Stability of stationary and rotating discs under edge load

The dominant dynamic instability mechanism in circular cutters, grinding wheels and the like is a moving load resonance excited by a constant transverse load at the “critical rotation speed.” The critical speed theory is extended here to include the effects of concentrated in-plane edge loads similar to loading occurring in engineering processes. The theoretical analysis shows the critical speed is only sensitive to edge load when the resonance mode and the mode of static edge load buckling are identical. This always occurs with large edge load, but it is not the case with the smaller edge loads typical of engineering processes. These analyses are confirmed through prediction and measurement of static buckling loads for centrally clamped discs with in-plane, concentrated loads inclined between 0° and 80° to the edge normal.     

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.     

Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered

In the conventional finite element method (FEM), the dynamic characteristics of a longitudinally vibrating rod with mass density ρ_r, Young's modulus E_r, cross-sectional area A_r and total length ℓ_r are considered to be the same as those of a helical spring with stiffness constant k_r=A_rE_r/ℓ_r and total mass m_r=ρ_rA_rℓ_r. For a lumped-mass model, the mass matrix of a rod element is a 2×2 diagonal one with each of its non-zero coefficients to be equal to one half of the total rod mass (i.e., 0.5m_r). Furthermore, the dynamic characteristics of a rod on the basis of last “lumped-mass” model have been found to be very close to those on the basis of “consistent-mass” model. Thus, one can easily take into account of the inertial effect of a helical spring using a massless one with “one half of its total mass”, respectively, concentrated at its two ends (in Method 2) instead of modeling it by an elastic rod with uniform mass per unit length (in Method 1). When one more spring-mass system is attached to the beam, the total number of unknown constants increases “one” in Method 2 and “two” in Method 1, thus, Method 2 will reduce more effort than Method 1 for studying the dynamic behaviors of a beam carrying a number of spring-mass systems with mass of each helical spring considered. In this paper, the formulations of Methods 1 and 2 are presented first and then the numerical examples are illustrated to confirm the reliability of the presented theory and the developed computer programs. Finally, the effect concerning mass of each helical spring of the spring-mass systems is studied.     

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.     

Uncertainty analysis of a laser calibration system for evaluating the positioning accuracy of a numerically controlled axis of coordinate measuring machines and machine tools

This paper presents an uncertainty analysis of a Positional Error Calibrator based on a laser interferometer system. This laser calibration system is capable of evaluating the positioning accuracy of a numerically controlled axis of machine tools and coordinate measuring machines (CMM) under dynamic conditions. In order to assess the measurement uncertainty of this calibrator, an analysis of the uncertainty components that make up the uncertainty budget of this calibrator has been carried out. These uncertainty components can be classified into three categories as follows: (1) uncertainties intrinsic to the laser system; (2) uncertainties due to environmental effects; (3) measuring uncertainties due to the installation. The procedure for evaluating the uncertainty of this calibrator follows GUM (“Guide to the Expression of Uncertainty in Measurement”). This uncertainty analysis was carried out when this calibrator was used to assess the positional errors of the “X” axis of a moving bridge type CMM.     

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.     

A simple approach to estimate failure surface of polymer and aluminum foams under multiaxial loads

From mechanical point of view, it is required to have a criterion for evaluating the failure of cellular solids (foams) under multiaxial loads. Well-documented experimental results in the literature show foams could fail by several mechanisms, e.g., elastic buckling, plastic yielding, brittle crushing or brittle fracture. In the previous years, both theoretical and phenomenological approaches have been applied to obtain the failure surface of various foams. The purpose of this paper is to present a simple approach to estimate the complete failure surface of “non-textured” foams. The predicted results of polymer and aluminum foams are compared with the experimental results reported in the literature. It is found that three selected tests will be sufficient to estimate the complete failure surface of a foam. The recommended testing stress states are σ₁=σ₂=σ₃>0, σ₁=σ₂=σ₃<0, and σ₁=−σ₂=−σ₃ (or σ₁=−σ₂, σ₃=0).     

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)].     

Paradoxical buckling behaviour of a thin cylindrical shell under axial compression

The widely accepted theory of buckling of thin cylindrical shells under axial compressive loading emphasises the sensitivity of the buckling load to the presence of initial imperfections. These imperfections are conventionally taken to be minor geometric perturbations of a shell which is initially stress-free. The original aim of the present study was to investigate the effect on the buckling load of imperfections in the form of local initial stress, which are probably more typical of practice than purely geometric ones. Experiments were performed on a vertical “melinex” cylinder of diameter 0.9 m and height 0.7 m, with radius/thickness ratio 1800. The upper and lower edges of the cylinder were clamped to end discs by means of circumferential belts — an arrangement that allowed states of self-stress to be introduced to the shell readily by means of local “uplift” at the base. The upper disc was made sufficiently heavy to buckle the shell, and it was supported by a vertical central rod under screw control. Many buckling tests were performed. Surprisingly, the buckling loads were generally at the upper end of the range of fractions of the classical buckling load that have been found in many previous experimental studies. Even when the local uplift at the base caused a local “dimple” to be formed before the shell was loaded, the buckling load was relatively high. A surface-scanning apparatus allowed the geometric form of the shell to be monitored, and the progress of such a dimple to be followed; and it was found that a dimple generally grew in size and migrated in a stable fashion up the shell as the load increased, until a point was reached when unstable buckling occurred. These unexpected and paradoxical features of the behaviour of the experimental shell may be attributed to the particular boundary conditions of the shell, which provide in effect statically determinate support conditions. This study raises some new issues in the field of shell buckling, both for the understanding of buckling phenomena and for the rational design of shells by engineers against buckling.     

