Surface effects on the bending,buckling and free vibration analysis of magneto-electro-elastic beams

Surface effects are responsible for the size dependence and should be taken into account for dielectric structures at nanoscale dimensions. By incorporating the effects of surface stress, surface piezoelectricity, surface elasticity and surface piezomagneticity, this paper investigates the bending, buckling and free vibration of magneto-electro-elastic (MEE) beams based on the Euler–Bernoulli beam theory. The governing differential equation and its corresponding boundary conditions are derived by Hamilton’s principle. The analytical solutions for the magneto-electro-elastic bending deflection, buckling magnetic potentials and frequency equations of MEE beams are obtained. In contrast to the previously published works, the positive surface stress is found to stiffen the MEE beams, as evidenced by the decrease in the deflections, the increase in the buckling magnet potentials and the increase in the resonant frequencies. Numerical studies show the importance of the surface effects, the electric and magnetic potentials and boundary conditions on the static and dynamic behavior of MEE beams. This work may be of special interest in the design and application of smart composite MEE beams.     

A single variable shear deformable nonlocal theory for transversely loaded micro- and nano-scale rectangular beams

In this paper, a simple single variable shear deformable nonlocal theory for bending of micro- and nano-scale rectangular beams is presented. To incorporate small size effects, the theory uses Eringen’s nonlocal differential constitutive relations. The theory has only one fourth-order governing differential equation involving a single unknown variable. The governing equation and the expressions for the bending moment and shear force of the present theory are strikingly similar to those of nonlocal Euler-Bernoulli Beam Theory (EBT) formulated based on Eringen’s nonlocal elasticity theory. The theory assumes that the axial and lateral displacements have bending and shear components such that the bending components do not contribute towards shear force, and the shear components do not contribute towards bending moment. Also, the chosen displacement functions of the theory give rise to a realistic parabolic transverse shear stress distribution across the beam cross-section. Efficacy of the proposed theory is demonstrated through bending of simply supported, cantilever and clamped-clamped micro- and nano-scale beams of rectangular cross-section. The numerical results obtained by using the present theory are compared with those predicted by other nonlocal first-order and higher-order shear deformation beam theories. The results obtained are quite accurate.     

Frequency equation and mode shape formulae for composite Timoshenko beams

Exact expressions for the frequency equation and mode shapes of composite Timoshenko beams with cantilever end conditions are derived in explicit analytical form by using symbolic computation. The effect of material coupling between the bending and torsional modes of deformation together with the effects of shear deformation and rotatory inertia is taken into account when formulating the theory (and thus it applies to a composite Timoshenko beam). The governing differential equations for the composite Timoshenko beam in free vibration are solved analytically for bending displacements, bending rotation and torsional rotations. The application of boundary conditions for displacement and forces for cantilever end condition of the beam yields the frequency equation in determinantal form. The determinant is expanded algebraically, and simplified in an explicit form by extensive use of symbolic computation. The expressions for the mode shapes are also derived in explicit form using symbolic computation. The method is demonstrated by an illustrative example of a composite Timoshenko beam for which some published results are available.     

Nonlocal theories for bending,buckling and vibration of beams

Various available beam theories, including the Euler–Bernoulli, Timoshenko, Reddy, and Levinson beam theories, are reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived, and variational statements in terms of the generalized displacements are presented. Analytical solutions of bending, vibration and buckling are presented using the nonlocal theories to bring out the effect of the nonlocal behavior on deflections, buckling loads, and natural frequencies. The theoretical development as well as numerical solutions presented herein should serve as references for nonlocal theories of beams, plates, and shells.     

