Tribology in Machine Design 2009 Part 11 pdf

288 Tribology in machine design Assuming further that there is a parabolic distribution of the velocity within the oil film the final form of eqn (8.15) is In order to derive the expression for the maximum velocity of the oil due to the action of centrifugal force, we assume that the tooth is immersed in the oil bath, x e =0.2m, below the dedendum and that the time of action of the centrifugal force is defined by the angle of rotating, a, and the angular velocity, co; thus, and finally Substituting eqn (8.18) into eqn (8.17) and rearranging gives where v is the kinematic viscosity in cSt. According to eqn (8.19) the allowable value of the pitch line velocity, F,, at which splash lubrication is still effective, is a function of the oil viscosity, the module, the angle between the point of immersion and the point of engagement, a, and, indirectly, the surface roughness of the tooth surface. Gear trains operating at high speeds and also power gear units are jet lubricated. Each pair of meshing gears should receive an amount of oil resulting from the expression where Q is the flow rate of oil per 1 cm of the tooth width, measured in [m 3 /(min cm)], m is the module in mm and V is the pitch line velocity in ms" 1 . For a very approximate estimate of the oil flow rate required, the following formula can be used: where N is the power transmitted in kW. 8.8. Efficiency of gears The power loss in properly lubricated spur, helical or similar types of gearing is usually very low, that is due to the tooth friction being of the Lubrication and efficiency of involute gears 289 order of only 1 per cent or less of the power transmitted at full load. To this, the losses due to oil churning and bearing friction have to be added. In such gears, there is inevitably, sliding at all points in the path of contact, except at the pitch point, and it can be deduced that the coefficient of friction is low and that lubrication must therefore be effective in spite of the extremely high contact pressures. In the case of skew, and more particularly worm, gearing, sliding occurs not only as in spur gears but, much more importantly, in a direction at right angles to this. In fact, we can obtain a sufficiently close approximation to the situation in a worm gear by ignoring the pressure angle of the thread and thinking of this thread as perpendicular to the axis. We can then regard the thread as an inclined plane which moves relatively to the surface of the worm wheel; and the analogous situation of a block being pushed up an inclined plane by a horizontal force is quite common in mechanics. Thus, the expression for efficiency can be written as where a is the inclination of the plane, or in this case the pitch angle of the worm, and (j> is the angle of friction. For the case of the worm-wheel driving the worm the expression for efficiency is Now, in the case of a single-tooth worm, a may be only a few degrees, and if the surfaces are dry or poorly lubricated ^ may well exceed a; in this case rj will be less than 0.5 and r\' will be negative. In other words, the drive will be irreversible. Such a gear has its uses, but would be unthinkable for power transmission. For multi-start worm gears, however, a can be made of the order of 45°, and if the gears are well lubricated, 0, under running conditions, particularly at high speeds, may well be of the order of 1° or less. The efficiency is then of the order 0.97-0.98, i.e. of the same order as that for spur gears. As far as power loss is concerned, the difference is probably negligible but it should be noted that the losses have to be dissipated as heat, and since the amount of heat which has to be expended is almost directly propor- tional to the effective coefficient of friction, it is vital to ensure that the best possible lubrication is maintained, and in the case of highly loaded gears that sufficient cooling is provided. 