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Solution of Eigenvalue Problems for Non-Proportionally Damped Systems with Multiple Frequencies

Fourth World Congress on Computational Mechanics Buenos Aires, Argentina June 30, 1998 Session V-C : Structural Dynamics I. Solution of Eigenvalue Problems for Non-Proportionally Damped Systems with Multiple Frequencies. In-Won Lee, Professor, PE

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Solution of Eigenvalue Problems for Non-Proportionally Damped Systems with Multiple Frequencies

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  1. Fourth World Congress on Computational Mechanics Buenos Aires, Argentina June 30, 1998 Session V-C : Structural Dynamics I Solution of Eigenvalue Problems for Non-Proportionally Damped Systems with Multiple Frequencies In-Won Lee, Professor, PE Structural Dynamics & Vibration Control Lab. Korea Advanced Institute of Science & Technology Korea

  2. OUTLINE • Introduction • Objectives and scope • Current methods • Proposed method • Newton-Raphson technique • Modified Newton-Raphson technique • Numerical examples • Grid structure with lumped dampers • Three-dimensional framed structure with lumped dampers • Conclusions Structural Dynamics & Vibration Control Lab., KAIST, Korea

  3. INTRODUCTION Objectives and Scope • Free vibration of proportional damping system (1) where : Mass matrix : Damping matrix : Stiffness matrix : Displacement vector Structural Dynamics & Vibration Control Lab., KAIST, Korea

  4. Eigenanalysis of proportional damping system where : Real eigenvalue : Natural frequency : Real eigenvector(mode shape) • Low in cost • Straightforward (2) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  5. (3) where (4) Let (5) • Free vibration analysis of non-proportionaldamping system , then (6) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  6. (7) where : Eigenvalue(complex conjugate) :Eigenvector(complex conjugate) (8) (9) : Orthogonality of eigenvector • Solution of Eq.(7) isvery expensive. • Therefore, an efficient eigensolution technique for • non-proportional damping system is required. Structural Dynamics & Vibration Control Lab., KAIST, Korea

  7. Current Methods • 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. where PROPOSED METHOD • Find p smallest multiple eigenpairs Solve Subject to For and : multiple Structural Dynamics & Vibration Control Lab., KAIST, Korea

  9. 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

  10. (12) • Note : If , from Eq.(13) (17) (18) • Introducing Eq.(10) into Eq.(7) (13) • Let where : Symmetric • Then (14) or (15) or (16) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  11. Newton-Raphson Technique (19) (20) (21) where (22) (23) : unknown incremental values Structural Dynamics & Vibration Control Lab., KAIST, Korea

  12. Introducing Eqs.(21) and (22) into Eqs.(19) and (20) and neglecting nonlinear terms (23) (24) : residual vector where • Matrix form of Eqs.(23) and (24) (25) Coefficient matrix: • Symmetric • Nonsingular Structural Dynamics & Vibration Control Lab., KAIST, Korea

  13. Modified Newton-Raphson Technique (25) Introducing modified Newton-Raphson technique (26) (21) (22) Coefficient matrix: • Symmetric • Nonsingular Structural Dynamics & Vibration Control Lab., KAIST, Korea

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

  15. Step 5: Multiple case or or • 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 Structural Dynamics & Vibration Control Lab., KAIST, Korea

  16. NUMERICAL EXAMPLES • Structures • Grid structure with lumped dampers • Three-dimensional framed structure with lumped dampers • Analysis methods • Proposed method • Subspace iteration method (Leung 1988) • Lanczos method (Chen 1993) Structural Dynamics & Vibration Control Lab., KAIST, Korea

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

  18. Grid Structure with Lumped Dampers 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 Structural Dynamics & Vibration Control Lab., KAIST, Korea

  19. Table 1. Results of proposed method for grid structure Structural Dynamics & Vibration Control Lab., KAIST, Korea

  20. Table 2. CPU time for twelve lowest eigenpairs of grid structure Structural Dynamics & Vibration Control Lab., KAIST, Korea

  21. Lanczos method (48 Lanczos vectors) Fig.3 Error norms of grid model by subspace iteration method Fig.4 Error norms of grid model by Lanczos method Fig.2 Error norms of grid model by proposed method Structural Dynamics & Vibration Control Lab., KAIST, Korea

  22. Three-Dimensional Framed Structure with Lumped Dampers Structural Dynamics & Vibration Control Lab., KAIST, Korea

  23. 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

  24. Table 3. Results of proposed method for three-dimensional framed structure Structural Dynamics & Vibration Control Lab., KAIST, Korea

  25. Table 4. CPU time for twelve lowest eigenpairs of three-dimensional framed structure Structural Dynamics & Vibration Control Lab., KAIST, Korea

  26. Lanczos method (48 Lanczos vectors) Fig.7 Error norms of 3-D. frame model by subspace iteration method Fig.8 Error norms of 3-D. frame model by Lanczos method Fig.6 Error norms of 3-D. frame model by proposed method Structural Dynamics & Vibration Control Lab., KAIST, Korea

  27. CONCLUSIONS • Proposed method • converges fast • guarantees nonsingularity of coefficient matrix Proposed method is efficient Structural Dynamics & Vibration Control Lab., KAIST, Korea

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

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