14-12 TAPERED ROLLER BEARINGS

The taper on the rollers of tapered roller bearings, evident in Figure 14-7, results in a load path different from that for the bearings discussed thus far. Figure 14-15 shows two tapered roller bearings supporting a shaft with a combination of a radial load and a thrust load. The design of the shaft is such that the thrust load is resisted by the left bearing.

But a peculiar feature of this type of bearing is that a radial load on one of the bearings creates a thrust on the opposing bearing, also; this feature must be considered in analysis of the bearing.

The location of the radial reaction must also be determined with care. Part (b) of Figure 14-15 shows a dimension a that is found by the intersection of a line perpendicular to the axis of the roller and the centerline of the shaft.

The radial reaction at the bearing acts through this point. The distance a is reported in the tables of data for the bearings.

The Anti-Friction Bearings Manufacturers' Association (AFBMA) recommends the following approach in computing the equivalent loads on a tapered roller bearing:

Equivalent Load for Tapered Roller Bearing

PA =0.4FrA+0.5YAFrB/YB+YATA (14-7)

PB=FB (14-8)

Where PA= equivalent radial load on bearing A; PB= equivalent radial load on bearing B; FrA

= applied radial load on bearing A; FrB= applied radial load on bearing B; TA= thrust load on bearing A; YA= thrust factor for bearing A from tables; YB= thrust factor for bearing B from tables

Table 14-6 shows an abbreviated set of data from a catalog to illustrate the method of computing equivalent loads.For the several hundred designs of standard tapered roller bearings available commercially, the value of the thrust factor varies from as small as 1.07 to as high as 2.26. In design problems, a trial-and-error procedure is usually necessary. Example Problem 14-5 illustrates one approach.

Example Problem 14-5The shaft shown in Figure 14-15 carries a transverse load of 6 800 lb and a thrust load of 2500 lb. The thrust is resisted by bearing A. The shaft rotates at 350 rpm and is to be used in a piece of agricultural equipment. Specify suitable tapered roller bearings for the shaft.SolutionThe radial loads on the bearings are

FrA = 6800(4 in/ 10 in) = 2720 lbFrB = 6 800(6 in/ 10 in) = 4 080 lbTA =2500lb

To use Equation (14-6), we must assume values of YA and YB. Let's use

YA = YB = 1.75. ThenPA = 0.40(2720) + 0.5(1.75)(4080)/

(1.75) + 1.75(2500) = 7503 lbPB = FrB = 4080 lb

Using Table 14-4 as a guide, let's select 4 000 h as a design life. Then the number of revolutions would beLd = (4000 h)(350 rpm)(60 min/h) = 8.4107 revThe required basic dynamic load rating can now be calculated from Equation (14-3), using k=3.33:CA = PA(Ld/106) 1/k

CA = 7 503(8.4107/106)0.30 = 28400 lb

Similarly,CB = 4080(8.4 107/106)0.30 = 15400 lb

From Table 14-6, we can choose the following bearings.Bearing A

d = 2.500 0 in D = 5.000 0 inC = 29 300 lb YA = 1.65

Bearing Bd= 1.7500 in D=4.0000 inC = 21400 lb YB = 1.50

We can now recompute the equivalent loads:

PA=0.40(2720)+0.51.654080/1.50+1.65(2500)=7457 lb

PB=FrB=4080 lbFrom these, the new values of CA =

28200 lb and CB = 15400 lb are still satisfactory for the selected bearings.

One caution must be observed in using the equations for equivalent loads for tapered roller bearings. If, from Equation (14-7), the equivalent load on bearing A is less than the applied radial load, the following equation should be used.If PA<FrA, then let

PA = FA

and compute PB.

PB=0.4FrB+0.5YBFA/YA-YBTA (14-9)

A similar analysis is used for angular contact ball bearings in which the design of the races results in a load path similar to that for tapered roller bearings. Figure 14-3 shows an angular contact bearing and the angle through the pressure center. This is equivalent to the line perpendicular to the axis of the tapered roller bearing. The radial reaction on the bearing acts through the intersection of this line and the axis of the shaft. Also, a

radial load on one bearing induces a thrust load on the opposing bearing, requiring the application of the equivalent load formulas of the type used in Equations (14-7) and (14-9). The angle of the load line in commercially available angular contact bearings ranges from 15 to 40.

