Modified Sturm Sequence Property for Damped Systems

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Presentation transcript:

Modified Sturm Sequence Property for Damped Systems The Fifth European Conference on Structural Dynamics Munich, Germany September 2-5, 2002 Modified Sturm Sequence Property for Damped Systems Ji-Seong Jo1, Man-Gi Ko2 & In-Won Lee3 Thank you, Mr. chairman. Good afternoon, Ladies and gentlemen. My name is Ji-Seong Jo. The title of my presentation is “Modified Sturm Sequence Property for Damped Systems”. 1 Ph. D. Candidate, KAIST, Korea 2 Professor, Kongju National University, Korea 3 Professor, KAIST, Korea Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. CONTENTS  Introduction  Previous Studies  Objective  Proposed Method  Numerical Examples I will present in the following sequence.  Conclusion Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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 K, C and M (1) The Dynamic Equations of Motion can be written as in Equation (1). Where M mass matrix C damping matrix K stiffness matrix U displacement vector F Load vector And n Order of K,C and M Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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. For Dynamic Analysis of structures, there are twp methods such as The Direct Integration Method and Mode Superposition Method. The Direct integration method is efficient for the case of loading of a short duration such as impulse loading. And Mode superposition method is for the case of loading of a long duration such as an earthquake. When the Mode Superposition Method is used, Required eigenpairs should be computed and also Missed eigenpairs should be checked. - Technique for checking missed eigenpairs is required. Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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) For Proportionally Damped System defined in this equation, The eigenvalue problem can be written as in equation(2). Ramda i is the i-th eigenvalue, Phi i is the i-th eigenvector. In this case, Eigenvalues and eigenvectors are real numbers and For Checking missed eigenpairs, Sturm sequence property is used. Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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) For non-proportionally Damped System defined in this equation, Such as Soil-structure interaction problem, Structural control problem, Composite structure and so on, The Eigenvalue problem can be written as in equation(3). In this case, Eigenvalues and eigenvectors are complex numbers and For Checking missed eigenpairs, a few method is used. Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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. Now I will explain Previous studies on Checking missed eigenvalues nonproportionally damped system. In 1993, Tsai and Chen proposed extended strum sequence property To determine the number of root of a polynomial on specified lines of the complex plane. However it is very difficult to find the specified line on the complex plane using this method, and Sturm sequence is not formed by factorizing the matrix in the field of complex arithmetic computation. Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. Modified Sturm Sequence Property for Damped Systems Previous Studies Jung and Lee (1999) - Based on augment principle (4) Imaginary axis Imaginary axis 6 5 7 4 S 8 3 3 Real axis 4 2 2 Real axis 1 5 1 8 6 7 f() Jung and Lee in 1999 proposed a method based on argument principle. A complex function f ramda can be written as in equation(4). and evaluated along an closed contour S on the complex plane, Then the number of rotations of f ramda about origin on the complex plane is equal to No. of eigenvalues inside an closed contour.  plane f() plane No. of Rotations = No. of eigenvalues inside an closed contour Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. Modified Sturm Sequence Property for Damped Systems Previous Studies 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. But this method has following Shortcomings Fisrt, Accuracy improved as the number of checking points increases. Second, It is difficult to find abrupt change in arguments And Third, Factorization processes are required at each checking point. Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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 So the Objective of this study is to develop an efficient technique for checking missed eigenpairs of a nonproportionally damped system. Structural Dynamics & Vibration Control Lab.

Modified Sturm Sequence Property for Damped Systems 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 In the proposed method, Using Chen’s algorithm, we formulate characteristic polynomial. And then, by Gleyse’s theorem, we can check the number of eigenvalues inside some open disks. Check the number of eigenvalues inside some open disks Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. Modified Sturm Sequence Property for Damped Systems Proposed Method Complex Eigenvalue Problem (3) - State space form ( ) (5) - Standard form (6) The Complex Eigenvalue Problem can be written as in equation (3). The state space form of the equation is as equation (5) or equation (6). Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. Modified Sturm Sequence Property for Damped Systems Proposed Method Chen’s Algorithm (7) Gauss elimination-like similarity transformations Give matrix A is transformed into the matrix A bar, Using Gauss elimination-like similarity transformations. (8) Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. Modified Sturm Sequence Property for Damped Systems Proposed Method - Characteristic polynomial (9) Then characteristic polynomial P ramda can be computed using next equations. Structural Dynamics & Vibration Control Lab.

