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

2. Structural Dynamics & Vibration Control Lab., KAIST, Korea 1 OUTLINE l Introduction l Method of analysis l Numerical examples l Conclusions

3. Structural Dynamics & Vibration Control Lab., KAIST, Korea 2 INTRODUCTION l Free vibration of proportional damping system where : Mass matrix : Proportional damping matrix : Stiffness matrix : Displacement vector (1)

4. Structural Dynamics & Vibration Control Lab., KAIST, Korea 3 l Eigenanalysis of proportional damping system where : Real eigenvalue : Natural frequency : Real eigenvector(mode shape) u Low in cost u Straightforward (2)

5. Structural Dynamics & Vibration Control Lab., KAIST, Korea 4 l Free vibration of non-proportional damping system (4) where (3) (5)Let (6), then and

6. Structural Dynamics & Vibration Control Lab., KAIST, Korea 5 Therefore, an efficient eigensolution technique is required. (7) (9) : Orthogonality of eigenvector : Eigenvalue(complex conjugate) : Eigenvector(complex conjugate) (8) where Solution of Eq.(7) is very expensive.

7. Structural Dynamics & Vibration Control Lab., KAIST, Korea 6 Current Methods l Transformation method: Kaufman (1974) l Perturbation method: Meirovitch et al (1979) l Vector iteration method: Gupta (1974; 1981) n Subspace iteration method: Leung (1995) n Lanczos method: Chen (1993) n Efficient Methods

8. Structural Dynamics & Vibration Control Lab., KAIST, Korea 7 l Proposed Lanczos algorithm u retains the n order quadratic eigenproblems u is one-sided recursion scheme u extracts the Lanczos vectors in real domain

9. Structural Dynamics & Vibration Control Lab., KAIST, Korea 8 METHOD OF ANALYSIS l Free vibration of non-proportional damping system where : Mass matrix : Non-proportional damping matrix : Stiffness matrix : Displacement vector (11)Let, then (10)

10. Structural Dynamics & Vibration Control Lab., KAIST, Korea 9 l Quadratic eigenproblem where : eigenvalue (complex conjugate) : independent eigenvector (complex conjugate) (12)

11. Structural Dynamics & Vibration Control Lab., KAIST, Korea 10 where : dependent eigenvector (13) l Orthogonality of the eigenvectors or (14)

12. Structural Dynamics & Vibration Control Lab., KAIST, Korea 11 Proposed Lanczos Algorithm l Assume that m independent and dependent Lanczos vectors are found l Calculate preliminary vectors and (15) (16)

13. Structural Dynamics & Vibration Control Lab., KAIST, Korea 12 l Preliminary vectors can be expressed as are the components of previous Lanczos vectors (real values), and is the pseudo length of and (17) (18) (19) (20) real where

14. Structural Dynamics & Vibration Control Lab., KAIST, Korea 13 Orthogonality conditions of Lanczos vectors (21) (22)  (23) (19) (20) where

15. Structural Dynamics & Vibration Control Lab., KAIST, Korea 14 l Coefficient the orthogonality conditions Eqs.(21) and (22) Eq.(17) + Eq.(18) and Applying Using Eqs.(15) and (16) u u (25) (24)

16. Structural Dynamics & Vibration Control Lab., KAIST, Korea 15 l Coefficients and (26) (27) (28) (29) Applying the orthogonality conditions Eqs.(21) and (22) Eq.(17) +Eq.(18) and u where

17. Structural Dynamics & Vibration Control Lab., KAIST, Korea 16 l Coefficients (30) Applying the orthogonality conditions Eqs.(21) and (22) Eq.(17) +Eq.(18) and u

18. Structural Dynamics & Vibration Control Lab., KAIST, Korea 17 l (m+1)th Lanczos vectors and (31) (32) where

19. Structural Dynamics & Vibration Control Lab., KAIST, Korea 18 Reduction to Tri-Diagonal System l Rewriting quadratic eigenproblem (33) where (34) l (35) (36) where

20. Structural Dynamics & Vibration Control Lab., KAIST, Korea 19 (37) Applying the orthogonality conditions Eqs.(21) and (22) Eq.(33) +Eq.(34) and u Unsymmetric (38) where : Real values

21. Structural Dynamics & Vibration Control Lab., KAIST, Korea 20 l Eigenvalues and eigenvectors of the system (39) (40) (41)

22. Structural Dynamics & Vibration Control Lab., KAIST, Korea 21 l Physical error norm(Bathe et al 1980) and : Acceptable eigenpair Error Estimation (42)

23. Structural Dynamics & Vibration Control Lab., KAIST, Korea 22 Comparison of Operations Proposed method Rajakumar’s method Chen’s method Method Initial operations (A) Operations in each row of T (B) Number of operations = A + p B p : Number of Lanczos vectors : Number of equations : Mean half bandwidths of K, M and C where

24. Structural Dynamics & Vibration Control Lab., KAIST, Korea 23 Example : Three-Dimensional Framed Structure Proposed method Rajakumar’s method Chen’s method Number of total operations Ratio p = 30 1,008 81 Method 38.27e+6 53.23e+06 61.38e+06 1.00 1.39 1.60

25. Structural Dynamics & Vibration Control Lab., KAIST, Korea 24 NUMERICAL EXAMPLES l Structures u Cantilever beam with lumped dampers u Three-dimensional framed structure with lumped dampers l Analysis methods u Proposed method u Rajakumar’s method (1993) u Chen’s method (1988)

26. Structural Dynamics & Vibration Control Lab., KAIST, Korea 25 l Comparisons u Solution time(CPU) u Physical error norm l Convex with 100 MIPS, 200 MFLOPS

27. Structural Dynamics & Vibration Control Lab., KAIST, Korea 26 Cantilever Beam with Lumped Dampers 123499100101 C 5  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

28. Structural Dynamics & Vibration Control Lab., KAIST, Korea 27 l Results of cantilever beam : Physical Error norm (number of Lanczos vectors=30)

29. Structural Dynamics & Vibration Control Lab., KAIST, Korea 28 l Results of cantilever beam : Physical Error norm (number of Lanczos vectors=60)

30. Structural Dynamics & Vibration Control Lab., KAIST, Korea 29 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

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

32. Structural Dynamics & Vibration Control Lab., KAIST, Korea 31 l Results of three-dimensional framed structure : Physical Error norm (number of Lanczos vectors=60)

33. Structural Dynamics & Vibration Control Lab., KAIST, Korea 32 An efficient solution technique! CONCLUSIONS l The proposed method u needs smaller storage space u gives better solutions u requires less solution time than other methods.

34. Structural Dynamics & Vibration Control Lab., KAIST, Korea 33 Thank you for your attention.

35. Structural Dynamics & Vibration Control Lab., KAIST, Korea 34 (A-1) (A-4) (A-5) (A-3) (A-2) where If,, To scale