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SOLUTION OF EIGENVALUE PROBLEM FOR NON-CLASSICALLY DAMPED SYSTEM WITH MULTIPLE EIGENVALUES

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.

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SOLUTION OF EIGENVALUE PROBLEM FOR NON-CLASSICALLY DAMPED SYSTEM WITH MULTIPLE EIGENVALUES

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  1. 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

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

  3. 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

  4. 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

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

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

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

  8. (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

  9. 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

  10. 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

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

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

  13. 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. - Get the following eigenpairs. Eigenvalues : Eigenvectors : (13) (10) Structural Dynamics & Vibration Control Lab., KAIST, Korea

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

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

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

  17. 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

  18. Step 4: Check the error norm. Error norm = • 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

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

  20. 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

  21. 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

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

  23. Cantilever Beam with Lumped Dampers (Distinct Case) Material Properties Tangential Damper :c = 0.3 Rayleigh Damping : =  = 0.001 Young’s Modulus :1000 Mass Density :1 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 1 2 3 4 99 100 101 C 5 Structural Dynamics & Vibration Control Lab., KAIST, Korea

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

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

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

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

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

  29. Grid Structure with Lumped Dampers (Multiple Case) Material Properties Tangential Damper :c = 0.3 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 Mean Half Bandwidths :14 100@0.1=10 100@0.1=10 Structural Dynamics & Vibration Control Lab., KAIST, Korea

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

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

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

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

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

  35. Three-Dimensional Framed Structure with Lumped Dampers(Close Case) 2@3.01=6.02 2@3=6 6@3.01=18.06 12@3=36 6@3=18 Structural Dynamics & Vibration Control Lab., KAIST, Korea

  36. Material Properties Lumped Damper :c = 12,000.0 Rayleigh Damping : =-0.1755  = 0.02005 Young’s Modulus :2.1E+11 Mass Density :7,850 Cross-section Inertia :8.3E-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

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

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

  39.  : 1st, 2nd eigenpairs  : 3rd, 4th eigenpairs  : 5th, 6th eigenpairs  : 7th, 8th eigenpairs  : 9th, 10th eigenpairs  : 11th, 12th eigenpairs Starting values of proposed method   Convergence by Lanczos method(Chen 1993) 3-D. framed structure (close)     Structural Dynamics & Vibration Control Lab., KAIST, Korea

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

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

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

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