1. ISEC-02 Second International Conference on Structural Engineering and Construction
Algebraic Method for Sensitivity Analysis of Eigensystems with Repeated Eigenvalues
Kang-Min Choi1), Sang-Won Cho2), Jong-Heon Lee and In-Won Lee3)
1), 2) Graduate Student, Department of Civil Engineering, KAIST
3) Professor, Department of Civil Engineering, Kyungil Univ.
3) Professor, Department of Civil Engineering, KAIST

2. OUTLINE
 INTRODUCTION
 PROPOSED METHOD
 NUMERICAL EXAMPLES
 CONCLUSIONS

3. INTODUCTION  Applications of sensitivity analysis are
● determination of the sensitivity of dynamic response
● optimization of natural frequencies and mode shapes
● optimization of structures subject to natural frequencies
 To find the derivatives of eigenvalues and eigenvectors
of damped systems with multiple eigenvalues according to
design variables.
 Typical structures have many multiple or nearly equal
eigenvalues, due to structural symmetries.

4. ♦ Problem Definition
● Eigenvalue problem of damped system
(1)

5. ● Objective
Given:
Find:
* represents the derivative of with respect
design variable α (length, area, moment of inertia, etc.)

6. PROPOSED METHOD ♦ Basic Equations ● Eigenvalue problem (2)
● Orthonormalization condition
(3)

7. ● Adjacent eigenvectors
(4)
where T is an orthogonal transformation matrix
and its order m
(5)

8. ♦ Rewriting Basic Equations
● Another eigenvalue problem
(6)
● Orthonormalization condition
(7)

9. Differentiating eq.(6) with respect to design parameter α
(8)
Differentiating eq.(7) with respect to design parameter α
(9)

10. Combining eq.(8) and eq.(9) into a single matrix
(10)
● It maintains N-space without use of state space equation.
● Eigenpair derivatives are obtained simultaneously.
● It requires only corresponding eigenpair information.
● Numerical stability is guaranteed.

11. ♦ Numerical Stability
● Determinant property
(11)

12. Then,
(12)
(13)

13. Arranging eq.(12)
(14)
Using the determinant property of partitioned matrix
(15)

14. Numerical Stability is Guaranteed.
Therefore
(16)
Numerical Stability is Guaranteed.

15. NUMERICAL EXAMPLES
♦ Cantilever Beam (proportionally damped system)

16. Eigenvalue derivative
● Results of Analysis
Mode Number
Eigenvalue
Eigenvalue derivative
Changed eigenvalue
Approximated
eigenvalue 1 2 3 4 5 6 7 8 9 10 11 12
-1.43e-03 – j5.25e+00
-1.43e-03 + j5.25e+00
-1.43e-03 – j5.25e+00
-5.42e-02 – j3.29e+01
-5.42e-02 + j3.29e+01
-4.24e-01 – j9.21e+01
-4.24e-01 + j9.21e+01
-8.67e-11 + j2.50e-10
-2.81e-10 – j3.53e-10
-2.76e-02 – j5.25e+01
-2.76e-02 + j5.25e+01
-6.63e-10 – j2.34e-10
-6.63e-10 + j2.16e-10
-1.08e+00 – j3.29e+02
-1.08e+00 + j3.29e+02
6.98e-10 + j7.80e-10
6.92e-10 – j6.96e-10
-8.47e+00 – j9.20e+02
-8.47e+00 + j9.20e+02
-1.46e-03 – j5.30e+00
-1.46e-03 + j5.30e+00
-5.52e-02 – j3.32e+01
-5.52e-02 + j3.32e+01
-4.33e-01 – j9.30e+01
-4.33e-01 + j9.30e+01

17. Mode Number Error of eigenvalue Error of eigenvector1 2 3 4 5 6 7 8 9 10 11 12
2.2283e-11
2.6622e-08
3.6872e-12
3.6899e-12
1.6763e-07
9.1485e-12
9.1432e-12
4.6508e-07
3.7376e-05
1.0000e-04
1.0001e-04
1.0002e-04
9.9041e-03

18. ♦ 5-DOF Non-proportional Damped System

19. Eigenvalue derivative
● Results of Analysis
Mode Number
Eigenvalue
Eigenvalue derivative
Changed eigenvalue
Approximated
eigenvalue 1 2 3 4 5 6 7 8 9 10
-4.33e-02 – j1.50e+00
-4.33e-02 + j1.50e+00
-2.40e-01 – j3.46e+00
-2.40e-01 + j3.46e+00
-3.52e-02 – j6.14e+00
-3.52e-02 + j6.14e+00
-2.45e-02 – j9.70e+00
-2.45e-02 + j9.70e+00
9.69e-07 – j1.80e-04
9.69e-07 + j1.80e-04
-1.63e-19 – j8.68e-04
-1.08e-19 + j8.68e-04
-7.89e-07 – j2.95e-05
-7.89e-07 + j2.95e-05
-1.80e-07 – j5.00e-06
-1.80e-07 + j5.00e-06
-4.32e-02 – j1.50e+00
-4.32e-02 + j1.50e+00
-2.40e-01 – j3.45e+00
-2.40e-01 + j3.45e+00
-3.52e-02 – j6.13e+00
-3.52e-02 + j6.13e+00

20. Mode Number Error of eigenvalue Error of eigenvector1 2 3 4 5 6 7 8 9 10
8.1631e-06
8.4309e-16
7.0672e-16
2.1632e-06
1.1763e-07
4.3893e-09
2.9463e-05
1.2945e-14
1.2770e-14
5.2014e-06
2.5394e-06
1.6332e-07

21. CONCLUSIONS An efficient eigensensitivity method for the
♦ Proposed Method
● is simple
● guarantees numerical stability
An efficient eigensensitivity method for the
damped system with multiple eigenvalues

22. Thank you for your attention.