Rotor dynamic instability analysis on hybrid air journal bearings

Hybrid air journal bearings with multi-array of 1, 2, 3, 4, or 5-row orifice feedings are analyzed for the problem of rotor dynamic instability. The bearing stiffness and damping coefficients are calculated numerically to determine threshold rotor mass under various operating conditions. The hybrid porous air journal bearings are also analyzed for comparison to investigate the similarities in dynamic characteristics between the multi-array of orifice feeding bearings and the porous bearings. The results show that the porous bearing is more stable than the orifice feeding bearing at lower rotation speeds (Λ<0.1) or at higher rotation speeds (Λ>1) with lower feeding parameters (λ_P<10⁻⁸). The 5-row orifice feeding bearing is more stable than the porous bearing at moderate speeds (0.3<Λ<0.6) with lower feeding parameters (λ₀<10⁻⁴).     

Closed-form solution for wedge cutting force through thin metal sheets

A simple kinematic model is developed which describes the main features of the process of the cutting of a plate by a rigid wedge. It is assumed in this model that the plate material curls up into two inclined cylinders as the wedge advances into the plate. This results in membrane stretching up to fracture of the material near the wedge tip, while the “flaps” in the wake of the cut undergo cylindrical bending. Self-consistent, single-term formulas for the indentation force and the energy absorption are arrived at by relating the “far-field” and “near-tip” deformation events through a single geometric parameter, the instantaneous rolling radius. Further analysis of this solution reveals a weak dependence on the wedge angle and a strong dependence on friction coefficient. The final equation for the approximate cutting force over a range of wedge semiangles 10° ≤ θ ≤ 30° and friction coefficients 0.1 ≤ μ ≤ 0.4 is: F = 3.28σ₀(δ_t)^0.2l^0.4t^1.6μ^0.4, which is identical in form and characteristics to the empirical results recently reported by Lu and Calladine [Int. J. Mech. Sci.32, 295–313 (1990)].This analysis is believed to resolve a controversy recently developed in the literature over the interpretation of plate cutting experiments.     

Elastic contact with a “donut”-shaped asperity

The laser-textured surfaces used for the touchdown area of computer hard-disks are sometimes covered with asperities consisting of a crater surrounded by a raised rim; contact with the read-head takes place over the rim of the crater, colloquially referred to as a “donut”. In order to analyse the load/compliance relation or the stiction to be expected in contact of hard disks, a number of authors have proposed load/compliance relations for contact between such a single doughnut and a plane, usually as simple modifications of the Hertz line contact equations. In this note simple, asymptotically correct, relations for a ring asperity are derived and verified by direct solutions. In particular, the relation between elastic deflection and load is approximately δ=(W/π²RE^*)[ln(16R/b)+0.5)].     

Pressure-viscosity effect on the solutions of a lubricated thick elastic bonded strip

Utilizing the numerical method developed by the author for isoviscous, fully flooded, elastohydrodynamic lubrication of a rigid cylinder rolling or sliding on an elastic strip (layer) which is attached to a rigid substrate (bonded strip), the influence on the pressure-viscosity coefficient, α, upon solutions is investigated. The present solutions are obtained for contacts operating in the transition region betweeen isoviscous-elastic and piezoviscous-elastic regimes where a pressure spike can be expected in some sense.New sets of results are presented for central and minimum film thicknesses in dimensionless form when the bonded strip is thick (0γ1, where γ = a/t is the ratio of the half contact width to strip thickness). It is shown that the film thickness depends not only on the values of α but also is influenced by Poisson's ratio, v.     

