共查询到20条相似文献,搜索用时 672 毫秒
1.
Nonlinear beam formulation incorporating surface energy and size effect: application in nano-bridges
A nonlinear beam formulation is presented based on the Gurtin-Murdoch surface elasticity and the modified couple stress theory. The developed model theoretically takes into account coupled effects of the energy of surface layer and microstructures sizedependency. The mid-plane stretching of a beam is incorporated using von-Karman nonlinear strains. Hamilton’s principle is used to determine the nonlinear governing equation of motion and the corresponding boundary conditions. As a case study, pull-in instability of an electromechanical nano-bridge structure is studied using the proposed formulation. The nonlinear governing equation is solved by the analytical reduced order method (ROM) as well as the numerical solution. Effects of various parameters including surface layer, size dependency, dispersion forces, and structural damping on the pullin parameters of the nano-bridges are discussed. Comparison of the results with the literature reveals capability of the present model in demonstrating the impact of nanoscale phenomena on the pull-in threshold of the nano-bridges. 相似文献
2.
An approximate analytical model for calculating the pull-in voltage of a stepped cantilever-type radio frequency(RF) micro electro-mechanical system(MEMS) switch is developed based on the Euler-Bernoulli beam and a modified couple stress theory, and is validated by comparison with the finite element results. The sensitivity functions of the pull-in voltage to the designed parameters are derived based on the proposed model. The sensitivity investigation shows that the pull-in voltage sensitivities increase/decrease nonlinearly with the increases in the designed parameters. For the stepped cantilever beam, there exists a nonzero optimal dimensionless length ratio, where the pull-in voltage is insensitive. The optimal value of the dimensionless length ratio only depends on the dimensionless width ratio, and can be obtained by solving a nonlinear equation. The determination of the designed parameters is discussed, and some recommendations are made for the RF MEMS switch optimization. 相似文献
3.
J.T. Katsikadelis G.C. Tsiatas 《Archive of Applied Mechanics (Ingenieur Archiv)》2005,74(11-12):790-799
In this paper, shape optimization is used to optimize the buckling load of a Euler–Bernoulli beam having constant volume. This is achieved by varying appropriately the beam cross section so that the beam buckles with the maximum or a prescribed buckling load. The problem is reduced to a nonlinear optimization problem under equality and inequality constraints as well as specified lower and upper bounds. The evaluation of the objective function requires the solution of the buckling problem of a beam with variable stiffness subjected to an axial force. This problem is solved using the analog equation method for the fourth-order ordinary differential equation with variable coefficients. Besides its accuracy, this method overcomes the shortcomings of a possible FEM solution, which would require resizing of the elements and recomputation of their stiffness properties during the optimization process. Several example problems are presented that illustrate the method and demonstrate its efficiency. 相似文献
4.
IntroductionSince 1 958,Levsionandseveralotherauthors[1]- [4 ]havemadeasubstantialstudyforexistenceandestimateofsolutionsforthethird_ordersingularlyperturbedboundaryvalueproblemwithtwopointboundaryconditionsx( 0 ) =A ,x′( 0 ) =B ,x′( 1 ) =C ,orslightlyextensivelylinearbo… 相似文献
5.
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. 相似文献
6.
基于应变梯度理论和哈密顿原理,并考虑卡西米尔力的影响,建立了静电激励纳米机电系统(NEMS)的尺寸效应模型,并得到模型的控制方程和边界条件。然后,引入广义微分求积法和拟弧长算法,得到模型的数值解。结果表明,当考虑卡西米尔力的影响时,系统两极的吸合电压有所减小。并且,当系统尺寸达到一个临界值时(即两电极间距小于“最小间距”,或可变形电极长度超过“拉起长度”),系统会在没有外加电压的作用下自动发生吸合,这将为NEMS的优化设计和定量分析提供理论基础。 相似文献
7.
The main aim of this paper is to contribute to the construction of Green’s functions for initial boundary value problems for fourth order partial differential equations. In this paper, we consider a transversely vibrating homogeneous semi-infinite beam with classical boundary conditions such as pinned, sliding, clamped or with a non-classical boundary conditions such as dampers. This problem is of important interest in the context of the foundation of exact solutions for semi-infinite beams with boundary damping. The Green’s functions are explicitly given by using the method of Laplace transforms. The analytical results are validated by references and numerical methods. It is shown how the general solution for a semi-infinite beam equation with boundary damping can be constructed by the Green’s function method, and how damping properties can be obtained. 相似文献
8.
