A simple explicit arc-length method using the dynamic relaxation method with kinetic damping

The explicit arc-length method is simulated to trace the post-buckling equilibrium path of structures by using dynamic relaxation method with kinetic damping. This method based on the cylindrical arc-length method does not require the computation and formulation of any tangent stiffness matrix to search the snap-through or snap-back problems. The convergence to the solution is achieved by using only vector equation with kinetic damping technique. Two approaches for cylindrical arc-length control are formulated with incremental and total displacement constraint. The merits of the explicit arc-length method, in tracing the post-buckling behavior of structures, are demonstrated by analyzing the numerical examples.     

On the optimal design of the shape of a buckled elastic beam

Analysis of stability, post-buckling bending and vibrations is performed for a beam (a spring element) having an optimal shape. A buckled pin-jointed spring element of a constant thickness and variable width is considered. The optimal shape of this beam is suggested to provide a uniform distribution of maximum bending stresses in its buckled equilibrium configuration for a given value of a supercritical axial force. Sensitivities of a critical force and a buckling mode to variations of the shape of a beam are calculated. A dependence of the static lateral deflection upon an axial force is analysed. Nonlinear equations of large-amplitude oscillations are derived by a use of the Hamilton principle. The natural frequencies of a spring element, compressed by a supercritical force are calculated. Received April 29, 1999     

Lateral buckling of locally buckled beams using finite element techniques

This paper deals with lateral-torsional buckling of beams which have already buckled locally before the occurrence of overall buckling. Due to the weakening effects of local buckling, the stiffness of the beam is reduced. As a result, overall lateral buckling takes place at a lower load than the member would carry in the absence of local buckling. The effective width concept is used in this investigation to account for the post-buckling strength in the buckled compression plate elements of the beam section. A finite element formulation in conjunction with effective width concept is presented. Due to the nonlinearity involved because of local buckling, an iterative procedure is necessary. Search techniques are used to find the load factor. The method combined with an analysis on nonlinear bending moment distribution can be used to analyze the lateral stability problem of locally buckled continuous structure. In this case, both elastic stiffness matrix and geometric stiffness matrix must be revised at each load level. A computer program has been prepared for an IBM 370/165 computer.     

不同脱层位置下脱层屈曲梁振动实验研究

通过实验对一端固定一端夹支脱层屈曲梁在轴向周期激励作用下的非线性动力响应进行了实验研究.利用位移时间历程图,相图和频谱图,对多组不同脱层位置下脱层屈曲梁的非线性动力响应进行了分析.实验表明脱层梁结构存在倍周期以及混沌运动等非线性动力学行为.同时实验还表明,在相同的脱层长度下,脱层位置对脱层梁的动力学特性有明显影响,即脱层区域中心越靠近梁结构的中心位置,脱层梁的一阶自然频率越低,且越容易在较低的激励频率和激励荷载下发生周期分叉和混沌等行为.     

Bifurcation and snap-through phenomena in asymmetric dynamic analysis of shallow spherical shells

The asymmetric dynamic behavior of clamped shallow spherical shells under a uniform step pressure of infinite duration is investigated. The solution of a linear eigenvalue problem yields the bifurcation paths and also the lower bound for the asymmetric dynamic snap-through buckling pressure. The asymmetric dynamic response of shells with a shape imperfection is studied. The asymmetric dynamic snap-through buckling load is defined to be the threshold value of the step pressure at which the asymmetric response shows significant growth rate. The snap-through buckling loads are obtained for a few shell parameters. The numerical results are compared with the available experimental results and they are in good agreement. Finally, a preliminary study of the phase planes is presented.     

On the role of geometrically exact and second-order theories in buckling and post-buckling analysis of three-dimensional beam structures

Several geometrically nonlinear beam models are evaluated with respect to their utility in the analysis of buckling and post-buckling behavior of three-dimensional beam structures. The first two models are based on the so-called geometrically exact beam theory capable of representing finite rotations and finite displacements. The principal difference between these models concerns only the chosen parameterization of finite rotations, with the orthogonal matrix used in the first and the rotation vector used in the second one. The third beam model based on the second-order approximation of finite rotations is also discussed along with its application to constructing a consistent formulation of the linear eigenvalue problem for computing an estimate of the critical load. Exact linearized forms, which are crucial for facilitating the buckling load computation and assuring a robust performance of a Newton-method-based continuation strategy, are presented for all three beam models. An elaborate set of numerical simulations of buckling and post-buckling analysis of beam structures is given in order to illustrate the performance of each of the presented models. Finally, some conclusions are drawn.     

