Dynamic stability of electrostatically actuated initially curved shallow micro beams

Micro and nano devices incorporating bistable structural elements have functional advantages including the existence of several stable configurations at the same actuation force, extended working range, and tunable resonant frequencies. In this work, after a short review of operational principles of bistable micro devices, results of a theoretical and numerical investigation of the transient dynamics of an initially curved, shallow, double-clamped micro beam, actuated by distributed electrostatic and inertial forces are presented. Due to the unique combination of mechanical and electrostatic nonlinearities, typically not encountered in large scale structures, the device exhibits sequential snap-through and electrostatic (pull-in) instabilities. A phase plane analysis, performed using a consistently derived lumped model along with the numerical results, indicate that critical voltages corresponding to the dynamic snap-through and pull-in instabilities are lower than their static counterparts, while the minimal curvature required for the appearance of the dynamic snap-through is higher than in the static case. The boundaries of the bistability region of a quasi-statically loaded beam are found in terms of the geometrical and loading parameters and are shown to be bounded from above by the dynamic pull-in instability. Some of the post-buckling states cannot be reached under suddenly applied or quasi-statically increasing voltages: specially tailored loading schemes are suggested for realization of these configurations often beneficial in applications.     

Non-linear response of geometrically imperfect sandwich curved panels under thermomechanical loading

This paper deals with the non-linear response of sandwich curved panels exposed to thermomechanical loadings. The mechanical loads consist of compressive/tensile edge loads, and a lateral pressure while the temperature field is assumed to exhibit a linear variation through the thickness of the panel. Towards obtaining the equations governing the postbuckling response, the Extended Galerkin’s Method is used. The numerical illustrations concern doubly curved, circular cylindrical and as a special case, flat panels, all the edges being simply supported. Moveable and immoveable tangential boundary conditions in the directions normal to the edges are considered and their implications upon the thermomechanical load-carrying capacity are emphasized. Effects of the radii of curvature and of initial geometric imperfections on the load-carrying capacity of sandwich panels are also considered and their influence upon the intensity of the snap-through buckling are discussed. It is shown that in special cases involving the thermomechanical loading and initial geometric imperfection, the snap-through phenomenon can occur also in the case of flat sandwich panels.     

Axisymmetric static and dynamics snap-through phenomena in a thermally postbuckled temperature-dependent FGM circular plate

Thermal postbuckling analysis and the axisymmetric static and dynamic snap-through phenomena due to static/sudden uniform lateral pressure in a thermally postbuckled functionally graded material circular plate are performed in this research. Plate is formulated using the first order shear deformation plate theory. Thermo-mechanical properties of the plate are assumed to be temperature dependent where dependency is described according to the higher order Touloukian representation. Two types of temperature loading are considered. Uniform temperature rise and heat conduction across the thickness direction. The one dimensional heat conduction equation in the thickness direction is obtained and discreted via the central finite difference method. The obtained system of equations is nonlinear since the thermal conductivity itself is a function of the unknown nodal temperatures. Using the von-Kármán assumptions, the governing equations of the plate are obtained in a matrix representation with the aid of the conventional Ritz method whose shape functions are developed using the Gram-Schmidt process. At first thermal postbuckling analysis is performed which is a nonlinear problem with respect to both temperature and displacements. Afterwards, response of the bulged thermally postbuckled plate is obtained under the static and dynamic uniform pressure. Snap-through phenomenon may be observed in both static and dynamic loading cases, due to the immovability of the edge of the plate and the initial deflection caused by postbuckling deflection. To capture the snapping phenomenon and trace the path beyond the limit loads, cylindrical arch-length technique is used. In dynamic snap-through analysis, the effect of structural damping is also included. Numerical results of this study reveal that the structure is sensitive to the initial deflection caused by thermal postbuckling load. Increasing the temperature prior to mechanical loads enhances the snap-through intensity and also increases both the upper and lower limit loads. As shown, dynamic snap-through loads are lower than the static ones, however dynamic snap-through intensity is more than the static snap-though intensity. Furthermore, structural damping enhances the dynamic buckling loads of the plate and decreases the dynamic postbuckling deflection of the plate.     

