* Hong-Ki Jo 1) , Kyu-Sik Park 2) , Hye-Rin Shin 3) and In-Won Lee 4)

*Hong-Ki Jo1), Kyu-Sik Park2), Hye-Rin Shin3) and In-Won Lee4)

1) ~ 3) Graduate Student, Department of Civil Engineering, KAIST 4) Professor, Department of Civil Engineering, KAIST

SIMPLIFIED ALGEBRAIC METHOD FOR COMPUTING EIGENPAIR SENSITIVITIES OF DAMPED SYSTEM

KKNN SeminarTaipei, Taiwan, Dec. 7-8, 2000

2Structural Dynamics and Vibration Control Lab., KAIST, Korea

OUTLINE

INTRODUCTION

PREVIOUS STUDIES

PROPOSED METHOD NUMERICAL EXAMPLE CONCLUSIONS


INTODUCTION• Objective of Study

• Applications of Sensitivity Analysis

- Determination of the sensitivity of dynamic responses

- Optimization of natural frequencies and mode shapes

- Optimization of structures subject to natural frequencies.

- To find efficient sensitivity method of eigenvalues and eigenvectors of damped systems.


)( 2 0KCM jjj

• Problem Definition

(1)

shape) (moder eigenvectocomplex th :

frequency) (natural eigenvaluecomplex th : definite-semi positive matrix, Stiffness :

damping classical-non matrix, Damping : definite positive matrix, Mass :

11

j

j

j

j

K

CKMKCMCM

n) ,2 1,( j

- Eigenvalue problem of damped system (N-space)


(2)

- Normalization condition

- State space equation (2N-space)

jj

jj

jj

j

00

0M

MCM

K

(3)1)2( 0

jiT

ijj

jT

ii

i

CM

MMC


jj , ,K ,C ,M K, C, M,

jj ,

Given:

Find:

- Objective

* indicates derivatives with respect to design variables (length, area, moment of inertia, etc.)

)(


PREVIOUS STUDIES

- many eigenpairs are required to calculate eigenvector derivatives. (2N-space)

,)( jjTjjλ BA

2/)(

)()(

])()([ )(

*

*

*

*

*

*

11

jTjjjjj

jj

Tjj

M

j

j

kj

Tkk

M

k

j

kj

Tkk

M

k

j

mjj

a

aa

ABAA

ABAAB

1M

0m

a

N

jk,1k

(4)

(5)

• Q. H. Zeng, “Highly Accurate Modal Method for Calculating Eigenvector Derivatives in Viscous Damping System,” AIAA Journal, Vol. 33, No. 4, pp. 746-751, 1995.


• Sondipon Adhikari, “Calculation of Derivative of Complex Modes Using Classical Normal Modes,” Computer & Structures, Vol. 77, No. 6, pp. 625-633, 2000.

- many eigenpairs are required to calculate eigenvector derivatives. (N-space) - applicable only when the elements of C are small.

N

k kjkj

kjTkj

jj

jTjjj

ji

kkkj

jiT

kkj

kj

jiTkkj

k

jjTjj

CiiC

where

FF

M

1*

)(*

*

))(()(

5.0

~)1(~)1(2

1

)(5.0

N

jk (6)


• I. W. Lee, D. O. Kim and G. H. Jung, “Natural Frequency and Mode Shape Sensitivities of Damped Systems: part I, Distinct Natural Frequencies,” Journal of Sound and Vibration, Vol. 223, No. 3, pp. 399-412, 1999.

• I. W. Lee, D. O. Kim and G. H. Jung, “Natural Frequency and Mode Shape Sensitivities of Damped Systems: part II, Multiple Natural Frequencies,” Journal of Sound and Vibration, Vol. 223, No. 3, pp. 413-424, 1999.


Lee’s method (1999)

jjjT

jj KCM 2

jjjT

j

jjjjjj

j

jT

j

jjjj

CMMKCMCM

CMCMKCM

25.0)()2(

00)2()2(

2

2

(7)

(8)

- the corresponding eigenpairs only are required. (N-space)- the coefficient matrix is symmetric and non-singular. - eigenvalue and eigenvector derivatives are obtained separately.


PROPOSED METHOD

)( 2 0KCM jjj n) ,2 1,( j

• Rewriting basic equations

1)2( jjTj CM

- Eigenvalue problem

- Normalization condition

(9)

(10)


• Differentiating eq.(9) with respect to design variable

jjj

jjjj

)(

)2( )(2

2

KCM

CMKCM


jjTj

jTjj

Tj

)2(5.0

)2(

CM

MCM

(11)

(12)

jj

jj


• Combining eq.(11) and eq.(12) into a single matrix

jjT

j

jjj

j

j

jT

jjT

j

jjjj

)2(5.0)(

)2()2(

2

2

CMKCM

MCMCMKCM

(13)

- the corresponding eigenpairs only are required. (N-space)- the coefficient matrix is symmetric and non-singular.- eigenpair derivatives are obtained simultaneously.eigenpair derivatives are obtained simultaneously.


