Forced oscillations and stability analysis of a nonlinear micro-rotating shaft incorporating a non-classical theory

In this paper, the stability and bifurcation analysis of symmetrical and asymmetrical micro-rotating shafts are investigated when the rotational speed is in the vicinity of the critical speed. With the help of Hamilton’s principle, nonlinear equations of motion are derived based on non-classical theories such as the strain gradient theory. In the dynamic modeling, the geometric nonlinearities due to strains, and strain gradients are considered. The bifurcations and steady state solution are compared between the classical theory and the non-classical theories. It is observed that using a non-classical theory has considerable effect in the steady-state response and bifurcations of the system. As a result, under the classical theory, the symmetrical shaft becomes completely stable in the least damping coefficient, while the asymmetrical shaft becomes completely stable in the highest damping coefficient. Under the modified strain gradient theory, the symmetrical shaft becomes completely stable in the least total eccentricity, and under the classical theory the asymmetrical shaft becomes completely stable in the highest total eccentricity. Also, it is shown that by increasing the ratio of the radius of gyration per length scale parameter, the results of the non-classical theory approach those of the classical theory.     

Size-dependent vibration characteristics of fluid-conveying microtubes

In this paper, a new theoretical model is developed, based on the modified couple stress theory, for the vibration analysis of fluid-conveying microtubes by introducing one internal material length scale parameter. Using Hamilton's principle, the equations of motion of fluid-conveying microtubes are derived. After discretization via the Differential Quadrature Method (DQM), the analytical model exhibits some essential vibration characteristics. For a microtube in which both ends are supported, it is found that the natural frequencies decrease with increasing internal flow velocities. It is also shown that the microtube will become unstable by divergence at a critical flow velocity. More significantly, when the outside diameter of the microtube is comparable to the material length scale parameter, the natural frequencies obtained using the modified couple stress theory are much larger than those obtained using the classical beam theory. It is not surprising, therefore, that the critical flow velocities predicted by the modified couple stress theory are generally higher than those predicted by the classical beam theory.     

基于等几何有限元法的功能梯度微板热力耦合屈曲预测

基于修正偶应力理论和Kirchhoff板理论,建立了功能梯度微板热力耦合屈曲等几何有限元模型。该模型仅包含一个材料尺度参数,能够描述尺度效应现象,且满足修正偶应力理论的高阶连续性要求。基于虚功原理推导了功能梯度微板热力耦合屈曲等几何有限元方程。通过对板的典型算例分析,讨论了材料尺度参数、边长比及梯度指数对板稳定性的影响。结果表明,本文模型预测的屈曲载荷总是大于宏观理论的结果,即捕捉到了尺度效应现象;随着临界屈曲力的增加,临界屈曲热载荷线性减少;此外,边长比和梯度指数也对微板的稳定性产生一定影响。     

基于修正偶应力和高阶剪切变形理论的变截面微梁的自由振动

基于修正偶应力和高阶剪切理论建立了仅含有一个尺度参数的Reddy变截面微梁的自由振动模型,研究了变截面微梁自由振动问题的尺度效应和横向剪切变形对自振频率计算的影响。基于哈密顿原理推导了动力学方程与边界条件,并采用微分求积法求解了各种边界条件下的自振频率。算例结果表明,基于偶应力理论预测的变截面微梁的自振频率均大于经典梁理论的预测结果,即捕捉到了尺度效应。另外,梁的几何尺寸与尺度参数越接近,尺度效应就越明显,而梁的长细比越小,横向剪切变形对自振频率的影响就越明显。     

Buckling and postbuckling of size-dependent cracked microbeams based on a modified couple stress theory

The elastic buckling analysis and the static postbuckling response of the Euler–Bernoulli microbeams containing an open edge crack are studied based on a modified couple stress theory. The cracked section is modeled by a massless elastic rotational spring. This model contains a material length scale parameter and can capture the size effect. The von Kármán nonlinearity is applied to display the postbuckling behavior. Analytical solutions of a critical buckling load and the postbuckling response are presented for simply supported cracked microbeams. This parametric study indicates the effects of the crack location, crack severity, and length scale parameter on the buckling and postbuckling behaviors of cracked microbeams.     

