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.

The Small Scale Effect on Linear Vibration of Beam-Like Nanostructures

2012 ◽  
Vol 446-449 ◽  
pp. 3432-3435
Author(s):  
Cheng Li ◽  
Lin Quan Yao
Keyword(s):  
Axial Load ◽  
Vibration Mode ◽  
Multiple Scales ◽  
Separation Of Variables ◽  
Classical Mechanics ◽  
Small Scale ◽  
Transverse Displacement ◽  
Linear Vibration ◽  
Small Scale Effect ◽  
Multiple Scales Analysis

Transverse free dynamics of a beam-like nanostructure with axial load is investigated. The effects of a small size at nano-scale unavailable in classical mechanics are presented. Explicit solutions for natural frequency, vibration mode and transverse displacement are obtained by separation of variables and multiple scales analysis. Results by two methods are in close agreement.

Self Adapting Mechanical Step Bearings for Variations in Load

Tribology ◽  
2005 ◽  
Author(s):  
Robert L. Jackson
Keyword(s):  
Friction And Wear ◽  
Multiple Scales ◽  
The Self ◽  
Dynamic Responses ◽  
External Control ◽  
Small Scale ◽  
Precision Control ◽  
Surface Profiles ◽  
Bearing Design ◽  
And Behavior

As the application of small-scale and precision technologies increases, the need will grow for bearings which are able to provide precision control of their location. At the micro and nano-scale there is a need for new bearing technologies to reduce friction and wear, and provide precision control of bearing motion. This control can be provided by electronically controlled actuators and sensors, but then the system is dependant on the reliability of the electronics. This work uses numerical methods to research the design and behavior of self adapting smart step bearings. These step bearings are designed to change their surface profiles to achieve an optimal or controlled behavior, without the use electronics or external control. The bearing changes its profile to control the film height of the bearing to a near constant value for different loads. The result is a self adapting step bearing design that may be applied at multiple scales for use in a wide variety of machine components. The numerical simulation shows that the self adapting step bearing is able to autonomously adapt in real time to dynamic loads and maintain a desired film thickness with a relatively small amount of deviation. The self adapting step bearing also exhibits smaller dynamic responses to transient loads in comparison to a conventional static geometry step bearing.

A Thermodynamic Consistent Model for Coupled Strain-Gradient Plasticity With Temperature

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Danial Faghihi ◽  
George Z. Voyiadjis
Keyword(s):  
Strain Gradient ◽  
Continuum Theory ◽  
Strain Gradient Plasticity ◽  
Bulge Test ◽  
Gradient Plasticity ◽  
Mechanical Responses ◽  
Nonlocal Continuum Theory ◽  
Thermodynamic Potentials ◽  
Metallic Compounds ◽  
Rate Of Dissipation

The mechanical responses of small volume metallic compounds are addressed in this work through developing a nonlocal continuum theory. In this regard, a thermodynamic-based higher-order strain-gradient plasticity framework for coupled thermoviscoplasticity modeling is presented. The concept of thermal activation energy and the dislocations interaction mechanisms are taken into consideration to describe the choice of thermodynamic potentials such as Helmholtz free energy and rate of dissipation. The theory is developed based on the decomposition of the thermodynamic conjugate forces into energetic and dissipative counterparts, which provides the constitutive equations to have both energetic and dissipative gradient length scales. The derived constitutive model is calibrated against the experimental data of bulge test conducted on thin films.

Nonlocal Continuum Theory Based Modeling of Carbon Nanotube Composites

2008 ◽  
Author(s):  
Ali Alavinasab ◽  
Goodarz Ahmadi ◽  
Ratneshwar Jha
Keyword(s):  
Finite Element ◽  
Carbon Nanotube ◽  
Continuum Theory ◽  
Nonlocal Theory ◽  
Total Force ◽  
Stress Distributions ◽  
Nonlocal Continuum ◽  
Element Analysis ◽  
Nonlocal Continuum Theory ◽  
Nanotube Composites

Analytical modeling of Carbon Nanotube (CNT) composite based on the nonlocal continuum theory is investigated. This approach accounts for nonlocal stress-strain relationships, that is, stress at any point in a structure is a function of strain in the entire structure. Finite element analysis of a representative volume element (RVE) of CNT composite is used to evaluate unknown constant in the nonlocal theory based solution. Stress distributions are obtained from finite element method (FEM), nonlocal theory, and standard (local) elasticity. Nonlocal theory and FEM stress distributions yield the same total force and first moment, whereas standard elasticity gives less accurate results.

