• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.19 no.6
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• pp.607-613
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• 2009

A new formulation of NDIF method to the algebraic eigenvalue problem is introduced to efficiently extract natural frequencies of arbitrarily shaped plates with the simply supported boundary condition. NDIF method, which was developed by the authors for the free vibration analysis of arbitrarily shaped membranes and plates, has the feature that it yields highly accurate natural frequencies compared with other analytical methods or numerical methods(FEM and BEM). However, NDIF method has the weak point that it needs the inefficient procedure of searching natural frequencies by plotting the values of the determinant of a system matrix in the frequency range of interest. A new formulation of NDIF method developed in the paper doesn't require the above inefficient procedure and natural frequencies can be efficiently obtained by solving the typical algebraic eigenvalue problem. Finally, the validity of the proposed method is shown in several case studies, which indicate that natural frequencies by the proposed method are very accurate compared to other exact, analytical, or numerical methods.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.26 no.5
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• pp.602-608
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• 2016

The paper proposes a practical meshless method for the free vibration analysis of clamped plates having arbitrary shapes by extending the non-dimensional dynamic influence function (NDIF) method, which was developed by the author in 1999. In the proposed method, the domain and boundary of the plate of interest are discretized using only nodes without elements unlike FEM and the system matrices are obtained by making domain nodes and boundary nodes satisfy the governing differential equation and boundary conditions, respectively. However, since the above system matrices are not square ones, the problem of free vibrations of clamped plates is not reduced to an algebraic eigenvalue problem. An additional theoretical treatment is considered to produce an algebraic eigenvalue problem. It is revealed from case studies that the proposed method is valid and accurate.

• Coupled systems mechanics
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• v.1 no.4
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• pp.345-360
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• 2012

In this article we have presented a unique representation for interval arithmetic. The traditional interval arithmetic is transformed into crisp by symbolic parameterization. Then the proposed interval arithmetic is extended for fuzzy numbers and this fuzzy arithmetic is used as a tool for uncertain finite element method. In general, the fuzzy finite element converts the governing differential equations into fuzzy algebraic equations. Fuzzy algebraic equations either give a fuzzy eigenvalue problem or a fuzzy system of linear equations. The proposed methods have been used to solve a test problem namely heat conduction problem along with fuzzy finite element method to see the efficacy and powerfulness of the methodology. As such a coupled set of fuzzy linear equations are obtained. These coupled fuzzy linear equations have been solved by two techniques such as by fuzzy iteration method and fuzzy eigenvalue method. Obtained results are compared and it has seen that the proposed methods are reliable and may be applicable to other heat conduction problems too.

• The Journal of Natural Sciences
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• v.8 no.2
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• pp.9-11
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• 1996

We apply a full mutigrid scheme to computing eigenvalues of the Laplace eigenvalue problem with Dirichlet boundary condition. We use finite difference method to get an algebraic equation and apply inverse power method to estimating the smallest eigenvalue. Our result shows that combined method of inverse power method and full multigrid scheme is very effective in calculating eigenvalue of the eigenvalue problem.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.21 no.5
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• pp.707-718
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• 1997

This paper presents an efficient numerical method for the computation of eigenpair derivatives for a real symmetric eigenvalue problem with distinct and multiple eigenvalues. The method has a very simple algorithm and gives an exact solution. Furthermore, it saves computer sotrage and CPU time. The algorithm preserves not only the symmetricity but also the band width of the matrices, allowing efficient computer storage and solution techniques. Results from the proposed method for calculating the eigenpair derivatives are compared with those from Rudisill and Chu's method and Nelson's method which is known efficient one in the case of distinct natural frequencies. As an example to demonstrate the efficiency of the proposed method in the case of distinct eigenvalues, a cantilever plate is considered. The design parameter of the cantilever plate is its thickness. For the eigenvalue problem with multiple natural frequencies, the adjacent eigenvectors are used in the algebraic equation as side conditions, lying adjacent to the multiplicity of multiple natural frequency distinct eigenvalues, which appear when design parameter varies. A cantilever beam is used to demonstrate the efficiency of the proposed method in the case of multiple natural frequencies. Results form the proposed method for calculating the eigenpair derivatives are compared with those from Dailey's method(an amendation of Ojalvo's work) which finds the exact eigenvector derivatives. The design parameter of the cantilever beam is its height. Data is presented showing the amount of CPU time used to compute the first ten eigenpair derivatives by each method. It is important to note that the numerical stability of the proposed method is proved.