Pull-in analysis of electrically actuated viscoelastic microbeams based on a modified couple stress theory

A new beam model is developed for the viscoelastic microbeam based on a modified couple stress model which contains only one material length scale parameter. The governing equations of equilibrium together with initial conditions and boundary conditions are obtained by a combination of the basic equations of modified couple stress theory and Hamilton’s principle. This new beam model is then used for an electrically actuated microbeam-based MEMS structure. The dynamic and quasi-static governing equations of an electrically actuated viscoelastic microbeam are firstly given where the axial force created by the midplane stretching effect is also considered. Galerkin method is used to solve above equation and this method is also validated by the finite element method (FEM) when our model is reduced into an elastic case. The numerical results show that the instantaneous pull-in voltage, durable pull-in voltage and pull-in delay time predicted by this newly developed model is larger (longer) than that predicted by the classical beam model. A comparison between the quasi-static model results and the dynamic model results is also given.     

Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges

This study investigates the size-dependent quasistatic response of a nonlinear viscoelastic microelectromechanical system (MEMS) under an electric actuation. To have this problem in view, the deformable electrode of the MEMS is modelled using cantilever and doubly-clamped viscoelastic microbeams. The modified couple stress theory in conjunction with Bernoulli–Euler beam theory are used for mathematical modeling of the size-dependent instability of microsystems in the framework of linear viscoelastic theory. Simultaneous effect of electrostatic actuation including fringing field, residual stress, mid-plane stretching and Casimir and van der Waals intermolecular forces are considered in the theoretical model. A single element of the standard linear solid element is used to simulate the viscoelastic behavior. Based on the extended Hamilton’s variational principle, the nonlinear governing integro-differential equation and boundary conditions are derived. Thereafter, a new generalized differential-integral quadrature solution for the nonlinear quasistatic response of electrically actuated viscoelastic micro/nanobeams under two different boundary conditions; doubly-clamped microbridge and clamped-free microcantilever. The developed model is verified and a good agreement is obtained. Finally, a comprehensive study is conducted to investigate the effects of various parameters such as material relaxation time, durable modulus, material length scale parameter, Casimir force, van der Waals force, initial gap and beam length on the pull-in response of viscoelastic microbridges and microcantilevers in the framework of viscoelasticity.     

Hopf bifurcation analysis of asymmetrical rotating shafts

The Hopf and double Hopf bifurcations analysis of asymmetrical rotating shafts with stretching nonlinearity are investigated. The shaft is simply supported and is composed of viscoelastic material. The rotary inertia and gyroscopic effect are considered, but, shear deformation is neglected. To consider the viscoelastic behavior of the shaft, the Kelvin–Voigt model is used. Hopf bifurcations occur due to instability caused by internal damping. To analyze the dynamics of the system in the vicinity of Hopf bifurcations, the center manifold theory is utilized. The standard normal forms of Hopf bifurcations for symmetrical and asymmetrical shafts are obtained. It is shown that the symmetrical shafts have double zero eigenvalues in the absence of external damping, but asymmetrical shafts do not have. The asymmetrical shaft in the absence of external damping has a saddle point, therefore the system is unstable. Also, for symmetrical and asymmetrical shafts, in the presence of external damping at the critical speeds, supercritical Hopf bifurcations occur. The amplitude of periodic solution due to supercritical Hopf bifurcations for symmetrical and asymmetrical shafts for the higher modes would be different, due to shaft asymmetry. Consequently, the effect of shaft asymmetry in the higher modes is considerable. Also, the amplitude of periodic solutions for symmetrical shafts with rotary inertia effect is higher than those of without one. In addition, the dynamic behavior of the system in the vicinity of double Hopf bifurcation is investigated. It is seen that in this case depending on the damping and rotational speed, the sink, source, or saddle equilibrium points occur in the system.     

基于新修正偶应力理论的Mindlin层合板稳定性分析

基于新的各向异性修正偶应力理论提出一个Mindlin复合材料层合板稳定性模型。该理论包含纤维和基体两个不同的材料长度尺度参数。不同于忽略横向剪切应力的修正偶应力Kirchhoff薄板理论,Mindlin层合板考虑横向剪切变形引入两个转角变量。进一步建立了只含一个材料细观参数的偶应力Mindlin层合板工程理论的稳定性模型。计算了正交铺设简支方板Mindlin层合板的临界载荷。计算结果表明该模型可以用于分析细观尺度层合板稳定性的尺寸效应。     

Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory

In this paper, a size-dependent first-order shear deformable shell model is developed based upon the modified strain gradient theory (MSGT) for the axial buckling analysis of functionally graded (FG) circular cylindrical microshells. It is assumed that the material properties of FG materials, which obey a simple power-law distribution, vary through the thickness direction. The principle of virtual work is utilized to formulate the governing equations and corresponding boundary conditions. Numerical results are presented for the axial buckling of FG circular cylindrical microshells subject to simply-supported end conditions and the effects of material length scale parameter, material property gradient index, length-to-radius ratio and circumferential mode number on the size-dependent critical buckling load are extensively studied. For comparison purpose, the critical buckling loads predicted by modified couple stress theory (MCST) and classical theory (CT) are also presented. Results show that the size effect plays an important role for lower values of dimensionless length scale parameter. Moreover, it is observed that the critical buckling loads obtained based on MSGT are greater than those obtained based on MCST and CT.     

