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Kang-Min Choi 1) , Woon-Hak Kim 2) and In-Won Lee 3)

The Fourteenth KKNN Semminar on Civil Engineering Kyoto International Conference Hall, Kyoto, Japan. Natural Frequencies and Mode Shape Sensitivities of Damped Systems with Multiple Natural Frequencies. Kang-Min Choi 1) , Woon-Hak Kim 2) and In-Won Lee 3)

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Kang-Min Choi 1) , Woon-Hak Kim 2) and In-Won Lee 3)

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  1. The Fourteenth KKNN Semminar on Civil Engineering Kyoto International Conference Hall, Kyoto, Japan Natural Frequencies and Mode Shape Sensitivities of Damped Systems with Multiple Natural Frequencies Kang-Min Choi1), Woon-Hak Kim2) and In-Won Lee3) 1) Graduate Student, Department of Civil Engineering, KAIST 2) Professor, Department of Civil Engineering, Hankyung 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. Therefore (16) Numerical Stability is Guaranteed.

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

  16. ●Results of Analysis

  17. ♦ 5-DOF Non-proportional Damped System

  18. ●Results of Analysis

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

  20. Thank you for your attention.

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