• Steel and Composite Structures
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• v.28 no.6
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• pp.655-670
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• 2018

A coupled method, that combines the Ritz method and the finite element (FE) method, is proposed to solve the vibration problem of rectangular thin and thick plates with general boundary conditions. The eigenvalue partial differential equation(s) of the plate is (are) first reduced to a set of eigenvalue ordinary differential equations by the application of the Ritz method. The resulting eigenvalue differential equations are then reduced to an eigenvalue algebraic equation system using the finite element method. The natural boundary conditions of the plate problem including the free edge and free corner boundary conditions are also implemented in a simple and accurate manner. Various boundary conditions including simply supported, clamped and free boundary conditions are considered. Comparisons with existing numerical and analytical solutions show that the proposed mixed method can produce highly accurate results for the problems considered using a small number of Ritz terms and finite elements. The proposed mixed Ritz-FE formulation is also compared with the mixed FE-Ritz formulation which has been recently proposed by the present author and his co-author. It is found that the proposed mixed Ritz-FE formulation is more efficient than the mixed FE-Ritz formulation for free vibration analysis of rectangular plates with Levy-type boundary conditions.

• Journal of applied mathematics & informatics
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• v.4 no.1
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• pp.243-248
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• 1997

A full multigrid scheme was used in computing some eigenvalues of the Laplace eigenvalues problem with the Dirichlet bound-ary condition. We get a system of algebraic equations with an aid of finite difference method and apply subspace iteration method to the system to compute first some eigenvalues. The result shows that this is very effective in calculating some eigenvalues of this problem.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.23 no.11
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• pp.1003-1011
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• 2013

A new formulation of the MNDIF method is introduced to extract highly accurate eigenvalues of concave acoustic cavities. Since the MNDIF method, which was introduced by the author, can be applicable for only convex acoustic cavities, a new approach of dividing a concave cavity into two convex domains and formulating an algebraic eigenvalue problem is proposed in the paper. A system matrix equation, which gives eigenvalues, is obtained from boundary conditions for each domain and the condition of continuity in the interface between the two domains. The validity and accuracy of the proposed method are shown through example studies.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.25 no.10
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• pp.653-659
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• 2015

A new formulation of the non-dimensional dynamic influence function method, which is a type of the meshless method, is introduced to extract highly accurate eigenvalues of clamped plates with arbitrary shape. Originally, the final system matrix equation of the method, which was introduced by the author in 1999, does not have a form of algebraic eigenvalue problem unlike FEM. As the result, the non-dimensional dynamic influence function method requires an inefficient process to extract eigenvalues. To overcome this weak point, a new approach for clamped plates is proposed in the paper and the validity and accuracy is shown in verification examples.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2011.10a
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• pp.124-129
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• 2011

An improved NDIF method is introduced to efficiently extract eigenvalues of two-dimensional, arbitrarily shaped acoustic cavities. The NDIF method, which was developed by the authors for the eigen-mode analysis of arbitrarily shaped acoustic cavities, membranes, and plates, has the feature that it yields highly accurate eigenvalues compared with other analytical methods or numerical methods (FEM and BEM). However, the NDIF method has the weak point that the system matrix of the NDIF method depends on the frequency parameter and, as a result, a final system equation doesn't take the form of an algebra eigenvalue problem. The system matrix of the improved NDIF method developed in the paper is independent of the frequency parameter and eigenvalues can be efficiently obtained by solving a typical algebraic eigenvalue problem. Finally, the validity and accuracy of the proposed method is verified in two case studies, which indicate that eigenvalues and mode shapes obtained by the proposed method are very accurate compared to the exact method, the NDIF method or FEM(ANSYS).