Nha Trang 2000 Nha Trang Vietnam Aug 14

Nha Trang 2000 Nha Trang, Vietnam, Aug. 14 -18, 2000 SOLUTION OF EIGENVALUE PROBLEM FOR NON-CLASSICALLY DAMPED SYSTEM WITH MULTIPLE EIGENVALUES * In-Won Lee: Professor, KAIST Man-Cheol Kim: Senior Researcher, KRRI Kyu-Hong Shim: Postdoctoral Researcher, KAIST

OUTLINE l PROBLEM DEFINITION l PROPOSED METHOD l NUMERICAL EXAMPLES l CONCLUSIONS Structural Dynamics & Vibration Control Lab. , KAIST, Korea 1

l PROBLEM DEFINITION • Dynamic Equation of Motion (1) where : Mass matrix, Positive definite : Damping matrix : Stiffness matrix, Positive semi-definite : Displacement vector : Load vector Structural Dynamics & Vibration Control Lab. , KAIST, Korea 2

• Methods of Dynamic Analysis • Step by step integration method • Mode superposition method • Mode Superposition Method • Free vibration analysis should be first performed Structural Dynamics & Vibration Control Lab. , KAIST, Korea 3

• Condition of Classical Damping (2) • Example : Rayleigh Damping Structural Dynamics & Vibration Control Lab. , KAIST, Korea 4

• Eigenproblem of classical damping systems (3) where : Real eigenvalue : Natural frequency : Real eigenvector(mode shape) - Low in cost - Straightforward Structural Dynamics & Vibration Control Lab. , KAIST, Korea 5

• Quadratic eigenproblem of non-classically damped systems (4) where : Complex eigenvalue : Complex eigenvector(mode shape) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 6

(5) where : Complex Eigenvalue : Complex Eigenvector (6) - Very expensive An efficient eigensolution technique of non-classically damped systems is required. Structural Dynamics & Vibration Control Lab. , KAIST, Korea 7

• Current Methods for Solving the Non-Classically Damped Eigenproblems • Transformation method: Kaufman (1974) • Perturbation method: Meirovitch et al (1979) • Vector iteration method: Gupta (1974; 1981) • Subspace iteration method: Leung (1995) • Lanczos method: Chen (1993) • Efficient Methods Structural Dynamics & Vibration Control Lab. , KAIST, Korea 8

l PROPOSED METHOD • Find p Smallest Eigenpairs Solve Subject to For and : multiple or close roots If p=1, then distinct root where Structural Dynamics & Vibration Control Lab. , KAIST, Korea 9

• Relations between and Vectors in the Subspace of (7) where (8) (9) • Let subspace of and respect to , then be the vectors in the be orthonormal with (10) (11) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 10

• Introducing Eq. (10) into Eq. (7) (12) • Let where (13) : Symmetric • Then (14) or (15) or (16) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 11

• Multiple or Close Eigenvalues • Multiple eigenvalues case : is a diagonal matrix. Eigenvalues : Eigenvectors : • Close eigenvalues case : is not a diagonal matrix. - Solve the small standard eigenvalue problem. (13) - Get the following eigenpairs. Eigenvalues : Eigenvectors : Structural Dynamics & Vibration Control Lab. , KAIST, Korea (10) 12

• Newton-Raphson Technique (17) (18) (19) where (20) (21) : unknown incremental values Structural Dynamics & Vibration Control Lab. , KAIST, Korea 13

• Introducing Eqs. (19) and (20) into Eqs. (17) and (18) and neglecting nonlinear terms (22) (23) where : residual vector • Matrix form of Eqs. (22) and (23) (24) Coefficient matrix : • Symmetric • Nonsingular Structural Dynamics & Vibration Control Lab. , KAIST, Korea 14

• Modified Newton-Raphson Technique (24) Introducing modified Newton-Raphson technique (25) (19) (20) Coefficient matrix : • Symmetric • Nonsingular Structural Dynamics & Vibration Control Lab. , KAIST, Korea 15

• Algorithm of Proposed Method • Step 1: Start with approximate eigenpairs • Step 2: Solve for and • Step 3: Compute Structural Dynamics & Vibration Control Lab. , KAIST, Korea 16

• Step 4: Check the error norm. Error norm = n If the error norm is more than the tolerance, then go to Step 2 and if not, go to Step 5. • Step 5: Check if is a diagonal matrix, go to Step 6, if not, go to Step 7. Structural Dynamics & Vibration Control Lab. , KAIST, Korea 17

• Step 6: Multiple case • Step 7: Close case • Go to step 8. • Step 8: Check the error norm. Error norm = • Stop ! Structural Dynamics & Vibration Control Lab. , KAIST, Korea 18

• Initial Values of the Proposed Method • Intermediate results of the iteration methods - Vector iteration method - Subspace iteration method • Results of the approximate methods - Static Condensation method - Lanczos method Structural Dynamics & Vibration Control Lab. , KAIST, Korea 19

l NUMERICAL EXAMPLES • Structures • Cantilever beam(distinct) • Grid structure(multiple) • Three-dimensional framed structure(close) • Analysis Methods • Proposed method • Subspace iteration method (Leung 1988) • Lanczos method (Chen 1993) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 20