Isogeometric analysis for the dynamic problem of curved structures including warping effects

Curved beam structures have been used in many civil, mechanical, aircraft, and aerospace constructions. The analysis is mainly based on solid and plate models due to the fact that traditional curved beam elements do not include nonuniform warping effects, especially in the dynamic analysis. In this article, independent warping parameters have been taken into account and the initial curvature effect is considered. Curved beam’s behavior becomes more complex, even for dead loading, due to the coupling between axial force, bending moments, and torque that curvature produces. In addition to these, the Isogeometric tools (b-splines or NURBS), either integrated in the Finite Element Method or in a Boundary Element–based Method called Analog Equation Method, have been employed in this contribution for the dynamic analysis of horizontally curved beams of open or closed (box-shaped) cross sections. Free vibration characteristics and responses of the stress resultants and displacements to moving loading have been studied.     

Fast approximations of dynamic stability boundaries of slender curved structures

Curved beams and panels can often be found as structural components in aerospace, mechanical and civil engineering systems. When curved structures are subjected to dynamic loads, they are susceptible to dynamic instabilities especially dynamic snap-through buckling. The identification of the dynamic stability boundary that separate the non-snap and post-snap responses is hence necessary for the safe design of such structures, but typically requires extensive transient simulations that may lead to high computation cost. This paper proposes a scaling approach that reveals the similarities between dynamic snap-through boundaries of different structures. Such identified features can be directly used for fast approximations of dynamic stability boundaries of slender curved structures when their geometric parameters or boundary conditions are varied. The scaled dynamic stability boundaries of half-sine arches, parabolic arches and cylindrical panels are studied.     

Stability of a shallow arch with one end moving at constant speed

In this paper we consider a shallow arch with rise parameter h, free of lateral loading, but subject to prescribed end motion e with constant speed c. Attention is focused on finding out whether dynamic snap-through will occur. Quasi-static analysis is first performed to identify all equilibrium configurations and their stability properties when e and h are specified. If the arch is stretched quasi-statically, it will be straightened up and no snap-through will occur. However, when the speed c is not negligible it is possible for the arch to snap to the other side dynamically. Careful analysis shows that the only possible situation when dynamic snap-through may occur is and . In this case, to prevent dynamic snap-through to occur the end speed c must not exceed a critical speed, which is a function of e and h. The minimum critical stretching speed is found to be 25.9 for all possible combinations of e and h.     

带集中质量的旋转柔性曲梁动力学特性分析

本文对带集中质量的平面内旋转柔性曲梁动力学特性进行了研究.基于绝对节点坐标法推导出曲梁单元,其中该曲梁单元采用Green-Lagrangian应变,并根据曲梁变形前后的曲率变化和曲率的精确表达式计算了曲梁单元弹性力所作的虚功.通过虚功原理,利用$\delta$函数和中心刚体与悬臂曲梁之间的固支边界条件,建立了带集中质量的旋转柔性曲梁非线性动力学模型.基于该模型,本文仿真计算了悬臂曲梁的纯弯曲问题和带有刚柔耦合效应的旋转柔性曲梁动力学响应问题,以此分别讨论了所提出曲梁单元的收敛性和动力学模型的正确性.进一步应用D'Alembert原理,将旋转曲梁等效为带离心力的无旋转曲梁,通过线性摄动处理得到系统的特征方程,以此分别研究了旋转角速度、初始曲率和集中质量对曲梁动力学特性的影响.最后重点分析了旋转曲梁的频率转向和振型切换问题,并阐述了两者之间的相互关系.研究结果表明:随着旋转角速度的增大,曲梁的频率特性与直梁的频率特性相近,以及曲梁拉伸变形占主导的模态振型会提前.      