Nonlinear,three-dimensional beam theory for dynamic analysis

For beams undergoing large motions but small strains, the displacement field can be decomposed into an arbitrarily large rigid-section motion and a warping field. When applying beam theory to dynamic problems, it is customary to assume that all inertial effects associated with warping are negligible. This paper examines this assumption in details. It is shown that inertial forces affect the beam’s dynamic response in two manners: (1) warping motion induces inertial forces directly, and (2) secondary warping arises that alters the beam’s constitutive laws. Numerical examples demonstrate the range of validity of the proposed approach for beams made of both homogeneous isotropic and heterogeneous anisotropic materials. For low-frequency warping, it is shown that inertial forces associated with warping and secondary warping resulting from inertial forces are negligible. To examine the dynamic behavior of beams over a wider range of frequencies, their dispersion curves, natural vibration frequencies, and mode shapes are evaluated using both one- and three-dimensional models; good correlation is observed between the two models. Applications of the proposed beam theory to multibody problems are also presented; here again, good correlation is observed between the prediction of beam models and of full three-dimensional analysis.     

Mechanical characterization of post-buckled micro-bridge beams by micro-tensile testing

A micro-tensile testing system has been developed to measure the mechanical properties of post-buckled silicon dioxide micro-bridge beams. A kind of vernier-groove carrier is presented to improve alignment precision and repeatability of the measurement, and the stiffness coefficient of the tensile system is calibrated in situ in order to obtain the deformation of the tensile beams. Through analyzing a series of stress states in the beam over film preparation, post-buckling and unfolding of the beam, the initial residual stress in the film is obtained from the original load–displacement curves. The residual stress of 354 MPa is consistent with that calculated from the theory of finite deflection of buckled beams. Young’s modulus and tensile fracture strength are also obtained from the load–displacement curves. The measured modulus and strength are 64.6 ± 3 GPa and 332–489 MPa respectively. The measured properties of the thermal silicon dioxide film are reasonably coherent with other reports.     

动态轴向载荷下谐振梁振动响应分析

在航空航天飞行控制中,为实现关键参数的高精度高动态测量,急需发展具有快速响应特性的谐振式传感器。谐振式传感器本质上是输入与谐振器振动状态之间的映射。这种映射一般通过跟随输入的轴向载荷调制谐振梁的固有频率实现。高动态应用中的核心问题是动态轴向载荷下谐振梁的振动响应。利用基本的微元力学平衡关系建立了动态轴向力作用下谐振梁振动行为的数学模型。此模型比Mathieu方程的适用面更广,在一般假设下更难以进行解析或数值求解。为此引入了等效电路方法进行模型求解。通过对等效电路的仿真,得到了谐振梁在多种典型动态载荷下的振动响应。动态轴向载荷对于谐振梁的作用具有强烈的非线性和独特的规律,值得进一步深入研究探讨。     

Analytical and experimental investigation of a parallel leaf spring guidance

The consequences for the static and dynamic system behaviour of misalignment in an overconstrained direction are analysed. Therefore, a relatively simple parallel leaf spring guidance, which is overdetermined only once, serves as a case to gain insight. A multibody program using flexible beam theory is used to determine the change in vibration mode frequencies and stiffnesses due to misalignment. A previously developed beam element for modelling the leaf springs is shown to be able to describe these phenomena with a limited number of elements. Buckling loads and associated buckling modes are also determined analytically. An instrument has been fabricated to measure the change of vibration mode frequencies due to a rotational misalignment in the overconstrained direction. Experiments using wire spark eroded leaf springs are in good agreement with the calculations. Differences between the experimental and calculated results are attributed to the imperfections in the hardware model, in particular residual stresses, and the assumptions used for the beam element in the numerical model. A small misalignment, in this case 0.8–5.8 mrad, causes strong change of static and dynamic system behaviour. The decreased stiffnesses perpendicular to the compliant direction are disturbing, because these are designed to be large to enhance precision manipulation. The negative effects of overconstrained design are largest if relatively thin and wide leaf springs are used. If the misalignment is kept below 50% of the angle of which a bifurcation occurs, the overconstrained design of the parallel leaf spring mechanism does not significantly influence the system natural frequencies and stiffness.     

