Mechanical Engineering-Tribology In Machine Design Episode 9 pot

Geometrical configuration and pressure generation In a simple plain journal bearing, the position of the journal is directly related to the external load.. The eccentricity e is measure

neglected, i.e the pad is assumed to be infinitely long Let

h =the mean thickness of the film,

h + e =the thickness at inlet,

h - e = the thickness at outlet,

;=the thickness at a section X-X, at a distance x from the centre of the breadth, so that

Adopting the same procedure as that used in fluid mechanics

1 dp ,I3

flow across X -X = 3 VA - - - -

p dx 12' Suppose x' is the value of x at which maximum pressure occurs, i.e where dpldx =0, then, for continuity of flow

so that

Similarly shear stress

so that

Integrating eqn (5.30), the pressure p at the section X-X is given by

where x' and k are regarded as constants As dA/dx = - 2e/B, this becomes

pB Bx' 1 B 1 ~~h 1

+ -

12pVe 4e A2 4e2A 8e2 z + ~

The constants x' and k are determined from the condition that p = O when

x = + +B, i.e when A = h - e and h + e respectively Hence

B2

k = and x'=- Be

1 88 Tribology in machine design

and the pressure equation becomes

For the maximum value of p write x = x' = Be/(2h), i.e II = (h2 - e2)/h Equation (5.33) then becomes

where a =e/h denotes the attitude of the bearing or pad surface

Referring to Fig 5.11, P is the load on the slider (per unit length measured perpendicular to the direction of motion) and F' is the pulling force equal and opposite to the tangential drag F Similarly Q and F, are the reaction forces on the oil film due to the bearing, so that the system is in equilibrium (a necessary condition is that the pad has sufficient freedom to adjust its slope so that equilibrium conditions are satisfied) under the action of the four forces, P, Q, F ' and F, Again, P and F' are equal and opposite to the resultant effects of the oil film on the slider, so that

For the former, eqn (5.33) gives

and writing a = e/h this reduces to

Similarly, for the tangential pulling force, eqns (5.31) and (5.36) give

and integrating between the limits + B/2, this reduces to

Iff is the virtual coefficient of friction for the slider we may write

F 1 = f P

Since a is very small we may write sin a w a and cos a = 1 Further, F , is very

small compared with Q, and so

P = Q and F' = Qa + F , (approximately),

F , = F ' - Pa (approximately) (5.40)

A critical value of a occurs when F , =0, i.e

where @ is the angle of friction for the slider When a > @ , F , becomes

negative This is caused by a reversal in the direction of flow of the oil film

adjacent to the surface of the pad The critical value of a is given by eqn

(5.39) Thus

therefore

and so

5.5.1 Geometrical configuration and pressure generation

In a simple plain journal bearing, the position of the journal is directly related to the external load When the bearing is sufficiently supplied with oil and the external load is zero, the journal will rotate concentrically within the bearing However, as the load is increased the journal moves to an

190 Tribology in machine design

increasingly eccentric position, thus forming a wedge-shaped oil film where load-supporting pressure is generated The eccentricity e is measured from

the bearing centre Oh to the shaft centre O j , as shown in Fig 5.12 The

maximum possible eccentricity equals the radial clearance c, or half the initial difference in diameters, c,, and it is of the order of one-thousandth of the diameter It will be convenient to use an eccentricity ratio, defined as

E =e/c Then E = O at no load, and E has a maximum value of 1.0 if the shaft should touch the bearing under extremely large loads

The film thickness h varies between h,,, = c(1 + e) and h,,, = c(1- E) A

Figure 5.12 sufficiently accurate expression for the intermediate values is obtained from

the geometry shown in Fig 5.12 In this figure the journal radius is r, the

bearing radius is r + c, and is measured counterclockwise from the position

of h,,, Distance OOj z OOh + e cos 0, or h + r = (r + c) + e cos 0, whence

4-

Figure 5.13

The rectilinear coordinate form of Reynolds' equation, eqn (5.7), is convenient for use here If the origin of coordinates is taken at any position

0 on the surface of the bearing, the X axis is a tangent, and the Z axis is

parallel to the axis of rotation Sometimes the bearing rotates, and then its

surface velocity is U , along the X axis The surface velocities are shown in

Fig 5.13 The surface of the shaft has a velocity Q2 making with the X axis

an angle whose tangent is ahldx and whose cosine is approximately 1.0 Hence components U2 = Q and V2 = U2(dh/dx) With substitution of these terms, Reynolds' equation becomes

where U = U , + U2 The same result is obtained if the origin of coordinates

is taken on the journal surface with X tangent to it Reynolds assumed an infinite length for the bearing, making dp/dz=O and endwise flow w=O Together with p constant, this simplifies eqn (5.43) to

