kang-min choi, graduate student, kaist, korea young-jong moon , graduate student, kaist, korea

Kang-Min Choi, Kang-Min Choi, Graduate Student, KAIST, Korea

Young-Jong MoonYoung-Jong Moon, Graduate Student, KAIST, Korea

Ji-Eun JangJi-Eun Jang, Graduate Student, KAIST, Korea

Woon-Hak Kim, Professor, Hankyong National University, Korea

In-Won LeeIn-Won Lee, Professor, KAIST, Korea

Algebraic Method for Eigenpair Derivatives of Damped System

with Repeated Eigenvalues

Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 22

OUTLINE

INTRODUCTION

PREVIOUS METHODS

PROPOSED METHOD

NUMERICAL EXAMPLE

CONCLUSIONS


INTRODUCTION

determination of the sensitivity of dynamic response optimization of natural frequencies and mode shapes optimization of structures subject to natural frequencies

Applications of sensitivity analysis are

Typical structures have many repeated or nearly equaleigenvalues, due to structural symmetry.

The second- and higher order derivatives of eigenpairsare important to predict the eigenpairs, which relieson the matrix Taylor series expansion.


)( 2 0KCM

Problem Definition Problem Definition

(1)

K

C

M

Eigenvalue problem of damped system

shape) (moder eigenvecto :

frequency) (natural eigenvalue :

definite semi-positive matrix, Stiffness :

matrix Damping :

definite positive matrix, Mass :


,

,,, K ,C ,M

K C, M,

,, ,

Given:

Find:

* represents the derivative of with respect design parameter α (length, area, moment of inertia, etc.)

,)( )(

Objective of this study: Objective of this study:

,, ,


PREVIOUS STUDIES

Damped system with distinct eigenvalues K. M. Choi, H. K. Jo, J. H. Lee and I. W. Lee, “Sensitivity

Analysis of Non-conservative Eigensystems,” Journal of Sound and Vibration, 2001. (Accepted)

jjj

jj

j

j

jjjj

jjjj

)CM2(5.0

)KCM(

M)CM2(

)CM2(KCM

,,

,,,

,

,2

T

T

TT

- The coefficient matrix is symmetric and non-singular.

- Eigenpair derivatives are obtained simultaneously.

- The algorithm is simple and guarantees stability.


Undamped system with repeated eigenvalues R. L. Dailey, “Eigenvector Derivatives with Repeated

Eigenvalues,” AIAA Journal, Vol. 27, pp.486-491, 1989.

- Introduction of Adjacent eigenvector

- Calculation derivatives of eigenvectors by the sum of homogenous solutions and particular solutions using Nelson’s algorithm

- Complicated algorithm and high time consumption


Second order derivatives of Undamped systemwith distinct eigenvalues

M. I. Friswell, “Calculation of Second and Higher Eigenvector Derivatives”, Journal of Guidance, Control and Dynamics, Vol. 18, pp.919-921, 1995.

j,jjjj,jjj

jjjjjj

)MMK()MMK(

)MMMK(

,,,,,,

,,,,,,,

TT

T

(3)

jjjj cv,

where j,jj,jjjj ,,,T )MK()MK(M5.0c

(4)

- Second eigenvector derivatives extended by Nelson’s algorithm


PROPOSED METHOD

First-order eigenpair derivatives of damped systemwith repeated eigenvalues

Second-order eigenpair derivatives of damped systemwith repeated eigenvalues

Numerical stability of the proposed method


First-order eigenpair derivatives of damped systemwith repeated eigenvalues

Basic Equations

• Eigenvalue problem

(5)

(6)

0KCM 2 mmmmm

mmmm I)CM2(T

• Orthonormalization condition

mm

m

seigenvalue repeated ofy mutiplicit ofnumber the:

rseigenvecto ofmatrix the

( seigenvalue ofmatrix the

:

)I: mmm


TX mm

mITTT

(7)

where T is an orthogonal transformation matrixand its order m

(8)

Adjacent eigenvectors


• Rearranging eq.(5) and eq.(6) using adjacent eigenvectors

0KXXCXM 2 mmmmm

mmmm IX)CM2(XT

(9)

