• 한국공간구조학회논문집
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• 제24권1호
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• pp.65-72
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• 2024

This paper aims to advance our understanding of extensible beams with multiple cracks by presenting a crack energy and motion equation, and mathematically justifying the energy functions of axial and bending deformations caused by cracks. Utilizing an extended form of Hamilton's principle, we derive a normalized governing equation for the motion of the extensible beam, taking into account crack energy. To achieve a closed-form solution of the beam equation, we employ a simple approach that incorporates the crack's patching condition into the eigenvalue problem associated with the linear part of the governing equation. This methodology not only yields a valuable eigenmode function but also significantly enhances our understanding of the dynamics of cracked extensible beams. Furthermore, we derive a governing equation that is an ordinary differential equation concerning time, based on orthogonal eigenmodes. This research lays the foundation for further studies, including experimental validations, applications, and the study of damage estimation and detection in the presence of cracks.

• 한국의학물리학회지:의학물리
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• 제7권1호
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• pp.53-64
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• 1996

목적 : 별모양무늬를 이용하여 원격 방사선치료기의 회전축의 정확성을 평가할 때 방사선의 진행방향을 고려해야 하는 이론적 근거와 방법을 개발하고, 방사선의 진행방향이 기록되지 않는 경우 길이 방향의 비대칭 조사면을 이용하여 흉내내는 방법을 개발하는 것이다. 방법 : 갠트리 회전축의 기계적 정확성을 평가하기 위해 방사선의 진행방향을 고려하였다. 좁은 조사면에 의해 별모양무늬를 만들어 회전축이라고 어림되는 교점에서 l0cm 떨어진 위치의 측방선량분포를 필름농도계로 측정하여 선축의 좌표를 구하고 하나의 선축에 있는 한 쌍의 좌표를 이용하여 선축의 식을 구한다. 선축과 일치하는 방사선 진행방향의 단위벡터 equation omitted를 구하고 가정된 회전축의 좌표에서 각 선축으로 향하는 벡터 equation omitted와 equation omitted의 벡터곱 equation omitted$\times$equation omitted을 구하여 평균을 취하고 평균에 대한 벡터곱의 최소자승법을 적용하여 회전축의 좌표를 구한다. 그 때 벡터곱의 최대치의 절대값이 구하는 회전축의 정확도이다. 방사선의 진행방향을 고려할 수 없는 콜리메이터와 치료대에 대해서는 진행방향에 대응하는 것으로 긴 방향이 비대칭인 조사면을 이용하였다. 결과 : 동일한 별모양 무늬에 대해 방사선의 진행방향을 고려할 때 회전축의 기계적 정확성이 진행방향을 무시할 때와 다르게 평가되었다. 결론 : 별모양 무늬를 이용하여 원격치료기의 기계적 정확성을 평가할 때는 방사선의 진행방향을 고려하거나 흉내내어 정량적으로 평가해야 한다.

• 한국소음진동공학회논문집
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• 제12권5호
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• pp.330-337
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• 2002

A frequency equation of beam subjected to the axial load and having ηthrough-the-width-splits is developed. The beam comprises of beam elements that are split into the upper and the lower part, and non-split beam elements. Equations of motion of each beam element are non-dimensionalized with respect to length. The frequency equation of beam is derived from that of each beam element, which satisfies the displacement of the longitudinal and transverse vibration and the boundary conditions between the beam elements. Numerical simulation and experimental work for the beam having several split beam elements are carried out to demonstrate the analytical development and its validity. The experimental results are in good agreement with those of the present frequency equation. The relationships between the split beam width and natural frequencies, and also the relationships between number of split and natural frequencies, in case that the total beam split length is same. are discussed.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2005년도 추계학술대회논문집
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• pp.608-611
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• 2005

In this paper the general equation of motion of damped sandwich beam including arbitrary viscoelastic material layer was derived based on the equation presented by Mead and Markus. The equation of motion of n-layered sandwich beam was represented by (n+3)th order ordinary differential equation. It was verified that the general equation of motion derived in this paper could represent the equations of motions for single-layered, three-layered, five-layered and multi-layered damped beam. Finite element method for the arbitrary-layered damped beam was formulated and programmed using higher order shape functions. Several numerical examples were implemented to show the effects of damped material.

