A Perturbation Approach for Lateral Excited Vibrations of a Beam-like Viscoelastic Microstructure Using the Nonlocal Theory

Cheng Li,Jianwei Yan,Suihan Sui,Chengxiu Zhu

doi:10.3390/app12010040

Cheng Li, Jianwei Yan + Show 2 more

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https://doi.org/10.3390/app12010040

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Abstract

In this paper, we investigate the lateral vibration of fully clamped beam-like microstructures subjected to an external transverse harmonic excitation. Eringen’s nonlocal theory is applied, and the viscoelasticity of materials is considered. Hence, the small-scale effect and viscoelastic properties are adopted in the higher-order mathematical model. The classical stress and classical bending moments in mechanics of materials are unavailable when modeling a microstructure, and, accordingly, they are substituted for the corresponding effective nonlocal quantities proposed in the nonlocal stress theory. Owing to an axial elongation, the nonlinear partial differential equation that governs the lateral motion of beam-like viscoelastic microstructures is derived using a geometric, kinematical, and dynamic analysis. In the next step, the ordinary differential equations are obtained, and the time-dependent lateral displacement is determined via a perturbation method. The effects of external excitation amplitude on excited vibration are presented, and the relations between the nonlocal parameter, viscoelastic damping, detuning parameter, and the forced amplitude are discussed. Some dynamic phenomena in the excited vibration are revealed, and these have reference significance to the dynamic design and optimization of beam-like viscoelastic microstructures.

Highlights

Microscaled materials and microscaled structures have attracted considerable interest as sensors, resonators, vibrators, and attenuators [1,2,3]
The lateral vibration dominates the beam-like microstructures with an oscillation, of which some can be modeled as the lateral excited vibrations of fully clamped microbeams according to the external excitation and boundary constraints
The microstructures are often subjected to time-dependent external excitation, and, the fully clamped boundary is one of the most common end restrictions, both of which are seen in micro-electromechanical systems (MEMS) and other devices at the microscale [5]

Summary

Introduction

Microscaled materials and microscaled structures have attracted considerable interest as sensors, resonators, vibrators, and attenuators [1,2,3]. Chwal and Muc [9] examined the free vibration of rectangular nanoplates using a refined shear deformation theory Both the nonlocal strain and nonlocal stress were considered, and the Rayleigh-Ritz method was employed to solve the governing equations. Behera and Chakraverty [25] reviewed recent advances on the nonlocal theory and its application in the vibration behaviors of carbon nanotubes using various microbeam models They summarized the research status of the nonlocal theory relating to different types of complicating effects, nonlinearity, functionally graded material, and beam theories. Only the first two terms in the effective nonlocal bending moment are considered and further substitute it into Equation (5), one can obtain the following nonlinear partial differential equation that governs the lateral vibration as EI. The following equation can be determined and shown as ω20π2

Perturbation Analyses

Results and Discussion

Conclusions