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

2. Contents Introduction Previous Studies Proposed Methods Numerical Example Conclusions

3. 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. 4 For asymmetric systems, few sensitivity technique has been developed Many real systems have asymmetric mass, damping and stiffness matrices. - moving vehicles on roads - ship motion in sea water - offshore structures

5. 5 Given: Find: Sensitivity Analysis Design parameter

6. 6 - 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 Conventional Modal Method for Symmetric System

7. 7 - 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. Modified Modal Method for Symmetric System

8. 8 - 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. Conventional Modal Method for Asymmetric System

9. 9 – Expand and as complex linear combinations of and (2) (1) Basic Idea of Modal Method 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. 10 (3) (4) - The derivatives of right eigenvectors - The derivatives of left eigenvectors From this idea, the eigenvector derivatives can be obtained

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

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

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

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

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

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

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

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

19. 19 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)

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

21. 21 (22) By the similar procedure

22. 22 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

23. 23 Equation of motion where

24. 24 Design parameter : L Material Property

25. 25 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

26. 26 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

27. 27 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)

28. 28 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)

29. 29 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