mechanical design an integrated approach Part 8 pot

As noted earlier, the rolling-element bearings are of two types: ball bearings and roller bearings.. CHAPTER 10 e BEARINGS AND LUBRICATION Figure 10.21 Some types of roller bearings:

Average air circulation 8.5

Average air circulation 11.3

projected area (i.e., 12.5.DL) It is to be emphasized that Eq (10.23) should be used only when “balipark” results are sufficient

Heat DEVELOPED

Under equilibrium conditions, the rate at which heat develops within a bearing is equal to

the rate at which heat dissipates:

{Wnri=H (10.24)

In this expression, f = coefficient of friction, W = load, r = journal radius, n = journal speed (as defined in Section 10.5), and H is given by Eq (10.22) A heat-balance computa- tion, involving finding average film temperature at the equilibrium, is a trial and error pro- cedure {14, 15]

10.10 MATERIALS FOR JOURNAL BEARINGS

The operating conditions for journal bearing materials are such that rather strict require- ments must be placed on the material to be used For instance, in thick-film lubrication, any material with sufficient compressive strength and a smooth surface is an adequate bearing material Small bushings and thrust bearings are often expected to run with thin-film lubri- cation Any foreign particles larger than the minimum film thickness present in the oil damage the shaft surface unless they can become imbedded in a relatively soft bearing

‘material In this section, we discuss some of the types of bearing materials in widespread usage Special uses are for many other materials, such as hard wood, glass, silver, ceram- ics, and sapphires [16-19]

ALLOYS

Babbitt alloys are the most commonly used materials, usually having a tin or lead base They posses low melting points, moduli of elasticity, yield strength, and good plastic flow In a bear- ing, the foregoing give good conformability and embeddability characteristics Comformabil- ity measures the capability of the bearing to adapt to shaft misalignment and deflection

Embeddability is the bearing’s capability to ingest harder, foreign particles Shafts for babbitt bearings should have a minimum hardness of 150-200 Bhn and a ground surface finish

Compressive and fatigue strengths of babbitts are low, particularly above about 77°C

Babbitts can rarely be used above about 121°C However, these shortcomings are improved

by using a thin internal babbitt surface on a steel (or aluminum) backing For small and

CHAPTER 10 © BEARINGS AND LUBRICATION ' Section A~B

Copper alleys are principally bronze and aluminum alloys They are generally stronger and harder, have greater load capacity and fatigue strength, but less compatible (.e., antiweld and andscoring) than babbitt bearings Owing to their thermal conductivity, corrosion resistance, and low cost, aluminum alloys are in widespread usage for bearings

in internal combustion engines A thin layer of babbitt is placed inside an aluminum bear- ing to improve its comformability and embeddability

SINTERED MATERIALS

Sintered materials, porous metal bearings or insertable powder-metallurgy bushings, have found wide acceptance These self-lubricated bearings have interconnected pores in which oil is stored in the factory The pores act as a reservoir for oil, expelling it when heated by shaft rubbing, reabsorbing it when inactive The low cost and lifetime use in a machine, without further lubrication, are their prime advantages

NONMETALLIC MATERIALS

A variety of plastics are used as bearing materials No corrosion, quiet operation, moldabil- ity, and excellent compatibility are their advantages The last characteristic often implies that no lubrication is required Carbon-graphite bearings can be used at high temperatures

They are chemically inert These bearings are useful in ovens and in pumps for acids and fuel oils Rubber and other elastomers are excellent bearing material for water pumps and propellers They are generally placed inside a noncorrodible metal shell and can provide vibration isolation, compensate for misalignment, and have good conformability

Recall from Section 10.1 that rolling-element bearings are also known as rolling bearings

or antifriction bearings The Anti-friction Bearing Manufacturing Association (AFBMA) and the ISO standardized bearing dimensions and the basis for their selection The load, speed, and operating viscosity of the lubricant affect the friction characteristics of a rolling bearing These bearings provide coefficients of friction between 0.001 and 0.002 The

designer must deal with such matters as fatigue, friction, heat, lubrication, kinematic prob-

lems, material properties, machining tolerances, assembly, use, and cost A complete his- tory of the rolling element bearings is given in [20]

COMPARISON OF ROLLING AND SLIDING BEARINGS

Some advantages of rolling-element bearings over the sliding or journal bearings are as follows:

Low starting and good operating friction torque

Ease of lubrication

Requiring less axial space

Generally, taking both radial and axial loads

Good low-temperature starting

The disadvantages of rolling-element bearings compared to sliding bearings include

Greater diametral space

More severe alignment requirements

Higher initial cost

Noisier normal operation

Finite life due to eventual failure by fatigùe

Ease of damage by foreign matter

Rolling bearings can carry radial, thrust or combinations of the two loads, depending on

