• Transactions of the Korean Society of Mechanical Engineers A
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• v.40 no.4
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• pp.363-372
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• 2016

When a system consisting of rigid and flexible bodies is optimized to improve its dynamic characteristics, its eigenfrequencies are typically maximized. While topology optimization formulations dealing with simultaneous design of a system of rigid and flexible bodies are available, studies on eigenvalue maximization of the system are rare. In particular, no work has solved for the case when the target frequency becomes one of the repeated eigenfrequencies. The problem involving repeated eigenfrequencies is solved in this study, and a topology optimization formulation and sensitivity analysis are presented. Further, several numerical case studies are considered to demonstrate the validity of the proposed formulation.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.12 no.3
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• pp.371-383
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• 1999

An efficient and numerically stable eigensolution method for structures with multiple natural frequencies is presented. The proposed method is developed by improving the well-known subspace iteration method with shift. A major difficulty of the subspace iteration method with shift is that because of singularity problem, a shift close to an eigenvalue can not be used, resulting in slower convergence. In this paper, the above singularity problem has been solved by introducing side conditions without sacrifice of convergence. The proposed method is always nonsingular even if a shift is on a distinct eigenvalue or multiple ones. This is one of the significant characteristics of the proposed method. The nonsingularity is proved analytically. The convergence of the proposed method is at least equal to that of the subspace iteration method with shift, and the operation counts of above two methods are almost the same when a large number of eigenpairs are required. To show the effectiveness of the proposed method, two numerical examples are considered.

• Journal of the Society of Naval Architects of Korea
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• v.33 no.2
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• pp.85-95
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• 1996

An iterative Kantorovich method is presented for the vibration analysis of rectangular isotopic and orthotropic thick plates. Mindlin plate characteristic functions are derived in general forms by the Kantorovich method initially starting with Timoshenko beam functions consistent with the boundary conditions of the plate. Through numerical calculations of natural pairs, i.e. natural frequencies and corresponding modes, and dynamic responses of appropriate models, it has been confirmed that the presented method is superior to the Rayleigh-Ritz analysis or the FEM analysis in accuracy and computational efficiency.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2004.11a
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• pp.721-726
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• 2004

A simplified method for the computation of first second and higher order derivatives of eigenvalues and eigenvectors derivatives associated with repeated eigenvalues is presented. Adjacent eigenvectors and orthonormal conditions are used to compose an algebraic equation whose order is (n+m)x(n+m), where n is the number of coordinates and m is the number of multiplicity of the repeated eigenvalues. The algebraic equation developed can be used to compute derivatives of both eigenvalues and eigenvectors simultaneously. Since the coefficient matrix in the proposed algebraic equation is non-singular, symmetric and based on N-space it is numerically stable and very efficient compared to previous methods. This method can be consistently applied to structural systems with structural design parameters and mechanical systems with lumped design parameters. To verify the effectiveness of the proposed method, the finite element model of the cantilever beam is considered.

• Proceedings of the Earthquake Engineering Society of Korea Conference
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• 2001.09a
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• pp.117-124
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• 2001

A simplified method fur the eigenpair sensitivities of damped system with multiple eigenvalues is presented. This approach employs a reduced equation to determine the sensitivities of eigenpairs of the damped vibratory systems with multiple natural frequencies. In the proposed method, adjacent eigenvectors and orthonormal conditions are used to compute an algebraic equation whose order is (n+m)x(n+m), where n is the number of coordinates and m the number of multiplicity of multiple natural frequencies. The proposed method is an improved Lee and Jung's method which was developed previously. Two equations are used to find eigenvalue derivatives and eigenvector derivatives in Lee and Jung's method. A significant advantage of this approach over Lee and Jung's method is that one algebraic equation newly developed is enough to compute such eigenvalue derivatives and eigenvector derivatives. This method can be consistently applied to both structural systems with structural design parameters and mechanical systems with lumped design parameters. To demonstrate the theory of the proposed method and its possibilities in the case of multiple eigenvalues, the finite element model of the cantilever beam and 5-DOF mechanical system in the case of a non-proportionally damped system are considered as numerical examples. The design parameter of the cantilever beam is its height. and that of the 5-DOF mechanical system is a spring.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2001.10a
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• pp.515-522
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• 2001

A simplified method is presented for the computation of eigenvalue and eigenvector derivatives associated with repeated eigenvalues. In the proposed method, adjacent eigenvectors and orthonormal conditions are used to compose an algebraic equation whose order is (n+m)x(n+m), where n is the number of coordinates and m is the number of multiplicity of the repeated eigenvalue. One algebraic equation developed can be computed eigenvalue and eigenvector derivatives simultaneously. Since the coefficient matrix of the proposed equation is symmetric and based on N-space, this method is very efficient compared to previous methods. Moreover the numerical stability of the method is guaranteed because the coefficient matrix of the proposed equation is non-singular, This method can be consistently applied to both structural systems with structural design parameters and mechanical systems with lumped design parameters. To verify the effectiveness of the proposed method, the finite element model of the cantilever beam and a 5-DOF mechanical system in the case of a non-proportionally damped system are considered as numerical examples. The design parameter of the cantilever beam is its width, and that of the 5-DOF mechanical system is a spring.