1. 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 International Conference on Numerical Methods & Computational Mechanics The University of Miskolc, Hungary August 26, 1998 Efficient Free Vibration Analysis of Large Structures with Proportional and Non-Proportional Dampers

2. Structural Dynamics & Vibration Control Lab., KAIST, Korea 1 n Problem Definition n Proposed Method n Numerical Examples n Conclusions OUTLINE

3. Structural Dynamics & Vibration Control Lab., KAIST, Korea 2 PROBLEM DEFINITION n Dynamic Equation of Motion where : Mass matrix, Positive definite : Damping matrix : Stiffness matrix, Positive semi-definite : Displacement vector : Load vector : Order of K, C and M ( = 1,000 ~ 100,000) (1)

4. Structural Dynamics & Vibration Control Lab., KAIST, Korea 3 n Methods of Dynamic Analysis u Step by step integration method u Mode superposition method n Mode Superposition Method u Free vibration analysis must be first performed. u Most of computation time is required for free vibration analysis. An efficient solution technique is required !!!

5. Structural Dynamics & Vibration Control Lab., KAIST, Korea 4 n Condition of Proportional Damping u Ex. : Rayleigh Damping

6. Structural Dynamics & Vibration Control Lab., KAIST, Korea 5 Eigenvalue Problem ( Proportionally Damped Case ) (3) : Orthogonality of eigenvector : ith eigenvalue(real) : ith eigenvector(real) : Number of eigenpairs to be sought where (2)

7. Structural Dynamics & Vibration Control Lab., KAIST, Korea 6 n Current Methods for Proportionally Damped Case u Subspace iteration method u Determinant search method u Householder-QR-inverse iteration method n Techniques Used by Commercial Programs u ABAQUS- Subspace iteration method u ADINA- Subspace iteration method - Determinant search method u ANSYS- Subspace iteration method - Householder-QR method u NASTRAN- Givens method - Inverse power method u SAP Series- Subspace iteration method

8. Structural Dynamics & Vibration Control Lab., KAIST, Korea 7 (4) (5) : Orthogonality of eigenvector : ith eigenvalue(complex conjugate) : ith eigenvector(complex conjugate) : Number of eigenpairs to be sought where Eigenvalue Problem ( Non-Proportionally Damped Case )

9. Structural Dynamics & Vibration Control Lab., KAIST, Korea 8 Current Methods for Non-Proportionally Damped Case  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

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

11. Structural Dynamics & Vibration Control Lab., KAIST, Korea 10 n For Proportionally Damped Case (real) n For Non-Proportionally Damped Case (complex conjugate)

12. Structural Dynamics & Vibration Control Lab., KAIST, Korea 11 n Relations between and Vectors in the Subspace of where (6) (7) (8) u Let be the vectors in the subspace of and be orthonormal with respect to, then (9) (10)

13. Structural Dynamics & Vibration Control Lab., KAIST, Korea 12 where : Symmetric u Let (12) u Introducing Eq.(9) into Eq.(6) (11) or u Then or (13) (14) (15)

14. Structural Dynamics & Vibration Control Lab., KAIST, Korea 13 n Multiple or Close Eigenvalues u Multiple eigenvalues case : is a diagonal matrix. Eigenvalues : Eigenvectors : u Close eigenvalues case : is not a diagonal matrix. n Solve the small standard eigenvalue problem. n Get the following eigenpairs. Eigenvalues : Eigenvectors : (12) (9)

15. Structural Dynamics & Vibration Control Lab., KAIST, Korea 14 n Find the Vectors in the Subspace of the Eigenvectors. n Rotate the Vectors in the Subspace to Find the Eigenvectors. Strategy

16. Structural Dynamics & Vibration Control Lab., KAIST, Korea 15 (16) (17) where : unknown incremental values (18) (19) (20) Newton-Raphson Technique

