Solution of Eigenvalue Problems for Non-Proportionally Damped Systems with Multiple Frequencies

In-Won Lee, Professor, PEIn-Won Lee, Professor, PE Structural Dynamics & Vibration Control Lab.Structural Dynamics & Vibration Control Lab. Korea Advanced Institute of Science & Technology Korea

Fourth World Congress on Computational MechanicsFourth World Congress on Computational MechanicsBuenos Aires, ArgentinaBuenos Aires, ArgentinaJune 30, 1998June 30, 1998 Session V-C : Structural Dynamics ISession V-C : Structural Dynamics I

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OUTLINE Introduction

Objectives and scope Current methods

Proposed method Newton-Raphson technique Modified Newton-Raphson technique

Numerical examples Grid structure with lumped dampers Three-dimensional framed structure with lumped dampers

Conclusions

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INTRODUCTION

Free vibration of proportional damping system

(1)

where : Mass matrix

: Damping matrix

: Stiffness matrix

: Displacement vector

0)()()( tuKtuCtuM

M

C

K)(tu

Objectives and Scope

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Eigenanalysis of proportional damping system

where : Real eigenvalue

: Natural frequency

: Real eigenvector(mode shape)

Low in cost Straightforward

niMK iii ,,2,1 (2)2ii

ii

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Free vibration analysis of non-proportional damping system

(4)

M

KA

0

0

where

0M

MCB

tu

tuz

0 tAztzB (3)

(5) tetu Let

tt eetz

(6)

, then

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Therefore, an efficient eigensolution technique fornon-proportional damping system is required.

nmiBA iii 2,,2,1 (7)

(9): Orthogonality of eigenvector

mjiB ijjTi ,,2,1,

: Eigenvalue(complex conjugate)

: Eigenvector(complex conjugate)

i

(8)

ii

ii

where

Solution of Eq.(7) is very expensive.

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Current Methods 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

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PROPOSED METHOD Find p smallest multiple eigenpairs

p 21

iii BA Solve

ijjTi B Subject to

For i iand pi ,,2,1 : multiple

BA

pT IB

where

p ,,, 21

pdiag ,,, 21

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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

X pxxxX ,,, 21

(10)

(11)

B

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ZDZ

where pdddD ,,, 21 : Symmetric

Let (13)

Introducing Eq.(10) into Eq.(7)

BXZAXZ (12)

BXDZAXZ

BXDAX

or piBXdAx ii ,,2,1

Then

or

(14)

(15)

(16)

Note : If , from Eq.(13)p 21

D

X

(17)(18)

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pidXBxA ki

kki ,,2,10)1()1()1(

pkTk IXMX )1()1( )(

(19)

(20)

)()()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

(21)

(22)

(23)

Newton-Raphson Technique

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)()()()( ki

kki

ki dXBAxr where : residual vector

)()()()()()( ki

ki

kki

kii

ki rdBXxBdxA

0)( )()( ki

Tk xBX

(23)

(24)

Introducing Eqs.(21) and (22) into Eqs.(19) and (20)

and neglecting nonlinear terms

Matrix form of Eqs.(23) and (24)

pi

r

d

x

X

BXBdA ki

ki

ki

Tk

kkii

,,2,1

00B)(

)(

)(

)(

)(

)()(

(25)

Coefficient matrix : • Symmetric• Nonsingular

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pi

r

d

x

X

BXBdA ki

ki

ki

Tk

kkii

,,2,1

00B)(

)(

)(

)(

)(

)()(

Coefficient matrix : • Symmetric• Nonsingular

(25)

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

(26)

(22)

(21)

Modified Newton-Raphson Technique

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Step

Step 2: Solve for and )(kix )(k

id

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(

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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: Multiple case

)(lim k

kX

)(lim k

kDi

kii

kd

)(lim

ik

ik

x

)(lim

or

or

2

)(2

)()(

ki

ki

kii

xA

xBdA

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NUMERICAL EXAMPLES

Structures Grid structure with lumped dampers Three-dimensional framed structure with lumped d

ampers Analysis methods

Proposed method Subspace iteration method (Leung 1988) Lanczos method (Chen 1993)

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Comparisons Solution time(CPU) Convergence

Error norm =

Convex with 100 MIPS, 200 MFLOPS

2

)(2

)()(

ki

ki

kii

xA

xBdA

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Grid Structure with Lumped Dampers 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