‘their design Accordingly, most rolling bearings are categorized in one of the three groups:

radial for carrying loads that are primarily radial, thrust or axial contact for supporting loads that are primarily axial, and angular contact for carrying combined axial and radial loads As noted earlier, the rolling-element bearings are of two types: ball bearings and roller bearings The latter are capable of higher speeds, and the former can take greater loads The rolling bearings are precise, yet simple machine elements They are made in a wide variety of types and sizes (Figure 10.18) Most bearing manufacturers provide engi- neering manuals and brochures containing descriptions of the various kinds available Only some common types are considered here

Figure 10.19 Ball bearing geametry and nomenclature (Courtesy of New Departure-Hyatt Division, General Motors Corporation)

403

(d) external self aligning; (e) thrust; (f) self-aligning

thrust (Courtesy of the Timken Company.)

Observe that the basic bearing consists of an inner ring, an outer ring, the balls, and the separator (also known as cage or retainer) To increase the contact area and hence permit larger loads to be carried, the balls run in curvilinear grooves in the rings called raceways

The radius of the raceway is very little larger than the radius of the ball

The deep-groove (Conrad-type) bearing (Figure 10.20a) can stand a radial load as well

as some thrust load The balls are inserted into grooves by moving the inner ring to an eccentric position: They are separated after loading, and then the retainers are inserted

Obviously, an increase in radial load capacity may be obtained by using rings with deep

ˆ grooves or by employing a double-row radial bearing (Figure 10.20b)

The angular-contact bearing (Figure 10.20c) has a two-shouldered ball groove in one ring and a single-shouldered ball groove in the other ring It can support greater thrust capacity in one direction as well as radial loads The cutaway shoulder allows bearing assembly and use of a one-piece machined cage The contact angle is defined in the fig- are Typical values of œ for angular ball bearings vary from 15° to 40°

The self-aligning bearing has an outer raceway ball path ground in a spherical shape

so it can accommodate large amounts of angular misalignments or shaft deflections

These bearings can support both radial afd axial loads and are available in two types:

self-aligning external (Figure 10.20d) and self-aligning internal Thrust bearings are designed to carry a pure axial load only, as shown in Figures 10.20e and 10.20f They are made exclusively for machinery with vertically oriented shafts and have modest speed capacity

CHAPTER 10 e BEARINGS AND LUBRICATION

Figure 10.21 Some types of roller bearings:

{a) straight cylindrical; (b) spherical; (c) tapered thrust; (d) needle; (e) tapered (Courtesy of the

Timken Company.)

ROLLER BEARINGS

A roller bearing uses straight, tapered, or contoured cylindrical rollers When shock and impact loads are present or when a large bearing is needed, these bearings usually are employed Roller bearings can support much higher static and dynamic (shock) loads than comparably sized ball bearings, since they have line contact instead of point contact A roller bearing generally consists of the same elements as a ball bearing These bearings can

be grouped into five basic types: cylindrical roller bearings, spherical roller bearings, tapered thrust roller bearings, needle roller bearings, and tapered roller bearings (Fig-

ure 10.21) Straight roller bearings provide purely radial load support in most applications;

they cannot resist thrust loads The spherical roller bearings have the advantage of accom- modating, some shaft misalignments in heavy-duty rolling mill and industrial gear drives

Needle bearings are in widespread usage where radial space is limited

Tapered roller bearings combine the advantages of ball and straight roller bearings, as they can stand either radial or thrust loads or any combination of the two The centerlines

of the conical roller intersect at a common apex on the centerline of rotation Tapered roller

bearings have numerous features that make them complicated [12], and space does not

permit their discussion in this text Note that pairs of single-row roller bearings are usually employed for wheel bearings and some other applications Double-row and four-row roller

types are used to support heavier loads Selection and analysis of most bearing types are identical to that presented in the following sections

405

406 PART IF | ©: ~ APPLICATIONS CHAPTER 10 @ BEARINGS AND LUBRICATION 407

Figure 10.22 Special bearings: (a) pillow block; (b) flange (Courtesy of Emerson Power Transmission, Sealmaster Load ratings (KN)

Bearings, Aurora, IL.) Bore op Width — Filletradius Deep groove Angular contact

SPECIAL BEARINGS 15 35 H 06 7.80 3.55 8.06 3.65

Rolling element bearings are available in many other types and arrangements Detailed in- 7 40 12 06 9.56 4.50 9.95 4.75

dimension series code The first and second digits represent the width series and the diam- 80 140 26 2.0 70.2 45.0 80.6 35.0

it is required to resort to tabulations Tables 10.3 and 10.4 furnish dimensions of some 02 90 160 30 20 95.6 62.0 106 235 and 03 series of ball and cylindrical roller bearings The load ratings of these bearings, 95 170 32 20 108 695 l2 850

discussed in the next section, are also included in the table More detailed information is

