Inverse approach for estimating rheological characteristics of adsorption layer in thin film EHL contacts

A modified Reynolds equation is derived for thin film elastohydrodynamic lubrication (TFEHL) by means of the viscous adsorption theory. This TFEHL theory can be used to explain the deviation between the measured film thickness and that predicted from the convenient elastohydrodynamic lubrication (EHL) theory under very thin film conditions. Results show that the thinner the film, the greater the ratio of the adsorption layer to the total film thickness becomes, and the greater the value of the pressure–viscosity index (z′). An inverse approach is proposed to estimate the pressure distribution based upon the film thickness measurement and to determine the pressure–viscosity index of oil film, and the thickness (δ) and the viscosity ratio (η^*) of the adsorption layer in TFEHL circular contacts. Based on TFEHL theory, the inverse approach can reduce z′ error, and provides a reasonably smooth curve of pressure profile by implementing the measurement error in the film thickness. This algorithm not only estimates the pressure, but also calibrates the film shape. Consequently, it predicts z′, η^*, and δ with very good accuracy. It can also be used to evaluate the lubrication performance from a film thickness map obtained from an optical EHL tester. Results show that the estimated value of z′ is in very good agreement with the experimental data.     

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.     

Externally pressurized conical step bearing with visco-elastic lubricant

A solution for an externally pressurized conical step bearing with visco-elastic lubricant is obtained by using a regular perturbation technique. The effects of the parameters S (elastic number), H = h₁/h₂ (the ratio of film thickness) and sinα (α, the semi-vertical angle) on the pressure, the load capacity and the ratio of flow flux have been studied and are presented graphically. The effect of elasticity on improving bearing performance is very small.     

Mixed film lubrication of strip rolling using O/W emulsions

A numerical study on the oil concentration effect of O/W emulsion in cold rolling operating in the mixed film lubrication regime has been carried out. The developed scheme is able to calculate oil concentration at any point within the inlet zone (IZ) and work zone (WZ), rolling pressure, film thickness, and contact ratio for various rolling speeds. Hence the intertwined effects of oil concentration of the supplied emulsion and rolling speed on strip rolling are discussed. The study encompasses mixed film regime with speeds S range from 10⁻⁵ to 10⁻³ and supplied emulsion's oil concentration levels λ_ds range from 5% to 90%. The result shows that a moderate rise in oil concentration occurs in the IZ followed by a rapid one at the beginning of the workzone. In most cases, the oil in the emulsion would have been transformed from disperse phase to continuous phase throughout the WZ. Notwithstanding further concentration, which depends on the oil concentration of the supplied emulsion, could still occur in the WZ. The effect of the concentration process is predominantly seen in the development of the lubricant pressure whilst its effect on the total pressure is less pronounced. The analysis of the results suggests that it is possible to lower the emulsion oil concentration without detrimental effects on the rolling process; and from the analysis of the outlet film thickness, it is shown that the variation of emulsions’ oil concentration could control the exit lubricant film thickness and consequently the strip surface quality.     

A numerical investigation of the sinusoidal model for elastic layers in line contact

Distributions of normal stresses and surface deformations, induced when an elastic layer of finite thickness is indented by a frictionless rough rigid flat or cylindrical indenter, are calculated numerically. It is assumed that the punch has a sinusoidal roughness superimposed on its nominal profile. Two cases will be examined, namely when the elastic layer is either bonded to a rigid backing or resting on a frictionless rigid backing (unbonded). Chebyshev polynomials of the first kind T_n(x) are utilized to model both the unknown pressure and the given deformation over the contact area. The governing elasticity equation is thereby reduced to a finite set of linear equations and hence a complete solution is found. The present numerical method is simple, accurate and valid in the full range of Poisson's ratio 0 v 0.5. Moreover, a set of semi-analytical solutions for the contact pressure is obtained for both thin unbonded and bonded elastic layers. The numerical results compared favourably with the asymptotic solutions. The effects of the layer thickness, layer compressibility and roughness amplitude parameters on the contact stresses and deformations are considered.     

