Rolling Bearing Analysis FIFTH EDITION Essential Concepts of Bearing Technology ß by Taylor & Francis Group, LLC. ß by Taylor & Francis Group. Файл формата pdf; размером 21,79 МБ The first of two books, Essential Concepts of Bearing Technology builds a basic understanding of. For the last four decades, Tedric Harris' Rolling Bearing Analysis has Essential Concepts of Bearing Technology DownloadPDF MB.
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Rolling Bearing Analysis. FIFTH EDITION. Essential Concepts of Bearing Technology. Tedric A. Harris. Michael N. Kotzalas. (• C\ Taylor & Francis. \0t*'.). We offer the downloading media like a pdf, word, ppt, txt, zip, rar, essential concepts of bearing technology pdf may not make exciting reading, but essential. Essential Concepts Of Bearing Technology Fifth Edition Harris Tedric A Kotzalas Mentoring It is presented with some downloading media such as a pdf.
Although liquid oxygen is a poor lubricant, it was adequate, since the service life of the pump was just a few hours. The operating environment and service needs are also important design considerations.
Some bearing assemblies require routine addition of lubricants, while others are factory sealed , requiring no further maintenance for the life of the mechanical assembly. Although seals are appealing, they increase friction, and in a permanently sealed bearing the lubricant may become contaminated by hard particles, such as steel chips from the race or bearing, sand, or grit that gets past the seal.
Contamination in the lubricant is abrasive and greatly reduces the operating life of the bearing assembly. Another major cause of bearing failure is the presence of water in the lubrication oil. Online water-in-oil monitors have been introduced in recent years to monitor the effects of both particles and the presence of water in oil and their combined effect. Designation[ edit ] Metric rolling-element bearings have alphanumerical designations, defined by ISO 15 , to define all of the physical parameters.
The main designation is a seven digit number with optional alphanumeric digits before or after to define additional parameters. Here the digits will be defined as: Any zeros to the left of the last defined digit are not printed; e. The third digit defines the "diameter series", which defines the outer diameter OD.
The diameter series, defined in ascending order, is: 0, 8, 9, 1, 7, 2, 3, 4, 5, 6. The fourth digit defines the type of bearing:  0. Ball radial single-row 1. Ball radial spherical double-row 2. Roller radial with short cylindrical rollers 3. Roller radial spherical double-row 4. Roller needle or with long cylindrical rollers 5.
Lin T, Lin W. Structure and motion analyses of the sails of Chinese great windmill. Mechanism and Machine Theory, , 48 1 : 29—40 Google Scholar In: Yan H, Ceccarelli M, eds.
Zhang B. Zhang J. Our rolling bearing industry. Machinery, , 1 8 : 5—6 in Chinese Google Scholar Li J, Huang K. History of Chinese Mechanical Industry. Beijing: Reform Press. Soviet Specialists at Luoyang Bearing Plant. LYC Bearing Company. History of Military Bearing Production — Lai Y. Ten years of designing practice in bearing factory. Jiang Q. Retrospect and rethinking of the development of bearing industry.
Bearing, , 36 6 : 39—41 in Chinese Google Scholar Lin Z. Xinhua, , 3 16 : 40—42 in Chinese Google Scholar Obviously, after a load is applied to the contacting bodies the point expands to an ellipse and the line to a rectangle in ideal line contact, that is, the bodies have equal length. When a roller of finite length contacts a raceway of greater length, the axial stress distribution along the roller is altered, as that in Figure 6.
Since the material in the raceway is in tension at the roller ends because of depression of the raceway outside of the roller ends, the roller end compressive stress tends to be higher than that in the center of contact.
To counteract this condition, cylindrical rollers or the raceways may be crowned as shown in Figure 1. The stress distribution is thereby made more uniform depending on the applied load. If the applied load is increased significantly, edge loading will occur once again.
Palmgren and Lundberg  have defined a condition of modified line contact for roller— raceway contact. If 2a 1. This condition may be ascertained approximately by the methods presented in Section 6. The analysis of the contact stress and deformation presented in this section is based on the existence of an elliptical area of contact, except for the ideal roller under load, which has a rectangular contact. As it is desirable to preclude edge loading and attendant high stress concentrations, roller bearing applications should be examined carefully according to the modified line contact criterion.
