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Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique

机译：正交技术振动分析裂纹纳米束

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This work concerns with free vibration analysis of cracked nanobeam problems. Based on Eringen's nonlocal elasticity theory, the governing equation of Euler-Bernoulli and Timoshenko nanobeams, are derived. It is assumed that strain at a certain point is a function of the strains at all points within the influence domain. The cracked beam is modeled as multi-segments connected by a rotational spring located at the cracked sections. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. Polynomial based differential quadrature method is employed to solve the problem. Derivatives of the field quantities are approximated as a weighted linear sum of the nodal values. For different supporting cases, the boundary conditions are directly substituted in the equation of motion, such that the problem is reduced to that of linear homogeneous algebraic system. This suggested numerical scheme accurately determined angular frequencies of the problem. A comparative study is tabulated to compare the obtained results with the previous ones. Further, a parametric study is introduced to investigate the influence of crack locations, crack severity and the nonlocal scale parameter on the obtained results. The obtained results recorded that frequency values decrease with the increasing of both of crack severity and the nonlocal scale parameter. The results of the proposed scheme may be applied for structural health monitoring.
机译：这项工作涉及裂纹纳米云问题的自由振动分析。基于eringen的非局部弹性理论，推导出欧拉伯努利和Timoshenko Nanobeams的控制方程。假设某个点的应变是影响结构域内的所有点的菌株的函数。裂纹光束被建模为通过位于裂纹部分的旋转弹簧连接的多段。由于弯曲，该模型促进了旋转位移的不连续性，其与由裂缝部分传递的弯曲力矩成比例。采用多项式基于差分正交方法来解决问题。场数量的衍生物近似为节点值的加权线性和。对于不同的支持情况，边界条件直接在运动方程中被替换，使得问题减少到线性均匀代数系统的问题。这表明数值方案精确确定了问题的角度频率。制表比较研究，以将所得结果与前一个研究结果进行比较。此外，引入了参数研究以研究裂缝位置，裂缝严重程度和非局部比例参数对所得结果的影响。所获得的结果记录了频率值随着裂缝严重程度和非识别尺度参数的增加而降低。拟议方案的结果可用于结构健康监测。

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