Transverse vibrations of an Euler–Bernoulli uniform beam carrying two particles in-span

Vibrations of beams carrying different combinations of particles, heavy bodies and spring-mass systems which are located on or off resilient supports have been tackled by several researchers. Most of the approaches were based mainly on various approximate methods. In this paper an analytical solution based on the classical beam eigenvalue technique is presented for the vibrations of a beam carrying two particles. For purpose of analysis, the beam was divided into a portion from one end to the first particle, a portion between particles and a portion from the second particle to the other end. The frequency equation is expressed in closed form as a 2nd order determinant equated to zero. Schemes are presented to compute the 4 elements of the determinant and to evaluate the roots of the frequency equation. Computational difficulties were not encountered in the implementation of the schemes. The first three natural frequency parameters are tabulated for 16 combinations of the classical boundary conditions and several combinations of the location and mass of the particles. The beam mode shape is the juxtaposition of the mode shapes of the three portions of the beam. Some examples of normalised beam mode shapes and location/s of node/s are also presented. The results may be used to judge the accuracy of values obtained by approximate methods.     

Vibration of an Euler–Bernoulli beam on elastic end supports and with up to three step changes in cross-section

Compared to the bibliography on the transverse vibration of Euler–Bernoulli beams with one step change in cross-section, publications on beams with more than one step changes is not extensive. In this paper an analytical method is proposed to calculate the frequencies of beams with up to three step changes in cross-section. Combinations of the classical clamped, pinned, sliding, free, ‘general’ and ‘degenerate’ types of elastic end supports are considered. The frequency equations of stepped beams were expressed as fourth order determinant equated to zero. A scheme to calculate the elements of the determinant and a scheme to evaluate the roots of the determinant are presented. Special consideration is given to three types of stepped beams frequently encountered in engineering applications. The first three frequency parameters of beams with two and three step changes in cross-section are tabulated for selected sets of system parameters and 45 types of end supports. Computational difficulties were not encountered. The method proposed may be extended to tackle beams with any number of step changes in cross-section. The tabulated results may be used to judge the frequencies calculated by numerical methods.     

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.     

A study of the friction and wear performance of MoSx thin films produced by ion beam enhanced deposition and magnetron sputtering

MoS_x thin films were deposited by ion beam enhanced deposition (IBED) and magnetron sputtering (MS) onto the surface of IBEN Si₃N₄ and TiN thin films. The friction and wear performances of thin films and 52100 steel were compared using an SRV model reciprocating testing machine. The results showed that all MoS_x films exhibit good tribological behavior. The MS MoS_x thin film has better wear resistance and the IBED MoS_x film has a longer wear life. The wear resistance of IBED Si₃N₄ and TiN thin film plus MoS_x film is 3–4 times and 8–20 times that of single IBED Si₃N₄ and TiN thin films and 52100 steel respectively. The analyses indicate that the difference in friction and wear performance between the two kinds of MoS_x thin film is determined by the x value of MoS_x, its microstructure and the atom mixing effect at the interface.     

A conversion method from the E system to the M system in measuring surface roughness

The relations between the surface roughness parameters measured by the envelope datum system (E system) and those measured by the mean line datum system (M system) are analysed theoretically. Experimental relations between the two systems are obtained by calculation from the surface roughness profiles of machined surfaces.

From the theoretical and experimental results it was found that the levelling depth R_P(E) in the E system is proportional to the mean arithmetic deviation R_a(M) in the M system and the ratio R_P(E)/R_a(M) increases with an increase in the tip radius of the skid. The mean roughness value R_a(E) in the E system is larger than R_a(M) by 10% at its maximum.     

