Development of instability in a rotating elastoplastic annular disk

The paper proposes a method, based on perfect-plasticity and perturbation theories, for instability analysis of an annular flat disk tightly set on a shaft with no interference fit. The perturbed elastoplastic state of the rotating disk is analyzed by determining the stress–strain state of a fixed elastic annular plate under in-plane loading. A characteristic equation of the first order for the critical radius of the plastic zone in the disk subject to internal pressure is derived. The critical rotation rate is calculated for different parameters of the disk     

弹性连接旋转柔性梁动力学分析

采用Chebyshev 谱方法对考虑根部连接弹性的平面内旋转柔性梁动力学特性进行研究. 基于Gauss-Lobatto 节点与Chebyshev 多项式方法对柔性梁变形场进行离散,通过投影矩阵法施加固定及弹性连接边界条件. 利用Chebyshev 谱方法获得了系统固有频率和模态振型数值解,通过与有限元方法及加权残余法的比较,验证了方法的有效性. 分析了弹性连接刚度、角速度比率、系统径长比及梁的长细比等参数对系统固有频率及模态振型的影响. 研究发现:由于系统弯曲模态、拉伸模态的频率随各参数的变化规律不一致,将出现频率转向与振型转换现象;随着弹性连接刚度、角速度比率及系统径长比的增大,低阶弯曲模态频率增大并超过高阶拉伸模态频率,随着梁的长细比的增大,低阶拉伸模态频率增大并超过高阶弯曲模态频率.      

Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory

Analytical solutions for bending, buckling, and vibration of micro-sized plates on elastic medium using the modified couple stress theory are presented. The governing equations for bending, buckling and vibration are obtained via Hamilton’s principles in conjunctions with the modified couple stress and Kirchhoff plate theories. The surrounding elastic medium is modeled as the Winkler elastic foundation. Navier’s method is being employed and analytical solutions for the bending, buckling and free vibration problems are obtained. Influences of the elastic medium and the length scale parameter on the bending, buckling, and vibration properties are discussed.     

Dynamic bifurcations analysis of a micro rotating shaft considering non-classical theory and internal damping

In this study, the dynamic bifurcation of a viscoelastic micro rotating shaft is investigated. The non-classical theory (the modified couple stress theory) and the Kelvin Voigt model are used for modeling the viscoelastic micro shaft. The transverse equations of motion are derived using the variational approach. The reduced order model of the system is obtained by the Galerkin method. Using the Routh–Hurwitz criteria the stability regions of the system are extracted in which the effect of the length scale parameter is significant. Using the center manifold theory and the normal form method the double zero eigenvalue bifurcation is analyzed. The results show that the internal and external damping coefficients, the rotational speed and the material length scale parameter influence the critical speed, amplitude, and phase of a non-trivial solution, and radius of limit cycle (periodic solution). Also, it is seen that by increasing the dimensionless length scale parameter (material length scale per radius of the shaft) the radius of the limit cycle is decreased, whereas the critical rotational speed and the rate of the phase are increased. However, the radius of the limit cycle concerning the classical theory is higher than that of regarding the modified couple stress theory. Furthermore, with an increase of the external damping coefficient the radius of the limit cycle is linearly decreased; however, the critical speed of the system is increased. Additionally, by decreasing length scale parameter the results of the modified couple stress theory approach the classical theory ones.     

Numerical simulation of contact problems in vane machinery by a parametric quadratic programming method

