Yeong-Jong Moon1), Jong-Heon Lee2) and In-Won Lee3)

1) Graduate Student, Department of Civil Engineering, KAIST

2) Professor, Department of Civil Engineering, Kyungil Univ.

3) Professor, Department of Civil Engineering, KAIST

Yeong-Jong Moon1), Jong-Heon Lee2) and In-Won Lee3)

1) Graduate Student, Department of Civil Engineering, KAIST

2) Professor, Department of Civil Engineering, Kyungil Univ.

3) Professor, Department of Civil Engineering, KAIST

Modified Modal Methods for Calculating

Eigenpair Sensitivity of Asymmetric Dam

ped Systems

Modified Modal Methods for Calculating

Eigenpair Sensitivity of Asymmetric Dam

ped Systems

Fifth European Conference of Structural DynamicsEURODYN 2002Munich, GermanySept. 2 - 5, 2002

Contents

Introduction

Previous Studies

Proposed Methods

Numerical Example

Conclusions

3 3

Introduction

• Applications of Sensitivity Analysis

- Determination of the sensitivity of dynamic response

- Optimization of natural frequencies and mode shapes

- Optimization of structures subject to natural frequencies

• Many sensitivity techniques for symmetric systems have

been developed

4 4

• For asymmetric systems, few sensitivity technique has

been developed

• Many real systems have asymmetric mass, damping

and stiffness matrices.

- moving vehicles on roads

- ship motion in sea water

- offshore structures

5 5

jjj yzs , ,

,K ,C ,M

K, C, M,

,,,

,,, , , jjj yzs

Given:

Find:

• Sensitivity Analysis

:

,,,

jj

jj

jj

yy

zz

sswhere

Design parameter

6 6

- Propose the modal method for sensitivity technique of

symmetric system

- The accuracy is dependent on the number of modes used

in calculation

• K. B. Lim and J. L. Junkins, “Re-examination of Eigenvector

Derivatives”, Journal of Guidance, 10, 581-587, 1987.

Previous Studies

• Conventional Modal Method for Symmetric System

7 7

- Modified modal method for symmetric system

- This method achieved highly accurate results using

only a few lower modes.

• Q. H. Zeng, “Highly Accurate Modal Method for Calculating

Eigenvector Derivative in Viscous Damping Systems”, AIAA

Journal, 33(4), 746-751, 1994.

• Modified Modal Method for Symmetric System

8 8

- Propose the modal method for sensitivity technique of

asymmetric system

- The accuracy is dependent on the number of modes used

in calculation

- The truncation error may become significant

• S. Adhikari and M. I. Friswell, “Eigenderivative Analysis of

Asymmetric Non-Conservative Systems”, International Journal

for Numerical Methods in Engineering, 51, 709-733, 2001.

• Conventional Modal Method for Asymmetric System

9 9

– Expand and as complex linear combinations of

and

N

kkjkj zaz

2

1,

N

kkjkj yby

2

1, (2

)

(1)

,jz ,jy

jz jy

• Basic Idea of Modal Method

where

,jz

: the j-th right eigenvector: the j-th left eigenvector: the derivatives of j-th right eigenvector: the derivatives of j-th left eigenvector,jy

jz

jy

10 10

jjjjjjj

Tjj

N

jkk kjk

Tkk

kjk

Tkk

j

zafsss

yz

sss

yz

sss

yzz

)(2

)(

)(2

)(

)(2

**

*

,1**

*

,

jjjjjjj

Tjj

N

jkk kjk

Tkk

kjk

Tkk

j

ybgsss

zy

sss

zy

sss

zyy

)(2

)(

)(2

)(

)(2

**

*

,1**

*

,

(3)

(4)

- The derivatives of right eigenvectors

- The derivatives of left eigenvectors

• From this idea, the eigenvector derivatives can be obtained

11 11

• Objective

- Develop the effective sensitivity techniques for

asymmetric damped systems

12 12

Proposed Methods

1. Modal Acceleration Method

2. Multiple Modal Acceleration Method

3. Multiple modal Acceleration Method

with Shifted Poles

1. Modal Acceleration Method

2. Multiple Modal Acceleration Method

3. Multiple modal Acceleration Method

with Shifted Poles

13 13

0)( zBsA

fzBsAAszBsA ,,,,)(

• Differentiate the Eq. (5) with a design parameter

0)( BsAyT

(5)

