미분변환법을 이용한 다중 크랙을 갖는 비균일 나노빔의 진동해석
Vibration Analysis of Multi Cracked Nonuniform Nanobeam by using Differential Transformation Method
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초록
본 연구에서는 비국지 탄성이론과 미분변환법을 이용하여 탄성매질속 다중 크랙을 가진 비균질 나노빔의 지배방정식을 유도하고,유도된 지배방정식과 경계조건에 미분변환법을 적용하여 나노빔의 축방향의 진동해석을 하며, 나노빔의 첫단과 끝단의 경계조건이 각각 고정단(Clamped end)과 자유단(Free end)의 경우에 대하여 수치해석을 수행하였다. 수치해석 결과를 기존 연구결과와 비교하여 타당성을 입증한후, 비국지 작은 척도효과 (Nonlocal small scale effect), 탄성매질의 강성, 크랙의 위치, 크랙의 강성 그리고 비국지 탄성이론의 나노빔에 대한진동해석 결과를 고찰하였다.
Abstract
In this study, the governing equations of motion for multi-cracked nonuniform nanobeam based on nonlocal elasticity theory and embeddedin an elastic medium were derived. DTM(differential transformation method) was applied to vibration analysis of multi-cracked nonuniform nanobeambased on nonlocal elasticity theory and embedded in an elastic medium. The non-dimensional natural frequencies of this nanobeam were obtained foreoe, crack stiffness and elastic medium stiffness with various boundary conditions. The results obtained by this method was compared with previousworks and showed the close agreement between two methods. The important conclusions obtained by this study are as follows : 1. As the length ofnanobeam is shorter, the effect of scale coefficient is greater. 2. The locations of crack change non-dimensional natural frequency, In the case of fixed-fixedends, the non-dimensional natural frequency is the biggest in the first crack location of 0.6L of nanobeam length, and the smallest in both ends. Inthe case of fixed-free ends, the closer the location of first crack go tho the free end, the bigger the non-dimensional natural frequency. 3. As the stiffnessof crack is greater, the non-dimensional natural frequency is smaller, And the effect of crack stiffness is similar on both fixed-free ends and fixed-fixedends. 4. The bigger the stiffness of elastic medium, the greater the non - dimensional natural frequency.
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