Experimental investigations of porous bearings under vertical sinusoidally fluctuating loads

There are many papers on the experimental investigations of porous bearings under static loads but there is no paper on the experimental investigations under dynamic loads. In the present paper, the results of experimental investigation of porous bearings under vertical sinusoidally fluctuating loads are presented. The friction force was measured under various conditions of fluctuating load/steady load ratio, journal frequency and load frequency. The investigations were carried out in the hydrodynamic lubrication regime in a specially designed and fabricated test rig. It was found that at any given rpm, as the fluctuating specific load/steady specific load ratio, P_f/P_s, increases, the mean coefficient of friction μ_m increases. It was also found that the mean coefficient of friction is not affected by the load frequency even when the load frequency is half of the journal frequency.     

Measurement of springback

Springback, the elastically-driven change of shape of a part after forming, has been measured under carefully-controlled laboratory conditions corresponding to those found in press-forming operations. Constitutive equations emphasizing low-strain behavior were generated for three automotive body alloys: drawing-quality silicon-killed steel; high-strength low-alloy steel; and 6022-T4 aluminum. Strip draw-bend tests were then conducted using a range of die radii (3<R/t<17), friction coefficients (0<μ<0.20), and controlled tensile forces (0.5<F_b/F_y<1.5). Springback angles and curvatures were measured for bend and bend–unbend areas of the specimen, the latter corresponding to the “sidewall curl” region, which dominates the geometric change and the dependence on process variables. Friction coefficient and R/t (die-radius-to-sheet-thickness) greater than 5 have modest but measurable effects over the ranges tested. As expected, strip tension dominates the springback sensitivity, with higher forces reducing springback. For 6022-T4, springback is dramatically reduced as the tensile stress approaches the yield stress, corresponding to the appearance of a persistent anticlastic curvature. The presence of this curvature, orthogonal to the principal curvature, violates the simple two-dimensional models of springback reported in the literature. The measured springback angles and curvatures are reported both in graphical summary and tabular form for use in assessing analytical models of springback.     

Strain-rate and inertia effects in the collapse of two types of energy-absorbing structure

The dynamic plastic collapse of energy-absorbing structures is more difficult to understand than the corresponding quasi-static collapse, on account of two effects which may be described as the “strain-rate factor” and the “inertia factor” respectively. The first of these is a material property whereby the yield stress is raised, while the second can affect the collapse mode, etc. It has recently been discovered [6,7]that structures whose load-deflection curve falls sharply after an initial “peak” are much more “velocity sensitive” than structures whose load-deflection curve is “flat-topped” (Fig. 1a); that is, when a given amount of energy is delivered by a moving mass, the final deflection depends more strongly on the impact velocity. In this paper we investigate strain-rate and inertia effects in these two types of structure by means of some simple experiments performed in a “drop hammer” testing machine, together with some simple analysis which enables us to give a satisfactory account of the experimental observations. The work is motivated partly by difficulties which occur in small-scale model testing of energy-absorbing structures, on account of the fact that the “strain-rate” and “inertia” factors not only scale differently in general, but also affect the two distinct types of structure differently.     

On the anomalous behaviour of anisotropic sheet metals

Using Hill's 1948 criterion [1] for anisotropic yielding and the strain ratio, r, it has been shown that the ratio of the balanced biaxial yield stress, σ_b, to the uniaxial tensile yield stress, σ_u, should be > 1 if r > 1 and < 1 if r < 1. Certain experimental results[2] showed that with commercial-purity aluminium, where r < 1, the ratio of σ_b to σ_u was always > 1 in that study. This was termed anomalous behaviour. Hill has proposed a new criterion[3] that not only appears to provide greater flexibility than does his earlier version but can also encompass anomalous behaviour which the earlier version cannot.Four simplified cases of the 1979 criterion have been proposed[3] and to date only one has been subjected to experimental assessment. However, the goals of those studies were not concerned with anomalous behaviour per se. In this paper, all four cases are analysed to determine the interrelationships of the parameters r and m (exponent in Hill's new criterion) required to encompass anomalous behaviour. It is found that for each of the four cases anomalous behaviour is predicted for a range of (m, r) combinations which are presented graphically in this paper.     

