Introduction 5 1. In any situation where the resultant of tangential forces is smaller than some force parameter specific to that particular situation, the friction force will be equal and opposite to the resultant of the applied forces and no tangential motion will occur. 2. When tangential motion occurs, the friction force always acts in a direction opposite to that of the relative velocity of the surfaces. 3. The friction force is proportional to the normal load. 4. The coefficient of friction is independent of the apparent contact area. 5. The static coefficient is greater than the kinetic coefficient. 6. The coefficient of friction is independent of sliding speed. Strictly speaking, none of these laws is entirely accurate. Moore indicated that laws (3), (4), (9, and (6) are reasonably valid for dry friction under the following conditions: For law (3), the normal load is assumed to be low compared to that causing the real area of contact to approach the apparent area. For law (4), the materials in contact are assumed to have a definite yield point (such as metals). It does not apply to elastic and viscoelastic materials. Law (5) does not apply for materials with appreciable viscoelastic properties. Law (6) is not valid for most materials, especially for elastomers where the viscoelastic behavior is very significant. A number of workers also found some exceptions to the first friction law. Rabinowicz [ 181 reported that Stevens [ 191, Rankin [20], and Courtney-Pratt and Eisner [21] had shown that when the tangential force Fis first applied, a very small displacement occurs almost instantaneously in the direction of F with a magnitude in the order of 10-5 or 10-6 cm. Seireg and Weiter [22] conducted experiments to investigate the load- displacement and displacement-time characteristics of friction contacts of a ball between two parallel flats under low rates of tangential load application. The tests showed that the frictional joint exhibited “creep” behavior at room temperatures under loads below the gross slip values which could be described by a Boltzmann model of viscoelasticity. They also investigated the frictional behaviors under dynamic excitation [23, 241. They found that under sinusoidal tangential forces the “break- away” coefficient of friction was the same as that determined under static conditions. They also found that the static coefficient of friction in Hertzian contacts was independent of the area of contact, the magnitude of the normal force, the frequency of the oscillatory tangential load, or the ratio 6 Chapter I of the static and oscillatory components of the tangential force. However, the coefficient of gross slip under impulsive loading was found to be approximately three times higher than that obtained under static or a vibra- tory load at a frequency of l00Hz using the same test fixture. Rabinowicz [25] developed a chart based on a compatibility theory which states that if two metals form miscible liquids and, after solidification, form solid solutions or intermetallic components, the metals are said to be compatible and the friction and wear between them will be high. If, how- ever, they are insoluble in each other, the friction and wear will be low. Accordingly two materials with low compatibility can be selected from the chart to produce low friction and wear. In the case of lubricated surfaces, Rabinowicz [26] found that the second law of friction was not obeyed. It was found that the direction of the instantaneous frictional force might fluctuate by one to three degrees from the expected direction, changing direction continuously and in a ran- dom fashion as sliding proceeded. The general mechanisms which have been proposed to explain the nature of dry friction are reviewed in numerous publications (e.g., Moore [17]). The following is a summary of the concepts on which dry friction theories are based: Mechanical interlocking. This was proposed by Amontons and de la Hire in 1699 and states that metallic friction can be attributed to the mechanical interlocking of surface roughness elements. This theory gives an explanation for the existence of a static coefficient of friction, and explains dynamic friction as the force required to lift the asperities of the upper surface over those of the lower surface. Molecular attraction. This was proposed by Tomlinson in 1929 and Hardy in 1936 and attributes frictional forces to energy dissipation when the atoms of one material are “plucked” out of the attraction range of their counterparts on the mating surface. Later work attributed adhesional friction to a molecular-kinetic bond rupture process in which energy is dissipated by the stretch, break, and relaxation cycle of surface and subsurface molecules. Efectrostatic forces. This mechanism was presented in 1961 and explains the stick-slip phenomena between rubbing metal surfaces by the initiation of a net flow of electrons. Welding, shearing and ploughing. This mechanism was proposed by