异性分形表面合成方法及表面特性研究

针对异性分形表面的建模,提出了一种以二维离散傅里叶变换为基础,以名义维数D_x,D_y为驱动参数的表面合成方法——合成维数法。通过对比名义维数与采用轮廓功率谱公式和数值分析法得到的表面计算维数的关系,验证了该方法的有效性。探讨了名义维数对x,y方向轮廓能量分布以及对表面及轮廓粗糙度的影响。一方面在轮廓功率谱分析的基础上定义了轮廓累计功率比,用于描述表面轮廓的能量分布特性,不仅促进了对名义维数和轮廓能量关系的定性理解,更能定量地解释不同维数值影响下频率和轮廓能量分布的关系。另一方面以能量分析为切入点,建立了名义维数与表面及轮廓粗糙度参数间的联系,并且依据能量守恒定律,证明并揭示了轮廓粗糙度R_qx,R_qy之间以及二者与面粗糙度S_q的关系。同时得出结论,D_x或D_y值越小,低频部分能量占比越高,反之亦然,但是x,y方向轮廓总功率始终相等,并随D_x,D_y增大而增大;D_x,D_y只影响S_q,R_qx,R_qy的绝对数值,二者的增大会引起粗糙度值增加,但并不影响R_qx,R_qy的相对关系。     

圆渐开线变截面涡旋压缩机几何性能综合分析

提出一种由不同基圆半径的圆渐开线组成的新型变截面涡旋压缩机型线,组成形式为圆渐开线Ⅰ+圆渐开线Ⅱ+圆渐开线Ⅰ。论述型线的生成方法,给出型线的一般方程,建立一系列圆渐开线变截面涡旋压缩机的几何模型。针对建立的几何模型,分析控制系数θ、φ^*、R_or对变截面涡旋压缩机几何性能的影响。以圆渐开线Ⅰ为基础,构建圆渐开线Ⅰ+高次曲线+圆渐开线Ⅰ的变截面涡旋压缩机的几何模型,综合分析两类变截面涡旋压缩机的几何性能。结果表明：θ取中值θ_M,φ^*,控制系数R_or取较大值,对应的几何性能较优。高次曲线变截面涡旋压缩机与中值θ_M对应的圆渐开线变截面涡旋压缩机相似,可相互替代。     

Method of caustics for the study of cracked plates made of viscoelastic materials

The optical method of caustics was used for the evaluation of the stress intensity factor at the crack tip in a viscoelastic plate made of polycarbonate of bisphenol A and containing a single external crack under conditions of generalized plane stress. It was assumed that the crack developed in a linear viscoelastic material. The reflected rays from both lateral faces, as well as the transmitted rays of a normally incident light beam were significantly dispersed at the close neighborhood of the crack tip because of the excessive lateral constraint and the strong variation of the refractive index, and concentrated along singular curves (caustics) at some distances from the plate. The reflected caustics formed two branches. While the internal branch, which was formed by reflections from the front face, depended only on the elastic modulus (E) and Poisson's ratio (v) of the material, the external branch, formed by reflections from the rear face, depended on E and v, as well as on the refractive index of the material (n). The transmitted rays formed the same type of caustic as that which was formed by reflections from the rear face, but with a different size.

By studying the relative position and shape of the two reflected branches and their variation with time, information was derived concerning the viscoelastic state of the material at fracture. The transmitted light caustic was used as a check on the results.

A series of viscoelastic tests in pure tension at various stress and strain levels yielded the variation of the mechanical (E, v) and optical (n, c_r, c_f) properties of the material with applied stress and strain. The values of these quantities at each stress level allowed the plotting of the relative positions of the two branches of the reflected caustics. Comparison of the actual position of the caustics at the crack tip with the caustic nomogram yielded the viscoelastic behavior of the material during the progress of loading. It was shown that by increasing the loading mode a strong viscoelastic behaviour was apparent which influenced considerably the fracture mode of the plate.     

Analytical Surface Roughness Parameters of a Theoretical Profile Consisting of Elliptical Arcs

The closed-form solutions of surface roughness parameters for a theoretical profile consisting of elliptical arcs are presented. Parabolic and simplified approximation methods are commonly used to estimate the surface roughness parameters for such machined surface profiles. The closed-form solution presented in this study reveals the range of errors of approximation methods for any elliptical arc size. Using both implicit and parametric methods, the closed-form solutions of three surface roughness parameters, R_t, R_a, and R_q, were derived. Their dimensionless expressions were also studied and a single chart was developed to present the surface roughness parameters. This research provides a guideline on the use of approximate methods. The error is smaller than 1.6% when the ratio of the feed and major semi-axis of the elliptical arc is smaller than 0.5. The closed-form expressions developed in this study can be used for the surface roughness modeling in CAD/CAM simulations.     

