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

2021 ◽  
Vol 12 (1) ◽  
pp. 40
Author(s):  
Cheng Li ◽  
Chengxiu Zhu ◽  
Suihan Sui ◽  
Jianwei Yan
Keyword(s):  
Lateral Displacement ◽  
Nonlocal Parameter ◽  
Nonlinear Partial Differential Equation ◽  
Excitation Amplitude ◽  
Nonlocal Theory ◽  
Harmonic Excitation ◽  
Small Scale ◽  
Perturbation Approach ◽  
Stress Theory ◽  
Excited Vibration

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.

scholarly journals The Size-Dependent Thermoelastic Vibrations of Nanobeams Subjected to Harmonic Excitation and Rectified Sine Wave Heating

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1128 ◽  
Author(s):  
Ahmed E. Abouelregal ◽  
Marin Marin
Keyword(s):  
External Force ◽  
Nonlocal Parameter ◽  
Generalized Thermoelasticity ◽  
Nonlocal Elasticity Theory ◽  
Sine Wave ◽  
Harmonic Excitation ◽  
Angular Frequency ◽  
Small Scale ◽  
Length Scale ◽  
Wave Heating

In this article, a nonlocal thermoelastic model that illustrates the vibrations of nanobeams is introduced. Based on the nonlocal elasticity theory proposed by Eringen and generalized thermoelasticity, the equations that govern the nonlocal nanobeams are derived. The structure of the nanobeam is under a harmonic external force and temperature change in the form of rectified sine wave heating. The nonlocal model includes the nonlocal parameter (length-scale) that can have the effect of the small-scale. Utilizing the technique of Laplace transform, the analytical expressions for the studied fields are reached. The effects of angular frequency and nonlocal parameters, as well as the external excitation on the response of the nanobeam are carefully examined. It is found that length-scale and external force have significant effects on the variation of the distributions of the physical variables. Some of the obtained numerical results are compared with the known literature, in which they are well proven. It is hoped that the obtained results will be valuable in micro/nano electro-mechanical systems, especially in the manufacture and design of actuators and electro-elastic sensors.

Nonlocal Modeling and Behavior of Carbon Nanotube-Based Sensors in Thermal Environment

2018 ◽  
Author(s):  
Seyedeh Sepideh Ghaffari ◽  
Samantha Ceballes ◽  
Abdessattar Abdelkefi
Keyword(s):  
Carbon Nanotube ◽  
Thermal Effects ◽  
Thermal Environment ◽  
Nonlocal Parameter ◽  
Nonlocal Theory ◽  
Small Scale ◽  
Mass Sensor ◽  
Buckling Behavior ◽  
Mass Detector ◽  
And Behavior

An exact solution that investigates the pre-buckling characteristics of nonlocal carbon nanotube (CNT)-based mass sensor subjected to thermal load under clamped-clamped boundary condition is determined. The uniform temperature rise is utilized to study thermal effects on the sensitivity of the mechanical resonator in pre-buckling configuration. Using Eringen’s nonlocal theory, along with the Hamilton’s principle, the governing equations considering small scale and geometric nonlinearity are derived. The influences of important parameters including nonlocal parameter, temperature change, length, and diameter of the CNT on the pre-buckling behavior and frequency shift of the CNT-based mass detector are also studied. Results show that these parameters have significant impact on the dynamic characteristics of the CNT-mass sensor.

Wave Propagation Analysis of Piezoelectric Nanoplates Based on the Nonlocal Theory

2018 ◽  
Vol 18 (04) ◽  
pp. 1850060 ◽  
Author(s):  
Li-Hong Ma ◽  
Liao-Liang Ke ◽  
Yi-Ze Wang ◽  
Yue-Sheng Wang
Keyword(s):  
Wave Propagation ◽  
Electric Potential ◽  
Mechanical Load ◽  
Tensile Force ◽  
Nonlocal Parameter ◽  
Nonlocal Theory ◽  
Wave Number ◽  
Small Scale ◽  
Axial Tensile ◽  
Propagation Analysis

Based on the nonlocal theory, this paper develops the Kirchhoff nanoplate and Mindlin nanoplate models for the wave propagation analysis of piezoelectric nanoplates. The effects of small scale parameter and thermo-electro-mechanical loads are incorporated in the nanoplate models. The Hamilton’s principle is employed to derive the governing equations of the nanoplate, which are solved analytically to obtain the dispersion relation for piezoelectric nanoplates. The results show that the nonlocal parameter, temperature change, mechanical load and external electric potential have significant influence on the wave propagation characteristics of the piezoelectric nanoplates. The cut-off wave number is observed to exist for piezoelectric nanoplates subjected to positive electric potential, axial tensile force and temperature rise.

