上海交通大学：《Design and Manufacturing》课程教学资源（参考文献）Shigley's Mechanical Engineering Design ch11 rolling contact bearing.pdf_大学文库

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 11.Rolling-Contact T©The McGraw-Hill Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 550 Mechanical Engineering Design The terms rolling-contact bearing,antifriction bearing,and rolling bearing are all used to describe that class of bearing in which the main load is transferred through elements in rolling contact rather than in sliding contact.In a rolling bearing the starting friction is about twice the running friction,but still it is negligible in comparison with the starting friction of a sleeve bearing.Load,speed,and the operating viscosity of the lubricant do affect the frictional characteristics of a rolling bearing.It is probably a mistake to describe a rolling bearing as"antifriction,"but the term is used generally throughout the industry. From the mechanical designer's standpoint,the study of antifriction bearings differs in several respects when compared with the study of other topics because the bearings they specify have already been designed.The specialist in antifriction-bearing design is confronted with the problem of designing a group of elements that compose a rolling bearing:these elements must be designed to fit into a space whose dimen- sions are specified;they must be designed to receive a load having certain character- istics;and finally,these elements must be designed to have a satisfactory life when operated under the specified conditions.Bearing specialists must therefore consider such matters as fatigue loading,friction,heat,corrosion resistance,kinematic prob- lems,material properties,lubrication,machining tolerances,assembly,use,and cost. From a consideration of all these factors,bearing specialists arrive at a compromise that,in their judgment,is a good solution to the problem as stated. We begin with an overview of bearing types;then we note that bearing life cannot be described in deterministic form.We introduce the invariant,the statistical distribution of life,which is strongly Weibullian.There are some useful deterministic equations addressing load versus life at constant reliability,and we introduce the catalog rating at rating life. The reliability-life relationship involves Weibullian statistics.The load-life-reliability relationship,combines statistical and deterministic relationships giving the designer a way to move from the desired load and life to the catalog rating in one equation. Ball bearings also resist thrust,and a unit of thrust does different damage per rev- olution than a unit of radial load,so we must find the equivalent pure radial load that does the same damage as the existing radial and thrust loads.Next,variable loading, stepwise and continuous,is approached,and the equivalent pure radial load doing the same damage is quantified.Oscillatory loading is mentioned. With this preparation we have the tools to consider the selection of ball and cylin- drical roller bearings.The question of misalignment is quantitatively approached. Tapered roller bearings have some complications,and our experience so far con- tributes to understanding them. Having the tools to find the proper catalog ratings,we make decisions (selec- tions),we perform a design assessment,and the bearing reliability is quantified.Lubri- cation and mounting conclude our introduction.Vendors'manuals should be consulted for specific details relating to bearings of their manufacture. 11-1 Bearing Types Bearings are manufactured to take pure radial loads,pure thrust loads,or a combination of the two kinds of loads.The nomenclature of a ball bearing is illustrated in Fig.11-1, which also shows the four essential parts of a bearing.These are the outer ring,the inner ring,the balls or rolling elements,and the separator.In low-priced bearings,the To completely understand the statistical elements of this chapter,the reader is urged to review Chap.20. Secs.20-1 through 20-3

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 11. Rolling−Contact Bearings © The McGraw−Hill 551 Companies, 2008 550 Mechanical Engineering Design The terms rolling-contact bearing, antifriction bearing, and rolling bearing are all used to describe that class of bearing in which the main load is transferred through elements in rolling contact rather than in sliding contact. In a rolling bearing the starting friction is about twice the running friction, but still it is negligible in comparison with the starting friction of a sleeve bearing. Load, speed, and the operating viscosity of the lubricant do affect the frictional characteristics of a rolling bearing. It is probably a mistake to describe a rolling bearing as “antifriction,” but the term is used generally throughout the industry. From the mechanical designer’s standpoint, the study of antifriction bearings differs in several respects