弹性曲梁几何非线性精确模型及其数值解

基于直法线假设,采用轴线可伸长梁的几何非线性理论,建立了弹性曲梁在任意荷载(保守和非保守)作用下的静态大变形数学模型。其中包含了轴线弧长、轴线位移、横截面转角、内力等七个独立未知函数。通过引进变形后的弧长为未知函数,使得问题的求解区间为未变形梁的轴线长度。该模型不仅考虑了轴线伸长,同时精确地考虑了梁的初始曲率对变形的影响以及轴向变形与弯曲变形之间的相互耦合效应。作为应用,采用打靶法计算了悬臂半圆形曲梁在沿轴线均布的切向随动载荷作用下的非线性平面弯曲问题,给出了随载荷参数大范围变化的平衡路径曲线及平衡构形。     

加载频率对悬臂梁振动疲劳特性的影响

研究了加载频率对悬臂梁振动疲劳特性的影响。首先,给三组相同的悬臂梁结构分别施加三种不同频率（悬臂梁的固有频率,略大于固有频率和略小于固有频率）的正弦激励,使其具有相同的初始应力,试验测得应力随循环次数的变化规律;其次,在试验测得应力历程的基础上,计算悬臂梁的疲劳损伤量,研究在相同初始应力下不同加载频率对同一悬臂梁振动疲劳特性的影响;最后,将预估结果与试验测得的固有频率下降量作了对比。结果表明：加载频率对振动疲劳寿命有较大的影响,文中给出的预估结果与试验结果比较吻合     

Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part I: theoretical formulations

The current work suggests mathematical models for the vibration of double-walled carbon nanotubes (DWCNTs) subjected to a moving nanoparticle by using nonlocal classical and shear deformable beam theories. The van der Waals interaction forces between atoms of the innermost and outermost tubes are modeled by an elastic layer. The equations of motion are derived for the nonlocal double body Euler?CBernoulli, Timoshenko and higher-order beams connected by a flexible layer under excitation of a moving nanoparticle. Analytical solutions of the problem are provided for the aforementioned nonlocal beam models with simply supported boundary conditions. The dynamical deflections and nonlocal bending moments of the innermost and outermost tubes are then obtained during the courses of excitation and free vibration. Finally, the critical velocities of the moving nanoparticle associated with the nonlocal beam theories are expressed in terms of small-scale effect parameter, geometry, and material properties of DWCNTs.     

On the determination of natural frequencies of a cantilever beam in free bending vibration: a rigid multibody approach

A new approximate method for the determination of natural frequencies of a cantilever beam in free bending vibration by a rigid multibody system is proposed. Uniform Euler-Bernoulli cantilever beams with and without a lumped mass at the tips are considered. The modelling method consists of two steps. In the first step, the cantilever beam is replaced by lumped masses interconnected by massless flexible beams. In the second step, the massless flexible beams are replaced by massless rigid beams connected through revolute and prismatic joints with corresponding springs in them. Elastic properties of the massless flexible beams are modelled by the springs introduced. The method proposed is compared with similar ones in the literature.     

基于Bessel和Meijer-G函数的楔形和锥形悬臂梁振动分析

基于欧拉-伯努利梁理论,利用Lagrange法建立了楔形和锥形截面梁在外激作用下的非线性微分方程.提出了一种基于Bessel函数和Meijer-G函数线性组合的无需迭代及近似截断的振型函数,且该振型函数不依赖于楔形和锥形变截面梁的弯曲振动的运动方程是否为标准的Bessel形式,该方法能快速求解线性基频和模态函数.随后将...     

Free vibration analysis of pre-twisted rotating FGM beams

The natural frequencies of vibration of a rotating pre-twisted functionally graded cantilever beam are investigated. Rotating cantilever beam with pre-twist made of a functionally gradient material (FGM) consisting of metal and ceramic is considered for the study. The material properties of the FGM beam symmetrically vary continuously in thickness direction from core at mid section to the outer surfaces according to a power-law form. Equations of motion for free vibration are derived using Lagrange’s equation and the natural frequencies are determined using Rayleigh–Ritz method. The effect of parameters such as the pre-twist angle, power law index, hub radius and rotational speed on the natural frequencies of rotating functionally graded pre-twisted cantilever beams are examined through numerical studies and comparison is made with the numerical results obtained using other methods reported in literature. The effect of coupling between chordwise and flapwise bending modes on the natural frequencies has also been investigated.     