Vibrations of point-supported rectangular plates with variable thickness using a set of static tapered beam functions

This paper studies the free vibrations of point-supported rectangular plates with variable thickness using the Rayleigh–Ritz method. The domain of the plate is bounded by x=αa′, a′ (0α<1); y=βb′, b′ (0β<1) in the Cartesian coordinate system. The thickness of the plate varies continuously and is represented by a power function (x/a′)^s(y/b′)^t. Varieties of tapered rectangular plates can be described by giving s and t different values. A set of static tapered beam functions which are the solutions of a tapered beam (a unit width strip taken from the particular plate under consideration in one or the other direction parallel to its edges) under a Taylor series of static loads, are developed as the admissible functions for the vibration analysis of point-supported rectangular plates with variable thickness in one or two directions. The eigenfrequency equation is derived through the Rayleigh–Ritz approach, supplemented by the zero deflection conditions at the point-supports. A very simple program in common use has been compiled. The convergence study shows a small computational cost and the comparison with known solutions for point-supported rectangular plates with uniform thickness demonstrates the accuracy of the present method. Finally, some new numerical results are given, which may serve as the benchmarks for future research on the aforementioned problem.     

A polar model for synovial fluid with reference to human joints

An analysis of “Boosted Lubrication” between two approaching solids, one of which is porous, is presented with reference to normally loaded living human joints. Micropolar fluid has been considered to represent the synovial fluid in the fluid film region between the approaching surfaces and the flow of viscous fluid in the porous matrix due to filtration through the porous material. Such a situation analysed in two regions separately using the slip flow model introduced by Beavers and Joseph. The effect of concentration, shape and size of the micro molecules on the bearing characteristics is discussed. The results are in accordance with those of Dowson et al.⁸     

Geometry of “developable cones”

When a thin disc is supported on the rim of a bowl, and its centre is pushed down by a finger, it adopts a characteristic conformation, known as a “developable cone”, and sketched in Fig. 1(a): the main, broadly conical, shape can only form if about one-quarter of the disc buckles upwards. There is a curved intersection between the two parts, which takes the form of a crescent-shaped “crease” near its apex, but with the flanking regions less tightly deformed. The “developable cone” is a recurring motif in a wide range of physical situations—crumpling, buckling, draping—and its mechanics provides a key to understand the phenomena, whether the disc deforms in the elastic or the plastic range. The task of this paper is to study only geometrical features of the “developable cone”. The first step is to replace the actual crease (Fig. 1(a)) by an idealised “sharp” crease (Fig. 1(b)). The second step is to study the apparently “large-rotation” problem of kinematics by means of an adaptation of the classical “yield-line” pattern of folding, but with a crucial added constraint that springs from Gauss's analysis of inextensional deformation. We illustrate the method via a graded sequence of examples, and we close with a discussion.     

Reference stresses and temperatures for cylinders and spheres under internal pressure with a steady heat flow in the radial direction

The paper examines the creep behavior of thick cylinders and spheres subjected to internal pressure and a negative temperature gradient in the radial direction. It is found that at stationary state the rate of radial displacement of the vessel wall is simply proportional to the material creep behavior associated with a single stress and temperature. Such “reference stresses” and “reference temperatures” are defined for spheres and cylinders of varying wall thicknesses. These reference stresses and reference temperatures are valid for any creep problem where the material behavior may be characterized by a function of the form exp (γT)σ^m. The extension of these results to variable pressure and temperature loading cases is discussed.     

A rate-dependent model for hot-rolling

A rate-dependent model for the plane-strain sheet-rolling problem is proposed. The governing equations are solved using an asymptotic scheme that assumes that the ratio δ of thickness of the sheet material at the entry to the roll-bite length is small. Both the relative-slip and no-slip sheet-roll interface conditions are considered. Depending on the magnitude of the friction, different regimes that correspond to different levels of shear deformation have been identified and asymptotic solutions are provided for each of these regimes. The effect of the reduction, the strain-rate hardening parameter and the magnitude of the friction on the field variables and the roll-speed is also studied. Further, it is shown that in the limit as the strain-rate hardening index n → ∞, the asymptotic solutions for the rate-dependent model are shown to approach those predicted by rigid perfectly-plastic theory. The theoretical predictions are compared with experimental results for a commercial purity aluminum. The comparisons indicate a reasonable agreement between theory and experiment.     

