Limit angular velocities of variable thickness rotating disks

Elastic and plastic limit angular velocities are calculated for rotating disks of variable thickness in power function form. Analytical solution is obtained and used to calculate elastic limit angular velocities of variable thickness rotating annular disks and annular disks with rigid inclusion. An efficient numerical solution procedure is designed and used to obtain the elastic limit angular velocities of variable thickness rotating solid disks. Von Mises yield criterion and its flow rule is used to estimate plastic limit angular velocities. Both linear and nonlinear hardening material behaviors are treated numerically. The results are verified by comparing with those of uniform thickness rotating solid disks available in the literature. Elastic and plastic limit angular velocities are found to increase with the reduction of the disk thickness at the edge as well as the reduction in the disk mass due to the shape of the profile.     

Elastoplastic analysis of nonlinearly hardening variable thickness annular disks under external pressure

Based on von Mises’ yield criterion, deformation theory of plasticity and Swift’s hardening law, elasto-plastic deformation of variable thickness annular disks subjected to external pressure is studied. A nonlinear shooting method using Newton’s iterations with numerically approximated tangent is designed for the solution of the problem. Considering a thickness profile in the form of a general parabolic function, the condition of occurrence of plastic deformation at the inner and outer edges of the annular disk is investigated. A critical disk profile is determined and the corresponding elastic–plastic stresses as well as the residual stress distribution upon removal of the applied pressure are computed and discussed.     

Numerical analysis of elastic–plastic rotating disks with arbitrary variable thickness and density

A unified numerical method is developed in this article for the analysis of deformations and stresses in elastic–plastic rotating disks with arbitrary cross-sections of continuously variable thickness and arbitrarily variable density made of nonlinear strain-hardening materials. The method is based on a polynomial stress–plastic strain relation, deformation theory in plasticity and Von Mises’ yield condition. The governing equation is derived from the basic equations of the rotating disks and solved using the Runge–Kutta algorithm. The proposed method is applied to calculate the deformations and stresses in various rotating disks. These disks include solid disks with constant thickness and constant density, annular disks with constant thickness and constant density, nonlinearly variable thickness and nonlinearly variable density, linearly tapered thickness and linearly variable density, and a combined section of continuously variable thickness and constant density. The computed results are compared to those obtained from the finite element method and the existing approaches. A very good agreement is found between this research and the finite element analysis. Due to the simplicity, effectiveness and efficiency of the proposed method, it is especially suitable for the analysis of various rotating disks.     

Elastoplastic deformations of rotating parabolic solid disks using Tresca''s yield criterion

Analytical solutions for the stress distribution in rotating parabolic solid disks are obtained. The analysis is based on Tresca's yield criterion, its associated flow rule and linear strain hardening. It is shown that, the deformation behavior of the convex parabolic disk is similar to that of the uniform thickness disk, but in the case of concave parabolic solid disk, it is different. In the latter, the plastic core consists of three different plastic regions with different mathematical forms of the yield criteria. Accordingly, three different stages of elastic–plastic deformation occur. All these stages of elastic–plastic deformation are studied in detail. It is also shown mathematically that in the limiting case the parabolic disk solution reduces to the solution of rotating uniform thickness solid disk.     

Elastic–plastic stresses in linearly hardening rotating solid disks of variable thickness

The distribution of stress, displacement and plastic strain in a rotating elastic–plastic solid disk of variable thickness in a power function form is investigated. The analysis is based on Tresca's yield condition, its associated flow rule and linear strain hardening material behavior. An analytical solution is obtained and numerical results are presented for different values of the geometric parameters. The validity of the solution is demonstrated by comparing the results with those for a uniform thickness disk available in the literature.     

Elastic-plastic stress distribution in the rotating annular disk with variable thickness

Summary The plane state of stress in a rotating annular disk with variable thickness is studied. The analysis is based on Tresca's yield condition, its associated flow rule and linear strain hardening.

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Die elastisch-pastische Spannungsverteilung in einem rotierenden Kreisring mit veränderlicher Dicke

Übersicht Der ebene Spannungszustand in einem rotierenden Kreisring mit veränderlicher Dicke und aus Material mit linearer Verfestigung wird unter Zugrundelegung der Trescaschen Fließbedingung und der zugehörigen Fließregel untersucht.