when compared with the study of other topics because the bearings they specify have already been designed. The specialist in antifriction-bearing design is confronted with the problem of designing a group of elements that compose a rolling bearing: these elements must be designed to fit into a space whose dimensions are specified; they must be designed to receive a load having certain characteristics; and finally, these elements must be designed to have a satisfactory life when operated under the specified conditions. Bearing specialists must therefore consider such matters as fatigue loading, friction, heat, corrosion resistance, kinematic problems, material properties, lubrication, machining tolerances, assembly, use, and cost. From a consideration of all these factors, bearing specialists arrive at a compromise that, in their judgment, is a good solution to the problem as stated. We begin with an overview of bearing types; then we note that bearing life cannot be described in deterministic form. We introduce the invariant, the statistical distribution of life, which is strongly Weibullian.1 There are some useful deterministic equations addressing load versus life at constant reliability, and we introduce the catalog rating at rating life. The reliability-life relationship involves Weibullian statistics. The load-life-reliability relationship, combines statistical and deterministic relationships giving the designer a way to move from the desired load and life to the catalog rating in one equation. Ball bearings also resist thrust, and a unit of thrust does different damage per revolution than a unit of radial load, so we must find the equivalent pure radial load that does the same damage as the existing radial and thrust loads. Next, variable loading, stepwise and continuous, is approached, and the equivalent pure radial load doing the same damage is quantified. Oscillatory loading is mentioned. With this preparation we have the tools to consider the selection of ball and cylindrical roller bearings. The question of misalignment is quantitatively approached. Tapered roller bearings have some complications, and our experience so far contributes to understanding them. Having the tools to find the proper catalog ratings, we make decisions (selections), we perform a design assessment, and the bearing reliability is quantified. Lubrication and mounting conclude our introduction. Vendors’ manuals should be consulted for specific details relating to bearings of their manufacture. 11–1 Bearing Types Bearings are manufactured to take pure radial loads, pure thrust loads, or a combination of the two kinds of loads. The nomenclature of a ball bearing is illustrated in Fig. 11–1, which also shows the four essential parts of a bearing. These are the outer ring, the inner ring, the balls or rolling elements, and the separator. In low-priced bearings, the 1 To completely understand the statistical elements of this chapter, the reader is urged to review Chap. 20, Secs. 20–1 through 20–3

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 11.Rolling-Contact T©The McGraw-Hill Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 552 Mechanical Engineering Design to an eccentric position.The balls are separated after loading,and the separator is then inserted.The use of a filling notch (Fig.11-2b)in the inner and outer rings enables a greater number of balls to be inserted,thus increasing the load capacity. The thrust capacity is decreased,however,because of the bumping of the balls against the edge of the notch when thrust loads are present.The angular-contact bearing(Fig. 11-2c)provides a greater thrust capacity. All these bearings may be obtained with shields on one or both sides.The shields are not a complete closure but do offer a measure of protection against dirt.A vari- ety of bearings are manufactured with seals on one or both sides.When the seals are on both sides,the bearings are lubricated at the factory.Although a sealed bearing is supposed to be lubricated for life,a method of relubrication is sometimes provided. Single-row bearings will withstand a small amount of shaft misalignment of deflec- tion,but where this is severe,self-aligning bearings may be used.Double-row bear- ings are made in a variety of types and sizes to carry heavier radial and thrust loads. Sometimes two single-row bearings are used together for the same reason,although a double-row bearing will generally require fewer parts and occupy less space.The one- way ball thrust bearings(Fig.11-2i)are made in many types and sizes. Some of the large variety of standard roller bearings available are illustrated in Fig.11-3.Straight roller bearings (Fig.11-3a)will carry a greater radial load than ball bearings of the same size because of the greater contact area.However,they have the disadvantage of requiring almost perfect geometry of the raceways and rollers.A slight misalignment will cause the rollers to skew and get out of line.For this reason, the retainer must be heavy.Straight roller bearings will not,of course,take thrust loads. Helical rollers are made by winding rectangular material into rollers,after which they are hardened and ground.Because of the inherent flexibility,they will take con- siderable misalignment.If necessary,the shaft and housing can be used for raceways instead of separate inner and outer races.This is especially important if radial space is limited. Figure 11-3 Types of roller bearings: (a)straight roller;(b)spherica roller,thrust;(c)tapered roller, thrust;(d)needle;le)tapered roller;(f)steepangle tapered roller.(Courtesy of The Timken Company.) (a) (b) (c) (d) (e) (f)

