Frequency equations of nonlocal elastic micro/nanobeams with the consideration of the surface effects

A nonlocal elastic micro/nanobeam is theoretically modeled with the consideration of the surface elasticity, the residual surface stress, and the rotatory inertia, in which the nonlocal and surface effects are considered. Three types of boundary conditions, i.e., hinged-hinged, clamped-clamped, and clamped-hinged ends, are examined. For a hinged-hinged beam, an exact and explicit natural frequency equation is derived based on the established mathematical model. The Fredholm integral equation is adopted to deduce the approximate fundamental frequency equations for the clamped-clamped and clamped-hinged beams. In sum, the explicit frequency equations for the micro/nanobeam under three types of boundary conditions are proposed to reveal the dependence of the natural frequency on the effects of the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia, providing a more convenient means in comparison with numerical computations.     

Enhancement of elementary beam theories in order to obtain exact solutions for elastic rectangular beams

Boley's method is utilized in order to show that the elementary Bernoulli–Euler beam theory can be enhanced such that exact solutions of the plane-stress theory of linear elasticity are obtained for force loaded rectangular beams. An equivalent enhancement is derived for the elementary Timoshenko theory of beams. The enhancement terms act analogous to thermal loadings; they follow from the force loading of the rectangular beam in an explicit form. The resulting boundary value problem of fourth order can be efficiently solved by means of symbolic computer codes. As an illustrative example, a redundant beam is studied, which is simply supported at one end, and which is clamped at the other end. Outcomes for three alternative clamped end boundary conditions are compared.     

Refined theories may be needed for vibration analysis of structures with overhang

The effect of shear deformation and rotary inertia terms on the free vibration of a beam with overhang was investigated. A recently proposed modified Timoshenko-type equations of motion were used to analyze the vibration of the structure. Two different sets of boundary conditions, with either a fixed or hinged end support, were studied. The results were compared with those obtained for the classical Bernoulli–Euler beam theory. The comparison shows that for a hinged end beam with very long overhang, where the span between the supports is less than one tenth of the overall beam length, the classical theory significantly overestimates the values of the fundamental natural frequencies, even for isotropic shear rigidity. On the other hand, the span effect is reversed for the clamped end beam, for which a relatively significant difference between the classical theory and shear theory results may occur only for a long span. For transversely isotropic beams, the refined theory predictions of the fundamental natural frequencies may be much smaller than those obtained through the rigid shear theory, especially for short span hinged end beams and long span clamped end beams.     

多孔功能梯度材料Timoshenko梁的自由振动分析

基于Timoshenko梁理论研究多孔功能梯度材料梁(FGMs)的自由振动问题.首先,考虑多孔功能梯度材料梁的孔隙率模型,建立了两种类型的孔隙分布.其次,基于Timoshenko梁变形理论,给出位移场方程、几何方程和本构方程,利用Hamilton原理推导多孔功能梯度材料梁的自由振动控制微分方程,并进行无量纲化,然后应用微分变换法(DTM)对无量纲控制微分方程及其边界条件进行变换,得到含有固有频率的等价代数特征方程.最后,计算了固定-固定(C-C)、固定-简支(C-S)和简支-简支(S-S)三种不同边界下多孔功能梯度材料梁自由振动的无量纲固有频率.将其退化为均匀材料与已有文献数据结果对照,验证了正确性.讨论了孔隙率、细长比和梯度指数对多孔功能梯度材料梁无量纲固有频率的影响.     

Analytical solutions for buckling of size-dependent Timoshenko beams

The inconsistences of the higher-order shear resultant expressed in terms of displacement(s) and the complete boundary value problems of structures modeled by the nonlocal strain gradient theory have not been well addressed. This paper develops a size-dependent Timoshenko beam model that considers both the nonlocal effect and strain gradient effect. The variationally consistent boundary conditions corresponding to the equations of motion of Timoshenko beams are reformulated with the aid of the weighted residual method. The complete boundary value problems of nonlocal strain gradient Timoshenko beams undergoing buckling are solved in closed forms. All the possible higher-order boundary conditions induced by the strain gradient are selectively suggested based on the fact that the buckling loads increase with the increasing aspect ratios of beams from the conventional mechanics point of view. Then, motivated by the expression for beams with simply-supported(SS) boundary conditions, some semiempirical formulae are obtained by curve fitting procedures.     

Asymptotic frequencies of various damped nonlocal beams and plates

A striking difference between the conventional local and nonlocal dynamical systems is that the later possess finite asymptotic frequencies. The asymptotic frequencies of four kinds of nonlocal viscoelastic damped structures are derived, including an Euler–Bernoulli beam with rotary inertia, a Timoshenko beam, a Kirchhoff plate with rotary inertia and a Mindlin plate. For these undamped and damped nonlocal beam and plate models, the analytical expressions for the asymptotic frequencies, also called the maximum or escape frequencies, are obtained. For the damped nonlocal beams or plates, the asymptotic critical damping factors are also obtained. These quantities are independent of the boundary conditions and hence simply supported boundary conditions are used. Taking a carbon nanotube as a numerical example and using the Euler–Bernoulli beam model, the natural frequencies of the carbon nanotubes with typical boundary conditions are computed and the asymptotic characteristics of natural frequencies are shown.     