Influences of equal biaxial tensile loads on the stress fields near the mixed mode crack

A hybrid method for photoelasticity is introduced and applied to the plane problems of isotropic polycarbonate plates with a central crack under uniaxial and equal biaxial tensile loads. Also, the influences of equal biaxial tensile loads on the isochromatic fringes, stress fields and stress intensity factors near the mixed mode crack-tip have been investigated. The results show that, when an equal lateral tensile load is added to the specimen under uniaxial tensile load, the asymmetric isochromatic fringes about the line of crack gradually become symmetric, and the slope of the isochromatic fringe loop near the crack-tip is inclined towards the crack surface according to the increasing of the inclined angle of crack. Furthermore, the shapes of distribution of all stress components are changed from asymmetric to symmetric. In the equal biaxial tensile load condition against the uniaxial tensile load condition, the values of stress intensity factors are changed little, and only the region of compressive stress of σ _x /σ _O is changed when β = 0°, but the values of K _I /K ₀ are increased and those of K _II /K ₀ become almost zero, namely, we have the mode I condition when β = 15°∼45°. This paper was recommended for publication in revised form by Associate Editor Chongdu Cho Dong-Chul Shin received the B.S., M.S. and Ph.D. degrees in Mechanical Engineering from Yeungnam University in 1995, 1997 and 2001, respectively. Dr. Shin studied at the University of Tokyo, Japan, for three years (from April, 2005 to January, 2008) as a Post-Doctoral fellow (supported by Korea Research Foundation (KRF) and Japan Society for the Promotion of Science (JSPS)). Dr. Shin is currently a Research Professor at the School of Mechanical Engineering at Pusan National University, Korea. His research interests include the static and dynamic fracture mechanics, stress analysis, and fracture criteria of piezoelectric ceramics, etc. Jai-Sug Hawong received a B.S. in Mechanical Engineering from Yeungnam University in 1974. Then he received his M.S. and Ph.D. degrees from Yeungnam University in Korea in 1976 and from Kanto Gakuin University in Japan in 1990, respectively. Prof. Hawong is currently a professor at the School of Mechanical Engineering at Yeungnam University, in Gyeongsan city, Korea. He is currently serving as vise-president of Korea Society Mechanical Engineering. His research interests are in the areas of static and dynamic fracture mechanics, stress analysis, experimental mechanics for stress analysis and composite material etc.     

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).     

Some observations on the CLNA model in fretting fatigue

Using the Atzori–Lazzarin criterion, the author has recently proposed a unified model for Fretting Fatigue denominated Crack-Like Notch Analogue—CLNA model, considering only two possible behaviours: either “crack-like” or “large blunt notch”. In a general FF condition, the former condition is treated with a single contact problem corresponding to the MIT Crack Analogue (CA) improved in some details also by the author. The latter, with a simple peak stress condition, i.e. a simple Notch Analogue model, simply stating that below the fatigue limit, infinite life is predicted for any size of contact. In the typical condition of constant normal load and in phase oscillating tangential and bulk loads, both limiting conditions are immediately written, and the CLNA model permits to collapse the effect of the contact loads on a single closed form equation (differently from many other models which do not permit this flexibility). For not too large contact areas (“crack-like” contact) no dependence at all on geometry is predicted, but only on 3 load factors (bulk stress, tangential load and average pressure) and size of the contact. Only in the “large blunt notch” region occurring typically only at very large sizes of contact does size-effect disappear, but the dependence on all other factors including geometry remains. The model compares favourably with some experimental results in the literature. In this paper, some aspects of the CLNA model are further elucidated.     

Cohesive and continuum damage models applied to fracture characterization of bonded joints

In this work, two different methods for simulating damage propagation are presented and applied to fracture characterization of bonded joints in pure modes I and II. The cohesive damage model is based on a special developed interface finite element including a linear softening damage process. In the continuum damage model the softening process is performed by including a characteristic length associated with a given Gauss point. The models were applied to the simulation of “double cantilever beam” (DCB) and “end notched flexure” (ENF) tests used to obtain the critical strain release rates in mode I and II of bonded joints. In mode I it was observed, under certain conditions, a good agreement between the results obtained by the two models with the reference value of critical strain energy release rate in mode I (G_Ic), which is an inputted parameter. However, in mode II some discrepancies on the obtained G_IIc values were observed between the two models. These inaccuracies can be explained by the simplifying assumptions inherent to the cohesive model. Better results were achieved considering the crack equivalent concept.     

An elastic–plastic stress analysis steel-reinforced thermoplastic composite cantilever beam

In the present study an analytical elastic–plastic stress analysis is carried out for a low-density homogeneous polyethylene thermoplastic cantilever beam reinforced by steel fibers. The beam is loaded by a constant single force at its free end. The expansion of the region and the residual stress component of σ_x are determined for 0°, 30°, 45°, 60° and 90° orientation angles. Yielding begins for 0° and 90° orientation angles at the upper and lower surfaces of the beam at the same distances from the free end. However, it starts first at the upper surface for 30° and 45° orientation angles. The elastic–plastic analysis is carried out for both the plastic region which spreads only at the upper surface and the plastic region which spreads at the upper and lower surfaces together. The residual stress components of σ_x and τ_xy are also determined. The intensity of the residual stress component is maximum at the upper and lower surfaces of the beam, but the residual stress component of τ_xy is maximum on or around the x-axis. The beam can be strengthened by using the residual stresses. The distance between the plastically collapsed point and the free end is calculated for the same load in the beam for 0°, 30°, 45°, 60° and 90° orientation angles.     

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.