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 11. Rolling−Contact Bearings © The McGraw−Hill 553 Companies, 2008 552 Mechanical Engineering Design (a) (d) (e) ( f ) (b) (c) Figure 11–3 Types of roller bearings: (a) straight roller; (b) spherical roller, thrust; (c) tapered roller, thrust; (d) needle; (e) tapered roller; (f ) steep-angle tapered roller. (Courtesy of The Timken Company.) to an eccentric position. The balls are separated after loading, and the separator is then inserted. The use of a filling notch (Fig. 11–2b) in the inner and outer rings enables a greater number of balls to be inserted, thus increasing the load capacity. The thrust capacity is decreased, however, because of the bumping of the balls against the edge of the notch when thrust loads are present. The angular-contact bearing (Fig. 11–2c) provides a greater thrust capacity. All these bearings may be obtained with shields on one or both sides. The shields are not a complete closure but do offer a measure of protection against dirt. A variety of bearings are manufactured with seals on one or both sides. When the seals are on both sides, the bearings are lubricated at the factory. Although a sealed bearing is supposed to be lubricated for life, a method of relubrication is sometimes provided. Single-row bearings will withstand a small amount of shaft misalignment of deflection, but where this is severe, self-aligning bearings may be used. Double-row bearings are made in a variety of types and sizes to carry heavier radial and thrust loads. Sometimes two single-row bearings are used together for the same reason, although a double-row bearing will generally require fewer parts and occupy less space. The oneway ball thrust bearings (Fig. 11–2i) are made in many types and sizes. Some of the large variety of standard roller bearings available are illustrated in Fig. 11–3. Straight roller bearings (Fig. 11–3a) will carry a greater radial load than ball bearings of the same size because of the greater contact area. However, they have the disadvantage of requiring almost perfect geometry of the raceways and rollers. A slight misalignment will cause the rollers to skew and get out of line. For this reason, the retainer must be heavy. Straight roller bearings will not, of course, take thrust loads. Helical rollers are made by winding rectangular material into rollers, after which they are hardened and ground. Because of the inherent flexibility, they will take considerable misalignment. If necessary, the shaft and housing can be used for raceways instead of separate inner and outer races. This is especially important if radial space is limited

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 11.Rolling-Contact T©The McGraw-Hil Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition Rolling-Contact Bearings 553 The spherical-roller thrust bearing(Fig.11-3b)is useful where heavy loads and misalignment occur.The spherical elements have the advantage of increasing their contact area as the load is increased. Needle bearings(Fig.11-3d)are very useful where radial space is limited.They have a high load capacity when separators are used,but may be obtained without sep- arators.They are furnished both with and without races. Tapered roller bearings (Fig.11-3e,f)combine the advantages of ball and straight roller bearings,since they can take either radial or thrust loads or any com- bination of the two,and in addition,they have the high load-carrying capacity of straight roller bearings.The tapered roller bearing is designed so that all elements in the roller surface and the raceways intersect at a common point on the bearing axis. The bearings described here represent only a small portion of the many available for selection.Many special-purpose bearings are manufactured,and bearings are also made for particular classes of machinery.Typical of these are: .Instrument bearings,which are high-precision and are available in stainless steel and high-temperature materials Nonprecision bearings,usually made with no separator and sometimes having split or stamped sheet-metal races Ball bushings,which permit either rotation or sliding motion or both Bearings with flexible rollers 11-2 