1. 한국지진공학회 추계학술발표회
IMPROVED SENSITIVITY METHOD FOR NATURAL FREQUENCY AND MODE SHAPE OF DAMPED SYSTEM
Hong-Ki Jo1), *Man-Gi Ko2) and In-Won Lee3)
1) Graduate Student, Department of Civil Engineering, KAIST
2) Professor, Department of Civil Engineering, KongJu National Univ.
3) Professor, Department of Civil Engineering, KAIST

2. OUTLINE INTRODUCTION PREVIOUS STUDIES PROPOSED METHOD
NUMERICAL EXAMPLE
CONCLUSIONS

3. INTODUCTION Objective of Study Applications of Sensitivity Analysis
- To find the derivatives of eigenvalues and eigenvectors of damped systems with respect to design variables.
Applications of Sensitivity Analysis
- Determination of the sensitivity of dynamic responses
- Optimization of natural frequencies and mode shapes
- Optimization of structures subject to natural frequencies.
- Stability of structures
- Reanalysis of modified structures

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

5. - State space equation
(2)
- Orthonormalization condition
(3)

6. - Objective
Given:
Find:
* indicates derivatives with respect to
design variables (length, area, moment of inertia, etc.)

7. PREVIOUS STUDIES
Z. Zimoch, “Sensitivity Analysis of Vibrating Systems,”
Journal of Sound and Vibration, Vol. 117, pp ,
1987.
(4)
- restricted to lumped systems with distinct eigenvalues.

8. Q. H. Zeng, “Highly Accurate Modal Method for
Calculating Eigenvector Derivatives in Viscous Damping System,” AIAA Journal, Vol. 33, No. 4, pp , 1995.
(5)
(6)
- many eigenvectors are required to calculate eigenvector
derivatives.

9. Sondipon Adhikari, “Calculation of Derivative of Complex Modes Using Classical Normal Modes,” Computer & Structures, Vol. 77, No. 6, pp , 2000.
(7)
- applicable only when the elements of C are small.

10. I. W. Lee, D. O. Kim and G. H. Jung, “Natural Frequency and Mode Shape Sensitivities of Damped Systems: part I, Distinct Natural Frequencies,” Journal of Sound and Vibration, Vol. 223, No. 3, pp , 1999.
I. W. Lee, D. O. Kim and G. H. Jung, “Natural Frequency and Mode Shape Sensitivities of Damped Systems: part II, Multiple Natural Frequencies,” Journal of Sound and Vibration, Vol. 223, No. 3, pp , 1999.

11. Lee’s method (1999)
(8)
(9)
- eigenvalue and eigenvector derivatives are obtained
separately.

12. PROPOSED METHOD Rewriting basic equations - Eigenvalue problem (10)
- Orthonormalization condition
(11)

13. Differentiating eq.(10) with respect to design variable
(12)
Differentiating eq.(11) with respect to design variable
(13)

14. - the coefficient matrix is symmetric and non-singular.
Combining eq.(12) and eq.(13) into a single matrix
(14)
- the coefficient matrix is symmetric and non-singular.
eigenpair derivatives are obtained simultaneously.
corresponding eigenpair only is required.

15. Numerical Stability
The determinant property
(15)

16. Then
(16)

17. Arranging eq.(16)
(17)
Using the determinant property of partitioned
matrix
(18)

18. Numerical Stability is Guaranteed.
Therefore
(19)
Numerical Stability is Guaranteed.

19. NUMERICAL EXAMPLE
Cantilever Beam

20. Analysis Methods Comparisons Lee’s method (1999) Proposed method
Solution time (CPU)

21. Eigenvalue derivative
Results of Analysis (Eigenvalue)
Mode
Number
Eigenvalue
Eigenvalue derivative
(Lee’s method)
(Proposed method) 1 i i 2 i i 3 i
-5.411e e+2i 4 i
-5.411e e+2i 5 i
4.770e e-8i 6 i
4.770e e-8i 7 i
-4.242e e+2i 8 i
-4.242e e+2i 9 i
-1.628e e+3i 10 i
-1.628e e+3i
Same

22. Eigenvector derivative
Results of Analysis (First eigenvector)
DOF
number
Eigenvector
Eigenvector derivative
(Lee’s method)
(Proposed method) 1 2 3
1.513e e-05i
-3.027e e-04i 4
1.204e e-04i i 5 
157
158
159 i i
160 i i
Same

23. CPU time for 160 Eigenpairs
Method
Ratio
(sec)
Lee’s method (1999)
223.33
1.00
Proposed method
164.89
0.74

24. Comparison for each operations
Total
Lee’s
method
Proposed
Method
Operations
CPU time
(sec)
33.89
61.01
47.09
81.34
223.33
53.62
40.60   
70.67
164.89

25. CONCLUSIONS  An efficient eigensensitivity technique !
Proposed method
- is simple
- guarantees numerical stability
- reduces the CPU time.
 An efficient eigensensitivity technique !

26. Future works
Proposed method is able to extend to
asymmetric non-conservative systems
- Needs for comparison with other method
in CPU time

27. Thank you for your attention.

28. APPENDIX Differentiating eq.(1) with respect to design variable (20)
Pre-multiplying each side of eq.(20) by gives eigenvalue derivative.
(21)

29. Lee’s method (1999)
Differentiating eq.(3) with respect to design variable
(22)
Combining eq.(20) and eq.(22) into a matrix gives eigenvector derivative.
(23)