ISEC-02 Second International Conference on Structural Engineering and Construction Algebraic Method for Sensitivity Analysis of Eigensystems with Repeated.

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Presentation transcript:

ISEC-02 Second International Conference on Structural Engineering and Construction Algebraic Method for Sensitivity Analysis of Eigensystems with Repeated Eigenvalues Kang-Min Choi1), Sang-Won Cho2), Jong-Heon Lee and In-Won Lee3) 1), 2) Graduate Student, Department of Civil Engineering, KAIST 3) Professor, Department of Civil Engineering, Kyungil Univ. 3) Professor, Department of Civil Engineering, KAIST

OUTLINE  INTRODUCTION  PROPOSED METHOD  NUMERICAL EXAMPLES  CONCLUSIONS

INTODUCTION  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  To find the derivatives of eigenvalues and eigenvectors of damped systems with multiple eigenvalues according to design variables.  Typical structures have many multiple or nearly equal eigenvalues, due to structural symmetries.

♦ Problem Definition ● Eigenvalue problem of damped system (1)

● Objective Given: Find: * represents the derivative of with respect design variable α (length, area, moment of inertia, etc.)

PROPOSED METHOD ♦ Basic Equations ● Eigenvalue problem (2) ● Orthonormalization condition (3)

● Adjacent eigenvectors (4) where T is an orthogonal transformation matrix and its order m (5)

♦ Rewriting Basic Equations ● Another eigenvalue problem (6) ● Orthonormalization condition (7)

Differentiating eq.(6) with respect to design parameter α (8) Differentiating eq.(7) with respect to design parameter α (9)

Combining eq.(8) and eq.(9) into a single matrix (10) ● 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.

♦ Numerical Stability ● Determinant property (11)

Then, (12) (13)

Arranging eq.(12) (14) Using the determinant property of partitioned matrix (15)

Numerical Stability is Guaranteed. Therefore (16) Numerical Stability is Guaranteed.

NUMERICAL EXAMPLES ♦ Cantilever Beam (proportionally damped system)

Eigenvalue derivative ● Results of Analysis Mode Number Eigenvalue Eigenvalue derivative Changed eigenvalue Approximated eigenvalue 1 2 3 4 5 6 7 8 9 10 11 12 -1.43e-03 – j5.25e+00 -1.43e-03 + j5.25e+00 -1.43e-03 – j5.25e+00 -5.42e-02 – j3.29e+01 -5.42e-02 + j3.29e+01 -4.24e-01 – j9.21e+01 -4.24e-01 + j9.21e+01 -8.67e-11 + j2.50e-10 -2.81e-10 – j3.53e-10 -2.76e-02 – j5.25e+01 -2.76e-02 + j5.25e+01 -6.63e-10 – j2.34e-10 -6.63e-10 + j2.16e-10 -1.08e+00 – j3.29e+02 -1.08e+00 + j3.29e+02 6.98e-10 + j7.80e-10 6.92e-10 – j6.96e-10 -8.47e+00 – j9.20e+02 -8.47e+00 + j9.20e+02 -1.46e-03 – j5.30e+00 -1.46e-03 + j5.30e+00 -5.52e-02 – j3.32e+01 -5.52e-02 + j3.32e+01 -4.33e-01 – j9.30e+01 -4.33e-01 + j9.30e+01

Mode Number Error of eigenvalue Error of eigenvector 1 2 3 4 5 6 7 8 9 10 11 12 2.2283e-11 2.6622e-08 3.6872e-12 3.6899e-12 1.6763e-07 9.1485e-12 9.1432e-12 4.6508e-07 3.7376e-05 1.0000e-04 1.0001e-04 1.0002e-04 9.9041e-03

♦ 5-DOF Non-proportional Damped System

Eigenvalue derivative ● Results of Analysis Mode Number Eigenvalue Eigenvalue derivative Changed eigenvalue Approximated eigenvalue 1 2 3 4 5 6 7 8 9 10 -4.33e-02 – j1.50e+00 -4.33e-02 + j1.50e+00 -2.40e-01 – j3.46e+00 -2.40e-01 + j3.46e+00 -3.52e-02 – j6.14e+00 -3.52e-02 + j6.14e+00 -2.45e-02 – j9.70e+00 -2.45e-02 + j9.70e+00 9.69e-07 – j1.80e-04 9.69e-07 + j1.80e-04 -1.63e-19 – j8.68e-04 -1.08e-19 + j8.68e-04 -7.89e-07 – j2.95e-05 -7.89e-07 + j2.95e-05 -1.80e-07 – j5.00e-06 -1.80e-07 + j5.00e-06 -4.32e-02 – j1.50e+00 -4.32e-02 + j1.50e+00 -2.40e-01 – j3.45e+00 -2.40e-01 + j3.45e+00 -3.52e-02 – j6.13e+00 -3.52e-02 + j6.13e+00

Mode Number Error of eigenvalue Error of eigenvector 1 2 3 4 5 6 7 8 9 10 8.1631e-06 8.4309e-16 7.0672e-16 2.1632e-06 1.1763e-07 4.3893e-09 2.9463e-05 1.2945e-14 1.2770e-14 5.2014e-06 2.5394e-06 1.6332e-07

CONCLUSIONS An efficient eigensensitivity method for the ♦ Proposed Method ● is simple ● guarantees numerical stability An efficient eigensensitivity method for the damped system with multiple eigenvalues

Thank you for your attention.