Bowden in 1950. It suggests that the pressure developed at the discrete contact spots causes local welding. The functions thus formed are subsequently sheared by relative sliding of the surfaces. Ploughing by the asperities of the harder surface through the matrix Introduction 7 of the softer material contributes the deformation component of friction. Dry rolling friction was first studied by Reynolds [27] in 1875. He found that when a metal cylinder rolled over a rubber surface, it moved forward a distance less than its circumference in each revolution of the cylinder. He assumed that a certain amount of slip occurred between the roller and the rubber and concluded that the occurrence of this slip was responsible for the rolling resistance. Palmgren [28] and Tabor [29] later repeated Reynolds’ experiment in more detail and found that the physical mechanism responsible for rolling friction was very different in nature than that suggested by Reynolds. Tabor’s experiments showed that interfacial slip between a rolling element and an elastic surface was in reality almost negligible and in any case quite inadequate to account for the observed friction losses. Thus he concluded that rolling resistance arose primarily from elastic-hysteresis losses in the material of the rolling element and the surface. 1.3 BOUNDARY LUBRICATION FRICTION Hardy [30] first used the term “boundary lubrication” to describe the sur- face frictional behavior of certain organic compounds derived from petro- leum products of natural origin such as paraffins, alcohols and fatty acids. Since then, boundary lubrication has been extended to cover other types of lubricants, e.g., surface films and solid mineral lubricants, which do not function hydrodynamically and are extensively used in lubrication. In the analysis of scoring of gear tooth surfaces, it has been fairly well established that welding occurs at a critical temperature which is reached by frictional heating of the surfaces. The method of calculating such a tem- perature was published by Blok [31], and his results were adapted to gears in 1952 [32]. Since then, some emphasis has been focused on boundary lubrica- tion. Several studies are available in the literature which deal with the boundary lubrication condition; some of them are briefly reviewed in the following. Sharma [33] used the Bowden-Leben apparatus to investigate the effects of load and surface roughness on the frictional behavior of various steels over a range of temperature and of additive concentration. The following observations are reported: Sharp rise in friction can occur but is not necessarily followed by scuff- Load affects the critical temperature quite strongly. ing and surface damage. 8 Chapter I Neither the smoother surface nor the rougher surface gives the max- imum absorption of heat, but there exists an optimum surface roughness. Nemlekar and Cheng [34] investigated the traction in rough contacts by solving the partial elastohydrodynamic lubrication (EHL) equations. It was found that traction approached dry friction as the ratio of lubricant film thickness to surface roughness approached zero, load had a great influ- ence on friction, and the roller radius had little influence on friction. Hirst and Stafford [35] examined the factors which influence the failure of the lubricant film in boundary lubrication. It is shown that substantial damage only occurs when a large fraction of the load becomes unsupported by hydrodynamic action. It is also shown that the magnitude of the surface deformation under the applied load is a major factor in breakdown. When the deformation is elastic, the solid surface film (e.g., oxide) remains intact and even a poor liquid lubricant provides sufficient protection against the build- up of the damage. The transition temperature is also much lower. They also discussed the effect of load and surface finish on the transition temperature. Furey and Appeldoorn [36] conducted an experiment to study the effect of lubricant viscosity on metallic contact and friction in the transition zone between hydrodynamic and boundary lubrication. The system used was one of pure sliding and relatively high contact stress, namely, a fixed steel ball on a rotating steel cylinder. It was found that increasing the viscosity of Newtonian fluids (mineral oils) over the range 2-1 100 centipoises caused a decrease in metallic contact. The effect became progressively more pro- nounced at higher viscosities. The viscosity here was the viscosity at atmo- spheric pressure and at the test temperature; neither pressure-viscosity nor temperature-viscosity properties appeared to be important factors. On the other hand, non-Newtonian fluids (polymer-thickened oils) gave more con- tact than their mineral oil counterparts. This suggested that shear-viscosity was important. However, no beneficial effects of viscoelastic properties were observed with these oils. Friction generally decreased with increasing visc- osity because