• Transactions of the Korean Society of Mechanical Engineers A
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• v.29 no.11 s.242
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• pp.1527-1533
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• 2005

Three methods for design sensitivity analysis such as finite difference method(FDM), direct differentiation method(DDM) and adjoint variable method(AVM) are well known. FDM and DDM for design sensitivity analysis cost too much when the number of design variables is too large. An AVM is required to compute adjoint variables from the simultaneous linear system equation, the so-called adjoint equation. Because the adjoint equation is independent of the number of design variables, an AVM is efficient for when number of design variables is too large. In this study, AVM has been extended to the eigenproblem of damped systems whose eigenvlaues and eigenvectors are complex numbers. Moreover, this method is implemented into a commercial finite element analysis program by means of the semi-analytical method to show applicability of the developed method into practical structural problems. The proposed_method is compared with FDM and verified its accuracy for analytical and practical cases.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 1998.04a
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• pp.283-290
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• 1998

A solution method is presented to solve the eigenproblem arising in tile dynamic analysis of non-proportional damping systems with symmetric matrices. The method is based on tile use of Lanczos method to generate a Krylov subspace of trial vectors, witch is then used to reduce a large eigenvalue problem to a much smaller one. The method retains the η order quadratic eigenproblem, without the need to the method of matrix augmentation traditionally used to cast the problem as a linear eigenproblem of order 2n. In the process, the method preserves tile sparseness and symmetry of the system matrices and does not invoke complex arithmetics, therefore, making it very economical for use in solving large problems. Numerical results are presented to demonstrate the efficiency and accuracy of the method.

• Journal of KSNVE
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• v.10 no.1
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• pp.64-73
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• 2000

We present finite element equations in the Laplace-domain for linear viscoelastic and viscoelstically damped structures governed by a constitutive equation involving factional order derivative opeartors. These equations yield a nonstandard eigenproblem consisted of frequency dependent stiffness matrix. To solve this nonstandard eigenproblem we suggest an eigenvalue iteration procedure in the Laplace-domain. Improved Zenor and GHM material function type constitutive equations in the Laplace-domain are also available for this procedure. From above equations, complex eigenvalues and complex eigenvectors are obtained. Using obtained eigenvalues and eigenvectors, time domain analysis is performed by means of mode superposition. Finally, finite element solutions of viscoelastic and viscoeleastically damped sandwich beam are presented as an example.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.7 no.2
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• pp.51-64
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• 2003

Analysis is given of construction and stability of complex symmetric analogues of Householder matrices, with applications to the eigenproblem for such matrices. We investigate numerical properties of the deflation of complex symmetric matrices by using complex symmetric Householder transformations. The proposed method is very similar to the well-known deflation technique for real symmetric matrices (Cf. [16], pp. 586-595). In this paper we present an error analysis of one step of the deflation of complex symmetric matrices.

• Structural Engineering and Mechanics
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• v.14 no.6
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• pp.679-698
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• 2002

In this paper some techniques for the dynamic analysis of non-classically damped linear systems are reviewed and compared. All these methods are based on a transformation of the governing equations using a basis of complex or real vectors. Complex and real vector bases are presented and compared. The complex vector basis is represented by the eigenvectors of the complex eigenproblem obtained considering the non-classical damping matrix of the system. The real vector basis is a set of Ritz vectors derived either as the undamped normal modes of vibration of the system, or by the load dependent vector algorithm (Lanczos vectors). In this latter case the vector basis includes the static correction concept. The rate of convergence of these bases, with reference to a parametric structural system subjected to a fixed spatial distribution of forces, is evaluated. To this aim two error norms are considered, the first based on the spatial distribution of the load and the second on the shear force at the base due to impulsive loading. It is shown that both error norms point out that the rate of convergence is strongly influenced by the spatial distribution of the applied forces.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2008.04a
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• pp.342-349
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• 2008

A selection method of primary degrees of freedom in dynamic condensation for nonproportional damping structures is proposed. Recently, many dynamic condensation schemes for complex eigenanalysis have been applied to reduce the number of degrees of freedom. Among them, iterative scheme is widely used because accurate eigenproperties can be obtained by updating the transformation matrix in every iteration. However, a number of iteration to enhance the accuracy of the eigensolutions may have a possibility to make the computation cost expensive. This burden can be alleviated by applying properly selected primary degrees of freedom. In this study, which method for selection of primary degrees of freedom is best fit for the iterative dynamic condensation scheme is presented through the results of a numerical experiment. The results of eigenanalysis of the proposed method is also compared to those of other selection schemes to discuss a computational effectiveness.