(10)


• Differentiating eq.(9) w.r.t. design parameter α

mmm

mmmmmm

X)KCM(

X)CM2(X)KCM(

,,,2

,,2

(11)

• Pre-multiplying at each side of eq.(11) by and

substituting

Tm

TX mm

,TDT m

mmmm )KCM(D ,,,2T

(12)

where



(13)

(14)

mmm

mmmm

X)KCM(

X)CM2()KCM(

,,,2

2

,Xm ,m


mmm

mmmm

X)CM2(X5.0

XMX )CM2(X

,,T

TT

,Xm ,m


mmm

mmm

m

m

mmmm

mmmm

X)CM2(X5.0

X)KCM(

X

XMX)CM2(X

X)CM2(KCM

,,T

,,,2

,

,

TT

2

(15)

- It maintains N-space without use of state space equation.


- It requires only corresponding eigenpair information.

- Numerical stability is guaranteed.

- It maintains N-space without use of state space equation.


- It requires only corresponding eigenpair information.

- Numerical stability is guaranteed.

• Combining eq.(13) and eq.(14) into a single equation


Second-order eigenpair derivatives of damped systemwith repeated eigenvalues

• Differentiating eq.(13) w.r.t. another design parameter β

(16)

(17)

mmmmmmmmm

mmmmmmmm

mmmm

MX2X)G~

G~

F~~

(

X)GF~

(X)GF~

(

X)CM2()KCM(

,,,,,,,

,,,,,,

2

,Xm ,m

• Differentiating eq.(14) w.r.t. another design parameter β

mmmmm

mmmm

mmmmmmm

mmmm

X)M2M2G~

(X5.0

X)M2G~

(X

X)M2G~

(XXGX

MXX)CM2(X

,,,,,T

,,,T

,,,T

,T

,

TT

,Xm ,m


• Combining eq.(16) and eq.(17) into a single equation

mmmmm

mmmmmmmmmmm

mmmmmm,mm,m

mmmmmmmm

m

m

mmmm

mmmm

X)M2M2G~

(X5.0

X)M2G~

(XX)M2G~

(XXGX

MX2X)G~

G~

F~~

(

X)GF~

(X)GF~

(

X

MXXCM2(X

X)CM2(KCM

,,,,,T

,,,T

,,,T

,T

,

,,,,,

,,,,,,

,

,

TT

2

(18)

where

,,,2

,

,,,2

,

2

KCMF~~

KCMF~

KCMF

mmm

mmm

mmm

,,,

,,,

CM2G~~

CM2G~

CM2G

mm

mm

mm

mmm

mmm

m

m

mmmm

mmmm

X)CM2(X5.0

X)KCM(

X

XMX)CM2(X

X)CM2(KCM

,,T

,,,2

,

,

TT

2


Numerical stability of the proposed method

)det()det()det()det( YAYYAY *T*T

Determinant property

x

]xxx[X]xxx

I0

0Y

A

m21m21mn2

m

r eigenvectoadjacent oft independen

be chosen to t vectorsindependenArbitrary :

when [

singular-non :

eq.(15) ofmatrix t coefficien The : where

i

1

*

(19)


00

0A~

)KCM( 2Tmm where

T

T

T

I

B~

)CM2(X

I

B~

X)CM2(

m

mm

m

mm

Then,

(20)

mmmm

mmmm

mmmmm

mmmm

m

XMX)CM2(X

X)CM2()KCM(

I

Ψ

XMX)CM2(X

X)CM2(KCM

IYAY

T

T2T

TT

2*T

T

(21)


• Arranging eq.(20)

mmm

m

MXXIB~

I00

B~

0A~

YAYTT

*T

0)A

~(det

B~

0A~

B~0

MXXI

I0det)A

~(det

Y)A(Ydet

1

TT

*T

mmm

m

(22)

• Using the determinant property of partitioned matrix

(23)


0)A(det *

Therefore

Numerical Stability is Guaranteed.Numerical Stability is Guaranteed.

