대한토목공학회 추계 학술발표회 대구 2003 년 10 월 24 일 t. x. nguyen,...

대한토목공학회 추계 학술발표회대구2003 년 10 월 24 일

T. X. Nguyen, 한국과학기술원 건설 및 환경공학과 박사과정김병완 , 한국과학기술원 건설 및 환경공학과 박사후연구원정형조 , 세종대학교 토목환경공학과 교수이인원 , 한국과학기술원 건설 및 환경공학과 교수

Application of Step Length Technique to an Eigensolution Method for Non-proportionally

Damped Systems

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Review Proposed method Numerical example Conclusion

Contents

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Review Eigenvalue problem receives much attention in dynamic analysis:

To avoid resonance To obtain dynamic characteristics

Problem statement: Free Vibration of a LTI system of order n

Quadratic Eigenvalue Problem

(1)

(2)M, C, K: (n x n) system matrices

u: (n x 1) displacement vector

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Transformation methodsQR (Moler and Stewart)LZ (Kaufman)Jacobi (Veselic)…

Determine all eigenpairs in arbitrary sequence Not efficient when only few lowest freq. are required Initial matrices are modified cannot fully take advantage of sparseness of matrices

Perturbation method(Meirovitch and Ryland, Cronin, Kwak, Peres-Da-Silva et al., Tang and Wang, …)

Sets the eigensolution of undamped system as zeros-order approximation and lets the high-order terms account for slight damping effect. Practical for eigenproblem with slight damping

Inverse Iteration + Sturm scheme(Gupta, Utku and Clement, …)

Preserves the banded nature of matrices Well suited for finding frequencies in a certain range Require many complex arithmetic operations for each eigenvalue sought

Subspace Iteration method(Bathe and Wilson, Chen and Taylor, Leung, …)

Combines Inverse Iteration, Simultaneous Iteration and Rayleigh-Ritz analysis More efficient than Inverse Iteration procedure Simultaneous solution minimum round-off error Require many complex arithmetic operations

Lanczos methods(Lanczos, Paige, Parlett and Scott, Simon, Kim and Craig, Rajakumar and Roger, Chen and Taylor, …)

Two-sided algorithm requires the generation of 2 sets of Lanczos vectors Symmetric algorithm uses a set of Lanczos vectors Only real arithmetic operations are used Possible serious breakdown; low accuracy (Zheng et al.)

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Objective

Reform (2) into

Proposed method

(3)

or(4)

where

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Newton-Raphson scheme(Robinson et al., Lee et al., …)

Residual vector after step kth

where is normalized

(5)

Initial solutions and are known

Increments

(6)

(7)

to be normalized (8)

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Equation

(10)

Residual vector after step (k+1)th is expected to be null

(9)

Modified Newton-Raphson scheme(Robinson et al., Lee et al., Kim, …)

(10’)

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Proposed modification

Increments(7’)

(k)

…

Minimize the norm of residual vector w.r.t.

(11)

Solve (11) for

(13)

(7)

where (12)

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Block diagram of the proposed method

Initial Solutions

andPerform 1st step by conventional method

Compute

Proposed

,

_ Conventional

Solve

method? method

Compute and as in (7) and (12)

Compute as in (7’)

for

Normalize

Compute as in (12)Check

Final Solutions

and+

+

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Numerical example Cantilever with multi-lumped viscous dampers

Material properties System dataE = 2*1011 N/ m2

= 8000 kg/m3

A = 3.0*10-4 m2

I = 2.25*10-8 m4

Ccon. = 0.1 N.s/m = 0.002, = 2.04*10-7

No. of nodes = 101No. of beam elements = 100No. of degrees of freedom = 200

1111Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea

Convergence

Convergence of the 14th eigenpair

Convergence of the 17th eigenpair

Proposed method Conventional

The total solution time to have 20 eigenpairs

211.2 (sec) 1.0

228.6 (sec) 1.08

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Conclusion

The convergence of the proposed method is improved by introducing the step length.

The algorithm of the proposed method is simple.

The efficiency of the method depends on the checking number. Further study on this checking number is being conducted.

Thank You for Your Coming and Listening!