the more viscous oils gave less metal-to-metal contact. The coefficient of friction was rather high: 0.13 at low viscosity, dropping to 0.08 at high viscosity. The oils having higher PVIs (pressure-viscosity index a) gave somewhat more friction which cannot be solely attributed to differ- ences in metallic contact. Furey [37] also investigated the surface roughness effects on metallic contact and friction in the transition zone between the hydrodynamic and boundary lubrications. He found that very smooth and very rough surfaces gave less metallic contact than surfaces with intermediate roughness. Friction was low for the highly polished surfaces and rose with increasing Introduction 9 surface roughness. The rise in friction continued up to a roughness of about 10 pin, the same general level at which metallic contact stopped increasing. However, whereas further increases in surface roughness caused a reduction in metallic contact, there was no significant effect on friction. Friction was found to be independent of roughness in the range of lopin center line average (CLA). He also used four different types of antiwearlantifriction additives (including tricresyl phosphate) and found that they reduced metal- lic contact and friction but had little effect on reducing surface roughness. The additives merely slowed down the wear-in process of the base oil. Thus he concluded that the “chemical polishing” mechanism proposed for explain- ing the antiwear behavior of tricresyl phosphate appeared to be incorrect. Freeman [38] studied several experimental results and summarized them as follows: An unnecessarily thick layer of boundary lubricant may give rise to excessive frictional resistance because shearing and ploughing of the lubricant film may become factors of importance. The bulk viscosity of a fluid lubricant appears to have no significance in its boundary frictional behavior. Coefficients of friction for effective boundary lubricants lie roughly in the range 0.02 to 0.1. The friction force is almost independent of the sliding velocity, provided the motion is insufficient to cause a rise in bulk temperature. If the motion is intermittent or stick-slip, the frictional behavior may be of a different and unpredictable nature. In general the variables that influence dry friction also influence bound- ary friction. Friction and surface damage depends on the chemical composition of the lubricant and/or the products of reaction between the lubricant and the solid surface. Lubricant layers only a few molecules thick can provide effective boundary lubrication. The frictional behavior may be influenced by surface roughness, tem- perature, presence of moisture, oxygen or other surface contami- nants. In general, the coefficient of friction tends to increase with surface roughness. 1.4 FRICTION IN FLUID FILM LUBRICATION Among the early investigations in fluid film lubrication, Tower’s experi- ments in 1883-1884 were a breakthrough which led to the development of 10 Chapter I lubrication theory [6, 71. Tower reported the results of a series of experi- ments intended to determine the best methods to lubricate a railroad journal bearing. Working with a partial journal bearing in an oil bath, he noticed and later measured the pressure generated in the oil film. Tower pointed out that without sufficient lubrication, the bearing oper- ates in the boundary lubrication regime, whereas with adequate lubrication the two surfaces are completely separated by an oil film. Petrov [5] also conducted experiments to measure the frictional losses in bearings. He con- cluded that friction in adequately lubricated bearings is due to the viscous shearing of the fluid between the two surfaces and that viscosity is the most important property of the fluid, and not density as previously assumed. He also formulated the relationship for calculating the frictional resistance in the fluid film as the product of viscosity, speed, and area, divided by the thickness of the film. The observations of Tower and Petrov proved to be the turning point in the history of lubrication. Prior to their work, researchers had concentrated their efforts on conducting friction drag tests on bearings. From Tower’s experiments, it was realized that an understanding of the pressure generated during the bearing operation is the key to perceive the mechanism of lubri- cation. The analysis of his work carried out by Stokes and Reynolds led to a theoretical explanation of Tower’s experimental results and to the theory of hydrodynamic lubrication. In 1886, Osborne Reynolds published a paper on lubrication theory [4] which is derived from the equations of motion, continuity equation, and Newton’s shear stress-velocity gradient relation. Realizing that the ratio of the film thickness to the bearing geometry is in the order of 10-3, Reynolds established the well-known theory using an order-of-magnitude analysis. The assumptions on which the theory is based can be listed as follows. The