Journal of the Korea Academia-Industrial cooperation Society (한국산학기술학회논문지)

The Korea Academia-Industrial cooperation Society (한국산학기술학회)
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Eigen-Frequency of a Cantilever Beam Restrained with Added Mass and Spring at Free End or a Node Point

자유단 혹은 노드점에 작용하는 스프링과 부가질량을 받는 일단 지지보의 고유진동수

  • Sim, Woo-Gun (Department of Mechanical Engineering, Hannam University)
  • 심우건 (한남대학교 기계공학과)
  • Received : 2018.09.17
  • Accepted : 2018.12.07
  • Published : 2018.12.31
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Abstract

In order to avoid excessive vibration, it is required to carry out a vibration analysis of heat-exchanger/nuclear-reactor at the design stage. Information of eigen-frequency in the vibration problem is required to evaluate safety of heat-exchange/nuclear reactor. This paper describes a numerical method, Galerkin's method, to solve the eigenvalue problem occurred in a cantilever beam. The beam is restrained with added mass and spring at the free end or a node point of a mode shape. The numerical results of eigen-frequency were compared with simple analytical and experimental results given by simple approach and simple test, respectively. It is found that Galerkin's method is applicable to estimate the eigen-frequency of the cantilever beam. The frequencies become lower with increasing the added mass and the frequencies increase with the spring force. It is shown the heavy added mass has a role of support on the flexible tube. The eigen-frequency of the first mode, for the system with the added mass mounted at the free end, can be calculated by the approximate analytical method existing with more or less accuracy.

열 교환기/원자로의 과도한 진동을 방지 하려면 진동해석을 설계 단계에서 수행해야 한다. 진동 문제에서 고유 진동수의 정보는 열 교환기/원자로의 안전성을 평가하기 위하여 요구된다. 본 논문은 일단 지지보에 발생되는 고유치 문제를 해석하기 위하여 수치해석 방법인 Galerkin의 방법을 기술하였다. 일단 지지보는 자유단 끝점 또는 모드의 노드 포인트에 부가 질량과 스프링에 의하여 구속되어 있다. 수치해석으로 구한 고유진동수는 간단한 해석 방법과 간단한 테스트에 의하여 각각 구한 결과와 비교 되었다. Galerkin의 방법을 사용하여 논의된 일단 지지보의 고유 진동수를 구할 수 있음을 보였다. 부가 질량 증가함에 따라 고유 주파수는 감소하며 스프링 힘의 증가에 따라 고유 주파수는 상승함을 보였다. 무거운 부가 질량은 가연성 배관의 지지대 역할을 함을 보였다. 일단 지지보의 끝단에 설치된 부가 질량의 경우에 개발된 기존의 어림적 해석 방법으로도 일차 모두의 고유 진동수를 비교적 정확하게 구할 수 있음을 알 수 있었다.

Keywords

SHGSCZ_2018_v19n12_32_f0001.png 이미지

Fig. 1. Schematic diagram of the clamped-free beam with the added mass at x=z(=ζL).

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Fig. 2. Equivalent spring.

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Fig. 3. Experiment apparatus for simple test.

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Fig. 4. Mode shapes, Φk(x/L) , of the clamped-free beam, EI/(mL4)=1, (a) without added mass and with added mass, M/mL=1, at (b) ζ = 1, (c) ζ = 0.783, (d) ζ = 0.502.

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Fig. 5. Mode shapes, Φk(x/L) , of the clamped-free beam, EI/(mL4)=1, (a) without added mass and with added mass, (M/mL=100), at (b) ζ = 1, (c) ζ = 0.783, (d) ζ = 0.502.

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Fig. 6. Eigen-frequencies for clamped-free beam, EI/(mL4)=1, with added mass, (M/mL=1).

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Fig. 7. Mode shapes, Φk(x/L) , of the clamped-free beam, EI/(mL4)=1, (a) without spring and with spring, KL/M=100, at (b) ζ = 1, (c) ζ = 0.783, (d) ζ = 0.502.

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Fig. 8. Eigen-frequencies for clamped-free beam, EI/(mL4)=1 & M/mL=1 with stiffness KL/M .

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Fig. 9. Power spectral density, given by FFT analyzer, for a clamped-free beam with added mass (M/L=5) located at free end, ζ = 1; (a) EI/(mL4)=380 (b) EI/(mL4)=92.

Table 1. Comparison of eigenfrequencies, βk , for clamped-free beam with added mass at the free end

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Table 2. Eigen-frequencies, βk , for clamped-free beam with added mass and spring at a nodal point, β .

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Table 3. Eigen-frequencies, βk , for clamped-free beam with added mass at free end or a nodal point, ζ.

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