Nanobeams are extensively used in the arena of modern science and technology. The dynamic model of nanobeam with steps and cracks is formed by the combination of classical beam theory and nonlocal theory of elasticity. Homotopy perturbation method is employed to solve the governing equations for two steps nanobeam. The results of this approximation method are compared with the results of exact solution technique. Our vital concern is to analyse the effect of rotatory inertia on the dynamic behaviour of nanobeam. The effect of crack severity, crack and step position, step height ratio, axial load on the vibration of nanobeam are also studied. The results show that the effect of rotatory inertia is very significant in the nanobeam and its influence increases with the increase of mode of frequency.

中文翻译：

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旋转惯性对裂纹和阶梯形纳米束固有频率的影响

纳米束被广泛用于现代科学和技术领域。结合经典梁理论和非局部弹性理论，形成了具有阶梯和裂纹的纳米束动力学模型。采用同伦摄动法求解两步纳米束的控制方程。将这种近似方法的结果与精确求解技术的结果进行比较。我们至关重要的问题是分析旋转惯性对纳米束动力学行为的影响。还研究了裂纹严重程度，裂纹和台阶位置，台阶高度比，轴向载荷对纳米束振动的影响。结果表明，旋转惯性的影响在纳米束中非常显着，其影响随着频率模式的增加而增加。