8.8.1. Analysis of friction losses In Fig. 8.7, one gear rotating clockwise drives another. Subscript 1 is used on the symbols for the driver and the subscript 2 is used on those for the driven gear. All the parameters used in the following analysis are clearly defined in Fig. 8.7. Using the assumption that when two or more pairs of teeth carry the load simultaneously, the normal pressure is shared equally 290 Tribology in machine design between them, it can be shown that the total friction loss and the power input to the driven gear during the engagement of one pair of mating teeth are the same as when one pair of mating teeth carry the entire load throughout their period of engagement. During approach, considering any position of contact as at x (see Fig. 8.7), the normal force W opposes the rotation of the driver, while the frictional force (fW) assists rotation. The torque exerted by the driver at any approach position is as follows: but thus Figure 8.7 The work output for the driver during approach is as follows During recess, the direction of sliding between the teeth is reversed, so that and the work output for the driver during recess Now, turning to the driven gear, during approach the normal force and the frictional force oppose one another. Thus Expressing the work output for the driven gear during approach is given by During recess, both the normal force and the tangential force assist the rotation of the driven gear, therefore but Lubrication and efficiency of involute gears 291 Case I: The coefficient of friction is constant The first case to be considered is that of the constant friction coefficient throughout the engagement. Integrating the equations describing the work output, gives The efficiency of the gears is therefore equal to where i is the gear ratio. The friction losses per minute are equal to The efficiency can be written more simply and almost exactly by considering the work input to be equal to hence 292 Tribology in machine design Cose II. The coefficient of friction considered as variable As a matter of fact, the friction coefficient is not constant but varies with different loads, speeds, lubricants and gear materials, as well as with different types of types of surface finish and many other factors. Actual tests carried out on gears have revealed that the form of the relationship between the average friction coefficients and the pitch line velocities is very much the same as in the case of journal bearings. At low speeds, the values of the friction coefficient are high, decreasing rapidly to a certain minimum value with increasing velocity, and then rising slowly with further increase in velocity. There is, however, one important difference in the lubrication mechanism operating in plain journal bearings and in gears. In the case of the journal bearings, hydrodynamic lubrication is usually a dominant type of lubrication while in gears, elastohydrodynamic lubrication is the main mechanism. It is known that the nature of sliding between involute gear teeth consists of sliding in one direction during approach, reducing to zero at the pitch point where the direction of sliding changes, and increasing again as the contact progresses through the recess action. This is shown, in a schematic way, in Fig. 8.8. Since the direction of sliding changes at the pitch point, we may conclude that the coefficient of friction will assume the value characteristic for a thick Figure 8.8 Lubrication and efficiency of involute gears 293 film lubrication regime during the period of engagement of a pair of mating teeth. It is possible to set up an efficiency equation in various ways. The chances are, however, that the most that can be determined by experiment is to establish some average values of friction coefficient for the approach action and similarly for the recess action. Experimental results suggest that at very low pitch line velocities (up to 1.5 m min~ 1 ) the friction of approach period appears to be approximately double that of the recess period on hobbed, milled and shaped gears made of cast-iron, soft-steel, bronze and aluminium. On hardened and ground steel gears, however, the difference between the friction of approach and the friction of recess is almost non-existent. When the contact passes through the pitch point, a significant increase in friction (about 150 per cent) takes place. Thus, introducing different