2001 Natural Frequencies and Mode Shape Sensitivities of

한국지진공학회 2001년도 추계학술발표회 Natural Frequencies and Mode Shape Sensitivities of Damped Systems with Multiple Natural Frequencies Choi, Kang-Min 1), *Ko, Man-Gi 2) and Lee In-Won 3) 1) Graduate Student, Department of Civil Engineering, KAIST 2) Professor, Department of Civil Engineering, Kong. Ju National Univ. 3) Professor, Department of Civil Engineering, KAIST

l OUTLINE • INTRODUCTION • PROPOSED METHOD • NUMERICAL EXAMPLES • CONCLUSIONS Structural Dynamics and Vibration Control Lab. , KAIST, Korea 2

l INTODUCTION • Objective of Study - To find the derivatives of eigenvalues and eigenvectors of damped systems with multiple eigenvalues with respect to design parameters. - Typical structures have many multiple or nearly equal eigenvalues, due to structural symmetries. Structural Dynamics and Vibration Control Lab. , KAIST, Korea 3

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

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

l PROPOSED METHOD • Basic Equations - Eigenvalue problem (2) - Orthonormalization condition (3) Structural Dynamics and Vibration Control Lab. , KAIST, Korea 6

- Adjacent eigenvectors (4) where T is an orthogonal transformation matrix and its order m (5) Structural Dynamics and Vibration Control Lab. , KAIST, Korea 7

• Rewriting Basic Equations - Another eigenvalue problem (6) - Orthonormalization condition (7) Structural Dynamics and Vibration Control Lab. , KAIST, Korea 8

Differentiating eq. (6) with respect to design parameter α (8) Differentiating eq. (7) with respect to design parameter α (9) Structural Dynamics and Vibration Control Lab. , KAIST, Korea 9

Combining eq. (8) and eq. (9) into a single matrix (10) - The coefficient matrix is symmetric and non-singular. - Eigenpair derivatives are obtained simultaneously. - Only corresponding eigenpair is required. Structural Dynamics and Vibration Control Lab. , KAIST, Korea 10

• Numerical Stability - Determinant property (11) Structural Dynamics and Vibration Control Lab. , KAIST, Korea 11

Then, (12) Structural Dynamics and Vibration Control Lab. , KAIST, Korea 12

Arranging eq. (12) (13) Using the determinant property of partitioned matrix (14) Structural Dynamics and Vibration Control Lab. , KAIST, Korea 13

Therefore (15) Numerical Stability is Guaranteed. Structural Dynamics and Vibration Control Lab. , KAIST, Korea 14

l NUMERICAL EXAMPLES • Cantilever Beam (proportionally damped system) Structural Dynamics and Vibration Control Lab. , KAIST, Korea 15

• Verification of results : approximated eigenvalue : approximated eigenvector Structural Dynamics and Vibration Control Lab. , KAIST, Korea 16

• Results of Analysis Mode Number Eigenvalue derivative Changed eigenvalue Approximated Eigenvalue 1 -1. 43 e-03 – j 5. 25 e+00 -8. 67 e-11 + j 2. 50 e-10 -1. 43 e-03 – j 5. 25 e+00 2 -1. 43 e-03 + j 5. 25 e+00 -2. 81 e-10 – j 3. 53 e-10 -1. 43 e-03 + j 5. 25 e+00 3 -1. 43 e-03 – j 5. 25 e+00 -2. 76 e-02 – j 5. 25 e+01 -1. 46 e-03 – j 5. 30 e+00 4 -1. 43 e-03 + j 5. 25 e+00 -2. 76 e-02 + j 5. 25 e+01 -1. 46 e-03 + j 5. 30 e+00 5 -5. 42 e-02 – j 3. 29 e+01 -6. 63 e-10 – j 2. 34 e-10 -5. 42 e-02 – j 3. 29 e+01 6 -5. 42 e-02 + j 3. 29 e+01 -6. 63 e-10 + j 2. 16 e-10 -5. 42 e-02 + j 3. 29 e+01 7 -5. 42 e-02 – j 3. 29 e+01 -1. 08 e+00 – j 3. 29 e+02 -5. 52 e-02 – j 3. 32 e+01 8 -5. 42 e-02 + j 3. 29 e+01 -1. 08 e+00 + j 3. 29 e+02 -5. 52 e-02 + j 3. 32 e+01 9 -4. 24 e-01 – j 9. 21 e+01 6. 98 e-10 + j 7. 80 e-10 -4. 24 e-01 – j 9. 21 e+01 10 -4. 24 e-01 + j 9. 21 e+01 6. 92 e-10 – j 6. 96 e-10 -4. 24 e-01 + j 9. 21 e+01 11 -4. 24 e-01 – j 9. 21 e+01 -8. 47 e+00 – j 9. 20 e+02 -4. 33 e-01 – j 9. 30 e+01 12 -4. 24 e-01 + j 9. 21 e+01 -8. 47 e+00 + j 9. 20 e+02 -4. 33 e-01 + j 9. 30 e+01 Structural Dynamics and Vibration Control Lab. , KAIST, Korea 17