Where that criterion is exceeded, redesign of roller and raceway curvatures may be necessitated. Second Volume of this handbook, or see Refs. Additionally, finite element methods FEMs have been employed  to perform the same analysis. In all cases, digital computation is required to solve even a single contact situation.
In a given roller bearing application, many contacts must be calculated. Note also the slight pressure increase where the roller crown blends into the roller end geometry. See Examples 6. As illustrated in Figure 6. Under light loads, a circular crowned profile does not enjoy full use of the roller length, somewhat negating the use of rollers in lieu of balls to carry heavier loads with longer endurance see Chapter Under light loads, the partially crowned roller of Figure 1.
When the roller axis is tilted relative to the bearing axis, both the fully crowned and partially crowned profiles tend to generate less edge stress under a given load as compared with the straight profile.
The profile is so named because it can be expressed mathematically as a special logarithmic function. Under misalignment, edge loading tends to be avoided under all but exceptionally heavy loads.
The roller ends are usually flat with corner radii blending into the crowned portion of the roller profile. The flange may also be a portion of a flat surface. This is the usual design in cylindrical roller bearings. When it is required to have the rollers carry thrust loads between the roller ends and the flange, sometimes the flange surface is designed as a portion of a cone.
In this case, the roller corners contact the flange. The angle between the flange and a radial plane is called the layback angle. Alternatively, the roller end may be designed as a portion of a sphere that contacts the flange. The latter arrangement, that is a sphere-end roller contacting an angled flange, is conducive to improved lubrication while sacrificing some flange—roller guidance capability.
In this case, some skewing control may have to be provided by the cage. From Reusner, H. With permission. For the case of rollers having spherical shape ends and angled flange geometry, the individual contact may be modeled as a sphere contacting a cylinder. For the purpose of calculation, the sphere radius is set equal to the roller sphere end radius, and the cylinder radius can be approximated by the radius of curvature of the conical flange at the theoretical point of contact.
By knowing the elastic contact load, roller—flange material properties, and contact geometries, the contact stress and deflection can be calculated. This approach is only approximate, because the roller end and flange do not meet the Hertzian half-space assumption. This method applies only to contacts that are fully confined to the spherical roller end and the conical portion of the flange.
It is possible that improper geometry or excessive skewing could cause the elastic contact ellipse to be truncated by the flange edge, undercut, or roller corner radius. Such a situation is not properly modeled by Hertz stress theory and should be avoided in design because high edge stresses and poor lubrication can result.
The case of a flat-end roller and angled flange contact is less amenable to simple contact stress evaluation.
The nature of the contact surface on the roller, which is at or near the intersection of the corner radius and end flat, is difficult to model adequately. The notion of an effective roller radius based on an assumed blend radius between roller corner and end flat is suitable for approximate calculations. A more precise contact stress distribution can be obtained by using FEM stress analysis technique if necessary. The model of a statically loaded bearing is somewhat distorted by the surface tangential stresses induced by rolling and lubricant actions.
However, under the effects of moderate to heavy loading, the contact stresses calculated herein are sufficiently accurate for the rotating bearing as well as the bearing at rest. The same is true with regard to the effect of edge stresses on roller load distribution and hence deformation. These stresses subtend a rather small area and therefore do not influence the overall elastic load-deformation characteristic.
In any event, from the simplified analytical methods presented in this chapter, a level of loading can be calculated against which to check other bearings at the same or different loads. The methods for calculation of elastic contact deformation are also sufficiently accurate, and these can be used to compare rolling bearing stiffness against the stiffness of other bearing types. Hertz, H. Timoshenko, S. Boussinesq, J. Brewe, D. Lundberg, G.
Palmgren, A. Thomas, H. Illinois Bull. Zwirlein, O. Johnson, K. Thesis, University of Manchester, Smith, J. Radzimovsky, E. Illinois Eng. Experiment Station Bull. Liu, C. Thesis, University of Illinois, June Bryant, M. Cattaneo, C. Roma Rend. Loo, T. Sinica, 7, —, Deresiewicz, H. Sayles, R.