• Comparisons • Solution time(CPU) • Convergence • Convex with 100 MIPS, 200 MFLOPS Structural Dynamics & Vibration Control Lab. , KAIST, Korea 21

Cantilever Beam with Lumped Dampers (Distinct Case) Material Properties 1 2 3 4 5 Tangential Damper : c = 0. 3 Rayleigh Damping : = = 0. 001 Young’s Modulus : 1000 99 100 101 Mass Density : 1 C Cross-section Inertia : 1 Cross-section Area : 1 System Data Number of Equations : 200 Number of Matrix Elements : 696 Maximum Half Bandwidths : 4 Mean Half Bandwidths : 4 Structural Dynamics & Vibration Control Lab. , KAIST, Korea 22

• Results of Cantilever Beam Structure (Distinct) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 23

• CPU Time for 10 Lowest Eigenpairs, Cantilever Beam Structural Dynamics & Vibration Control Lab. , KAIST, Korea 24

Starting values of proposed method : 1 st, 2 nd eigenpairs : 3 rd, 4 th eigenpairs : 5 th, 6 th eigenpairs : 7 th, 8 th eigenpairs : 9 th, 10 th eigenpairs Convergence by Lanczos method(Chen 1993) Cantilever beam (distinct) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 25

: Proposed Method : Subspace Iteration Method (q=2 p) Convergence of the 1 st eigenpair Cantilever beam (distinct) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 26

: Proposed Method : Subspace Iteration Method (q=2 p) Convergence of the 5 th eigenpair Cantilever beam (distinct) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 27

Grid Structure with Lumped Dampers (Multiple Case) Material Properties Tangential Damper : c = 0. 3 100@0. 1=10 Rayleigh Damping : = = 0. 001 Young’s Modulus : 1, 000 Mass Density : 1 Cross-section Inertia : 1 Cross-section Area : 1 System Data Number of Equations : 590 Number of Matrix Elements : 8, 115 Maximum Half Bandwidths : 15 100@0. 1=10 Mean Half Bandwidths : 14 Structural Dynamics & Vibration Control Lab. , KAIST, Korea 28

• Results of Grid Structure (Multiple) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 29

• CPU Time for 10 Lowest Eigenpairs, Grid Structure Structural Dynamics & Vibration Control Lab. , KAIST, Korea 30

Starting values of proposed method : 1 st, 3 rd eigenpairs : 2 nd, 4 th eigenpairs : 5 th, 7 th eigenpairs : 6 th, 8 th eigenpairs : 9 th, 11 th eigenpairs : 10 th, 12 th eigenpairs Convergence by Lanczos method(Chen 1993) Grid structure (multiple) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 31

: Proposed Method : Subspace Iteration Method (q=2 p) Convergence of the 2 nd eigenpair Grid structure (multiple) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 32

: Proposed Method : Subspace Iteration Method (q=2 p) Convergence of the 9 th eigenpair Grid structure (multiple) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 33

Three-Dimensional Framed Structure with Lumped Dampers(Close Case) 2@3. 01=6. 02 2@3=6 6@3. 01=18. 06 6@3=18 12@3=36 Structural Dynamics & Vibration Control Lab. , KAIST, Korea 34

Material Properties Lumped Damper : c = 12, 000. 0 Rayleigh Damping : =-0. 1755 = 0. 02005 Young’s Modulus : 2. 1 E+11 Mass Density : 7, 850 Cross-section Inertia : 8. 3 E-06 Cross-section Area : 0. 01 System Data Number of Equations : 1, 128 Number of Matrix Elements : 135, 276 Maximum Half Bandwidths : 300 Mean Half Bandwidths : 120 Structural Dynamics & Vibration Control Lab. , KAIST, Korea 35

• Results of Three-Dimensional Frame Structure (Close) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 36

• CPU Time for 12 Lowest Eigenpairs, 3 -D. Frame Structural Dynamics & Vibration Control Lab. , KAIST, Korea 37

: 1 st, 2 nd eigenpairs : 3 rd, 4 th eigenpairs : 5 th, 6 th eigenpairs : 7 th, 8 th eigenpairs Starting values of proposed method : 9 th, 10 th eigenpairs : 11 th, 12 th eigenpairs Convergence by Lanczos method(Chen 1993) 3 -D. framed structure (close) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 38

: Proposed Method : Subspace Iteration Method (q=2 p) Convergence of the 9 th eigenpair 3 -D. framed structure (close) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 39

l CONCLUSIONS • The proposed method • is simple • guarantees numerical stability • converges fast. An efficient Eigensolution technique ! Structural Dynamics & Vibration Control Lab. , KAIST, Korea 40

Thank you for your attention. Structural Dynamics & Vibration Control Lab. , KAIST, Korea 41