Assessments of the kinetic and dynamic loads sustained by stationary tools during high rate plastic forming

A systematic method for evaluating the kinetic and dynamic loads sustained by stationary tools (as opposed to moving tools for which methods already exist) during high rate plastic forming is examined and exemplified by examples. It is essentially based on the momentum theorem for continua for incompressible flow, utilizing kinematically admissible velocity fields. In steady state forming processes (such as rolling, wire drawing, etc.), the difference between the active load (imposed or calculated a priori) and the reactive load, is formulated rigorously, whereas for non-steady processes (forging, impact extrusion, etc.) the formulation gives merely an approximation to the dynamic effects on the tools. The resulting velocity-dependent reactions on the tools are given in terms of two nondimensional numbers, namely, the “kinetic head” (u₀²/σ₀) (called the Euler Number) and the “dynamic head” (ú₀L/σ₀), which includes the machine speed (u₀), machine acceleration ( ), material density , yield strength ₀ and a characteristic dimension of the product, L. The same two non-dimensional heads emerged previously from energy-balance consideration in Ref. [1], while approximating dynamic loads on moving tools, hence a consistency is demonstrated. These heads are unavoidably multiplied by geometrical functions, which typify the specific process under consideration and may amplify (or diminish) the intensity of the dynamic effects. The present work is focussed on quantifying, by the above method, the inherent difference between the reactive load sustained by the non-moving tool (say, a die) and the acting load carried by the moving tool (piston, ram, etc.) In particular cases of very slow processes, these loads are equal by static equilibrium. In some practical processes (like rolling) their difference appears to be relatively small, whereas in others (like impact extrusion) it appears extremely large.     

Mechanical analysis of the scratching of metals and polymers with conical indenters at moderate and large strains

Scratch test provides a convenient mean to study the surface mechanical properties and the tribological performances of materials. The representative strain of the material in this test increases with the attack angle β of the indenter and so for a conical indenter increases as its apical angle 2θ decreases. But the mechanical analysis of this test by analytic models is very intricate. First we perform a preliminary discussion of the various aspects of the problem by considering the plane strain scratching of materials by wedges. After we present the conditions of the numerical simulations of the scratch test with conical indenters with a three-dimensional (3D) finite element code. These simulations provide the scratch geometry (contact surface, elastic recovery), the plastic strain map and the volume average plastic strain, the scratch hardness and the force ratio, the apparent friction coefficient μ₀=F_t/W. So we compare the behaviour of polymeric and metallic materials in scratch test at low and large strain and relate their difference in scratching resistance to their rheological properties. Polymers develop more higher elastic strains than metals a phenomenon which is characterised at low strain by the yield stress to Young's modulus ratio ε_e=σ_y/E. For θ=70.3° where pure ploughing occurs we study the scratching of elastic perfectly plastic solids with various values of ε_e under zero friction. Some comparisons with the behaviour in indentation are performed and we study the influence of friction in the scratching of workhardened steel with the same cone. At high strain the main rheological difference is the workhardening behaviour: it is described by a power law for metals and an exponential law for polymers. For θ decreasing from 70.3 to 20° we compare the behaviour of a cold worked steel to the behavour of polycarbonate, a thermoplastic polymer: a transition from ploughing to ploughing–cutting occurs only for steel.     

Forming analysis of porous materials

A procedure for estimating the applied traction required for forming processes of porous material is suggested. It carries the general structure of the upper-bound approach in the sense that static equilibrium is not enforced and a detailed mechanism of pore closure is not needed. A velocity field for compressible, porosity dependent, plastic continua, is solved for plane and axisymmetric converging flow problems. By employing, in conjunction with the velocity field, Gurson's model (to characterize the effects of the mean stress and the porosity on the yielding of the material), account is taken of the work expenditure due to the dilatancy of the material, beside the other customary dissipation rate terms. The tractions are then evaluated by the limit analysis scheme, leaving the porosity at the exit as a free parameter to be fixed by minimization of the plastic dissipation. It has been found that the dissipation terms depend on the mean normal stresses and the porosity in a highly nonlinear way. In cases where the porosity and the mean stress vary moderately in the domain of the flow, closed-form expressions for the various dissipation rates are provided. They exhibit, in a systematic fashion, the exact features of the corresponding solutions of non-porous materials, but are multiplied by a set of “weight functions” which depend merely on the averaged values of the porosity and the mean normal stress.Parametric study of the presented solutions for several classical processes (wire drawing, extrusion, ironing and rolling) indicates the role of the average porosity in decreasing or increasing the loads (or moments) needed to operate the forming processes, compared to materials with zero porosity. Comparisons to various experiments, numerical solutions (FEM) and to few exact (asymptotic) solutions, validate the feasibility of the suggested scheme for extending the upper-bound analysis to metal forming of porous materials.The fact that forming loads of non-porous material are not necessarily upper bounds to forming loads of porous materials (under the same geometrical and influx speeds) may have valuable consequences.     

