Simplified Algebraic Method Fifth European Conference of Structural Dynamics EURODYN 2002 Munich, Germany Sept. 2 - 5, 2002 Simplified Algebraic Method for Computing Eigenpair Sensitivities of Damped System with Repeated Eigenvalues Kang-Min Choi, Graduate Student, KAIST, Korea Woon-Hak Kim, Professor, Hankyong National University, Korea In-Won Lee, Professor, KAIST, Korea

CONTENTS  INTRODUCTION  PREVIOUS METHODS  PROPOSED METHOD  NUMERICAL EXAMPLES  CONCLUSIONS Structural Dynamics & Vibration Control Lab., KAIST, Korea

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 equal eigenvalues, due to structural symmetry. The second- and higher order derivatives of eigenpairs are important to predict the eigenpairs, which relies on the matrix Taylor series expansion. Structural Dynamics & Vibration Control Lab., KAIST, Korea

Problem Definition Eigenvalue problem of damped system (1) Structural Dynamics & Vibration Control Lab., KAIST, Korea

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

PREVIOUS STUDIES Damped system with distinct eigenvalues H. K. Jo, J. H. Lee and I. W. Lee, “Sensitivity Analysis of Non-conservative Eigensystems: Part I, Symmetric Systems,” Journal of Sound and Vibration, 2001. (submitted) (2) 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

Undamped system with repeated eigenvalues R. L. Dailey, “Eigenvector Derivatives with Repeated Eigenvalues,” 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

Second order derivatives of Undamped system with repeated 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

PROPOSED METHOD First-order eigenpair derivatives of damped system with repeated eigenvalues Second-order eigenpair derivatives of damped system with repeated eigenvalues Numerical stability of the proposed method Structural Dynamics & Vibration Control Lab., KAIST, Korea

First-order eigenpair derivatives of damped system with repeated eigenvalues Basic Equations Eigenvalue problem (5) Orthonormalization condition (6) Structural Dynamics & Vibration Control Lab., KAIST, Korea

Adjacent eigenvectors (7) where T is an orthogonal transformation matrix and its order m (8) Structural Dynamics & Vibration Control Lab., KAIST, Korea

Rearranging eq.(5) and eq.(6) using adjacent eigenvectors (9) (10) Structural Dynamics & Vibration Control Lab., KAIST, Korea

Differentiating eq.(9) w.r.t. design parameter α (11) Pre-multiplying at each side of eq.(11) by and substituting (12) where Structural Dynamics & Vibration Control Lab., KAIST, Korea

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

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

Second-order eigenpair derivatives of damped system with 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

Combining eq.(16) and eq.(17) into a single equation (18) where Structural Dynamics & Vibration Control Lab., KAIST, Korea

Numerical stability of the proposed method Determinant property (19) Structural Dynamics & Vibration Control Lab., KAIST, Korea

Then, (20) (21) Structural Dynamics & Vibration Control Lab., KAIST, Korea

Using the determinant property of partitioned matrix Arranging eq.(20) (22) Using the determinant property of partitioned matrix (23) Structural Dynamics & Vibration Control Lab., KAIST, Korea

Numerical Stability is Guaranteed. Therefore (24) Numerical Stability is Guaranteed. Structural Dynamics & Vibration Control Lab., KAIST, Korea

NUMERICAL EXAMPLES Cantilever beam - Proportionally damped system 5-DOF mechanical system - Non-proportionally damped system Structural Dynamics & Vibration Control Lab., KAIST, Korea

Cantilever beam (proportionally damped system) Structural Dynamics & Vibration Control Lab., KAIST, Korea

First derivatives of eigenvalues Second derivatives of eigenvalues Results of analysis (eigenvalues) Mode number Eigenvalues First derivatives of eigenvalues Second derivatives of eigenvalues 1, 2 -1.4279e-03 ±j5.2496e+00 -2.8057e-10 j3.5347e-10 4.3916e-09 ±j1.0285e-08 3, 4 -2.2756e-02 ±j5.2494e+01 -2.7553e-01 j6.1102e-02 5, 6 -5.4154e-02 ±j3.2895e+01 -6.6265e-10 ±j2.3445e-10 1.0084e-08 j2.4918e-09 7, 8 -1.0818e+00 ±j3.2886e+02 -1.0391e-08 j2.6913e+00 9, 10 -4.2409e-01 ±j9.2090e+01 6.9247e-10 j6.9600e-10 ±j1.1514e-08 11, 12 -8.4753e+00 ±j9.2029e+02 -8.4535e+01 j1.8358e+01 ± ± ± ± ± ± Structural Dynamics & Vibration Control Lab., KAIST, Korea