17. Structural Dynamics & Vibration Control Lab., KAIST, Korea 16 where : residual vector (21) (22) u Introducing Eqs.(18) and (19) into Eqs.(16) and (17) and neglecting nonlinear terms u Matrix form of Eqs.(21) and (22) (23) Coefficient matrix : Symmetric Nonsingular

18. Structural Dynamics & Vibration Control Lab., KAIST, Korea 17 Modified Newton-Raphson Technique Coefficient matrix : Symmetric Nonsingular (24) (19) (18)

19. Structural Dynamics & Vibration Control Lab., KAIST, Korea 18 n Intermediate results by u Subspace iteration method : Proportionally damped case u Determinant search method n Results by Approximate Solution Methods such as u Static or dynamic condensation method  Lanczos method : Non-Proportionally damped case Starting Eigenpairs

20. Structural Dynamics & Vibration Control Lab., KAIST, Korea 19 Step u Step 2: Solve for and u Step 3: Compute u Step 1: Start with approximate eigenpairs

21. Structural Dynamics & Vibration Control Lab., KAIST, Korea 20 u 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. u Step 5: Check if is a diagonal matrix, go to Step 6, if not, go to Step 7.

22. Structural Dynamics & Vibration Control Lab., KAIST, Korea 21 u Step 7: Close case u Step 6: Multiple case n Go to step 8. u Step 8: Check the error norm. Error norm = u Stop !

23. Structural Dynamics & Vibration Control Lab., KAIST, Korea 22 NUMERICAL EXAMPLES: Proportionally Damped Case n Structures u Three-dimensional framed structure(distinct) u Simply-supported rectangular plate(multiple & close) u Cooling tower(multiple) n Analysis Methods u Proposed method u Subspace iteration method u Determinant search method n Comparisons u CPU time u Convergence n IRIS4D20-S17 with 10 MIPS, 0.9 MFLOPS

24. Structural Dynamics & Vibration Control Lab., KAIST, Korea 23 Three-Dimensional Framed Structure (Distinct Case) Elevation Plan  Material Property Young’s modulus : 2.068E10 Pa Mass density : 5.154E2 kg/m 3 - Column in Front Building I : 8.631E-3 m 4, A : 0.2787 m 2 - Column in Rear Building I : 10.787E-3 m 4, A : 0.3716 m 2 - All Beams into x-Direction I : 6.473E-3 m 4, A : 0.6906 m 2 - All Beams into y-Direction I : 8.631E-3 m 4, A : 0.2787 m 2  System Data Number of equations : 468 Number of matrix elements : 42498 Maximum half-bandwidth : 138 Mean half-bandwidth : 91

25. Structural Dynamics & Vibration Control Lab., KAIST, Korea 24 Eigenvalues (Distinct), 3-D. Frame

26. Structural Dynamics & Vibration Control Lab., KAIST, Korea 25 Starting values : Subspace iteration method Relative error = 10 -1 Relative error = Error norm = p = No. of eigenpairs Solution Time (sec), 3-D. Frame

27. Structural Dynamics & Vibration Control Lab., KAIST, Korea 26 Convergence of the 12th eigenpair 3-D. framed structure (distinct)  : Proposed Method  : Subspace Iteration Method (q=2p)  : Determinant Search Method Error Limit   

28. Structural Dynamics & Vibration Control Lab., KAIST, Korea 27 Simply-Supported Rectangular Plate  Material Properties Young’s Modulus: 2.0E11 Pa Mass Density: 7.850E3 kg/m 3 Poisson Ratio: 0.3 Thickness: 0.01m  System Data Number of Equations: 701 Number of Matrix Elements: 62,301 Maximum Half Bandwidths: 133 Mean Half Bandwidths: 89 (a) Multiple eigenvalues (b) Close eigenvalues

29. Structural Dynamics & Vibration Control Lab., KAIST, Korea 28 Eigenvalues (Multiple), Square Plate

30. Structural Dynamics & Vibration Control Lab., KAIST, Korea 29 Starting values : Subspace iteration method Relative error = 10 -1 Relative error = Error norm = p = No. of eigenpairs Solution Time (sec), Square Plate

