Abstract Free flexural vibrations of nanobeams with non-rigid edge supports are studied by means of the stress-driven nonlocal elasticity model and Euler-Bernoulli kinematics. The elastic deformations of the supports are modelled by transversal and flexural springs, so that, in the limit conditions when the springs stiffnesses tend to zero or infinity, the classical free, pinned, and clamped boundary conditions may be recovered. An analytical procedure is used to derive the closed form solution of the spatial differential equation. The problem of finding the natural frequencies is then reduced to find the roots of the determinant of a matrix, whose elements are explicitly given. The proposed technique, then, avoids the numerical instabilities usually arising when the numerical techniques are used to obtain the solution. The effects of both non-rigid supports elastic deformations and nonlocal parameter on the natural frequencies are studied also for higher vibrations modes. The comparison between the solutions of the proposed model and those available in the literature shows an excellent agreement, and new insightful results and discussions are presented.

中文翻译：

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使用应力驱动的非局部模型在非经典边界条件下纳米梁的自由弯曲振动

摘要 利用应力驱动的非局部弹性模型和欧拉-伯努利运动学研究了非刚性边缘支撑纳米梁的自由弯曲振动。支撑的弹性变形由横向和弯曲弹簧建模，因此，在弹簧刚度趋于零或无穷大的极限条件下，可以恢复经典的自由、固定和夹紧边界条件。分析程序用于导出空间微分方程的封闭形式解。然后将求自然频率的问题简化为求矩阵行列式的根，矩阵的元素已明确给出。然后，所提出的技术避免了在使用数值技术获得解时通常会出现的数值不稳定性。还研究了非刚性支撑弹性变形和非局部参数对固有频率的影响，适用于更高的振动模式。所提出模型的解决方案与文献中可用的解决方案之间的比较显示出极好的一致性，并提出了新的有见地的结果和讨论。