14-13 PRACTICAL CONSIDERATIONS IN THE APPLICATION OF BEARINGS

This section will discuss lubrication of bearings, installation, preloading, stiffness, operation under varying loads, sealing, limiting speeds, and standards and bearing tolerance classes related to the manufacture and application of bearings.

LubricationThe functions of lubrication in a bearing

unit are as follows:1. To provide a low-friction film between

the rolling elements and the races of the bearing and at points of contact with cages, guiding surfaces, retainers, and so on.

2. To protect the bearing components from corrosion.

3. To help dissipate heat from the bearing unit.

4. To carry heat away from the bearing unit.

5. To help dispel contaminants and moisture from the bearing.Rolling contact bearings are usually lubricated with either grease or oil. Under normal ambient temperatures (approximately 700F) and relatively slow speeds (under 500

rpm), grease is satisfactory. At higher speeds or higher ambient temperatures, oil lubrication applied in a continuous flow is required, possibly with external cooling of the oil. Oils used in bearing lubrication are usually clean, stable mineral oils. Under lighter loads and lower speeds, light oil is used. Heavier loads and/or higher speeds require heavier oils up to SAE 30. A recommended upper limit for lubricant

temperature is 1600F. The choice of the correct oil or grease depends on many factors, so each application should be discussed with the bearing manufacturer. In general, a viscosity of 70 to 100 Savbolt Universal Seconds (SUS) should be maintained at the operating temperature of the lubricant in the bearing.

In some critical applications such as bearings in jet engines and very-high-

speed devices, lubricating oil is pumped under pressure to an enclosed housing for the bearing where the oil is directed at the rolling elements themselves. A controlled return path is also provided. The temperature of the oil in the sumo is monitored and controlled with heat exchangers or refrigeration to maintain oil viscosity within acceptable limits. Such systems provide reliable lubrication and ensure the removal of heat from the bearing.

See Section 14-14 for additional discussion about the importance of oil film thickness in bearings. Reference 7 contains an extensive amount of information on this subject. Greases used in bearings are mixtures of lubricating oils and thickening agents, usually soaps such as lithium or barium. The soaps act as carriers for the oil which is drawn out at the point of need within the bearing. Additives to resist

corrosion or oxidation of the oil itself are sometimes added. Classifications of greases specify the operating temperatures to which the greases will be exposed, as defined by the Anti-Friction Bearing Manufacturers' Association (AFBMA), and outlined in Table 14-7.

InstallationIt has already been stated that most bearings should be installed with a light interference fit between the bore of the bearing and the shaft to preclude the possibility of rotation of the inner race of the bearing with respect to the shaft. Such a condition would result in uneven wear of the bearing elements and early failure. To install the bearing then requires rather heavy forces applied

axially. Care must be exercised so that the bearing is not damaged during installation. The installation force should be applied directly to the inner race of the bearing.

If the-force were a2plied through the outer race, the load would be transferred through the rolling elements to the inner race. Because of the small contact area, it is likely that such transfer of forces would overstress some element,

exceeding the static load capacity. Brinelling would result, along with the noise and rapid wear that accompany this condition. For large bearings it may be necessary to heat the bearing to expand its diameter in order to keep the installation forces within reason. Removal of bearings intended for reuse requires similar precautions. Bearing pullers are available to facilitate this task.

PreloadingSome bearings are made with internal clearances that must be taken up in a particular direction to ensure satisfactory operation. In such cases, preloading must be provided, usually in the axial direction. On horizontal shafts, springs are typically used, with axial adjustment of the spring deflection sometimes provided to adjust the amount of preload. When space is limited, the use of

Belleville washers is desirable because they provide high forces with small deflections. Shims can be used to adjust the actual deflection and preload obtained (see Chapter 19). On vertical shafts the weight of the shaft assembly itself may be sufficient to provide the required preload.

Bearing StiffnessStiffness is the deflection that a given bearing undergoes when carrying a given load. Usually the radial stiffness is most important because the dynamic behavior of the rotating shaft system is affected. The critical speed and the mode of vibration are both function of the bearing stiffness. Generally speaking, the softer the bearing (low stiffness), the lower the critical speed of the shaft assembly.

Stiffness is measured in the units used for springs, such as pounds per inch or newtons per millimeter. Of course, the stiffness values are quite high, with values of 500 000 to 1000 000 lb/in reasonable. The manufacturer should be consulted when such information is needed, because it is rarely included in standard catalogs.