Linear stability of short journal bearings with consideration of flow rheology and surface roughness

The combined effects of surface roughness and flow rheology on the linear stability of a rigid rotor supported on short-length journal bearings are analyzed. The modified Reynolds equation and the rotor motion equation are linearized about the equilibrium position and the fluid film is modeled as spring and damping elements. From the characteristic equation, the instability threshold is then obtained with various surface roughness parameters (standard derivation, σ, and Peklenik number, γ), flow rheology (power-law index, n) and eccentricity ratios (). The results show that the size of the stability regions of different roughness patterns has the following characteristics: longitudinal (γ>1)>isotropic (γ=1)>smooth>transverse (γ<1). The stability of the bearing system deteriorates with decreasing power-law index. Moreover, there are crossing points in the vicinity of =0.45 among the curves of dimensionless speed parameter ( ) associated with various power-law indices and surface roughness parameters.     

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 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.     

Welding wave on the contact spot of solids

In this paper the theoretical model of adhesion between clean solid surfaces is presented. Material behaviour is described by hydrodynamic equations for viscous liquid and by the Gruneisen equation of state. This approach is quite suitable for quantitative analysis of mechanical processes running under high pressures. The model states that growth of a bonded area can occur in a regime of self-propagating welding wave (WW). WW originates at any active point where initial bonding takes place and further moves at velocity U_wwγ/η (γ is surface energy and η is viscosity of material). According to the model presented surface energy plays the key role in adhesion.     

Transverse vibration of a circular plate with unidirectional quadratic thickness variation

Free transverse vibration of a circular plate with thickness varying as (1 + α× + βx²) has been studied when the edge of the plate is clamped or simply-supported. The Rayleigh—Ritz method with suitable choice of basis functions satisfying the essential boundry conditions, has been employed to find the fundamental frequency and the associated mode shapes for various values of α and β with different boundary conditions. The convergence of results is ensured by working out several approximations till the results converge to the desired accuracy. The results for uniform thickness, linear and parabolic variation of thickness have been obtained as special cases. Tables and graphs are given for frequency, mode shapes and for depicting the effect of parameters α and β on the frequency.     

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.     

On the hollowness ratio effect on the dynamics of a spinning Rayleigh beam

The analytical solutions of a spinning Rayleigh beam with rotatory moment inertia and gyroscopic effect are presented in this paper. The critical speeds can be written analytically in a function of the length-to-radius ratio (l) defined by the beam's length over its outer radius and the hollowness ratio (α) defined by the hollow area over the total area of the cross section. The sensitivity analyses show that the critical speed is decreasing with l, but increasing with α. Moreover, α is more sensitive to the critical speeds. The design of a spinning beam should therefore be emphasized more on the hollowness factor. Contrary to common belief, only finite critical speeds exist and the number is independent of the boundary conditions. It increases monotonically with l, but decreases with α. The steady state unbalanced response can therefore be expressed analytically by the finite precessional modes and the corresponding generalized coordinates.     

Soft-tribology: Lubrication in a compliant PDMS–PDMS contact

We investigate the influence of surface roughness and hydrophobicity on the lubrication of a soft contact, consisting of a poly(dimethylsiloxane) (PDMS) sphere and a flat PDMS disk. The full Stribeck curves, showing boundary, mixed and elasto-hydrodynamic (EHL) lubrication, are presented for varying surface roughness and hydrophobicity. It is found that neither surface roughness nor hydrophobicity influence the friction coefficient (μ) within the EHL regime. However, increasing surface roughness decreases μ in the boundary regime, while extending the limits of the boundary and mixed lubrication regimes to larger values of the product of velocity and lubricant viscosity (Uη). The transition from the mixed lubrication to EHL regime is found to take place at lower values of the film thickness parameter Λ for increasingly rough surfaces. We found Λ=0.7 in the case of a root mean square (r.m.s.) surface roughness of 3.6 μm, suggesting that the effective surface roughness in a compliant compressed tribological contact is lower than that at ambient pressures. Rendering the PDMS surface hydrophilic promotes full-film lubrication and dramatically lowers μ in the boundary regime by more than an order of magnitude. This influence of surface wetting is also displayed when examining a range of lubricants using hydrophobic tribopairs, where the boundary μ decreases with decreasing lubricant–substrate contact angle. Implications of these measurements are discussed in terms of the creation of model surfaces for biotribological applications.     

