A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses

A frequency analysis of an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented. The dimensionless frequency equation for classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions associated to the masses. The resulting transcendental equations are numerically solved for the eigenvalue. On the other hand, the eigenvalue can be predicted merely from the individual beam system carrying a single mass, by virtue of the Dunkerley's formula. A parametric study on the effects of the two masses and their locations is presented for the beam with different boundary conditions. It is found that the Dunkerley's expression can generally yield good approximation if compared with the result associated with the original characteristic equation. The computation time saved owing to the modified Dunkerley method is also illustrated in a comparison. The Dunkerley's method is recommended for the beam carrying more than two masses at different positions, owing to its good approximation and the saving in computational time.     

Comparisons of experimental and numerical frequencies for classical beams carrying a mass in-span

A frequency analysis of an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented. The dimensionless frequency equation for ten combinations of classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions. The resulting transcendental frequency equations are numerically solved. A parametric study on the effects of the mass and its location for each respective case is presented. To verify the validity of the transcendental equations, the results for the fixed-fixed cases are compared with those obtained experimentally. On the other hand, approximate results are given, using the Rayleigh’s method with two static deflection shape functions. The effects of the position and magnitude of the mass, as well as comparisons of the different results obtained analytically, are investigated and discussed. The comparisons clearly show that the eigenfrequencies of the beam–mass system can be accurately predicted by solving the transcendental equation, whereas the closed-form Rayleigh’s expression is suggested for a quick estimation of fundamental frequency.     

任意连续梁内力包络图的自动绘制法

根据内力包络图设计梁的断面,可以保证满足安全和经济两方面的要求。但是按传统方法绘制梁的内力包络图手续十分繁琐,对于连续梁之类的多跨梁尤其如此。介绍一种利用奇异函数对连续梁进行力学分析的新方法及其智能分析软件。该方法利用奇异函数与拉普拉斯变换相结合的方法导出连续梁变形和内力的普遍表达式,利用变形和内力的边界条件确定约束反力,然后再在内力普遍表达式的基础上建立一种内力包络图的自动绘制方法。该方法和程序对承受任意恒载和活载、具有任意跨数的连续梁具有普遍的适用性。     

Natural frequencies of a beam–mass system in transverse vibration: Rayleigh estimation versus eigenanalysis solutions

A frequency analysis of a beam carrying multiple point masses at various locations are presented by using an eigenanalysis and the Rayleigh's estimation. In the eigenanalysis, the frequency equation is generated by satisfying all boundary and mass-loading conditions. As for Rayleigh's method, the frequency is obtained by solving an algebraic expression involving a specified shape function. As compared with the general expression derived by virtue of Rayleigh method, the eigenfrequency equation is lengthy with products of transcendental functions. The Rayleigh's results with each respective shape function are compared with the eigenfrequencies to verify the effectiveness of Rayleigh's method with assumed shape functions. It is found that the Rayleigh method with the simple shape functions can provide good approximation and can thus replace the solving of complicated eigenfrequency equations. The time saving by using Rayleigh's method becomes more significant as the number of masses carried by the beam increases.     

On the methods to derive frequency equations of beams carrying multiple masses

Two commonly used methods to generate the eigenfrequency equation of a beam carrying multiple concentrated masses at an arbitrary location are of main concern in this paper. Both the methods of frequency determinant and the method of Laplace transform are considered and compared here for the effectiveness of each derivation for the frequency equation of the same system. The dimensionless frequency equation for a clamped–clamped beam carrying two different masses is obtained for comparison. The computational times to generate the same expression by using the two methods are noted. It is found that the processing and computational time by using the Laplace transform is longer although the resultant system equations are often claimed to be more compact than those by the method of frequency determinant.     

