Abstract Transverse vibration of a circular cross sectional micro-rod subjected to a new kind of boundary constraints with elastic torsional springs is presented based on nonlocal elasticity. A nonlocal strengthening beam model is utilized and the effect of temperature changing is taken into consideration. The variational method and Hamilton’s principle are applied to derive the governing equation of motion and corresponding boundary conditions. A higher-order partial differential equation that is a typical characteristic of nonlocal strengthening model is developed, and the boundary conditions contain not only classical conditions but also non-classical higher-order conditions. Unlike previous studies which were only concerned with some conventional boundary constraints, we consider more general boundary conditions named elastic torsional spring supports. Such boundary conditions are between the simply supported and clamped ones, and they are closer to the actual constraints of existing engineering structures. Natural frequencies of micro-rods with new boundary constraints are determined via an eigenvalue method and compared with other results in the literature. It is shown that the nonlocal scale factor, thermal parameter, rigidity parameter and torsional spring coefficient play significant roles in free vibration of micro-rods. The research can provide a reference for a large class of boundary conditions ranging from simply supported to clamped micro-rods.

中文翻译：

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考虑非局部强化和温度效应的经典和非经典新边界条件下自由振动微棒的变分方法

摘要 基于非局部弹性，提出了一种新型弹性扭转弹簧边界约束下圆形截面微杆的横向振动。采用非局部强化梁模型，并考虑了温度变化的影响。应用变分方法和哈密顿原理推导出运动控制方程和相应的边界条件。建立了具有非局部强化模型典型特征的高阶偏微分方程，边界条件不仅包含经典条件，还包含非经典高阶条件。与以往仅关注一些常规边界约束的研究不同，我们考虑称为弹性扭转弹簧支撑的更一般的边界条件。这种边界条件介于简支和夹钳之间，更接近现有工程结构的实际约束条件。具有新边界约束的微杆的固有频率通过特征值方法确定，并与文献中的其他结果进行比较。结果表明，非局部比例因子、热参数、刚度参数和扭转弹簧系数对微杆自由振动起重要作用。该研究可为从简支到夹紧微杆的一大类边界条件提供参考。它们更接近现有工程结构的实际约束。具有新边界约束的微杆的固有频率通过特征值方法确定，并与文献中的其他结果进行比较。结果表明，非局部比例因子、热参数、刚度参数和扭转弹簧系数对微杆自由振动起重要作用。该研究可为从简支到夹紧微杆的一大类边界条件提供参考。它们更接近现有工程结构的实际约束。具有新边界约束的微杆的固有频率通过特征值方法确定，并与文献中的其他结果进行比较。结果表明，非局部比例因子、热参数、刚度参数和扭转弹簧系数对微杆自由振动起重要作用。该研究可为从简支到夹紧微杆的一大类边界条件提供参考。刚度参数和扭转弹簧系数在微杆自由振动中起重要作用。该研究可为从简支到夹紧微杆的一大类边界条件提供参考。刚度参数和扭转弹簧系数在微杆自由振动中起重要作用。该研究可为从简支到夹紧微杆的一大类边界条件提供参考。