10.12 ROLLING BEARING LIFE

When the ball or roller of an antifriction bearing rolls into a loading region, contact (i.e., Hertzian) stresses occur on the raceways and on the rolling element Owing to these stresses, which are higher than the endurance limit of the material, the bearing has a limited life Ifa bearing is well maintained and operating at moderate temperatures, metal fatigue is cause of failure alone Failure consists of pitting, spalling, or chipping load carrying surfaces, as dis- cussed in Section 8.15,

Practically, the life of an individual bearing or any one group of identical bearings cannot be accurately predicted Hence, the AFBMA established the following definitions associated with the life of a bearing We note that bearing life is defined as the

number of revolutions or hours at some uniform speed at which the bearing operates until

fatigue failure

Rating life Ly refers to the number of evolutions (or hours at a uniform speed) that 90% of a group of identical roller bearings will complete or exceed before the first evi- dence of fatigue develops The term minimum life is also used to denote the rating life

Median life refers to the life that 50% of the group of bearings would complete or exceed

Test results show that the median life is about five times the L 19 life

CHAPTER 10 ° BEARINGS AND LUBRICATION Basic dynamic load rating C is the constant radial load that a group of apparently identical bearings can take for a rating life of 1 million (i-e., 10°) revolutions of the inner ring in a stationary load (outer ring does not rotate)

Basic static load rating C, refers to the maximum allowable static load that does not impair the running characteristics of the bearing

The basic load ratings for different types of bearings are listed in Tables 10.3 and 10.4 The value of C, depends on the bearing material, number of rolling elements per row, the bearing contact angle, and the ball or roller diameter Except for an additional pa- rameter relating the load pattern, the value of € is based on the same factors that determine C,

409

10.13 EQUIVALENT RADIAL LOAD

Catalog ratings are based on only the radial load However, with the exception of thrust bearings, bearings are usually operated with some combined radial and axial loads It is then necessary to define an equivalent radial load that has the same effect on bearing life as

the applied loading The AFBMA recommends, for rolling bearings, the maximum of the

values of these two equations:

P=XVFh+Yr, (10.25)

BESVP (10.26)

where

P = equivalent radial load

F, = applied radial load

F, = applied axial load (thrust)

VY = a rotation factor

= {ts (for inner-ring rotation)

1.2 (for outer-ring rotation)

X = aradial factor

Y = a thrust factor The equivalent load factors X and Y depend on the geometry of the bearing, includ- ing the number of balls and the ball diameter The AFBMA recommendations are based

on the ratio of the axial load F, to the basic static load rating C, and a variable reference value e For deep-groove (single-row and double-row) and angular-contact ball bearings, the values of X and Y are given in Tables 10.5 and 10.6 Straight cylindrical roller bearings are very limited in their thrust capacity because axial loads produce sliding

friction at the roller ends So, the equivalent load for these bearings can also be estimated

using Eq (10.26)

Table 10.6 Factors for commonly used angular-contact ball bearings

angle («) T£ ga" x Y x Y x Y ñnish, in roundness of rolling elements, and so on Consequently, no two bearings within

0.40 0.029 140 137 2.28 rolling bearings is often made from tables of standard types and sizes containing data on

0.56 0.44 L00 112 1.63 The basic dynamic load rating C enters directly into the process of selecting a bear-

if

where

Lip = rating life, in 10° revolutions

C = basic load rating (from Tables 10.3 and 10.4)

P = equivalent radial load (from Section 10.13)

3 (for bail bearings) a= to (for roller bearings)

We note that the load C is simply a reference value (see Section 10.12) that permits bear- ing life to be predicted at any level of actual load applied Alternatively, the foregoing equation may be written in the form

{10.30}

in which Lio = rating life, in hours

n = rotational speed, in rpm When two groups of identical bearings are run with different loads P; and P», the ratio

of their rating lives Ly and L{, by Eq (10.29), is

Using the general Weibull equation [11, 24, 25] together with extensive experimental data, the AFBMA formulated recommended life adjustment factors, K,, plotted in Figure 10.24 This curve can be applied to both bail and roller bearings but is restricted to reliabilities no greater than 99% The expected bearing life is the product of the rating life

and the adjustment factor Combining this factor with Eq (10.29), we have

ee

The quantity Ls represents the rating life for any given reliability greater than 90%

