• 전산구조공학
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• 제8권4호
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• pp.181-187
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• 1995

동적해석에 대한 최근의 연구는 구조물의 자유도보다 적은 모우드 형상들을 사용하여 구조물을 해석하는 효과적인 방법을 찾는데 있다. Ritz알고리즘과 모우드가속도법은 모우드중첩법을 개선하고자 개발되었는데, Ritz알고리즘은 하중의 공간적 특성을 포함하지만, 계산과정에서 유용한 직교성을 잃는 경향이 있으며, 모우드가속도법은 만족할 만한 해를 얻기 위해 많은 수의 모우드 형상들을 고려해야 하는 단점이 있다. 또한 앞의 두 방법을 조합한 방법이 개발되었으나 너무 많은 계산과정과 시간을 필요로 한다. 이 연구의 목적은 Lanczos알고리즘을 이용하여 Ritz알고리즘의 효율성과 정확성을 보완하고 이를 프로그램화하여 검증하는데 있다. 본 연구의 결과로부터 Modified Ritz알고리즘을 이용한 동적해석방법이 합리적임이 증명되었다.

• Steel and Composite Structures
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• 제28권6호
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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 Advanced Marine Engineering and Technology
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• 제29권8호
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• pp.877-882
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• 2005

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. Enen if the solution of Bessel's function exists. as Bessel function is a series function. we must got the solution by numerical method Hereby the author Proposes the ununiform beam solution of the matrix method by Ritz's method. and Proposes a new deflection shape function.

• 전산구조공학
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• 제6권1호
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• pp.99-105
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• 1993

A Stepped beam with immovable ends under forced vibrations with large amplitude is investigated by using the finite element method and the Ritz vectors. Unlike the Eigen vectors, the Ritz vectors are generated by a simple recurrence relation. Moreover the Ritz vectors yield much faster convergence with respect to the number of vectors used than the use of Eigen vectors. The computer program is developed for nonlinear analysis using Ritz vectors instead of Eigen vectors and numerical examples are analysed for deflections and natural frequencies of stepped beam under various support conditions. Results show that the proposed method is valid and efficient.

• Structural Engineering and Mechanics
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• 제39권5호
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• pp.703-716
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• 2011

The accurate dynamic analysis of structures is usually performed by a fine finite element discretization with very large number of degrees of freedom. Apart from modal analysis, one can reduce the number of final equations by assuming the deformed shape of the structure as a linear combination of independent Ritz vectors. The efficiency of this method relies heavily on the vectors selected. In this paper, a new set of Ritz vectors is proposed. It is primarily proved that these vectors are linearly independent. Subsequently, various two and three-dimensional examples are analyzed based on the proposed method. In each case, the results are compared with the ones obtained based on usual Ritz and modal analysis methods. It is finally concluded that the proposed method is very effective and efficient method for dynamic analysis of structures in frequency domain.

• 한국자동차공학회논문집
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• 제17권4호
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• pp.108-117
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• 2009

Leissa claimed in his article that the Rayleigh method is not the same as the Ritz method for determining natural frequencies and its corresponding mode shapes and contended that Rayleigh's name should not be attached to the method. The present article examines the methods in viewpoint of admissible functions and its minimization process, and of the historical developments. It concludes that Leissa's assertion is relevant, although Rayleigh did apply a conceptual theory systematized from the Lagrange method, and given 38 years earlier than Ritz's 'masterly exposition of theory'.

• 한국지진공학회논문집
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• 제23권5호
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• pp.279-288
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• 2019

Safety-related cabinets and their electrical parts, such as relays and switches in nuclear power plants, should maintain continuous functioning, as well as structural safety according to the nuclear regulatory guidelines. Generally, an electrical part is qualified if its functioning is maintained without abnormality during excitement by motion compatible with the test response spectrum, which is larger than its in-cabinet response spectrum (ICRS). ICRS can be determined by shake-table test or dynamic analysis. Since existing cabinets in use can hardly be stopped and moved, dynamic analysis is preferred over shake-table test in determining ICRS. The simple method, suggested by the Electric Power Research Institute (EPRI) to determine ICRS, yields conservative or non-conservative results from time to time. In order to determine that the ICRS is better than EPRI method in a simple way, Ritz method considering global and local plate behaviors was suggested by Gupta et al. In this paper, the Ritz method is modified in order to consider the rocking and frame behaviors simultaneously, and it is applied to a simple numerical example for verification. ICRS is determined by Ritz method and compared with the results by finite element method (FEM). Based on this numerical example, recommendations for using Ritz method are suggested.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 1989년도 봄 학술발표회 논문집
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• pp.33-37
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• 1989

