In the framework of the two-phase theory of nonlocal elasticity we study free high-frequency vibrations of nanobeams with varying cross section. Considering the Helmholtz kernel in the constitutive equation, we reduce the original integro-differential equation to an equivalent six-order differential equation with variable coefficients and derive the pair of additional boundary conditions accounting for the nonlocal edge effects. Applying WKB method, a solution of the boundary-value problem is constructed in the form of superposition of the outer expansion and edge effect integrals with high variability along the beam axis. As an example, a clamped nanobeam with a shape matching to the shape of a slab waveguide in a photonic crystal beam laser is considered. The derived relations for natural frequencies and performed calculations revealed strong effect of both the nonlocal model fraction and the cross-section variability on the correction to natural frequencies calculated within the classical beam theory

在两阶段非局部弹性理论的框架内，我们研究了具有可变横截面的纳米束的自由高频振动。考虑到本构方程中的亥姆霍兹核，我们将原始积分微分方程简化为具有可变系数的等效六阶微分方程，并得出考虑了非局部边缘效应的一对附加边界条件。应用WKB方法，通过沿光束轴的高度扩展和外部扩展与边缘效应积分的叠加形式构造了边值问题的解决方案。作为示例，考虑具有与光子晶体束激光器中的平板波导的形状匹配的形状的被夹持的纳米束。