CHAPTER 10 ° BEARINGS AND LUBRICATION

10 0.9 0.8 0.7 0.6 0.5

04 0.3

Table 10.8 Representative rolling bearing design lives

Instruments and apparatus for infrequent use Up to 0.5

Machines used intermittently Service interruption is minor importance 4-8 Reliability is of great importance 8-14 Machines used in an 8-hour service working day

413

Determination of the Median Life of a Deep-Groove Ball Bearing

A 50 am bore (02-series) deép-groove ball bearing carries’ a‘combined load of 9 KN radially and

GO KN axially at 1200 rpm Calcilate

_ @) the equivalent radial load

® ` the median life ip hours:

Assumptions: ‘The inner ring rotates and the’ load'is steady

EXAMPLE 10.4

(a) To obtain the values of the radial load factors X and Y, it is necessary to obtain š : J

oe =0.206, — = xay = 0.667 Eq (10.32) with a = 3.and Li, = 1.25145 gives 00.)

ộ Through the use of Eg (10.26); P = V7, = 1) = 9 kN Comment: A reduction of the load to about-93% of its initial value causes a 25% incréase in the

ml EXAMPLE 10.5 The Median Life of a Deep-Groove Ball Bearing under Moderate Shock Comment: To improve the reliability of the bearing in Example 10.4 from 90 to 94%, a reduction

Table-10:5 shows that still F,/V F, > e; therefore, X = 0.56 and Y = 1.13, as before

From Eq (10.28); P = KV Fy = 2(1.2 x 9) = 21.6 KN

(b)." Inasmuch as: 25.66 > 21.6 KN, we use the larger value for calculating the rating life 1725 rpm Rating life is 30 khr

Through the use of Eq (10.30),

Figure 10.25 Flanged ball

416 PARTI @ APPLICATIONS

deep groove and subjected to light-shock loading The

inner ring rotates

Solution: Sce Figures 9.16, £0.25, and 11.20; Tables 10.3, 10.5, and 10.7

Referring to Figure 9.16b, the forces acting on bear-

F, = 0, the equivalent radial load is

2.69 KN, and the bearing is quite satisfactory Following

Here this procedure, other bearings for the shafts in the gear

Kye LS (from Table 10.7)

Comment: The final selection of the bearings would

X= 10, yr=0 (by Table 10.5) be made on the basis of standard shaft and housing Vel (from Section 10.13) dimensions

Most balls and rings are made from high-carbon chromium steel (SAE 52100), heat treated

‘to high strength and hardness, and the surfaces smoothly ground and polished Separators are usually made of low-carbon steel and copper alloy, such as bronze Unlike ball bearings, roller bearings are often fabricated of case-hardened steel alloys, Modern steel manufactur- ing processes have resulted in bearing steels with reduced level of impurities

As pointed out in Section 10.3, elastohydrodynamic lubrication occurs in rolling

bearings in which deformation of the parts must be taken into account as well as increased

viscosity of the oil owing to the high pressure This small elastic flattening of parts, together with the increase in viscosity, provides a filma, although very thin, that is much thicker than would prevail with complete rigid parts [26-28] In addition to providing a film between the sliding and rolling parts, a lubricant may help distribute and dissipate heat, prevent corrosion

of the bearing surfaces, and protect the parts from the entrance of foreign particles

Depending on the load, speed, and temperature requirements, bearing lubricants are either greases or oils Where bearing speeds are higher or loading is severe, oil is preferred

CHAPTER 10 @ BEARINGS AND LUBRICATION

Synthetic and dry lubricants are also widely used for special applications Greases are suit- able for low-speed operation and permit bearings to be prepacked

417

10.16 MOUNTING AND CLOSURE

OF ROLLING BEARINGS

Rolling-element bearings are generally mounted with the rotating inner or outer ring with

a press fit Then the stationary ring is mounted with a push fit Bearing manufacturers’ lit- erature contain extensive information and illustrations on mountings Here, we discuss

only the basic principle of mounting ball bearings properly

Figure 10.26 shows a common method of mounting, where the inner rings are backed

up against the shaft shoulders and held in position by round nuts threaded into the shaft

As noted, the outer ring of the left-hand bearing is backed up against a housing shoulder

and retained in position, but the outer ring of the right-hand bearing floats in the housing

This allows the outer ring to slide both ways in its mounting to avoid thermal-expansion- induced axial forces on the bearings, which would seriously shorten their life An alterna- tive bearing mounting is illustrated in Figure 10.27 Here the inner ring is backed up against the shaft shoulder, as before, however, no retaining device is needed; threads are eliminated With this assembly, the outer rings of both bearings are completely retained

As a result, accurate dimensions in axial direction or the use of adjusting devices is required

Duplexing of angular contact ball bearings arises when maximum stiffness and resistance to shaft misalignment is required, such as in machine tools and instruments