Frictionally excited thermoelastic instability in viscoelastic and poroelastic media

Friction materials commonly used in sliding applications, such as clutches and brakes, can be poroelastic and exhibit a viscoelastic behaviour. To the author's knowledge, there are no comprehensive analysis of the influence of poroelastic and viscoelastic material properties on the onset of the phenomenon of frictionally excited thermoelastic instability in sliding systems. This issue is here analysed in some details. Firstly, a linear standard model for the friction material is adopted, introducing an effective complex dynamic modulus E=|E|e^jδ and individuating three independent parameters, E₁, E₂/E₁ and c₂/E₁, that fully describe its viscoelastic behaviour. Subsequently, a similarity between viscoelastic and poroelastic formulation is presented and the three independent parameters introduced are related to the viscosity of the fluid μ_f, the permeability k_p and elastic properties M, α_B of the porous material.The linear elastic formulation proposed by Decuzzi et al. (ASME J. Tribiol. 2001;123:865) has been modified in order to take account of the new constitutive model and the variation of the critical sliding speed with the wave parameter, and viscoelastic/poroelastic properties of the material are examined.It has been found that the susceptibility towards thermoelastic instability increases by increasing both the elastic E₂/E₁ and viscoelastic c₂/E₁ parameters, or by increasing the Biot modulus M and effective stress coefficient α_B, the viscosity μ_f of the fluid, and by reducing the permeability k_p of the porous skeleton. It has been shown that for porous friction materials employed in wet clutches which are weakly viscoelastic, the neglect of its poroelastic behaviour leads to an overestimation of the critical speed smaller than 10%. However, much larger variations are predicted for elastomeric and porous materials with more pronounced viscoelastic behaviour.     

Anisotropic plasticity with application to sheet metals

Hill’s 1948 anisotropic theory of plasticity is extended to include the concept of isotropic–kinematic hardening. The “anomalous” effect can be accounted for by kinematic hardening. It is shown that the quadratic yield function can be used for sheet metals irrespective of its plastic strain ratio R. It is further shown that effects of thickness reduction due to further rolling may be accounted for by kinematic hardening.     

Identification of yield stress and plastic hardening parameters from a spherical indentation test

Assuming plastic hardening of metals are specified by the stress–strain curve in the form , the material parameters σ₀, k and m are identified from spherical indentation tests by measuring compliance moduli in loading and unloading of the load–penetration curve. The curve P(h_p) is analytically described by a two term expression, each with different exponents. Here, ε_p and h_p denote the plastic strain and permanent penetration. The proposed identification method is illustrated by specific examples including numerical and physical identification tests.     

An efficient approach to characterize pseudo-merohedral twins by precession electron diffraction: Application to the LaGaO3 perovskite

Pseudo-merohedral twins are frequently observed in crystals displaying pseudo-symmetry. In these crystals, many [u v w] zone axis electron diffraction patterns are very close and can only be distinguished from intensity considerations. On conventional diffraction patterns (selected-area electron diffraction or microdiffraction), a strong dynamical behaviour averages the diffracted intensities so that only the positions of the reflections on a pattern can be considered. On precession electron diffraction patterns, the diffracted beams display an integrated intensity and a “few-beam” or “systematic row” behaviour prevails which strongly reduces the dynamical interactions. Therefore the diffracted intensity can be taken into account. A procedure based on observation of the weak extra-reflections connected with the pseudo-symmetry is given to identify without ambiguity any zone axis. It is successfully applied to the identification and characterization of {1 2 1} reflection twins present in the LaGaO₃ perovskite.     

