Bending of axially constrained beams on elastic foundation

Approximate analysis of bending of a pin-pointed beam under distributed load resting on elastic foundation is considered. The supports are assumed to be immovable, thereby giving rise to the axial stretching of the neutral surface under lateral loads. The resulting problem is nonlinear since the axial force of constraint is dependent on the lateral displacements. Quantitative results of axial force, benidng moment and deflections are given in terms of two dimensionless parameters which depend on the load intensity, foundation modulus and properties of the beam.     

Experimental behaviour of a fixed ended beam under simulated uniformly distributed load

The behaviour of an aluminium beam of constant rectangular cross section which is fully restrained against translation and rotation and subjected to a simulated uniformly distributed load is investigated. It is shown experimentally and supported analytically that the axial end restraint has a significant stiffening effect on the post-elastic behaviour when the central plastic zone spreads towards the beam ends. The engineering bending theory which includes axial force effects gives an accurate prediction of the pre-yield behaviour and the rigid-plastic theories are shown to give a close approximation to the load/central deflection response for a real fixed ended beam in the plastic range.     

纤维增强FGM梁的自由振动和临界屈曲载荷分析

基于经典梁理论（CBT）研究轴向力作用下纤维增强功能梯度材料（FGM）梁的横向自由振动和临界屈曲载荷问题。首先考虑由混合律模型来表征纤维增强FGM梁的材料属性,其次利用Hamilton原理推导轴向力作用下纤维增强FGM梁横向自由振动和临界屈曲载荷的控制微分方程,并应用微分变换法（DTM）对控制微分方程及边界条件进行变换,计算了纤维增强FGM梁在固定-固定（C-C）、固定-简支（C-S）和简支-简支（S-S）3种边界条件下横向自由振动的无量纲固有频率和无量纲临界屈曲载荷。退化为各向同性梁和FGM梁,并与已有文献结果进行对比,验证了本文方法的有效性。最后讨论在不同边界条件下纤维增强FGM梁的刚度比、纤维体积分数和无量纲压载荷对无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

轴向载荷作用下细长柔性旋转梁横向振动响应分析

将水平井柔性钻柱简化为轴向载荷作用下的细长柔性梁,建立了考虑几何大变形的非线性动力学方程,并且利用多重尺度法得到了柔性梁横向振动的平均方程。因钻柱转速和轴向力是影响钻柱振动的两个主要参数,故用改变转速的方法来对平均方程进行非线性动力学响应分析,给出了一阶、二阶模态下钻柱随转速变化的分叉图和不同转速下的相图。分析结果表明,系统响应经历了从周期到混沌再到二倍周期的变化过程,振动幅值有跳变现象出现,在共振条件下,柔性旋转梁振动幅值是有限值,并非无穷大,当轴向力发生变化时,可以通过控制转速变化达到定性、定量地控制柔性钻柱横向振动的目的。     

Parametric studies on buckling loads and critical speeds of microdrill bits

Transverse bending vibrations of the spinning microdrill bit subjected to a compressive axial load are developed based on the Timoshenko beam theory. The system equations of motion are discretized into the form of time-dependent ordinary differential equations by the finite element method. Two types of eigenvalue problems are formulated and utilized to study the effects of the drill helix angle, flute length and diameter on the buckling load and critical speed of microdrill bits with different supported ends. Equivalent formulae similar to those of untwisted Euler beams are established to predict critical buckling loads and critical speeds for microdrills and provide results with sufficient accuracy. The effect of rotational speed on the buckling load, and the influence of thrust force on critical speed are also investigated. A Galef-type equation associated with critical speed, thrust force and buckling load is formulated.     

两端固支微开关梁中的轴力效应分析

根据考虑轴力的两端固支梁挠曲线微分方程,将受电场力作用的两端固支微开关梁的挠度求解问题归结为求解含有未知边界条件的微分方程问题,通过将微分方程的数值解法和一种三重二分迭代求解方法相结合,实现对含有未知边界条件的挠度微分方程问题的数值求解,再通过和优化方法相结合实现失稳挠度及吸合电压的自动计算与识别.给出的部分计算结果包括,不同的梁高下轴力对吸合电压的影响、轴力对失稳挠度的影响、不同的初始驱动电极间隙下轴力对吸合电压的影响、梁高对失稳时轴力引起的应力的影响等.     

