1. Modified Sturm Sequence Property for Damped Systems
The Fifth European Conference on Structural Dynamics
Munich, Germany
September 2-5, 2002
Modified Sturm Sequence Property for Damped Systems
Ji-Seong Jo1, Man-Gi Ko2 & In-Won Lee3
Thank you, Mr. chairman.
Good afternoon, Ladies and gentlemen.
My name is Ji-Seong Jo.
The title of my presentation is “Modified Sturm Sequence Property for Damped Systems”.
1 Ph. D. Candidate, KAIST, Korea
2 Professor, Kongju National University, Korea
3 Professor, KAIST, Korea
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2. Structural Dynamics & Vibration Control Lab.
CONTENTS
 Introduction
 Previous Studies
 Objective
 Proposed Method
 Numerical Examples
I will present in the following sequence.
 Conclusion
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3. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Introduction
 Introduction
Dynamic Equations of Motion
M : Mass matrix of order n
C : Damping matrix of order n
K : Stiffness matrix of order n
u(t) : Displacement vector
f(t) : Load vector
n : Order of K, C and M
(1)
The Dynamic Equations of Motion can be written as in Equation (1).
Where
M mass matrix
C damping matrix
K stiffness matrix
U displacement vector
F Load vector
And n Order of K,C and M
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4. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Introduction
Methods of Dynamic Analysis
- Direct integration method :
Loading of a short duration such as impulse loading
- Mode superposition method :
Loading of a long duration such as an earthquake
Mode Superposition Method
- Required eigenpairs should be computed.
- Technique for checking missed eigenpairs is required.
For Dynamic Analysis of structures, there are twp methods such as
The Direct Integration Method and Mode Superposition Method.
The Direct integration method is efficient for the case of loading of a short duration such as impulse loading.
And Mode superposition method is for the case of loading of a long duration such as an earthquake.
When the Mode Superposition Method is used,
Required eigenpairs should be computed and also
Missed eigenpairs should be checked.
- Technique for checking missed eigenpairs is required.
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5. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Introduction
Proportionally Damped System
i : i-th eigenvalue
i : i-th eigenvector
- Eigenvalues and eigenvectors : real numbers
- Checking missed eigenpairs : Sturm sequence property
(2)
For Proportionally Damped System defined in this equation,
The eigenvalue problem can be written as in equation(2).
Ramda i is the i-th eigenvalue, Phi i is the i-th eigenvector.
In this case,
Eigenvalues and eigenvectors are real numbers and
For Checking missed eigenpairs, Sturm sequence property is used.
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6. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Introduction
Nonproportionally Damped System
(Soil-structure interaction problem, Structural
control problem, Composite structure and so on)
- Eigenvalues and eigenvectors : complex numbers
- Checking missed eigenpairs : a few methods
(3)
For non-proportionally Damped System defined in this equation,
Such as Soil-structure interaction problem, Structural control problem, Composite structure and so on,
The Eigenvalue problem can be written as in equation(3).
In this case,
Eigenvalues and eigenvectors are complex numbers and
For Checking missed eigenpairs, a few method is used.
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7. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Previous Studies
 Previous Studies
Tsai and Chen (1993)
- Extended Sturm sequence property to determine
the number of roots of a polynomial on specified
lines of the complex plane
- It is very difficult to find the specified line on the
complex plane.
- Sturm sequence is not formed by factorizing the
matrix in the field of complex arithmetic computation.
Now I will explain Previous studies on Checking missed eigenvalues nonproportionally damped system.
In 1993, Tsai and Chen proposed extended strum sequence property
To determine the number of root of a polynomial on specified lines of the complex plane.
However it is very difficult to find the specified line on the complex plane using this method,
and Sturm sequence is not formed by factorizing the matrix in the field of complex arithmetic computation.
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8. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Previous Studies
Jung and Lee (1999)
- Based on augment principle
(4)
Imaginary axis
Imaginary axis 6 5 7 4 S 8 3 3
Real axis 4 2 2
Real axis 1 5 1 8 6 7
f()
Jung and Lee in 1999 proposed a method based on argument principle.