Experimental and numerical study of O/W emulsion lubricated strip rolling in mixed film regime

An experimental and numerical study of cold rolling lubricated by O/W emulsion has been carried out. The strip rolling experiment was carried out on a Hille experimental rolling mill with a view to study the performance of emulsion lubrication in terms of practical rolling parameters. Accordingly, rolling parameters such as rolling force and torque were measured. The experimental measurements compare favourably with the computed results from a numerical scheme developed by the authors. The scheme, based on a two-phase lubricant model, is capable of calculating the oil concentration at any point within the inlet zone and work zone, rolling pressure, film thickness, and fractional contact area ratio associated with strip rolling under mixed film lubrication at different rolling speeds. Using this scheme, the intertwined effects of an emulsion’s parameters such as: oil concentration, mean oil droplet size, and rolling speed on strip rolling were investigated. The numerical study encompassed the mixed film regime for speed, S ranges from 10⁻⁴ to 10⁻², supply oil concentration level λ_ds from 1 to 10%, and oil droplet size D _S from 5 to 10. Experimentally, the differences between water, oil and emulsion-lubricated rolling are not discernible except for film thickness. At a low speed of 10 RPM, force and torque of water-lubricated rolling are marginally higher than oil- or emulsion-lubricated ones. However, the difference between emulsion and neat oil is not apparent. The numerical results show the occurrence of a moderate oil concentration increase in the inlet zone followed by a sharp one at the beginning of the work zone. The effect of the concentration process is predominantly seen in the film thickness and the lubricant pressure whilst its effect on the total pressure is less pronounced. The analysis of the results suggests that it is possible to lower the emulsion oil concentration without any adverse effect on the rolling process. This principle can be used to control the outlet lubricant film thickness and hence the surface quality of the rolled strip.     

A three-parameter model for fatigue behaviour of circumferential surface flaws in pipes

A three-parameter model, recently proposed by the authors for part-through-cracked round bars, is employed herein to examine the shape evolution of a circumferential external surface defect in a metallic round pipe under fatigue loading. The elliptical-arc flaw presents an aspect ratio α=a_el/b_el (a_el,b_el=ellipse semi-axes) and a relative crack depth ξ=a/t, where a and t are the depth of the deepest point on the crack front and the pipe wall thickness, respectively. The third parameter of the model is the ellipse shifting s=a_el/a, which defines the distance of the ellipse centre from the external boundary of the pipe cross-section. Thick- and thin-walled pipes are considered. Fatigue crack propagation paths in the diagram of α against s and ξ are numerically obtained for different initial flaw configurations under cyclic bending loading.     

Optimum design of dynamic absorber for a random-excited machine mounted on a platelike structure foundation

The optimum design of a dynamic absorber for a machine mounted on a floor system is presented. The floor is considered to be a platelike structure. The transfer function is derived in closed-form. Based on the band-limited white-noise excitation, the optimum tuning and damping ratios of the absorber are determined by minimising the variance of response of the machine. Since the variance cannot be calculated directly by integrating the transfer function over the band-limited frequency range, the steepest descent method is used for determining these optimum parameters by iteration. The same procedure can be extended to deal with the cases of other multi degrees-of-freedom systems. The effects of mass ratios (i.e. absorber/machine and machine/floor), primary damping, frequency ratio and the thickness of plate on the design parameters are examined. The results show that the values of optimum tuning are much different from those of the rigid foundation case. To have a small variance, it is better to keep the frequency ratio ω₁ (defined as the frequency of primary system divided by half of the lowest natural frequency of the floor) in the range 1 < ω₁ < 1.5, and have the mass ratio μ and the primary damping ratio ζ₁ are both greater than 0.1.     