Bearing Life When the ball or roller of rolling-contact bearings rolls,contact stresses occur on the inner ring,the rolling element,and on the outer ring.Because the curvature of the contacting elements in the axial direction is different from that in the radial direction, the equations for these stresses are more involved than in the Hertz equations pre- sented in Chapter 3.If a bearing is clean and properly lubricated,is mounted and sealed against the entrance of dust and dirt,is maintained in this condition,and is operated at reasonable temperatures,then metal fatigue will be the only cause of fail- ure.Inasmuch as metal fatigue implies many millions of stress applications success- fully endured,we need a quantitative life measure.Common life measures are Number of revolutions of the inner ring (outer ring stationary)until the first tangible evidence of fatigue Number of hours of use at a standard angular speed until the first tangible evidence of fatigue The commonly used term is bearing life,which is applied to either of the measures just mentioned.It is important to realize,as in all fatigue,life as defined above is a sto- chastic variable and,as such,has both a distribution and associated statistical parame- ters.The life measure of an individual bearing is defined as the total number of revo- lutions (or hours at a constant speed)of bearing operation until the failure criterion is developed.Under ideal conditions,the fatigue failure consists of spalling of the load- carrying surfaces.The American Bearing Manufacturers Association(ABMA)standard states that the failure criterion is the first evidence of fatigue.The fatigue criterion used by the Timken Company laboratories is the spalling or pitting of an area of 0.01 in2 Timken also observes that the useful life of the bearing may extend considerably beyond this point.This is an operational definition of fatigue failure in rolling bearings

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 11. Rolling−Contact Bearings 554 © The McGraw−Hill Companies, 2008 Rolling-Contact Bearings 553 The spherical-roller thrust bearing (Fig. 11–3b) is useful where heavy loads and misalignment occur. The spherical elements have the advantage of increasing their contact area as the load is increased. Needle bearings (Fig. 11–3d) are very useful where radial space is limited. They have a high load capacity when separators are used, but may be obtained without separators. They are furnished both with and without races. Tapered roller bearings (Fig. 11–3e, f ) combine the advantages of ball and straight roller bearings, since they can take either radial or thrust loads or any combination of the two, and in addition, they have the high load-carrying capacity of straight roller bearings. The tapered roller bearing is designed so that all elements in the roller surface and the raceways intersect at a common point on the bearing axis. The bearings described here represent only a small portion of the many available for selection. Many special-purpose bearings are manufactured, and bearings are also made for particular classes of machinery. Typical of these are: • Instrument bearings, which are high-precision and are available in stainless steel and high-temperature materials • Nonprecision bearings, usually made with no separator and sometimes having split or stamped sheet-metal races • Ball bushings, which permit either rotation or sliding motion or both • Bearings with flexible rollers 11–2 Bearing Life When the ball or roller of rolling-contact bearings rolls, contact stresses occur on the inner ring, the rolling element, and on the outer ring. Because the curvature of the contacting elements in the axial direction is different from that in the radial direction, the equations for these stresses are more involved than in the Hertz equations presented in Chapter 3. If a bearing is clean and properly lubricated, is mounted and sealed against the entrance of dust and dirt, is maintained in this condition, and is operated at reasonable temperatures, then metal fatigue will be the only cause of failure. Inasmuch as metal fatigue implies many millions of stress applications successfully endured, we need a quantitative life measure. Common life measures are • Number of revolutions of the inner ring (outer ring stationary) until the first tangible evidence of fatigue • Number of hours of use at a standard angular speed until the first tangible evidence of fatigue The commonly used term is bearing life, which is applied to either of the measures just mentioned. It is important to realize, as in all fatigue, life as defined above is a stochastic variable and, as such, has both a distribution and associated statistical parameters. The life measure of an individual bearing is defined as the total number of revolutions (or hours at a constant speed) of bearing operation until the failure criterion is developed. Under ideal conditions, the fatigue failure consists of spalling of the loadcarrying surfaces. The American Bearing Manufacturers Association (ABMA) standard states that the failure criterion is the first evidence of fatigue. The fatigue criterion used by the Timken Company laboratories is the spalling or pitting of an area of 0.01 in2. Timken also observes that the useful life of the bearing may extend considerably beyond this point. This is an operational definition of fatigue failure in rolling bearings