scholarly journals Dynamic Behavior of Multi-Layered Viscoelastic Nanobeam System Embedded in a Viscoelastic Medium with a Moving Nanoparticle

2016 ◽  
Vol 33 (5) ◽  
pp. 559-575 ◽  
Author(s):  
Sh. Hosseini Hashemi ◽  
H. Bakhshi Khaniki
Keyword(s):  
Dynamic Behavior ◽  
Viscoelastic Medium ◽  
Scale Effects ◽  
Nonlocal Parameter ◽  
Nonlocal Theory ◽  
Function Technique ◽  
Small Scale ◽  
Transform Method ◽  
Damping Parameter ◽  
Moving Nanoparticle

AbstractIn this paper, dynamic behavior of multi-layered viscoelastic nanobeams resting on a viscoelastic medium with a moving nanoparticle is studied. Eringens nonlocal theory is used to model the small scale effects. Layers are coupled by Kelvin-Voigt viscoelastic medium model. Hamilton's principle, eigen-function technique and the Laplace transform method are employed to solve the governing differential equations. Analytical solutions for transverse displacements of double-layered is presented for both viscoelastic nanobeams embedded in a viscoelastic medium and without it while numerical solution is achieved for higher layered nanobeams. The influences of the nonlocal parameter, stiffness and damping parameter of medium, internal damping parameter and number of layers are studied while the nanoparticle passes through. Presented results can be useful in analysing and designing nanocars, nanotruck moving on surfaces, racing nanocars etc.

A Nonlinear Approach for Dynamic Responses of a Nano-Beam Based on a Strain Gradient Nonlocal Theory

2014 ◽  
Vol 609-610 ◽  
pp. 1483-1488
Author(s):  
Cheng Li ◽  
Shuang Li
Keyword(s):  
Strain Gradient ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Continuum Theory ◽  
Nonlocal Theory ◽  
Dynamic Responses ◽  
Transverse Motion ◽  
Small Scale ◽  
Transverse Displacement ◽  
Nonlocal Continuum Theory

The transverse nonlinear vibration of a nanobeam fully clamped at both two ends was investigated using a strain gradient type of nonlocal continuum theory. The small scale effect was considered to the mechanical model at nanoscale. The axial elongation of the nanobeam was taken into account and the nonlinear partial differential equation governing the transverse motion was derived. Subsequently, a perturbation method was applied to the nonlinear governing equation. The dynamical responses of the nanobeam such as transverse displacement and resonant angular frequency were obtained and they were compared with those by a numerical method. The comparison indicated the validity of the present nonlinear model and the multiple-scales analysis method.

Determination of Transverse Amplitude of Microbeam Vibrating Accounting for Nonlinear Effects

2012 ◽  
Vol 446-449 ◽  
pp. 1190-1193
Author(s):  
Cheng Li ◽  
Wei Guo Huang
Keyword(s):  
Nonlinear Partial Differential Equation ◽  
Nonlinear Effects ◽  
Nonlocal Theory ◽  
Basic Element ◽  
Small Scale ◽  
Problem Model ◽  
Axial Tension ◽  
Small Scale Effect ◽  
Mechanical Devices

The nonlinear dynamics of a microbeam with initial axial tension is presented. The nonlocal theory with a small scale effect is applied to the problem model. Considering the axial protraction due to the transverse deformation of the microbeam, a nonlinear partial differential equation that governs the dynamic motion is derived. Explicit solution of transverse amplitude is obtained by the method of multiscale analysis. Both nonlinear and nonlocal effects are found to play significant roles in the vibration behaviors of a microbeam. The results may be helpful for the application and design of various micro-electronic-mechanical devices, where a microbeam acts as a basic element.

scholarly journals Differential Quadrature and Differential Transformation Methods in Buckling Analysis of Nanobeams

2019 ◽  
Vol 6 (1) ◽  
pp. 68-76 ◽  
Author(s):  
Subrat Kumar Jena ◽  
S. Chakraverty
Keyword(s):  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Nonlocal Parameter ◽  
Transformation Method ◽  
Nonlocal Theory ◽  
Buckling Analysis ◽  
Differential Quadrature ◽  
Load Parameter ◽  
Computationally Efficient ◽  
Differential Transformation