Bearings for duplex mounting have their rings ground with an offset, so that, when a pair of bearings is rigidly assembled, a controlled axial preload is automatically achieved [14, 15] Figures 10.28a and 10.28b show a face-to-face (DF) and back-to- back (DB) mounting arrangements, respectively, that take heavy radial and thrust loads

mounting (Courtesy of the Timken Company.) Note: The outer rings of

both bearings are held in position by

devices (not shown)

Figure 10.26 = A common bearing mounting

(Courtesy of the Timken Company.) Note: The

outer ring of the left-hand bearing is held in position by a device (not shown)

Figure 10.28 Mounting arangements of angular ball bearings: (a) lace

to face; (b) back to back; (c) tandem (Courtesy of the Timken Company.)

from either direction The latter has greater mounting stiffness Clearly, a tandem (DT) mounting arrangement is employed when the thrust is in the same direction (Fig- ure 10.28c) Single-row ball bearings are often loaded by the axial load built in during assembly, as shown in Figure 10.26 Preloading helps to remove the internal clear-

ance often found in bearings to increase the fatigue life and decrease the shaft slope at

the bearings

Note that the majority of bearings may be supplied with side shields The shields are not complete closures, but they offer a measure of protection against dust or dirt A sealed bearing is generally to be lubricated for life The roller bearings are’ not often supplied in a sealed and self-lubricated form, as are most ball bearing types

REFERENCES

1 Booser, BE R., ed CRC Handbook of Lubrication, vols | and UW Boca Raton, FL: CRC Press,

1983 and 1984

Lansdown, A R Lubrication Oxford: Pergamon Press, 1982

Hamrock, B J Fundamentals of Fluid Film Lubrication New York: McGraw-Hill, 1993

Cameron,’A Basic Lubrication Theory New York: Wiley, 1976

Szeri, A Z Tribology New York: McGraw-Hill, 1980

Wills, J G Lubrication Fundamentals New York: Marcel Decker, 1980

Avallone, A B., and T Beaumeister III Mark's Standard Handbook for Mechanical Engineers, 10th ed., New York: McGraw-Hill, 1996

8 Reynolds, O “On the Theory of Lubrication and Its Application to Mr Beauchamp Tower's Experiments.” Philosophical Transactions of the Royal Society (London) 1774886),

pp 157-234

9 Oevirk, F W “Short Bearing Approximation for Full Journal Bearings.” TN 2208, NACA, 1952

Also see Dubois, G B., and F W Ocvirk “The Short Bearing Approximation for Plain Journal

Bearings.” Transactions of the ASME 77 (1955), pp 1173-178; Ocvirk, F W., and G B Dubois

“Surface Finish and Clearance Effects on Journal-Bearing Load Capacity and Friction.”

Transactions of the ASME 81 (1959), Journal of Basic Engineering, p 245

10 Raimondi, A A., and J Boyd “A Solution for Finite Journal Bearings and its Application to Analysis and Design.” Parts I, Il, and IIL Transactions of the ASLE 1, no 1, pp 159-209;

reprinted in Lubrication Science and Technology New York: Pergamon Press, 1958

li Juvinall, R E., and M Marshek Fundamentals of Machine Component Design, 3rd ed New York: Wiley, 2000

CHAPTER 10 e BEARINGS AND LuBRICATION

12 Shigley, J E., and C R Mischke Mechanical Engineering Design, 6th ed New York: McGraw- Hill, 2001

13 Rothbart, H A., ed Mechanical Design and Systems Handbook, 2nd ed, New York: McGraw- Hill, 1985

14 Hamrock, B J., B, Jacobson, and S R Schmid Fundamentals of Machine Elements New York:

McGraw-Hill, 1999

15 Deutschman, A D., and C E Wilson Machine Design, New York: Macmillan, 1975

16 Burr, A H., and J B Cheatham Mechanical Analysis and Design, 2nd ed Upper Saddle River,

NJ: Prentice Hail, 1995

17 Fuller, D D Theory and Practice of Lubrication for Engineers, 2nd ed New York: Wiley, 1984

18 O’Conner, J J., and J Boyd Standard Handbook of Lubrication Engineering New York:

McGraw-Hill, 1968

19 Norton, R E Machine Design—An Integrated Approach, 2nd ed Upper Saddle River, NIJ:

Prentice Hall, 2000

20, Hamrock, J., and D Dowson Ball Bearing Lubrication—The Elastohydrodynamics of Elliptical

Contacts New York: Wiley, 1981

21 Standards of the Anti-friction Bearing Manufacturing Association New York: 1990

22, Ball Bearing Genera] Catalog Sandusky, OH: New Departure-Hyatt Bearing Division, General