Timoshenko beam-based stability and natural frequency analysis for heavy load mechanical spindles

The heavy load mechanical spindle is an important functional component in a 5-axis computer numerical control (CNC) machine tool, which is used to process large and complex free-form surfaces. It is necessary to obtain the natural frequency and analyze the spindle stability for improving the machining precision. In this paper, Timoshenko beam theory is introduced to model the mechanical spindle shaft, where the centrifugal force and gyroscopic effects are considered. Stability of the heavy load mechanical spindle shaft is analyzed, and the buckling load of the spindle shaft is obtained under different rotational speeds. The natural frequency of spindle is investigated in a freedom and restraint state, respectively. Comparing the proposed method with the simplified hollow cylinder and shaft prototype in the freedom state, the results show that they are highly correlated with experimental results. For the restraint state, the axial load, rotational speed, gyroscopic effect, and centrifugal force are discussed, and all of these parameters affect the natural frequency. The proposed modeling approach can be used for spindle design and optimization in a given machining process and can be easily extended to other spindle design.     

Effect of compressive axial load on forced transverse vibrations of a double-beam system

Based on Bernoulli–Euler beam theory, the forced transverse vibrations of an elastically connected simply supported double-beam system under compressive axial load are investigated. It is assumed that the two beams of the system are continuously joined by a Winkler elastic layer. The dynamic responses of the system caused by arbitrarily distributed continuous loads are obtained. The effects of compressive axial load on the forced vibrations of the double-beam system are discussed for two cases of particular excitation loadings. The properties of the forced transverse vibrations of the system are found to be significantly dependent on the compressive axial load.     

Transverse vibration of a uniform Euler-Bernoulli beam under varying axial force using differential transformation method

This paper presents the application of techniques of differential transformation method (DTM) to analyze the transverse vibration of a uniform Euler-Bernoulli beam under varying axial force. The governing differential equation of the transverse vibration of a uniform Euler-Bernoulli beam under varying axial force is derived and verified. The varying axial force was extended to the more general case which was high polynomial consisted of many terms. The concepts of DTM were briefly introduced. Numerical calculations are carried out and compared with previous published results. The accuracy and the convergence in solving the problem by DTM are discussed.     

点焊薄壁构件受集中载荷或均布载荷作用时的翘曲扭转研究

基于点焊薄壁构件翘曲扭转理论，分析和研究点焊双帽箱型横截面薄壁构件中部有集中载荷和整个构件长度上有均匀分布载荷作用时的扭转变形特性。进行数值仿真发现，在构件中部受集中力偶作用时，扭转刚度比与载荷作用点有关，在集中力偶作用附近流过焊点的剪应力很小，并作用点左右两侧的剪应力方向相反；在整个构件长度上受均匀分布力偶作用时，只要两端不约束翘曲，扭转刚度比保持不变等。     

偏航、变桨轴承疲劳寿命的计算

计算了偏航、变桨轴承在轴向力、径向力和倾覆力矩共同作用下的最大钢球载荷,并根据Hertz接触理论计算了钢球组载荷分布的轴向和径向分量,将两者合成得到了钢球组的实际载荷分布,根据Lundberg-Palmgren理论,计算了内、外圈沟道的当量动载荷和额定动载荷,通过计算偏航、变桨轴承内、外圈的疲劳寿命得到整个轴承的疲劳寿命。     

Effects of shear deformation and rotary inertia on the natural frequencies of axially loaded beams

The purpose of this study is to investigate how the axial load in beams influences the relationships between the natural frequencies and the effects of shear deformation and rotary inertia. Four beam theories are considered in this study. Finite element equations of motion for the beams under a tensile load are formulated to allow the application of various axial loads as well as to impose any type of boundary conditions. The results demonstrate that the stiffening effect by a tensile load may not reduce the frequency error of the Euler beam theory, unlike the results reported in other studies.     

斗轮堆取料机回转支承螺栓连接疲劳寿命分析

常规螺栓连接中,螺栓承受的预紧力和工作载荷是拉力,而斗轮堆取料机回转支承联接螺栓承受轴向压力而产生压缩,轴向工作载荷和变形不同于常规螺栓连接。本文在对受偏心轴向压力的回转支承高强度螺栓群进行受力分析的基础上,对螺栓群疲劳寿命进行了分析。     

Vibration and stability of an Euler–Bernoulli beam with up to three-step changes in cross-section and in axial force

The paper treats the vibration of beams with up to three-step changes in cross-section and in which the axial force in each portion is constant but different. The system parameters are the step positions, the flexural rigidity, the mass per unit length and the axial force in the beam portions—all of which were normalized. The frequency equation for 16 combinations of classical boundary conditions are expressed as fourth-order determinants equated to zero. The first three frequency parameters are tabulated for sets of system parameters (arbitrarily chosen and which includes a stepped beams under tensile or compressive axial end force). Critical compressive end force which causes a stepped beam to buckle are tabulated. Buckling under a system of axial forces, one of which is critical is discussed and several critical combinations of the system parameters are tabulated. Beams of constant depth and step change in breadth, of constant breadth and step change in depth and shafts with step change in diameter are considered. It is shown that stepped shafts are inferior machine elements if dynamic properties are the prime consideration.     