31. Structural Dynamics & Vibration Control Lab., KAIST, Korea 30 Convergence of the 8th eigenpair Square plate (multiple)  : Proposed Method  : Subspace Iteration Method (q=2p)  : Determinant Search Method Error Limit  

32. Structural Dynamics & Vibration Control Lab., KAIST, Korea 31 Eigenvalues (Close), Plate

33. Structural Dynamics & Vibration Control Lab., KAIST, Korea 32 Starting values : Subspace iteration method Relative error = 10 -1 Relative error = Error norm = p = No. of eigenpairs Solution Time (sec), Plate

34. Structural Dynamics & Vibration Control Lab., KAIST, Korea 33 Convergence of the 8th eigenpair Plate (close)  : Proposed Method  : Subspace Iteration Method (q=2p)  : Determinant Search Method Error Limit   

35. Structural Dynamics & Vibration Control Lab., KAIST, Korea 34  Material Properties Young’s Modulus: 4.32E8 lb/ft 2 Mass Density: 4.66 slug/ft 3 Poisson Ratio: 0.15 Shell Thickness: 0.583 ft  System Data Number of Equations: 2,448 Number of Matrix Elements: 490,572 Maximum Half Bandwidths: 2,358 Mean Half Bandwidths: 201 Cooling Tower(Multiple Case) Elevation Plan

36. Structural Dynamics & Vibration Control Lab., KAIST, Korea 35 Eigenvalues (Multiple), Cooling Tower

37. Structural Dynamics & Vibration Control Lab., KAIST, Korea 36 Starting values : Subspace iteration method Relative error = 10 -1 Relative error = Error norm = p = No. of eigenpairs Solution Time (sec), Cooling Tower

38. Structural Dynamics & Vibration Control Lab., KAIST, Korea 37 Convergence of the 10th eigenpair Cooling tower (multiple)  : Proposed Method  : Subspace Iteration Method (q=2p) Error Limit 

39. Structural Dynamics & Vibration Control Lab., KAIST, Korea 38 NUMERICAL EXAMPLES: Non-Proportionally Damped Case n Structures u Cantilever beam(distinct) u Grid structure(multiple) u Three-dimensional framed structure(close) n Analysis Methods u Proposed method u Subspace iteration method (Leung 1988) u Lanczos method (Chen 1993) n Comparisons u Solution time(CPU) u Convergence n Convex with 100 MIPS, 200 MFLOPS

40. Structural Dynamics & Vibration Control Lab., KAIST, Korea 39 Cantilever Beam with Lumped Dampers (Distinct Case) 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

41. Structural Dynamics & Vibration Control Lab., KAIST, Korea 40 Results of Cantilever Beam Structure (Distinct)  Number of Lanczos vectors = 20

42. Structural Dynamics & Vibration Control Lab., KAIST, Korea 41 CPU Time for 10 Lowest Eigenpairs, Cantilever Beam

43. Structural Dynamics & Vibration Control Lab., KAIST, Korea 42 Convergence by Lanczos method(Chen 1993) Cantilever beam (distinct) Starting values of proposed method       : 1st, 2nd eigenpairs  : 3rd, 4th eigenpairs  : 5th, 6th eigenpairs  : 7th, 8th eigenpairs  : 9th, 10th eigenpairs

44. Structural Dynamics & Vibration Control Lab., KAIST, Korea 43 Convergence of the 1st eigenpair Cantilever beam (distinct)  : Proposed Method  : Subspace Iteration Method (q=2p)  

45. Structural Dynamics & Vibration Control Lab., KAIST, Korea 44 Convergence of the 5th eigenpair Cantilever beam (distinct)  : Proposed Method  : Subspace Iteration Method (q=2p)  

46. Structural Dynamics & Vibration Control Lab., KAIST, Korea 45 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

47. Structural Dynamics & Vibration Control Lab., KAIST, Korea 46 Results of Grid Structure (Multiple)  Number of Lanczos vectors = 48