Motors Corporation

23 Timken Engineering Journal Canton, OH: The Timken Roller Bearing Company

24, Bamberger, E N., et al “Life Adjustment Factors for Ball and Roller Bearings.” ASME

Engineering Design Guide, 1971

25, Harris, T, A Rolling Bearing Analysis 3rd ed New York: Wiley, 1991

26 Cheng, H S “Elastohydradynamic Lubrication.” In E R Booser, ed., Handbook of Lubrication

Boca Raton, FL: CRC Press, 1983, pp 155-60

27 Dowson, D., and G Higginson “A Numerical Solution to the Elastohdrodynamic Problems.”

Journal of Mechanical Engineering Science 1, no | (1959), p 6

28 Hamrock, B., and D Dowson “Isothermal Elastohydrodynamic Lubrication of Point

Contacts Part HI, Fully Flooded Results.” ASME Journal of Lubrication Technology 99 (1977), pp 264-76

419

PROBLEMS

Sections 10.1 through 10.5 10.1 A lightly loaded journal bearing 220 mm in length and 160 mm in diameter consumes 2 hp in

friction when running at 1200 rpm Diametral clearance is 0.18 mm and SAE 30 oil is used

Find the temperature of the oil film

10.2 A journal bearing has a 4-in length, a 3-in diameter, a c/r ratio of 0.002, carries a 500-lb

radial load at 24,000 rpm, and is supplied with an oil having a viscosity of 0.6 jreyns Using

the Petroff approach, estimate (a) The frictional torque developed

(b) The frictional horsepower

(c) The coefficient of friction

A Petroff bearing has a 120-mm length, a 120-mm diameter, a 0,05-mm radial clearance, a

speed of 600 rpm, and a radial load of 8 KN Assume that the coefficient of friction is 0.01 and

the average oil-film temperature is 70°C Determine (a) The viscosity of the oil

(b) The approximate SAE grade of the oil

A journal bearing having a 125-mm diameter, a 125-mm length, and c/r ratio of 0.0004 carries a radial load of 12 KN A frictional force of 80 N is developed at a speed of 240 rpm

What is the viscosity of the oil according to the Petroff approach?

A journal bearing 6 in in diameter and 1.5 in long carries a radial load of 500 Ib at

1500 rpm; ¢/r = 0.001 It is lubricated by SAE 30 oil at 180°F Estimate, using the Petroff

approach, (a) The bearing coefficient of friction

(b) The friction power loss

Sections 10.6 through 10.10 10.6

10.7 10,8 10.9

10.10

10.11

A4-in, diameter x 2-in long bearing turns at 1800 rpm; ¢c/r = 0.001; hy = 0.001 in SAE

30 oil is used at 200°F Through the use of the design charts, find the load W

Redo Problem 10.4 employing the design charts

Resolve Problem 10.5 using the design charts

A 4-in diameter shaft is supported by a bearing 4 in long with a minimum oil-film thick-

ness of 0.001 in and radial clearance of 0.0025 in It is lubricated by SAE 20 oil The

bearing carries a load of 100 psi of projected area at 900 rpm Employing the design charts, determine

(a) The temperature of the oil film

(6) The coefficient of friction

(c) The friction power

A 25-mm diameter x 25-mm long bearing carries a radial load of 1.5 KN at 1000 rpm;

cfr == 0.0008; 7 = 50 mPa - s Using the design charts, determine (a) The minimum oil-film thickness

(6) The friction power loss

A 80-mm diameter x 40-mm long bearing supports a radial load of 4 KN at 600 rpm;

c/r == 0,002 SAE 40 oil is used at 65°C Employing the design charts, determine

(a) The minimum oil-film thickness

(b) The maximum oil pressure

CHAPTER10 © BEARINGS AND LUBRICATION

A 50-mm diameter x 50-mm long bearing having a ¢/r ratio of 0.001 consumes 0.16 hp in friction at an operating speed_of 1630 rpm It is lubricated by SAE 20 oil at 83°C (Hint: Try

S = 0.03.) Using the design charts, determine (a) The radial load for the bearing

(b) The minimum oil-film thickness

{c) The eccentricity ratio

A journal bearing having an L/D ratio of 1/2, a 100-mm diameter, a c/r ratio of 0.0015, and

an operating speed of 900 rpm carries a radial load of 8 KN The minimum oil-film thickness

is to be 0.025 mm Using the design charts, determine (a) The viscosity of the oil

(6) The friction force and power developed

A 100-mm diameter x 50-mm long ring-oiled bearing supports a radial load 6 KN at 300 rpm

in still air; c/r = 0.001; and » = 20 mPa -s If the temperature of the surrounding air of the

housing is 20°C, estimate the average film temperature

Redo Problem 10.14 for an ojl-bath lubrication system in an average air circulation condition when the temperature of the air surrounding air of the housing is t, = 30°C