Interference fits are widely used for connecting impeller and shaft assembly that are forced together slowly by pressing. The interference fit design ensures stable balance behavior and allows for positive contact between the impeller and shaft assembly throughout the range of operating speeds. In addition to maintaining radial contact, sufficient net radial interface pressure must remain in order to transmit torque when the rotational speed is very high. Therefore, the interference fit between the impeller and the shaft assembly is one of the most important factors influencing the performance of the turbo unit in the design of turbocharger compressor. A suitable fit tolerance needs to be considered in the structural design. A locomotive-type turbocharger compressor with 24 blades under combined centrifugal and interference fit loading is used for the analysis. The finite-element (FE) parametric quadratic programming (PQP) method developed based on the parametric variational principle (PVP) is used for the analysis of the stress distribution in the three-dimensional (3D) contact problem of impeller. The advantages of the parametric programming method compared with conventional approaches are that the penalty factors can be canceled and that solutions can be obtained directly without tedious iterative procedures such as the general incremental iterative method. To save time in the computation, a~multi-substructure technique is adopted for structural modeling. This not only simplifies the calculation, but also provides a convenient service for process computer-aided design (CAD) by means of FE simulation. The effects of the fit tolerance, coefficient of friction and rotational speed (centrifugal force), wall thickness of the shaft sleeve and the contact stress on the interference-fitting surfaces are studied in detail in the numerical computation. It is found that a nonuniform initial amount of interference in the structural design avoids the relative displacement generated and ensures uniformity of the contact stress. To assure quality of press-fitting, the amount of interference between the shaft sleeve and shaft should be strictly controlled to avoid the rapid increase of the contact stress. The numerical results demonstrate the high accuracy and good convergence of the algorithm presented here, which provides an effective approach that achieves more-reliable interference-fitted connections and more-precise assembly accuracy with lower manufacturing cost in the structural design.     

Relection of elastic waves at the elastically supported boundary of a couple stress elastic half-space

The relection elastic waves at the elastically supported boundary of a couple stress elastic half-space are studied in this paper. Different from the classical elastic solid, there are three kinds of elastic waves in the couple stress elastic solid, and two of them are dispersive. The boundary conditions of a couple stress elastic half-space include the couple stress vector and the rotation vector which disappear in the classical elastic solids. These boundary conditions are used to obtain a linear algebraic equation set, from which the amplitude ratios of relection waves to the incident wave can be determined. Then, the relection coeficients in terms of energy lux ratios are calculated numerically, and the normal energy lux conservation is used to validate the numerical results. Based on these numerical results,the inluences of the boundary parameters, which relect the mechanical behavior of elastic support, on the relection energy partition are discussed. Both the incident longitudinal wave(the P wave) and incident transverse wave(the SV wave) are considered.     

The photoelastic-coating technique for plastic-elastic contact

Steel models coated with thin birefringent layers representing transverse sections through British-Standardinch keyed shafts and hubs have been loaded in torsion. Models with two different steel shafts and clearance fits between key and keyways and between shaft and hub have been analyzed over a large range of torques. Large plastic strains occurred at six contact positions where there are discontinuities of profile and surface contact of the chamfered key or the shaft. Contact strains have been measure, corrected for shear lag and normalized. Plastic-elastic SCF's are lower than corresponding elastic values.     

Optimal design of a compound plate weakened by a periodic crack system

We use the minimax criterion to perform atheoretical analysis of determining the optimal interference for fitting elastic inclusions into holes in an isotropic elastic plate weakened by a periodic system of rectilinear cracks. We construct a closed system of algebraic equations that allows us to minimize the fracture parameters depending on the geometric and mechanical characteristics of the inclusions. The obtained fitting interference for the inclusions ensures an increase in the bearing strength of the compound plate under bending.     

Reduction of elastic stress concentrations in end-milled keyed connections

Frozen-stress, three-dimensional models were used to study the elastic stress distribution in both metric and inch rectangular, parallel, standard keys, shafts and hubs with end-milled keyways, loaded in torsion. The coefficient of friction between the shaft and the hub was measured and the surface conditions were arranged to simulate typical prototype friction. Positions and magnitudes of the peak shear stresses were determined in the prismatic part and at the ends of keys and keyways. The axial position of torque application to the hub, the key length and key-end shape were altered to investigate their effects on the stress distributions. The maximum shear stresses in the shaft occur in the keyway end if the key is not chamfered. The maximum stress in the standard metric shaft is 25 percent greater than in the inch standard. Changing the axial position of torque application from the middle to the front increases the peak stresses by about 13 percent. Doubling the length of hub and key reduces the stresses by about 25 percent. Filing off parts of the key to prevent contact at the keyway end reduces the maximum stresses by 11 percent.     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

