International Conference on Numerical Methods Computational Mechanics The

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 In-Won Lee, Professor, PE Structural Dynamics & Vibration Control Lab. Korea Advanced Institute of Science & Technology

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

PROBLEM DEFINITION n Dynamic Equation of Motion (1) 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) Structural Dynamics & Vibration Control Lab. , KAIST, Korea 2

n Methods of Dynamic Analysis u Step by step integration method u Mode n superposition method 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 !!! Structural Dynamics & Vibration Control Lab. , KAIST, Korea 3

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

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

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 u ADINA u ANSYS u NASTRAN u SAP Series - Subspace iteration method Determinant search method Subspace iteration method Householder-QR method Givens method Inverse power method Subspace iteration method Structural Dynamics & Vibration Control Lab. , KAIST, Korea 6

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

n Current Methods for Non-Proportionally Damped Case u Transformation method: Kaufman (1974) u Perturbation method: Meirovitch et al (1979) u Vector iteration method: Gupta (1974; 1981) u Subspace iteration method: Leung (1995) u Lanczos method: Chen (1993) u Efficient Methods Structural Dynamics & Vibration Control Lab. , KAIST, Korea 8

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

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

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

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

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

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

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

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

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

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

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

u Step 4: Check the error norm. Error norm = n 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. Structural Dynamics & Vibration Control Lab. , KAIST, Korea 20

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

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 IRIS 4 D 20 -S 17 with 10 MIPS, 0. 9 MFLOPS Structural Dynamics & Vibration Control Lab. , KAIST, Korea 22

Three-Dimensional Framed Structure (Distinct Case) Material Property Young’s modulus : 2. 068 E 10 Pa Mass density : 5. 154 E 2 kg/m 3 - Column in Front Building I : 8. 631 E-3 m 4 , A : 0. 2787 m 2 - Column in Rear Building I : 10. 787 E-3 m 4 , A : 0. 3716 m 2 - All Beams into x-Direction I : 6. 473 E-3 m 4 , A : 0. 6906 m 2 - All Beams into y-Direction I : 8. 631 E-3 m 4 , A : 0. 2787 m 2 Elevation Plan System Data Number of equations : 468 Number of matrix elements : 42498 Maximum half-bandwidth : 138 Mean half-bandwidth : 91 Structural Dynamics & Vibration Control Lab. , KAIST, Korea 23

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

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

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

Simply-Supported Rectangular Plate (a) Multiple eigenvalues Material Properties (b) Close eigenvalues System Data Young’s Modulus: 2. 0 E 11 Pa Number of Equations: 701 Mass Density: 7. 850 E 3 kg/m 3 Number of Matrix Elements: 62, 301 Poisson Ratio: 0. 3 Maximum Half Bandwidths: 133 Thickness: 0. 01 m Mean Half Bandwidths: 89 Structural Dynamics & Vibration Control Lab. , KAIST, Korea 27

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

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

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

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

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

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

Cooling Tower(Multiple Case) Material Properties Young’s Modulus: 4. 32 E 8 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 Elevation Plan Maximum Half Bandwidths: 2, 358 Mean Half Bandwidths: 201 Structural Dynamics & Vibration Control Lab. , KAIST, Korea 34

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

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

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

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 Structural Dynamics & Vibration Control Lab. , KAIST, Korea 38

Cantilever Beam with Lumped Dampers (Distinct Case) Material Properties 1 2 3 4 5 Tangential Damper : c = 0. 3 Rayleigh Damping : = = 0. 001 Young’s Modulus : 1000 99 100 101 Mass Density : 1 C 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 Structural Dynamics & Vibration Control Lab. , KAIST, Korea 39

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

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

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

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

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

Grid Structure with Lumped Dampers (Multiple Case) Material Properties Tangential Damper : c = 0. 3 100@0. 1=10 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 : 15 100@0. 1=10 Mean Half Bandwidths : 14 Structural Dynamics & Vibration Control Lab. , KAIST, Korea 45

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

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

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

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

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

Three-Dimensional Framed Structure with Lumped Dampers(Close Case) 2@3. 01=6. 02 2@3=6 6@3. 01=18. 06 6@3=18 12@3=36 Structural Dynamics & Vibration Control Lab. , KAIST, Korea 51

Material Properties Lumped Damper : c = 12, 000. 0 Rayleigh Damping : =-0. 1755 = 0. 02005 Young’s Modulus : 2. 1 E+11 Mass Density : 7, 850 Cross-section Inertia : 8. 3 E-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 52

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

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

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

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

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

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

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

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

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

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

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

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

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