*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

3Structural Dynamics and Vibration Control Lab., KAIST, Korea

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.

4Structural Dynamics and Vibration Control Lab., KAIST, Korea

)( 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)

5Structural Dynamics and Vibration Control Lab., KAIST, Korea

(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

6Structural Dynamics and Vibration Control Lab., KAIST, Korea

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

jj ,

Given:

Find:

- Objective

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

)(

7Structural Dynamics and Vibration Control Lab., KAIST, Korea

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.

8Structural Dynamics and Vibration Control Lab., KAIST, Korea

• 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)

9Structural Dynamics and Vibration Control Lab., KAIST, Korea

• 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.

10Structural Dynamics and Vibration Control Lab., KAIST, Korea

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.

11Structural Dynamics and Vibration Control Lab., KAIST, Korea

PROPOSED METHOD

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

• Rewriting basic equations

1)2( jjTj CM

- Eigenvalue problem

- Normalization condition

(9)

(10)

12Structural Dynamics and Vibration Control Lab., KAIST, Korea

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

jjj

jjjj

)(

)2( )(2

2

KCM

CMKCM

• Differentiating eq.(10) with respect to design variable

jjTj

jTjj

Tj

)2(5.0

)2(

CM

MCM

(11)

(12)

jj

jj

13Structural Dynamics and Vibration Control Lab., KAIST, Korea

• 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.

14Structural Dynamics and Vibration Control Lab., KAIST, Korea

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

15Structural Dynamics and Vibration Control Lab., KAIST, Korea

• Analysis Methods

• Zeng’s method (1995)

• Lee’s method (1999)

• Proposed method

• Comparisons

• Solution time (CPU)

16Structural Dynamics and Vibration Control Lab., KAIST, Korea

• 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

17Structural Dynamics and Vibration Control Lab., KAIST, Korea

• 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

18Structural Dynamics and Vibration Control Lab., KAIST, Korea

• 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

19Structural Dynamics and Vibration Control Lab., KAIST, Korea

: 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

20Structural Dynamics and Vibration Control Lab., KAIST, Korea

� : 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

21Structural Dynamics and Vibration Control Lab., KAIST, Korea

CONCLUSIONS

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

An efficient eigen-sensitivity technique !

22Structural Dynamics and Vibration Control Lab., KAIST, Korea

Thank you for your attention.

23Structural Dynamics and Vibration Control Lab., KAIST, Korea

• 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

24Structural Dynamics and Vibration Control Lab., KAIST, Korea

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

25Structural Dynamics and Vibration Control Lab., KAIST, Korea

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)

26Structural Dynamics and Vibration Control Lab., KAIST, Korea

0A)(det

Therefore

Numerical Stability is Guaranteed.Numerical Stability is Guaranteed.

(18)

27Structural Dynamics and Vibration Control Lab., KAIST, Korea

• Lee’s method (1999)

• Differentiating eq.(1) with respect to design variable

(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)

28Structural Dynamics and Vibration Control Lab., KAIST, Korea

• Differentiating eq.(3) with respect to design variable

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)