Micro Cantilever Beam Theory for Transverse Dynamics Using a Continuum Mechanics Model

2011 ◽  
Vol 415-417 ◽  
pp. 760-763
Author(s):  
Cheng Li ◽  
Wei Guo Huang ◽  
Lin Quan Yao
Keyword(s):  
Scale Parameter ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Beam Model ◽  
Cantilever Beams ◽  
Mechanics Model ◽  
Axial Tension ◽  
Vibrational Characteristics ◽  
Micro Cantilever

The vibrational characteristics of cantilever beams with initial axial tension were studied using a nonlocal continuum Euler-Bernoulli beam model. Small size effects are essential to nanotechnology and it can not be ignored in micro or nano scale. Nonlocal elasticity theory has been proved to work well in nanomechanics and it is considered into the governing equation which can be transformed into a fourth-order ordinary differential equation together with a dispersion relation. Boundary conditions are applied so as to determine the analytical solutions of vibrational mode shape and transverse deformation through a numerical method. Relations between natural frequency and the small scale parameter are obtained, including the fundamental and the second order frequencies. It is found that both the small scale parameter and dimensionless initial axial tension play remarkable roles in dynamic behaviors of micro cantilever beams and their effects are analyzed and discussed in detail.

Nonlocal Size Dependence of a Softness Nanobeam with Large Axial Tension under Various Boundary Conditions

2012 ◽  
Vol 490-495 ◽  
pp. 3226-3230
Author(s):  
Cheng Li ◽  
Wei Guo Huang
Keyword(s):  
Natural Frequency ◽  
Scale Parameter ◽  
Separation Of Variables ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Problem Model ◽  
Various Boundary Conditions ◽  
Size Dependent ◽  
Axial Tension ◽  
Dependent Theory

The transverse dynamical behaviors of softness Euler-Bernoulli nanobeams subjected to a biggish initial axial force based on nonlocal elasticity theory are investigated in this paper. The size-dependent theory is considered and a small intrinsic length scale parameter unavailable in classical continuum mechanics is adopted into the problem model as a size parameter. The linear partial differential governing equation is derived from the Newton’s second law and the ordinary equation and its dispersion relation are gained from by the method of separation of variables. Five sets of supporting conditions are presented respectively including simply supported, fully clamped, flexible fixed ends, sliding supports ends and completely free ends. Vibration frequencies are obtained approximately and correlations between the natural frequency and the dimensionless small scale parameter are also analyzed and discussed in detail. It shows that an increase in small scale parameter and dimensionless initial axial tension causes natural frequency to increase, while an increase in the dimensionless stiffness of nanostructures causes natural frequency to decrease, or the nanostructural bending stiffness is enhanced when nonlocal effects are considered.

On thermo-mechanical vibration analysis of multi-scale hybrid composite beams

2018 ◽  
Vol 25 (4) ◽  
pp. 933-945 ◽  
Author(s):  
Farzad Ebrahimi ◽  
Ali Dabbagh
Keyword(s):  
Hybrid Composite ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Beam Theory ◽  
Composite Beams ◽  
Effect Of Temperature ◽  
Correction Coefficient ◽  
Beam Model ◽  
Multi Scale ◽  
Vibrational Characteristics

This article is primarily organized to analyze the thermo-elastic vibrational characteristics of multi-scale hybrid composite beams according to a refined beam model. In this novel type of composites, multi-scale reinforcing elements, carbon fiber (CF) and carbon nanotube (CNT) in particular, are presumed to be dispersed in an initial resin. The homogenization process is carried out employing a mixture of the Halpin–Tsai model and the rule of mixture. The effect of temperature and its gradient on the mechanical properties of CNTs and epoxy resin is rendered to present a more reliable thermal analysis. On the other hand, a refined trigonometric shear deformable beam theory is extended to derive the kinematic relations of the beam needless of any external shear correction coefficient. On the basis of Hamilton's principle, the partial differential equations of motion are developed. Thereafter, the natural frequencies are achieved by the means of Galerkin's method for both simply supported and fully clamped edge conditions. Then, the validity of the presented model is shown by comparing these results with those of previously published researches. Finally, effects of different parameters on the natural frequency of composite beams are rendered in the framework of some numerical case studies. It can be found that multi-scale hybrid composite beams can satisfy higher frequencies once compared with each of the CF- or CNT-reinforced composite beams.