A complex function f ramda can be written as in equation(4).
and evaluated along an closed contour S on the complex plane,
Then the number of rotations of f ramda about origin on the complex plane is
equal to No. of eigenvalues inside an closed contour.
 plane f() plane
No. of Rotations = No. of eigenvalues inside
an closed contour
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9. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Previous Studies
Shortcomings
Accuracy improves as the number of checking points
increases.
- Difficult to find abrupt change in arguments
Factorization processes are required at each checking
point.
But this method has following Shortcomings
Fisrt, Accuracy improved as the number of checking points increases.
Second, It is difficult to find abrupt change in arguments
And Third, Factorization processes are required at each checking point.
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10. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Objective
 Objective
To develop an efficient technique for checking
missed eigenpairs of nonproportionally damped
systems with distinct or multiple eigenvalues
So the Objective of this study is
to develop an efficient technique
for checking missed eigenpairs of a nonproportionally damped system.
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11. Modified Sturm Sequence Property for Damped Systems
Modified Sturm Sequence Property for Damped Systems Proposed Method
 Proposed Method
Modified Sturm Sequence Property for Damped Systems
Complex eigenvalue problem
Chen’s algorithm
Characteristic polynomial
Gleyse’s theorem
In the proposed method,
Using Chen’s algorithm, we formulate characteristic polynomial.
And then, by Gleyse’s theorem, we can check the number of eigenvalues inside some open disks.
Check the number of eigenvalues inside some open disks
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12. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Proposed Method
Complex Eigenvalue Problem
(3)
- State space form ( )
(5)
- Standard form
(6)
The Complex Eigenvalue Problem can be written as in equation (3).
The state space form of the equation is as equation (5) or equation (6).
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13. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Proposed Method
Chen’s Algorithm
(7)
Gauss elimination-like similarity transformations
Give matrix A is transformed into the matrix A bar,
Using Gauss elimination-like similarity transformations.
(8)
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14. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Proposed Method
- Characteristic polynomial
(9)
Then characteristic polynomial P ramda can be computed using next equations.
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15. Gleyse’s Theorem - Characteristic polynomial - Schur Cohn matrix T:
Modified Sturm Sequence Property for Damped Systems Proposed Method
Gleyse’s Theorem
- Characteristic polynomial
(10)
- Schur Cohn matrix T:
(11)
Gleyse’s theorem is that
Given the characteristic polynomial defined in equation (10),
We can construct the following Schur Cohn matrix T,
using equations (11) and (12).
(12)
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16. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Proposed Method
The number of eigenvalues inside an unit open disk
(13)
N : the number of eigenvalues inside an unit open disk
2n : degree of the characteristic polynomial P
S[1, d1, d2, ···, d2n]: the number of sign changes in the
sequence (1, d1, d2,···, d2n)
Imaginary axis 1
Then, the number of eigenvalues inside an unit open disk can be found using equation (13).
N the number of eigenvalues inside an unit open disk,
2n degree of the characteristic polynomial P
S the number of sign changes in the sequence
And D I determinant of T i -1 1
Real axis -1
unit disk
 plane
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17. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Numerical Examples
 Numerical Examples
Simple Spring-Mass-Damper System
To apply the proposed method to the distinct eigenvalue system 1 2 10 k k m m m
Mass: m = 1.0
Stiffness: k = 1.0
Rayleigh damping:  = 0.05,  = 0.05
Number of d.o.f:
For numerical examples Simple Spring-Mass-Damper System is considered
to apply the proposed method to the distinct eigenvalue system.
Shape and Properties are shown here.
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18. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Numerical Examples
- Exact eigenvalues
(14)
(15)
(16)
The exact eigenvalues can be computed using equation (14).
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Modified Sturm Sequence Property for Damped Systems Numerical Examples
- Calculated eigenvalues
19,20
17,18
15,16
13,14
11,12
9,10
7,8
5,6
3,4
1,2
1.9777
1.9111
1.8019
1.6525
1.4661
1.2470
1.0000
0.7307
0.4450
0.1495
Radii
Mode No.
Eigenvalues
1.7046
1.6651
1.5959
1.4932
1.3540
1.1763
0.9614
0.7133
0.4388
0.1463
Real
Imaginary
The calculated eigenvalues are shown here.
