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Modified Sturm Sequence Property for Damped Systems

This paper presents a novel method for efficiently computing eigenvalues of damped systems, with a focus on nonproportional damping scenarios. It introduces a complex eigenvalue algorithm based on Chen's approach and Gleyse's theorem to verify missed eigenpairs. Various numerical examples are discussed to illustrate the efficacy of the proposed method.

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Modified Sturm Sequence Property for Damped Systems

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  1. The Second China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems Busan, Korea November 4-8, 2002 Modified Sturm Sequence Property for Damped Systems Ji-Seong Jo1,Byoung-Wan Kim2, Man-Gi Ko3 & In-Won Lee3 1,2 Ph. D. Candidate, KAIST, Korea 3 Professor, Kongju National University, Korea 3 Professor, KAIST, Korea Structural Dynamics & Vibration Control Lab.

  2. CONTENTS Introduction  Previous Studies  Objective  Proposed Method  Numerical Examples  Conclusion Structural Dynamics & Vibration Control Lab.

  3. Modified Sturm Sequence Property for Damped Systems Introduction  Introduction • Dynamic Equations of Motion M : Mass matrix of order n C : Damping matrix of order n K : Stiffness matrix of order n u(t) : Displacement vector f(t) : Load vector n : Order of M, C and K (1) Structural Dynamics & Vibration Control Lab.

  4. Modified Sturm Sequence Property for Damped Systems Introduction • Methods of Dynamic Analysis - Direct integration method : Loading of a short duration such as impulse loading - Mode superposition method : Loading of a long duration such as an earthquake • Mode Superposition Method - Required eigenpairs should be computed. - Technique for checking missed eigenpairs is required. Structural Dynamics & Vibration Control Lab.

  5. Modified Sturm Sequence Property for Damped Systems Introduction • Proportionally Damped System i : i-th eigenvalue i : i-th eigenvector - Eigenvalues and eigenvectors : real numbers - Checking missed eigenpairs : Sturm sequence property (2) Structural Dynamics & Vibration Control Lab.

  6. Modified Sturm Sequence Property for Damped Systems Introduction • Nonproportionally Damped System (Soil-structure interaction problem, Structural control problem, Composite structure and so on) - Eigenvalues and eigenvectors : complex numbers - Checking missed eigenpairs : a few methods (3) Structural Dynamics & Vibration Control Lab.

  7. Modified Sturm Sequence Property for Damped Systems Previous Studies  Previous Studies • Tsai and Chen (1993) - Extended Sturm sequence property to determine the number of roots of a polynomial on specified lines of the complex plane - It is very difficult to find the specified line on the complex plane. - Sturm sequence is not formed by factorizing the matrix in the field of complex arithmetic computation. Structural Dynamics & Vibration Control Lab.

  8. 6 5 7 2 4 3 Modified Sturm Sequence Property for Damped Systems Previous Studies • Jung and Lee (1999) - Based on augment principle (4) Imaginary axis Imaginary axis S 8 3 4 Real axis Real axis 2 1 5 8 1 6 7 f()  plane f() plane No. of Rotations = No. of eigenvalues inside an closed contour Structural Dynamics & Vibration Control Lab.

  9. Modified Sturm Sequence Property for Damped Systems Previous Studies • Shortcomings • Accuracy is improved asthe number of checking points • increases. - Difficult to find abrupt change in arguments • Factorization processes are required at each checking • point. Structural Dynamics & Vibration Control Lab.

  10. Modified Sturm Sequence Property for Damped Systems Objective  Objective To develop an efficient technique for checking missed eigenpairs of nonproportionally damped systems with distinct or multiple eigenvalues Structural Dynamics & Vibration Control Lab.

  11. Modified Sturm Sequence Property for Damped Systems Proposed Method  Proposed Method Modified Sturm Sequence Property for Damped Systems Complex eigenvalue problem Chen’s algorithm Characteristic polynomial Gleyse’s theorem Check the number of eigenvalues inside some open disks Structural Dynamics & Vibration Control Lab.

  12. Modified Sturm Sequence Property for Damped Systems Proposed Method • Complex Eigenvalue Problem (3) - State space form ( ) (5) - Standard form (6) Structural Dynamics & Vibration Control Lab.

  13. Modified Sturm Sequence Property for Damped Systems Proposed Method • Chen’s Algorithm (7) Gauss elimination-like similarity transformations (8) Structural Dynamics & Vibration Control Lab.

  14. Modified Sturm Sequence Property for Damped Systems Proposed Method - Characteristic polynomial (9) Structural Dynamics & Vibration Control Lab.

  15. Modified Sturm Sequence Property for Damped Systems Proposed Method • Gleyse’s Theorem - Characteristic polynomial (10) - Schur Cohn matrix T: (11) (12) Structural Dynamics & Vibration Control Lab.

  16. Modified Sturm Sequence Property for Damped Systems Proposed Method The number of eigenvalues inside an unit open disk (13) N : the number of eigenvalues inside an unit open disk 2n : degree of the characteristic polynomial P S[1, d1, d2, ···, d2n]: the number of sign changes in the sequence (1, d1, d2,···, d2n) Imaginary axis 1 1 -1 Real axis -1 unit disk  plane Structural Dynamics & Vibration Control Lab.