Three-dimensional vibration analysis of thick, tapered rods and beams with circular cross-section

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, tapered rods and beams with circular cross-section. Unlike conventional rod and beam theories, which are mathematically one-dimensional (1-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_r, u_θ, and u_z in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the rods and beams are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four- digit exactitude is demonstrated for the first five frequencies of the rods and beams. Novel numerical results are tabulated for nine different tapered rods and beams with linear, quadratic, and cubic variations of radial thickness in the axial direction using the 3-D theory. Comparisons are also made with results for linearly tapered beams from 1-D classical Euler–Bernoulli beam theory.     

电子注分析系统中三坐标精密工作台的设计与控制

提出一种利用收集极在三维空间内通过束流采样进行电子注分析的测量方法,介绍了电子注分析系统中三坐标精密移动工作台的设计与控制。该方法可对电子注发射系统的结构参数、工作性能进行全面测量和分析,为超高频大功率电子器件的研制提供实验手段。     

Correlations between microstructural parameters, micromechanical properties and wear resistance of plasma sprayed ceramic coatings

The micromechanical integrity of a ceramic plasma sprayed (PS) coating is determined by the size and distribution of the defects found in the coating, such as porosity, the inter-lamellar microcrack density, the intra-lamellar microcrack density as well as the lamellar, or splat, dimensions. In this work, several micromechanical tests were used to advance our understanding of the relationships between the different microstructural parameters found in PS ceramic coatings. The tests included depth sensing indentation, micro and macrohardness testing, and controlled scratch testing. Abrasive and erosive wear tests were performed on the same set of coatings, including plasma sprayed alumina and chromia, as well as sintered alumina as a reference material. The best correlations were found between the material hardness (H), the level of porosity (P) and the abrasive wear volume (W). Knoop hardness measurements provided the best correlation with wear data, followed by scratch hardness and Vickers hardness. An exponential function of the type W=k/Hⁿ was found, where k and n are constants. A similar function describes the correlation of wear volume with the elastic modulus of the coating. Fracture toughness could only be correlated with wear volume when combined with hardness in a function of the type W=k/H^0.5K_c^0.5. The incorporation into this function of a “microstructural factor” M=Pⁿ improves the correlation.     

Compliance of rough cylinders in compression

Compression of a rough turned cylinder between two hard, smooth, flat plates has been analysed with the aid of a mathematical model based on statistical analysis. It is assumed that the asperity peak heights follow Gaussian or normal and beta distribution functions and that the loaded asperities comply as though they are completely isolated from the neighbouring ones. Equations have been developed for the loadcompliance relation of the real surface using a simplified relation of the form W₀ = K₁ δⁿ for the load-compliance of a single asperity. Parameters K₁ and n have considerable influence on the load-compliance curve and they depend on the material, tip angle of the asperity, standard deviation of the asperity peak height distribution and the density of the asperities.     

X轴分离式高速原子力显微镜系统设计

为了提高原子力显微镜(Atomic Force Microscope,AFM)的成像速度,本文提出了一种新的AFM结构设计方案并搭建了相应的实验系统。在该方案中,Y、Z扫描器集成于测头内驱动探针进行慢轴扫描和形貌反馈;X扫描器与测头分离,驱动样品做快轴扫描。X扫描器采用高刚性的独立一维纳米位移台,能够承载尺寸和质量较大的样品高速往复运动而不易发生共振;同时Z扫描器的载荷实现最小化,固有频率得以显著提高。为了避免测头的扫描运动引起检测光束与探针相对位置的偏差,设计了一种随动式光杠杆光路;为了便于装卸探针以及精确调整激光在探针上的反射位置,设计了基于磁力的探针固定装置和相应的光路调节方案。对所搭建的AFM系统的初步测试结果表明,该系统在采用三角波驱动和简单PID控制算法的情况下,可搭载尺寸达数厘米且质量超过10g的较大样品实现13μm×13μm范围50Hz行频的高速成像。     

