Flexibility of rectangular beams with abrupt changes of section

The usual strength of materials approach to the calculation of deflexions of stepped beams can incur errors if applied to stepped beams with large and abrupt changes of section. After initial comparison with experimental results, the finite element displacement method was used in an investigation into the properties of such beams, particular attention being paid to the relationships between the geometry and the centre line deflexion curves. Data are presented so that for a beam of any geometry and any loading condition, the centre line deflexion curve can be calculated by using simple bending relationships in conjunction with an equivalent variable second moment of area.     

Detection and localisation of multiple cracks in a shaft system: An experimental investigation

Experimental verification of an algorithm for detection and localisation of multiple cracks in a simple shaft system is presented. Cracks in a shaft cause the slope discontinuity in the shaft elastic line. The algorithm is based upon detecting the slope discontinuity due to cracks. Two simply supported non-rotating shafts are used for the experimentation. Both the shafts are tested with single as well as double artificially introduced open cracks. Transverse deflections of the shaft due to sine-sweep excitation force through an exciter are measured at regular axial locations of the shaft. The slope discontinuity is obtained by fitting shaft deflections at regular axial locations in a quadratic polynomial. The algorithm uses shaft deflections at several excitation frequencies to incorporate adequate modal information in the response to improve the crack identification. A scheme is proposed to improve the working of the algorithm in low signal to noise ratio conditions. The algorithm identifies the presence of crack and successfully locates the crack along the shaft length.     

On anticlastic curvature with special reference to plastic bending: A literature survey and some experimental investigations

The literature on the anticlastic curvature developed by beams and plates when subject to elastic and plastic bending is reviewed.

The paper also reports some experimental investigations into the anticlastic curvature of plastically bent plates of varying widths and constant thickness for the two materials, a mild steel and an aluminium alloy. Profiles of the anticlastic surfaces after various degrees of bending are shown, and the differences between the deflections of the edge and the centre line of the plates are compared with the values predicted by the elastic theory for anticlastic deflections. Curves of the bending moment per unit width against curvature are compared with each other for the aluminium alloy plates; the bending moments for the mild steel plates were not measurable.     

阶梯轴弯曲变形的普遍表达式

利用奇异函数与拉普拉斯变换相结合的方法,导出了阶梯轴弯曲变形的普遍表达式。由试式可以方便,准确地计算复杂阶梯轴任意截面的挠度和转角,对阶梯轴弯曲刚度校核及优化设计有实用价值。     

复杂载荷下传动轴强度与刚度的程序化求解

用传统方法求解有复杂载荷作用的传动轴的强度与刚度,需要十分繁重的计算工作量。利用奇异函数描述轴的弯矩方程、扭矩方程和抗弯截面模量方程,推出在任何载荷作用下传动轴所受应力、转角和挠度的通用方程式,采用一维优化方法,借助计算机技术求解传动轴在复杂载荷作用下的强度与刚度,突破了传统求解方法,便于计算机编程处理,适应范围广,对受不同载荷以及含有不同几何形状的阶梯轴具有通用性,且可提高机械工程设计效率,具有较高的工程实用价值。     

阻尼分段阶梯传动轴主共振的分析

阶梯传动轴的主共振与轴的运动稳定性和振幅突变性等有关。用质心运动定理、Heaviside函数建立了非惯性系下倾斜两端支承阶梯传动轴的非线性弯曲振动偏微分动力学方程，并根据相邻轴段的衔接面处位移相等的条件，得到阶梯轴的当量轴及其动力学方程；用Galerkin法得到当量轴的弯曲运动方程，用多尺度法求得稳态下当量轴的主共振的一次近似定常解，分析了影响阶梯传动轴主共振的因素、运动稳定性、振幅突变性等。提出了新的求解阶梯轴非线性弯曲振动分段偏微分动力学方程的方法及当量轴模型，得到了阶梯传动轴与当量轴的主共振特性相同的结论。利用当量轴可方便地分析阶梯传动轴的主共振。     

阶梯轴弯曲固有振动分析及应用

给出阶梯轴弯曲固有振协精确的分析方法,该法亦可用于工程中阶梯梁弯曲固有振动分析。我们用该方法计算了硅压阻式加速度传感顺五阶梯硅梁固频率和振型。     

Stiffness identification of laminated composite shafts

Distributions of bending stiffness along the spans of laminated composite shafts are determined via a non-destructive evaluation approach. The finite element method formulated on the assumption of uniform bending stiffness within each element is used in the deflection analysis of the shafts. Differences between measured and theoretically predicted deflections at any two points on a shaft are used to construct an error function for deflection. The identification of bending stiffness is formulated as a minimization problem in which the elemental bending stiffness are determined to make the error function a global minimum. A global minimization technique and a bounding method for establishing side constraints are presented to solve the above minimization problem. Experiments are performed to study the feasibility and applications of the proposed method.     