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 11.Rolling-Contact T©The McGraw--Hill Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 554 Mechanical Engineering Design The rating life is a term sanctioned by the ABMA and used by most manufactur- ers.The rating life of a group of nominally identical ball or roller bearings is defined as the number of revolutions (or hours at a constant speed)that 90 percent of a group of bearings will achieve or exceed before the failure criterion develops.The terms minimum life,Lio life,and Bio life are also used as synonyms for rating life.The rating life is the 10th percentile location of the bearing group's revolutions-to-failure distribution. Median life is the 50th percentile life of a group of bearings.The term average life has been used as a synonym for median life,contributing to confusion.When many groups of bearings are tested,the median life is between 4 and 5 times the Lio life. 11-3 Bearing Load Life at Rated Reliability When nominally identical groups are tested to the life-failure criterion at different loads,the data are plotted on a graph as depicted in Fig.11-4 using a log-log trans- formation.To establish a single point,load Fi and the rating life of group one(Lio) are the coordinates that are logarithmically transformed.The reliability associated with this point,and all other points,is 0.90.Thus we gain a glimpse of the load-life func- tion at 0.90 reliability.Using a regression equation of the form FLI/a =constant (11-1) the result of many tests for various kinds of bearings result in ·a=3 for ball bearings .a=10/3 for roller bearings(cylindrical and tapered roller) A bearing manufacturer may choose a rated cycle value of 10 revolutions (or in the case of the Timken Company,90(10)revolutions)or otherwise,as declared in the manufacturer's catalog to correspond to a basic load rating in the catalog for each bearing manufactured,as their rating life.We shall call this the catalog load rating and display it algebraically as Cio.to denote it as the 10th percentile rating life for a particular bearing in the catalog.From Eq.(11-1)we can write F LVe FaLila (11-2) and associate load F with Cio,life measure LI with Lio,and write C1oL=FLe where the units of L are revolutions. Figure 11-4 log F Typical bearing load-ife log-log curve. 0 logL

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 11. Rolling−Contact Bearings © The McGraw−Hill 555 Companies, 2008 554 Mechanical Engineering Design log L log F 0 Figure 11–4 Typical bearing load-life log-log curve. The rating life is a term sanctioned by the ABMA and used by most manufacturers. The rating life of a group of nominally identical ball or roller bearings is defined as the number of revolutions (or hours at a constant speed) that 90 percent of a group of bearings will achieve or exceed before the failure criterion develops. The terms minimum life, L10 life, and B10 life are also used as synonyms for rating life. The rating life is the 10th percentile location of the bearing group’s revolutions-to-failure distribution. Median life is the 50th percentile life of a group of bearings. The term average life has been used as a synonym for median life, contributing to confusion. When many groups of bearings are tested, the median life is between 4 and 5 times the L10 life. 11–3 Bearing Load Life at Rated Reliability When nominally identical groups are tested to the life-failure criterion at different loads, the data are plotted on a graph as depicted in Fig. 11–4 using a log-log transformation. To establish a single point, load F1 and the rating life of group one (L10)1 are the coordinates that are logarithmically transformed. The reliability associated with this point, and all other points, is 0.90. Thus we gain a glimpse of the load-life function at 0.90 reliability. Using a regression equation of the form F L1/a = constant (11–1) the result of many tests for various kinds of bearings result in • a = 3 for ball bearings • a = 10/3 for roller bearings (cylindrical and tapered roller) A bearing manufacturer may choose a rated cycle value of 106 revolutions (or in the case of the Timken Company, 90(106) revolutions) or otherwise, as declared in the manufacturer’s catalog to correspond to a basic load rating in the catalog for each bearing manufactured, as their rating life. We shall call this the catalog load rating and display it algebraically as C10, to denote it as the 10th percentile rating life for a particular bearing in the catalog. From Eq. (11–1) we can write F1L1/a 1 = F2L1/a 2 (11–2) and associate load F1 with C10, life measure L1 with L10, and write C10L1/a 10 = F L1/a where the units of L are revolutions