According to the Rayleigh-Ritz approximation method, a solution can be represented as a finite series consisting of space-dependent functions, which satisfy all the geometric boundary conditions of the problem and appropriate smoothness requirement in the interior of the domain. In this paper, an efficient formulation for solving structural dynamics systems in frequency domain is presented. A general procedure called Ritz modes (or vectors) generation algorithm is used to generate the admissible functions, i.e. Ritz modes, Then, the use of direct superposition of the Ritz modes is utilized to reduce the size of the problem in spatial dimension via geometric coordinates projection. For the reduced system, the frequency domain approach is applied. Finally, a numerical example is presented to illustrate the effectiveness of the proposed method.

• Coupled systems mechanics
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• 제7권6호
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• pp.755-774
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• 2018

The Ritz method is known as very successful strategy for discretizing continuous problems, but it has never been used for solving systems of algebraic equations. The Iterated Ritz Method (IRM) is a novel iterative solver based on the discretized Ritz procedure applied at each iteration step. With an appropriate choice of coordinate vectors, the method may be efficient in linear, nonlinear and optimization problems. Additionally, some iterative methods can be explained as special cases of this approach, which helps to understand advantages and limitations of these methods and gives motivation for their improvement in sense of IRM. In this paper, some ideas for generation of efficient coordinate vectors are presented. The algorithm was developed and tested independently and then implemented into the open source program FEAP. Method has been successfully applied to displacement based (even ill-conditioned) models of structural engineering practice. With this original approach, a new iterative solution strategy has been opened.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 2011년도 정기 학술대회
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• pp.685-688
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• 2011

본 논문에서는 다양한 음향 가진에 따른 음향 응답을 유한 요소법을 통하여 효과적으로 계산하기 위한 새로운 모델 축소법을 제안한다. 일반적인 유한 요소법을 통한 기계구조물의 응답을 구하기 위해서는 음향 방정식의 강성 및 행렬을 구한 뒤 이들의 조합을 통한 동적 강성행렬을 구한 뒤 역행렬을 구하여 다양한 주파수 응답을 구하게 된다. 현재 컴퓨터 하드웨어의 발전과 소프트 웨어의 발전에 의하여 더 많은 유한 요소를 사용할 수 있게 되었고 이로 인하여 더욱 정확하고 넓은 대역의 음향 응답을 구할 수 있게 되었다. 그러나, 아직까지도 아주 복잡한 구조물의 음향 응답을 구하기 위하여 유한 요소를 무한정으로 증가할 수 없는 경우가 많다. 이를 해결하기 위하여 일반적으로 모델 축소법(Model order reduction) 기법을 사용한다. 이 모델 축소법은 기본적으로 전체 행렬을 아주 작지만 효율적인 작은 행렬로 바꾸어 응답을 예측하는 기법으로 mode superposition method, ritz vector method, quasi-static ritz vector method등이 있다. 기존의 모델 축소법은 기본적으로 질량 및 강성행렬이 가진 주파수에 영향을 받지 않는 행렬이라 가정한다. 그렇기 때문에 경계조건이나 다공성 재료를 모델링할 경우 가진 주파수에 영향을 받는 강성행렬과 질량행렬이 만들어지게 되어 기존의 모델 축소법은 효과적이지 못하게 된다. 이런 문제점을 해결하기 위하여 이 논문에서는 Quasi-static ritz vector method의 기본적인 개념을 확장하여 여러 개의 중심 주파수(Center frequency)에서 기저를 계산하고 이를 동시에 이용하는 Multi-frequency quasi-static ritz vector method를 제안한다.