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Solution of Eigenproblem of Non-Proportional Damping Systems by Lanczos Method

The Fourth International Conference on Computational Structures Technology Edinburgh, Scotland 18th-20th August 1998. Solution of Eigenproblem of Non-Proportional Damping Systems by Lanczos Method. In-Won Lee, Professor, PE Structural Dynamics & Vibration Control Lab.

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Solution of Eigenproblem of Non-Proportional Damping Systems by Lanczos Method

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  1. The Fourth International Conference on Computational Structures Technology Edinburgh, Scotland 18th-20th August 1998 Solution of Eigenproblem of Non-Proportional Damping Systems by Lanczos Method In-Won Lee, Professor, PE Structural Dynamics & Vibration Control Lab. Korea Advanced Institute of Science & Technology

  2. OUTLINE • Introduction • Method of analysis • Numerical examples • Conclusions Structural Dynamics & Vibration Control Lab., KAIST, Korea

  3. INTRODUCTION • Free vibration of proportional damping system where : Mass matrix : Proportional damping matrix : Stiffness matrix : Displacement vector (1) 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. Free vibration of non-proportionaldamping system (3) where and (4) Let (5) , 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 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. Proposed Lanczos algorithm • retains the n order quadratic eigenproblems • is one-sided recursion scheme • extracts the Lanczos vectors in real domain Structural Dynamics & Vibration Control Lab., KAIST, Korea

  9. METHOD OF ANALYSIS • Free vibration of non-proportional damping system where : Mass matrix : Non-proportional damping matrix : Stiffness matrix : Displacement vector (10) Let (11) , then Structural Dynamics & Vibration Control Lab., KAIST, Korea

  10. Quadratic eigenproblem where : eigenvalue (complex conjugate) : independent eigenvector (complex conjugate) (12) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  11. Orthogonality of the eigenvectors where : dependent eigenvector (13) or (14) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  12. Proposed Lanczos Algorithm • Assume that m independent and dependent Lanczos vectors are found • Calculate preliminary vectors and (15) (16) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  13. Preliminary vectors can be expressed as (17) (18) where (19) (20) is the pseudo length of and , and real are the components of previous Lanczos vectors (real values) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  14. Orthogonality conditions of Lanczos vectors (21) (22)  where (23) (19) (20) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  15. Coefficient Eq.(17) + Eq.(18) and Applying the orthogonality conditions Eqs.(21) and (22) (24) Using Eqs.(15) and (16) (25) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  16. (26) (27) Eq.(17) + Eq.(18) and Applying the orthogonality conditions Eqs.(21) and (22) • Coefficients and where (28) (29) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  17. Coefficients Eq.(17) + Eq.(18) and Applying the orthogonality conditions Eqs.(21) and (22) (30) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  18. (m+1)th Lanczos vectors and (31) (32) where Structural Dynamics & Vibration Control Lab., KAIST, Korea

  19. Reduction to Tri-Diagonal System • Rewriting quadratic eigenproblem (33) where (34) • (35) (36) where Structural Dynamics & Vibration Control Lab., KAIST, Korea

  20. Eq.(33) + Eq.(34) and Applying the orthogonality conditions Eqs.(21) and (22) (37) where Unsymmetric (38) : Real values Structural Dynamics & Vibration Control Lab., KAIST, Korea

  21. Eigenvalues and eigenvectors of the system (39) (40) (41) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  22. Error Estimation • Physical error norm(Bathe et al 1980) and : Acceptable eigenpair (42) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  23. Proposed method Rajakumar’s method Chen’s method : Number of equations : Mean half bandwidths of K, M and C Comparison of Operations Initial operations (A) Operations in each row of T (B) Method Number of operations = A + pB where p : Number of Lanczos vectors Structural Dynamics & Vibration Control Lab., KAIST, Korea

  24. 1,008 81 p = 30 Example : Three-Dimensional Framed Structure Method Number of total operations Ratio 38.27e+6 Proposed method 1.00 Rajakumar’s method 53.23e+06 1.39 61.38e+06 1.60 Chen’s method Structural Dynamics & Vibration Control Lab., KAIST, Korea

  25. NUMERICAL EXAMPLES • Structures • Cantilever beam with lumped dampers • Three-dimensional framed structure with lumped dampers • Analysis methods • Proposed method • Rajakumar’s method (1993) • Chen’s method (1988) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  26. Comparisons • Solution time(CPU) • Physical error norm • Convex with 100 MIPS, 200 MFLOPS Structural Dynamics & Vibration Control Lab., KAIST, Korea

  27. Cantilever Beam with Lumped Dampers 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

  28. Results of cantilever beam : Physical Error norm (number of Lanczos vectors=30) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  29. Results of cantilever beam : Physical Error norm (number of Lanczos vectors=60) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  30. Three-Dimensional Framed Structure with Lumped Dampers Material Properties Tangential Damper :c = 1,000 Rayleigh Damping : = -0.92  = 0.106 Young’s Modulus: 2.1E+11 Mass Density: 7,850 Cross-section Inertia: 8.3E-06 Cross-section Area: 001 System Data Number of Equations: 1,008 Number of Matrix Elements :80,784 Maximum Half Bandwidths : 150 Mean Half Bandwidths : 81 Structural Dynamics & Vibration Control Lab., KAIST, Korea

  31. Results of three-dimensional framed structure : Physical Error norm (number of Lanczos vectors=30) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  32. Results of three-dimensional framed structure : Physical Error norm (number of Lanczos vectors=60) Structural Dynamics & Vibration Control Lab., KAIST, Korea

  33. CONCLUSIONS • The proposed method • needs smaller storage space • gives better solutions • requires less solution time than other methods. An efficient solution technique! Structural Dynamics & Vibration Control Lab., KAIST, Korea

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

  35. (A-2) (A-3) (A-1) To scale (A-4) If where , (A-5) , If Structural Dynamics & Vibration Control Lab., KAIST, Korea

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