• Structural Engineering and Mechanics
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• 제53권3호
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• pp.537-573
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• 2015

Multiple-step beams carrying intermediate lumped masses with/without rotary inertias are widely used in engineering applications, but in the literature for free vibration analysis of such structural systems; Bernoulli-Euler Beam Theory (BEBT) without axial force effect is used. The literature regarding the free vibration analysis of Bernoulli-Euler single-span beams carrying a number of spring-mass systems, Bernoulli-Euler multiple-step and multi-span beams carrying multiple spring-mass systems and multiple point masses are plenty, but that of Timoshenko multiple-step beams carrying intermediate lumped masses and/or rotary inertias with axial force effect is fewer. The purpose of this paper is to utilize Numerical Assembly Technique (NAT) and Differential Transform Method (DTM) to determine the exact natural frequencies and mode shapes of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and/or rotary inertias. The model allows analyzing the influence of the shear and axial force effects, intermediate lumped masses and rotary inertias on the free vibration analysis of the multiple-step beams by using Timoshenko Beam Theory (TBT). At first, the coefficient matrices for the intermediate lumped mass with rotary inertia, the step change in cross-section, left-end support and right-end support of the multiple-step Timoshenko beam are derived from the analytical solution. After the derivation of the coefficient matrices, NAT is used to establish the overall coefficient matrix for the whole vibrating system. Finally, equating the overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the differential equations of the motion. The calculated natural frequencies of Timoshenko multiple-step beam carrying intermediate lumped masses and/or rotary inertias for the different values of axial force are given in tables. The first five mode shapes are presented in graphs. The effects of axial force, intermediate lumped masses and rotary inertias on the free vibration analysis of Timoshenko multiple-step beam are investigated.

• 한국지진공학회논문집
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• 제23권4호
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• pp.231-238
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• 2019

A conventional lumped-mass stick model is based on the tributary area method to determine the masses lumped at each node and used in earthquake engineering due to its simplicity in the modeling of structures. However the natural frequencies of the conventional model are normally not identical to those of the actual structure. To solve this problem, recently an updated lumped-mass stick model is developed to provide the natural frequencies identical to actual structure. The present study is to investigate the seismic response accuracy of the updated lumped-mass stick model, comparing with the response results of the shaking table test. For the test, a small size four-story steel frame structure is prepared and tested on shaking table applying five earthquake ground motions. From the comparison with shaking table test results, the updated model shows an average error of 3.65% in the peak displacement response and 9.68% in the peak acceleration response. On the other hand, the conventional model shows an average error of 5.15% and 27.41% for each response.

• 전산구조공학
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• 제10권3호
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• pp.217-223
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• 1997

본 논문은 집중 질량 구조물의 천이응답에 대한 시간영역 민감도 해석의 기본 개념을 설명한다. 외부 가진에 따른 구조물의 응답에 미치는 설계변수 변화의 영향을 구하기 위해 시간영역 민감도 함수를 구하는 방법을 제시하였다. 시간영역에서 구조물의 설계변수 민감도는 1차 표준 민감도 함수와 백분율 민감도 함수를 통해 확인하였다. 이러한 민감도 함수와 그 계산은 설계변수에 대한 시스템 상태변수의 편미분에 의한 것이다. 또한, 직접 미분법에 의한 해석적 방법의 편미분 결과와 수치적 방법에 의한 결과를 비교하였다.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2008년도 춘계학술대회논문집
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• pp.389-392
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• 2008

This paper predicts the changes of natural frequencies due to the changes of mass at different point mass stations by using iterative calculation Transfer Matrices Method for different boundary conditions of a single beam structure (fixed-free and fixed-fixed beam). Firstly, the first three natural frequencies of an original beam are obtained using Transfer Matrices Method to verify the accuracy of the obtained results. The results are then compared with the exact solutions before purposely changing the parameter of mass. Both beams are modeled as discrete continuous systems with six-lumped-mass system. A single beam is broken down into a point mass and a massless beam which represent a single station and expressed in matrix form. The assembled matrices are used to determine the value of natural frequencies using numerical interpolation method corresponding to their mode number by manipulating some elements in the assembled matrix.

• Structural Engineering and Mechanics
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• 제22권6호
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• pp.701-717
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• 2006

In the existing reports regarding free transverse vibrations of the Euler-Bernoulli beams, most of them studied a uniform beam carrying various concentrated elements (such as point masses, rotary inertias, linear springs, rotational springs, spring-mass systems, ${\ldots}$, etc.) or a stepped beam with one to three step changes in cross-sections but without any attachments. The purpose of this paper is to utilize the numerical assembly method (NAM) to determine the exact natural frequencies and mode shapes of the multiple-step Euler-Bernoulli beams carrying a number of lumped masses and rotary inertias. First, the coefficient matrices for an intermediate lumped mass (and rotary inertia), left-end support and right-end support of a multiple-step beam are derived. Next, the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method (FEM). Finally, the exact natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions, respectively. The effects of distribution of lumped masses and rotary inertias on the dynamic characteristics of the multiple-step beam are also studied.

