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Kang-Min Choi, Graduate Student, KAIST, Korea Young-Jong Moon , Graduate Student, KAIST, Korea

Algebraic Method for Eigenpair Derivatives of Damped System with Repeated Eigenvalues. Kang-Min Choi, Graduate Student, KAIST, Korea Young-Jong Moon , Graduate Student, KAIST, Korea Ji-Eun Jang , Graduate Student, KAIST, Korea

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Kang-Min Choi, Graduate Student, KAIST, Korea Young-Jong Moon , Graduate Student, KAIST, Korea

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  1. Algebraic Method for Eigenpair Derivatives of Damped System with Repeated Eigenvalues Kang-Min Choi, Graduate Student, KAIST, Korea Young-Jong Moon, Graduate Student, KAIST, Korea Ji-Eun Jang, Graduate Student, KAIST, Korea Woon-Hak Kim, Professor, Hankyong National University, Korea In-Won Lee, Professor, KAIST, Korea

  2. OUTLINE INTRODUCTION  PREVIOUS METHODS  PROPOSED METHOD  NUMERICAL EXAMPLE  CONCLUSIONS Structural Dynamics & Vibration Control Lab., KAIST, Korea

  3. INTRODUCTION • 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 • Typical structures have many repeated or nearly equaleigenvalues, due to structural symmetry. • The second- and higher order derivatives of eigenpairsare important to predict the eigenpairs, which relieson the matrix Taylor series expansion. Structural Dynamics & Vibration Control Lab., KAIST, Korea

  4. Problem Definition • Eigenvalue problem of damped system (1) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  5. * represents the derivative of with respect design parameter α(length, area, moment of inertia, etc.) • Objective of this study: Given: Find: Structural Dynamics & Vibration Control Lab., KAIST, Korea

  6. PREVIOUS STUDIES • Damped system with distinct eigenvalues • K. M. Choi, H. K. Jo, J. H. Lee andI. W. Lee, “Sensitivity Analysis of Non-conservative Eigensystems,” Journal of Sound and Vibration, 2001. (Accepted) • The coefficient matrix is symmetric and non-singular. • Eigenpair derivatives are obtained simultaneously. • The algorithm is simple and guarantees stability. Structural Dynamics & Vibration Control Lab., KAIST, Korea

  7. Undamped system with repeated eigenvalues • R. L. Dailey, “Eigenvector Derivatives with RepeatedEigenvalues,” AIAA Journal, Vol. 27, pp.486-491, 1989. • Introduction of Adjacent eigenvector • Calculation derivatives of eigenvectors by the sum of homogenous solutions and particular solutions using Nelson’s algorithm • Complicated algorithm and high time consumption Structural Dynamics & Vibration Control Lab., KAIST, Korea

  8. Second order derivatives of Undamped systemwith distinct eigenvalues • M. I. Friswell, “Calculation of Second and Higher Eigenvector Derivatives”, Journal of Guidance, Control and Dynamics, Vol. 18, pp.919-921, 1995. (3) (4) where - Second eigenvector derivatives extended by Nelson’s algorithm Structural Dynamics & Vibration Control Lab., KAIST, Korea

  9. PROPOSED METHOD • First-order eigenpair derivatives of damped systemwith repeated eigenvalues • Second-order eigenpair derivatives of damped systemwith repeated eigenvalues • Numerical stability of the proposed method Structural Dynamics & Vibration Control Lab., KAIST, Korea

  10. First-order eigenpair derivatives of damped systemwith repeated eigenvalues • Basic Equations • Eigenvalue problem (5) • Orthonormalization condition (6) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  11. Adjacent eigenvectors (7) where T is an orthogonal transformation matrix and its order m (8) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  12. Rearranging eq.(5) and eq.(6) using adjacent eigenvectors (9) (10) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  13. Differentiating eq.(9) w.r.t. design parameter α (11) • Pre-multiplying at each side of eq.(11) by andsubstituting (12) where Structural Dynamics & Vibration Control Lab., KAIST, Korea

  14. Differentiating eq.(9) w.r.t. design parameter α (13) • Differentiating eq.(10) w.r.t. design parameter α (14) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  15. Combining eq.(13) and eq.(14) into a single equation (15) - 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. Structural Dynamics & Vibration Control Lab., KAIST, Korea

  16. Second-order eigenpair derivatives of damped systemwith repeated eigenvalues • Differentiating eq.(13) w.r.t. another design parameter β (16) • Differentiating eq.(14) w.r.t. another design parameter β (17) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  17. Combining eq.(16) and eq.(17) into a single equation (18) where Structural Dynamics & Vibration Control Lab., KAIST, Korea

  18. Numerical stability of the proposed method • Determinant property (19) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  19. Then, (20) (21) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  20. Arranging eq.(20) (22) • Using the determinant property of partitioned matrix (23) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  21. Therefore (24) Numerical Stability is Guaranteed. Structural Dynamics & Vibration Control Lab., KAIST, Korea

  22. NUMERICAL EXAMPLE • Cantilever beam Structural Dynamics & Vibration Control Lab., KAIST, Korea

  23. Results of analysis (eigenvalues) ± ± ± ± ± ± Structural Dynamics & Vibration Control Lab., KAIST, Korea

  24. Results of analysis (first eigenvectors) … … … … Structural Dynamics & Vibration Control Lab., KAIST, Korea

  25. Results of analysis (errors of approximations) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  26. CONCLUSIONS • Proposed Method • is an efficient eigensensitivity method for the damped system with repeated eigenvalues • guarantees numerical stability • gives exact solutions of eigenpair derivatives • can be extended to obtain second- and higher order derivatives of eigenpairs Structural Dynamics & Vibration Control Lab., KAIST, Korea

  27. An efficient eigensensitivity method for the damped system with repeated eigenvalues Thank you for your attention! Structural Dynamics & Vibration Control Lab., KAIST, Korea

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