Geometry and thickness dependant anomalous mechanical behavior of fabricated SU-8 thin film micro-cantilevers

2020 ◽  
Vol 3 (2) ◽  
pp. 113-120
Author(s):  
Aviru Kumar Basu ◽  
Anup Basak ◽  
Shantanu Bhattacharya
Keyword(s):  
Beam Theory ◽  
Contradictory Result ◽  
Cantilever Beams ◽  
Bending Tests ◽  
Cantilever Arrays ◽  
Scale Thickness ◽  
Peak Loads ◽  
Intrinsic Length ◽  
Micro Cantilever ◽  
Similar Peak

SU-8 micro-cantilever arrays consisting of V- and M-shaped structures fabricated using a simplified single hard mask step. Bending tests were performed under similar peak loads (ranging 2–10 µN), with thickness ranging between micron (3.5 µm) and sub-micron (0.2 µm) scales. Various mechanical properties such as stiffness and hysteresis are determined from the load versus deflection curves. When the thickness of the V-shaped beam is decreased from 2 µm to 0.2 µm, the stiffness increases by a factor of 2.7, which is in contradiction with the classical beam theory according to which the stiffness for 0.2 µm beam should be three orders of magnitude less than that of 2 µm beam. Micropolar elasticity theory with a variable-intrinsic length scale (thickness dependant) is used to explain such an anomalous response. Experimentally obtained stiffness of two M-shaped beams of thickness 2 µm and 0.2 µm are almost identical. Reason behind this contradictory result is that the thicker beam has a residual strain with a large plastic deformation which usually increases the cross-linking network density, leading to increase in elastic modulus, hardness and thus stiffness of polymers. But the thinner beam has undergone an elastic deformation. The size effect of V- and M-shaped cantilever beams is discussed.

Bernoulli–Euler Dielectric Beam Model Based on Strain-Gradient Effect

2013 ◽  
Vol 80 (4) ◽  
Author(s):  
Xu Liang ◽  
Shuling Hu ◽  
Shengping Shen
Keyword(s):  
Electric Field ◽  
Strain Gradient ◽  
Gradient Elasticity ◽  
Beam Theory ◽  
Field Strain ◽  
Beam Model ◽  
Strain Gradient Elasticity ◽  
Cantilever Beams ◽  
Size Dependent ◽  
Classical Beam Theory

The theoretical investigation of the size dependent behavior of a Bernoulli–Euler dielectric nanobeam based on the strain gradient elasticity theory is presented in this paper. The variational principle is utilized to derive the governing equations and boundary conditions, in which the coupling between strain and electric field, strain gradient and electric field, and strain gradient and strain gradient are taken into account. Different from the classical beam theory, the size dependent behaviors of dielectric nanobeams can be described. The static bending problems of elastic, pure dielectric (nonpiezoelectric), and piezoelectric cantilever beams are solved to show the effects of the electric field-strain gradient coupling and the strain gradient elasticity. Comparisons between the classical beam theory and the strain gradient beam theory are given in this study. It is found that the beam deflection predicted by the strain gradient beam theory is smaller than that by the classical beam theory when the beam thickness is comparable to the internal length scale parameters and the external applied voltage obviously affects the deflection of the dielectric and piezoelectric nanobeam. The presented model is very useful for understanding the electromechanical coupling in nanoscale dielectric structures and is very helpful for designing devices based on cantilever beams.

Free Vibration of Edge Cracked Functionally Graded Microscale Beams Based on the Modified Couple Stress Theory

2017 ◽  
Vol 17 (03) ◽  
pp. 1750033 ◽  
Author(s):  
Şeref Doğuşcan Akbaş
Keyword(s):  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Beam Model ◽  
Cracked Beam ◽  
Stress Theory ◽  
Modified Couple Stress

In this study, the free vibration analysis of edge cracked cantilever microscale beams composed of functionally graded material (FGM) is investigated based on the modified couple stress theory (MCST). The material properties of the beam are assumed to change in the height direction according to the exponential distribution. The cracked beam is modeled as a modification of the classical cracked-beam theory consisting of two sub-beams connected by a massless elastic rotational spring. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new nonclassical beam model reduces to the classical one when the length scale parameter is zero. The problem considered is investigated using the Euler–Bernoulli beam theory by the finite element method. The system of equations of motion is derived by Lagrange’s equations. To verify the accuracy of the present formulation and results, the frequencies obtained are compared with the results available in the literature, for which good agreement is observed. Numerical results are presented to investigate the effect of crack position, beam length, length scale parameter, crack depth, and material distribution on the natural frequencies of the edge cracked FG microbeam. Also, the difference between the classical beam theory (CBT) and MCST is investigated for the vibration characteristics of the beam of concern. It is believed that the results obtained herein serve as a useful reference for research of similar nature.

Scale Effect on Dynamic Analysis of Electrostatically Actuated Nano Beams Using the Nonlocal-Gradient Elasticity Theory

2013 ◽  
Vol 411-414 ◽  
pp. 1859-1862
Author(s):  
Liu Yang ◽  
Jian She Peng
Keyword(s):  
Dynamic Analysis ◽  
Elasticity Theory ◽  
Scale Effect ◽  
Gradient Elasticity ◽  
Nonlocal Elasticity Theory ◽  
Strain Gradient Theory ◽  
Small Scale ◽  
Gradient Theory ◽  
Beam Model ◽  
Electrostatically Actuated

A nonlocal-gradient elasticity beam model with two independent gradient coefficients based on the classical nonlocal elasticity theory and strain gradient theory is used to study dynamic analysis of electrostatically actuated nanobeams. The numerical results show that the deflection response and frequency response of nanobeams are all affected by the small scale. And the two independent gradient coefficients play the different roles in it. This paper broadens the way of studying scale effect.

