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Yeong-Jong Moon 1) , Jong-Heon Lee 2) and In-Won Lee 3)

Fifth European Conference of Structural Dynamics EURODYN 2002 Munich, Germany Sept. 2 - 5, 2002. Modified Modal Methods for Calculating Eigenpair Sensitivity of Asymmetric Damped Systems. Yeong-Jong Moon 1) , Jong-Heon Lee 2) and In-Won Lee 3)

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Yeong-Jong Moon 1) , Jong-Heon Lee 2) and In-Won Lee 3)

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  1. Fifth European Conference of Structural Dynamics EURODYN 2002 Munich, Germany Sept. 2 - 5, 2002 Modified Modal Methods for Calculating Eigenpair Sensitivity of Asymmetric Damped Systems Yeong-Jong Moon1), Jong-Heon Lee2) and In-Won Lee3) 1) Graduate Student, Department of Civil Engineering, KAIST 2) Professor, Department of Civil Engineering, Kyungil Univ. 3) Professor, Department of Civil Engineering, KAIST

  2. Contents • Introduction • Previous Studies • Proposed Methods • Numerical Example • Conclusions

  3. Introduction • Applications of Sensitivity Analysis - Determination of the sensitivity of dynamic response - Optimization of natural frequencies and mode shapes - Optimization of structures subject to natural frequencies • Many sensitivity techniques for symmetric systems have been developed

  4. Many real systems have asymmetric mass, damping and stiffness matrices. - moving vehicles on roads - ship motion in sea water - offshore structures • For asymmetric systems, few sensitivity technique has been developed

  5. Sensitivity Analysis Given: Find: Design parameter

  6. Previous Studies Conventional Modal Method for Symmetric System • K. B. Lim and J. L. Junkins, “Re-examination of Eigenvector Derivatives”, Journal of Guidance, 10, 581-587, 1987. • Propose the modal method for sensitivity technique of symmetric system • The accuracy is dependent on the number of modes used in calculation

  7. Modified Modal Method for Symmetric System • Q. H. Zeng, “Highly Accurate Modal Method for Calculating Eigenvector Derivative in Viscous Damping Systems”, AIAA Journal, 33(4), 746-751, 1994. • Modified modal method for symmetric system • This method achieved highly accurate results using only a few lower modes.

  8. Conventional Modal Method for Asymmetric System • S. Adhikari and M. I. Friswell, “Eigenderivative Analysis of Asymmetric Non-Conservative Systems”, International Journal for Numerical Methods in Engineering, 51, 709-733, 2001. • Propose the modal method for sensitivity technique of asymmetric system • The accuracy is dependent on the number of modes used in calculation • The truncation error may become significant

  9. Basic Idea of Modal Method – Expand and as complex linear combinations of and (1) (2) where : the j-th right eigenvector : the j-th left eigenvector : the derivatives of j-th right eigenvector : the derivatives of j-th left eigenvector

  10. From this idea, the eigenvector derivatives can be obtained - The derivatives of right eigenvectors (3) - The derivatives of left eigenvectors (4)

  11. Objective - Develop the effective sensitivity techniques for asymmetric damped systems

  12. Proposed Methods 1. Modal Acceleration Method 2. Multiple Modal Acceleration Method 3. Multiple modal Acceleration Method with Shifted Poles

  13. 1. Modal Acceleration Method (MA) • The general equation of motion for asymmetric systems (5) (6) • Differentiate the Eq. (5) with a design parameter (7)

  14. • Separate the response into and (8) where (9) (10)

  15. • Substituting the Eq. (9) and (10) into the Eq. (8) (11) • By the similar procedure, the left eigenvector derivatives can be obtained (12)

  16. 2. Multiple Modal Acceleration Method (MMA) • Separate the response into and (13) where (14) (15)

  17. • Therefore the right eigenvector derivatives are given as (16) • By the similar procedure, (17)

  18. • Based on the similar procedure, we can obtain the higher order equations (18) (19)

  19. Multiple Modal Acceleration with Shifted-Poles (MMAS) • For more high convergence rate, the term is expanded in Taylor’s series at the position (20)

  20. • Using the Eq. (20), we can obtain the following equation (21)

  21. • By the similar procedure (22)

  22. Y y M x X Z, z L • Numerical Example L. Meirovitch and G. Ryland, “A Perturbation Technique for Gyroscopic Systems with Small Internal and External Damping,” Journal of Sound and Vibration, 100(3), 393-408, 1985. Figure 1. The whirling beam

  23. Equation of motion where

  24. Material Property Design parameter : L

  25. Eigenvalues and their derivatives of system

  26. First right eigenvector and its derivative

  27. Errors of modified modal methods using six modes (%) • MA : Modal Acceleration Method • MMA : Multiple Modal Acceleration Method (M=2) • MMAS : Multiple Modal Accelerations with Shifted Poles (M=2, =eigenvalue –1)

  28. Errors of MMAS method using 2, 4 and 6 lower modes (%) (M=2, =eigenvalue –1)

  29. Conclusions • The modified modal methods for the eigenpair derivatives of asymmetric damped systems is derived • In the proposed methods, the eigenvector derivatives of asymmetric systems can be calculated by using only a few lower modes • Multiple modal acceleration method with shifted poles is the most efficient technique of proposed methods

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