Theoretical analysis on elastic buckling of nanobeams based on stress-driven nonlocal integral model

Several studies indicate that Eringen's nonlocal model may lead to some inconsistencies for both Euler-Bernoulli and Timoshenko beams, such as cantilever beams subjected to an end point force and fixed-fixed beams subjected a uniform distributed load. In this paper, the elastic buckling behavior of nanobeams, including both EulerBernoulli and Timoshenko beams, is investigated on the basis of a stress-driven nonlocal integral model. The constitutive equations are the Fredholm-type integral equations of the first kind, which can be transformed to the Volterra integral equations of the first kind. With the application of the Laplace transformation, the general solutions of the deflections and bending moments for the Euler-Bernoulli and Timoshenko beams as well as the rotation and shear force for the Timoshenko beams are obtained explicitly with several unknown constants. Considering the boundary conditions and extra constitutive constraints, the characteristic equations are obtained explicitly for the Euler-Bernoulli and Timoshenko beams under different boundary conditions, from which one can determine the critical buckling loads of nanobeams. The effects of the nonlocal parameters and buckling order on the buckling loads of nanobeams are studied numerically, and a consistent toughening effect is obtained.     

Surface effect on the biaxial buckling and free vibration of FGM nanoplate embedded in visco-Pasternak standard linear solid-type of foundation

In this study, nonlocal elasticity theory in conjunction with Gurtin–Murdoch elasticity theory is employed to investigate biaxial buckling and free vibration behavior of nanoplate made of functionally graded material (FGM) and resting on a visco-Pasternak standard linear solid-type of the foundation. The material characteristics of simply supported FGM nanoplates are assumed to be varied continuously as a power law function of the plate thickness. Hamilton’s principle is implemented to derive the non-classical governing equations of motion and related boundary conditions, which analytically solved to obtain the explicit closed-form expression for complex natural frequencies and buckling loads. Finally, attention is focused on considering the influences of various parameters on variation of damped natural frequency and buckling load ratio such as nonlocal parameter, surface effects, geometric parameters, power law index and properties of visco-Pasternak foundation and it is clearly demonstrated that these factors highly affect on vibration and buckling behavior.     

Stability of fluid conveying nanobeam considering nonlocal elasticity

In this study, the nonlocal Euler–Bernoulli beam theory is employed in the vibration and stability analysis of a nanobeam conveying fluid. The nanobeam is assumed to be traveling with a constant mean velocity along with a small harmonic fluctuation. In the considered analysis, the effects of the small-scale of the nanobeam are incorporated into the equations. By utilizing Hamilton’s principle, the nonlinear equations of motion including stretching of the neutral axis are derived. Damping effect is considered in the analysis. The closed form approximate solution of nonlinear equations is solved by using the multiple scale method, a perturbation technique. The effects of the different value of the nonlocal parameters, mean speed value and ratios of fluid mass to the total mass as well as effects of the simple–simple and clamped–clamped boundary conditions on the linear and nonlinear frequencies, stability, frequency–response curves and bifurcation point are presented numerically and graphically. The solvability conditions are obtained for the three distinct cases of velocity fluctuation frequency. For all cases, the stability areas of system are constructed analytically.     

Vibration of a variable cross-section beam

Vibration of an isotropic beam which has a variable cross-section is investigated. Governing equation is reduced to an ordinary differential equation in spatial coordinate for a family of cross-section geometries with exponentially varying width. Analytical solutions of the vibration of the beam are obtained for three different types of boundary conditions associated with simply supported, clamped and free ends. Natural frequencies and mode shapes are determined for each set of boundary conditions. Results show that the non-uniformity in the cross-section influences the natural frequencies and the mode shapes. Amplitude of vibrations is increased for widening beams while it is decreased for narrowing beams.     

Transverse free vibration of non uniform rotating Timoshenko beams with elastically clamped boundary conditions

In the present paper the Differential Quadrature Method, DQM, and the domain decomposition are used to carry out the free transverse vibration analysis of non-uniform multi-span rotating Timoshenko beams with perfect and not perfect boundary conditions. The cross section could vary in a continuous or discontinuous fashion along the beam length. The material of the beam could be different in each beam span. The influence of elastically clamped boundary conditions at hub end are studied and discussed. The effect of an arbitrary hub radius is considered. The governing differential equations of motion for rotating Timoshenko beams come from the derivation of Hamilton’s principle. The first six natural frequencies of vibration are obtained for many particular situations and for some of them the mode shapes are also available. The examples of applications of the method indicated its effectiveness. The results for particular cases are in excellent agreement with published results and results obtained by means of the finite element method.     