Vibration analysis of a rotating tapered Timoshenko beam using DTM

In this study, free vibration analysis of a rotating, tapered Timoshenko beam that undergoes flapwise bending vibration is performed. Derivation of the equations of motion of a rotating, uniform Timoshenko beam was made step by step in a previous work of the authors. Therefore, differential equations of motion are given directly without making any derivations in this paper. The parameters for the hub radius, rotational speed, taper ratio, rotary inertia, shear deformation and slenderness ratio are incorporated into the equations of motion. In the solution part, an efficient mathematical technique called the Differential Transform Method, DTM, is used. Finally, using the computer package Mathematica, the natural frequencies are calculated and the effects of the incorporated parameters are examined. Moreover, numerical examples are solved to make comparisons with the existing results in open literature and it is observed that the agreement between the results is very good.     

Mode I stress intensity factor solutions for spot welds in lap-shear specimens

The analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens are investigated based on the classical Kirchhoff plate theory for linear elastic materials. First, closed-form solutions for an infinite plate containing a rigid inclusion under counter bending conditions are derived. The development of the closed-form solutions is then used as a guide to develop approximate closed-form solutions for a finite square plate containing a rigid inclusion under counter bending conditions. Based on the J integral, the closed-form solutions are used to develop the analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens of large and finite sizes. The analytical solutions of the mode I stress intensity factor based on the solutions for infinite and finite square plates with an inclusion are compared with the results of the three-dimensional finite element computations of lap-shear specimens with various ratios of the specimen half width to the nugget radius. The results indicate that the mode I stress intensity factor solution based on the finite square plate model with an inclusion agrees well with the computational results for lap-shear specimens for the ratio of the half specimen width to the nugget radius between 4 and 15. Finally, a set of the closed-form stress intensity factor solutions for lap-shear specimens at the critical locations are proposed for future applications.     

大展弦比机翼中液压管路的抗大变形设计与优化方法

提出了一种大展弦比机翼管路的抗大变形设计与优化方法。首先建立了大展弦比机翼平板-不同布局管路的装配简化模型;然后分析了在机翼大变形下,管道的弯曲位置、弯曲半径、横向距离、弯曲角度等几种不同布局参数,对管路根部应力、最大应力和卡箍处变形的影响关系。结果表明:弯曲位置与横向距离对应力有较大影响,弯曲位置靠近机翼根部可以降低管道根部应力,但是最大应力显著增加,横向距离的增加可以降低管路根部应力以及最大应力;弯曲位置和弯曲半径对卡箍处变形有较大影响,随着弯曲位置从机翼板根部向变形处移动,卡箍处变形量均先减小后增加,弯曲半径的增加会降低卡箍处变形量。采用遗传算法得到在机翼大变形下最优的管形布局,结果表明,卡箍附近最大应力比直管降低了51%。     

Optimization of straight-sided spline design

Spline connection of shaft and hub is commonly applied when large torque capacity is needed together with the possibility of disassembly. The designs of these splines are generally controlled by different standards. In view of the common use of splines, it seems that few papers deal with splines and the subject of improving the design. The present paper concentrates on the optimization of splines and the predictions of stress concentrations, which are determined by finite element analysis (FEA). Using different design modifications, that do not change the spline load carrying capacity, it is shown that large reductions in the maximum stress are possible. Fatigue life of a spline can be greatly improved with up to a 25% reduction in the maximum stress level. Design modifications are given as simple analytical functions (modified super elliptical shape) with only two active design parameters and the designs are practical realizable. Some generality is found in the design parameters.     