Kalker, J. Ahmadi, N. Harris, T. Kunert, K. Ingenieurwes, 27 6 , —, Reusner, H. Fredriksson, B. To do this it is first necessary to develop load—deflection relationships for rolling elements contacting raceways. By using Chapter 2 and Chapter 6, these load—deflection relationships can be developed for any type of rolling element contacting any type of raceway.
Hence, the material presented in this chapter is completely dependent on the previous chapters, and a quick review might be advantageous at this point.
Most rolling bearing applications involve steady-state rotation of either the inner or outer raceway; sometimes both raceways rotate. In most applications, however, the speeds of rotation are usually not so great as to cause ball or roller inertial forces of sufficient magnitude to significantly affect the distribution of applied load among the rolling elements.
Moreover, in most applications the frictional forces and moments acting on the rolling elements also do not significantly influence this load distribution. Therefore, in determining the distribution of rolling element loads, it is usually satisfactory to ignore these effects in most applications.
Furthermore, before the general use of digital computation, relatively simple and effective means were developed to assist in the analyses of these load distributions. In this chapter, load distributions in statically loaded ball and roller bearings will be investigated using these simple and effective methods of analysis. The total normal approach between two raceways under the load separated by a rolling element is the sum of the approaches between the rolling element and each raceway.
Figure 7. Equation 7.
From Equation 7. This is given in Table 7. For a given bearing with a given clearance under a given load, Equation 7. If Equation 7. For bearings supporting light loads, however, Equation 7. See Examples 7. For thrust ball bearings whose contact angles are nominally less than , the contact angle in the loaded bearing is greater than the initial contact angle a8 that occurs in the nonloaded bearings.
The phenomenon is discussed in detail in the next sections. A thrust load Fa applied to the inner ring as shown in Figure 7. This axial deflection is a component of a normal deflection along the line of contact such that from Figure 7.
Jones , however, defines an axial deflection constant K as follows: The axial deflection constant K is related to Kn as follows: The axial deflection da corresponding to dn may also be determined from Figure 7. Substituting dn from Equation 7. See Example 7. Rumbarger, J. February 15, Considering Equation 7. Table 7. For static equilibrium to exist, the summation of rolling element forces in each direction must equal the applied load in that direction: Note that the contact angle a is assumed identical for all loaded balls or rollers.
Thus, the values of the integrals are approximate; however, they are sufficiently accurate for most calculations. It is further clear from Equation 7. Under these speed conditions, effects of rolling element centrifugal forces and gyroscopic moments are negligible. At high rotational speeds, these body forces become significant, tending to alter contact angles and internal clearance and can affect the internal load distribution to a great extent.
In the foregoing discussion, relatively simple calculation techniques were used to determine the internal load distribution. Together with the tabular and graphical data provided, hand calculation devices may be employed to achieve the calculated results. In subsequent chapters in the Second Volume of this handbook, to evaluate the effects of loading in three or five degrees of freedom in ball and roller bearings, the effects of misalignment and thrust loading in roller bearings, and nonrigid bearing rings, digital computation must be used.
Nevertheless, for many applications the relatively simple methods demonstrated in this chapter may be used effectively.
It has been demonstrated in this chapter that bearing radial and axial deflections are functions of the internal load distribution. Further, since the contact stresses in a bearing depend on the load, maximum contact stress in a bearing is also a function of load distribution. Consequently, bearing fatigue life, which is governed by stress level, is significantly affected by the rolling element load distribution. Stribeck, R. ASME 29, —, For bearings with rigidly supported rings, the elastic deflection of a bearing as a unit depends on the maximum elastic contact deformation in the direction of the applied load or in the direction of interest to the application designer.
Because the maximum elastic contact deformation depends on the rolling element loads, it is necessary to analyze the load distribution occurring within the bearing before determination of the bearing deflection. Chapter 7 demonstrated methods for evaluating the load distribution among the rolling elements for bearings with rigidly supported rings subjected to a relatively simple statically applied loading.
In these methods, the variables dr and da, the principal bearing deflections, were utilized. These deflections may be critical in determining system stability, dynamic loading on other components, and accuracy of system operation in many applications and are discussed in this chapter. In lieu of a more rigorous approach to the determination of bearing deflections, Palmgren  provided a series of formulas to calculate the bearing deflection for specific conditions of applied loading.