average values for the friction coefficients of the approach and recess, gives 8.8.2. Summary of efficiency formulae In order to collate the material presented in the previous section the following summary is made: when N l5 N 2 is the number of teeth on driver and driven gear, respectively, i is the gear ratio, /? a , /? r is the arc of approach and recess on the driver, respectively,/is the average coefficient of friction,/ a is the average coefficient of friction during the approach period and/ r is the 294 Tribology in machine design average coefficient of friction during the recess period, then, for the constant friction coefficient and for different average friction coefficients during the approach and recess periods References tO Chapter 8 1. H. M. Martin. Lubrication of gear teeth. Engineering, 102, (1916), 16-19. 2. D. W. Dudley. Practical Gear Design. New York: McGraw-Hill, 1954. 3. K. F. Martin. The efficiency of involute spur gears. ASME Technical Paper, No. 80-C2/DET-16, 1980. 4. D. W. Dudley. Gear Handbook. New York: McGraw-Hill, 1962. 5. D. Dowson and G. R. Higginson. A Theory of Involute Gear Lubrication. Gear Lubrication Symposium. London: Inst. of Petroleum, 1964. 6. D. W. Dudley. Information sheet Gear scoring design guide for aerospace spur and helical power gears. Washington, D.C.: AGMA, 1965. 7. H. Blok. The postulate about the constancy of scoring temperature. Inter- disciplinary Approach to the Lubrication of Concentrated Contacts, NASA SP- 237, 1970. Index Abrasive wear, 19, 20 Concave surface, 67 Acoustic emission, 268 Concentrated force, 65 Addendum, 10 Cone clutch, 114 Adhesive interaction, 15 Conformal surfaces, 2 Adhesive junction, 14, 15 Conjunction temperature, 75 Adhesive wear, 19 Connecting-rod bearing, 213 Adhesive wear equation, 39 Contact mechanics, 64 Aerosol lubrication, 264 Convex surface, 67 Angle of lap, 129, 133 Copper-lead alloy, 221 Apparent area of contact, 14 Cornering of tyre, 152 Asperity, 14 Counterformal surfaces, 2 Attitude angle, 192 Crankshaft bearing, 213 Attitude of journal, 57 Creep of tyre, 152 Axially loaded bearing, 123 Critical slope, 188 Critical temperature, 82, 280 Ball bearing, 7 Critical temperature hypothesis, 11 Band and block brake, 144 Curvature factor, 281 Band brake, 136 Curved brake block, 138 Basic dynamic capacity, 7 Cylinder liner, 8 Bearing clearance, 54 Bearing eccentricity, 54 Dedendum, 10 Bearing materials, 220 Deformations in rolling-contact Belt drive, 128 bearing, 254 Belt power transmission rating, 132 Diametral clearance, 190, 195 Big-end bearing, 213 Differential sliding, 249 Blistering, 167 Distributed force, 65 Blok theory, 75, 280 Driven rolling, 156 Boundary lubricated bearing, 121 Dynamic hydroplaning, 158 Brake design, 136 Dynamically loaded journal bearing, Braking of vehicle, 145 212 Bulk temperature, 79 Eccentricity ratio, 190, 192, 203 Cam, 9 Efficiency of involute gears, 273, 288 Cam-follower, 9, 246 Elastic contact, 14 Centrifugal clutch, 120 Elastic extension of belt, 131 Chemical wear, 19 Elastic hysteresis, 251 Coefficient of adhesion, 146 Elasticity parameter, 241 Coefficient of viscosity, 48 Elastohydrodynamic lubrication, 3 Collar bearing, 124 Elliptical bearing, 206 296 Index Energy dissipation, 18 Engineering design, 1 Equivalent cylinder, 95 Equivalent speed method, 214 Extreme pressure oil, 11 Externally pressurized bearing, 18 1 Fatigue wear equation, 40 Film lubrication, 48 Flash temperature, 75, 83, 280 Flat pivot, 184 Fleming-Suh model, 45 Fluid film, 3, 6, 210 Four lobe bearing, 206 Fractional film defect, 34 Fracture mechanics and wear, 45 Fracture of adhesive junction, 16 Fracture toughness, 17 Free rolling, 156 Friction angle, 98 Friction circle, 122 Friction coefficient, 13 Friction drive, 10, 127 Friction due to adhesion, 15 Friction due to deformation, 17 Friction due to ploughing, 16 Friction in slideways, 98 Friction losses, 289 Friction stability, 100 Friction torque, 249 Frictional force, 13 Frictional traction. 