     

Analysis of functionally graded rotating disks with variable thickness

Elastic solutions for axisymmetric rotating disks made of functionally graded material with variable thickness are presented. The material properties and disk thickness profile are assumed to be represented by two power-law distributions. In the case of hollow disk, based on the form of the power-law distribution for the mechanical properties of the constituent components and the thickness profile function, both analytical and semi-analytical solutions are given under free–free and fixed-free boundary conditions. For the solid disk, only semi-analytical solution is presented. The effects of the material grading index and the geometry of the disk on the stresses and displacements are investigated. It is found that a functionally graded rotating disk with parabolic or hyperbolic convergent thickness profile has smaller stresses and displacements compared with that of uniform thickness. It is seen that the maximum radial stress for the solid functionally graded disk with parabolic thickness profile is not at the centre like uniform thickness disk. Results of this paper suggest that a rotating functionally graded disk with parabolic concave or hyperbolic convergent thickness profile can be more efficient than the one with uniform thickness.     

Displacement Field and Strain Distribution in a Rotating Annular Disk

Abstract

The displacement field and strain distribution in a thin rotating disk with constant thickness and density are found based on Mises’ yield criterion and its associated flow rule. The material of the disk is elastic-perfectly plastic and the assumption of plane stress is adopted. The solution is illustrated by an example.     

内压作用下弯管的塑性极限载荷分析

在变壁厚椭圆截面弯管应力分析的基础上,运用Tresca 和von Mises 屈服准则,对承受内压作用的钢制弯管进行了极限载荷分析,推导出考虑弯管截面壁厚变化和弯管椭圆度的变壁厚椭圆弯管的塑性极限压力计算式. 弯管的极限载荷随着弯管的壁厚和弯管的椭圆度的不同而变化.     

Rotor oscillation and stability in complex motion

Precession vibration of a rigid disk with unequal axial moments of inertia is considered when the axis of rotation turns; the disk is located asymmetrically on a flexible axle. Periodic solutions of the equations of motion and the amplitude-frequency relations are obtained for various values of the angular velocity of the axis of rotation. The critical rotational velocities of disks with various moments of inertia are defined in terms of the gyroscopic forces. The stability of motion is analyzed for various angular velocities of the rotating axis. State Technical University of Building and Architecture, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 7, pp. 104–107, July, 1999.     

加速旋转变厚度锥壳和圆盘中的应力

通过求解球坐系中的弹性力学平衡方程，得到线性变厚度锥壳和圆盘在角加速旋转中位移和应力的封闭形式解，从而发现锥壳式圆盘中的剪应力只与几何尺寸、材料密度和角加速度有关，而与材料的弹性常数无关，文末的数字结果表明，变厚度圆盘的变形和应力与等厚度圆盘相比有较大的不同。     

Elastic stress analysis of non-linear variable thickness rotating disks subjected to thermal load and having variable density along the radius

An analytical procedure for evaluation of elastic stresses and strains in non-linear variable thickness rotating disks, either solid or annular, subjected to thermal load, and having a fictitious density variation along the radius is presented. Thickness variation of disks is described by means of a power of linear function, which can be used to describe a fourfold infinity of actual disk profiles. The procedure is based on two independent integrals of the hypergeometric differential equation describing the displacement field; this theoretical procedure is just general and does not present limitations and drawbacks of the approaches as the one found in technical literature. General unpublished relations of stress state and displacement field in non-linear variable thickness disks subjected, under elastic conditions, to thermal gradient, and featuring a density variation along the radius are defined. Particular consideration is given to some industrial example of turbine rotors carrying hub and rim with buckets on periphery or radial blades on lateral surfaces. The analytical results obtained by using the new general relations perfectly match those obtained by FEA and overlap those concerning the special cases of tapered conical disks found in literature.     

Critical state of imbalanced rotating anisotropic disks with small radial and shear moduli

Critical speeds and the mass center movement of an imbalanced, circumferentially stiff, radially compliant, rotating annular disk on a stiff shaft and bearings are evaluated. It is demonstrated that recently developed hoop-wound composite material disks having elastomeric resin and carbon fibers can enter a critical state before reaching the circumferential strength limit of the material if certain material and geometric relationships are met.     

RV减速器主支撑角接触球轴承混合润滑分析

针对RV减速器角接触球轴承承受预紧力、轴向力和径向力等联合外载荷作用的工况,分析得出了内、外圈滚道接触界面的接触区几何参数和接触载荷.在此基础上,综合考虑了角接触球轴承的接触区宏观几何、接触载荷、真实表面粗糙度、瞬态效应等因素,建立了角接触球轴承混合润滑数学模型,分析了在不同工况下角接触轴承的润滑状况及表面以下应力分布.结果表明:随着载荷的不断增加,钢球与内圈沟道之间的油膜厚度会不断减少,导致干接触面积迅速扩大,接触点表面以下最大应力增大;转速的增加会使油膜变厚,干接触面积缩小.该结果对角接触球轴承的实际工程应用具有重要借鉴意义.     

