KAISTKyoto Univ Joint Seminar on Earthquake Engineering KAIST

KAIST-Kyoto Univ. Joint Seminar on Earthquake Engineering KAIST, Korea February 25, 2002 Modified Sturm Sequence Property for Damped Systems Ji-Seong Jo*, Byoung-Wan Kim & In-Won Lee Korea Advanced Institute of Science & Technology Structural Dynamics & Vibration Control Lab. 1

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

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

Modified Sturm Sequence Property for Damped Systems Introduction l 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 l Mode Superposition Method - Required eigenpairs should be computed. - Technique for checking missed eigenpairs is required. Structural Dynamics & Vibration Control Lab. 4

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

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

Modified Sturm Sequence Property for Damped Systems Previous Study Previous Studies l 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. 7

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

Modified Sturm Sequence Property for Damped Systems Previous Study l Shortcomings - Accuracy improves as the 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. 9

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

Modified Sturm Sequence Property for Damped Systems 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. 11

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

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

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

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

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 2 n : degree of the characteristic polynomial P S[1, d 2, ···, d 2 n]: the number of sign changes in the sequence (1, d 2, ···, d 2 n) Imaginary axis 1 1 -1 -1 Real axis unit disk plane Structural Dynamics & Vibration Control Lab. 16

Modified Sturm Sequence Property for Damped Systems Numerical Examples l Simple Spring-Mass-Damper System To apply the proposed method to the distinct eigenvalue system k 1 k m Mass: Stiffness: Rayleigh damping: Number of d. o. f: Structural Dynamics & Vibration Control Lab. 2 10 m m m = 1. 0 k = 1. 0 = 0. 05, = 0. 05 10 17

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

Modified Sturm Sequence Property for Damped Systems - Calculated eigenvalues Mode No. Eigenvalues Numerical Examples 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 Structural Dynamics & Vibration Control Lab. 19

Imaginary axis Real axis eigenvalues Structural Dynamics & Vibration Control Lab. 20

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

Modified Sturm Sequence Property for Damped Systems Numerical Examples - Radius = 2. 0 > | 20| = 1. 978 i sign(di) S 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + i sign(di) S 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + Structural Dynamics & Vibration Control Lab. 0 = 20 22

Modified Sturm Sequence Property for Damped Systems Numerical Examples l. 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. 23

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: Number of nodes: Number of DOF: Structural Dynamics & Vibration Control Lab. 12 14 18 24

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. 25

Modified Sturm Sequence Property for Damped Systems Mode No. Eigenvalues Numerical Examples 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. 26

Imaginary axis Real axis eigenvalues Structural Dynamics & Vibration Control Lab. 27

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

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 + Structural Dynamics & Vibration Control Lab. 0 = 36 29

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

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