Mode Number Error of eigenvalue Error of eigenvector 1 2. 2283 e-11 3. 7376 e-05 2 2. 2283 e-11 3. 7376 e-05 3 2. 6622 e-08 1. 0000 e-04 4 2. 6622 e-08 1. 0000 e-04 5 3. 6872 e-12 3. 7376 e-05 6 3. 6899 e-12 3. 7376 e-05 7 1. 6763 e-07 1. 0001 e-04 8 1. 6763 e-07 1. 0001 e-04 9 9. 1485 e-12 3. 7376 e-05 10 9. 1432 e-12 3. 7376 e-05 11 4. 6508 e-07 1. 0002 e-04 12 4. 6508 e-07 9. 9041 e-03 Structural Dynamics and Vibration Control Lab. , KAIST, Korea 18

• 5 -DOF Non-proportional Damped System Structural Dynamics and Vibration Control Lab. , KAIST, Korea 19

• Results of Analysis Mode Number Eigenvalue derivative Eigenvalue Changed eigenvalue Approximated Eigenvalue 1 -4. 33 e-02 – j 1. 50 e+00 9. 69 e-07 – j 1. 80 e-04 -4. 32 e-02 – j 1. 50 e+00 2 -4. 33 e-02 + j 1. 50 e+00 9. 69 e-07 + j 1. 80 e-04 -4. 32 e-02 + j 1. 50 e+00 3 -2. 40 e-01 – j 3. 46 e+00 0 -2. 40 e-01 – j 3. 45 e+00 4 -2. 40 e-01 + j 3. 46 e+00 0 -2. 40 e-01 + j 3. 45 e+00 5 -2. 40 e-01 – j 3. 46 e+00 -1. 63 e-19 – j 8. 68 e-04 -2. 40 e-01 – j 3. 46 e+00 6 -2. 40 e-01 + j 3. 46 e+00 -1. 08 e-19 + j 8. 68 e-04 -2. 40 e-01 + j 3. 46 e+00 7 -3. 52 e-02 – j 6. 14 e+00 -7. 89 e-07 – j 2. 95 e-05 -3. 52 e-02 – j 6. 13 e+00 8 -3. 52 e-02 + j 6. 14 e+00 -7. 89 e-07 + j 2. 95 e-05 -3. 52 e-02 + j 6. 13 e+00 9 -2. 45 e-02 – j 9. 70 e+00 -1. 80 e-07 – j 5. 00 e-06 -2. 45 e-02 – j 9. 70 e+00 10 -2. 45 e-02 + j 9. 70 e+00 -1. 80 e-07 + j 5. 00 e-06 -2. 45 e-02 + j 9. 70 e+00 Structural Dynamics and Vibration Control Lab. , KAIST, Korea 20

Mode Number Error of eigenvalue Error of eigenvector 1 8. 1631 e-06 2. 9463 e-05 2 8. 1631 e-06 2. 9463 e-05 3 8. 4309 e-16 1. 2945 e-14 4 7. 0672 e-16 1. 2770 e-14 5 2. 1632 e-06 5. 2014 e-06 6 2. 1632 e-06 5. 2014 e-06 7 1. 1763 e-07 2. 5394 e-06 8 1. 1763 e-07 2. 5394 e-06 9 4. 3893 e-09 1. 6332 e-07 10 4. 3893 e-09 1. 6332 e-07 Structural Dynamics and Vibration Control Lab. , KAIST, Korea 21

l CONCLUSIONS • Proposed Method - is simple - guarantees numerical stability An efficient eigensensitivity method for the damped system with repeated eigenvalues Structural Dynamics and Vibration Control Lab. , KAIST, Korea 22

• Future Work - Proposed method is able to extend to second-order sensitivity of the damped systems with distinct eigenvalue and multiple eigenvalue. Structural Dynamics and Vibration Control Lab. , KAIST, Korea 23

Thank you for your attention. Structural Dynamics and Vibration Control Lab. , KAIST, Korea 24