Analytical method of predicating the instabilities of a micro arch-shaped beam under electrostatic loading

An arch-shaped beam with different configurations under electrostatic loading experiences either the direct pull-in instability or the snap-through first and then the pull-in instability. When the pull-in instability occurs, the system collides with the electrode and adheres to it, which usually causes the system failure. When the snap-through instability occurs, the system experiences a discontinuous displacement to flip over without colliding with the electrode. The snap-through instability is an ideal actuation mechanism because of the following reasons: (1) after snap-through the system regains the stability and capability of withstanding further loading; (2) the system flips back when the loading is reduced, i.e. the system can be used repetitively; and (3) when approaching snap-through instability the system effective stiffness reduces toward zero, which leads to a fast flipping-over response. To differentiate these two types of instability responses for an arch-shaped beam is vital for the actuator design. For an arch-shaped beam under electrostatic loading, the nonlinear terms of the mid-plane stretching and the electrostatic loading make the analytical solution extremely difficult if not impossible and the related numerical solution is rather complex. Using the one mode expansion approximation and the truncation of the higher-order terms of the Taylor series, we present an analytical solution here. However, the one mode approximation and the truncation error of the Taylor series can cause serious error in the solution. Therefore, an error-compensating mechanism is also proposed. The analytical results are compared with both the experimental data and the numerical multi-mode analysis. The analytical method presented here offers a simple yet efficient solution approach by retaining good accuracy to analyze the instability of an arch-shaped beam under electrostatic loading.     

Post-buckling behavior of cold-formed thin-walled stainless steel beams

A numerical approach for predicting the post-buckling responses of cold-formed thin-walled stainless steel beams is presented. In the analysis, the nonlinear and unsymmetrical stress-strain relations in tension and compression for both flat and corner materials are considered. The effective width concept is used to account for the post-buckling strength of the thin compression flange of the beam. Using this concept, the value of the effective width varies with the stress borne by the flange edge. In view of the nonlinearities introduced by considering the material characteristics and the post-buckling behavior of the individual elements, an iterative procedure is employed to establish the nonlinear moment-curvature relation of a section. In a beam subjected to bending, the effective moment of inertia of the section varies from point to point along the length of the beam depending upon the magnitude of the moment at the point considered. In this analysis, the continuously flexible beam is replaced with a finite number of discrete rigid elements connected by flexible joints at which the continuously varied curvature is lumped through numerical integration. In the case of a continuous beam, it is further complicated by the fact that the bending moment distribution along the beam is not known a priori and its determination is part of the solution to the problem. An iterative algorithm is developed to obtain the approximate true moment distribution considering the compatibility conditions at the support. A computer program following th e above solution scheine has been developed and is capable of predicting the nonlinear responses of most of the common thin-walled sections encountered in light-gage steel design. Comparisons with some available experimental data are provided.     

隔振对象重量变化对准零刚度隔振器隔振性能的影响

本文主要研究隔振对象重量变化对一类准零刚度隔振器隔振性能的影响,并给出了新的研究结果.文中使用欧拉屈曲梁构建负刚度调节结构并设计了隔振系统的平衡位置可调机构.假设系统有轻微的过载和超载,推导了系统的动力学方程并进行求解,定义了非线性隔振系统的力传递率及位移传递率来评价系统的隔振性能.对线性隔振系统,超载会让隔振频率略微降低,共振放大峰略微增大.对于准零刚度隔振系统,力传递率和线性系统类似,但是对于位移传递率,过载会导致系统固有频率和共振放大峰均升高,反而不利于隔振.研究结果可以对此类隔振系统的使用场合以及对过载和轻载的选择有工程指导意义.     

Vibration and local instability of thermally stressed plates

The vibration and buckling of a double wedge square cantilever plate has been investigated. It is shown that the free vibration modes, which occur at ΔT_ref = 0, transition into the buckled modes which occur at ΔT_ref = ΔT_refcr for the respective mode. ΔT_refcr for a particular mode is defined as the magnitude of thermal load at which the frequency of the particular mode vanishes. The analysis, in which no assumption whatsoever is made about the shape of the vibration modes, about the vibration frequencies, about the shape of the buckled modes, or about the magnitude of the critical loads, yields the same number of buckling eigenvalues and buckling modes as there are vibration eigenvalues and vibration modes. Gradual application of the load in the analysis permits the change in each vibration frequency of interest and its associated mode to be followed up to the load at which the frequency of the mode becomes zero. This constitutes the limit of linear theory. Only linear theory is used in this paper; thus, no post buckled behavior is considered. As the load is increased, the thin edges of the plate begin to duform during vibration. This local deformation, which begins in the vibration mode, is shown to transition into the phenomena of local edge buckling at ΔT_refcr for the mode.     