48. Structural Dynamics & Vibration Control Lab., KAIST, Korea 47 CPU Time for 12 Lowest Eigenpairs, Grid Structure

49. Structural Dynamics & Vibration Control Lab., KAIST, Korea 48 Convergence by Lanczos method(Chen 1993) Grid structure (multiple)  : 1st, 3rd eigenpairs  : 2nd, 4th eigenpairs  : 5th, 7th eigenpairs  : 6th, 8th eigenpairs  : 9th, 11th eigenpairs  : 10th, 12th eigenpairs       Starting values of proposed method

50. Structural Dynamics & Vibration Control Lab., KAIST, Korea 49 Convergence of the 2nd eigenpair Grid structure (multiple)  : Proposed Method  : Subspace Iteration Method (q=2p)  

51. Structural Dynamics & Vibration Control Lab., KAIST, Korea 50 Convergence of the 9th eigenpair Grid structure (multiple)  : Proposed Method  : Subspace Iteration Method (q=2p) 

52. Structural Dynamics & Vibration Control Lab., KAIST, Korea 51 Three-Dimensional Framed Structure with Lumped Dampers(Close Case) 2@3.01=6.02 6@3=18 2@3=6 6@3.01=18.06 12@3=36

53. Structural Dynamics & Vibration Control Lab., KAIST, Korea 52  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

54. Structural Dynamics & Vibration Control Lab., KAIST, Korea 53 Results of Three-Dimensional Framed Structure (Close)  Number of Lanczos vectors = 48

55. Structural Dynamics & Vibration Control Lab., KAIST, Korea 54 CPU Time for 12 Lowest Eigenpairs, 3-D. Framed Structure

56. Structural Dynamics & Vibration Control Lab., KAIST, Korea 55 Convergence by Lanczos method(Chen 1993) 3-D. framed structure (close)  : 1st, 2nd eigenpairs  : 3rd, 4th eigenpairs  : 5th, 6th eigenpairs  : 7th, 8th eigenpairs  : 9th, 10th eigenpairs  : 11th, 12th eigenpairs       Starting values of proposed method

57. Structural Dynamics & Vibration Control Lab., KAIST, Korea 56 Convergence of the 9th eigenpair 3-D. framed structure (close)  : Proposed Method  : Subspace Iteration Method (q=2p) 

58. Structural Dynamics & Vibration Control Lab., KAIST, Korea 57 CONCLUSIONS n The proposed method u is simple u guarantees numerical stability u converges fast. An efficient solution technique !

59. Structural Dynamics & Vibration Control Lab., KAIST, Korea 58 Thank you for your attention.

60. Structural Dynamics & Vibration Control Lab., KAIST, Korea 59 Convergence of the 3rd eigenpair Cantilever beam (distinct)  : Proposed Method  : Subspace Iteration Method 

61. Structural Dynamics & Vibration Control Lab., KAIST, Korea 60 Convergence of the 7th eigenpair Cantilever beam (distinct)  : Proposed Method  : Subspace Iteration Method 

62. Structural Dynamics & Vibration Control Lab., KAIST, Korea 61 Convergence of the 9th eigenpair Cantilever beam (distinct)  : Proposed Method  : Subspace Iteration Method 

63. Structural Dynamics & Vibration Control Lab., KAIST, Korea 62 Convergence of the 10th eigenpair Grid structure (multiple)  : Proposed Method  : Subspace Iteration Method 

64. Structural Dynamics & Vibration Control Lab., KAIST, Korea 63 Convergence of the 5th eigenpair 3-D. framed structure (close)  : Proposed Method  : Subspace Iteration Method 

65. Structural Dynamics & Vibration Control Lab., KAIST, Korea 64 Convergence of the 7th eigenpair 3-D. framed structure (close)  : Proposed Method  : Subspace Iteration Method 

66. Structural Dynamics & Vibration Control Lab., KAIST, Korea 65 Convergence of the 11th eigenpair 3-D. framed structure (close)  : Proposed Method  : Subspace Iteration Method 