The pull-in instability of a cantilever nano-actuator model incorporating the effects of the surface, the fringing field, and the Casimir attraction force is investigated. A new quartic polynomial is proposed as the shape function of the beam during the deflection, satisfying all of the four boundary values. The Gaussian quadrature rule is used to treat the involved integrations, and the design parameters are preserved in the evaluated formulas. The analytic expressions are derived for the tip deflection and pull-in parameters of the cantilever beam. The micro-electromechanical system (MEMS) cantilever actuators and freestanding nanoactuators are considered as two special cases. It is proved that the proposed method is convenient for the analyses of the effects of the surface, the Casimir force, and the fringing field on the pull-in parameters. 相似文献
9.
In this study, stability and bimodal optimization of clamped beam elastically restrained against translation on one end subjected to a constant axially load are analyzed. The beam is positioned on elastic Winkler type foundation. The Euler method of adjacent equilibrium configuration is used in deriving the nonlinear governing equations. The critical load parameters, axial force and stiffness of foundation, are obtained for beam with the unit cross-sectional area.The shape of the beam stable against buckling that has minimal volume is determined by using Pontryagin’s maximum principle. The optimality conditions for the case of bimodal optimization are derived. The cross-sectional area for optimally designed beam is found from the solution of a nonlinear boundary value problem. New numerical results are obtained. A first integral (Hamiltonian) is used to monitor accuracy of integration. It is shown that there is the saving in material for the same buckling force. 相似文献
10.
11.
Steady flow of a viscous incompressible fluid in a channel, driven by suction or injection of the fluid through the channel walls, is investigated. The velocity equation of this problem is reduced to nonlinear ordinary differential equation with two boundary conditions by appropriate transformation and convert the two‐point boundary‐value problem for the similarity function into an initial‐value problem in which the position of the upper channel. Then obtained differential equation is solved analytically using differential transformation method and compare with He's variational iteration method and numerical solution. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
12.
SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEM FOR A KIND OF VOLTERRA TYPE FUNCTIONAL DIFFERENTIAL EQUATION 总被引:2,自引:0,他引:2
鲁世平 《应用数学和力学(英文版)》2003,24(12):1441-1449
IntroductionThereweresomeresultsofstudyingonboundaryvalueproblemsforfunctionaldifferentialequation[1~6 ]byemployingthetoplolgicaldegreetheoryandsomefixedpointprinciplesinrecentyears.Buttheworktostudyboundaryvalueproblemsfordelaydifferentialequationwithsmallparameterbymeansofthetheoryofsingularperturbationrarelyappeared[7~11].Thereasonforitisthattheworktoconstructtheuppersolutionandlowersolutionforthecaseofdifferentialequationwithdelayisdifficult.Theauthorhasstudiedakindofboundaryvalueproblem… 相似文献
13.
An investigation is performed into the nonlinear pull-in behavior of a cantilever-type nano-mechanical electrostatic actuator. In performing the analysis, the actuator is modeled as an Euler–Bernoulli beam and the influence of surface effects, the fringing field effect and the Casimir force effect are taken into explicit account. In general, analyzing the dynamic behavior of nanoscale electrostatic devices is challenging due to the nonlinear coupling of the electrostatic force and Casimir force. In the present study, this problem is resolved by using a hybrid computational scheme comprising the differential transformation method and the finite difference approximation technique. The feasibility of the proposed approach is demonstrated by the two cantilever-type micro-beams when actuated by a DC voltage. The numerical results show that the present results for the pull-in voltage deviate by no more than 1.47% from those presented in the literature using a different scheme. In addition, it is shown that surface effects play a significant role in determining the static deflection and pull-in voltage of the cantilever beam nano-beam. In general, the results confirm that the hybrid differential transformation/finite difference approximation method provides an accurate and computationally efficient means of simulating the nonlinear electrostatic behavior of nanostructure systems. 相似文献
14.
An exact analytical solution is presented for the laminar boundary-layer flow over a semi-infinite flat plate subjected to a type of similarity preserving suction. The solution is developed for the case of a plate immersed in either a uniform compressible stream with viscosity proportional to temperature or a uniform incompressible stream with constant viscosity. The problem is formulated in Crocco's variables. It is described by a second-order, non-linear, ordinary differential (and singular) boundary-value problem for the shear stress as a function of the velocity in the boundary layer. A unique solution is shown to exist and to possess a power series representation for all magnitudes of suction. The series is constructed explicitly and provides a transcendental equation for the shear stress at the plate (the important skin friction) which can be solved to any desired accuracy. Examples of upper and lower bounds for the wall shear are presented for several magnitudes of suction and confirm the reasonable accuracy of results obtained heretofore only by numerical solutions of the problem. In addition to the intrinsic value of the technique developed, it can be the basis of accurate checks for the numerical solution of more complex problems. 相似文献
15.