Nonlinear bending of size-dependent circular microplates based on the modified couple stress theory

The present study proposes a nonclassical Kirchhoff plate model for the axisymmetrically nonlinear bending analysis of circular microplates under uniformly distributed transverse loads. The governing differential equations are derived from the principle of minimum total potential energy based on the modified couple stress theory and von Kármán geometrically nonlinear theory in terms of the deflection and radial membrane force, with only one material length scale parameter to capture the size-dependent behavior. The governing equations are firstly discretized to a set of nonlinear algebraic equations by the orthogonal collocation point method, and then solved numerically by the Newton–Raphson iteration method to obtain the size-dependent solutions for deflections and radial membrane forces. The influences of length scale parameter on the bending behaviors of microplates are discussed in detail for immovable clamped and simply supported edge conditions. The numerical results indicate that the microplates modeled by the modified couple stress theory causes more stiffness than modeled by the classical continuum plate theory, such that for plates with small thickness to material length scale ratio, the difference between the results of these two theories is significantly large, but it becomes decreasing or even diminishing with increasing thickness to length scale ratio.     

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.     

A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model

A new modified couple stress theory for anisotropic elasticity is proposed. This theory contains three material length scale parameters. Differing from the modified couple stress theory, the couple stress constitutive relationships are introduced for anisotropic elasticity, in which the curvature (rotation gradient) tensor is asymmetric and the couple stress moment tensor is symmetric. However, under isotropic case, this theory can be identical to modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The differences and relations of standard, modified and new modified couple stress theories are given herein. A detailed variational formulation is provided for this theory by using the principle of minimum total potential energy. Based on the new modified couple stress theory, composite laminated Kirchhoff plate models are developed in which new anisotropic constitutive relationships are defined. The First model contains two material length scale parameters, one related to fiber and the other related to matrix. The curvature tensor in this model is asymmetric; however, the couple stress moment tensor is symmetric. Under isotropic case, this theory can be identical to the modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The present model can be viewed as a simplified couple stress theory in engineering mechanics. Moreover, a more simplified model of couple stress theory including only one material length scale parameter for modeling the cross-ply laminated Kirchhoff plate is suggested. Numerical results show that the proposed laminated Kirchhoff plate model can capture the scale effects of microstructures.     

VIBRATION ANALYSIS OF MICROSCALE PLATES BASED ON MODIFIED COUPLE STRESS THEORY

A non-classical Kirchhoff plate model is developed for the dynamic analysis of microscale plates based on the modified couple stress theory in which an internal material length scale parameter is included. Unlike the classical Kirchhoff plate model, the newly developed model can capture the size effect of microscale plates. Two boundary value problems of rectangular micro- plates are solved and the size effect on the lowest two natural frequencies is investigated. It is shown that the natural frequencies of the microscale plates predicted by the current model are size-dependent when the plate thickness is comparable to the material length scale parameter.     

Investigation of the size effects in Timoshenko beams based on the couple stress theory

In this paper, a size-dependent Timoshenko beam is developed on the basis of the couple stress theory. The couple stress theory is a non-classic continuum theory capable of capturing the small-scale size effects on the mechanical behavior of structures, while the classical continuum theory is unable to predict the mechanical behavior accurately when the characteristic size of structures is close to the material length scale parameter. The governing differential equations of motion are derived for the couple-stress Timoshenko beam using the principles of linear and angular momentum. Then, the general form of boundary conditions and generally valid closed-form analytical solutions are obtained for the axial deformation, bending deflection, and the rotation angle of cross sections in the static cases. As an example, the closed-form analytical results are obtained for the response of a cantilever beam subjected to a static loading with a concentrated force at its free end. The results indicate that modeling on the basis of the couple stress theory causes more stiffness than modeling by the classical beam theory. In addition, the results indicate that the differences between the results of the proposed model and those based on the classical Euler–Bernoulli and classical Timoshenko beam theories are significant when the beam thickness is comparable to its material length scale parameter.     