pressure is constant across the thickness of the film. The radius of curvature of bearing surface is large compared with film The lubricant behaves as a Newtonian fluid. Inertia and body forces are small compared with viscous and pressure There is no slip at the boundaries. Both bearing surfaces are rigid and elastic deformations are neglected. thickness. terms in the equations of motion. Since then the hydrodynamic theory based on Reynolds’ work has attracted considerable attention because of its practical importance. Most initial investigations assumed isoviscous conditions in the film to simplify the ana- lysis. This assumption provided good correlation with pressure distribution Introduction I1 under a given load but generally failed to predict the stiffness and damping behavior of the bearing. A model which predicts bearing performance based on appropriate thermal boundaries on the stationary and moving surfaces and includes a pointwise variation of the film viscosity with temperature is generally referred to as the thermohydrodynamic (THD) model. The THD analyses in the past three decades have drawn considerable attention to the thermal aspects of lubrication. Many experimental and theoretical studies have been undertaken to shed some light on the influence of the energy generated in the film, and the heat transfer within the film and to the surroundings, on the generated pressure. In 1929, McKee and McKee [39] performed a series of experiments on a journal bearing. They observed that under conditions of high speed, the viscosity diminished to a point where the product of viscosity and rotating speed is a constant. Barber and Davenport [40] investigated friction in several journal bearings. The journal center position with respect to the bearing center was determined by a set of dial indicators. Information on the load-carrying capacity and film pressure was presented. In 1946, Fogg [41] found that parallel surface thrust bearings, contrary to predictions by hydrodynamic theory, are capable of carrying a load. His experiments demonstrated the ability of thrust bearings with parallel sur- faces to carry loads of almost the same order of magnitude as can be sustained by tilting pad thrust bearings with the same bearing area. This observation, known as the Fogg effect, is explained by the concept of the “thermal wedge,” where the expansion of the fluid as it heats up produces a distortion of the velocity distribution curves similar to that produced by a converging surface, developing a load-carrying capacity. Fogg also indi- cated that this load-carrying ability does not depend on a round inlet edge nor the thermal distortion of the bearing pad. Cameron [42], in his experiments with rotating disks, suggested that a hydrodynamic pressure is created in the film between the disks due to the variation of viscosity across the thickness of the film. Viscoelastic lubricants in journal bearing applica- tions were studied by Tao and Phillipoff [43]. The non-Newtonian behavior of viscoelastic liquids causes a flattening in the pressure profile and a shift of the peak film pressure due to the presence of normal stresses in the lubricant. Dubois et al. [44] performed an experimental study of friction and eccen- tricity ratios in a journal bearing lubricated with a non-Newtonian oil. They found that a non-Newtonian oil shows a lower friction than a corresponding Newtonian fluid under the same operating conditions. However, this phenomenon did not agree with their analytical work and could not be explained. 12 Chapter I Maximum bearing temperature is an important parameter which, together with the minimum film thickness, constitutes a failure mechanism in fluid film bearings. Brown and Newman [45] reported that for lightly loaded bearings of diameter 60 in. operating under 6000 rpm, failure due to overheating of the bearing material (babbitt) occurred at about 340°F. Booser et al. [46] observed a babbitt-limiting maximum temperature in the range of 266 to 392°F for large steam turbine journal bearings. They also formulated a one-dimensional analysis for estimating the maximum temperature under both laminar and turbulent conditions. In a study of heat effects in journal bearings, Dowson et al. [47] in 1966 conducted a major experimental investigation of temperature patterns and heat balance of steadily loaded journal bearings. Their test apparatus was capable of measuring the pressure distribution, load, speed, lubricant flow rate, lubricant inlet and outlet temperatures, and temperature distribution within the stationary bushing and rotating shaft. They found that the heat flow patterns in the bushing are a combination of both radial flows and a significant amount of circumferential flow traveling from the hot region in the vicinity of the minimum film thickness to the cooler region near the oil inlet. The test results showed that the cyclic variation in shaft surface tem- perature is small and the shaft can be treated