In-plane and out-of-plane buckling of arches made of FGM

The mechanical buckling of curved beams made of functionally graded materials is studies in this paper. The equilibrium and stability equations of curved beams under mechanical loads are derived. Using proper approximate functions for the displacement components, the stability equations are employed to obtain the related eigenvalues associated with the buckling load of the curved beam. Closed-form solutions are obtained for mechanical buckling of curved beams with doubly symmetric cross section subjected to uniform distributed radial load and pure bending moment. The results are validated with the known data in the literature for beams with isotropic materials.     

Parameters estimation of sandwich beam model with rigid polyurethane foam core

In this work, the physical parameters of sandwich beams made with the association of hot-rolled steel, Polyurethane rigid foam and High Impact Polystyrene, used for the assembly of household refrigerators and food freezers are estimated using measured and numeric frequency response functions (FRFs). The mathematical models are obtained using the finite element method (FEM) and the Timoshenko beam theory. The physical parameters are estimated using the amplitude correlation coefficient and genetic algorithm (GA). The experimental data are obtained using the impact hammer and four accelerometers displaced along the sample (cantilevered beam). The parameters estimated are Young's modulus and the loss factor of the Polyurethane rigid foam and the High Impact Polystyrene.     

钢筋混凝土简支梁超载后动力特性试验研究

基于采用动态法评估桥梁结构构件超载后损伤的目的,设计和开展了试验研究和相应理论分析。通过对钢筋混凝土简支梁进行反复超载试验,测出构件从完整到破损,以及不同超载幅值和次数对应损伤状态下的频率和阻尼比等模态参数,考查了其与构件承载力的关系,为采用结构动力参数评估结构损伤分析了频率和阻尼比随损伤状态不同的变化规律,根据试验结果得到频率和荷载等级关系的经验公式,为采用动态法评估既有桥梁提供了理论依据     

用于光束整形的表面等离子体双光栅结构

为了对双边布拉格反射波导半导体激光器的远场双瓣特性进行整形,使之合并成为单瓣出光远场,在布拉格反射波导的出光腔面上制作了表面等离子体双光栅结构。利用Au-SiO₂光栅结构对表面等离子体的耦合效应和表面等离子体的透射增强现象将双瓣远场耦合成为单瓣出射,然后通过Au-Si₃N₄光栅结构将透射的表面等离子体耦合成为光子进行准直出射,最终得到单瓣准直的远场光斑。计算结果表明：当Au-SiO₂光栅厚度为50 nm,填充因子为0.5,光栅周期为350 nm;Au-Si₃N₄光栅厚度为70 nm,填充因子为0.5,光栅周期为660 nm时可以得到远场发散角压缩到6.1°的整形光斑,比没有双光栅结构的发散角缩小了3.6倍;其远场透射光功率达到了模式光源的62%,是没有双光栅结构单瓣出射功率的1.59倍;同时腔面反射率也降低到12.4%,是没有双光栅结构的0.53倍。结果显示,提出的双光栅结构优化了布拉格反射波导半导体激光器的出光远场特性。     

考虑翘曲影响的Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀薄壁梁单元弯扭耦合振动的运动偏微分方程，推导其自由振动时的精确动态传递矩阵。采用考虑翘曲影响的Bernoulli-Euler梁理论，且假定薄壁梁单元的横截面是单对称的。动态传递矩阵可以用于计算薄壁梁集合体的精确固有频率和模态形状。针对两个薄壁梁算例，采用自动Muller法和结合频率扫描法的二分法求解频率特征方程，并讨论翘曲刚度对弯扭耦合：Bernoulli-Euler薄壁梁固有频率的影响。数值结果验证了本文方法的精确性和有效性，并指出翘曲刚度可以显著改变薄壁开口截面梁的固有频率。