含黏弹性夹芯FG-GRC后屈曲梁的低速冲击响应

研究了含黏弹性夹芯的功能梯度石墨烯增强复合材料(functionally graded graphene reinforced composite, FG-GRC)后屈曲梁在低速跌落冲击下的跳跃振荡行为.采用修正Halpin-Tsai细观模型预测FG-GRC的材料宏观属性.使用赫兹点接触模型确定冲击器和梁之间的接触力.提出了考虑轴向预应力的复合材料层本构关系和阻尼层的Kelvin型黏弹性本构.通过一种广义高阶剪切变形锯齿梁模型建立夹芯梁的非线性位移场. 基于Hamilton 能量变分原理, 推导了动力学控制方程组. 通过两步分析,首先获得弹性后屈曲平衡路径作为冲击问题的初始状态. 随后, 结合四阶龙格库塔法,拓展了两步摄动-伽辽金法计算接触力的时程曲线以及后屈曲梁的位移时程曲线.研究了后屈曲梁在单次和两次撞击下双稳态大幅振荡过程的动力学特征.讨论了轴向载荷、冲击速度、黏弹性阻尼特性、冲击器材料等因素对于碰撞接触力以及后屈曲梁动力响应的影响规律.结果表明, 接触力仅对冲击速度较为敏感,一定的结构碰撞参数设计可以在接触力变化不大的情况下,使得后屈曲梁由单势能阱运动转变为双阱大幅振荡.      

Chaotic and asymmetrical beam response to impulsive load

Both symmetrical and asymmetrical final displacements are observed for elastic–plastic beams under symmetrical impulsive loading. A three-degree-of-freedom Shanley-type model is developed in this study, which is capable of revealing chaotic and asymmetrical responses of an elastic–plastic beam by introducing initial imperfections. To identify the asymmetrical displacement, the beam response is decomposed into three vibration modes. Corresponding modal participation factors are derived based on the displacement of the three-degree-of-freedom beam model. Phase plane trajectories, Poincaré maps and power spectral density diagrams are derived to illustrate both the symmetrical and asymmetrical chaotic vibrations. Numerical simulations using a general-purpose FE code LS-DYNA are carried out for an elastic–plastic beam subjected to impulsive load. The simulation results indicate that the elastic–plastic beam demonstrates chaotic and asymmetrical vibration when the applied impulsive load exceeds a critical value, which agrees with experimental observations.     

Snap-through under unilateral displacement control with constant velocity

When the side of a beverage can or the domed lid of a jar is pushed inward, all or part of the structure may suddenly snap into an inverted configuration. The velocity of the pushing motion affects this instability. Most previous analyses of snap-through have considered force control (increasing the pushing force, e.g., a weight). Snap-through under dynamic, unilateral displacement control is investigated here, with the indentor moving at constant velocity (as in a universal testing machine) until snap-through occurs. Shallow elastic arches with immovable pinned ends are analyzed. Attention is focused on the critical height of the indentor at which snap-through is initiated. The effects of the indentor velocity, indentor location along the span, initial arch height, and damping magnitude are investigated. In addition, experiments are conducted on shallow buckled beams, which behave similarly to arches. Usually, the higher the indentor velocity, the further the indentor must move before snap-through occurs.     

Relationship between Euler buckling and unstable equilibria of buckled beams

Classic snap-through of curved beams, plates, and shells has long been an object of attention in structural engineering. Euler buckling under axial loading is perhaps an even more entrenched part of the canon of engineering education and practice. In this paper we introduce a relationship between the two phenomena, that to our knowledge has not been directly addressed before. The relationship shows that Euler buckling configurations are connected by the force–displacement curve under transverse loading. The results are used to develop a very simple metric to estimate the number of unstable static equilibria of a buckled structure based only on its geometry with no need for static or dynamic solvers. The study is focused on beams as this allows for an unambiguous discussion of the idea on the simplest possible structure.     