It can be seen from Figure 8. Hence, it would be advantageous with regard to minimizing bearing deflection under load to operate above the knee of the load—deflection curve. This condition can be realized by axially preloading angular-contact ball bearings. This is usually done, as shown in Figure 8. Figure 8. Suppose that two identical angular-contact ball bearings are placed back-to-back or faceto-face on a shaft as shown in Figure 8.
Each bearing experiences an axial deflection dp due to preload Fp. The shaft is thereafter subjected to thrust load Fa, as shown in Figure 8. The inner-ring faces are ground to provide a specific axial gap. The contact angles have increased. In this case it is the outer-ring faces that are ground to provide the required gap.
The convergent contact angles increase under preloading. As the load increases, the rate of increase of deflection decreases; therefore, preloading top line tends to reduce the bearing deflection under additional loading. Fa F1 F2 Bearing no. The computation for the resulting deflection is complicated by the fact that the preload at bearing 1 is increased by load Fa while the preload at bearing 2 is decreased.
Equation 8. Subsequent substitution of a1 and a2 into Equation 7. The data pertaining to the selected preload Fp may be obtained from the following equations: Note that deflection is everywhere less than that for a nonpreloaded bearing up to the load at which preload is removed. Thereafter, the unit acts as a single bearing under thrust load and assumes the same load—deflection characteristics as those given by the single-bearing curve.
The point at which bearing 2 loses load may be determined graphically by inverting the single-bearing load—deflection curve about the preload point. This is shown in Figure 8. As roller bearing deflection is almost linear with respect to load, there is not much advantage to be gained by axially preloading tapered or spherical roller bearings; hence, this is not a universal practice as it is for ball bearings.
See Example 8. If it is desirable to preload ball bearings that are not identical, Equation 8. As before, Equation 7. To reduce axial deflection still further, more than two bearings can be locked together axially as shown in Figure 8. The disadvantages of this system are increased space, weight, and cost. More data on axial preloading are given in Ref. Instead, its purpose is generally to obtain a greater number of rolling elements under load and thus reduce the maximum rolling element load.
It is also used to prevent skidding. Methods used to calculate maximum radial rolling element load are given in Chapter 7.
In other words, a load in either the axial or radial direction should cause identical deflections ideally. This necessity for isoelasticity in the ball bearings came with the development of the highly accurate, low drift inertial gyroscopes for navigational systems, and for missile and space guidance systems.
Such inertial gyroscopes usually have a single degree of freedom tilt axis and are extremely sensitive to error moments about this axis. This arrangement provides an even higher axial stiffness and longer bearing life than with a duplex set, but requires more space.
Consider a gyroscope in which the spin axis Figure 8. The tilt axis is perpendicular to the paper at the Origin O, and the center of gravity of the spin mass is acted on by a disturbing force F in the xz-plane and directed at an oblique angle f to the x-axis; this force will tend to displace the spin mass center of gravity from O to O0.
If, as shown in Figure 8. To minimize M and subsequent drift, d0z must be as nearly equal to d0x as possible—a requirement for pinpoint navigation or guidance. Also, from Figure 8. In most radial ball bearings, the radial rate is usually smaller than the axial rate. This is best overcome by increasing the bearing contact angle, which reduces the axial yield rate and increases the radial yield rate.
One-to-one ratios can be obtained by using bearings with contact angles that are or higher. At these high angles, the sensitivity of the axial-to-radial yield rate ratio to the amount of preload is quite small. It is, however, necessary to preload the bearings to maintain the desired contact angles. The latter condition results in severe stress concentration and attendant rapid fatigue failure of the bearing.
One object of preloading is to remove this clearance during assembly. The situation in which the balls override the land will be examined first. From Figure 8. Both the inner and outer ring lands must be considered. It is frequently desirable to obtain isoelasticity in bearings in which the displacement in any direction is in line with the disturbing force. Combining Equation 7. Having calculated a, it is then possible to determine the limiting thrust load Fao for the ball overriding the outer land from Equation 7.
This consideration is important in high-speed systems such as aircraft gas turbines. The bearing radial deflection in this case can contribute to the system eccentricity. In other applications, such as inertial gyroscopes, radiotelescopes, and machine tools, minimization of bearing deflection under load is required to achieve system accuracy or accuracy of manufacturing. That the bearing deflection is a function of bearing internal design, dimensions, clearance, speeds, and load distribution has been indicated in the previous chapters.
