* In-Won Lee 1), Sun-Kyu Park 2) and Hong-Ki Jo 3) 1) Professor, Department of Civil Engineering, KAIST 2) Professor, Department of Civil Engineering, SungKyunKwan Univ. 3) Graduate Student, Department of Civil Engineering, KAIST SIMPLIFIED ALGEBTAIC METHOD FOR COMPUTING EIGENPAIR SENSITIVITIES OF DAMPED SYSTEM ECCOMAS 2000 Barcelona, Spain, Sep , 2000 Structural Dynamics and Vibration Control Lab., KAIST, Korea 2 OUTLINE INTRODUCTION PREVIOUS STUDIES PROPOSED METHOD NUMERICAL EXAMPLE CONCLUSIONS Structural Dynamics and Vibration Control Lab., KAIST, Korea 3 INTODUCTION Objective of Study Applications of Sensitivity Analysis - Determination of the sensitivity of dynamic responses - 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 respect to design variables. Structural Dynamics and Vibration Control Lab., KAIST, Korea 4 Problem Definition (1) - Eigenvalue problem of damped system Structural Dynamics and Vibration Control Lab., KAIST, Korea 5 (2) - Orthonormalization condition - State space equation (3) Structural Dynamics and Vibration Control Lab., KAIST, Korea 6 Given: Find: - Objective * indicates derivatives with respect to design variables (length, area, moment of inertia, etc.) Structural Dynamics and Vibration Control Lab., KAIST, Korea 7 PREVIOUS STUDIES Z. Zimoch, Sensitivity Analysis of Vibrating Systems, Journal of Sound and Vibration, Vol. 117, pp , restricted to lumped systems with distinct eigenvalues. (4) Structural Dynamics and Vibration Control Lab., KAIST, Korea 8 Q. H. Zeng, Highly Accurate Modal Method for Calculating Eigenvector Derivatives in Viscous Damping System, AIAA Journal, Vol. 33, No. 4, pp , many eigenvectors are required to calculate eigenvector derivatives. (5) (6) Structural Dynamics and Vibration Control Lab., KAIST, Korea 9 Sondipon Adhikari, Calculation of Derivative of Complex Modes Using Classical Normal Modes, Computer & Structures, Vol. 77, No. 6, pp , applicable only when the elements of C are small. (7) Structural Dynamics and Vibration Control Lab., KAIST, Korea 10 I. W. Lee, D. O. Kim and G. H. Jung, Natural Frequency and Mode Shape Sensitivities of Damped Systems: part I, Distinct Natural Frequencies, Journal of Sound and Vibration, Vol. 223, No. 3, pp , I. W. Lee, D. O. Kim and G. H. Jung, Natural Frequency and Mode Shape Sensitivities of Damped Systems: part II, Multiple Natural Frequencies, Journal of Sound and Vibration, Vol. 223, No. 3, pp , 1999. Structural Dynamics and Vibration Control Lab., KAIST, Korea 11 Lees method (1999) (8) (9) - eigenvalue and eigenvector derivatives are obtained separately. Structural Dynamics and Vibration Control Lab., KAIST, Korea 12 PROPOSED METHOD Rewriting basic equations - Eigenvalue problem - Orthonormalization condition (10) (11) Structural Dynamics and Vibration Control Lab., KAIST, Korea 13 Differentiating eq.(10) with respect to design variable Differentiating eq.(11) with respect to design variable (12) (13) Structural Dynamics and Vibration Control Lab., KAIST, Korea 14 Combining eq.(12) and eq.(13) into a single matrix (14) - the coefficient matrix is symmetric and non-singular. - eigenpair derivatives are obtained simultaneously. simultaneously. Structural Dynamics and Vibration Control Lab., KAIST, Korea 15 NUMERICAL EXAMPLE Cantilever Beam Structural Dynamics and Vibration Control Lab., KAIST, Korea 16 Analysis Methods Lees method (1999) Proposed method Comparisons Solution time (CPU) Structural Dynamics and Vibration Control Lab., KAIST, Korea 17 Results of Analysis (Eigenvalue) Mode Number Eigenvalue Eigenvalue derivative (Lees method) Eigenvalue derivative (Proposed method) i i i i i-5.411e e+2i i-5.411e e+2i i4.770e e-8i i4.721e e-7i i-4.242e e+2i i-4.242e e+2i i-1.628e e+3i i-1.628e e+3i Same Structural Dynamics and Vibration Control Lab., KAIST, Korea 18 Results of Analysis (First eigenvector) Eqn. number Eigenvector Eigenvector derivative (Lees method) Eigenvector derivative (Proposed method) e e-05i-3.027e e-04i e e-04i i 5000 i i i i Same Structural Dynamics and Vibration Control Lab., KAIST, Korea 19 CPU time for 160 Eigenpairs Method CPU time Ratio Lees method Proposed method (sec) Structural Dynamics and Vibration Control Lab., KAIST, Korea 20 Comparison for each operations Total Lees method Proposed method Method Operations CPU time (sec) Total Structural Dynamics and Vibration Control Lab., KAIST, Korea 21 CONCLUSIONS P roposed method - is simple - guarantees numerical stability - reduces the CPU time. An efficient eigensensitivity technique ! Structural Dynamics and Vibration Control Lab., KAIST, Korea 22 Thank you for your attention. Structural Dynamics and Vibration Control Lab., KAIST, Korea 23 Numerical Stability The determinant property (15) APPENDIX Structural Dynamics and Vibration Control Lab., KAIST, Korea 24 Then (16) Structural Dynamics and Vibration Control Lab., KAIST, Korea 25 Arranging eq.(16) (17) Using the determinant property of partitioned matrix (18) Structural Dynamics and Vibration Control Lab., KAIST, Korea 26 Therefore Numerical Stability is Guaranteed. (19) Structural Dynamics and Vibration Control Lab., KAIST, Korea 27 Lees method (1999) Differentiating eq.(1) with respect to design variable (20) Pre-multiplying each side of eq.(20) by gives eigenvalue derivative. (21) Structural Dynamics and Vibration Control Lab., KAIST, Korea 28 Differentiating eq.(3) with respect to design variable (22) Combining eq.(20) and eq.(22) into a matrix gives eigenvector derivative. (23)