All eigenvalues are distinct.
the radii is the distance of an eigenvalue form the origin on the complex plane.
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20. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Numerical Examples
Imaginary axis
Real axis
This figure shows the calculated eigenvalues on the complex plane.
eigenvalues
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21. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Numerical Examples
- Radius  = 1.1 > |8 | = 1.0
i sign(di) S
i sign(di) S 8 9 1 10 2  3 4 11 12 5
First, when  is a little bit larger than the ramda sixteen.
The calculated dii, sign di and sign change S are as shown here.
The number of sign change is 4. And the number of eigenvalue N inside open disk of radius , is 4.
This agrees with the calculated result. 6 7
 = 8
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22. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Numerical Examples
- Radius  = 2.0 > |20| = 1.978
i sign(di) S
i sign(di) S 
And third, when  is a little bit larger than the ramda twentieth .
The calculated dii, sign di and sign change S are as shown here.
The number of sign change is 0. And the number of eigenvalue N inside open disk of radius , is 20.
This agrees with the calculated result.
 = 20
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23. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Numerical Examples
Plane Frame Structure with Lumped Dampers
To apply the proposed method to the multiple eigenvalue system y v L x
The time is running out, so I will stop here.
As the Second example Plane Frame Structure with Lumped Dampers is considered
To apply the proposed method to the multiple eigenvalue system.
This system has a following shapes. L
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24. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Numerical Examples
- Geometric and material properties
Damping
Concentrated:
Rayleigh:  = 0.001,  = 0.001
Young’s modulus:
Mass density:
Cross-sectional inertia: 1.0
Cross-sectional area:
Span length:
- System data
And Geometric and material properties and System data are as followings.
The Number of elements is twelve, the Number of nodes is fourteen and the number of DOF is eighteen 18.
Number of elements:
Number of nodes:
Number of DOF:
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25. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Numerical Examples
- Calculated eigenvalues
Mode No.
Eigenvalues
Radii
Real
Imaginary
1,2
-1.137
46.219
46.233
3,4
-1.137
46.219
46.233
5,6
-1.373
51.133
51.152
7,8
-1.373
51.133
51.152
9,10
-3.390
81.078
81.149
11,12
-3.390
81.078
81.149
13,14
-3.941
87.477
87.566
The calculated eigenvalues are shown here.
As you can see, eigenvalues are multiple.
The radii is the distance of an eigenvalue form the origin on the complex plane.
15,16
-3.941 
87.566
17,18
-8.164 
19,20
-8.164 
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Modified Sturm Sequence Property for Damped Systems Numerical Examples
Mode No.
Eigenvalues
Radius
Real
Imaginary
21,22 
23,24 
25,26 
27,28 
29,30 
31,32 
33,34 
35,36 
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27. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Numerical Examples
Imaginary axis
Real axis
This figure shows the calculated eigenvalues on the complex plane.
eigenvalues
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28. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Numerical Examples
- Radius  = 82.0 > |12| =
i sign(di) S
i sign(di) S 1 2 18 3 4 19 5 6 20 7 21  8 9 10 11 22 12 23 13
First, when  is a little bit larger than the ramda twelfth .
The calculated dii, sign di and sign change S are as shown here.
The number of sign change is 24. And the number of eigenvalue N inside open disk of radius , is 12.
This agrees with the calculated result. 14 24 15 16 17
 = 12
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29. Structural Dynamics & Vibration Control Lab.
Modified Sturm Sequence Property for Damped Systems Numerical Examples
- Radius  = 220 > |36| =
i sign(di) S
i sign(di) S 
And third, First, when  is a little bit larger than the ramda thirty sixth.
The calculated dii, sign di and sign change S are as shown here.
The number of sign change is 0. And the number of eigenvalue N inside open disk of radius , is .
This agrees with the calculated result.
 = 36
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Modified Sturm Sequence Property for Damped Systems Conclusion
 Conclusion
The proposed method is more efficient than the previous methods for checking missed eigenpairs of damped systems !!
Conclusion
The proposed method is more efficient than the previous methods for checking missed eigenpairs of damped systems.
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31. Thank you for your attention!!
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