The paradox of abnormal frequencies exists for free vibration of nonlocal cantilever beams. This work is concerned with solving this paradox within the framework of nonlocal strain gradient theory. The inconsistences of the reported boundary conditions are updated by using the weighted residual method. Then, the closed-form solutions for free vibration of cantilever beams is obtained in terms of the inconsistences of the reported boundary value problems (BVPs) and the reformulated BVPs. The softening phenomenon is captured by the updated numerical results. A method for calibrating the size-effect parameters is firstly proposed in this work. The numerical results show that the present model is capable for capturing the size-dependent mechanical properties of materials, which exhibit either the stiffness-hardening effect or the stiffness-softening effect, depending upon the relative values of the two size-effect parameters.

对于非局部悬臂梁的自由振动，存在异常频率的悖论。这项工作涉及在非局部应变梯度理论的框架内解决这一悖论。通过使用加权残差法更新了所报告边界条件的不一致性。然后，根据所报告的边值问题（BVP）和重新制定的BVP的不一致性，获得了悬臂梁自由振动的封闭形式解。通过更新的数值结果可以捕获软化现象。在这项工作中，首先提出了一种校准尺寸效应参数的方法。数值结果表明，该模型能够捕获与尺寸有关的材料机械性能，