The influence of corner stress singularities on the vibration characteristics of rhombic plates with combinations of simply supported and free edges

This paper offers accurate flexural vibration solutions for rhombic plates with simply supported and free edge conditions. A cornerstone here is that the analysis explicitly considers the bending stress singularities that occur in the two opposite, hinged–hinged and/or hinged–free corners having obtuse angles of the rhombic plates. These singularities become significant to the vibration solution as the rhombic plate becomes highly skewed (i.e. the obtuse angles increase). The classical Ritz method is employed with the assumed normal displacement field constructed from a hybrid set of (1) admissible and mathematically complete algebraic polynomials, and (2) comparison functions (termed here as “corner functions”) which account for the bending stress singularities at the obtuse hinged–hinged and/or hinged–free corners. It is shown that the corner functions accelerate the convergence of solutions, and that these functions are required if accurate solutions are to be obtained for highly skewed plates. Accurate nondimensional frequencies and normalized contours of the vibratory transverse displacement are presented for rhombic plates having a large enough skew angle of 75° (i.e. obtuse angles of 165°), so that the influence of the stress singularities is large. Frequencies and mode shapes of isosceles triangular, hinged–free plates are also available from the data presented.     

Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses

Exact solutions are presented for the free vibration and buckling of rectangular plates having two opposite edges (x=0 and a) simply supported and the other two (y=0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress σ_x=−N₀[1−α(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement (w) to vary as sin(mπx/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (the method of Frobenius). Applying the clamped boundary conditions at y=0 and b yields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters α=0,0.5,1,1.5,2, for which α=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for α=0,1,2 obtained by the method of integration of the differential equation (α=0) or the method of energy (α=1,2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b=0.5,1,2 subjected to three types of loadings (α=0,1,2), with load intensities N₀/N_cr=0,0.5,0.8,0.95,1, where N_cr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.     

A universal slip-line model with non-unique solutions for machining with curled chip formation and a restricted contact tool

A universal slip-line model and the corresponding hodograph for two-dimensional machining which can account for chip curl and chip back-flow when machining with a restricted contact tool are presented in this paper. Six major slip-line models previously developed for machining are briefly reviewed. It is shown that all the six models are special cases of the universal slip-line model presented in this paper. Dewhurst and Collins's matrix technique for numerically solving slip-line problems is employed in the mathematical modeling of the universal slip-line field. A key equation is given to determine the shape of the initial slip-line. A non-unique solution for machining processes when using restricted contact tools is obtained. The influence of four major input parameters, i.e. (a) hydrostatic pressure (P_A) at a point on the intersection line of the shear plane and the work surface to be machined; (b) ratio of the frictional shear stress on the tool rake face to the material shear yield stress (τ/k); (c) ratio of the undeformed chip thickness to the length of the tool land (t₁/h); and (d) tool primary rake angle (γ₁), upon five major output parameters, i.e. (a) four slip-line field angles (θ, η₁, η₂, ψ); (b) non-dimensionalized cutting forces (F_c/kt₁w and F_t/kt₁w); (c) chip thickness (t₂); (d) chip up-curl radius (R_u); and (e) chip back-flow angle (η_b), is theoretically established. The issue of the “built-up-edge” produced under certain conditions in machining processes is also studied. It is hoped that the research work of this paper will help in the understanding of the nature and the basic characteristics of machining processes.     

An approximate analysis for studying the plane strain deformation of strain rate sensitive materials

The paper presents a numerical method for analyzing the plane strain deformation of rate sensitive materials. A rate of energy functional is introduced which is thought to take adequate account of the strain rate sensitivity of the material. In the numerical technique the functional is minimized with respect to a kinematically admissible velocity field and used in a discretized form in a finite element analysis.To serve as an illustration the frictionless, plane-strain, side extrusion process was considered. To simulate actual side extrusion processes friction was incorporated into the analysis by assuming a constant fraction, α, of the current shear stress of the material.Data were available from some preliminary experiments on the side extrusion of a superplastic tin-lead alloy. The theoretically predicted forming pressure, taking α = 0·3, showed reasonably good agreement with the experimental values.