Maximum Half Bandwidths :15

Mean Half Bandwidths :14

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Proposed MethodMode

Number

Error Norm ofStarting Eigenpair(Lanczos method)

Number ofIterations

Eigenvalue Error Norm

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

111100001111

-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

0.824521E-100.805278E-100.824521E-100.805278E-100.170965E-060.823644E-060.170965E-060.823644E-060.344167E-100.432693E-090.344167E-100.432693E-09

Table 1. Results of proposed method for grid structure

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Method CPU time in seconds Ratio

Proposed method(Lanczos method + Iteration scheme)

Subspace Iteration Method

872.67(214.28 + 658.39)

3,096.62

1.00

3.55

Table 2. CPU time for twelve lowest eigenpairs of grid structure

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0 1 2 3

Iteration Number

1.0E-11

1.0E-10

1.0E-9

1.0E-8

1.0E-7

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

Err

or

No

rm

Error L im it

Lanczos method(48 Lanczos vectors)

Fig.2 Error norms of grid model by proposed method

0 5 10 15 20 25 30 35 40 45 50

Iteration Number

1.0E-8

1.0E-7

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

Err

or

No

rm

Error L im it

Fig.3 Error norms of grid model by subspace iteration method

12 24 36 48 60 72 84 96 108

Num ber of Generated Lanczos Vectors

1.0E-10

1.0E-9

1.0E-8

1.0E-7

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

Err

or

No

rm

E rror L im it

1st, 3rd eigenpairs

2nd, 4th eigenpairs

5th, 7th eigenpairs

6th, 8th eigenpairs

9th, 11th eigenpairs

10th, 12th eigenpairs

Fig.4 Error norms of grid model by Lanczos method

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Three-Dimensional Framed Structure with Lumped Dampers

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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

Maximum Half Bandwidths :300

Mean Half Bandwidths :120

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Proposed MethodMode

Number

Error Norm ofStarting Eigenpair(Lanczos method)

Number ofIterations

Eigenvalue Error Norm

123456789101112

0.334872E-070.602910E-060.334872E-070.602910E-060.256380E-050.497414E-040.256380E-050.497414E-040.117746E-060.374128E-040.117746E-060.374128E-04

000022221111

-0.13811 + j 3.09308-0.13811 + j 3.09308-0.13811 - j 3.09308-0.13811 - j 3.09308-3.53017 + j 2.20867-3.53017 - j 2.20867-0.24297 + j 4.16980-0.24297 - j 4.16980-1.65509 + j 7.04244-1.65509 + j 7.04244-1.65509 - j 7.04244-1.65509 - j 7.04244

0.334872E-070.602910E-060.334872E-070.602910E-060.308387E-070.132855E-060.308387E-070.132855E-060.482385E-110.390433E-060.482385E-110.390433E-06

Table 3. Results of proposed method for three-dimensional framed structure

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Method CPU time in seconds Ratio

Proposed method(Lanczos method + Iteration scheme)

Subspace Iteration Method

7,641.94(1,006.73 + 6,635.21)

8,337.60

1.00

1.09

Table 4. CPU time for twelve lowest eigenpairs of three-dimensional framed structure

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0 1 2 3 4

Iteration Number

1.0E-12

1.0E-11

1.0E-10

1.0E-9

1.0E-8

1.0E-7

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

Err

or

No

rm

Error L im it

Lanczos method(48 Lanczos vectors)

Fig.6 Error norms of 3-D. frame model by proposed method

0 4 8 12 16 20 24 28

Iteration Number

1.0E-8

1.0E-7

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

Err

or

No

rm

Error L im it

Fig.7 Error norms of 3-D. frame model by subspace iteration method

12 24 36 48 60 72 84 96 108

Num ber of Generated Lanczos Vectors

1.0E-12

1.0E-11

1.0E-10

1.0E-9

1.0E-8

1.0E-7

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

Err

or

No

rm

Error L im it

1st, 3rd eigenpairs

2nd, 4th eigenpairs

5th, 7th eigenpairs

6th, 8th eigenpairs

9th, 11th eigenpairs

10th, 12th eigenpairs

Fig.8 Error norms of 3-D. frame model by Lanczos method

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CONCLUSIONS Proposed method

converges fast guarantees nonsingularity of coefficient matrix

Proposed method is efficient

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Thank you for your attention.