• Structural Engineering and Mechanics
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• 제52권5호
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• pp.997-1017
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• 2014

According to the structure characteristics of the non-uniform beam bridge, a practical model for calculating the vibration equation of the non-uniform beam bridge is given and the application scope of the model includes not only the beam bridge structure but also the non-uniform beam with added masses and elastic supports. Based on the Bernoulli-Euler beam theory, extending the application of the modal perturbation method and establishment of a semi-analytical method for solving the vibration equation of the non-uniform beam with added masses and elastic supports based is able to be made. In the modal subspace of the uniform beam with the elastic supports, the variable coefficient differential equation that describes the dynamic behavior of the non-uniform beam is converted to nonlinear algebraic equations. Extending the application of the modal perturbation method is suitable for solving the vibration equation of the simply supported and continuous non-uniform beam with its arbitrary added masses and elastic supports. The examples, that are analyzed, demonstrate the high precision and fast convergence speed of the method. Further study of the timesaving method for the dynamic characteristics of symmetrical beam and the symmetry of mode shape should be developed. Eventually, the effects of elastic supports and added masses on dynamic characteristics of the three-span non-uniform beam bridge are reported.

• 대한수학회지
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• 제34권3호
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• pp.609-622
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• 1997

We investigate the existence of solutions of the nonlinear beam equation under the Dirichlet boundary condition on the interval $-\frac{2}{\pi}, \frac{2}{\pi}$ and periodic condition on the varible t, $Lu + bu^+ -au^- = f(x, t)$, when the jumping nonlinearity crosses the first positive eigenvalue.

• Korean Journal of Mathematics
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• 제17권2호
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• pp.211-218
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• 2009

We investigate the multiple solutions for a suspending beam equation with jumping nonlinearity crossing three eigenvalues, with Dirichlet boundary condition and periodic condition. We show the existence of at least six nontrivial periodic solutions for the equation by using the finite dimensional reduction method and the geometry of the mapping.

• Korean Journal of Mathematics
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• 제17권3호
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• pp.299-314
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• 2009

We investigate the multiple nontrivial solutions of the asymmetric beam equation $u_{tt}+u_{xxxx}=b_1[{(u + 2)}^+-2]+b_2[{(u + 3)}^+-3]$ with Dirichlet boundary condition and periodic condition on t. We reduce this problem into a two-dimensional problem by using variational reduction method and apply the Mountain Pass theorem to find the nontrivial solutions of the equation.

• 한국소음진동공학회논문집
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• 제21권1호
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• pp.9-17
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• 2011

In this paper, the general equation of motion of damped sandwich beam with multi-viscoelastic material layer was derived based on the equation presented by Mead and Markus. The viscoelastic layer, which has characteristics of complex shear modulus, was assumed to be dominantly under shear deformation. The equation of motion of n-layered damped sandwich beam in bending could be represented by (n+3)th order ordinary differential equation. Finite element model for the n-layered damped sandwich beam was formulated and programmed using higher order shape functions. Several numerical examples were implemented to show the effects of damped material.

• 한국공간구조학회논문집
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• 제19권4호
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• pp.61-68
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• 2019

This paper discusses the influence of transverse reinforcement spacing and support width of concrete wide beam on shear performance. In order to evaluate the shear performance, a total of thirteen specimens were constructed and tested. The transverse reinforcement spacing, the number of legs and support width were considered as variables. From the test results, the shear strength equation of concrete wide beam is proposed for prediction of shear strength of concrete wide beam to consider the transverse reinforcement spacing and support width. It is shown that the proposed equation is able to predict shear strength reasonably well for concrete wide beam.