NUMERICAL EXAMPLE• Cantilever beam with lumped dampers

1 : (A) areasection -Cross1 : (I) inertiasection -Cross

1 : )(density Mass1000 :(E) Modulus sYoung'0.3 :(c)damper Tangential

Design parameter : depth of beam

Material Properties System Data

Number of elements : 20

Number of nodes : 21

Number of DOF : 40

v1

v2

1 2 3 4 2119 20


• Analysis Methods

• Zeng’s method (1995)

• Lee’s method (1999)

• Proposed method

• Comparisons

• Solution time (CPU)


• Results of Analysis (Eigenvalue)

Modenumber

Eigenvalue Eigenvalue derivative

1 -0.0035 - 1.0868i 0.0010 - 0.2997i2 -0.0035 + 1.0868i 0.0010 + 0.2997i3 -0.0203 - 6.0514i 0.0072 - 1.3173i4 -0.0203 + 6.0514i 0.0072 + 1.3173i5 -0.0422 - 14.7027i 0.0140 - 2.4536i6 -0.0422 + 14.7027i 0.0140 + 2.4536i7 -0.0719 - 24.7343i 0.0189 - 3.1194i

8 -0.0719 + 24.7343i 0.0189 + 3.1194i

9 -0.1106 - 35.3632i 0.0213 - 3.4203i

10 -0.1106 + 35.3632i 0.0213 + 3.4203i


• Results of Analysis (First eigenvector)

DOFnumber Eigenvector Eigenvector derivative

1 0.0013 + 0.0013i -0.0004 - 0.0004i2 0.0050 + 0.0050i -0.0015 - 0.0015i

3 0.0049 + 0.0049i -0.0015 - 0.0015i

4 0.0096 + 0.0096i -0.0029 - 0.0029i

5 0.0108 + 0.0108i -0.0033 - 0.0032i6 0.0139 + 0.0139i -0.0042 - 0.0042i7 0.0188 + 0.0188i -0.0056 - 0.0056i8 0.0179 + 0.0178i -0.0054 - 0.0053i9 0.0287 + 0.0286i -0.0086 - 0.0085i

10 0.0215 + 0.0215i -0.0064 - 0.0064i


• CPU time for 40 Eigenpairs

Method CPU time Ratio

Lee’s method 2.21 1.4

Proposed method 1.59 1.0

(sec)

Zeng’s method 184.05 115.8


: Zeng’s method (Using full modes(40), exact solution)

: Zeng’s method (Using two modes(2), 5% error)

� : Lee’s method (Exact solution) : Proposed method(Exact solution)

Fig 1. Comparison with previous method

Δ

5 10 15 20 25 30 35 400

50

100

150

200

Modes

CPU

tim

e (s

ec)

Δ Δ Δ ΔΔ Δ

184.05

61.47Improvement about 99%

Δ 2.21

1.59


� : Lee’s method (Exact solution) : Proposed method(Exact solution)

Fig 2. Comparison with Lee’s method

Δ

5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

Modes

CPU

tim

e (s

ec)

Δ

ΔΔ

ΔΔ

ΔΔ

Improvement about 25% 2.21

1.59


CONCLUSIONS

• Proposed method- is composed of simple algorithm- guarantees numerical stability - reduces the CPU time.

An efficient eigen-sensitivity technique !


Thank you for your attention.


• Numerical Stability

)det()det()det()det( YAYYAY TT

• The determinant property

), ..., n-, i( oft independen be chosen to t vectorsindependenArbitary :

nn: ]....[

singular-Non:

eq.(13) ofmatrix t coefficien The : where

j

i

jn

121

1

1321

Ψ0

0ΨY

A

(14)

APPENDIX


rnonsingula , )1()1(:~

,0

~)( where 2

nn

jj

A

00AKCMT

1n : ~ ,1~)2(

,1

~)2(

bbCM

bCM

T

T

jTj

jj

Then

(15)

jT

jjT

j

jjjj

jT

jjT

j

jjjj

MΨCM

CMΨΨKCMΨ

ΨMCM

CMKCMΨYAYT

)2(

)2()(

1)2()2(

1

T2T

2T


Arranging eq.(15)

MT1~

10

~~

T

T

b0

b0AYAY

0 )A~(det

~~~1

10det)A~(det

Y)A(Ydet

1

T

bA

bM T

T

(16)

Using the determinant property of partitionedmatrix

(17)


0A)(det

Therefore

Numerical Stability is Guaranteed.Numerical Stability is Guaranteed.

(18)


• Lee’s method (1999)


(19)

• Pre-multiplying each side of eq.(19) by gives eigenvalue derivative.

jjjT

jj KCM 2

Tj

jjjjjj

jjj

)()2(

)(2

2

KCMCM

KCM

(20)



jjjTj

jjTj

CMM

CM

)(25.0

)2((21)

jjjT

j

jjjjjj

j

jT

j

jjjj

CMMKCMCM

CMCMKCM

25.0)()2(

00)2()2(

2

2

• Combining eq.(19) and eq.(21) into a matrix gives eigenvector derivative.

(22)

Documents

* Hong-Ki Jo 1) , Kyu-Sik Park 2) , Hye-Rin Shin 3) and In-Won Lee 4)