56 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 11.Rolling-Contact T©The McGraw-Hill Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition Rolling-Contact Bearings 555 Further,we can write C10(LRnR60)1/a Fp(LDnp60)1/a catalog rating,Ibf or kN- desired speed,rev/min rating life in hours desired life,hours rating speed,rev/min desired radial load,Ibf or kN Solving for Cio gives LDnD60 l/a C10=FD LRnR60 (11-3) EXAMPLE 11-1 Consider SKF,which rates its bearings for 1 million revolutions,so that Lio life is 60LgnR=106 revolutions.The LRnR60 product produces a familiar number.Timken, for example,uses 90(10)revolutions.If you desire a life of 5000 h at 1725 rev/min with a load of 400 Ibf with a reliability of 90 percent,for which catalog rating would you search in an SKF catalog? Solution From Eq.(11-3), C10=FD LDnD60 I/a =400 5000(1725)607/3 106 =32111bf=14.3kN LRIR60 If a bearing manufacturer rates bearings at 500 h at 33 rev/min with a reliability of 0.90,then LRng60=500(33)60=106 revolutions.The tendency is to substitute 10 for LgnR60 in Eq.(11-3).Although it is true that the 60 terms in Eq.(11-3)as displayed cancel algebraically,they are worth keeping,because at some point in your keystroke sequence on your hand-held calculator the manufacturer's magic number (10 or some other number)will appear to remind you of what the rating basis is and those manufacturers'catalogs to which you are limited.Of course,if you evaluate the bracketed quantity in Eq.(11-3)by alternating between numerator and denominator entries,the magic number will not appear and you will have lost an opportunity to check. 11-4 Bearing Survival:Reliability versus Life At constant load,the life measure distribution is right skewed as depicted in Fig.11-5. Candidates for a distributional curve fit include lognormal and Weibull.The Weibull is by far the most popular,largely because of its ability to adjust to varying amounts of skewness.If the life measure is expressed in dimensionless form asx=L/L1o,then the reliability can be expressed as [see Eq.(20-24),p.970] b R=exp (= (11-4) where R=reliability x life measure dimensionless variate,L/L1o xo=guaranteed,or"minimum,"value of the variate

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 11. Rolling−Contact Bearings 556 © The McGraw−Hill Companies, 2008 Rolling-Contact Bearings 555 Further, we can write C10(L RnR60) 1/a = FD(L DnD60) 1/a catalog rating, lbf or kN desired speed, rev/min rating life in hours desired life, hours rating speed, rev/min desired radial load, lbf or kN Solving for C10 gives C10 = FD L DnD60 L RnR60 1/a (11–3) EXAMPLE 11–1 Consider SKF, which rates its bearings for 1 million revolutions, so that L10 life is 60L RnR = 106 revolutions. The L RnR60 product produces a familiar number. Timken, for example, uses 90(106) revolutions. If you desire a life of 5000 h at 1725 rev/min with a load of 400 lbf with a reliability of 90 percent, for which catalog rating would you search in an SKF catalog? Solution From Eq. (11–3), C10 = FD L DnD60 L RnR60 1/a = 400 5000(1725)60 106 1/3 = 3211 lbf = 14.3 kN If a bearing manufacturer rates bearings at 500 h at 331 3 rev/min with a reliability of 0.90, then L RnR60 = 500(331 3 )60 = 106 revolutions. The tendency is to substitute 106 for L RnR60 in Eq. (11–3). Although it is true that the 60 terms in Eq. (11–3) as displayed cancel algebraically, they are worth keeping, because at some point in your keystroke sequence on your hand-held calculator the manufacturer’s magic number (106 or some other number) will appear to remind you of what the rating basis is and those manufacturers’ catalogs to which you are limited. Of course, if you evaluate the bracketed quantity in Eq. (11–3) by alternating between numerator and denominator entries, the magic number will not appear and you will have lost an opportunity to check. 11–4 Bearing Survival: Reliability versus Life At constant load, the life measure distribution is right skewed as depicted in Fig. 11–5. Candidates for a distributional curve fit include lognormal and Weibull. The Weibull is by far the most popular, largely because of its ability to adjust to varying amounts of skewness. If the life measure is expressed in dimensionless form as x = L/L10, then the reliability can be expressed as [see Eq. (20–24), p. 970] R = exp − x − x0 θ − x0 b (11–4) where R = reliability x = life measure dimensionless variate, L/L10 x0 = guaranteed, or “minimum,’’ value of the variate