Sections 10.11 through 10.16 10.W

10.16

10.17 10.18 10.19 10.20 10.21

10.22

Use the website at www.grainger.com to conduct a search for roller bearings Locate a thrust bail bearing ‡ in bore, 1$ in OD, and 5 in width List the manufacturer and description

A 25-mm (02-series) deep-groove ball bearing carries a combined load of 2 KN radially and

3 KN axially at 1500 rpm The outer ring rotates and the load is steady, Determine the rating life in hours

Resolve Problem 10.16, for a single-row, angular-contact ball bearing having 35° contact

angle

Redo Problem 10.16, if the inner ring rotates and the bearing is subjected to a light shock load

What percentage change in loading of a bail bearing causes the expected life be doubled?

Resolve Problem 10.19 for a roller bearing

A 60-mm bore (02-series) double-row, angular-contact ball bearing has a 15° contact angle

The outer ring rotates, and the bearing carries a combined steady load of 5 KN radially and

1.5 kN axially at 1000 rpm Calculate the median life in hours

Determine the expected rating lives in hours of a 35-mm bore (02- and 03-series) straight

cylindrical bearings operating at 2400 rpm Radial load is 5 kN, with heavy shock, and the

outer rings rotate

421

422 PARTHI.:- ®- - ÁPPLICATIONS

10.23: Calculate the median lives in hours of a 75-mm bore (02- and 03-series) straight cylindrical bearings operating at 2000 rpm Radial load is 25 KN, with light shock, and inner rings rotate, 10.24 ° Select two (02- and 03-series) straight cylindrical bearings for an industrial machine intended

for a rating life of 24-hour operation at 2400 rpm The radial load is 12.5 kN, with extreme shock, and the inner rings rotate

10.25 Select a (02-series) deep-groove ball bearing for a machine intended for a median life of

40 hours operation at 900 rpm The bearing is subjected to a radial load of 8 KN, with heavy

shock, and the outer ring rotates

10.26 Determine the expected rating life of the deep-groove bal! bearing in Problem 10.16, if only

a 5% probability of failure can be permitted at 1200 rpm

10.27 Calculate the expected median life of the straight cylindrical bearing in Problem 10.22, if only a 2% probability of failure can be permitted

Compared to various other means of power transmission (¢.g., belts and chains), gears are the most rugged and durable They have transmission efficiency as high as 98% However, gears are generally more costly than belts and chains As we shall see, two modes of failure affect gear teeth: fatigue fracture owing to fluctuating bending stress at the root of the tooth and fatigue (wear) of the tooth surface Both must be checked when designing the gears The shapes and sizes of the teeth are standardized by the American Gear Manufacturers Associa- tion (AGMA) The methods of AGMA are widely employed in design and analysis of gear- ing Selection of the proper materials to obtain satisfactory strength, fatigue, and wear prop- erties is important The AGMA approach requires extensive use of charts and graphs accompanied by equations that facilitate application of computer-aided design Gear design strength and life rating equations have been computer modeled and programmed by most gear suppliers It is not necessary for designers to create their own computer programs [1-4]

There are four principal types of gearing: spur, helical, bevel, and worm gears (Figure 11.1) Note that spur and helical gears have teeth parallel and inclined to the axis of

Figure 11.1 A variety of gears, including spur gears, rack

and pinion, helical gears, bevel gears, worm, and worm gear

(Courtesy of Quality Transmission Components.)

CHAPTER 11 @ Spur Gears

rotation, respectively Bevel gears have teeth on conical surfaces The geometry of a worm

is similar to that of a screw Of all types, the spur gear is the simplest Here, we introduce the general gearing terminology, develop fundamental geometric relationships of the tooth form, and deal mainly with spur gears A review of the nomenclature and kinematics is fol-

lowed by a detailed discussion of the stresses and a number of factors influencing gear de-

sign The basis of the AGMA method and its use are illustrated Other gear types are dealt with in the next chapter For general information on gear types, gear drives, and gearboxes,

see the website at www.machinedesign.com The site at www.powertransmission.com lists

websites for numerous manufacturers of gears and gear drives

425

41.2 GEOMETRY AND NOMENCLATURE

Consider two virtual friction cylinders (or disks) having no slip at the point of contact,

represented by the circles in Figure 11.2a A friction cylinder can be transformed into spur gear by placing teeth on it that run parallel to the axis of the cylinder The surfaces of the rolling cylinders, shown by the dashed lines in the figures then become the pitch circles