AbstractIn this paper, two computationally efficient techniques viz. Differential Quadrature Method (DQM) and Differential Transformation Method (DTM) have been used for buckling analysis of Euler-Bernoulli nanobeam incorporation with the nonlocal theory of Eringen. Complete procedures of both the methods along with their mathematical formulations are discussed, and MATLAB codes have been developed for both the methods to handle the boundary conditions. Various classical boundary conditions such as SS, CS, and CC have been considered for investigation. A comparative study for the convergence of DQM and DTM approaches are carried out, and the obtained results are also illustrated to demonstrate the effects of the nonlocal parameter, aspect ratio (L/h) and the boundary condition on the critical buckling load parameter.

scholarly journals Double mode model of size-dependent chaotic vibrations of nanoplates based on the nonlocal elasticity theory

2021 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Grzegorz Kudra ◽  
Olga Mazur
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Scale Effects ◽  
Plate Theory ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Phase Portraits ◽  
Small Scale ◽  
Governing System ◽  
Account Due

AbstractIn this paper vibrations of the isotropic micro/nanoplates subjected to transverse and in-plane excitation are investigated. The governing equations of the problem are based on the von Kármán plate theory and Kirchhoff–Love hypothesis. The small-size effect is taken into account due to the nonlocal elasticity theory. The formulation of the problem is mixed and employs the Airy stress function. The two-mode approximation of the deflection and application of the Bubnov–Galerkin method reduces the governing system of equations to the system of ordinary differential equations. Varying the load parameters and the nonlocal parameter, the bifurcation analysis is performed. The bifurcations diagrams, the maximum Lyapunov exponents, phase portraits as well as Poincare maps are constructed based on the numerical simulations. It is shown that for some excitation conditions the chaotic motion may occur in the system. Also, the small-scale effects on the character of vibrating regimes are illustrated and discussed.

Wave Propagation in Two-Dimensional Nonlinear Periodic Lattices

2009 ◽  
Author(s):  
Raj K. Narisetti ◽  
Massimo Ruzzene ◽  
Michael J. Leamy
Keyword(s):  
Wave Propagation ◽  
Linear Models ◽  
Periodic Structures ◽  
Harmonic Wave ◽  
Excitation Amplitude ◽  
Pass Band ◽  
Perturbation Approach ◽  
Nonlinear Stiffness ◽  
Two Dimensional ◽  
Low Amplitude

This paper investigates wave propagation in two-dimensional nonlinear periodic structures subject to point harmonic forcing. The infinite lattice is modeled as a springmass system consisting of linear and cubic-nonlinear stiffness. The effects of nonlinearity on harmonic wave propagation are analytically predicted using a novel perturbation approach. Response is characterized by group velocity contours (derived from phase-constant contours) functionally dependent on excitation amplitude and the nonlinear stiffness coefficients. Within the pass band there is a frequency band termed the “caustic band” where the response is characterized by the appearance of low amplitude regions or “dead zones.” For a two-dimensional lattice having asymmetric nonlinearity, it is shown that these caustic bands are dependent on the excitation amplitude, unlike in corresponding linear models. The analytical predictions obtained are verified via comparisons to responses generated using a time-domain simulation of a finite two-dimensional nonlinear lattice. Lastly, the study demonstrates amplitude-dependent wave beaming in two-dimensional nonlinear periodic structures.

Vibration of Single- and Double-Layered Graphene Sheets

2011 ◽  
Vol 2 (1) ◽  
Author(s):  
Behrouz Arash ◽  
Quan Wang
Keyword(s):  
Molecular Dynamics ◽  
Boundary Conditions ◽  
Molecular Dynamics Simulations ◽  
Nonlocal Parameter ◽  
Small Scale ◽  
Graphene Sheets ◽  
Elastic Model ◽  
Resonant Frequencies ◽  
Nonlocal Continuum Theory ◽  
Dynamics Simulations

Free vibration of single- and double-layered graphene sheets is investigated by employing nonlocal continuum theory and molecular dynamics simulations. Results show that the classical elastic model overestimated the resonant frequencies of the sheets by a percentage as high as 62%. The dependence of small-scale effects, sizes of sheets, boundary conditions, and number of layers on vibrational characteristic of single- and double-layered graphene sheets is studied. The resonant frequencies predicted by the nonlocal elastic plate theory are verified by the molecular dynamics simulations, and the nonlocal parameter is calibrated through the verification process. The simulation results reveal that the calibrated nonlocal parameter depends on boundary conditions and vibrational modes. The nonlocal plate model is found to be indispensable in vibration analysis of grapheme sheets with a length less than 8 nm on their sides.

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