• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.17 no.10
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• pp.946-949
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• 2007

This paper discusses the lateral vibration of a beam with boundary condition of one end fixed and the other end free. The uniform beam has a solution by summation of some simple exponential functions. But if its shape is not uniform, its solution could be by Bessel's function or mathematical solution could not exist. Even if the solution of Bessel's function exists, as Bessel function is a series function, we must get the solution by numerical method. Author had proposed the solution of the matrix method by Ritz's method and a new mode shape function, and had earned the good results for a wedge beam. Hereby a vibration analysis for the tapered beam with circle cross section was executed, and so good results were showed.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.19 no.8
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• pp.71-77
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• 2018

In order to estimate behavior of soil mass which is located straight up of reinforced concrete culvert, Ritz method and FEM were applied and arching effects between the soil mass and adjacent soil were considered for the analyses. Analysis results obtained from the Ritz method and finite element method were compared with analytical solution. In the case of estimating nodal forces considered in FEM, caution is needed that shear stress depending on depth from ground surface should be reflected regardless of local coordinate system. Comparing the displacements computed from Ritz method with those of the analytic solution, it is seen that as the power of assumed displacement function increases, differences between the computed displacements and those of analytic solution decreases. It seems that displacements of FEM becomes closer to those of analytical solution as the number of elements are increased. It is seen that stresses computed from the Ritz method don't get closer to those of the analytic solution as the power of assumed displacement function. Stresses from FEM become closer to those of analytic solution as the number of elements are increased. Comparing the analysis results from the Ritz method and FEM with those of analytic solution, it can be seen that FEM is more reliable than Ritz method.

• Journal of applied mathematics & informatics
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• v.9 no.1
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• pp.321-330
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• 2002

We apply an Intermediate Problem Method to compute eigenvalues of an anharmonic oscillator. The method produces lower bounds to the eigenvalues while the Rayleigh-Ritz method yields upper bounds. We show the convergence rate of the Intermediate Problem Method is the same as the rate of the Rayleigh-Ritz method.

• Journal of KSNVE
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• v.3 no.1
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• pp.77-82
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• 1993

In general, the dynamic analysis with FEM(Finite Element Method) of large structures requires large computer memory space and long computational time. For the purpose of economical dynamic analysis of large structures, most of engineers want to use an efficient solution algorithm. This paper reports the modified CMS(Component Mode Synthesis) method which uses more efficient algorithm than the classical CMS method. In this paper, it is shown that Ritz vector sets can play the role of normal mode vector sets of substurctures in the CMS algorithm. The modified CMS method has good convergence performance compared with that of the classical CMS method.

• Proceedings of the Earthquake Engineering Society of Korea Conference
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• 1997.10a
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• pp.151-158
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• 1997

The main concern of numerical dynamic analysis of large structures is to find an acceptable solution with fewer mode shapes and less computational efforts. component mode method utilizes substructure technique to reduce the degrss of freedom but have a disadvantage to not consider the dynamic characteristics of loads. Ritz Vector method consider the load characteristics but requires many integrations and errors are accumulated. In this study, to prove the effectiveness of component mode method, Lanczos algorithm are introduced. To prove the effectiveness of this method, example structures areanalyzed and the results are compared with SAP90.

• Structural Engineering and Mechanics
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• v.81 no.4
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• pp.411-428
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• 2022

This paper presents the Campbell diagram analysis of the rotordynamic system using the full order model (FOM) and the reduced order model (ROM) techniques to determine the critical speeds, identify the stability and reduce the computational time. Due to the spin-speed-dependent matrices (e.g., centrifugal stiffening matrix), several model order reduction (MOR) techniques may be considered, such as the modal superposition (MS) method and the Krylov subspace-based MOR techniques (e.g., Ritz vector (RV), quasi-static Ritz vector (QSRV), multifrequency quasi-static Ritz vector (MQSRV), multifrequency/ multi-spin-speed quasi-static Ritz vector (MMQSRV) and the combined Ritz vector & modal superposition (RV+MS) methods). The proposed MMQSRV method in this study is extended from the MQSRV method by incorporating the rotational-speed-dependent stiffness matrices into the Krylov subspace during the MOR process. Thus, the objective of this note is to respond to the question of whether to use the MS method or the Krylov subspace-based MOR technique in establishing the Campbell diagram of the shaft-disc-blade assembly systems in three-dimensional (3D) finite element analysis (FEA). The Campbell diagrams produced by the FOM and various MOR methods are presented and discussed thoroughly by computing the norm of relative errors (ER). It is found that the RV and the MS methods are dominant at low and high rotating speeds, respectively. More precisely, as the spinning velocity becomes large, the calculated ER produced by the RV method is significantly increased; in contrast, the ER produced by the MS method is smaller and more consistent. From a computational point of view, the MORs have substantially reduced the time computing considerably compared to the FOM. Additionally, the verification of the 3D FE rotordynamic model is also provided and found to be in close agreement with the existing solutions.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2003.11a
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• pp.984-987
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• 2003

This paper discusses the lateral vibration of a bar which has its tip free. The uniform bar has a solution by summation of some simple exponential functions. But if its shape is not uniform, its solution could be by Bessel's function, or mathematical solution could not be existed. Even if the solution of Bessel's function exists. as Bessel function is a series function, we must get the solution by numerical method, Hereof the author proposes the solution of the matrix method by Ritz's method, and proposes a new deflection shape

• Coupled systems mechanics
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• v.9 no.6
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• pp.563-573
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• 2020

Dynamic responses of a laminated composite cantilever beam under a harmonic are investigated in this study. The governing equations of problem are derived by using the Lagrange procedure. The Timoshenko beam theory is considered and the Ritz method is implemented in the solution of the problem. The algebraic polynomials are used with the trivial functions for the Ritz method. In the solution of dynamic problem, the Newmark average acceleration method is used in the time history. In the numerical examples, the effects of load parameter, the fiber orientation angles and stacking sequence of laminas on the dynamic responses of the laminated beam are investigated.

• Structural Engineering and Mechanics
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• v.14 no.4
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• pp.401-416
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• 2002

This paper presents a domain decomposition method for buckling analysis of rectangular Kirchhoff plates subjected to uniaxial inplane load and with mixed edge support conditions. A plate is decomposed into two rectangular subdomains along the change of the discontinuous support conditions. The automated Ritz method is employed to derive the governing eigenvalue equation for the plate system. Compatibility conditions are imposed for transverse displacement and slope along the interface of the two subdomains by modifying the Ritz trial functions. The resulting Ritz function ensures that the transverse displacement and slope are continuous along the entire interface of the two subdomains. The validity and accuracy of the proposed method are verified with convergence and comparison studies. Buckling results are presented for several selected rectangular plates with various combination of mixed edge support conditions.

• Structural Engineering and Mechanics
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• v.2 no.3
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• pp.285-294
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• 1994

Based on a fast and accurate method for the stationary random seismic response analysis for discretized structures(Lin 1992, Lin et al. 1992), a Ritz method for dealing with such responses of continuous systems in developed. This method is studied quantitatively, using cantilever shear beams for simplicity and clarity. The process can be naturally extended to deal with various boundary conditions as well as non-uniform Bernoulli-Euler beams, or even Timoshenko beams. Algorithms for both proportionally and non-proportionally damped responses are described. For all of such damping cases, it is not necessary to solve for the natural vibrations of the beams. The solution procedure is very simple, and equally efficient for a white or a non-white ground excitation spectrum. Two examples are given where various power spectral density functions, variances, covariances and second spectral moments of displacement, internal force response, and their derivatives are calculated and analyses. Some Ritz solutions are compared with "exact" CQC solutions.