Buckling Analysis of Nanobeams Based on Nonlocal Timoshenko Beam Model by the Method of Initial Values

2019 ◽  
Vol 19 (04) ◽  
pp. 1950036 ◽  
Author(s):  
Erol Demirkan ◽  
Reha Artan
Keyword(s):  
Timoshenko Beam ◽  
Beam Theory ◽  
Small Scale ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Initial Values ◽  
Nonlocal Timoshenko Beam Theory ◽  
Small Scale Effect ◽  
Carry Over ◽  
Euler Bernoulli Beam

Investigated herein is the buckling of nanobeams based on a nonlocal Timoshenko beam model by the method of initial values within the framework of nonlocal elasticity. Since the nonlocal Timoshenko beam theory is of higher order than the nonlocal Euler–Bernoulli beam theory, it is known to be superior in predicting the small-scale effect. The buckling determinants and critical loads for bars with various kinds of supports are presented. The Carry-Over matrix (Transverse Matrix) is presented and the priorities of the method of initial values are depicted. To the best of the researchers’ knowledge, this is the first work that investigates the buckling of nonlocal Timoshenko beam with the method of initial values.

Pull-In Instability of Electrostatically Actuated Nano-Beams Using the Nonlocal-Gradient Elasticity Theory

2014 ◽  
Vol 488-489 ◽  
pp. 1256-1259
Author(s):  
Jian She Peng ◽  
Liu Yang
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Gradient Elasticity ◽  
Nonlocal Elasticity Theory ◽  
Strain Gradient Theory ◽  
Small Scale ◽  
Gradient Theory ◽  
Beam Model ◽  
Electrostatically Actuated ◽  
Gradient Elasticity Theory

A nonlocal-gradient elasticity beam model with two independent gradient coefficients based on the classical nonlocal elasticity theory and strain gradient theory is used to study the pull-in instability of electrostatically actuated nanobeams. The numerical results show that the pull-in voltage of nanobeams are affected by the small scale. And the two independent gradient coefficients play the different roles in it. This paper broadens the way of studying scale effect.

scholarly journals Stress Analysis of Suspended Pipe Partially Buried in Linear Elastic Soil

2016 ◽  
Vol 10 (1) ◽  
pp. 161-169
Author(s):  
Kun Huang ◽  
Xia Li ◽  
Yiheng Zhang
Keyword(s):  
Beam Theory ◽  
Bending Moment ◽  
Critical Length ◽  
Beam Model ◽  
Linear Elastic ◽  
Limit Value ◽  
Axial Tension ◽  
Calculation Results ◽  
Stress And Deformation ◽  
Elastic Soil

Based on the small deflection beam theory, bending equation with axial tension of suspended pipe partially buried in the linear elastic soil is established. And the corresponding boundary conditions are given according to the stress and deformation characteristics of suspended section and buried section. Then deflection equation for the suspended section is deduced. Afterwards, the stress and critical length of a suspended pipeline are calculated and analyzed. The results show that the tensile stress and bending stress on the endpoint of the suspended section meet the requirement of first strength theory and the critical suspended length is greater than the real suspended length, which is consistent with the actual situation. When the stiffness of soil tends to approach infinity, both the limit value of axial tension and endpoint bending moment agree well with the calculation results of fixed-fixed supported beam model.

scholarly journals Thermo-Mechanical Vibration of Short Carbon Nanotubes Embedded in Pasternak Foundation Based on Nonlocal Elasticity Theory

2013 ◽  
Vol 20 (4) ◽  
pp. 821-832 ◽  
Author(s):  
B. Amirian ◽  
R. Hosseini-Ara ◽  
H. Moosavi
Keyword(s):  
Carbon Nanotubes ◽  
High Temperature ◽  
Aspect Ratio ◽  
Elastic Medium ◽  
Scale Effects ◽  
Mechanical Vibration ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Beam Model ◽  
Short Carbon

This study is concerned with the thermal vibration analysis of a short single-walled carbon nanotube embedded in an elastic medium based on nonlocal Timoshenko beam model. A Winkler- and Pasternak-type elastic foundation is employed to model the interaction of short carbon nanotubes and the surrounding elastic medium. Influence of all parameters such as nonlocal small-scale effects, high temperature change, Winkler modulus parameter, Pasternak shear parameter, vibration mode and aspect ratio of short carbon nanotubes on the vibration frequency are analyzed and discussed. The present study shows that for high temperature changes, the effect of Winkler constant in different nonlocal parameters on nonlocal frequency is negligible. Furthermore, for all temperatures, the nonlocal frequencies are always smaller than the local frequencies in short carbon nanotubes. In addition, for high Pasternak modulus, by increasing the aspect ratio, the nonlocal frequency decreases. It is concluded that short carbon nanotubes have the higher frequencies as compared with long carbon nanotubes.

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