• Structural Engineering and Mechanics
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• 제21권6호
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• pp.713-735
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• 2005

Due to the complexity of mathematical expressions, the literature concerning the free vibration analysis of plates carrying multiple three-degree-of-freedom (dof) spring-mass systems is rare. In this paper, the three degrees of freedom (dof's) for a spring-mass system refer to the translational motion of its lumped mass in the vertical ($\bar{z}$) direction and the two pitching motions of its lumped mass about the two horizontal ($\bar{x}$ and $\bar{y}$) axes. The basic concept of this paper is to replace each three-dof spring-mass system by a set of equivalent springs, so that the free vibration characteristics of a rectangular plate carrying any number of three-dof spring-mass systems can be obtained from those of the same plate supported by the same number of sets of equivalent springs. Since the three dof's of the lumped mass for each three-dof spring-mass system are eliminated to yield a set of equivalent springs, the total dof of the entire vibrating system is not affected by the total number of the spring-mass systems attached to the rectangular plate. However, this is not true in the conventional finite element method (FEM), where the total dof of the entire vibrating system increases three if one more three-dof spring-mass system is attached to the rectangular plate. Hence, the computer storage memory required by using the presented equivalent spring method (ESM) is less than that required by the conventional FEM, and the more the total number of the three-dof spring-mass systems attached to the plate, the more the advantage of the ESM. In addition, since manufacturing a spring with the specified stiffness is much easier than making a three-dof spring-mass system with the specified spring constants and mass magnitude, the presented theory of replacing a three-dof spring-mass system by a set of equivalent springs will be also significant from this viewpoint.

• 한국해양공학회지
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• 제33권5호
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• pp.400-410
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• 2019

In this study, the numerical code for the 3D nonlinear dynamic analysis of an SLWR (Steel Lazy Wave Riser) was developed using the lumped mass line model in a FORTRAN environment. Because the lumped mass line model is an explicit method, there is no matrix operation. Thus, the numerical algorithm is simple and fast. In the lumped mass line model, the equations of motion for the riser were derived by applying the various forces acting on each node of the line. The applied forces at the node of the riser consisted of the tension, shear force due to the bending moment, gravitational force, buoyancy force, riser/ground contact force, and hydrodynamic force based on the Morison equation. Time integration was carried out using a Runge-Kutta fourth-order method, which is known to be stable and accurate. To validate the accuracy of the developed numerical code, simulations using the commercial software OrcaFlex were carried out simultaneously and compared with the results of the developed numerical code. To understand the nonlinear dynamic characteristics of an SLWR, dynamic simulations of SLWRs excited at the hang-off point and of SLWRs in regular waves were carried out. From the results of these dynamic simulations, the displacements at the maximum bending moments at important points of the design, like the hang-off point, sagging point, hogging points, and touch-down point, were observed and analyzed.

• 소음진동
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• 제9권1호
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• pp.206-213
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• 1999

선박의 보-유추 진동해석에 있어 2차원 부가수질량의 3차원 효과를 고려하기 위해서 3차원 수정계수(J-factor)를 계산해야 하는데 광폭선의 경우에는 J-factor의 계산이 부정확하고 번거롭다. 이 논문에서는 이를 개선하기 위해 새로운 선박의 보-유추 접수진동해석 방법을 소개하였다. 이 방법은 선박에 접수된 유체에 대해 BEM 기법을 이용하여 3차원 유체력을 직접 계산하고 이를 일정 간격으로 나눈 각 스트립에 집중질량으로 평가한 후에 선체의 보모델과 결합하여 보-유추 진동해석을 수행하는 방법이다. 오픈탑 컨테이너선의 모델에 대해 기존의 보-유추 진동해석방법과 이 논문에서 제시한 새로운 진동해석방법을 이용하여 진동해석을 수행하고 가진 실험에 의한 진동계측결과와 상호 비교함으로써 새로운 방법의 유용성을 검증하였다.

• 한국해양공학회지
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• 제23권1호
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• pp.54-59
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• 2009

In this paper, the effect of the position and size of a lumped mass on the structural stability of a high speed underwater vehicle is presented. For simplicity, a real vehicle was modeled as a follower force subjected beam that was resting on an elastic foundation, and the lumped mass effect was simplified as an elastic intermediate support. The stability of the simplified model was numerically analyzed based on the Finite element method (FEM). This numerical simulation revealed that flutter type instability or divergence type instability occurs, depending on the position and stiffness of the elastic intermediate support, which implies that the instability of the real model is affected by the position and size of the lumped mass.

• 한국소음진동공학회논문집
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• 제11권6호
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• pp.208-215
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• 2001

The paper presents both theoretical and experimental study fur dynamic instabilities of a vortical cantilevered pipe with two attached lumped masses conveying fluid. The two attached lumped masses can be considered as valves or some mechanical paras in real pipe systems. Eigenvalue behaviors depending on the flow velocity are investigated for the change of Positions and magnitudes of an attached lumped mass and a tip mass. In order to verify appropriate of numerical solutions, experiments were accomplished. Theoretical predictions have a good agreement with experimental ones.