非驱动轮毂轴承单元法兰盘轴颈静强度计算及试验

针对轮毂单元静强度计算及试验验证问题,根据第三代非驱动轮毂轴承单元的受力情况,提供了一种用CAE仿真分析方法以外的简便数值分析计算方法,即在车辆受到侧向1.2 g的临界侧向加速度下,将轮毂轴承单元受到的外部径向力和轴向力,通过轮毂轴承单元内部受力分析,转化为轴承内部双列滚道的两个载荷中心的径向力和轴向力;并将轮毂轴承单元法兰盘内侧受力轴颈简化等效为实心悬臂梁的处理方法,来计算法兰盘轴颈静强度;并用侧向静强度试验方法,通过构建适当的试验方案,来验证轮毂轴承单元法兰盘静强度,为设计优化提供简便的分析验证方法。研究结果表明,该简化模型计算结果能够与侧向静强度试验结果保持一致,为笔者利用简化计算方法来简便快速地定量选取轴径和轴颈r角大小提供了可能;该方法计算简便有效,结果可靠。     

杆件轴力的一种识别方法

针对杆系结构中的轴力杆或索缆结构中的短索,提出了一种解析方法识别边界不确定杆件的绝对轴力,重点探讨了杆件抗弯刚度的影响.基于振动测试提取杆件的固有频率、位移模态或应变模态,以此构建特征方程进而识别其绝对轴力.数值算例分析表明,本方法在采用不同阶次的模态参数以及不同的杆件边界条件下都可取得相对误差3％以下的高精度识别结果,且在不测量杆件两端节点的有效长度、仅已知传感器之间的相对位置时即可实施.试验证实,基于加速度测量得到的前5阶模态参数都可基本准确地识别不同加载工况下杆件的绝对轴力,除小荷载外的多数工况都取得了相对误差10％以下的良好识别效果.     

Vibration suppression of atomic-force microscopy cantilevers covered by a piezoelectric layer with tensile force

In this paper, vibration suppression of a micro-beam covered by a piezoelectric layer is studied. The micro-beam is modeled with the specific attention to its application in AFM. The AFM micro-beam is a cantilever one which is stimulated close to its natural frequency by applying a harmonic voltage to the piezoelectric layer. The beam is an Euler-Bernoulli beam which abbeys Kelvin-Voigt model. Using such model supplies the comparison between elastic and viscoelastic beams; and one of the most important properties of viscoelastic materials, damping effect can readily be investigated. The pump provides an axial load with the result that it suppresses the vibrations. First, the vibration equations are extracted using Lagrangian and extended Hamiltonian method in vertical, longitudinal, as well as torsional directions and are discretized by exploiting the Galerkin mode summation approach. The discretized time-domain equations are solved by the aid of the Runge-Kutta method. The viscoelastic beam is compared with the elastic one, and the effects of damping ratio on vibration responses are presented. Additionally, the effects of micro-pump load, excitation voltage, and initial twist angle are investigated on the amplitude of vibration and natural frequency of system. It is observed that viscoelasticity of beam and axial load of the pump reduce vibrations and provide uniform time-domain responses without beatings.     

Dynamics of a rotating Rayleigh beam subject to a repetitively travelling force

The dynamics of a rotating Rayleigh beam subject to a force travelling at a constant speed along the axial direction is studied. The beam is chosen as a simple model of the workpiece in the lathing process. A technique is developed for modeling the repetitive cutting force on the workpiece. The amplitude of the cutting force is chosen to be either constant or dependent on the motion of the beam. The discretized equations of motion of the rotating beam are obtained by Galerkin’s method. The time response of the rotating beam subject to the external force is discussed. The possible resonant conditions resulting in divergent solutions are studied. The stability of the response, due to a travelling motion-dependent force, is determined by the method of multiple scales. The effects of varying the rotating frequency, the travelling speed of the external force, and the movement of the force are also investigated.     

Post-yield behavior of a beam with partial end fixity

A simply supported beam can undergo a relative axial displacement proportional to the induced axial force. Complete formulas and curves are obtained for transverse displacements of the order of the beam thickness.