as an isothermal component. The experiments also indicated that the axial temperature gradients within the bushing are negligible. Viscosity is generally considered to be the single most important prop- erty of lubricants, therefore, it represents the central parameter in recent lubricant analyses. By far the easiest approach to the question of viscosity variation within a fluid film bearing is to adopt a representative or mean value viscosity. Studies have provided many suggestions for calculations of the effective viscosity in a bearing analysis [48]. When the temperature rise of the lubricant across the bearing is small, bearing performance calcula- tions are customarily based on the classical, isoviscous theory. In other cases, where the temperature rise across the bearing is significant, the classical theory loses its usefulness for performance prediction. One of the early applications of the energy equation to hydrodynamic lubrication was made by Cope [49] in 1948. His model was based on the assumptions of negligible temperature variation across the film and negligible heat conduction within the lubrication film as well as into the neighboring solids. The consequence of the second assumption is that both the bearing and the shaft are isothermal components, and thus all the generated heat is carried out by the lubricant. As indicated in a review paper by Szeri [50], the belief, that the classical theory on one hand and Cope’s adiabatic model on the other, bracket bearing performance in lubrication analysis, was widely accepted for a while. A thermohydrodynamic hypothesis was Introduction 13 later introduced by Seireg and Ezzat [51] to rationalize their experimental findings. An empirical procedure for prediction of the thermohydrodynamic behavior of the fluid film was proposed in 1973 by Seireg and Ezzat. This report presented results on the load-carrying capacity of the film from extensive tests. These tests covered eccentricity ratios ranging from 0.6 to 0.90, pressures of up to 750 psi and speeds of up to 1650 ftlmin. The empirical procedure applied to bearings submerged in an oil bath as well as to pump-fed bearings where the outer shell is exposed to the atmosphere. No significant difference in the speed-pressure characteristics for these two conditions was observed when the inlet temperature was the same. They showed that the magnitudes of the load-carrying capacity obtained experi- mentally may differ considerably from those predicted by the insoviscous hydrodynamic theory. The isoviscous theory can either underestimate or overestimate the results depending on the operating conditions. It was observed, however, that the normalized pressure distribution in both the circumferential and axial directions of the journal bearing are almost iden- tical to those predicted by the isoviscous hydrodynamic theory. Under all conditions tested, the magnitude of the peak pressure (or the average pres- sure) in the film is approximately proportional to the square root of the rotational speed of the journal. The same relationship between the peak pressure and speed was observed by Wang and Seireg [52] in a series of tests on a reciprocating slider bearing with fixed film geometry. A compre- hensive review of thermal effects in hydrodynamic bearings is given by Khonsari [53] and deals with both journal and slider bearings. In 1975, Seireg and Doshi [54] studied nonsteady state behavior of the journal bearing performance. The transient bushing temperature distribu- tion in journal bearing appears to be similar to the steady-state temperature distribution. It was also found that the maximum bushing surface tempera- ture occurs in the vicinity of minimum film thickness. The temperature level as well as the circumferential temperature variation were found to rise with an increase of eccentricity ratio and bearing speed. Later, Seireg and Dandage [55] proposed an empirical thermohydrodynamic procedure to calculate a modified Sommerfeld number which can be utilized in the stan- dard formula (based on the isoviscous theory) to calculate eccentricity ratio, oil flow, frictional loss, and temperature rise, as well as stiffness and damp- ing coefficients for full journal bearings. In 1980, Barwell and Lingard [56] measured the temperature distribu- tion of plain journal bearings, and found that the maximum bearing tem- perature, which is encountered at the point of minimum film thickness, is the appropriate value for an estimate of effective viscosity to be used in load capacity calculation. Tonnesen and Hansen [57] performed an experiment 14 Chapter 1 on a cylindrical fluid film bearing to study the thermal effects on the bearing performance. Their test bearings were cylindrical and oil was supplied through either one or two holes or through two-axial grooves, 180" apart. Experiments were conducted with three types of turbine oils. Both viscosity and oil inlet geometry were found to have a significant effect on the operat- ing