Analytical Modeling for the Bending Resonant Frequency of Sensors Based on Micro and Nanoresonators With Complex Structural Geometry

Micro- and nanoresonator sensors have important applications such as in chemical and biological sensing, environmental control, monitoring of viscosity and magnetic fields, and inertial forces detection. However, most of these resonators are designed as complex structures that complicate the estimation of their resonant frequencies (generally of the bending or torsional mode). In this paper, we present an analytical model to estimate the resonant frequency of the first bending mode of micro- and nanoresonators based on a beam system under different load types. This system is constructed of beams with different cross sections joined through a series-parallel arrangement. The analytical model is derived using the Rayleigh and Macaulay methods, as well as the Euler–Bernoulli beam theory. In addition, we determined the deflection function of the beam system, which can be used to establish its bending structural response under several load types. We applied the model to both a silicon microresonator (with a thickness of 5
$\mu$
m) for an experimental magnetic field sensor developed in our laboratory and for a polycrystalline silicon nanoresonator (with a thickness of 160 nm) of a mass sensor reported in the literature. The results of our analytical model have a comparable agreement with those obtained from the finite-element models (FEMs) and with the experimental measurements. Our analytical model can be useful in the mechanical design of micro- and nanoresonators with complex structural configurations.