Michaël Peigney 《Journal of the mechanics and physics of solids》2008,56(2):360-375
New upper bounds are proposed for a generic problem of geometric compatibility, which covers the problem of bounding the effective recoverable strains in composite shape memory alloys (SMAs), such as polycrystalline SMAs or rigidly reinforced SMAs. Both the finite deformation and infinitesimal strain frameworks are considered. The methodology employed is a generalization of a homogenization approach introduced by Milton and Serkov [2000. Bounding the current in nonlinear conducting composites. J. Mech. Phys. Solids 48, 1295-1324] for nonlinear composites in infinitesimal strains. Some analytical and numerical examples are given to illustrate the method. 相似文献
16.
ABSTRACT The problem of controlling the dynamic response of a beam by means of displacement and velocity feedback is solved. The objective of the control is to prevent the maximum deflection and/or velocity of a beam from exceeding given upper and lower bounds. The control is The theory is illustrated by two numerical examples that involve displacement and velocity feedback control. An assessment is made of the effectiveness of the proposed control by defining a performance measure. It is observed that the dynamic response of the beam can be kept within specified bounds by applying a large enough control force, which also depends on the extent of the deadband region. A measure of force spent in the control process is defined and plotted against the dynamic response, which is observed to decrease rapidly as the bounds approach the values set by the uncontrolled beam. 相似文献
17.
An accurate nonlinear model for electrostatically actuated beams made of nanocrystalline materials is proposed accounting for the beam material structure and the beam size effects. Two sets of measures are incorporated in the context of the proposed model to account for the inherent properties (the material structure related properties) and the acquired properties (the size dependent properties) of the beam. The inherent properties of the beam are modeled via a micromechanical model while the acquired properties are modeled via a non-classical continuum beam theory. The micromechanical model for nanocrystalline materials is proposed where the necessary measures to account for the effects of the grain size, the voids percent and size, and the interface (grain boundary) are incorporated. All the measures presented in the micromechanical model are related to the material structure to correctly model the structure of nanocrystalline materials. According to the classical couple stress and Gurtin-Murdoch surface elasticity theories, a size-dependent Euler-Bernoulli beam model is developed to model the mechanics of electrostatically actuated nano-beams. For the first time, the impacts of the beam material structure along with the beam size on the nonlinear dynamics and pull-in instability behaviors of electrostatically actuated nano-beams are intensively studied. The performed analyses through the present effort reflect the great impacts of the beam material structure and the beam size on the static pull-in, the natural frequencies, the dynamic pull-in, and nonlinear dynamics of electrostatically actuated nano-beams. 相似文献
18.
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. 相似文献
19.
R.C. Batra M. Porfiri D. Spinello 《International Journal of Solids and Structures》2008,45(11-12):3558-3583
We consider the von Kármán nonlinearity and the Casimir force to develop reduced-order models for prestressed clamped rectangular and circular electrostatically actuated microplates. Reduced-order models are derived by taking flexural vibration mode shapes as basis functions for the transverse displacement. The in-plane displacement vector is decomposed as the sum of displacements for irrotational and isochoric waves in a two-dimensional medium. Each of these two displacement vector fields satisfies an eigenvalue problem analogous to that of transverse vibrations of a linear elastic membrane. Basis functions for the transverse and the in-plane displacements are related by using the nonlinear equation governing the plate in-plane motion. The reduced-order model is derived from the equation yielding the transverse deflection of a point. For static deformations of a plate, the pull-in parameters are found by using the displacement iteration pull-in extraction method. Reduced-order models are also used to study linear vibrations about a predeformed configuration. It is found that 9 basis functions for a rectangular plate give a converged solution, while 3 basis functions give pull-in parameters with an error of at most 4%. For a circular plate, 3 basis functions give a converged solution while the pull-in parameters computed with 2 basis functions have an error of at most 3%. The value of the Casimir force at the onset of pull-in instability is used to compute device size that can be safely fabricated. 相似文献
20.
A nonlinear frequency-amplitude relation is developed to investigate the vibrational amplitude effect on the dynamic pull-in instability of double-sided-actuated nano-torsional switches. The governing equation of a nano-electro-mechanical system pre-deformed by an electric field contains the quintic nonlinear term. The influences of basic parameters on the pull-in instability and natural frequency are investigated using a powerful analytical approach called the homotopy perturbation method. It is demonstrated that two terms in series expansion are sufficient to produce an acceptable solution. The numerical results obtained have verified the soundness of the asymptotic procedure. The phase portraits of the double-sided nano-torsionalactuator exhibit periodic, homoclinic and heteroclinic orbits. 相似文献