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 11.Rolling-Contact T©The McGraw-Hill 559 Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 558 Mechanical Engineering Design As useful as Eq.(11-6)is,one's attention to keystrokes and their sequence on a hand- held calculator strays,and,as a result,the most common error is keying in the inappropriate logarithm.We have the opportunity here to make Eq.(11-6)more error- proof.Note that 1 In =In- RD 1=n(l+py+=p时=1-Rn 1-Pf where pf is the probability for failure.Equation(11-6)can be written as 71/a C10÷fFD XD Lx0+(0-x0)1-RD)b R≥0.90 (11-7刀 Loads are often nonsteady.so that the desired load is multiplied by an applica- tion factor af.The steady load af Fp does the same damage as the variable load Fp does to the rolling surfaces.This point will be elaborated later. EXAMPLE 11-3 The design load on a ball bearing is 413 Ibf and an application factor of 1.2 is appro- priate.The speed of the shaft is to be 300 rev/min,the life to be 30 kh with a reliability of 0.99.What is the Cio catalog entry to be sought (or exceeded)when searching for a deep-groove bearing in a manufacturer's catalog on the basis of 106 revolutions for rat- ing life?The Weibull parameters are xo =0.02.(-xo)=4.439,and b=1.483. Solution XD= L=60Lnnp=60(30000)300 L10 60LRBR =540 106 Thus,the design life is 540 times the Lio life.For a ball bearing,a=3.Then,from Eq.(11-7). 540 1/3 Answer C10=(1.2)(413) =66961bf 0.02+4.439(1-0.99)1/1.483 We have learned to identify the catalog basic load rating corresponding to a steady radial load Fp.a desired life Lp,and a speed np. Shafts generally have two bearings.Often these bearings are different.If the bear- ing reliability of the shaft with its pair of bearings is to be R,then R is related to the individual bearing reliabilities RA and Rg by R=RARB First,we observe that if the product RARg equals R,then,in general,RA and Rg are both greater than R.Since the failure of either or both of the bearings results in the shutdown of the shaft,then A or B or both can create a failure.Second,in sizing bear- ings one can begin by making RA and Rg equal to the square root of the reliability goal,R.In Ex.11-3,if the bearing was one of a pair,the reliability goal would be 0.99,or 0.995.The bearings selected are discrete in their reliability property in your problem,so the selection procedure "rounds up,"and the overall reliability exceeds the goal R.Third,it may be possible,if RA>R,to round down on B yet have the product RARg still exceed the goal R

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 11. Rolling−Contact Bearings © The McGraw−Hill 559 Companies, 2008 558 Mechanical Engineering Design As useful as Eq. (11–6) is, one’s attention to keystrokes and their sequence on a handheld calculator strays, and, as a result, the most common error is keying in the inappropriate logarithm. We have the opportunity here to make Eq. (11–6) more errorproof. Note that ln 1 RD = ln 1 1 − pf = ln(1 + pf +···) . = pf = 1 − RD where pf is the probability for failure. Equation (11–6) can be written as C10 . = FD xD x0 + (θ − x0)(1 − RD)1/b 1/a R ≥ 0.90 (11–7) Loads are often nonsteady, so that the desired load is multiplied by an application factor af . The steady load af FD does the same damage as the variable load FD does to the rolling surfaces. This point will be elaborated later. EXAMPLE 11–3 The design load on a ball bearing is 413 lbf and an application factor of 1.2 is appropriate. The speed of the shaft is to be 300 rev/min, the life to be 30 kh with a reliability of 0.99. What is the C10 catalog entry to be sought (or exceeded) when searching for a deep-groove bearing in a manufacturer’s catalog on the basis of 106 revolutions for rating life? The Weibull parameters are x0 = 0.02, (θ − x0) = 4.439, and b = 1.483. Solution xD = L L10 = 60L DnD 60L RnR = 60(30 000)300 106 = 540 Thus, the design life is 540 times the L10 life. For a ball bearing, a = 3. Then, from Eq. (11–7), Answer C10 = (1.2)(413) 540 0.02 + 4.439(1 − 0.99)1/1.483 1/3 = 6696 lbf We have learned to identify the catalog basic load rating corresponding to a steady radial load FD, a desired life L D, and a speed nD. Shafts generally have two bearings. Often these bearings are different. If the bearing reliability of the shaft with its pair of bearings is to be R, then R is related to the individual bearing reliabilities RA and RB by R = RA RB First, we observe that if the product RA RB equals R, then, in general, RA and RB are both greater than R. Since the failure of either or both of the bearings results in the shutdown of the shaft, then A or B or both can create a failure. Second, in sizing bearings one can begin by making RA and RB equal to the square root of the reliability goal, √ √ R. In Ex. 11–3, if the bearing was one of a pair, the reliability goal would be 0.99, or 0.995. The bearings selected are discrete in their reliability property in your problem, so the selection procedure “rounds up,” and the overall reliability exceeds the goal R. Third, it may be possible, if RA > √R, to round down on B yet have the product RA RB still exceed the goal R