非稳态非线性油膜力作用下柔性轴裂纹转子的动力特性分析

分析了在动载轴承非稳态非线性油膜力作用下，具有横向裂纹柔性轴Jeffcott转子在非线性涡动影响下的动力特性。通过数值计算表明，在油膜失稳转速前，随着裂纹轴刚度变化比的增大，系统在低转速区域内具有丰富的非线性动力行为，出现倍周期分叉及混沌现象，涡动振幅随转速升高而减小，直到非稳态非线性油膜失稳，在无裂纹转子油膜临界失稳点处发现了类Hopf分叉现象，系统运动由平衡变为拟周期运动；裂纹转子在油膜临界失稳时的系统运动亦为拟周期运动，裂纹转子轴刚度变化对油膜失稳点及油膜失稳之后转子的运动影响不大，转子系统作拟周期运动。     

基于各向异性修正偶应力理论的Reddy型层合板的自由振动

本文基于各向异性修正偶应力理论建立了只含一个尺度参数的Reddy型复合材料层合板的自由振动模型。同见诸于文献的细观尺度Kirchhoff薄板偶应力模型相比,本文提出的新模型能够更精确的预测细观尺度下的中、厚层合板的自振频率。基于Hamilton原理推导了细观尺度下Reddy型复合材料层合板的运动微分方程以及边界条件,并以正交铺设的四边简支复合材料层合方板为例进行了解析求解,分析了尺度参数对自振频率的影响并对比了Kirchhoff、Mindlin和Reddy等三种板模型计算结果的异同。算例结果表明本文所给出的模型能够捕捉到复合材料层合板自由振动问题的尺度效应。另外,在细观尺度下Kirchhoff板模型所预测的自振频率相对于Mindlin板模型和Reddy板模型总是过高,且越接近厚板三者的差别就越大,这与经典理论中三种板模型的对比情况是一致的。     

Size-dependent vibration of functionally graded curved microbeams based on the modified strain gradient elasticity theory

On the basis of the modified strain gradient elasticity theory, the free vibration characteristics of curved microbeams made of functionally graded materials (FGMs) whose material properties vary in the thickness direction are investigated. A size-dependent first-order shear deformation beam model is developed containing three internal material length scale parameters to incorporate small-scale effect. Through Hamilton’s principle, the higher-order governing equations of motion and boundary conditions are derived. Natural frequencies of FGM curved microbeams corresponding to different mode numbers are evaluated for over a wide range of material property gradient index, dimensionless length scale parameter and aspect ratio. Moreover, the results obtained via the present non-classical first-order shear deformation beam model are compared with those of degenerated beam models based on the modified couple stress and the classical theories. It is found that the difference between the natural frequencies predicted by the various beam models is more significant for lower values of dimensionless length scale parameter and higher values of mode number.     

Influence of size effect and elastic boundary condition on the pull-in instability of nano-scale cantilever beams immersed in liquid electrolytes

In this study, the static pull-in instability of nanocantilever beams immersed in a liquid electrolyte is theoretically investigated. In modeling the nanocantilever beam, the effects of van der Waals forces, elastic boundary condition and size dependency are considered. The modified couple stress theory, containing material length scale parameter, is used to interpret the size effect which appears in micro/nanoscale structures. The modified Adomian decomposition (MAD) method is used to gain an approximate analytical expression for the critical pull-in parameters which are essential for the design of micro/nanoactuators. The results show that the beam can deflect upward or downward, based on the values of the non-dimensional parameters. It is found that the size effect greatly influences the beam deflection and is more noticeable for small thicknesses. Neglecting size effect overestimates the deflection of the nanobeam. The findings reveal that the increase of ion concentration increases the pull-in voltage but decreases the pull-in deflection. Furthermore, an increase in ion concentration increases the influence of size-dependent effect on pull-in voltage.     

A size-dependent Reddy–Levinson beam model based on a strain gradient elasticity theory

A size-dependent Reddy–Levinson beam model is developed based on a strain gradient elasticity theory. Governing equations and boundary conditions are derived by using Hamilton’s principle. The model contains three material length scale parameters, which may effectively capture the size effect in micron or sub-micron. This model can degenerate into the modified couple stress model or even the classical model if two or all material length scale parameters are taken to be zero respectively. In addition, the present model recovers the micro scale Timoshenko and Bernoulli–Euler beam models based on the same strain gradient elasticity theory. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Reddy–Levinson beam are solved respectively; the results are compared with the reduced models. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Reddy–Levinson models are getting larger as the beam thickness is comparable to the material length scale parameters. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. This study may be helpful to characterize the mechanical properties of small scale beam-like structures for a wide range of potential applications.