A computer model of mixed lubrication in point contacts

A numerical model of mixed lubrication is presented in this paper. The idea introduced here is that asperity contact may be viewed as a result of a continuous decrease in film thickness, so that the transition between contact and non-contact is continuous and the same mathematical model should work for both regions. The pressure over the thin films is assumed to obey the Reynolds equation, and the solution of the equation, under the condition of h→0, is expected to be the same as that predicted by the theory of elasticity. To achieve convergent and stable solutions, the left-hand side terms of the Reynolds equation are switched off when the local film thickness approaches zero, leading to a reduced Reynolds equation. Pressure distributions over the entire computation domain are thus obtained through solving a unified equation system without identifying hydrodynamic or asperity contact regions. Computations were conducted for several example cases and results show that convergent solutions are achievable on different types of roughness, over a wide range of λ ratios (0.01 to infinity), and for different slide-to-roll ratios (0.0–2.0).     

Numerical construction of axisymmetric slip-line fields for indentation of thick blocks by rigid conical indenters and friction at the tool-metal interface

A general form of axisymmetric slip-line field solutions using numerical methods is given for cases of axisymmetric smooth and partially rough conical punch indentation of a rigid-perfectly plastic ductile material of semi-infinite thickness, where the cone semi-angle α varies from 15° to 90°. The basic problem of indetermination of the field and pressures near the apex of the conical punch experienced earlier by Locket (J. Mech. Phys. Solids 11, 345 (1963), [1]) for small cone angles was overcome by treating small segments of the curvilinear slip lines as arcs of circles of varying radii and assuming a straight boundary to the deformed lip. The numerical method outlined in detail by Chitkara and Butt (Int. J. Mech. Sci. 34, 833 (1992), [2]) allows the construction of axisymmetric slip-line fields for cone indentation, with varying frictional conditions at the tool-metal interface. The computer program developed requires only the cone semi-angle α and the frictional conditions at the tool-metal interface as the data set. It includes graphical plotting subroutines to plot the calculated slip-line field on a Hewlett-Packard 7550 plotter. A computographic plot of the associated velocity field for a typical case of conical punch indentation is also included.     

Asymptotic analysis of extrusion of porous metals

Plane strain extrusion of fully dense and porous metals is analysed using asymptotic techniques. The extrusion die is assumed to taper gradually down the extrusion axis. The asymptotic expansions are based on a small parameter ε which is defined as the ratio of the total reduction of the original cross-section to the length of the reduction region. Coulomb's law is used to model the frictional forces that develop along the metal-die interface and the coefficient of friction is assumed to be of order ε. Analytical solutions for the first two terms in the expansions are obtained. In the case of the fully dense metals, it is shown that the leading order [O(1)] solution involves “slab flow.” It is also shown that the next term in the expansion of the solution is O(ε²), and this provides a theoretical justification for the use of the so-called “slab methods” of analysis for dies of moderate slope. An asymptotic analysis of the extrusion of porous metals with dilute concentration of voids is also carried out. Gurson's plasticity model is used to describe the constitutive behavior of the material. The leading order solution is the same as that of the fully dense material and the effects of porosity enter as an O(ε) correction. In order to verify the asymptotic solutions developed, detailed finite element calculations are carried out for both the fully dense and the porous material. The asymptotic solutions agree well with the results of the finite element calculations.     

Measurements of friction in strip drawing under thin film lubrication

Strip drawing is used to investigate the friction behaviour under thin film lubrication in metal forming with plastic deformation. Friction coefficients are measured under a wide range of tribological conditions. The surface roughness is measured on an interferometric profilometer. The results show that the friction coefficient decreases with increasing oil film thickness h_w, as estimated using a formula appropriate for smooth tool and workpiece. Measurements of the surface topography show that change in friction is associated with a change in contact ratio between the tool and strip. The effect of strip reduction, strip roughness and die roughness on the friction coefficient is also investigated.