The diameters are the pitch diameters, and the cylinders represent the pitch cylinders The

teeth, which lie in axial paths on the cylinder, are arranged to extend both outside and in- side the pitch circles (Figure 11.2b) All calculations are based on the pitch circle Note that spur gears are used to transmit rotary motion between parallel shafts

A pinion is the smaller of the two mating gears, which is also referred to as a pair of gears or gearset The larger is often called the gear In most applications, the pinion is the driving element whereas the gear is the driven element This reduces speed, but increases torque, from the power source (engine, motor, turbine): Machinery being driven runs slower In some cases, gears with teeth cut on the inside of the rim are needed, Such a gear

is known as an internal gear or an annulus (Figure 11.3a) A rack (Figure 11.3b) can be thought of as a segment of an internal gear of infinite diameter

PROPERTIES OF GEAR TOOTH

The face and flank portion of the tooth surface are divided by the pitch cylinder The circular pitch p is the distance, on the pitch circle, from a point on one tooth to a

Figure 11.4 ‘Nomenclature of the spur gear teeth

corresponding point on the next This leads to the definition

where

p = circular pitch, in

d = pitch diameter, in

N = number of teeth The diametral pitch P is defined as the number of teeth in the gear per inch of pitch diam- eter Therefore,

(11.2)

This measure is used in the U.S specification of gears The units of P are teeth/in or in.7!

Both circular and diametral pitches prescribe the tooth size The latter is a more convenient definition Combining Eqs (11.1) and (11.2) yields the useful relationship

For two gears to mesh, they must have the same pitch,

In SI units, the size of teeth is specified by the module (denoted by m) measured ia millimeters We have

It is measured in teeth/mm, or mm™', Note that metric gears are not interchangeable with

U.S gears, as the standards for tooth size are different

The addendum a is radial distance between the top land and the pitch circle as shown

in Fig 11.4 The dedendum 6, represents the radial distance from the bottom land to the pitch circle The face width b of the tooth is measured along the axis of the gear The whole

depth h is the sum of the addendum and dedendum The clearance circle represents a circle

tangent to the addendum circle of the mating gear The clearance f represents the amount

by which the dedendum in a given gear exceed the addendum of the mating gear Clearance

is required to prevent the end of the tooth of one gear from riding on the bottom of the mat- ing gear The difference between the whole depth and clearance represents the working

depth h, The distance between the centers of the two gears in mesh is called the center dis-

tance c Using Eq (11.2) with d = 2r,

Ni Mã

Here, subscripts | and 2 refer to driver and driven gears, respectively

The width of space between teeth must be made slightly larger than the gear tooth thickness t, both measured on the pitch circle Otherwise, the gears cannot mesh without jamming The difference between the foregoing dimensions is known as backlash That is, the backlash is the gap between mating teeth measured along the circumference of the pitch circle Manufacturing tolerances preclude a 0 backlash, since all teeth cannot be exactly the same dimensions and all must mesh without jam- ming The amount of backlash must be limited to the minimum amount necessary to

427

Basic LAW OF GEARING

For quiet, vibrationless operation, the velocities of two mating gears must be the same at all'times This condition is satisfied when the pitch circle of the driver is moving with constant velocity, the velocity of the pitch circle of the driven gear neither increases nor decreases at any instant while the two teeth are touching The basic law of gearing states that as the gears rotate, the common normal at the point of contact between the teeth must always pass through a fixed point on the line of centers, The fixed point is called the pitch point P (Figure 11.2) If two gears in mesh satisfy the basic law, the gears are said to produce conjugate action

According to the fundamental law, when two gears are in mesh, their pitch circles roll

on one another without slipping Denoting the pitch radii by r, and ra and angular velocities

as w; and øœ¿, respectively, the pitch line velocity is then

rs = speed or velocity ratio

@ = angular velocity, rad/sec

n = speed, rpm

N = number of teeth

d = pitch circle diameter Subscripts 1 and 2 refer to the driver and driven gears, respectively

INVOLUTE TOOTH FORM

To obtain conjugate action, most gear profiles are cut to conform to an involute curve (5] Our discussions are limited to toothed wheel gearing of the involute form The

CHAPTER 11 @ SpuR GEARS

involute curve may be generated graphically by wrapping a string around a fixed cylinder, then tracing the path a point on the string (kept taut) makes as the string is unwrapped from the cylinder When the involute is applied to gearing, the cylinder around which the string is wrapped is defined as the base circle (Figure 11.5) Gear teeth are cut in the shape of an involute curve between the base and the addendum circles, while that part of the tooth between the base and dedendum circles is generally

a radial line Figure 11.6 shows two involutes, on separate cylinders in mesh, representing the gear teeth Note especially that conjugate involute action can occur

\

Pressure line

Figure 11.6 — Involute gear teeth contact form and pressure angle