(6)

(7)

1. Modal Acceleration Method (MA)

• The general equation of motion for asymmetric systems

14 14

00, ds ddz

fBd s1

0

fYs

s

sssZ

fBfBsAd

T

kkk

d

)(2

1

)( 110

where

(8)

(9)

(10)

• Separate the response into and,z 0sd 0dd

15 15 jjjj

jjj

Tjj

j

j

N

jkk kjk

Tkk

k

j

kjk

Tkk

k

jTj

ybgsss

zy

s

s

sss

zy

s

s

sss

zy

s

sBy

)(2

)(

)(2

)(

)(2)(

**

*

*

,1**

*

*1

,

jjjjjjj

Tjj

j

j

N

jkk kjk

Tkk

k

j

kjk

Tkk

k

jj

zafsss

yz

s

s

sss

yz

s

s

sss

yz

s

sBz

)(2

)(

)(2

)(

)(2

**

*

*

,1**

*

*1

,

• Substituting the Eq. (9) and (10) into the Eq. (8)

• By the similar procedure, the left eigenvector derivatives can be obtained

(11)

(12)

k

j

s

s

*k

j

s

s

k

j

s

s

*k

j

s

s

16 16

2. Multiple Modal Acceleration Method (MMA)

11, ds ddz

fsABIBd s ][ 111

fYs

s

sssZ

dfBsAdzd

T

kkk

ssd

2

11

1,1

)(2

1

)(

where

(13)

(14)

(15)

• Separate the response into and,z1sd 1dd

17 17

• Therefore the right eigenvector derivatives are given as

jjjjjjj

Tjj

j

j

kjk

Tkk

k

j

N

jkk kjk

Tkk

k

jjj

zafsss

yz

s

s

sss

yz

s

s

sss

yz

s

sABsIBz

)(2

)(

)(2

)(

)(2)(

**

*2

***

*2

*

,1

2

11,

• By the similar procedure,

jjjjjjj

Tjj

j

j

kjk

Tkk

k

j

N

jkk kjk

Tkk

k

jTTj

Tj

ybgsss

zy

s

s

sss

zy

s

s

sss

zy

s

sBAsIBy

)(2

)(

)(2

)(

)(2)(

**

*2

***

*2

*

,1

2

,

(16)

(17)

2

k

j

s

s

2

*

k

j

s

s

2

k

j

s

s

2

*

k

j

s

s

18 18

jjjjjjj

Tjj

M

j

j

kjk

Tkk

M

k

j

N

jkk kjk

Tkk

M

k

jM

m

mjj

zafsss

yz

s

s

sss

yz

s

s

sss

yz

s

sABsBz

)(2

)(

)(2

)(

)(2)(

**

*

***

*

*

,1

1

0

11,

• Based on the similar procedure, we can obtain the higher order equations

jjjjjjj

Tjj

M

j

j

kjk

Tkk

M

k

j

N

jkk kjk

Tkk

M

k

jM

m

mTTj

Tj

ybgsss

zy

s

s

sss

zy

s

s

sss

zy

s

sBAsBy

)(2

)(

)(2

)(

)(2)(

**

*

***

*

*

,1

1

0,

(18)

(19)

M

k

j

s

s

M

k

j

s

s

*

M

k

j

s

s

M

k

j

s

s

*

19 19

1

0

11

111

11

])()([)(B

]))(([)(B

)])(([()(

M

m

mj

j

jj

ABAsA

AABsIA

AsABBAs

3. Multiple Modal Acceleration with Shifted-Poles (MMAS)

• For more high convergence rate, the term is expanded in Taylor’s series at the position

(20)

1)( BAs j

20 20

• Using the Eq. (20), we can obtain the following equation

(21)

jjjjjjj

Tjj

M

j

j

kjk

Tkk

M

k

j

N

jkk kjk

Tkk

M

k

j

M

m

mjj

zafsss

yz

s

s

sss

yz

s

s

sss

yz

s

s

ABAsABz

)(2

)(

)(2

)(

)(2

])()([)(

**

*

*

**

*

*

,1

1

0

11,

M

k

j

s

s

M

k

j

s

s

*

21 21

jjjjjjj

Tjj

M

j

j

kjk

Tkk

M

k

j

N

jkk kjk

Tkk

M

k

j

M

m

mTTj

Tj

ybgsss

zy

s

s

sss

zy

s

s

sss

zy

s

s

ABAsABy

)(2

)(

)(2

)(

)(2

])()([)(

**

*

*

**

*

*

,1

1

0,

(22)

