• Proceedings of the Computational Structural Engineering Institute Conference
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• 2011.04a
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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.

• 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.

• Computational Structural Engineering
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• v.25 no.4
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• pp.28-39
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• 2012

• Computational Structural Engineering
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• v.8 no.4
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• pp.181-187
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• 1995

Recent researches in dynamics are focused on finding effective methods to analyze the dynamic behavior of structures by fewer mode shapes their number of dgrees of freedom. Ritz algorithm and mode acceleration method were developed to improved the mode superposition. Ritz algorithm can include distribution of external loads but be apt to lose the orthogonality condition, which is useful properties in the analysis. Also mode acceleration method should consider a large number of mode shapes to get a satisfactory results. Another method, combining previous two method, was developed but too much computational efforts and times were required. The purpose of this study is to develop and evaluate the Ritz algorithm modified with the lanczos algorithm to improve the efficiency and accuracy. As a result of !this study, dynamic analysis using modified Ritz algorithm was proved to be the rational analysis 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.

• Computational Structural Engineering
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• v.9 no.2
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• pp.115-120
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• 1996

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 degree 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 improve the effectiveness of component mode method, Lanczos algorithm is introduced. To prove the effectiveness of this method, example structure are analyzed and the results are compared with SAP90.

• 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.14 no.6
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• pp.679-698
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• 2002

In this paper some techniques for the dynamic analysis of non-classically damped linear systems are reviewed and compared. All these methods are based on a transformation of the governing equations using a basis of complex or real vectors. Complex and real vector bases are presented and compared. The complex vector basis is represented by the eigenvectors of the complex eigenproblem obtained considering the non-classical damping matrix of the system. The real vector basis is a set of Ritz vectors derived either as the undamped normal modes of vibration of the system, or by the load dependent vector algorithm (Lanczos vectors). In this latter case the vector basis includes the static correction concept. The rate of convergence of these bases, with reference to a parametric structural system subjected to a fixed spatial distribution of forces, is evaluated. To this aim two error norms are considered, the first based on the spatial distribution of the load and the second on the shear force at the base due to impulsive loading. It is shown that both error norms point out that the rate of convergence is strongly influenced by the spatial distribution of the applied forces.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2009.04a
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• pp.139-145
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• 2009

Reduction scheme is remarkably useful in the case requiring the repeated calculation procedure. Recently, the efficiency of the reduction scheme has been improved by combining scheme of sub-domain method. But, when the global domain is partitioned into a few sub-domains, sub-domains without constraints can be produced. it is needed to extract the ritz vector from each sub-domain to construct the reduced system of each sub-domain. it is easy to extract the ritz vector from sub-domain with constraint. on the other hand, pseudo inverse method should be employed to extract the ritz vector from sub-domain without constraint. generally, the pseudo inverse takes a large number of computing time to obtain a reduced system of a sub-domain without boundary condition. This trouble can be overcome by the reduced pseudo inverse scheme which proposed in this study. This scheme is based on the static condensation that is not related with selection of the primary degrees of freedom. Numerical examples demonstrate that present method saves computational cost effectively and predicts the accurate eigenvalues.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.22 no.2
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• pp.173-179
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• 2009

Reduction scheme is remarkably useful in the case requiring the repeated calculation procedure. Recently, the efficiency of the reduction scheme has been improved by combining scheme of sub-domain method. But, when the global domain is partitioned into a few sub-domains, sub-domains without constraints can be produced. it is needed to extract the ritz vector from each sub-domain to construct the reduced system of each sub-domain. it is easy to extract the ritz vector from sub-domain with constraint. on the other hand, pseudo inverse method should be employed to extract the ritz vector from sub-domain without constraint. generally, the pseudo inverse takes a large number of computing time to obtain a reduced system of a sub-domain without boundary condition. This trouble can be overcome by the reduced pseudo inverse scheme which proposed in this study. This scheme is based on the static condensation that is not related with selection of the primary degrees of freedom. Numerical examples demonstrate that present method saves computational cost effectively. In addition, it is shown that the reduced system based on the proposed scheme predicts the accurate eigenvalues of global system.