Man-Cheol Kim, Hyung-Jo Jung, Sun-Kyu Park and Man-Cheol Kim, Hyung-Jo Jung, Sun-Kyu Park and In-Won Lee In-Won Lee Structural Dynamics & Vibration Control

**Man-Cheol Kim, Hyung-Jo Jung, Sun-Kyu Park and Man-Cheol Kim, Hyung-Jo Jung, Sun-Kyu Park and

In-Won LeeIn-Won Lee Structural Dynamics & Vibration Control Lab.Structural Dynamics & Vibration Control Lab. Korea Advanced Institute of Science & Technology

Free Vibration Analysis of Non-Proportionally Damped Structures with Multiple or Close Frequencies

Tokyo Univ.-KAIST Tokyo Univ.-KAIST

1st Bilateral Seminar on Construction Technology1st Bilateral Seminar on Construction TechnologyKAIST, September 28-29, 1998KAIST, September 28-29, 1998

Structural Dynamics & Vibration Control Lab., KAIST, Korea

2

Problem Definition

Proposed Method

Numerical Examples

Conclusions

OUTLINE


3

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


4

Methods of Dynamic Analysis Step by step integration method Mode superposition method

Mode Superposition Method Free vibration analysis should be first performed.


5

Condition of Proportional Damping

Example : Rayleigh Damping

CMKKMC 11

KMC

(2)


6

Eigenproblem of proportional damping systems

where : Real eigenvalue

: Natural frequency

: Real eigenvector(mode shape)

Low in cost Straightforward

niMK iii ,,2,1 (3)

2ii

ii


7

Quadratic eigenproblem of non-proportionally damped systems

niKCM iiiii ,,2,102 (4)

where : Complex eigenvalue

: Complex eigenvector(mode shape)i

i


8

niBA iii 2,,2,1 (5)

: Complex Eigenvector (6)

ii

ii

where

An efficient eigensolution technique of An efficient eigensolution technique of non-proportionally damped systems is required.non-proportionally damped systems is required.

: Complex Eigenvalue

Very expensive

M

KA

0

0

0M

MCB

i


9

Current Methods for Solving the Non-Proportionally 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


10

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


11

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


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

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)


14

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


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


17

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


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


21

NUMERICAL EXAMPLES 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)


22

Comparisons Solution time(CPU) Convergence

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)


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


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

1 2 3 4 5

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

Convergence of the 1st eigenpairCantilever beam (distinct)

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


28

1 2 3 4 5 6 7 8 9

Ite ra tio n N u m b er

1 .0 E -11

1 .0E -1 0

1 .0E -9

1 .0 E -8

1 .0 E -7

1 .0 E -6

1 .0E -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 rror L im it

Convergence of the 5th eigenpairCantilever beam (distinct)



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)


31




872.67(214.28 + 658.39)

3,096.62

1.00

3.55

CPU Time for 12 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)


1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 13 1 4 1 5


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)


1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 2 7 2 9 3 1 3 3 3 5 3 7 3 9 4 1 4 3 4 5 4 7 4 9


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 Framed 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. Framed 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)


1 3 5 7 9 1 1 1 3 1 5 1 7 19 2 1 23 2 5 27 2 9

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

CONCLUSIONS

The proposed method is simple guarantees numerical stability converges fast.

An efficient Eigensolution technique !


42

Thank you for your attention.

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