1 2 3 4 5 6

Itera tio n N u m b er

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rro r L im it


28

Convergence of the 5th eigenpairCantilever beam (distinct)


0 1 2 3 4 5 6 7 8 9 1 0 1 1


1 .0 E -1 1

1 .0 E -1 0

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rro r L im it


29

Grid Structure with Lumped Dampers (Multiple Case)

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


Mean Half Bandwidths :[email protected]=10

[email protected]=

10


30

Proposed MethodMode

Number


Number ofIterations


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

010100001111

-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

0.467621E-070.805278E-100.467621E-070.805278E-100.170965E-060.823644E-060.170965E-060.823644E-060.344167E-100.432693E-090.344167E-100.432693E-09

• Results of Grid Structure (Multiple)• Results of Grid Structure (Multiple)


31




872.67(214.28 + 658.39)

3,096.62

1.00

3.55

• CPU Time for 10 Lowest Eigenpairs, Grid Structure• CPU Time for 10 Lowest Eigenpairs, Grid Structure


32

24 3 6 4 8 6 0 7 2 84 96 10 8

N u m b er o f G en era ted L a n czos V ec to rs

1 .0 E -10

1 .0E -9

1 .0E -8

1 .0 E -7

1 .0E -6

1 .0 E -5

1 .0 E -4

1 .0E -3

1 .0 E -2

1 .0E -1

1 .0E + 0E

rror

Nor

m

E rro r L im it

Convergence by Lanczos method(Chen 1993)Grid structure (multiple)

: 1st, 3rd eigenpairs : 2nd, 4th eigenpairs : 5th, 7th eigenpairs : 6th, 8th eigenpairs : 9th, 11th eigenpairs : 10th, 12th eigenpairs



33

Convergence of the 2nd eigenpairGrid structure (multiple)


0 2 4 6 8 1 0 1 2 1 4 1 6 1 8


1 .0 E -1 1

1 .0 E -1 0

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rro r L im it


34

Convergence of the 9th eigenpairGrid structure (multiple)


0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0


1 .0 E -1 1

1 .0 E -1 0

1 .0 E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0 E + 0E

rror

Nor

m

E rro r L im it


35

Three-Dimensional Framed Structure with Lumped Dampers(Close Case)

[email protected]=6.02

6@3=

18

2@3=6

[email protected]=18.06

12@3=

36


36

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


Mean Half Bandwidths :120


37

Proposed MethodMode

Number


Number ofIterations


123456789101112

0.579542E-060.579542E-060.567549E-060.567549E-060.929150E-050.929150E-050.767207E-030.767207E-030.910611E-000.910611E-000.920451E-000.920451E-00

00001133

11111111

-0.13763 + j 3.08907-0.13763 - j 3.08907-0.13803 + j 3.09109-0.13803 - j 3.09109-3.52574 + j 2.20649-3.52574 - j 2.20649-0.24236 + j 4.16556-0.24236 – j 4.16556-1.64294 + j 7.02958-1.64294 - j 7.02958-1.65070 + j 7.03590-1.65070 - j 7.03590

0.579542E-060.579542E-060.567549E-060.567549E-060.421992E-090.421992E-090.614880E-060.614880E-060.624657E-060.624657E-060.660196E-060.660196E-06

• Results of Three-Dimensional Frame Structure (Close)• Results of Three-Dimensional Frame Structure (Close)


38




8,335.20(918.15 + 7417.05)

9,644.75

1.00

1.16

• CPU Time for 12 Lowest Eigenpairs, 3-D. Frame Structure• CPU Time for 12 Lowest Eigenpairs, 3-D. Frame Structure


39

Convergence by Lanczos method(Chen 1993)3-D. framed structure (close)

: 1st, 2nd eigenpairs : 3rd, 4th eigenpairs : 5th, 6th eigenpairs : 7th, 8th eigenpairs

: 9th, 10th eigenpairs : 11th, 12th eigenpairs


2 4 3 6 4 8 6 0 7 2 8 4 96 10 8

N u m b er o f G en era ted L a n czos V ecto rs

1 .0 E -8

1 .0E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0 E -2

1 .0 E -1

1 .0E + 0E

rror

Nor

m

E rror L im it


40

Convergence of the 9th eigenpair3-D. framed structure (close)


0 2 4 6 8 1 0 12 1 4 1 6 1 8 20 2 2 24 2 6 2 8 3 0 3 2

Ite ra tio n N u m b er

1 .0 E -7

1 .0 E -6

1 .0 E -5

1 .0 E -4

1 .0 E -3

1 .0E -2

1 .0E -1

1 .0 E + 0E

rror

Nor

m

E rror L im it


41

An efficient Eigensolution technique !

CONCLUSIONS

• The proposed method

• is simple

• guarantees numerical stability

• converges fast.

• The proposed method

• is simple

• guarantees numerical stability

• converges fast.


42

Thank you for your attention.

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SOLUTION OF EIGENVALUE PROBLEM FOR NON-CLASSICALLY DAMPED SYSTEM WITH MULTIPLE EIGENVALUES

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