• By the similar procedure

M

k

j

s

s

M

k

j

s

s

*

22 22

M

L

Y

X

Z, z

x

y

t

Figure 1. The whirling beam

L. Meirovitch and G. Ryland, “A Perturbation Technique for Gyroscopic Systems with Small Internal and External Damping,” Journal of Sound and Vibration, 100(3), 393-408, 1985.

Numerical Example

23 23

)()()()()()( tttt FuHKuGCuM

matrix ycirculator :H

matrix gyroscopic :G

0H

H0H,

K0

0KK

,0G

G0G,

C0

0CC,

M0

0MM

12

12

22

11

12

12

22

11

22

11

• Equation of motion

where

24 24

5 ,rad6.21

,5/9EI ,5/4EI ,4/1

,20/KK ,5 ,10M ,/10

1

2232231

2210

ps

NmLNmLNsmhc

NmLmLkgmkgm

yx

Design parameter : L

• Material Property

25 25

Mode Number Eigenvalues Derivatives

1-8.4987e-03

+2.3563e+00i1.3251e-03

+1.5799e+00i

2-2.7151e-03

+6.3523e+01i2.2533e-03

+8.5934e-01i

31.6771e-02

+1.0548e+01i3.3394e-03

+3.4034e-01i

8-5.8579e-02

+1.8650e+01i -3.7909e-03 -3.3918e-01i

9-4.7285e-02

+2.2774e+01i -2.2533e-03 -8.2215e-01i

10-3.6890e-02

+2.6214e+01i -1.2833e-03

-1.0644e+00i

• Eigenvalues and their derivatives of system

26 26

DOF Number Eigenvector Derivative

16.0874e-03

-6.2442e-06i 6.3118e-04

+6.3342e-06i

20.0000e+00

+0.0000e+00i 0.0000e+00

+0.0000e+00i

3-7.4415e-03

+6.7358e-06i -7.6005e-04 -7.1917e-06i

8+1.4785e-05 -1.4677e-02i

-1.2799e-05 +5.9162e-03i

90.0000e+00

+0.0000e+00i 0.0000e+00

+0.00005e+00i

108.3733e-05

-5.7187e-02i -3.7941e-05

+1.6957e-02i

• First right eigenvector and its derivative

27 27

DOF Number

Error (%)

MA MMA MMAS

1 0.831 0.202 0.072

2 0.000 0.000 0.000

3 38.258 1.506 0.478

4 0.000 0.000 0.000

5 4.631 0.121 0.035

6 0.080 0.053 0.012

7 0.000 0.000 0.000

8 1.679 0.588 0.118

9 0.000 0.000 0.000

10 0.520 0.157 0.030

• Errors of modified modal methods using six modes (%)

• MA : Modal Acceleration Method

• MMA : Multiple Modal

Acceleration Method (M=2)

• MMAS : Multiple Modal Accelerations

with Shifted Poles

(M=2, =eigenvalue –1)

28 28

DOF Number

Error (%)

6 modes 4 modes 2 modes

1 0.072 3.406 2.090

2 0.000 0.000 0.000

3 0.478 0.454 3.140

4 0.000 0.000 0.000

5 0.035 0.035 0.052

6 0.012 0.626 0.383

7 0.000 0.000 0.000

8 0.118 0.114 0.542

9 0.000 0.000 0.000

10 0.030 0.030 0.038

• Errors of MMAS method using 2, 4 and 6 lower modes (%)

(M=2, =eigenvalue –1)

29 29

• The modified modal methods for the eigenpair derivatives of asymmetric damped systems is derived

• In the proposed methods, the eigenvector derivatives of

asymmetric systems can be calculated by using only a few

lower modes

• Multiple modal acceleration method with shifted poles

is the most efficient technique of proposed methods

Conclusions