Investigation of erosion–corrosion processes using electrochemical noise measurements

This paper presents an example-based discussion of erosion–corrosion and flow corrosion processes that have been identified using electrochemical noise measurements. Various single and dual phase corrosion and erosion–corrosion experiments on austenitic stainless steels and various thermally sprayed coatings using jet impingement and pipe flow rigs are discussed. Localised corrosion events, metastable and propagating pitting, passive and general corrosion processes have been identified under various flow conditions of NaCl solutions. Oscillations in the electrochemical potential noise signals have been related to an erosion-enhanced corrosion synergistic effect. Electrochemical noise measurements show responses to electrolyte permeation of the coating, coating erosion penetration and substrate activity under erosion–corrosion conditions.     

基于不可压缩光滑粒子流体动力学的单螺杆挤出三维流动数值模拟

基于不可压缩光滑粒子流体动力学(Incompressible smoothed particle hydrodynamics, ISPH)算法,利用FORTRAN语言编程,开发一套模拟三维非牛顿流程序。通过模拟平板Poiseuille流验证ISPH算法计算非牛顿流的正确性。指出在模拟非牛顿流时,采用镜像粒子边界法处理壁边界较采用静态粒子边界法更适合。并利用该程序计算单螺杆挤出三维流场。提出两种入口出口边界设置方法：周期边界法及压力边界法。分析计算结果发现：选取周期边界较压力边界更合理;选取适当粒子分布密度可以同时满足计算效率与计算精度要求;螺槽横截面上形成环流;切应力分布梯度主要集中在左上角和右上角;沿螺槽方向的流动很平稳,然而,随着反向压力梯度的增加,流动受到反向压力梯度的阻碍更加明显。为进一步采用ISPH方法模拟几何形状更加复杂的三维螺杆挤出打下基础。     

考虑温度等多因素影响的拱桥吊杆张力识别

基于Euler-Bernoulli梁理论，综合考虑了抗弯刚度、长度、减振垫、附加质量、弹性支承、黏性护套和环境温度等因素的影响，建立了吊杆张力与振动频率的解析表达式。首先，提出了边界影响系数法，针对不同边界条件计算拱桥吊杆边界影响系数，给出了边界影响吊杆张力实用表达式；然后，以刘江黄河大桥为例进行验证，利用在综合考虑多因素影响的解析表达式计算吊杆张力时，长吊杆张力计算最大误差为1.10%，中、短吊杆张力计算最大误差不超过1.12%；最后，利用边界影响实用表达式计算吊杆张力时，最大相对误差不超过2.38%。结果表明:考虑温度影响时，采用复杂边界条件下的解析表达式计算精度提高了11.39%，采用实用表达式计算时精度提高了11.38%。边界影响系数法建立的实用表达式计算精度较高，且为显式形式，实用性强。     

Free vibration of Timoshenko beam with finite mass rigid tip load and flexural–torsional coupling

In this paper, the free vibration of a cantilever Timoshenko beam with a rigid tip mass is analyzed. The mass center of the attached mass need not be coincident with its attachment point to the beam. As a result, the beam can be exposed to both torsional and planar elastic bending deformations. The analysis begins with deriving the governing equations of motion of the system and the corresponding boundary conditions using Hamilton's principle. Next, the derived formulation is transformed into an equivalent dimensionless form. Then, the separation of variables method is utilized to provide the frequency equation of the system. This equation is solved numerically, and the dependency of natural frequencies on various parameters of the tip mass is discussed. Explicit expressions for mode shapes and orthogonality condition are also obtained. Finally, the results obtained by the application of the Timoshenko beam model are compared with those of three other beam models, i.e. Euler–Bernoulli, shear and Rayleigh beam models. In this way, the effects of shear deformation and rotary inertia in the response of the beam are evaluated.     

An improved model for the frequency estimate of mass-loaded plates by a combined use of equivalent center mass and stiffness factors

In this paper, an improved model is developed to determine the frequency of vibrating plates carrying multiple masses at various positions. By virtue of an equivalent-center method, the frequencies of the plates carrying multiple masses can be predicted by using merely the data obtained earlier for the same plate with each corresponding mass component. Further accounting the change of the strain energy into the model and by introducing respective stiffness ratios, one can obtain a quick and better estimation of the loaded plates with various boundary conditions. The proposed model is validated through experimentation of a rectangular clamped plate carrying concentrated masses. Analytical and experimental results for the plate system are compared and discussed. It is found that the change in the strain energy should be incorporated in the model, especially for cases of large masses placed on a thin plate, in order to predict well the natural frequency of the amalgamated system from those of the component systems. The excellent performance of the proposed model is due to the fact that the effects of both the different masses and their locations on the frequency have been accounted for, implicitly, in the respective equivalent center mass and stiffness factors. Although the model considered is rather simple, the proposed methodology can be extended to plates with other geometry and configurations.     