temperatures. The shaft temperature was found to increase with increas- ing loads when a high-viscosity lubricant was used. At the end of the paper, they concluded that even a simple geometry bearing exhibits over a broad range small but consistent discrepancies when correlated with existing theory. In 1983, Ferron et al. [58] conducted an experiment on a finite-length journal bearing to study the performance of a plain bearing. The pressure and the temperature distributions on the bearing wall were measured, along with the eccentricity ratio and the flow rate, for different speeds and loads. All measurements were performed under steady-state conditions when ther- mal equilibrium was reached. Good agreement was found with measure- ments reported for pressure and temperature, but a large discrepancy was noted between the predicted and measured values of eccentricity ratios. In 1986, Boncompain et al. [59] showed good agreement between their theo- retical and experimental work on a journal bearing analysis. However, the measured journal locus and calculated values differ. They concluded that the temperature gradient across and along the fluid film is the most impor- tant parameter when evaluating the bearing performance. 1.5 FRICTIONAL RESISTANCE IN ELASTOHYDRODYNAMIC CONTACTS In many mechanical systems, load is transmitted through lubricated con- centrated contacts where rolling and sliding can occur. For such conditions the pressure is expected to be sufficiently high to cause appreciable deforma- tion of the contacting bodies and consequently the surface geometry in the loaded area is a function of the generated pressure. The study of the beha- vior of the lubricant film with consideration of the change of film geometry due to the elasticity of the contacting bodies has attracted considerable attention from tribologists over the last half century. Some of the studies related to frictional resistance in this elastohydrodynamic (EHD) regime are briefly reviewed in the following with emphasis on effect of viscosity and temperature in the film. Dyson [60] interpreted some of the friction results in terms of a model of viscoelastic liquid. He divided the experimental curves of frictional traction versus sliding speed into three regions: the linear region, the nonlinear [...]... Surfaces 2 Figure 2 .1 Concentrated load on a semi-infinite elastic solid a = horizontal stress at any point , z - v2 (E~ 3 ( , 2 ~ 2 ) - 5 / 2 ) - ” ~ ] - 3r2Z(r2+ 2 2 ) - 5 / 2 2n = P( (1- 2 v ) [ $ 2n az = vertical stress at any point I - 3p 23 (, .2~ 2)-5 /2 2n rrz = shear stress at any point -_ ,z2( ,2 +z2)-5 /2 - 3p 2n where v = Poisson’s ratio The resultant principal stress passes through the origin and. .. Technol., 19 81, Vol 10 3, pp 10 7 -1 14 Ferron, J., Frene, J., and Boncompain, R., “A Study of the Thermohydrodynamic Performance of a Plain Journal Bearing, Comparison Between Theory and Experiments,” ASME J Lubr Technol., 19 83, Vol 10 5, pp 422 428 Boncompain, R., Fillon, M., and Frene, J., “Analysis of Thermal Effects in Hydrodynamic Bearings,” J Tribol., 19 86, Vol 10 8, pp 21 9 -22 4 20 60 61 62 63 64 65... Surfaces Table 2 .1 27 Values of Factor k 0.95 0.94 0. 92 0.88 0. 82 0. 71 0.37 1 1.5 2 3 5 10 10 0 Case 4: A Rigid Circular Cylinder Pressed Against a Semi-Infinite Solid In this case, which is shown in Fig 2. 4, the displacement of the rigid cylinder is calculated from: P (1 - ”*) w=2aE P Figure 2. 4 Rigid cylinder over a semi-infinite elastic solid 28 Chapter 2 where p = total load on the cylinder a = radius... coefficient of friction in this regime of lubrication Introduction I7 REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 MacCurdy, E., Leonardo da Vinci Notebooks, Jonathan Cape, London, England, 19 38 Amontons, G., Histoire de 1 AcadCmie Royale des Sciences avec Les Memoires de Mathematique et de Physique, Paris, 16 99 Coulomb, C A., Memoires de Mathematique et de Physique de 1 Academie Royale... 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Impulsive Loading,” Wear, 19 65, Vol 8, pp 20 8- 21 9 “Designing for Zero Wear - Or a Predictable Minimum,” Prod Eng., August 15 , 19 66, pp 41- 50 Rabinowicz, E., “Variation of Friction and Wear of Solid Lubricant Films with Film Thickness,” ASLE Trans., Vol 10 , n .1, 19 67, pp 1- 7 Reynolds, O., Phil Trans., 18 75, p 16 6 Palmgren, A., “Ball and Roller Bearing Engineering,” S.H Burbank, Philadelphia, PA, 19 45 Tabor, . z v2 (E 2n ~3( ,2~ 2)-5 /2) -”~] - 3r2Z(r2 + 22 )-5 /2 az = vertical stress at any point - 3p 23 (, .2~ 2)-5 /2 2n rrz = shear stress at any point - -_ 3p ,z2( ,2 + z2)-5 /2 2n where. on Gear Lubrication, ” Bull. JSME, 19 61, Vol. 4 (14 ), p. 3 82. 26 6( 1 17 0), pp. 1- 33. p. 57. 2 15 -23 6. 17 (4), pp. 2 71- 27 9. Introduction 21 77. 78. 79. 80. 81. 82. 83. 84 Proc. Roy. Soc. 19 57, Vol. A238, pp. 529 -550. 18 Chapter 1 22 . 23 . 24 . 25 . 26 . 27 . 28 . 29 . 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. Seireg, A., and Weiter, E.