Underwater blast loading of sandwich beams: Regimes of behaviour

Finite element (FE) calculations are used to develop a comprehensive understanding of the dynamic response of sandwich beams subjected to underwater blast loading, including the effects of fluid–structure interaction. Design maps are constructed to show the regimes of behaviour over a broad range of loading intensity, sandwich panel geometry and material strength. Over the entire range of parameters investigated, the time-scale associated with the initial fluid–structure interaction phase up to the instant of first cavitation in the fluid is much smaller than the time-scales associated with the core compression and the bending/stretching responses of the sandwich beam. Consequently, this initial fluid–structure interaction phase decouples from the subsequent phases of response. Four regimes of behaviour exist: the period of sandwich core compression either couples or decouples with the period of the beam bending, and the core either densifies partially or fully. These regimes of behaviour are charted on maps using axes of blast impulse and core strength. The simulations indicate that continued loading by the fluid during the core compression phase and the beam bending/stretching phase cannot be neglected. Consequently, analyses that neglect full fluid–structure interaction during the structural responses provide only estimates of performance metrics such as back face deflection and reaction forces at the supports. The calculations here also indicate that appropriately designed sandwich beams undergo significantly smaller back face deflections and exert smaller support forces than monolithic beams of equal mass. The optimum transverse core strength is determined for minimizing the back face deflection or support reactions at a given blast impulse. Typically, the transverse core strength that minimizes back face deflection is 40% below the value that minimizes the support reaction. Moreover, the optimal core strength depends upon the level of blast impulse, with higher strength cores required for higher intensity blasts.     

Symmetry breaking in an initially curved pre-stressed micro beam loaded by a distributed electrostatic force

The symmetric and asymmetric buckling of an initially curved micro beam subjected to an axial pre-stressing load and transversal distributed electrostatic force is studied. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition with buckling modes of a straight beam used as the base functions. The criteria of symmetric limit point buckling and of non-symmetric bifurcation are derived in terms of the geometric parameters of the beam and the axial load. Two symmetry breaking conditions, defining the relations between the axial load and the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one, are obtained. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric response curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the symmetric response curve. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict symmetric and non-symmetric buckling in electrostatically actuated curved pre-stressed micro beams. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. The nature of the latter and the relations between it and the symmetry breaking criteria are found to go through a prominent qualitative change as the initial distance between the beam and the electrode, characterizing the electrostatic force, changes.     

端头带有质量块的悬臂梁在冲击载荷下的剪切失效

以飞射物撞击构件引起破坏为工程背景,研究端头带有质量块的悬臂梁受到冲击载荷作用后发生剪切失效的可能性。分析表明:在初始速度间断面上是否发生失效取决于无量纲初始动能和质量块尺寸与梁厚之比,而与梁的长度无关;质量块的转动惯量对于剪切失效具有不可忽略的影响。     

含初缺陷裂纹损伤梁的冲击动力屈曲

由Hamilton原理导出考虑初始缺陷及横向剪切变形时裂纹梁的动力屈曲控制方程;应用断裂力学中常用的线弹簧模型将裂纹引入到屈曲控制方程中;基于B-R动力屈曲判断准则,采用数值方法求解了受轴向冲击载荷作用时裂纹梁的动力屈曲;对比讨论了不同冲击速度、初始几何缺陷大小以及分布形式等因素对梁冲击动力屈曲的影响。     

Thermoelastic buckling of steel columns with load-dependent supports

The in-plane elastic buckling of a steel column with load-dependent supports under thermal loading is investigated. Two elastic rotational springs at the column ends are used to model the restraints which are provided by adjacent structural members or elastic foundations. The temperature is assumed to be linearly distributed across the section. Based on a nonlinear strain–displacement relationship, both the equilibrium and buckling equations are obtained by using the energy method. Then the limits for different buckling modes and the critical temperature of columns with different cases are studied. The results show that the proposed analytical solution can be used to predict the critical temperature for elastic buckling. The effect of thermal loading on the buckling of steel columns is significant. Furthermore, the thermal gradient plays a positive role in improving the stability of columns, and the effect of thermal gradients decreases while decreasing the modified slenderness ratios of columns. It can also be found that rotational restraints can significantly affect the column elastic buckling loads. Increasing the initial stiffness coefficient α or the stiffening rate β of thermal restraints will increase the critical temperature.     