对偶边界元法中对数奇异积分的计算

采用三次多项式坐标变换计算边界元积分的方法，对对偶边界法中含对数奇异积分的计算进行研究。几个解析积分算例和运用对偶边界元法以二给平面裂纹问题裂尖应力强度因子的计算结果表明，多项式坐标变换法对于计算对数奇异积分是一种简单有效的方法。     

某重型自卸车车架的有限元分析与改进设计

运用UG软件建立起某重型自卸车车架总成的三维模型,并利用有限元分析方法,采用壳单元进行网格划分以及合理的模型简化和连接方式,建立起某重型自卸车车架基于Nastran的有限元模型.对应车架在纯弯曲和扭转两种工况下,施加相应的边界条件和载荷条件,进行刚强度分析,得到车架在纯弯曲、扭转工况下的应力、应变图,找出车架中的薄弱部位并对薄弱部位进行结构优化.通过优化前后对比分析为后续车架结构改进提供理论依据.     

Development of the meshless Hermite–Cloud method for structural mechanics applications

In this work, a novel true meshless numerical technique is proposed. It is termed the Hermite–Cloud method and is based on the classical reproducing kernel particle method except that a fixed reproducing kernel approximation is used instead. Another distinction is that the point collocation technique is used for the discretization of the governing partial differential equations. In this method, the Hermite theorem is employed for the construction of the interpolation functions. Through the constructed Hermite-type interpolation functions, we are able to generate the expressions of approximate solutions of both the unknown functions and the first-order derivatives, in a direct manner. A set of auxiliary conditions have also been developed so as to construct a complete set of PDEs with mixed Dirichlet and Neumann boundary conditions. Through several structural analysis examples, it is shown that the numerical results at the scattered discrete points generated by the Hermite–Cloud method are distinctly improved, for both the approximate solutions as well as the first-order derivatives.     

Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions

An analytical approach for crack identification procedure in uniform beams with an open edge crack, based on bending vibration measurements, is developed in this research. The cracked beam is modeled as two segments connected by a rotational mass-less linear elastic spring with sectional flexibility, and each segment of the continuous beam is assumed to obey Timoshenko beam theory. The method is based on the assumption that the equivalent spring stiffness does not depend on the frequency of vibration, and may be obtained from fracture mechanics. Six various boundary conditions (i.e., simply supported, simple–clamped, clamped–clamped, simple–free shear, clamped–free shear, and cantilever beam) are considered in this research. Considering appropriate compatibility requirements at the cracked section and the corresponding boundary conditions, closed-form expressions for the characteristic equation of each of the six cracked beams are reached. The results provide simple expressions for the characteristic equations, which are functions of circular natural frequencies, crack location, and crack depth. Methods for solving forward solutions (i.e., determination of natural frequencies of beams knowing the crack parameters) are discussed and verified through a large number of finite-element analyses. By knowing the natural frequencies in bending vibrations, it is possible to study the inverse problem in which the crack location and the sectional flexibility may be determined using the characteristic equation. The crack depth is then computed using the relationship between the sectional flexibility and the crack depth. The proposed analytical method is also validated using numerical studies on cracked beam examples with different boundary conditions. There is quite encouraging agreement between the results of the present study and those numerically obtained by the finite-element method.     

