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

Abstract

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 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 microand nanoresonators with complex structural configurations.

Tunable Micro- and Nanomechanical Resonators

“…Although micro- and nanoresonators have significant applications in many fields, most of these resonators are designed as complex structures that complicate the estimation of their resonant frequencies [ 106 ]. The resonant frequency changes should be determined by variations of geometrical variables and mechanical properties of the resonators [ 107 ].…”

“…Pasini [ 79 ] developed an interesting model applied to obtain the resonant frequency of multilayered microresonators with different shaped cross section, symmetry and number of layers, and materials. Considering different loading such as concentrated and uniformly distributed loads, and bending moments acting on the beam at the same time, Herrera-May [ 106 ] developed an analytical model for estimating the resonant frequency of micro- and nanoresonators. Zhang et al .…”

Dynamic Characteristics of Electrostatically Actuated Shape Optimized Variable Geometry Microbeam

“…Then the time of one period is discretized into points. In order to solve the obtained ordinary differential equations, finite differential method is used to compute the velocity and the acceleration of the microbeam and subsequently make the results satisfy (15); finally utilizing the Levenberg-Marquardt optimization method, the deflection with respect to and can be obtained:…”

“…The goal of their research is to enhance the static and dynamic pull-in ranges of electrostatically actuated microbeams, but they did not discuss the impact of the resulting optimized shape on the dynamic response of the microbeam. Herrera-May et al [14][15][16] not only studied the resonant characteristic of the single layer variable cross-section microbeam but also researched the bending resonant frequency of multilayered microresonators with variable cross-section. So far, the research on the dynamic characteristic of the optimized microbeam is fewer.…”

A Thermoacoustic Model for High Aspect Ratio Nanostructures

“…New materials that are rigid and have high aspect ratio structures, such as Ag 2 Ga metallic nanoneedles [7] and ZnO nanowires and nanobelts [26], may enable better materials for nano-actuation and SPM applications. While many experiments have reported such nanoresonators, only a few theoretical reports exist on the modeling and characterization of nanoresonators' natural frequency and loss mechanisms [3,[27][28][29][30].…”