Operation under Varying LoadsThe load/life relationships used thus far assume that the load is reasonably constant in magnitude and direction. If the load varies considerably, an effective mean load must be used for determining the expected life of the bearing. (See References 5 and 13.) Oscillating loads also require special analysis because only a few of the rolling elements share the load. See Section 14-15 for additional

information about life prediction under varying loads.SealingWhen the bearing is to operate in dirty or moist environments, special shields and seals a usually specified. They can be provided on either or both sides of the rolling elements Shields are typically metal and are fixed to the stationary race, but they remain clear of the rotating race. Seals are made of

elastomeric materials and do contact the rotating race. Bearings fitted with both seals and shields and precharged at the factory with grease are sometimes called permanently lubricated. Although such bearings are likely to give many years of satisfactory service, extreme conditions can produce a degradation of the lubricating properties of the grease. The presence of seals also increases the friction in a bearing.

Sealing can be provided outside the bearing in the housing or at the shaft/housing interface. On high-speed shafts, a labyrinth seal, consisting of a noncontacting ring around the shaft with a few thousandths of an inch radial clearance, is frequently used. Grooves, sometimes in the form of a thread, are machined in the ring; the relative motion of the shaft with respect to the ring creates the sealing action.

Limiting SpeedsMost catalogs list limiting speeds for each bearing. Exceeding these limits may result in excessively high operating temperatures due to friction between the cages supporting the rolling elements. Generally, the limiting speed is lower for larger bearings than for smaller bearings. Also, a given bearing will have a lower limiting speed as loads increase. With special care, either in the fabrication of

the bearing cage or in the lubrication of the bearing, bearings can be operated at higher speeds than those listed in the catalog. The manufacturer should be consulted in such applications. The use of ceramic rolling elements with their lower mass can result in higher limiting speeds.StandardsSeveral groups are involved in standard setting for the bearing industry. A partial list is given here:

Anti-Friction Bearing Manufacturers' Association (AFBMA)Annular Bearing Engineers Committee (ABEC)Roller Bearing Engineers Committee (RBEC)Ball Manufacturers Engineers Committee (BMEC)American National Standards Institute (ANSI)International Standards Organization (ISO)

Many standards are co-listed by the ANSI and AFBMA organizations. The catalog of ANSI lists 22 different standards related to bearings. Three of them are listed here:Terminology and Definitions for Ball and Roller Bearings and Parts, ANSI B3.7Load Ratings and Fatigue Life for Ball Bearings, ANSI/AFBMA 9Load Ratings and Fatigue Life for Roller Bearings, ANSI/AFBMA 11

TolerancesSeveral different tolerance classes are recognized in the bearing industry to accommodate the needs of the wide variety of equipment using rolling contact bearings. In general, of course, all bearings are precision machine elements and should be treated as such. As noted before, the general range of tolerances is of the order of a few ten-thousandths of an inch. The standard

tolerance classes are defined by ABEC, as identified below.ABEC 1: Standard radial ball and roller bearingsABEC 3: Semiprecision instrument ball bearingsABEC 5: Precision radial ball and roller bearingsABEC 5P: Precision instrument ball bearings

ABEC 7: High-precision radial ball bearingsABEC 7P: High-precision instrument ball bearings

Most machine applications would use ABEC 1 tolerances, the data for which are usually provided in the catalogs. Machine tool spindles requiring extra smooth and accurate running would use ABEL 5 or ABEC 7 classes.

14-14 IMPORTANCE OF OIL FILM I-HICKNESS IN BEARINGSIn bearings operated at heavy loads or high speeds, it is critical to maintain a film of lubricating oil at the surface of the rolling elements. A steady supply of clean lubricant with an adequate viscosity is required. With careful analysis of the geometry of the bearing, the speed of rotation, and the properties of the lubricant, it is possible to estimate

the thickness of the oil film between the rolling elements and the races. Although the film thickness may be only a few microinches, it has been shown that oil starvation of the contact area is a chief cause of re failure of rolling contact bearings. Conversely, if a film thickness significantly greater than the height of surface roughness features can be maintained, the expected life will be several times greater than published

data from catalogs. (See References 7 and 15.)