Evaluation of multiaxial theories for room-temperature plasticity and elevated-temperature creep and relaxation of several metals

Based on the assumption that the material satisfies the condition of isotropic hardening for either a von Mises or a Tresca material, finite-strain theories are derived for solid circular torsion members for the conditions that the inelastic deformations are either time independent or time dependent. In the latter case, both creep and relaxation theories are derived. At room temperature the theories are evaluated for each of eight metals using finite-strain data from tension, compression and torsion members. Of the six metals that are found to satisfy the condition required for the isotropic-hardening model, two are von Mises, one is Tresca, and the other three are between von Mises and Tresca. At elevated temperatures, the theories are evaluated for each of five of the latter six metals, using data from tension and torsion members. Material properties obtained from the tension specimens are used to predict creep and relaxation curves for the torsion members. Contrary to the results at room temperature, creep curves for the torsion members do not all fall within the region bounded by von Mises and Tresca theories. In the case of relaxation, either excellent agreement is obtained between the von Mises strain-hardening theory and experimental data or the theory is conservative.     

Flow due to parallel disks rotating about non-coincident axis with one of them oscillating in its plane

An exact solution of the Navier–Stokes equations is obtained for the flow between two eccentric disks rotating with the same angular velocity and one of them executing non-torsional oscillations. An analytical solution describing the flow at large and small times after the start is given. The solutions depend on the ratio of the frequency of oscillation to the angular velocity of the disks and the ratio of the amplitude of oscillation to the angular velocity of the disks and to the distance between the axes of rotation, and the Reynolds number based on the distance between the disks and the angular velocity of the disks. The solutions for three cases when the angular velocity is greater than the frequency of oscillation or it is smaller than the frequency or it is equal to the frequency are discussed.     

Jeffery solution for an elastic disk containing a sliding eccentric circular inclusion assembled by interference fit

An analytic solution is presented for stresses induced in an elastic and isotropic disk by an eccentric press-fitted circular inclusion. The disk is also subject to uniform normal stress applied at its outer border. The inclusion is assumed to be of the same material as the annular disk and both elements are in a plane stress or plane strain state. A frictionless contact condition is assumed between the two members. The solution is obtained by using the general expression for a biharmonic stress function in bipolar coordinates. The results show that the maximum of the von Mises effective stress due to the inclusion interference occurs in the ligament for large eccentricity, but it deviates from the symmetry axis for small eccentricity. Moreover, along the border of the circular inclusion the hoop stress locally coincides with the contact pressure, in agreement with a similar classical result valid for a half plane.     

Axisymmetric flows of an incompressible fluid between movable rotating disks

Within the Karman family of exact solutions of the Navier-Stokes equations, some non-selfsimilar solutions are considered to the problem of unsteady incompressible flow between two rotating disks one of which moves along a common rotation axis. Three classes of the flow regimes are studied: (i) a flow between the non-rotating disks, (ii) a flow between the disks rotating with identical angular velocities, and (iii) a flow between the disks rotating with opposite velocities. Examples of exact rotationally symmetric solutions for the inviscid-fluid equations, satisfying the no-slip conditions, are given.     

Formation of a regular sequence of vortex loops around a rotating disk in stratified fluid

A regular system of vortex loops in annular flow behind the edge of a disk of 6–11 cm in diameter rotating in stratified fluid is first visualized using the shadowgraph techniques (classical method with the Foucault knife and the “filament-in-focus” method). Clearly outlined vortex loops are observed in strongly and weakly stratified fluids over a wide range of the angular disk rotation velocities. The dimensions of the vortex flow region depend on the stratification (buoyancy period), the angular velocity of rotation, and the disk diameter. Extended lengthy filaments which form spiral and irregular patterns inside the loops are clearly expressed in the fine flow structure. The filaments connect the neighboring loops. The trajectories of motion of the upper edges of the loops, i.e., the sources of short internal waves, are traced.     

Resonant and Stationary Waves in Rotating Disks

The analytical conditions for resonant and stationary waves inrotating disks are presented. These conditions are derived from anonlinear plate theory pertaining to initial configurations and areapplicable to rotating disks with initial waviness and/or undergoinglarge-amplitude displacements. The rotational speeds at which theresonant and stationary waves occur for a 3.5-inch diameter computermemory disk are computed. The resonant waves for linear and nonlinear,rotating disks are simulated numerically. It is found that some diskmodes exhibit a hardening effect under which the rotational speeds forthe resonant and stationary waves increase with increasing waveamplitude, while other modes experience a softening effect with thoserotational speeds decreasing with increasing wave amplitude. Therotating-disk resonant spectrum presented in this paper is relevant tothe disk drive industry for determining the range of operationalrotation speed.