  17. Modified Sturm Sequence Property for Damped Systems Numerical Examples Numerical Examples • Simple Spring-Mass-Damper System To apply the proposed method to the distinct eigenvalue system 1 2 10 k k m m m Mass: m = 1.0 Stiffness: k = 1.0 Rayleigh damping:  = 0.05,  = 0.05 Number of d.o.f: 10 Structural Dynamics & Vibration Control Lab.

  18. Modified Sturm Sequence Property for Damped Systems Numerical Examples - Exact eigenvalues (14) (15) (16) Structural Dynamics & Vibration Control Lab.

  19. Modified Sturm Sequence Property for Damped Systems Numerical Examples Mode No. Eigenvalues Radii Real Imaginary 1,2 -0.0306 0.1463 0.1495 3,4 -0.0745 0.4388 0.4450 5,6 -0.1585 0.7133 0.7307 7,8 -0.2750 0.9614 1.0000 9,10 -0.4137 1.1763 1.2470 11,12 -0.5624 1.3540 1.4661 13,14 -0.7077 1.4932 1.6525 15,16 -0.8368 1.5959 1.8019 17,18 -0.9381 1.6651 1.9111 19,20 -1.0028 1.7046 1.9777 - Calculated eigenvalues Structural Dynamics & Vibration Control Lab.

  20. Modified Sturm Sequence Property for Damped Systems Numerical Examples Imaginary axis Real axis eigenvalues Structural Dynamics & Vibration Control Lab.

  21. Modified Sturm Sequence Property for Damped Systems Numerical Examples 10 1 9 7 8 12 3 4 5 6 11 2 - Radius  = 1.1 > |8 | = 1.0 i sign(di) S 11 - 12 + 13 - 14 + 15 + 16 + 17 - 18 + 19 + 20 + i sign(di) S 0 + 1 + 2 + 3 - 4 + 5 - 6 + 7 + 8 + 9 - 10 +   12 = 8 Structural Dynamics & Vibration Control Lab.

  22. Modified Sturm Sequence Property for Damped Systems Numerical Examples - Radius  = 2.0 > |20| = 1.978 i sign(di) S 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + i sign(di) S 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 +   0 = 20 Structural Dynamics & Vibration Control Lab.

  23. Modified Sturm Sequence Property for Damped Systems Numerical Examples • Plane Frame Structure with Lumped Dampers To apply the proposed method to the multiple eigenvalue system y v L x L Structural Dynamics & Vibration Control Lab.

  24. Modified Sturm Sequence Property for Damped Systems Numerical Examples - Geometric and material properties Damping Concentrated: 0.3 Rayleigh:  = 0.001,  = 0.001 Young’s modulus: 1000 Mass density: 1.0 Cross-sectional inertia: 1.0 Cross-sectional area: 1.0 Span length: 6.0 - System data Number of elements: 12 Number of nodes: 14 Number of DOF: 18 Structural Dynamics & Vibration Control Lab.

  25. Modified Sturm Sequence Property for Damped Systems Numerical Examples - Calculated eigenvalues Mode No. Eigenvalues Radii Real Imaginary 1,2 -1.137 46.219 46.233 3,4 -1.137 46.219 46.233 5,6 -1.373 51.133 51.152 7,8 -1.373 51.133 51.152 9,10 -3.390 81.078 81.149 11,12 -3.390 81.078 81.149 13,14 -3.941 87.477 87.566 15,16 -3.941  87.477 87.566 17,18 -8.164 127.439 127.701 19,20 -8.164 127.439 127.701 Structural Dynamics & Vibration Control Lab.

  26. Modified Sturm Sequence Property for Damped Systems Numerical Examples Mode No. Eigenvalues Radius Real Imaginary 21,22 -10.263 142.837 143.205 23,24 -10.263 142.837 143.205 25,26 -14.862 171.730 172.372 27,28 -14.862 171.730 172.372 29,30 -20.537 201.625 202.668 31,32 -20.537 201.625 202.668 33,34 -23.770 216.733 218.033 35,36 -23.770 216.733 218.033 Structural Dynamics & Vibration Control Lab.

  27. Modified Sturm Sequence Property for Damped Systems Numerical Examples Imaginary axis Real axis eigenvalues Structural Dynamics & Vibration Control Lab.

  28. Modified Sturm Sequence Property for Damped Systems Numerical Examples 24 23 1 3 4 5 6 7 8 10 11 9 13 12 22 21 2 19 20 17 16 15 14 18 - Radius  = 82.0 > |12| = 81.149 i sign(di) S 0 + 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10 + 11 - 12 + 13 - 14 + 15 - 16 - 17 + 18 - i sign(di) S 19 - 20 - 21 + 22 + 23 - 24 - 25 + 26 - 27 - 28 - 29 - 30 + 31 - 32 - 33 + 34 + 35 + 36 +   24 = 12 Structural Dynamics & Vibration Control Lab.

  29. Modified Sturm Sequence Property for Damped Systems Numerical Examples - Radius  = 220 > |36| = 218.033 i sign(di) S 0+ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + i sign(di) S 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 +   0 = 36 Structural Dynamics & Vibration Control Lab.

  30. Modified Sturm Sequence Property for Damped Systems Conclusion Conclusion The proposed method is more efficient than the previous methods for checking missed eigenpairs of damped systems !! Structural Dynamics & Vibration Control Lab.

  31. Thank you for your attention!! Structural Dynamics & Vibration Control Lab.

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