Adaptive enrichment meshfree simulation and experiment on buckling and post-buckling analysis in sheet metal forming

Sheet buckling, a form of instability, is one of the major considerations in the design of part shape, die geometry and processing parameters of sheet metal forming. In this study, an adaptive enrichment meshfree method is developed to capture wrinkling and post-buckling behavior in sheet metal forming. A three-dimensional meshfree continuum approach is applied to the large deformation of plate/shell structures. A stress-based wrinkling predictor is used to predict the onset of buckling within effective compressive regions. Enrichment particles with a proper enrichment function are inserted/deleted in those regions to capture the buckling mode and therefore post-buckling behavior. For verification of the simulation results, a high-resolution wedge strip test is designed to study the onset and post-buckling behavior of a sheet under different boundary conditions.     

Post-buckling analysis of spatial structures by vector iteration methods

The present study is concerned with the application of two vector iteration methods in the investigation of the large deflection behavior of spatial structures. The dynamic relaxation and the first order conjugate gradient belong to this category of methods which do not require the computation or formulation of any tangent stiffness matrix. The convergence to the solution is achieved by using only vectorial quantities and no stiffness matrix is required in its overall assembled form. In an effort to evaluate the merits of the methods, extensive numerical studies were carried out on a number of selected structural systems. The advantages of using these vector iteration methods, in tracing the post-buckling behavior of spatial structures, are demonstrated.     

Snap-Through and Pull-In Instabilities of an Arch-Shaped Beam Under an Electrostatic Loading

The snap-through and pull-in instabilities of the micromachined arch-shaped beams under an electrostatic loading are studied both theoretically and experimentally. The pull-in instability that results in a system collision with an electrode substrate may lead to a system failure and, thus, limits the system maximum displacement. The beam/plate structure with a flat initial configuration under an electrostatic loading can only experience the pull-in instability. With the different arch configurations, the structure may experience either only the pull-in instability or the snap-through and pull-in instabilities together. As shown in our computation and experiment, those arch-shaped beams with the snap-through instability have the larger maximum displacement compared with the arch-shaped beams with only the pull-in stability and those with the flat initial configuration. The snap-through occurs by exerting a fixed load, and the structure experiences a discontinuous displacement jump without consuming power. Furthermore, after the snap-through jump, the structures are demonstrated to have the capacity to withstand further electrostatic loading without pull-in. Those properties of consuming no power and increasing the structure deflection range without pull-in is very useful in microelectromechanical systems design, which can offer better sensitivity and tuning range.     

Dynamic behavior of resonant piezoelectric cantilever as liquid level detection sensor

Resonant Piezoelectric-excited Millimeter-sized Cantilevers (PEMC), has attracted many researchers’ interest in the applications such as liquid level and density sensing. As in these applications, the PEMC are partially immersed in liquid, an appropriate analytical model is needed to predict the dynamic behavior of these devices. In this work, a PEMC has been fabricated for liquid level sensing. An analytical model based on Euler–Bernoulli beam theory and energy method is developed and applied to evaluate the performance of this device with respect to different tip immersion depth. To validate the proposed model, the theoretical results are compared with the experimental results for the tip immersion depth from 5 to 15?mm in water. The simulation results are in almost good agreement with experimental data. Using the proposed model, the two key parameters of sensor performance: sensitivity and working range have been examined for different mode shapes of PEMC vibration.     

Numerical solution of large deformation problems involving stability and unilateral constraints

The present study is concerned with the numerical treatment of large deformation beam problems where stability as well as post-buckling behaviour is coupled with frictional contact constraints. The flexible beams are described according to a nonlinear rod-type theory which accounts for both finite rotations and large deformations. The contact conditions are introduced via a penalty function method. From these conditions we obtain a linear complementary problem (LCP) resulting from the variational inequality formulation. For the examination of the post-buckling behaviour the displacement control method is applied. Particular attention is paid to the development of the linear complementary problem combining with the computational strategy for tracing limit points. Finally, the modification algorithms of the linear complementary problem, in which the penalty factors have been eliminated, are proposed. The numerical techniques not only allow some limit points to be passed, but also guarantee the computational stability characteristics during the Newton-Raphson's iterative process. Numerical examples are presented that illustrate the performances of the proposed algorithms.     

Recent Researches on Nonlocal Elasticity Theory in the Vibration of Carbon Nanotubes Using Beam Models: A Review

Understanding dynamic behavior of carbon nanotubes has been of interest to researchers because of its practical applications. Recent studies show that nonlocal elasticity theory gives better results in the vibration of carbon nanotubes. The necessity of nonlocal elasticity theory, calibration of nonlocal parameter and application of nonlocal elasticity theory in various studies related to vibration of carbon nanotubes are discussed. This review emphasizes the application of nonlocal elasticity theory in the vibration of carbon nanotubes considering various types of complicating effects, nonlinearity, functionally graded material and different beam theories.