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.

ANALYTIC SOLUTIONS TO THE OSCILLATORY BEHAVIOR AND PRIMARY RESONANCE OF ELECTROSTATICALLY ACTUATED MICROBRIDGES

2011 ◽  
Vol 11 (06) ◽  
pp. 1119-1137 ◽  
Author(s):  
M. MOJAHEDI ◽  
M. MOGHIMI ZAND ◽  
M. T. AHMADIAN ◽  
M. BABAEI
Keyword(s):  
Differential Equation ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Axial Stress ◽  
Nonlinear Partial Differential Equation ◽  
Primary Resonance ◽  
Dynamic Responses ◽  
Design Parameters ◽  
Electrostatically Actuated ◽  
Homotopy Perturbation

In this paper, the vibration and primary resonance of electrostatically actuated microbridges are investigated, with the effects of electrostatic actuation, axial stress, and mid-plane stretching considered. Galerkin's decomposition method is adopted to convert the governing nonlinear partial differential equation to a nonlinear ordinary differential equation. The homotopy perturbation method (a special case of homotopy analysis method) is then employed to find the analytic expressions for the natural frequencies of predeformed microbridges, by which the effects of the voltage, mid-plane stretching, axial force, and higher mode contribution on the natural frequencies are studied. The primary resonance of the microbridges is also investigated, where the microbridges are predeformed by a DC voltage and driven to vibrate by an AC harmonic voltage. The methods of homotopy perturbation and multiple scales are combined to find the analytic solution for the steady-state motion of the microbeam. In addition, the effects of the design parameters and damping on the dynamic responses are discussed. The results are shown to be in good agreement with the existing ones.

An analytical approach for vibration analysis of laminated orthotropic beam based on nonlocal theory

2018 ◽  
Vol 233 (10) ◽  
pp. 3633-3648
Author(s):  
Saeed Khadem Moshir ◽  
Hamidreza Eipakchi
Keyword(s):  
Numerical Solutions ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Nonlocal Theory ◽  
Free Vibrations ◽  
Small Scale ◽  
Laminated Beam ◽  
Analytical Perturbation

On the basis of the first-order shear deformation beam theory, free vibrations and dynamic response of orthotropic laminated beam subjected to transient and harmonic loading have been studied based on Eringen’s nonlocal elasticity theory. Three coupled nonlinear governing partial differential equations of motion are derived using Hamilton’s principle. The purely analytical perturbation method as well as the method of multiple scales are used for the solution. A parametric study is carried out to realize the effect of small-scale and axial static load on the natural frequencies, transient, and harmonic responses. In addition, the obtained results have been compared with numerical solutions and literature.

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 Critical Temperatures for Vibrations and Buckling of Magneto-Electro-Elastic Nonlocal Strain Gradient Plates

Nanomaterials ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 87
Author(s):  
Giovanni Tocci Monaco ◽  
Nicholas Fantuzzi ◽  
Francesco Fabbrocino ◽  
Raimondo Luciano
Keyword(s):  
Strain Gradient ◽  
Thermal Environment ◽  
Scale Effects ◽  
Nonlocal Theory ◽  
Elastic Plates ◽  
Critical Temperatures ◽  
Buckling Behavior ◽  
Linear Vibrations ◽  
Nonlocal Strain Gradient ◽  
Plate Kinematics

An analytical method is presented in this work for the linear vibrations and buckling of nano-plates in a hygro-thermal environment. Nonlinear von Kármán terms are included in the plate kinematics in order to consider the instability phenomena. Strain gradient nonlocal theory is considered for its simplicity and applicability with respect to other nonlocal formulations which require more parameters in their analysis. Present nano-plates have a coupled magneto-electro-elastic constitutive equation in a hygro-thermal environment. Nano-scale effects on the vibrations and buckling behavior of magneto-electro-elastic plates is presented and hygro-thermal load outcomes are considered as well. In addition, critical temperatures for vibrations and buckling problems are analyzed and given for several nano-plate configurations.

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