First derivatives of eigenvectors Second derivatives of eigenvectors Results of analysis (first eigenvectors) DOF number Eigenvectors First derivatives of eigenvectors Second derivatives of eigenvectors 1 2 3 -6.6892e+05 -j6.6892e+05 3.3446e-04 +j3.3446e-04 -5.0169e+03 -j5.0169e+03 4 -2.6442e+04 -j2.6442e+04 1.3221e-03 +j1.3221e-03 -1.9596e+02 -j1.9596e+02 77 78 79 -1.5577e+02 -j1.5577e+02 7.7887e-02 +j7.7887e-02 -1.1683e+00 -j1.1683e+00 80 -2.1442e+03 -j2.1442e+03 1.0721e-02 +j1.0721e-02 -1.6082e+01 -j1.6082e+01 … … … … Structural Dynamics & Vibration Control Lab., KAIST, Korea

Results of analysis (errors of approximations) Mode number Actual Eigenvalues Approximated eigenvalues Variations of eigenpairs Errors of approximations Eigenvectors 1, 2 -1.4279e-03 ±j5.2496e+00 2.2281e-11 4.9628e-03 2.2283e-11 3.7376e-05 3, 4 -1.4556e-03 ±j5.3021e+00 -1.4555e-03 1.0000e-02 9.9010e-03 2.6622e-08 1.0000e-04 5, 6 -5.4154e-02 ±j3.2895e+01 3.7084e-12 3.6899e-12 7, 8 -5.5241e-02 ±j3.3224e+01 -5.5236e-02 9.9997e-04 9.9023e-03 1.6763e-07 1.0001e-04 9, 10 -4.2409e-01 ±j9.2090e+01 9.1400e-12 9.1432e-12 11, 12 -4.3261e-01 ±j9.3010e+01 -4.3256e-01 9.9936e-03 9.9041e-03 4.6508e-07 1.0002e-04 Structural Dynamics & Vibration Control Lab., KAIST, Korea

5-DOF mechanical system (Non-proportionally damped system) Structural Dynamics & Vibration Control Lab., KAIST, Korea

First derivatives of eigenvalues Second derivatives of eigenvalues Results of analysis (eigenvalues) Mode number Eigenvalues First derivatives of eigenvalues Second derivatives of eigenvalues 1, 2 -4.3262e-02 ±j1.5023e+00 9.6943e-07 ±j1.7995e-04 -1.4634e-08 j2.4680e-07 3, 4 -2.4000e-01 ±j3.4558e+00 0.0000e+00 ±j0.0000e+00 5, 6 ±j8.6811e-04 1.6409e-08 j1.4913e-07 7, 8 -3.5202e-02 ±j6.1354e+00 -7.8926e-07 ±j2.9526e-05 -1.7067e-09 ±j1.4301e-08 9, 10 -2.4535e-02 ±j9.7000e+00 -1.8017e-07 ±j5.0001e-06 -6.8945e-11 ±j8.4803e-10 ± ± Structural Dynamics & Vibration Control Lab., KAIST, Korea

First derivatives of eigenvectors Second derivatives of eigenvectors Results of analysis (first eigenvectors) DOF number Eigenvectors First derivatives of eigenvectors Second derivatives of eigenvectors 1 1.0851e-02 -j1.0743e-02 -4.1482e-07 +j4.2149e-07 3.9185e-10 -j2.7842e-10 2 1.0334e-02 -j1.0284e-02 -5.1501e-07 +j5.1709e-07 5.2496e-10 -j4.2633e-10 3 9.4601e-03 -j9.5112e-03 -6.7729e-07 +j6.7328e-07 7.4423e-10 -j6.7172e-10 4 0.0000e+00 +j0.0000e+00 5.4601e-06 -j6.2004e-06 -6.5680e-09 +j8.1097e-09 5 -5.4901e-06 +j6.2004e-06 6.5680e-09 -j8.1097e-09 Structural Dynamics & Vibration Control Lab., KAIST, Korea

Results of analysis (errors of approximations) Mode number Actual Eigenvalues Approximated eigenvalues Variations of eigenpairs Errors of approximations Eigenvectors 1, 2 -4.3243e-02 ±j1.5040e+00 -4.3253e-02 ±j1.5041e+00 1.1893e-03 4.6721e-03 8.1631e-07 2.9463e-05 3, 4 -2.4000e-01 ±j3.4558e+00 0.0000e+00 5, 6 ±j3.4645e+00 2.5039e-03 1.5461e-03 2.1632e-06 5.2014e-06 7, 8 -3.5210e-02 ±j6.1357e+00 -3.5210e-02 ±j6.1357e+00 4.8257e-05 1.0987e-03 1.1763e-07 2.5394e-06 9, 10 -2.4537e-02 ±j9.7000e+00 5.1624e-06 1.9422e-04 4.3893e-09 1.6332e-07 Structural Dynamics & Vibration Control Lab., KAIST, Korea

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

Thank you for your attention! An efficient eigensensitivity method for the damped system with repeated eigenvalues Thank you for your attention! Structural Dynamics & Vibration Control Lab., KAIST, Korea