弯曲剪应力对轴强度的影响

分析了弯曲剪应力对弯扭组合变形轴主应力强度的影响，推出了考虑该因素影响时第三、第四强度理论的相当应力，并给出了设计公式。结合实例进行了分析比较，指出当轴中传动点距支承较近且扭矩比弯矩大时，剪应力对轴的强度有较大的影响。     

中小型轴类零件旋转弯曲矫直工艺数值模拟研究

直线度是轴类零件重要的质量参数之一,矫直是轴生产过程中的一道必要工序。针对现有矫直工艺的局限性,提出中小型轴类零件旋转弯曲矫直工艺。基于通用有限元软件LS-DYNA,使用Y-U强化模型对工艺过程进行了数值模拟研究,揭示其矫直机理,探究各因素对矫直效果的影响。研究结果表明,矫直过程中材料任意质点应力均经历多次的正负交替变化,在逐渐卸载过程中使轴向任意微段的曲率逐渐趋近于零,实现其矫直。残余挠度随着弯曲量的增加而降低,当弹区比达到0.2时,残余挠度达到了最小。随着旋转速度的增加,残余挠度先减小再增大,当旋转速度与弯曲量的比值为125r/rad时,矫直效果较好。长径比、初始挠度(大小及挠曲形式)以及初始残余应力对矫直效果影响较小,采用相同的工艺参数均取得了良好的矫直效果。数值模拟结果验证了工艺的可行性,无需经过复杂的初始挠度测量,光轴和阶梯轴矫直后残余挠度均在0.1mm以内。     

台阶轴校直工艺计算方法及实验研究

在研究热处理后台阶轴轴肩处发生弯曲变形的基础上，根据弹塑性力学理论，探讨应力集中对校直过程的影响，提出台阶轴校直工艺计算方法。实验研究表明该方法能有效应用于台阶轴的校直过程，对制定齿轮轴等台阶轴零件校直工艺有一定的指导意义。     

Spindle deflections in high-speed machine tools — Modelling and simulation

This research attempts to develop spindle deflection error models for high-speed machining systems. A model for determining total spindle deflection at the tool-end is presented. The model incorporates spindle bearing characteristics, shifts in ball contact angles, and centrifugal force and gyroscopic moment effects at high speeds. It uses the transfer matrix method to determine the total deflections at the tool-end based upon the point contact deformations at the individual balls of an angular contact ball-bearing assembly. A simulator is also developed for simulating spindle end deflections for various spindle rotational speeds. The results of the simulation show contact angle variations and peak deflections at particular spindle rotational speeds. Important research issues are also presented.Nomenclature AF final position, inner raceway groove centre - RF initial position, inner raceway groove centre - W final position of ball centre - V initial position of ball centre - D ball diameter, mm - r _o inner raceway groove radius, mm - r _i inner raceway groove radius, mm - M gyroscopic moment, N-mm - FO r _o/D - FI r _i/D - P bearing pitch diameter, mm - K _o outer race load-deflection constant, N/mm^1.5 - K _i inner race load-deflection constant, N/mm^1.5 - CF centrifugal force, N - J mass moment of inertia, N.mm² - l length of spindle, mm - E modulus of elasticity, N/mm² - I moment of inertia of spindle, mm⁴ - Y deflection of spindle alongy-direction, mm - z deflection of spindle alongz-direction, mm - M moment at spindle end, N.mm - V shear force at spindle end, N - m spindle mass, kg - material density - _o outer race contact angle - _i inner race contact angle - nominal contact angle - _i inner race deformation - _o outer race deformation - angle between ball centre of rotation and the horizontal - mis-alignment (in degrees) of shaft assembly measured in a plane perpendicular to shaft axis (x-direction) - W1 ball and raceway angular raceway velocity ratio for outer raceway control - W2 ball orbital and angular raceway velocity ratio for rotating inner raceway and outer raceway control - circumferential ball position - raceway control parameter     

