1. 한국지진공학회 춘계 학술발표회
서울대학교 호암교수관
2003년 3월 14일
Application of Step Length Technique to an Eigensolution Method for Non-proportionally Damped Systems
T. X. Nguyen, 한국과학기술원 건설 및 환경공학과 박사과정
김병완, 한국과학기술원 건설 및 환경공학과 박사과정
정형조, 세종대학교 토목환경공학과 교수
이인원, 한국과학기술원 건설 및 환경공학과 교수
안녕하십니까?
Good afternoon,
My name is T.X. Nguyen. I am doing my Ph.D. at the Structural Dynamics and Vibration Control Lab., in KAIST.
I am pleased for this first chance that I could present our research at this Conference of Earthquake Engineering Society of Korea.
The title of my presentation today is ‘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
First of all, I will begin with the Review.
Next is the Proposed method.
The method’s efficiency is demonstrated by a numerical example.
Conclusion comes last in my speech today.
Structural Dynamics & Vibration Control Lab., KAIST, Korea

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)
The eigenvalue problem should be solved a priori for two main reasons: - To avoid the resonance and To obtain the dynamic characteristics of the system.
The Equation of Free Vibration of a Linear Time-Invariant System of order n is shown in Equation 1, where M, C, K are the (n by n) mass, damping, and stiffness matrices. u is the (n by 1) displacement vector. The damping can be either proportional, as defined by Caughey and Kelly 1965, or (click) non-proportional.
From this Equation, we have the Equation for the Quadratic Eigenvalue Problem as shown in Equation (2).
M, C, K: (n x n) system matrices
u: (n x 1) displacement vector
(2)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

4. Structural Dynamics & Vibration Control Lab., KAIST, Korea
Transformation methods
QR (Moler and Stewart)
LZ (Kaufman)
Jacobi (Veselic) …
Perturbation method
(Meirovitch and Ryland, Cronin, Kwak, Peres-Da-Silva et al., Tang and Wang, …)
Inverse Iteration + Sturm scheme
(Gupta, Utku and Clement, …)
Subspace Iteration method
(Bathe and Wilson, Chen and Taylor, Leung, …)
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.)
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
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
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
Preserves the banded nature of matrices
Well suited for finding frequencies in a certain range
Require many complex arithmetic operations for each eigenvalue sought
Researchers have proposed many eigensolution methods. Among them are:
The Transformation methods such as, (… reading …)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

5. Structural Dynamics & Vibration Control Lab., KAIST, Korea
Proposed method
Objective
Reform (2) into
(3) or
(4)
The method presented by Lee et al. in 1998 is considered as an efficient one dealing with eigenproblem of Non-proportionally damped systems. Our proposed method is an extension of it.
Let’s consider the Objective of the problem. Equation (2) is reformed into the first companion form, Equation (3), or, in a short form, we can write it as in Equation (4).
where
Structural Dynamics & Vibration Control Lab., KAIST, Korea

6. Structural Dynamics & Vibration Control Lab., KAIST, Korea
Newton-Raphson scheme
(Robinson et al., Lee et al., …)
Initial solutions and are known
Residual vector after step kth
(5)
where is normalized
(6)
Increments
The method is based on the Newton-Raphson scheme. Assume that the initial solutions ‘lambda-zero’ and ‘psi-zero’ are known. The residual vector after step k is defined as in Equation (5), where ‘psi-k’ is normalized with respect to matrix B, Equation (6). Let’s call the increments of eigenvalue and eigenvector from step k to step (k+1) are ‘Delta-lambda-k’ and ‘Delta-psi-k’, we have Equation (7). Here, ‘psi-k-plus-one’ is again normalized with respect to matrix B, as shown in Equation (8).
(7)
to be normalized
(8)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

7. Structural Dynamics & Vibration Control Lab., KAIST, Korea
Residual vector after step (k+1)th is expected to be null
(9)
Equation
(10)
Modified Newton-Raphson scheme
(Robinson et al., Lee et al., Kim, …)
We expect that the residual vector after step ‘k-plus-one’ be null vector as in Equation (9). Based on these above Equations, after some arrangement and neglecting high-order terms, we have the Equation for unkowns ‘Delta-psi-k’ and ‘Delta-lamda-k’. In Equation ‘ten’, the coefficient matrix is always nonsingular.
To accelerate the convergence, one can use the modified Newton-Raphson scheme. Equation ‘ten-prime’ is used instead of Equation ‘ten’. This part of the coefficient matrix (pointing) is computed just once, so it can save time.
(10’)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

8. Structural Dynamics & Vibration Control Lab., KAIST, Korea
Proposed modification …
Increments
(7’)
(7)
 (k)
Minimize the norm of residual vector w.r.t. 
(11)
(12)
where
Our research proposes a modification in the above procedure. Instead of using Equation (7), we use Equation ‘seven-prime’ with introduction of ‘alpha-k’. The norm of residual vector is the minimized with respect to ‘alpha-k’, thus we can have the value of ‘alpha-k’ as given in Equation (13).
Solve (11) for 
(13)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

9. Block diagram of the proposed method
Obtain Initial Solutions
and
Perform 1st step by conventional method _
Proposed
Conventional +
method?
method
Solve
Compute ,
for
Compute
and
as in (7) and (12)
Here is the diagram of the proposed method. (Stay silently for a while the jump to the next slide.)
Compute
as in (7’)
Normalize
Final Solutions +
Check
and
Compute
as in (12)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

10. Structural Dynamics & Vibration Control Lab., KAIST, Korea
Numerical example
Cantilever with multi-lumped viscous dampers
Material properties
System data
E = 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 = 101
No. of beam elements = 100
No. of degrees of freedom = 200
A numerical example demonstrates the efficiency of the proposed method. The system is a cantilever with multi-lumped viscous dampers. The material properties and system data are given here. (If time is up, cut and jump to next slide, otherwise talk a little bit about concentrate damping.)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

11. Structural Dynamics & Vibration Control Lab., KAIST, Korea
Convergence
Convergence of the
17th eigenpair
Convergence of the
14th eigenpair
With the draft initial solutions obtained by the Lanczos method, the representative results are shown here. This slide shows the convergence of the ‘fourteenth’ eigenpair. The initial error norm is ‘one’. The error limit is ‘ten exponent by minus six’. Two methods converge just after 2 steps. In this case, in the view of computing time, there is NO advantage of the proposed method since it takes time to compute ‘alpha’, while the conventional method does not. However, in the worse case of initial solutions, for example, the 17th eigenpair, the proposed method converges after 5 steps but it takes 6 steps for the conventional method to converge. In addition, the final error norm of the result obtained by the proposed method is much lower than that of the conventional method. The propose method obviously shows its advantage.
Proposed method
Conventional
The total solution time to have
211.2 (sec)
228.6 (sec)
20 eigenpairs
1.0
1.08
Structural Dynamics & Vibration Control Lab., KAIST, Korea

12. Conclusion Thank You for Your Coming and Listening!
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.
These are conclusions of my presentation today. First, (reading…). Second, (reading…). Third, (reading…).
Thank you for your coming and Listening!
감사합니다!
Thank You for Your Coming and Listening!
Structural Dynamics & Vibration Control Lab., KAIST, Korea