The natural frequency of nanotubes with consideration of surface effects is studied using the nonlocal Timoshenko beam theory. This theory with long-range interactions revealing the nanoscale effect can be applied to study the vibration behavior of nanostructures. When the nonlocal effect is neglected in the analysis, the result is identical to the previous study which used a modified Timoshenko beam theory. According to the analysis, the application of the modified Timoshenko beam theory for nanotubes would lead to an underprediction of the natural frequency.
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Lee et al. (2010) studied this question.
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