A micro scale Timoshenko beam model based on strain gradient elasticity theory

A micro scale Timoshenko beam model is developed based on strain gradient elasticity theory. Governing equations, initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle. The new model incorporated with Poisson effect contains three material length scale parameters and can consequently capture the size effect. This model can degenerate into the modified couple stress Timoshenko beam model or even the classical Timoshenko beam model if two or all material length scale parameters are taken to be zero respectively. In addition, the newly developed model recovers the micro scale Bernoulli–Euler beam model when shear deformation is ignored. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Timoshenko beam are solved respectively. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Timoshenko models are large as the beam thickness is comparable to the material length scale parameter. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. In addition, Poisson effect on the beam deflection, rotation and natural frequency possesses an interesting “extreme point” phenomenon, which is quite different from that predicted by the classical Timoshenko beam model.     

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.     

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.     

Winkler-Pasternak弹性地基梁自由振动的二维弹性分析Two-dimensional elastic analysis for free vibration of beams set on winkler-pasternak elastic foundations

基于二维线弹性理论,应用Hamilton原理,获得Winkler-Pasternak弹性地基梁自由振动的控制微分方程,应用微分求积法(DQM)数值研究了梁自由振动的无量纲频率特性。计算结果与已有的结果(Bernoulli-Euler梁和Timoshenko梁)比较表明,本文的分析方法对弹性地基长梁和短梁自由振动的研究都有效。最后考虑了几何参数对梁频率的影响,以及不同边界条件下地基系数对频率的影响和收敛性。     

Nonlocal Thermo-Electro-Mechanical coupling vibrations of axially moving piezoelectric nanobeams

This work is concerned with the thermo-electro-mechanical coupling transverse vibrations of axially moving piezoelectric nanobeams which reveal potential applications in self-powered components of biomedical nano-robot. The nonlocal theory and Euler piezoelectric beam model are employed to develop the governing partial differential equations of the mathematical model for axially moving piezoelectric nanobeams. The natural frequencies of nanobeams under simply supported and fully clamped boundary constraints are numerically determined based on the eigenvalue method. Subsequently, some detailed parametric studies are presented and it is shown that the nonlocal nanoscale effect and axial motion effect contribute to reduce the bending rigidity of axially moving piezoelectric nanobeam and hence its natural frequency decreases within the framework of nonlocal elasticity. Moreover, the natural frequency decreases with increasing the positive external voltage, axial compressive force and change of temperature, while increases with increasing the axial tensile force. The critical speed and critical axial compressive force are determined and the dynamical buckling behaviors of axially moving piezoelectric nanobeams are indicated. It is concluded the nonlocal nanoscale parameter plays a remarkable role in the size-dependent natural frequency, critical speed and critical axial compressive force.     

Correction of local elasticity for nonlocal residuals: application to Euler–Bernoulli beams

Complications exist when solving the field equation in the nonlocal field. This has been attributed to the complexity of deriving explicit forms of the nonlocal boundary conditions. Thus, the paradoxes in the existing solutions of the nonlocal field equation have been revealed in recent studies. In the present study, a new methodology is proposed to easily determine the elastic nonlocal fields from their local counterparts without solving the field equation. This methodology depends on the iterative-nonlocal residual approach in which the sum of the nonlocal fields is treaded as a residual field. Thus, in this study the corrections of the local linear and nonlinear elastic fields for the nonlocal residuals in materials are presented. These corrections are formed based on the general nonlocal theory. In the context of the general nonlocal theory, two distinct nonlocal parameters are introduced to form the constitutive equations of isotropic elastic continua. In this study, it is demonstrated that the general nonlocal theory outperforms Eringen’s nonlocal theory in accounting for the impacts of the material’s Poisson’s ratio on its mechanics. To demonstrate the effectiveness of the proposed approach, the corrections of the local static bending, vibration, and buckling characteristics of Euler–Bernoulli beams are derived. Via these corrections, bending, vibration, and buckling behaviors of simple-supported nonlocal Euler–Bernoulli beams are determined without solving the beam’s equation of motion.     

Application of nonlocal beam models for carbon nanotubes

Investigations of wave and vibration properties of single- or multi-walled carbon nanotubes based on nonlocal beam models have been reported recently. However, there are numerous inconsistencies in the handling of the governing equations, applied forces, and boundary conditions based on some of the reported nonlocal beam models. In this paper, the consistent equations of motion for the nonlocal Euler and Timoshenko beam models are provided, and some issues on the nonlocal beam theories are discussed. The models are then applied to the studies of wave properties of single- and double-walled nanotubes. The wave and vibration properties of the nanotubes based on the presented nonlocal beam equations are studied, and scale effects are discussed.     

Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories

The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.     

Winkler-Pasternak弹性地基梁自由振动的二维弹性分析

基于二维线弹性理论,应用Hamilton原理,获得Winkler-Pasternak弹性地基梁自由振动的控制微分方程,应用微分求积法(DQM)数值研究了梁自由振动的无量纲频率特性。计算结果与已有的结果(Bernoulli-Euler梁和Timoshenko梁)比较表明,本文的分析方法对弹性地基长梁和短梁自由振动的研究都有效。最后考虑了几何参数对梁频率的影响,以及不同边界条件下地基系数对频率的影响和收敛性。