Mechanical Engineering-Tribology In Machine Design Episode 13 pps

Lubrication and efficiency of involute gears 287 pinion, V is the pitch line velocity in m/s and I// is the helix angle for spur gears I// = 0.. According to eqn 8.19 the allowable valu

Lubrication and efficiency of involute gears 287

pinion, V is the pitch line velocity in m/s and I// is the helix angle (for spur gears I// = 0)

As a general rule, the lower values of V are used in the case of single-stage gear trains and the higher values of V are recommended for multi-stage gear trains Slow running gears are usually splash lubricated Thedepth to which the gear is immersed in the oil bath is given by

where m is the module in mm

In multi-stage gear trains it is difficult to obtain a proper immersion of all the gearwheels in the oil bath This is always the case when there are substantial differences in the diameters of the gearwheels The usual solution is to install an auxiliary oil tank for each gearwheel in order to achieve the required depth of immersion

Splash lubrication is effective up to a certain clearly defined pitch line velocity This velocity can be determined in the way illustrated schemati- cally in Fig 8.6 The centrifugal force, acting on an element of oil having a mass dm will cause the motion of the element in the radial direction This will be counteracted by the force required to shear the oil film formed on the surface of the tooth Using symbols from Fig 8.6 we can write

Figure 8.6

ds

w2r dm = A -dy

dy But

and

therefore, eqn (8.13) can be written as

Assuming that the thickness of the oil film formed on the tooth surface is h,

and integrating, eqn (8.14) gives

288 Tribology in machine design

Assuming further that there is a parabolic distribution of the velocity within the oil film

Vmax

V = ~ ~ ~ ~ - ( j - h , ) ~ -

h,2 the final form of eqn (8.15) is

Vmax

h,2

In order to derive the expression for the maximum velocity of the oil due to the action ofcentrifugal force, we assume that the tooth is immersed in the oil bath, x, =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, m; 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, Vl,

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 [m3/(min cm)], m is the module in mm and V is the pitch line velocity in ms-'

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 T o 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 ofcontact, 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

tan a

= tan(a + 4 ) '

where a is the inclination of the plane, or in this case the pitch angle of the worm, and 4 is the angle of friction For the case of the worm-wheel driving the worm the expression for efficiency is

tan(a - 4 )

tan a

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 4 may well exceed a ; in this case q will be less than 0.5 and q' 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 of45", and if the gears are well lubricated, 4 , under runningconditions, particularly at high speeds, may well be of the order of lo o r less The efficiency is then oft he order 0.97 -0.98, i.e oft he 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 lasses

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 CfW) assists rotation The torque exerted by the driver at

any approach position is as follows:

but

thus

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 La, in the form

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 29 1

The first case t o be considered is that o f the constant friction coefficient throughout the engagement Integrating the equations describing the work output gives

The efficiency of the gears is therefore equal t o

W22+Wr2 - (Pal+Prl)-.fr(tan4)(Pa1-Prl)-(f'/2i)(P~1+PZ1)

w,, I + wr 1 ( P a 1 + P r 1 ) - f r ( t a n +)(Pa 1 - P r 1 ) + (fr/2)(Pz1 + P : 1 ) '

(8.36)

where i is the gear ratio

The friction losses per minute are equal t o

The efficiency can be written more simply and almost exactly by considering the work input t o be equal t o

hence

efficiency = 1 - 1; (Pil + P : I ) *

292 Tribology in machine design

Case 11 The coejjicient 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 oftypes ofsurface finish and many other factors Actual tests carried out on gears have revealed that the form ofthe 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

a) - Approach b) p i t c h p i n t

V = V = c o n s t

pn qn v = v - v = o

P t q t s V*= v p t - "qt

v , v

Pt g t

C l neceSS

vpn= Vgn: c o n s t

v = v

3 p t - 'Jqt

Figure 8.8 gt"k

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- ' ) 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, gves