(24)


80 : DOF ofNumber

20 : elements ofNumber

21 : nodes ofNumber

Data System

0.001 :dampingRayleigh

kg/m 10.887 :density Mass

N/m 102.10E : Modulus sYoung'33

211

Properties Material

)01.0w/ww ( : parameter Design*

z

y

z

mL 10 mw 5.0

mh 05.0x

x

z

y

1

2

3 4

Cantilever beam

NUMERICAL EXAMPLE


Results of analysis (eigenvalues)

Modenumber

EigenvaluesFirst derivatives

of eigenvaluesSecond derivatives

of eigenvalues

1, 2-1.4279e-03

±j5.2496e+00-2.8057e-10

j3.5347e-104.3916e-09

±j1.0285e-08

3, 4-1.4279e-03

±j5.2496e+00-2.2756e-02

±j5.2494e+01-2.7553e-01

j6.1102e-02

5, 6-5.4154e-02

±j3.2895e+01-6.6265e-10±j2.3445e-10

1.0084e-08 j2.4918e-09

7, 8-5.4154e-02

±j3.2895e+01-1.0818e+00±j3.2886e+02

-1.0391e-08 j2.6913e+00

9, 10-4.2409e-01

±j9.2090e+016.9247e-10

j6.9600e-10-1.0391e-08±j1.1514e-08

11, 12-4.2409e-01

±j9.2090e+01-8.4753e+00±j9.2029e+02

-8.4535e+01 j1.8358e+01

±

±

±

±

±

±


Results of analysis (first eigenvectors)

DOFnumber

EigenvectorsFirst derivatives of eigenvectors

Second derivatives of eigenvectors

1 0 0 0

2 0 0 0

3-6.6892e+05-j6.6892e+05

3.3446e-04+j3.3446e-04

-5.0169e+03-j5.0169e+03

4-2.6442e+04-j2.6442e+04

1.3221e-03+j1.3221e-03

-1.9596e+02-j1.9596e+02

77 0 0 0

78 0 0 0

79-1.5577e+02-j1.5577e+02

7.7887e-02+j7.7887e-02

-1.1683e+00-j1.1683e+00

80-2.1442e+03-j2.1442e+03

1.0721e-02+j1.0721e-02

-1.6082e+01-j1.6082e+01

… … … …


Results of analysis (errors of approximations)

Mode number

ActualEigenvalues

Approximated eigenvalues

Variations of eigenpairs Errors of approximations

Eigenvalues Eigenvectors Eigenvalues Eigenvectors

1, 2-1.4279e-03

±j5.2496e+00-1.4279e-03

±j5.2496e+002.2281e-11 4.9628e-03 2.2283e-11 3.7376e-05

3, 4-1.4556e-03

±j5.3021e+00-1.4555e-03

±j5.3021e+001.0000e-02 9.9010e-03 2.6622e-08 1.0000e-04

5, 6-5.4154e-02

±j3.2895e+01-5.4154e-02

±j3.2895e+013.7084e-12 4.9628e-03 3.6899e-12 3.7376e-05

7, 8-5.5241e-02

±j3.3224e+01-5.5236e-02

±j3.3224e+019.9997e-04 9.9023e-03 1.6763e-07 1.0001e-04

9, 10-4.2409e-01

±j9.2090e+01-4.2409e-01

±j9.2090e+019.1400e-12 4.9628e-03 9.1432e-12 3.7376e-05

11, 12-4.3261e-01

±j9.3010e+01-4.3256e-01

±j9.3010e+019.9936e-03 9.9041e-03 4.6508e-07 1.0002e-04


CONCLUSIONS

Proposed Method Proposed Method

is an efficient eigensensitivity method for the damped system with repeated eigenvalues

guarantees numerical stability

gives exact solutions of eigenpair derivatives

can be extended to obtain second- and higher order derivatives of eigenpairs


An efficient eigensensitivity method for the An efficient eigensensitivity method for the

damped system with repeated eigenvaluesdamped system with repeated eigenvalues

Thank you for your attention!Thank you for your attention!

kang-min choi, graduate student, kaist, korea young-jong moon , graduate student, kaist, korea

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