*Man-Cheol Kim, Hyung-Jo Jung and In-Won Lee *Man-Cheol Kim, Hyung-Jo Jung and In-Won Lee Structural Dynamics & Vibration Control Lab. Structural Dynamics & Vibration Control Lab. Korea Advanced Institute of Science & Technology Solution of Eigenvalue Problem for Non-Classically Damped System with Multiple Eigenvalues PSSC 1998 Fifth Pacific Structural Steel Conference Seoul, Korea, October 1998

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

Structural Dynamics & Vibration Control Lab., KAIST, Korea 2 PROBLEM DEFINITION n Dynamic Equation of Motion (1) where : Mass matrix, Positive definite : Damping matrix : Stiffness matrix, Positive semi-definite : Displacement vector : Load vector

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 should be first performed.

Structural Dynamics & Vibration Control Lab., KAIST, Korea 4 n Condition of Classical Damping u Example : Rayleigh Damping (2)

Structural Dynamics & Vibration Control Lab., KAIST, Korea 5 l Eigenproblem of classical damping systems where : Real eigenvalue : Natural frequency : Real eigenvector(mode shape) u Low in cost u Straightforward (3)

Structural Dynamics & Vibration Control Lab., KAIST, Korea 6 l Quadratic eigenproblem of non-classically damped systems (4) where : Complex eigenvalue : Complex eigenvector(mode shape)

Structural Dynamics & Vibration Control Lab., KAIST, Korea 7 (5) : Complex Eigenvector (6) where An efficient eigensolution technique of non-classically damped systems is required. : Complex Eigenvalue u Very expensive

Structural Dynamics & Vibration Control Lab., KAIST, Korea 8 Current Methods for Solving the Non-Classically Damped Eigenproblems  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 9 PROPOSED METHOD n Find p Smallest Eigenpairs Solve Subject to Forand : multiple or close roots where If p=1, then distinct root

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

Structural Dynamics & Vibration Control Lab., KAIST, Korea 11 where : Symmetric u Let (13) u Introducing Eq.(10) into Eq.(7) (12) or u Then or (14) (15) (16)

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 eigenvalues case : is not a diagonal matrix. n Solve the small standard eigenvalue problem. n Get the following eigenpairs. Eigenvalues : Eigenvectors : (13) (10)

Structural Dynamics & Vibration Control Lab., KAIST, Korea 13 (17) (18) where : unknown incremental values (19) (20) (21) Newton-Raphson Technique

Structural Dynamics & Vibration Control Lab., KAIST, Korea 14 where : residual vector (22) (23) u Introducing Eqs.(19) and (20) into Eqs.(17) and (18) and neglecting nonlinear terms u Matrix form of Eqs.(22) and (23) (24) Coefficient matrix : Symmetric Nonsingular

Structural Dynamics & Vibration Control Lab., KAIST, Korea 15 Coefficient matrix : Symmetric Nonsingular (24) Introducing modified Newton-Raphson technique (25) (20) (19) Modified Newton-Raphson Technique

Structural Dynamics & Vibration Control Lab., KAIST, Korea 16 Algorithm of Proposed Method u Step 2: Solve for and u Step 3: Compute u Step 1: Start with approximate eigenpairs

Structural Dynamics & Vibration Control Lab., KAIST, Korea 17 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.

Structural Dynamics & Vibration Control Lab., KAIST, Korea 18 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 !

Structural Dynamics & Vibration Control Lab., KAIST, Korea 19 n Initial Values of the Proposed Method u Intermediate results of the iteration methods l Vector iteration method l Subspace iteration method u Results of the approximate methods l Static Condensation method l Lanczos method

Structural Dynamics & Vibration Control Lab., KAIST, Korea 20 NUMERICAL EXAMPLES 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)

Structural Dynamics & Vibration Control Lab., KAIST, Korea 21 n Comparisons u Solution time(CPU) u Convergence n Convex with 100 MIPS, 200 MFLOPS

Structural Dynamics & Vibration Control Lab., KAIST, Korea 22 Cantilever Beam with Lumped Dampers (Distinct Case) 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 23 Results of Cantilever Beam Structure (Distinct)

Structural Dynamics & Vibration Control Lab., KAIST, Korea 24 CPU Time for 10 Lowest Eigenpairs, Cantilever Beam

Structural Dynamics & Vibration Control Lab., KAIST, Korea 25 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

Structural Dynamics & Vibration Control Lab., KAIST, Korea 26 Convergence of the 1st eigenpair Cantilever beam (distinct)  : Proposed Method  : Subspace Iteration Method (q=2p)  

Structural Dynamics & Vibration Control Lab., KAIST, Korea 27 Convergence of the 5th eigenpair Cantilever beam (distinct)  : Proposed Method  : Subspace Iteration Method (q=2p)  

Structural Dynamics & Vibration Control Lab., KAIST, Korea 28 Grid Structure with Lumped Dampers (Multiple Case)  Material Properties Tangential Damper :c = 0.3 Rayleigh Damping :  =  = 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

Structural Dynamics & Vibration Control Lab., KAIST, Korea 29 Results of Grid Structure (Multiple)

Structural Dynamics & Vibration Control Lab., KAIST, Korea 30 CPU Time for 12 Lowest Eigenpairs, Grid Structure

Structural Dynamics & Vibration Control Lab., KAIST, Korea 31 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

Structural Dynamics & Vibration Control Lab., KAIST, Korea 32 Convergence of the 2nd eigenpair Grid structure (multiple)  : Proposed Method  : Subspace Iteration Method (q=2p)  

Structural Dynamics & Vibration Control Lab., KAIST, Korea 33 Convergence of the 9th eigenpair Grid structure (multiple)  : Proposed Method  : Subspace Iteration Method (q=2p) 

Structural Dynamics & Vibration Control Lab., KAIST, Korea 34 Three-Dimensional Framed Structure with Lumped Dampers(Close Case)

Structural Dynamics & Vibration Control Lab., KAIST, Korea 35  Material Properties Lumped Damper :c = 12,000.0 Rayleigh Damping :  =  = 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

Structural Dynamics & Vibration Control Lab., KAIST, Korea 36 Results of Three-Dimensional Framed Structure (Close)

Structural Dynamics & Vibration Control Lab., KAIST, Korea 37 CPU Time for 12 Lowest Eigenpairs, 3-D. Framed Structure

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

Structural Dynamics & Vibration Control Lab., KAIST, Korea 39 Convergence of the 9th eigenpair 3-D. framed structure (close)  : Proposed Method  : Subspace Iteration Method (q=2p) 

Structural Dynamics & Vibration Control Lab., KAIST, Korea 40 CONCLUSIONS n The proposed method u is simple u guarantees numerical stability u converges fast. An efficient Eigensolution technique !

Structural Dynamics & Vibration Control Lab., KAIST, Korea 41 Thank you for your attention.