Reynolds obtained a solution in series, which was published in 1886 In

1904 Sommerfeld found a suitable substitution that enabled him to make

an integration to obtain a solution in a closed form The result was

This result has been widely used, together with experimentally determined end-leakage factors, to correct for finite bearing lengths It will be referred

to as the Sommerfeld solution or the long-bearing folution Modern bearings are generally shorter than those used many years ago The length-

parabolic, pressure-induced flow portion of the U velocity, obtained the

Reynolds equation in the same form as proposed by Michell and Cardullo, but with greater justification This form is

Unlike eqn (5.44), eqn (5.46) is easily integrated, and it leads to the load number, a non-dimensional group of parameters, including length, which is useful in design and in plotting experimental results It will be used here in the remaining derivations and discussion of the principles involved It is known as the Ocvirk solution or the short-bearing approximation Ifthere

is no misalignment of the shaft and bearing, then h and ahlax are independent of z and eqn (5.46) may be integrated twice t o give

From the boundary conditions aplaz =O at z =O and p =O at z = kt This is shown in Fig 5.14 Thus

The slope ahlax = ah/a(rO) = ( l / r ) a h / d O and from eqn (5.42),

ahlax = - (CE sin O ) / r

192 Tribology in machine design

Substitution into eqn (5.47) gives

3~ sin O P=$(:-Z2)(, +Ecos,,3-

This equation indicates that pressures will be distributed radially and axially somewhat as shown in Fig 5.14; the axial distribution being parabolic The peak pressure occurs in the central plane z = 0 at an angle

and the value of p,,, may be found by substituting Om into eqn (5.48)

5.5.2 Mechanism of load transmission

Figure 5.14 shows the forces resulting from the hydrodynamic pressures developed within a bearing and acting on the oil film treated as a free body These pressures are normal to the film surface along the bearing, and the elemental forces d F =pr d O dz can all be translated to the bearing centre Ob and combined into a resultant force Retranslated, the resultant P shown

acting on the film must be a radial force passing through Ob Similarly, the resultant force of the pressures exerted by the journal upon the film must pass through the journal centre Oj These two forces must be equal, and they must be in the opposite directions and parallel In the diverging half of the film, beginning at the 0 = I T position, a negative (below atmospheric) pressure tends to develop, adding to the supporting force This can never be very much, and it is usually neglected The journal exerts a shearing torque

T j upon the entire film in the direction ofjournal rotation, and a stationary bearing resists with an opposite torque Tb However, they are not equal A

summation of moments on the film, say about Oj, gives T j = Tb + Pe sin 4

where 4 , the attitude angle, is the smaller of the two angles between the line

of force and the line of centres If the bearing instead of the journal rotates, and the bearing rotates counterclockwise, the direction of Tb and T j reverses, and Tb = T j + Pe sin 4

Hence, the relationship between torques may be stated more generally as

where T, is the torque from the rotating member and T, is the torque from the stationary member

Load P and angle 4 may be expressed in terms of the eccentricity ratio E

by taking summations along and normal t o the line ObOj, substituting for p from eqn (5.48) and integrating with respect to O and z Thus

The fractioncontaining the many eccentricity terms ofeqn (5.51) is equal

to P c 2 / ( p ~ 1 3 ) , and although it is not obvious, the eccentricity, like the fraction, increases non-linearly with increases in P and c and with decreases

in p, I, U and the rotational speed n' It is important to know the direction of the eccentricity, so that parting lines and the holes or grooves that supply lubricant from external sources may be placed in the region of the diverging film, or where the entrance resistance is low The centre Ob is not always fixed, e.g at an idler pulley, the shaft may be clamped, fixing Oj, and the pulley with the bearing moves to an eccentric position A rule for determining the configuration is to draw the fixed circle, then t o sketch the movable member in the circle, such that the wedge or converging film lies between the two force vectors P acting upon it The wedge must point in the direction of the surface velocity of the rotating member This configuration should then bechecked by sketching in the vectors of force and torque in the directions in which they act on the film If the free body satisfies eqn (5.50) the configuration is correct

Oil holes or axial grooves should be placed so that they feed oil into the diverging film or into the region just beyond where the pressure is low This should occur whether the load is low or high, hence, the hole should be at least in the quadrant 90"-180°, and not infrequently, in the quadrant 135"-225" beyond where the load P i s applied to the film The 180" position

is usually used for the hole or groove since it is good for either direction of rotation, and it is often a top position and accessible The shearing force d F

on an element ofsurface (r d O ) dz is (r d O ) dzp(au/ay),= H, where either zero

or h must be substituted for H The torque is rdF If it is assumed that the entire space between the journal and bearing is filled with the lubricant,

