1. Modified Sturm Sequence Property for Damped Systems
2001년도 한국지진공학회 학술발표회
Modified Sturm Sequence Property for Damped Systems
조지성*: 한국과학기술원 토목공학과 박사과정
김병완 : 한국과학기술원 토목공학과 박사과정
이인원 : 한국과학기술원 토목공학과 교수

2. Contents 1. Introduction 2. Proposed Method 3. Numerical Example
4. Conclusions

3. 1. Introduction  modal transformation
- dynamic equations of motion  modal equations
- transformation matrix : incomplete eigenvector set  inexact dynamic response
- checking technique of missed eigenpairs is required.

4.  proportionally damped system
- eigenvalues and eigenvectors : real numbers
- checking technique : sturm sequence property
 nonproportionally damped system
(soil-structure interaction problem, structural control
problem, composite structure and so on)
- eigenvalues and eigenvectors : complex numbers
- checking technique : not developed yet.
(1)
(2)

5.  objective
development of an effective checking technique of
missed eigenpairs applicable to nonproportionally
damped system

6.  complex eigenvalue problem
2. Proposed Method
 complex eigenvalue problem
(3)
 linearized form ( )
(4)
(5)

7. - characteristic polynomial
- given matrix A
(6)
- characteristic polynomial
,P()=0,   eigenvalue
(7)

8. - change of the matrix form
 Chen’s algorithm
- change of the matrix form
(8)

9.  Gauss elimination-like procedures
(9)

10. tij : i-th row, j-th column element of T
 Schur-Cohn Matrix T
(10)
tij : i-th row, j-th column element of T
(11)

11. the number of poles in the unit open disk:
 Gleyse’s theorem
imaginary axis 1 -1 1
Real axis -1
unit disk
the number of poles in the unit open disk:
(12)

12. n : degree of the characteristic polynomial P S[k0, k1, k2, ···, kn ]: the number of sign changes
in the sequence (ki, i=0, 1, ···, n)
(di , i=1, ··· , n) : determinant of the leading principal
submatrices of order i in the Schur-
Cohn matrix T

13. - T=LDLT factorization
 calculation of di
- T=LDLT factorization
(13)

14.  the number of zeros in the open disk radius >0
(14)
let
(15)

15.  plane frame structure with lumped dampers
3. Numerical Example
 plane frame structure with lumped dampers L v u L

16. Given properties System data Damping Concentrated: 0.3
Rayleigh:  = 0.001,  = 0.001
Young’s modulus:
Mass density:
Cross-sectional inertia: 1.0
Cross-sectional area:
Span length:
System data
Number of elements:
Number of nodes:
Number of DOF:

17.  calculated eigenvalues10 9 8 7 6 5 4 3 2 1
81.149
51.152
46.233
Radius
Mode No.
Eigenvalues
-3.390
-1.373
-1.137
Real
Imaginary

18. 20 19 18 17 16 15 14 13 12 11
87.566
81.149
Radius
Mode No.
Eigenvalues
-8.164
-3.941
-3.390
Real
Imaginary

19. 30 29 28 27 26 25 24 23 22 21
Radius
Mode No.
Eigenvalues
Real
Imaginary

20. Mode No.
Eigenvalues
Radius
Real
Imaginary 31 32 33 34 35 36

21. S  = 36-24 = 12  O.K [ + - + - + - + - + - + - + - + - - + - - [ ] S
= = 12  O.K

22. S  = 36- 4 = 32  O.K [ + + + + + + + + + + + + + + + + + + + +[ ] S
= = 32  O.K

23. S  = 36- 0 = 36  O.K [ + + + + + + + + + + + + + + + + + + + +[ ] S
= = 36  O.K

24. 4. Conclusions A technique of calculating the number of
eigenvalues inside an open disk of arbitrary
radius was given.
Comparing to the recently technique by Jung,
the proposed method needs no iterations, so
more effective than that technique.