10 Gas bearing, 210 Gear lubrication, 286 Gear tribodesign, 273 Gear wear, 285 Grease lubrication, 26 1 Grubin approximation, 245 Gyroscopic spin, 250 Heat of adsorption of lubricant, 35 Helical seal, 163 Hertzian area, 2 Hertzian stress, 9 Higher kinematic pair, 232 Hydrodynamically lubricated bearing, Hydrostatic bearing, 178 Hydrostatic thrust bearing, 225 Hypoid gears, 11 174, 204 Hysteresis losses, 234 Hysteresis loss factor, 234 Inlet zone temperature, 244 Interfacial adhesive bonds, 15 Interfacial shear strength, 16 Involute gears, 10, 273 Jet lubrication, 262 Journal bearing, 189, 204 Journal bearing with: fixed non-preloaded pads, 205 fixed preloaded pads, 205 movable pads, 207 special geometric features, 207 Junction growth, 15 Kinematics of rolling-contact bearing, Kinetic friction, 98 Kingsbury, 186 256 Labyrinth seals, 164 Lambda ratio, 26, 29, 260, 265, 281 Line contact, 242 L life, 7, 267 Load bearing capacity, 196 Load number, 197, 215 Load sharing, 37 Load transmission, 1 Loading factor, 282 Lower kinematic pair, 97 Lubricant contamination, 266 Lubricant factor, 28 1 Lubricant filtration, 266 Lubricant viscosity, 33 Lubricated contact, 3 1 Lubrication effect on fatigue life, 265 Lubrication of cylinders, 238 Lubrication of rolling-contact Lubrication of seals, 172 Lubrication of involute gears, 273 Lubrication regimes, 275 bearings, 259 Marangoni effect, 162 Mechanical seal, 160 Michell, 186 Michell bearing, 223 Micro-slip, 236 Misalignment, 6 Mist lubrication, 264 Index 297 Nonconforming contact, 22 Ocvirk number, 197 Ocvirk solution, 19 1 Offset factor, 206, 207 Oil film thickness, 214 Oil flow, 194 Pad pivot, 207 Palmgren, 255 Peclet number, 76 Petroff law, 50 Piston, 8 Piston ring, 8 Pitting, 10, 12, 72, 282 Pivot bearing, 124 Plastic deformation, 15 Plasticity index, 14, 30 Plate clutch, 11 1 Ploughing, 14 Pneumatic tyres, 1 5 1 Point-contact lubrication, 245 Preload factor, 206, 208 Pressure gradient, I75 Pressure-viscosity coefficient, 33, 241, Propulsion of vehicle, 145 Protective layer, 4 PV limit, 230 242, 281 Radial clearance, 190 Rayleigh step, 163 Real area of contact, 14 Recess, 180 Reynolds equation, 174, 177, 239 Reynolds hypothesis, 249 Reynolds number, 163, 238 Reynolds theory, 51 Rim clutch, 116 Roller bearing, 7 Rolling contact bearing, 7, 248 Rolling friction, 235, 248 Rolling of tyre, 155 Rope drive, 134 Run-in, 11 Screw jack, 105 Scuffing, 9, 11, 12, 278 Self-lubricating bearing, 226 Short-bearing approximation, 19 1 Short-bearing theory, 203 Sliding bearing, 6, 174 Sliding of tyre, 155 Solid film lubrication, 260 Sommerfeld diagram, 60 Sommerfeld number, 2 15 Sommerfeld solution, 190 Square thread, 103 Squeeze-film lubrication, 18 I Standard deviation, 27 Static load rating, 7 Subcase fatigue, 73 Subsurface fatigue, 11 Surface active additives, 12 Surface failure, 71, 265 Surface fatigue, 73 Surface fatigue wear, 19, 21 Surface finish, 9 Surface peak, 90 Surface roughness, 2 Surface temperature, 74 Surface tension, 162, 240 Surface topography, 14 Surface traction, 233 Taper rolling bearing, 7 Thermal bulging, 80 Thermal correction factor, 244 Thermal effects, 74 Thermal loading factor, 244 Three lobe bearing, 206 Thrust bearing, 183, 221 Tilting pad bearing, 207, 223 Tin-aluminium alloy, 220 Traction effort, 146 Tractive resistance, 150 Tread pattern, 155 Triangular thread, 109 Tribodesign, 1 Tribology, 1 Tyre performance, 157 Tyre surface, 154 Unloaded bearing, 53 Variance, 27 V-belt drive, 134 Velocity factor, 282 Virtual coefficient of friction, 59 Viscosity, 180, 202, 204, 21 5 Viscosity parameter, 241 Viscosity-temperature coefficient, 245 Viscous flow, 50 [...]...298 Index Viscous hydroplaning, 158 Viscous shear, 210 Wear rate, 13 Worm gears, 12 Wear in mechanical seals, 164 Yield strength, 14, 21 . rising slowly with further increase in velocity. There is, however, one important difference in the lubrication mechanism operating in plain journal bearings and in gears. In. sliding between involute gear teeth consists of sliding in one direction during approach, reducing to zero at the pitch point where the direction of sliding changes, and increasing again . clutch, 116 Roller bearing, 7 Rolling contact bearing, 7, 248 Rolling friction, 235, 248 Rolling of tyre, 155 Rope drive, 134 Run -in, 11 Screw jack, 105 Scuffing, 9, 11, 12, 278