대한토목공학회 추계 학술발표회 대구 2003 년 10 월 24 일 T. X. Nguyen, 한국과학기술원 건설 및 환경공학과 박사과정 김병완, 한국과학기술원 건설 및 환경공학과 박사후연구원 정형조, 세종대학교 토목환경공학과 교수 이인원, 한국과학기술원 건설 및 환경공학과 교수 Application of Step Length Technique to an Eigensolution Method for Non- proportionally Damped Systems

2 Structural Dynamics & Vibration Control Lab., KAIST, Korea Review Proposed method Numerical example Conclusion Contents

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

4 Structural Dynamics & Vibration Control Lab., KAIST, Korea Transformation methods QR (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.)

5 Structural Dynamics & Vibration Control Lab., KAIST, Korea Objective Reform (2) into Proposed method (3) or (4) where

6 Structural Dynamics & Vibration Control Lab., KAIST, Korea Newton-Raphson scheme (Robinson et al., Lee et al., …) Residual vector after step k th where is normalized (5) Initial solutions and are known Increments (6) (7) to be normalized (8)

7 Structural Dynamics & Vibration Control Lab., KAIST, Korea 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’)

8 Structural Dynamics & Vibration Control Lab., KAIST, Korea Proposed modification Increments (7’)  (k) (k) … Minimize the norm of residual vector w.r.t.  (11) Solve (11) for  (13) (7) where (12)

9 Structural Dynamics & Vibration Control Lab., KAIST, Korea Block diagram of the proposed method Initial Solutions and Perform 1 st step by conventional method Compute Proposed, _ Conventional Solve method? method Computeandas in (7) and (12) Computeas in (7’) for Normalize Computeas in (12)Check Final Solutions and + +

10 10 Structural Dynamics & Vibration Control Lab., KAIST, Korea Numerical example Cantilever with multi-lumped viscous dampers Material propertiesSystem data E = 2*10 11 N/ m 2  = 8000 kg/m 3 A = 3.0*10 -4 m 2 I = 2.25*10 -8 m 4 C con. = 0.1 N.s/m  = 0.002,  = 2.04*10 -7 No. of nodes = 101 No. of beam elements = 100 No. of degrees of freedom = 200

11 11 Structural Dynamics & Vibration Control Lab., KAIST, Korea Convergence Convergence of the 14 th eigenpair Convergence of the 17 th eigenpair Proposed methodConventional The total solution time to have 20 eigenpairs (sec) (sec) 1.08

12 12 Structural Dynamics & Vibration Control Lab., KAIST, Korea 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!