1. Hong-Ki Jo 1), Man-Gi Ko 2) and * In-Won Lee 3) 1) Graduate Student, Dept. of Civil Engineering, KAIST 2) Professor, Dept. of Civil Engineering, Kongju National University 3) Professor, Dept. of Civil Engineering, KAIST S IMPLIFIED A LGEBRAIC M ETHOD FOR C OMPUTING E IGENPAIR S ENSITIVITIES OF D AMPED S YSTEMS ISEC-01 Honolulu, Hawaii January 24-26, 2001

2. Structural Dynamics and Vibration Control Lab., KAIST, Korea 2 OUTLINE INTRODUCTION PREVIOUS STUDIES PROPOSED METHOD NUMERICAL EXAMPLE CONCLUSIONS

3. Structural Dynamics and Vibration Control Lab., KAIST, Korea 3 INTODUCTION Objective of Study 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. - To find efficient sensitivity method of eigenvalues and eigenvectors of damped systems.

4. Structural Dynamics and Vibration Control Lab., KAIST, Korea 4 Problem Definition (1) - Eigenvalue problem of damped system (N-space)

5. Structural Dynamics and Vibration Control Lab., KAIST, Korea 5 (2) - Normalization condition - State space equation (2N-space) (3)

6. Structural Dynamics and Vibration Control Lab., KAIST, Korea 6 Given: Find: - Objective * indicates derivatives with respect to design variables (length, area, moment of inertia, etc.)

7. Structural Dynamics and Vibration Control Lab., KAIST, Korea 7 PREVIOUS STUDIES - many eigenpairs are required to calculate eigenvector derivatives. (2N-space) (4) (5) Q. H. Zeng, “Highly Accurate Modal Method for Calculating Eigenvector Derivatives in Viscous Damping System,” AIAA Journal, Vol. 33, No. 4, pp. 746-751, 1995. Structural Dynamics and Vibration Control Lab., KAIST, Korea

8. 8 Sondipon Adhikari, “Calculation of Derivative of Complex Modes Using Classical Normal Modes,” Computer & Structures, Vol. 77, No. 6, pp. 625-633, 2000. - many eigenpairs are required to calculate eigenvector derivatives. (N-space) - applicable only when the elements of C are small. (6) Structural Dynamics and Vibration Control Lab., KAIST, Korea

9. 9 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. 399-412, 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. 413-424, 1999.

10. Structural Dynamics and Vibration Control Lab., KAIST, Korea 10 Lee’s method (1999) (7) (8) - the corresponding eigenpairs only are required. (N-space) - the coefficient matrix is symmetric and non-singular. - eigenvalue and eigenvector derivatives are obtained separately.

11. Structural Dynamics and Vibration Control Lab., KAIST, Korea 11 PROPOSED METHOD Rewriting basic equations - Eigenvalue problem - Normalization condition (9) (10)

12. Structural Dynamics and Vibration Control Lab., KAIST, Korea 12 Differentiating eq.(9) with respect to design variable Differentiating eq.(10) with respect to design variable (11) (12)

13. Structural Dynamics and Vibration Control Lab., KAIST, Korea 13 Combining eq.(11) and eq.(12) into a single matrix (13) - the corresponding eigenpairs only are required. (N-space) - the coefficient matrix is symmetric and non-singular. - eigenpair derivatives are obtained simultaneously.

14. Structural Dynamics and Vibration Control Lab., KAIST, Korea 14 NUMERICAL EXAMPLE Cantilever beam with lumped dampers Design parameter : depth of beam Material PropertiesSystem Data Number of elements : 20 Number of nodes : 21 Number of DOF : 40 v1v1 v2v2

15. Structural Dynamics and Vibration Control Lab., KAIST, Korea 15 Analysis Methods Zeng’s method (1995) Lee’s method (1999) Proposed method Comparisons Solution time (CPU)

