• 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.

• 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.

• 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.

• 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.

• 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.

• 한국산학기술학회논문지
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• 제19권8호
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• pp.71-77
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• 2018

철근 콘크리트 암거 상부지반에 있어 아칭효과에 의한 지반의 상대적 변형 때문에 발생하는 침하거동을 평가하기 위해 Ritz 방법과 유한요소법을 적용하고 해석결과를 해석 해에 의한 결과와 상호 비교해 보았다. 유한요소법을 적용하여 절점외력을 구할 경우 요소별 국부좌표계에 관계없이 지표면으로부터의 깊이의 함수로 표현되는 전단응력이 반영되도록 주의할 필요가 있다. 암거 상부지반의 지표면에서의 변위는 해석방법에 상관없이 동일한 값을 보였다. Ritz 방법을 통해 구한 변위를 해석 해와 비교해 볼 때 가정한 변위의 차수를 증가시킬수록 해석 해에 근접한 결과를 보임을 알 수 있었다. 유한요소법에 의한 변위 또한 요소를 세분화 할수록 해석 해에 근접한 결과를 보임을 알 수 있었다. Ritz 방법에 의한 해석결과에 따르면 계산된 응력값이 해석 해에 근접한 결과를 보이지는 않았다. 유한요소법에 의한 응력의 경우 요소를 세분화 할수록 해석 해에 근접한 결과를 보였다. 본 연구에서 고려한 Ritz 방법과 유한요소법에 의한 해석결과를 해석 해와 비교한 결과 유한요소법에 의한 해석결과가 해석 해에 안정적으로 근접하는 결과를 보임을 알 수 있었다.

• 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'.

• 한국소음진동공학회논문집
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• 제17권10호
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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.

• 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.