Presentation is loading. Please wait.

Presentation is loading. Please wait.

* 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,

Published byGwendoline Farmer Modified over 9 years ago

Similar presentations

Presentation on theme: "* 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,"— Presentation transcript:

1 * 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. 11-14, 2000

2 Structural Dynamics and Vibration Control Lab., KAIST, Korea 2 OUTLINE INTRODUCTION PREVIOUS STUDIES PROPOSED METHOD NUMERICAL EXAMPLE CONCLUSIONS

3 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.

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

5 Structural Dynamics and Vibration Control Lab., KAIST, Korea 5 (2) - Orthonormalization condition - State space equation (3)

6 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.)

7 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. 447-458, 1987. - restricted to lumped systems with distinct eigenvalues. (4)

8 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. 746-751, 1995. - many eigenvectors are required to calculate eigenvector derivatives. (5) (6)

9 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. 625-633, 2000. - applicable only when the elements of C are small. (7)

10 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. 399-412, 1999. 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. 413-424, 1999.

11 Structural Dynamics and Vibration Control Lab., KAIST, Korea 11 Lee’s method (1999) (8) (9) - eigenvalue and eigenvector derivatives are obtained separately.

12 Structural Dynamics and Vibration Control Lab., KAIST, Korea 12 PROPOSED METHOD Rewriting basic equations - Eigenvalue problem - Orthonormalization condition (10) (11)

13 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)

14 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.

15 Structural Dynamics and Vibration Control Lab., KAIST, Korea 15 NUMERICAL EXAMPLE Cantilever Beam

16 Structural Dynamics and Vibration Control Lab., KAIST, Korea 16 Analysis Methods Lee’s method (1999) Proposed method Comparisons Solution time (CPU)

17 Structural Dynamics and Vibration Control Lab., KAIST, Korea 17 Results of Analysis (Eigenvalue) Mode Number Eigenvalue Eigenvalue derivative (Lee’s method) Eigenvalue derivative (Proposed method) 1-0.001 - 2.625i-0.014 -52.496i 2-0.001 + 2.625i-0.014 +52.496i 3-0.014 -16.449i-5.411e-1 -3.290e+2i 4-0.014 +16.449i-5.411e-1+3.290e+2i 5-0.035 -26.236i4.770e-7 +2.970e-8i 6-0.035 +26.236i4.721e-7 +1.549e-7i 7-0.106 -46.056i-4.242e+0 -9.210e+2i 8-0.106 +46.056i-4.242e+0+9.210e+2i 9-0.407 -90.244i-1.628e+1 -1.804e+3i 10-0.407 +90.244i-1.628e+1+1.804e+3i Same

18 Structural Dynamics and Vibration Control Lab., KAIST, Korea 18 Results of Analysis (First eigenvector) Eqn. number Eigenvector Eigenvector derivative (Lee’s method) Eigenvector derivative (Proposed method) 1000 2000 31.513e-05 +1.513e-05i-3.027e-04 -3.027e-04i 41.204e-04 +1.204e-04i-0.002 - 0.002i 5000  157000 158000 1590.014 + 0.014i-0.279 - 0.279i 1600.002 + 0.002i-0.038 - 0.038i Same

19 Structural Dynamics and Vibration Control Lab., KAIST, Korea 19 CPU time for 160 Eigenpairs Method CPU time Ratio Lee’s method223.33 1.00 Proposed method 164.890.74 (sec)

20 Structural Dynamics and Vibration Control Lab., KAIST, Korea 20 Comparison for each operations Total Lee’s method Proposed method Method Operations CPU time (sec) 33.89 61.01 47.09 81.34 223.33 53.62 40.60 70.67 164.89 Total

21 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 !

22 Structural Dynamics and Vibration Control Lab., KAIST, Korea 22 Thank you for your attention.

23 Structural Dynamics and Vibration Control Lab., KAIST, Korea 23 Numerical Stability The determinant property (15) APPENDIX

24 Structural Dynamics and Vibration Control Lab., KAIST, Korea 24 Then (16)

25 Structural Dynamics and Vibration Control Lab., KAIST, Korea 25 Arranging eq.(16) (17) Using the determinant property of partitioned matrix (18)

26 Structural Dynamics and Vibration Control Lab., KAIST, Korea 26 Therefore Numerical Stability is Guaranteed. (19)

27 Structural Dynamics and Vibration Control Lab., KAIST, Korea 27 Lee’s method (1999) Differentiating eq.(1) with respect to design variable (20) Pre-multiplying each side of eq.(20) by gives eigenvalue derivative. (21)

28 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)

Download ppt "* 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,"

Similar presentations

1 Department of Civil and Environmental Engineering Sungkyunkwan University 비대칭 박벽보의 개선된 해석이론 및 방법 An Improved Theory and Analysis Procedures of Nonsymmetric.

1 Department of Civil and Environmental Engineering Sungkyunkwan University 비대칭 박벽보의 개선된 해석이론 및 방법 An Improved Theory and Analysis Procedures of Nonsymmetric.
MDOF SYSTEMS WITH DAMPING General case Saeed Ziaei Rad.