Symmetrical PolyMUMPs-Based Piezoresistive Microcantilever Sensors With On-Chip Temperature Compensation for Microfluidics Applications

Microelectromechanical systems (MEMS)-based cantilever beam sensors for microfluidics applications with on-chip temperature sensors for temperature drift compensation were developed. The stress induced on gold surface with polysilicon piezoresistive sensing is demonstrated. In principle, adsorption of biochemical species on a functionalized surface of the microfabricated cantilever will cause surface stress and, consequently, cantilever bending. The sensing mechanism relies on the piezoresistive properties of the doped polysilicon wire encapsulated in the beam. The beam is constructed through multiusers MEMS Process (PolyMUMPs) foundry with postprocessing silicon etching. Bending analysis is performed so that the beam tip deflection can be predicted. The piezoresistor designs on the beams were varied, within certain constraints, so that the sensitivity of the sensing technique could be measured by external read-out circuit. The mass detection of 0.0058-0.0110 g is measured by the beam resistor series as a balanced Wheatstone bridge configuration. The voltage output of the bridge is directly proportional to the amount of bending in the MEMS cantilever. The temperature dependency and sensor performance have been characterized in experiments. Compensation by resisters on the substrate significantly reduces the temperature dependence.     

非均匀热载荷作用下功能梯度梁的非线性静态响应

由于功能梯度材料结构沿厚度方向的非均匀材料特性,使得夹紧和简支条件的功能梯度梁有着相当不同的行为特征。该文给出了热载荷作用下,功能梯度梁非线性静态响应的精确解。基于非线性经典梁理论和物理中面的概念导出了功能梯度梁的非线性控制方程。将两个方程化简为一个四阶积分-微分方程。对于两端夹紧的功能梯度梁,其方程和相应的边界条件构成微分特征值问题;但对于两端简支的功能梯度梁,由于非齐次边界条件,将不会得到一个特征值问题。导致了夹紧与简支的功能梯度梁有着完全不同的行为特征。直接求解该积分-微分方程,得到了梁过屈曲和弯曲变形的闭合形式解。利用这个解可以分析梁的屈曲、过屈曲和非线性弯曲等非线性变形现象。最后,利用数值结果研究了材料梯度性质和热载荷对功能梯度梁非线性静态响应的影响。     

Application of nonlocal continuum models to nanotechnology

A version of nonlocal elasticity theory is employed to develop a nonlocal Benoulli/Euler beam model. Some representative problems are solved to illustrate the magnitude of predicted nonlocal effects. Particular attention is paid to cantilever beams which are often used as actuators in small scale systems.     

The modified couple stress functionally graded Timoshenko beam formulation

In this paper, a size-dependent formulation is presented for Timoshenko beams made of a functionally graded material (FGM). The formulation is developed on the basis of the modified couple stress theory. The modified couple stress theory is a non-classic continuum theory capable to capture the small-scale size effects in the mechanical behavior of structures. The beam properties are assumed to vary through the thickness of the beam. The governing differential equations of motion are derived for the proposed modified couple-stress FG Timoshenko beam. The generally valid closed-form analytic expressions are obtained for the static response parameters. As case studies, the static and free vibration of the new model are respectively investigated for FG cantilever and FG simply supported beams in which properties are varying according to a power law. The results indicate that modeling beams on the basis of the couple stress theory causes more stiffness than modeling based on the classical continuum theory, such that for beams with small thickness, a significant difference between the results of these two theories is observed.     

多晶硅梁的加速疲劳实验

MEMS器件在循环振动载荷作用下,器件可能会发生断裂、软化等疲劳失效现象.本文中选取了以表面工艺加工的多晶硅结构—固支梁与悬臂梁作为实验研究对象,并在微结构梁上利用光刻的方法对两个被测结构分别引入了凹槽和切口两种缺陷形式,并在其上加载循环静电载荷,进行加速疲劳实验.实验利用激光多普勒测振仪测量谐振频率的变化,来表征微梁结构等效弹性模量的改变.实验结果表明,无论固支梁或者悬臂梁,其谐振频率都发生了明显的偏移:固支梁结构初始频率为170.749 kHz,实验后谐振频率增大,偏移量达到15.618 kHz,其相对变化量为9.15%,而悬臂梁结构初始频率为112.357 kHz,实验后谐振频率变小,减小量达到1.342 kHz,相对偏移为1.34%,器件性能发生明显退化.     

Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model

Double-walled carbon nanotubes (DWCNTs) are being investigated for use as latent materials for drug carriers. However, the surface effects cannot be ignored when drugs or other functional materials, such as nickel or silver, adhere to the surface of the outer tube of a DWCNT. In this paper, the vibrational frequency of DWCNTs, while accounting for surface effects, is studied using the nonlocal Timoshenko beam model. The influence of the surface elasticity modulus, residual surface stress, nonlocal parameter, axial half-wave number and aspect ratio are investigated in detail. The results show that the vibrational frequency is significantly affected by the surface material, nonlocal parameter, vibration mode and aspect ratio. In short DWCNTs on condition of higher vibrational modes, the influences of the surface and nonlocal effects on vibration are more pronounced.     

Nonlinear analysis of forced vibration of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic nanobeams

This paper deals with the forced vibration behavior of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic (METE) nanobeams based on the nonlocal elasticity theory in conjunction with the von Kármán geometric nonlinearity. The METE nanobeam is assumed to be subjected to the external electric potential, magnetic potential and constant temperature rise. Based on the Hamilton principle, the nonlinear governing equations and corresponding boundary conditions are established and discretized using the generalized differential quadrature (GDQ) method. Thereafter, using a Galerkin-based numerical technique, the set of nonlinear governing equations is reduced into a time-varying set of ordinary differential equations of Duffing type. The pseudo-arc length continuum scheme is then adopted to solve the vectorized form of nonlinear parameterized equations. Finally, a comprehensive study is conducted to get an insight into the effects of different parameters such as nonlocal parameter, slenderness ratio, initial electric potential, initial external magnetic potential, temperature rise and type of boundary conditions on the natural frequency and forced vibration characteristics of METE nanobeams.     

Size-dependent dynamic modeling and vibration analysis of MEMS/NEMS-based nanomechanical beam based on the nonlocal elasticity theory

Accurate modeling and analysis of micro-/nanoelectromechanical systems (MEMS/NEMS) has an immense contribution in identification and improvement of the performance of such systems. This article investigates a nonclassical formulation for dynamicmodeling and vibration analysis of a piezo-actuatedmicrocantilever considering the Euler–Bernoulli beam model. Regarding the size effects in micro- to nanoscales, the size-dependent nonlocal continuum theory is employed to derive dynamic equations of the nonclassical microbeam taking into account the beam discontinuities. The nonlocal formulation of the beam and piezoelectric actuator is taken into consideration. Furthermore, the size effects on the resonant vibration characteristics of the beam are studied and some results are obtained. The results illustrate the size-dependent behavior of the beam particularly at higher resonant modes of vibrations. Also, it is indicated that the nonlocality and piezoelectric characteristics have a significant influence on the resonance behavior of the beam. However, this effect is more significant at higher resonant modes of vibrations.     

Differential quadrature method for nonlocal nonlinear vibration analysis of a boron nitride nanotube using sinusoidal shear deformation theory

This article presents a nonlocal sinusoidal shear deformation beam theory (SDBT) for the nonlinear vibration of single-walled boron nitride nanotubes (SWBNNTs). The surrounding elastic medium is simulated based on nonlinear Pasternak foundation. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the SWBNNTs are derived using Hamilton's principle. Differential quadrature method (DQM) for the nonlinear frequency is presented, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory (TBT). The effects of nonlocal parameter, vibrational modes, length, and elastic medium on the nonlinear frequency of SWBNNTs are considered.     

Effect of nonlocal elasticity on the performance of a flexoelectric layer as a distributed actuator of nanobeams

The paper is concerned with the development of finite element model for the static analysis of smart nanobeams integrated with a flexoelectric layer on its top surface, using nonlocal elastic theory. The flexoelectric layer acts as a distributed actuator of the nanobeam. A layerwise displacement theory has been used to derive the element stiffness matrices from variational principles incorporating nonlocal effects. The finite element model for nonlocal response of the beams has been validated with the exact solution for the case of a simply supported standalone flexoelectric layer. Also, the finite element model of the simply supported smart beam has been validated with exact solutions and numerical models for the local elastic case. The performance of the flexoelectric actuator has been compared for different values of nonlocal parameters and different combinations of nonlocal and local elastic substrate and flexoelectric layer. Further, the model developed has been utlized for investigating the performance of the active flexoelectric layer in case of cantilever beam, for which the exact solutions are not available.