The friction losses per minute are given by

w, =

P a l +Prl

Considering the work input to be equal to

then the efficiency is given by

1 + (l/i) efficiency = 1 - ] +; P:l)

a 1 + P r l

8.8.2 Summary of efficiency formulae

In order to collate the material presented in the previous section the following summary is made: when N1, N2 is the number of teeth on driver and driven gear, respectively, i is the gear ratio, Pa, Pr is the arc of approach and recess on the driver, respectively, f is the average coefficient of friction, fa

is the average coefficient of friction during the approach period andf, is the

294 Tribology in machine design

average coefficient of friction during the recess period, then, for the constant friction coefficient

efficiency = 1 - [y::) 1; (pi + I?)

and for different average friction coefficients during the approach and recess periods

1 + (l/i)

f i c e c y = - -1 P a + B r + ; p: 1

References to Chapter 8 1 H M Martln Lubrication of gear teeth Erlyinrerirlg 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 AS ME 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 oflnrolute 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-

di.sciplinar?~ Approuc11 ro rhe Lubrication qf Concentrated Contacrs, NASA SP-

237 1970

Index

Abrasive wear, 19, 20

Acoustic emission, 268

Addendum, 10

Adhesive interaction, 15

Adhesive junction, 14, 15

Adhesive wear, 19

Adhesive wear equation, 39

Aerosol lubrication, 264

Angle of lap, 129, 133

Apparent area of contact, 14

Asperity, 14

Attitude angle, 192

Attitude of journal, 57

Axially loaded bearing, 123

Ball bearing, 7

Band and block brake, 144

Band brake, 136

Basic dynamic capacity, 7

Bearing clearance, 54

Bearing eccentricity, 54

Bearing materials, 220

Belt drive, 128

Belt power transmission rating, 132

Big-end bearing, 2 1 3

Blistering, 167

Blok theory, 75, 280

Boundary lubricated bearing, 12 1

Brake design, 136

Braking of vehicle, 145

Bulk temperature, 79

Cam, 9

Cam-follower, 9, 246

Centrifugal clutch, 120

Chemical wear, 19

Coefficient of adhesion, 146

Coefficient of viscosity, 48

Collar bearing, 124

Concave surface, 67 Concentrated force, 65 Cone clutch, 1 14 Conformal surfaces, 2 Conjunction temperature, 75 Connecting-rod bearing, 2 1 3 Contact mechanics, 64 Convex surface, 67 Copper-lead alloy, 22 1

Cornering of tyre, 152 Counterformal surfaces, 2 Crankshaft bearing, 21 3 Creep of tyre, 152 Critical slope, 188 Critical temperature, 82, 280 Critical temperature hypothesis, 1 1

Curvature factor, 28 1

Curved brake block, 138 Cylinder liner, 8 Dedendum, 10 Deformations in rolling-contact bearing, 254

Diametral clearance, 190, 195 Differential sliding, 249 Distributed force, 65 Driven rolling, 156 Dynamic hydroplaning, 158 Dynamically loaded journal bearing,

212 Eccentricity ratio, 190, 192, 203 Efficiency of involute gears, 273, 288 Elastic contact, 14

Elastic extension of belt, 13 1

Elastic hysteresis, 25 1

Elasticity parameter, 241 Elastohydrodynamic lubrication, 3 Elliptical bearing, 206

296 Index

Energy dissipation, 18 Hysteresis losses, 234 Engineering design, 1 Hysteresis loss factor, 234 Equivalent cylinder, 95

Equivalent speed method, 2 14 Inlet zone temperature, 244 Extreme pressure oil, 1 1 Interfacial adhesive bonds, 15 Externally pressurized bearing 18 1 Interfacial shear strength,

Involute gears, 10, 273 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, 2 10 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,

174, 204 Hydrostatic bearing, 178 Hydrostatic thrust bearing, 225 Hypoid gears, I I

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,

256 Kinetic friction, 98 Kingsbury, 186 Labyrinth seals, 164 Lambda ratio, 26, 29, 260, 265, 281 Line contact, 242

L life, 7, 267 Load bearing capacity, 196 Load number, 197, 2 15 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 bearings, 259

Lubrication of seals, 172 Lubrication of involute gears, 273 Lubrication regimes, 275

Marangoni effect, 1 62 Mechanical seal, 160 Michell, 186 Michell bearing, 223 Micro-slip, 236 Misalignment, 6 Mist lubrication, 264