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

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

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)

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)

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

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.

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

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

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)

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

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

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)

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

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

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)

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

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

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

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

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

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

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

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

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, Method 38.27e e e

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)

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

Structural Dynamics & Vibration Control Lab., KAIST, Korea 26 Cantilever Beam with Lumped Dampers C 5  Material Properties Tangential Damper :c = 0.3 Rayleigh Damping :  =  = 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

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

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

Structural Dynamics & Vibration Control Lab., KAIST, Korea 29 Three-Dimensional Framed Structure with Lumped Dampers  Material Properties Tangential Damper :c = 1,000 Rayleigh Damping :  =  = 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 30 l Results of three-dimensional framed structure : Physical Error norm (number of Lanczos vectors=30)

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

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.

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

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