대한토목공학회 추계 학술발표회 대구 2003 년 10 월 24 일 t. x. nguyen,...
Post on 06-Jan-2018
234 Views
Preview:
DESCRIPTION
TRANSCRIPT
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
22Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
Review Proposed method Numerical example Conclusion
Contents
33Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural 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
44Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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.)
55Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
Objective
Reform (2) into
Proposed method
(3)
or(4)
where
66Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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)
77Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural 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’)
88Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
Proposed modification
Increments(7’)
(k)
…
Minimize the norm of residual vector w.r.t.
(11)
Solve (11) for
(13)
(7)
where (12)
99Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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+
+
1010Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
1212Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural 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!
top related