Gleyse’s Theorem - Characteristic polynomial - Schur Cohn matrix T: Modified Sturm Sequence Property for Damped Systems Proposed Method Gleyse’s Theorem - Characteristic polynomial (10) - Schur Cohn matrix T: (11) Gleyse’s theorem is that Given the characteristic polynomial defined in equation (10), We can construct the following Schur Cohn matrix T, using equations (11) and (12). (12) Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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 Then, the number of eigenvalues inside an unit open disk can be found using equation (13). N the number of eigenvalues inside an unit open disk, 2n degree of the characteristic polynomial P S the number of sign changes in the sequence And D I determinant of T i -1 1 Real axis -1 unit disk  plane Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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 For numerical examples Simple Spring-Mass-Damper System is considered to apply the proposed method to the distinct eigenvalue system. Shape and Properties are shown here. Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. Modified Sturm Sequence Property for Damped Systems Numerical Examples - Exact eigenvalues (14) (15) (16) The exact eigenvalues can be computed using equation (14). Structural Dynamics & Vibration Control Lab.

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

Structural Dynamics & Vibration Control Lab. Modified Sturm Sequence Property for Damped Systems Numerical Examples Imaginary axis Real axis This figure shows the calculated eigenvalues on the complex plane. eigenvalues Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. Modified Sturm Sequence Property for Damped Systems Numerical Examples - Radius  = 1.1 > |8 | = 1.0 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 + 8 9 1 10 2  3 4 11 12 5 First, when  is a little bit larger than the ramda sixteen. The calculated dii, sign di and sign change S are as shown here. The number of sign change is 4. And the number of eigenvalue N inside open disk of radius , is 4. This agrees with the calculated result. 6 7  12 = 8 Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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 +  And third, when  is a little bit larger than the ramda twentieth . The calculated dii, sign di and sign change S are as shown here. The number of sign change is 0. And the number of eigenvalue N inside open disk of radius , is 20. This agrees with the calculated result.  0 = 20 Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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 The time is running out, so I will stop here. As the Second example Plane Frame Structure with Lumped Dampers is considered To apply the proposed method to the multiple eigenvalue system. This system has a following shapes. L Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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 And Geometric and material properties and System data are as followings. The Number of elements is twelve, the Number of nodes is fourteen and the number of DOF is eighteen 18. Number of elements: 12 Number of nodes: 14 Number of DOF: 18 Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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 The calculated eigenvalues are shown here. As you can see, eigenvalues are multiple. The radii is the distance of an eigenvalue form the origin on the complex plane. 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.

Structural Dynamics & Vibration Control Lab. 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.

Structural Dynamics & Vibration Control Lab. Modified Sturm Sequence Property for Damped Systems Numerical Examples Imaginary axis Real axis This figure shows the calculated eigenvalues on the complex plane. eigenvalues Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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 - i sign(di) S 19 - 20 - 21 + 22 + 23 - 24 - 25 + 26 - 27 - 28 - 29 - 30 + 31 - 32 - 33 + 34 + 35 + 36 + 1 2 18 3 4 19 5 6 20 7 21  8 9 10 11 22 12 23 13 First, when  is a little bit larger than the ramda twelfth . The calculated dii, sign di and sign change S are as shown here. The number of sign change is 24. And the number of eigenvalue N inside open disk of radius , is 12. This agrees with the calculated result. 14 24 15 16 17  24 = 12 Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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 +  And third, First, when  is a little bit larger than the ramda thirty sixth. The calculated dii, sign di and sign change S are as shown here. The number of sign change is 0. And the number of eigenvalue N inside open disk of radius , is . This agrees with the calculated result.  0 = 36 Structural Dynamics & Vibration Control Lab.

Structural Dynamics & Vibration Control Lab. 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 !! Conclusion The proposed method is more efficient than the previous methods for checking missed eigenpairs of damped systems. Structural Dynamics & Vibration Control Lab.

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