Flapwise bending vibration analysis of a rotating double-tapered Timoshenko beam

In this study, free vibration analysis of a rotating, double-tapered Timoshenko beam that undergoes flapwise bending vibration is performed. At the beginning of the study, the kinetic- and potential energy expressions of this beam model are derived using several explanatory tables and figures. In the following section, Hamilton’s principle is applied to the derived energy expressions to obtain the governing differential equations of motion and the boundary conditions. The parameters for the hub radius, rotational speed, shear deformation, slenderness ratio, and taper ratios are incorporated into the equations of motion. In the solution, an efficient mathematical technique, called the differential transform method (DTM), is used to solve the governing differential equations of motion. Using the computer package Mathematica the effects of the incorporated parameters on the natural frequencies are investigated and the results are tabulated in several tables and graphics.     

Strength analysis of an elastic plane containing a square lattice of circular holes under mechanical loading

The strength characteristics of an elastic plane weakened by a square lattice of circular holes are considered. The stress concentrations in three distinct lattices under the conditions of uniaxial tension/compression in various directions are studied. The minimum and maximum values of the stress concentrations are calculated, and the stress fields in various lattices are considered. We show that under the compression conditions fracture can occur inside the material rather than on the hole boundaries. It is demonstrated that in dense lattices the stress concentrations exhibit power-law dependence on the structure parameter equal to the ratio of the length of the interval between the holes to the hole radius.     

基于平板振动精确化方程求解动应力集中问题

基于文献[8]给出的平板弯曲振动精确化方程,对含圆孔平板中弹性波散射与动应力集中问题进行了研究.文中给出了分别基于Mindlin板与精确化板方程在不同参数下圆孔动弯矩集中系数的数值结果,并对结果进行了对比分析和讨论.结果表明:在较低频率和薄板情况下,基于文献[8]的方程与基于Mindlin板理论得到的动弯矩结果是基本一致的;在较高频率和厚板情况下,基于文献[8]的方程与基于Mindlin板理论的动弯矩结果相差较大,最大值超出可达16%.由于文献[8]给出的平板振动精确化方程是在没有任何工程假设条件下得到的,因此其分析计算结果更精确一些.      

含孔von Karman板中非线性波散射与边值问题

基于ｖｏｎ Ｋａｒｍａｎ板大挠度弯曲理论,利用小参数摄动法,分析研究了含孔ｖｏｎＫａｒｍａｎ板的非线性波散射与动应力集中问题,其中一类可看成是薄板弯曲波动问题的控制方程。当有单频波入射时,由于弯曲应力与膜应力状态的非线性耦合,孔洞会产生高次谐波散射现象。建立了求解本问题的边界积分方程法,利用积分方程法交替求求这两类问题,最终可获得问题的近似分析解。     

用高阶剪切变形理论研究双模量梁的弯曲

利用高阶剪切变形理论研究了双模量梁的弯曲变形问题,推导出了双模量梁的挠曲线方程及弯曲正应力公式．讨论分析了翘曲函数的指数n对挠度、正应力的影响．研究结果表明：拉压弹性模量的差异对梁的弯曲应力有较大影响．把高阶剪切变形理论的计算结果与弹性理论计算结果进行比较,可知该方法计算精度非常高．     

Size-dependent thermoelastic initially stressed micro-beam due to a varying temperature in the light of the modified couple stress theory

The bending of the Euler-Bernoulli micro-beam has been extensively modeled based on the modified couple stress (MCS) theory. Although many models have been incorporated into the literature, there is still room for introducing an improved model in this context. In this work, we investigate the thermoelastic vibration of a micro-beam exposed to a varying temperature due to the application of the initial stress employing the MCS theory and generalized thermoelasticity. The MCS theory is used to investigate the material length scale effects. Using the Laplace transform, the temperature, deflection, displacement, flexure moment, and stress field variables of the micro-beam are derived. The effects of the temperature pulse and couple stress on the field distributions of the micro-beam are obtained numerically and graphically introduced. The numerical results indicate that the temperature pulse and couple stress have a significant effect on all field variables.