The nature of the lubrication at the interface between rolling elements and the races is called elastohydrodynamic lubrication because it depends on the specific elastic deformation of the mating surfaces under the influence of high contact stresses and the creation of a pressurized film of lubricant by the dynamic action of the rolling elements

The data needed to evaluate film thickness for ball bearings are given in the following lists:Bearing Geometry FactorsBall diameterNumber of ballsRadius of curvature of the inner race groove in both the circumferential and the axial directionsPitch diameter of the bearing; the average of the bore diameter and the outside diameter

Surface roughness of the balls and the racesContact angle for angular contact bearingsBearing Material FactorsModulus of elasticity of balls and racesPoisson's ratioLubricant FactorsDynamic viscosity at the operating temperature within the bearingPressure coefficient of viscosity; the change of viscosity with pressure

Operational FactorsRotational speed of both the inner and the outer racesRadial loadThrust loadThe details of the analysis are given in Reference 7. A simplified approach with helpful charts to ease the determination of critical parameters is given in Reference 15.

The results of the analysis include the following:The minimum film thickness of the lubricant, h0

The composite roughness of the balls and the races, SThe ratio A = h0lSThe service life of the bearing is dependent on the value of A:

If A < 0.90, a life less than the manufacturer's rated life is to be expected because of surface damage due to inadequate lubricant film.If A is in the range of 0.90 to 1.50, rated service life can be expected.If A is in the range of 1.50 to 3.0, an increase of life of up to three times the rated life is possible.If A is greater than 3.0, a life of up to six times the rated life is possible.

General Recommendations for Achieving Long Life for Bearings

1. Choose a bearing with an adequate rated life using the procedures outlined in this chapter.

2. Ensure that the bearing has a fine surface finish and is not damaged by rough handling, poor installation practices, corrosion, vibration, or exposure to electrical current flow.

3. Ensure that operating loads are within the design values.

4. Supply the bearing with a copious flow of clean lubricant at an adequate viscosity at the operating temperature within the bearing according to the bearing manufacturer's recommendations. Provide external cooling for the lubricant if necessary. For an existing system, this is the factor over which you have the most control without a significant redesign of the system itself.

5. If redesign is possible, design the system to operate at as low a speed as possible.

14-15 PFE PREDICTION UNDER VARYING LOADSThe design and analysis procedures used so far in this book assumed that the bearing would operate at a single design load throughout its life. It is possible to predict the life of the bearing under such conditions fairly accurately using manufacturers' data published in catalogs. If loads vary with time, a modified procedure is required.

One procedure that is recommended by bearing manufacturers is called the Palmgren-Miner rule, or sometimes simply Miner's rule. References 11 and 12 describe their work, and Reference 10 discusses a modified approach more tailored to bearings.

The basis of Miner's rule is that if a given bearing is subjected to a series of loads of different magnitudes for known lengths of time, each load contributes to the eventual failure of the bearing in proportion to the ratio of the load to the expected life that the bearing would have under that load. Then the cumulative effect of the series of loads must account for all such contributions to failure.

A similar approach, described in Reference 7, introduces the concept of mean effective load, Fm:

Mean Effective Load under Varying Loadsp

i ip

i

m N

NFF

/1)(

where Fi = individual load among a seri

es of i loads; Ni = number of revolutions at

which Fi operates; N =total number of revol

utions in a complete cycle

(14-10)

p = exponent on the load/life relationship; p = 3 for ball bearings, and p = 10/3 for rollersAlternatively, if the bearing is rotating at a constant speed, and because the number of revolutions is proportional to the time of operation, Ni can be the number of minutes of operation at Fi and N is the sum of the number of minutes in the total cycle. That is,

N=N1+N2+ ...+NiThen the total expected life, in millions of

revolutions of the bearing, would bep

mF

CL

Example Problem 14-8A single-row, deep-groove ball bearing number 6308 is subjected to the following set of loads for the given times:

(14-11)

Condition Fi Time1 650 lb 30 min2 750 lb 10 min3 250 lb 20 min

This cycle of 60 min is repeated continuously throughout the life of the bearing. The shaft carried by the bearing rotates at 600 rpm. Estimate the total life of the bearing.

SolutionUsing Equation (14-10), we have

p

i ip

i

m N

NFF

/1)(

(14-10)

lbFm 597201030

)250(20)750(10)650(303/1333

Now use Equation (14-11):

p

mF

CL

From Table 14-3, for the 6308 bearing, we find that C = 7 050 lb. Then

revmillion 1647597

70503

L

At a rotational speed of 600 rpm, the number of hours of life would be

(14-11)

Note that this is the same bearing used in Example Problem 14-3 which was selected to operate at least 30 000 h when carrying a steady load of 650 lb.

h 45745min 60

h

rev 600

min

1

rev101647 6

L