Investigation of Size-Dependency in Free-Vibration of Micro-Resonators Based on the Strain Gradient Theory

R Vatankhah,M.H Kahrobaiyan

doi:10.1590/1679-78252430

Abstract

This paper investigates the vibration behavior of micro-resonators based on the strain gradient theory, a non-classical continuum theory capable of capturing the size effect appearing in micro-scale structures. The micro-resonator is modeled as a clamped-clamped micro-beam with an attached mass subjected to an axial force. The governing equations of motion and both classical and non-classical sets of boundary conditions are developed based on the strain gradient theory. The normalized natural frequency of the micro-resonator is evaluated and the influences of various parameters are assessed. In addition, the current results are compared to those of the classical and modified couple stress continuum theories.

Highlights

Today, micro-scale structures such as microbeams, microplates, microbars, etc are widely used in Micro-Electro-Mechanical-Systems (MEMS) such as micro-actuators (Padoina et al, 2015), microswitches (Joglekar, and Pawaskar, 2011), Atomic Force Microscopes (AFMs) (Kahrobaiyan et al, 2010), micro-resonators (Hassanpour et al, 2007, Ghanbari et al, 2015) and etc
The normalized natural frequency of the micro-resonator is respectively delineated as a function of the normalized axial load, mass, attached mass position and gyration radius in Figures 2-5 for various values of the ratio of the beam thickness to the material length scale parameter h / l
Since the attempts of the classical continuum theory to capture the size-dependency happening in the micro-scale structures have been in vain, utilizing the non-classical continuum theory to investigate the mechanical behavior of such structures seems to be crucial

Summary

INTRODUCTION

Micro-scale structures such as microbeams, microplates, microbars, etc are widely used in Micro-Electro-Mechanical-Systems (MEMS) such as micro-actuators (Padoina et al, 2015), microswitches (Joglekar, and Pawaskar, 2011), Atomic Force Microscopes (AFMs) (Kahrobaiyan et al, 2010), micro-resonators (Hassanpour et al, 2007, Ghanbari et al, 2015) and etc. A modified couple stress theory has been proposed by Yang et al (2002) in which a new higher-order equilibrium equation, i.e. the equilibrium equation of moments of couples, is considered in addition to the classical equilibrium equations of forces and moments of forces This theory has been employed to formulate the size-dependent static and dynamic behavior of linear and nonlinear Euler-Bernoulli and Timoshenko microbeams (Park and Gao, 2006, Ma et al, 2008, Asghari et al, 2010, Liang et al, 2015), linear homogenous Kirchhoff microplates (Tsiatas, 2009, Ansari et al, 2014), buckling of composite laminated beams (Abadi and Daneshmehr, 2014), and nonlinear three dimensional curved microtubes (Tang et al, 2014). The results are compared to those of the modified couple stress and classical theories and the effects of different parameters such as the length scale parameter and the design parameters like the gyration radius of the attached mass are assessed on the frequency of microresonators

PRELIMINARIES

MODELING

RESULTS AND DISCUSSION