1. Yeong-Jong Moon*: Graduate Student, KAIST, Korea Kang-Min Choi: Graduate Student, KAIST, Korea Hyun-Woo Lim: Graduate Student, KAIST, Korea Jong-Heon Lee: Professor, Kyungil University, Korea In-Won Lee: Professor, KAIST, Korea Modified Modal Methods for Calculating Eigenpair Sensitivity of Asymmetric Damped Systems EASEC-9, Bali, Indonesia 16-18, December, 2003

2. Structural Dynamics & Vibration Control Lab., KAIST, Korea 2 Contents Introduction Previous Studies Proposed Methods Numerical Example Conclusions

3. Structural Dynamics & Vibration Control Lab., KAIST, Korea 3 Recently Adhikari and M. I. Friswell proposed a modal method for asymmetric damped systems. Many real systems have asymmetric mass, damping and stiffness matrices. - moving vehicles on roads - ship motion in sea water - offshore structures Introduction

4. Structural Dynamics & Vibration Control Lab., KAIST, Korea 4 Given: Find: Sensitivity Analysis Design parameter

5. Structural Dynamics & Vibration Control Lab., KAIST, Korea 5 - Propose the modal method for sensitivity technique of symmetric system - The accuracy is dependent on the number of modes used in calculation K. B. Lim and J. L. Junkins, “Re-examination of Eigenvector Derivatives”, Journal of Guidance, 10, 581-587, 1987. Previous Studies

6. Structural Dynamics & Vibration Control Lab., KAIST, Korea 6 - Modified modal method for symmetric system - This method achieved highly accurate results using only a few lower modes. Q. H. Zeng, “Highly Accurate Modal Method for Calculating Eigenvector Derivative in Viscous Damping Systems”, AIAA Journal, 33(4), 746-751, 1994.

7. Structural Dynamics & Vibration Control Lab., KAIST, Korea 7 - 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 S. Adhikari and M. I. Friswell, “Eigenderivative Analysis of Asymmetric Non-Conservative Systems”, International Journal for Numerical Methods in Engineering, 51, 709-733, 2001.

8. Structural Dynamics & Vibration Control Lab., KAIST, Korea 8 – Expand and as complex linear combinations of and (2) (1) Modal Method for Asymmetric System 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

9. Structural Dynamics & Vibration Control Lab., KAIST, Korea 9 (3) (4) - The derivatives of right eigenvectors - The derivatives of left eigenvectors From this idea, the eigenvector derivatives can be obtained

10. Structural Dynamics & Vibration Control Lab., KAIST, Korea 10 Objective - Develop the effective sensitivity techniques for asymmetric damped systems

11. Structural Dynamics & Vibration Control Lab., KAIST, Korea 11 Proposed Methods 1. Modal Acceleration Method 2. Multiple Modal Acceleration Method 3. Multiple modal Acceleration Method with Shifted Poles

12. Structural Dynamics & Vibration Control Lab., KAIST, Korea 12 Differentiate the Eq. (5) with a design parameter (5) (6) (7) 1. Modal Acceleration Method (MA) The general equation of motion for asymmetric systems

13. Structural Dynamics & Vibration Control Lab., KAIST, Korea 13 where (8) (9) (10) Separate the response into and

14. Structural Dynamics & Vibration Control Lab., KAIST, Korea 14 Substituting the Eq. (9) and (10) into the Eq. (8) By the similar procedure, the left eigenvector derivatives can be obtained (11) (12)

15. Structural Dynamics & Vibration Control Lab., KAIST, Korea 15 2. Multiple Modal Acceleration Method (MMA) where (13) (14) (15) Separate the response into and

16. Structural Dynamics & Vibration Control Lab., KAIST, Korea 16 Therefore the right eigenvector derivatives are given as By the similar procedure, (16) (17)

17. Structural Dynamics & Vibration Control Lab., KAIST, Korea 17 Based on the similar procedure, we can obtain the higher order equations (18) (19)

18. Structural Dynamics & Vibration Control Lab., KAIST, Korea 18 3.Multiple Modal Acceleration with Shifted-Poles (MMAS) For more high convergence rate, the term is expanded in Taylor’s series at the position (20)

19. Structural Dynamics & Vibration Control Lab., KAIST, Korea 19 Using the Eq. (20), we can obtain the following equation (21)

20. Structural Dynamics & Vibration Control Lab., KAIST, Korea 20 (22) By the similar procedure

21. Structural Dynamics & Vibration Control Lab., KAIST, Korea 21 M L Y X Z, z x y Figure 1. The whirling beam 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. Numerical Example

22. Structural Dynamics & Vibration Control Lab., KAIST, Korea 22 Equation of motion where

23. Structural Dynamics & Vibration Control Lab., KAIST, Korea 23 Design parameter : L Material Property

24. Structural Dynamics & Vibration Control Lab., KAIST, Korea 24 Mode NumberEigenvaluesDerivatives 1 -8.4987e-03 +2.3563e+00i 1.3251e-03 +1.5799e+00i 2 -2.7151e-03 +6.3523e+01i 2.2533e-03 +8.5934e-01i 3 1.6771e-02 +1.0548e+01i 3.3394e-03 +3.4034e-01i 8 -5.8579e-02 +1.8650e+01i -3.7909e-03 -3.3918e-01i 9 -4.7285e-02 +2.2774e+01i -2.2533e-03 -8.2215e-01i 10 -3.6890e-02 +2.6214e+01i -1.2833e-03 -1.0644e+00i Eigenvalues and their derivatives of system

25. Structural Dynamics & Vibration Control Lab., KAIST, Korea 25 DOF NumberEigenvectorDerivative 1 6.0874e-03 -6.2442e-06i 6.3118e-04 +6.3342e-06i 2 0.0000e+00 +0.0000e+00i 0.0000e+00 +0.0000e+00i 3 -7.4415e-03 +6.7358e-06i -7.6005e-04 -7.1917e-06i 8 +1.4785e-05 -1.4677e-02i -1.2799e-05 +5.9162e-03i 9 0.0000e+00 +0.0000e+00i 0.0000e+00 +0.00005e+00i 10 8.3733e-05 -5.7187e-02i -3.7941e-05 +1.6957e-02i First right eigenvector and its derivative

26. Structural Dynamics & Vibration Control Lab., KAIST, Korea 26 DOF Number Error (%) MA MMAMMAS 10.8310.2020.072 20.000 338.2581.5060.478 40.000 54.6310.1210.035 60.0800.0530.012 70.000 81.6790.5880.118 90.000 100.5200.1570.030 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)

27. Structural Dynamics & Vibration Control Lab., KAIST, Korea 27 DOF Number Error (%) 6 modes 4 modes2 modes 10.0723.4062.090 20.000 30.4780.4543.140 40.000 50.035 0.052 60.0120.6260.383 70.000 80.1180.1140.542 90.000 100.030 0.038 Errors of MMAS method using 2, 4 and 6 lower modes (%) (M=2, =eigenvalue –1)

28. Structural Dynamics & Vibration Control Lab., KAIST, Korea 28 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 Conclusions