16. Structural Dynamics and Vibration Control Lab., KAIST, Korea 16 Results of Analysis (Eigenvalue) Mode number Eigenvalue Eigenvalue derivative 1-0.0035 - 1.0868i 0.0010 - 0.2997i 2-0.0035 + 1.0868i 0.0010 + 0.2997i 3-0.0203 - 6.0514i 0.0072 - 1.3173i 4-0.0203 + 6.0514i 0.0072 + 1.3173i 5-0.0422 - 14.7027i 0.0140 - 2.4536i 6-0.0422 + 14.7027i 0.0140 + 2.4536i 7-0.0719 - 24.7343i 0.0189 - 3.1194i 8-0.0719 + 24.7343i 0.0189 + 3.1194i 9-0.1106 - 35.3632i0.0213 - 3.4203i 10-0.1106 + 35.3632i0.0213 + 3.4203i

17. Structural Dynamics and Vibration Control Lab., KAIST, Korea 17 Results of Analysis (First eigenvector) DOF number Eigenvector Eigenvector derivative 1 0.0013 + 0.0013i -0.0004 - 0.0004i 20.0050 + 0.0050i-0.0015 - 0.0015i 30.0049 + 0.0049i-0.0015 - 0.0015i 40.0096 + 0.0096i-0.0029 - 0.0029i 5 0.0108 + 0.0108i-0.0033 - 0.0032i 60.0139 + 0.0139i -0.0042 - 0.0042i 7 0.0188 + 0.0188i-0.0056 - 0.0056i 80.0179 + 0.0178i-0.0054 - 0.0053i 9 0.0287 + 0.0286i-0.0086 - 0.0085i 10 0.0215 + 0.0215i-0.0064 - 0.0064i

18. Structural Dynamics and Vibration Control Lab., KAIST, Korea 18 CPU time for 40 Eigenpairs Method CPU time Ratio Lee’s method2.21 1.4 Proposed method 1.591.0 (sec) Zeng’s method 184.05115.8

19. Structural Dynamics and Vibration Control Lab., KAIST, Korea 19 ď   : Zeng’s method (Using full modes(40), exact solution)  : Zeng’s method (Using two modes(2), 5% error)  : Lee’s method (Exact solution)  : Proposed method(Exact solution) Fig 1. Comparison with previous method   Δ    510152025303540 0 50 100 150 200 Modes CPU time (sec)            Δ ΔΔΔΔΔ    184.05 61.47 Improvement about 99% 2.21 1.59 Structural Dynamics and Vibration Control Lab., KAIST, Korea

20. 20  : Lee’s method (Exact solution)  : Proposed method(Exact solution) Fig 2. Comparison with Lee’s method Δ 510152025303540 0 0.5 1 1.5 2 2.5 Modes CPU time (sec)         Δ Δ Δ Δ Δ Δ Δ Improvement about 25% 2.21 1.59 Structural Dynamics and Vibration Control Lab., KAIST, Korea

21. 21 CONCLUSIONS P roposed method - is composed of simple algorithm - guarantees numerical stability - reduces the CPU time.  An efficient eigen-sensitivity technique !

22. Structural Dynamics and Vibration Control Lab., KAIST, Korea 22 Thank you for your attention.

23. Structural Dynamics and Vibration Control Lab., KAIST, Korea 23 Numerical Stability The determinant property (14) APPENDIX

24. Structural Dynamics and Vibration Control Lab., KAIST, Korea 24 Then (15)

25. Structural Dynamics and Vibration Control Lab., KAIST, Korea 25 Arranging eq.(15) (16) Using the determinant property of partitioned matrix (17)

26. Structural Dynamics and Vibration Control Lab., KAIST, Korea 26 Therefore Numerical Stability is Guaranteed. (18)

27. Structural Dynamics and Vibration Control Lab., KAIST, Korea 27 Lee’s method (1999) Differentiating eq.(1) with respect to design variable (19) Pre-multiplying each side of eq.(19) by gives eigenvalue derivative. (20)

28. Structural Dynamics and Vibration Control Lab., KAIST, Korea 28 Differentiating eq.(3) with respect to design variable (21) Combining eq.(19) and eq.(21) into a matrix gives eigenvector derivative. (22)