194 Tribology in machine design

integration must be made from zero to 271, thus

The short bearing approximation assumes a linear velocity profile such that (duldy ), = = (duldy ), , ,,.Use of this approximation in eqn (5.53) will give but one torque, contrary to the equilibrium condition of eqn (5.50) How- ever, the result has been found to be not too different from the experi-

mentally determined values of the stationary member torque T, Hence

we use eqn (5.53), with h from eqn (5.42), integrating and substituting

c = cd/2, r =dl2 and U, - U2 = xd(n; - n;) where n2 and n1 are the rotation-

al velocities in r.p.s; the results are

Dimensionless torque ratios are obtained by dividing T , or T , by the no- load torque To given by the formula

and first setting n' = n; - n; Thus

The amount of oil flowing out at the end of a journal bearing, i.e the oil loss

at plane z =+ or z = - + may be determined by integration of eqn (5.3b) over the pressure region of the annular exit area, substituting r d O for dx Thus, since W1 = W2 = O

T o determine the flow QH out of the two ends of the converging area or the hydrodynamic film, dpldz is obtained from eqn (5.48), h from eqn (5.42), and QH=2Q from eqn (5.56) The limits of integration may be O 1 = 0 and

O 2 = 71, or the extent may be less in a partial bearing However, QH is more easily found from the fluid rejected in circumferential flow With the linear velocity profiles ofthe short bearing approximation, shown in Fig 5.16, and

fi-.::<\ , ., \ with eqn (5.42), the flow is seen to be

\ \ Q H =) Uhmaxl - 4 Uhmi,l = f UI[(c + e) - (c - e)]

~ 8 ' Q H = ule=-= Ulccd d ( n ; - n;)lecd

-4

Figure 5.16 where cd is the diametral clearance Although it is not directly evident from

this simple result, the flow is an increasing function of the load and a decreasing function of viscosity, indicated by the eccentricity term E At the ends of the diverging space in the bearing, negative pressure may draw in some of the oil previously forced out However, if a pump supplies oil and distribution grooves keep the space filled and under pressure, there is an outward flow This occurs through a cylindrical slot of varying thickness, which is a function of the eccentricity The flow is not caused by journal or bearing motion, and it is designated film flow Qf It is readily determined

whether a central source of uniform pressure po may be assumed, as from a

pump-fed partial annular groove Instead of starting with eqn (5.56), an elemental flow q, from one end may be obtained from the flat slot, eqn

(5.17), by writing r d 0 for h and (I - a)/2 for I, where the new I is the bearing

length and a is the width of the annular groove Then Qf = 2

0, and e2 define the appropriate angular positions, such as n and 2n

respectively Additional flow may occur through the short slots which close the ends of an axial groove or through a small triangular slot formed by chamfering the plane surfaces at the joint in a split bearing

Oil flow and torque are closely related to bearing and film temperature and, thereby, to oil viscosity, which in turn affects the torque Oil temperature may be predicted by establishing a heat balance between the

heat generated and the heat rejected Heat H , is generated by the shearing action on the oil, heat H , is carried away in oil flowing out of the ends of the

bearing, and by radiation and convection, heat H,, is dissipated from the

bearing housing and attached parts, and heat H , from the rotating shaft In

equation form

The heat generation rate H , is the work done by the rotating member per

unit time (power loss) Thus, if torque T , is in Nm and n' in r.p.s., the heat generation rate is

Now, T , and T o vary as d 3 and n' Therefore, H , varies as d3 and

approximately as (n')' Hence a large diameter and high speed bearings generally require a large amount of cooling, which may be obtained by a liberal flow of oil through the space between the bearing and the journal Flowing out of the ends of the bearing, the oil is caught and returned to a

196 Tribology in machine design

sump, where it is cooled and filtered before being returned The equation for the heat removed by the oil per unit of time is

where c, is the specific heat of the oil, and y is the specific weight of the oil The flow Q in eqn (5.60) may consist of the hydrodynamic flow Q,,, eqn (5.57), film flow Qf, and chamfer flow Q, as previously discussed, or any others which may exist The heat lost by radiation and convection may often be neglected in well-flushed bearings

