1 / 28

* Hong-Ki Jo 1) , Kyu-Sik Park 2) , Hye-Rin Shin 3) and In-Won Lee 4)

KKNN Seminar Taipei, Taiwan, Dec. 7-8, 2000. SIMPLIFIED ALGEBRAIC METHOD FOR COMPUTING EIGENPAIR SENSITIVITIES OF DAMPED SYSTEM. * Hong-Ki Jo 1) , Kyu-Sik Park 2) , Hye-Rin Shin 3) and In-Won Lee 4) 1) ~ 3) Graduate Student, Department of Civil Engineering, KAIST

gay
Download Presentation

* Hong-Ki Jo 1) , Kyu-Sik Park 2) , Hye-Rin Shin 3) and In-Won Lee 4)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. KKNN Seminar Taipei, Taiwan, Dec. 7-8, 2000 SIMPLIFIED ALGEBRAIC METHOD FOR COMPUTING EIGENPAIR SENSITIVITIES OF DAMPED SYSTEM *Hong-Ki Jo1), Kyu-Sik Park2), Hye-Rin Shin3) and In-Won Lee4) 1) ~ 3) Graduate Student, Department of Civil Engineering, KAIST 4) Professor, Department of Civil Engineering, KAIST

  2. OUTLINE • INTRODUCTION • PREVIOUS STUDIES • PROPOSED METHOD • NUMERICAL EXAMPLE • CONCLUSIONS

  3. INTODUCTION • Objective of Study - To find efficient sensitivity method of eigenvalues and eigenvectors of damped systems. • 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.

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

  5. - State space equation (2N-space) (2) - Normalization condition (3)

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

  7. PREVIOUS STUDIES • 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. (4) (5) -many eigenpairs are required to calculate eigenvector derivatives. (2N-space)

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

  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. 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. PROPOSED METHOD • Rewriting basic equations - Eigenvalue problem (9) - Normalization condition (10)

  12. Differentiating eq.(9) with respect to design variable (11) • Differentiating eq.(10) with respect to design variable (12)

  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. v1 v2 • NUMERICAL EXAMPLE • Cantilever beam with lumped dampers Material Properties System Data Number of elements : 20 Number of nodes : 21 Number of DOF : 40 Design parameter : depth of beam

  15. Analysis Methods • Zeng’s method (1995) • Lee’s method (1999) • Proposed method • Comparisons • Solution time (CPU)

  16. 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 -0.1106 + 35.3632i 9 -0.1106 - 35.3632i 0.0213 - 3.4203i 10 0.0213 + 3.4203i • Results of Analysis (Eigenvalue)

  17. DOF number Eigenvector Eigenvector derivative 1 0.0013 + 0.0013i -0.0004 - 0.0004i 2 0.0050 + 0.0050i -0.0015 - 0.0015i 3 0.0049 + 0.0049i -0.0015 - 0.0015i 4 0.0096 + 0.0096i -0.0029 - 0.0029i 5 0.0108 + 0.0108i -0.0033 - 0.0032i 6 0.0139 + 0.0139i -0.0042 - 0.0042i 7 0.0188 + 0.0188i -0.0056 - 0.0056i 8 0.0179 + 0.0178i -0.0054 - 0.0053i 9 0.0287 + 0.0286i -0.0086 - 0.0085i 10 0.0215 + 0.0215i -0.0064 - 0.0064i • Results of Analysis (First eigenvector)

  18. CPU time Method Ratio (sec) Zeng’s method 184.05 115.8 Lee’s method 2.21 1.4 Proposed method 1.59 1.0 • CPU time for 40 Eigenpairs

  19. 200  184.05  150   CPU time (sec) 100   Improvement about 99% 61.47   50       2.21        Δ    Δ Δ Δ Δ Δ Δ Δ 0 1.59 5 10 15 20 25 30 35 40 Modes  : 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

  20. 2.5 2.21  Improvement about 25% 2  Δ 1.59 1.5  Δ CPU time (sec)  Δ Δ 1  Δ  Δ 0.5  Δ  Δ 0 5 10 15 20 25 30 35 40 Modes Fig 2. Comparison with Lee’s method  : Lee’s method (Exact solution)  : Proposed method(Exact solution)

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

  22. Thank you for your attention.

  23. APPENDIX • Numerical Stability • The determinant property (14)

  24. Then (15)

  25. Arranging eq.(15) (16) Using the determinant property of partitioned matrix (17)

  26. Therefore (18) Numerical Stability is Guaranteed.

  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. Differentiating eq.(3) with respect to design variable (21) • Combining eq.(19) and eq.(21) into a matrix gives eigenvector derivative. (22)

More Related