基于多向挤压的超长异形轴径向挤压工艺研究

传统锻压方法成形阶梯轴,多采用轴向正挤压方法,所加工轴的长度受水压机开口高度的限制。为此,本文基于多向挤压技术提出了阶梯轴径向挤压成形方法,来加工“超长”阶梯轴,并通过数值模拟的方法对新工艺与传统工艺进行了比较,其优势为可以实现采用小吨位水压机加工“超长”异形阶梯轴;可以有效减小毛坯轴的轴肩过度体积,减少材料浪费。     

On the large deformations of C-shaped springs

Large deflections of C-shaped springs are studied using the theory of the Elastica. The resulting nonlinear differential equation is integrated by a new homotopy method. It is found that the force-displacement curve is decidely nonlinear and depends heavily on the free shape of the spring. The maximum moment is also obtained.     

某石油钻井绞车链轮传动轴断裂原因

某石油钻井绞车在运行过程中链轮传动轴突然断裂,通过断口形貌观察、化学成分分析、力学性能测试、显微组织观察等方法,研究了链轮传动轴断裂的原因.结果表明:链轮传动轴的断裂属于早期疲劳失效.裂纹在链轮传动轴?210 mm与?200 mm轴台阶处的粗糙过渡圆角根部应力集中处萌生,在交变弯扭应力作用下扩展,最终导致链轮传动轴的疲...     

轴向摩擦双盘转子的振动分析

为研究轴向摩擦对转子运动的影响,利用弯曲变形和微小角位移矢量合成理论,推导出圆盘与封严圈之间的轴向碰撞力和摩擦力计算公式.根据力矩相等原则,将轴向碰撞力转化为作用于盘心的横向力.建立简支单跨双盘转、静子轴向碰摩的弯扭耦合振动方程,并数值模拟分析弯曲共振时,不同的转、静子轴向间隙对转子运动的影响.结果表明,随着间隙改变,转子横向和扭转运动特征都会发生相应变化,且扭转运动更能反映轴向摩擦的程度.     

Influence of pre-buckling displacements on the elastic critical loads of thin walled bars of open cross section

Nonlinear expressions for the strains occurring in thin walled bars of open cross section, when subjected to axial, flexural and torsional displacements, are incorporated in a general instability analysis based on the vanishing of the second variation of the total potential energy. It is shown that the influence of the pre-buckling displacements is automatically included in the analysis. A closed form solution for the lateral buckling of a simply supported beam subjected to uniform bending agrees exactly with a solution based on the governing differential equations. Solutions obtained using numerical methods are also presented. The significance of the second order axial strains induced by rotation about the shear centre, is investigated by considering the instability of an inverted T-beam subjected to uniform bending.     

Transfer matrix for rotor coupler with parallel misalignment

A transfer matrix for shafts coupler with parallel misalignment (offset) was derived. The responses of a rotor system composed of flexible shafts, unbalanced disks, elastic supports and shafts coupler with misalignment were then investigated. Through the derivation, the boundary shears induced by a rotating shaft were first discovered to be coupled in two perpendicular directions. These coupling shears might reduce the first critical speed up to 50% in the free-free case. The studies showed that the shafts coupler altered the rotor’s critical speeds and the misalignment played as an external load resulting through the whole driven shaft. The combined effects of disk unbalance and shaft misalignment showed that the misalignment predominated the response in most of the rotation speeds, but the unbalance could become significant at high speed. The whirling orbits before and after the misalignment were illustrated as well, and numerical results showed that the two ends of the misalignment whirled asynchronously as the rotation fell into some regions. This paper was recommended for publication in revised form by Associate Editor Seong-Wook Hong Shyh-Chin Huang received a M.S. degree in Mechanical Engineering from University of Iowa in 1984, and a Ph.D. degree in Mechanical Engineering from Purdue University in 1987. Dr. Huang is currently a distinguished professor and researcher at National Taiwan University of Science and Technology. His research interests are in the area of vibration controls, smart materials and electric shunt damping design. Dr. Huang is currently an editor of Research Letters in Materials Science. Chao-Yang Tsai received a M.S. degree in Mechanical Engineering from National Taiwan University of Science and Technology in 1997.Now he is studying at the same department for the Doctor Program. He is currently an instructor at the Army Academy R.O.C, teaching basic technological science including Engineering Mechanics, Mechanics Materials and so on.     

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.