MDOF SYSTEMS WITH DAMPING General case Saeed Ziaei Rad.
Scissor-Jack-Damper System for Reduction of Stay Cable

Scissor-Jack-Damper System for Reduction of Stay Cable
Mar 08 2002Numerical approach for large-scale Eigenvalue problems 1 Definition Why do we study it ? Is the Behavior system based or nodal based? What are.

Mar Numerical approach for large-scale Eigenvalue problems 1 Definition Why do we study it ? Is the Behavior system based or nodal based? What are.
M M S S V V 0 Free vibration analysis of a circular plate with multiple circular holes by using addition theorem and direct BIEM Wei-Ming Lee 1, Jeng-Tzong.

M M S S V V 0 Free vibration analysis of a circular plate with multiple circular holes by using addition theorem and direct BIEM Wei-Ming Lee 1, Jeng-Tzong.
Unit 6: Structural vibration An Introduction to Mechanical Engineering: Part Two Structural vibration Learning summary By the end of this chapter you should.

Unit 6: Structural vibration An Introduction to Mechanical Engineering: Part Two Structural vibration Learning summary By the end of this chapter you should.
Modal Testing and Analysis

Modal Testing and Analysis
1 Adjoint Method in Network Analysis Dr. Janusz A. Starzyk.

1 Adjoint Method in Network Analysis Dr. Janusz A. Starzyk.
TWO DEGREE OF FREEDOM SYSTEM. INTRODUCTION Systems that require two independent coordinates to describe their motion; Two masses in the system X two possible.

TWO DEGREE OF FREEDOM SYSTEM. INTRODUCTION Systems that require two independent coordinates to describe their motion; Two masses in the system X two possible.
Differential Equations

Differential Equations
Solution of Eigenproblem of Non-Proportional Damping Systems by Lanczos Method In-Won Lee, Professor, PE In-Won Lee, Professor, PE Structural Dynamics.

Solution of Eigenproblem of Non-Proportional Damping Systems by Lanczos Method In-Won Lee, Professor, PE In-Won Lee, Professor, PE Structural Dynamics.
Structural Dynamics & Vibration Control Lab 1 December 20. 2005 Department of Civil & Environmental Engineering K orea A dvanced I nstitute of S cience.

Structural Dynamics & Vibration Control Lab 1 December Department of Civil & Environmental Engineering K orea A dvanced I nstitute of S cience.
Yeong-Jong Moon*: Graduate Student, KAIST, Korea Kang-Min Choi: Graduate Student, KAIST, Korea Hyun-Woo Lim: Graduate Student, KAIST, Korea Jong-Heon Lee:

Yeong-Jong Moon*: Graduate Student, KAIST, Korea Kang-Min Choi: Graduate Student, KAIST, Korea Hyun-Woo Lim: Graduate Student, KAIST, Korea Jong-Heon Lee:
한국강구조학회 2001 년도 학술발표대회 Three Dimensional Finite Element Analysis of Structures under Wind Loads *Byoung-Wan Kim 1), Woon-Hak Kim 2) and In-Won Lee 3) 1)

한국강구조학회 2001 년도 학술발표대회 Three Dimensional Finite Element Analysis of Structures under Wind Loads *Byoung-Wan Kim 1), Woon-Hak Kim 2) and In-Won Lee 3) 1)
1 PSSC 1998. 10. 13. Mode Localization in Multispan Beams with Massive and Stiff Couplers on Supports Dong-Ok Kim and In-Won Lee Department of Civil Engineering.

1 PSSC Mode Localization in Multispan Beams with Massive and Stiff Couplers on Supports Dong-Ok Kim and In-Won Lee Department of Civil Engineering.
1 SIMULATION OF VIBROACOUSTIC PROBLEM USING COUPLED FE / FE FORMULATION AND MODAL ANALYSIS Ahlem ALIA presented by Nicolas AQUELET Laboratoire de Mécanique.

1 SIMULATION OF VIBROACOUSTIC PROBLEM USING COUPLED FE / FE FORMULATION AND MODAL ANALYSIS Ahlem ALIA presented by Nicolas AQUELET Laboratoire de Mécanique.
Sang-Won Cho* : Ph.D. Student, KAIST Sang-Won Cho* : Ph.D. Student, KAIST Dong-Hyawn Kim: Senior Researcher, KORDI Dong-Hyawn Kim: Senior Researcher, KORDI.

Sang-Won Cho* : Ph.D. Student, KAIST Sang-Won Cho* : Ph.D. Student, KAIST Dong-Hyawn Kim: Senior Researcher, KORDI Dong-Hyawn Kim: Senior Researcher, KORDI.
1 Efficient Mode Superposition Methods for Non-Classically Damped System Sang-Won Cho, Graduate Student, KAIST, Korea Ju-Won Oh, Professor, Hannam University,

1 Efficient Mode Superposition Methods for Non-Classically Damped System Sang-Won Cho, Graduate Student, KAIST, Korea Ju-Won Oh, Professor, Hannam University,
In-Won Lee, Professor, PE In-Won Lee, Professor, PE Structural Dynamics & Vibration Control Lab. Structural Dynamics & Vibration Control Lab. Korea Advanced.

In-Won Lee, Professor, PE In-Won Lee, Professor, PE Structural Dynamics & Vibration Control Lab. Structural Dynamics & Vibration Control Lab. Korea Advanced.
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the.

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the.

Similar presentations

Ads by Google