Impulsive loading of clamped monolithic and sandwich beams over a central patch

An analytical model is developed for the response of clamped monolithic and sandwich beams subjected to impulse loading over a central loading patch. A number of topologies of sandwich core are investigated, including the honeycomb core, pyramidal core, prismatic diamond core and metal foam. The various cores are characterised by their dependencies of through-thickness compressive strength and longitudinal tensile strength upon relative density. Closed-form expressions are derived for the deflection of the beam when the ratio r of length of loading patch to the beam span exceeds 0.5. In contrast, an ordinary differential equation needs to be solved numerically for the choice r<0.5. Explicit finite element calculations show that most practical shock loadings can be treated as impulsive and the accuracy of the impulsive analytical predictions is confirmed. The analytical formulae are employed to determine optimal geometries of the sandwich beams that maximise the shock resistance of the beams for a given mass. The optimisation reveals that sandwich beams have a superior shock resistance relative to monolithic beams of the same mass, with the prismatic diamond core sandwich beam providing the best performance. Further, the optimal sandwich beam designs are only mildly sensitive to the length of the loading patch.     

Stability analysis of gradient elastic beams by the method of initial value

The buckling of a bar is studied analytically on the basis of a simple linear theory of gradient elasticity in the frame of the method of initial values. The method of initial values provides the values of the displacements and stress resultants throughout the bar once the initial displacements and initial stress resultants are known. We use probably for the first time the method of initial values to get critical loads of a strain gradient beam under completely different boundary conditions at the two end faces of the beam. Exact carryover matrix is presented for the classical beam and gradient beam analytically. The first mode shapes of classical beam and gradient beam are plotted. The method of initial values is also applied to the beams with variable cross-section. The priorities of the method of initial values are depicted. The variational approach gives a sixth-order ordinary differential equation for a beam in buckling. The additional boundary conditions are used to obtain critical loads. It is observed that critical loads increase dramatically for increasing values of the gradient coefficient.     

Large deflection of curved elastic beams made of Ludwick type material

The large deflection of an axially extensible curved beam with a rectangular cross-section is investigated. The elastic beam is assumed to satisfy the Euler-Bernoulli postulation and be made of the Ludwick type material. Through reasonably simplified integration, the strain and curvature of the axis of the beam are presented in implicit formulations. The governing equations involving both geometric and material nonlinearities of the curved beam are derived and solved by the shooting method. When the initial curvature of the beam is zero, the curved beam is degenerated into a straight beam,and the predicted results obtained by the present model are consistent with those in the open literature. Numerical examples are further given for curved cantilever and simply supported beams, and the couplings between elongation and bending are found for the curved beams.     

Symmetry breaking in an initially curved micro beam loaded by a distributed electrostatic force

The asymmetric buckling of a shallow initially curved stress-free micro beam subjected to distributed nonlinear deflection-dependent electrostatic force is studied. In order to highlight the symmetry breaking phenomenon and the approach to its analysis, the subsidiary simplified problem of a curved beam attached to a linearly elastic foundation, and subjected to uniformly distributed “mechanical” load, which is independent of deflections, is addressed first. The analysis is based on a two degrees of freedom reduced order (RO) model resulting from the Galerkin decomposition with linear undamped eigenmodes of a straight beam used as the base functions. Simple approximate expressions are derived defining the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric limit point buckling curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the curve. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict non-symmetric buckling in electrostatically actuated bistable micro beams.