Use of an CXP8 X-ray streak camera for measuring the time characteristics of X rays emitted by a laser plasma

The time dependence of the X-ray intensity from targets exposed to single-shot or successive picosecond laser pulses at the NEODIM facility was obtained by direct measurements. The measurements were taken using an CXP8 streak camera (produced by the Research Institute of Pulse Techniques) with a picosecond time resolution under different operating conditions of the facility. The observed X-ray pulses had a half-height duration of 15–50 ps and a rise time of 5 ps at a ∼1.5-ps duration of the effecting laser pulse. Under actual experimental conditions, it was ascertained that time dependences of the X-ray intensity could be recorded for more than 200 spatial elements on a single record at a sweep duration of 0.5 ns per frame and a time resolution of <5 ps.     

Vibrations of a deformable ball rolling on a lubricated rigid half-space

The problem of vertical vibrations of a deformable ball rolling on a lubricated rigid half-space has been stated and numerically solved. It is assumed that the force, which causes the ball’s elastic deformation in the zone of lubricated concentrated contact, is applied to the ball. The problem’s mathematical model is reduced to the system of nonstationary integrodifferential equations and inequalities with initial and boundary conditions. The computational algorithm of the problem’s solution was based on Newton’s method and application of an implicit scheme to approximate nonstationary equations. Vibrational solutions, which describe the behavior of the system consisting of the ball and the lubrication layer both under excitation of the initial conditions and under the excitation force, have been obtained. It is shown that as opposed to the traditional elastohydrodynamic lubrication theory approach, in which contacting bodies are considered as massless ones, taking the ball’s mass and corresponding inertia into account makes it possible to extend the category of obtained solutions and investigate vibrational processes in the contact zone. The results may be used to analyze processes occurring in lubricated concentrated contacts of gears and rolling bearings.     

含集中参数弹性梁振动特性解析与实验识别

基于Laplace变换推导出含集中质量与集中刚度弹性梁的振型函数和典型边界条件对应的频率方程,针对有一个集中质量与一个集中刚度的悬臂梁,求解出其固有特性,并利用基于自然激励技术(natural excitation technique,简称NExT)的特征系统实现算法(eigensystem realization algorithm,简称ERA),即NExT-ERA法对相应结构系统进行了模态识别。通过对比解析结果和实验结果,分别讨论了集中质量与集中刚度大小变化和位置变化时对梁振动特性的影响,得出了集中质量和集中刚度在悬臂梁上位置和大小变化时,悬臂梁固有频率的相应变化规律,为工程中具有集中质量和集中刚度等直弹性梁的振动分析方法和集中参数布置设计提供了参考。     

基于迭代距离分类与轨迹关联检测空间弱小目标

为了实现高效自动目标检测,提出了一种可用于低信噪比条件下的空间可见光弱小目标检测算法。首先,建立空间光学图像模型,利用恒虚警率(CFAR)方法确定分割系数对单帧图像背景进行分割;然后,基于恒星结构稳定特性构建距离特征空间,并针对特征空间构造分类准则函数,使用迭代最优化分类方法提取出候选目标点;最后,依据目标运动轨迹的连续性建立空间目标轨迹关联、合并以及虚假目标轨迹删除规则,进行轨迹处理,实现空间可见光弱小目标的检测。文中还提出了单帧检测率、虚警率与序列检测率、虚警率相结合的评价方法。实验结果表明:在低信噪比条件下(SNR≤3),得到的序列检测率达到96.02%以上,序列虚警率达到4.4%以下。该方法在低信噪比条件下显著提高了目标检测率,并有效抑制了虚警。     

An analytical solution of a finite bearing with an inclined journal

A solution of Reynolds' equation for a finite bearing with an inclined journal is attempted. A solution (pressure) is expanded as a series of circular trigonometric functions of β′, the coordinate corresponding to circumferential distances, with coefficients as functions of z, the coordinate corresponding to axial distances. Substitution of such a series for pressure in the equation gives an infinite set of coupled ordinary differential equations with varying coefficients. Forms of solutions of this set of equations are assumed in terms of arbitrary constants and with restrictions of continuity at zero values of the inclinations. The arbitrary constants are evaluated using differential equations and boundary conditions. The integrations of pressure for forces and torques are converted into single integrations which are readily evaluated by digital computer.