Nano and micromechanical mass sensing using cantilever oscillators of different length-scales has been an established approach. The main principle underpinning this technique is the shift in the resonance frequency caused by the additional mass in the dynamic system. While the mass of an object to be sensed is useful information, some idea about the shape of the object would be an additional benefit. The shape information may be used to make a distinction between two different objects of the same mass. This paper establishes the conceptual framework for simultaneous sensing of the mass as well as the rotary inertia of an object attached to a vibrating cantilever beam. The rotary inertia of an object gives additional insight into its shape, which is a key motivation of this work. It is shown that by using two modes it is possible to formulate two coupled nonlinear equations, which in turn can be solved to obtain the mass and the rotary inertia simultaneously from the frequency shifts of first two vibration modes. Euler-Bernoulli beam theory and an energy approach are used to derive closed-form expressions for the identified mass and rotary inertia from the measured frequency shifts. Analytical expressions are validated using high fidelity finite element simulation results.

使用不同长度尺度的悬臂振荡器的纳米和微机械质量感测已被确定。支撑此技术的主要原理是动态系统中附加质量引起的共振频率偏移。尽管要感测的物体的质量是有用的信息，但是有关物体形状的一些想法将是额外的好处。形状信息可以用于在相同质量的两个不同物体之间进行区分。本文建立了同时感测质量以及附着到振动悬臂梁的物体的旋转惯性的概念框架。物体的旋转惯性使人们对其形状有了更多的了解，这是这项工作的主要动机。结果表明，通过使用两种模式，可以公式化两个耦合的非线性方程，进而可以求解这些非线性方程，从而从前两个振动模式的频移中同时获得质量和转动惯量。Euler-Bernoulli束理论和能量方法用于从测得的频移中得出识别出的质量和转动惯量的闭式表达式。使用高保真有限元模拟结果验证了解析表达式。