The outlet temperature to represents an average film temperature that may be used to determine oil viscosity for bearing calculations, at least in large bearings with oil grooves that promote mixing The average film temperature is limited to 70 "C or 80 "C in most industrial applications, although it may be higher in internal combustion engines Higher temperatures occur beyond the place of minimum film thickness and maximum shear They may be estimated by an equation based on experimental results The maximum temperatures are usually limited by the softening temperature of the bearing material or permissible lubricant temperature

In self-contained bearings, those lubricated internally as by drip, waste packing, oil-ring feed or oil bath (immersion ofjournal), dissipation of heat occurs only by radiation and convection from the bearing housing, connected members and the shaft Experimental studies have been directed towards obtaining overall dissipation coefficients K for still air and for moving air These dissipation coefficients are used in an equation of the form

H, = KA(tb - t,),

where A is some housing or bearing surface area or projected area, tb is the temperature of its surface, and t, is the ambient temperature

5.5.4 Design for load bearing capacity

It is convenient to convert eqn (5.51) into a non-dimensional form One substitution is a commonly used measure of the intensity of bearing loading, the unit load or nominal contact pressure, p, which is the load divided by the projected bearing area (I x d), thus

where 1 is the bearing length, d is the nominal bearing diameter, and p has the same units as pressure

The surface velocity sum, U = U1 + UZ, is replaced by nd(n; + n; ) = n dn',

where n' = n; + n; is the sum of the rotational velocities Also, c may be expressed in terms of the more commonly reported diametral clearance, cd,

where c = cd/2 Let the non-dimensional fraction containing E in eqn (5.51)

be represented by E Transposition of eqn (5.51), followed by substitution gives

The load number determines the dimensions and parameters over which the designer has a choice Equation (5.62) indicates that all combinations of unit load p or Plld, viscosity p, speed sum n', length-diameter ratio, l/d and clearance ratio cd/d that give the same value of N, will give the same eccentricity ratio Eccentricity ratio is a measure of the proximity to failure

of the oil film since its minimum thickness is hmi, = c(1- E) Hence the load number is a valuable design parameter

5.5.5 Unconventional cases of loading

Reynolds equation, eqn (5.43), which describes the process of pressure generation, contains the sum of surface velocities of the oil film,

U = U, + U2 In the derivation of the load number, N,, the substitution

, ,

U = n dn' was made, where n' = n, + n,, the sum of the rotational velocities

of the two surfaces The load number was found to be inversely propor-

, *

tional to n' and hence to n, +n2 It is important to note that velocities U1 and U2 were taken relative to the line of action of load P, which was considered fixed If the load rotates, the same physical relationships occur between the film surfaces and the load, provided that the surface velocities are measured relative to the line of action of the load Thus all previous equations apply, if the rotational velocities n; and n; are measured relative

to the rotational velocity n' of the load Several examples are given in Fig 5.17 Member 1 is fixed and represents the support of the machine At least one other member is driven at a rotational speed cu relative to it The load source and the two fluid films are not necessarily at the same axial position along the central member Thus case (e) in Fig 5.17, for instance, may illustrate a type of jet engine which has a central shaft with bladed rotors turning at one speed in the bearings supporting it, and a hollow spool with additional rotors turning at a different speed and supported on the first shaft by another set of bearings The sum (n') a multiple o f u , is a measure of the load capacity, which equals u in a standard bearing arrangement, case (a) If n' = 20, as for the inner film of case (d), then for the same load number

198 Tribology in machine design

and eccentricity, the load capacities p and P a r e doubled The zero capacity

of the bearing in case (c) represents a typical situation for the crankpin bearings of four-stroke-cycle engines The same is true in the case of the bushing of an idler gear and the shaft that supports it, if they turn with opposite but equal magnitude velocities relative to a non-rotating load on the gear The analyses discussed give some ideas on relative capacities that can be attained and indicate the care that must be taken in determining n' for substitution in the load number equation However, it should be noted that the load numbers and actual film capacities are not a function of n'

alone

The diameters d and lengths I of the two films may be different, giving different values t o p = P/ld and to (d/l)' in the load number, but they may be adjusted to give the same load number Also, a load rotating with the shaft, case (b), appears to give the bearing the same capacity as the bearing illustrated by case (a) However, unless oil can be fed through the shaft to a hole opposite the load, it will probably be necessary to feed oil by a central annular groove in the bearing so that oil is always fed to a space at low pressure With pressure dropping to the oil-feed value at the groove in the converging half, the bearing is essentially divided into two bearings of approximately half the l/d ratio Since dl1 is squared in the load-number equation, each half of the bearing has one-fourth and the whole one-halfthe capacity of the bearing in case (a)

Another way to deal with the problem of the rotating load vector is shown in Fig 5.18 Let o1 and o, be the angular velocities ofthe shaft or the