1 / 27

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

matthew-snow
Download Presentation

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

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. 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

More Related