However, for applications in which speeds are slow and extreme accuracy is not required, the simplified equations presented in this chapter are sufficient to estimate bearing deflection. To minimize deflection, axial or radial preloading may be employed. Care must be exercised, however, not to excessively preload rolling bearings since this can cause increased friction torque, resulting in bearing overheating and reduction in endurance.
Units N lb mm in. Bearing steel loaded in compression behaves in a similar manner. Thus, when a loaded ball is pressed on a bearing raceway, an indentation may remain in the raceway and the ball may exhibit a flat spot after the load is removed.
These permanent deformations, if they are sufficiently large, can cause excessive vibration and possibly stress concentrations of considerable magnitude.
Figure 9. Thus, it is probable that the compressive yield strength is exceeded locally and both surfaces are somewhat flattened and polished in operation.
According to Palmgren , this flattening has little effect on the bearing operation because of the extremely small magnitude of deformation.
It may be detected by a slight change in reflection of light from the surface. As the load between the surfaces is increased, the deformation gradually departs from that depicted in Equation 6. From Sayles, R. Ground M. RMS surface roughness 1. On the basis of empirical data for bearing quality steel hardened between For ball—raceway contact, Equation 9.
For roller—raceway point contact, the following equation obtains: The foregoing formulas are valid for permanent deformation in the vicinity of the compressive elastic limit yield point of the steel.
See Example 9. For line contact between roller and raceway, the following formula may be used to ascertain permanent deformation with the same restrictions as earlier: Palmgren  stated that of the total permanent deformation, approximately two thirds occur in the ring and one third in the rolling element.
Later, some of these tests were repeated using modern measurement devices. The following conclusions were reached: The amount of total permanent indentation occurring due to an applied load Q between a ball and a raceway appears to be less than that given in Equation 9.
The amount of permanent deformation that occurs in the ball surface is virtually equal to that occurring in the raceway, when balls have not been work hardened. Accordingly, it can be stated that permanent deformations calculated according to Equation 9. Moreover, experience has demonstrated that rolling bearings do not generally fracture under normal operating loads. Further, experience has shown that permanent deformations have little effect on the operation of the bearing if the magnitude at any given contact point is limited to a maximum of 0.
If the deformations become much larger, the cavities formed in the raceways cause the bearing to vibrate and become noisier, although bearing friction does not appear to increase significantly.
The bearing operation is usually not impaired in any other manner; however, indentations together with conditions of marginal lubrication can lead to surface-initiated fatigue. The basic static load rating of a rolling bearing is defined as the load applied to a nonrotating bearing that will result in a permanent deformation of 0.
In other words, in Equation 9. This concept of an allowable amount of permanent deformation consistent with smooth minimal vibration and noise operation of a rolling bearing continues to be the basis of the ISO standard  and ANSI standards [5,6].
In the latest revision of the ISO standard , it is stated that contact stresses at the center of contact at the maximum loaded rolling elements as shown in Table 9. The ANSI standards [5,6] use the same criteria. Substituting for Qmax in Equation 9. The corresponding formula for radial roller bearings as taken from Ref. A graph of Vickers hardness versus Rockwell C hardness is shown in Figure 9. Equation 9. The values of h1 depend on the type of contact and are given in Table 9.
A theoretical calculation of this load may be made in accordance with the methods of Chapter 7. In lieu of the more rigorous approach, for bearings subjected to combined radial and thrust loads, the static equivalent load may be calculated as follows: Table 9.
Data in Table 9. Double-row bearings are presumed to be symmetrical. Rockwell C hardness. For radial roller bearings, the values of Table 9. TABLE 9.
Values of Yo for intermediate contact angles are obtained by linear interpolation. In this manner, the permanent deformations that occur are uniformly distributed over the raceways and rolling elements, and the bearing retains satisfactory operation. If, on the other hand, the load is of short duration, unevenly distributed deformations may develop even when the bearing is rotating at the instant when shock occurs.
For this situation, it is necessary to use a bearing whose basic static load rating exceeds the maximum applied load. When the load is of longer duration, the basic static load rating may be exceeded without impairing the operation of the bearing. According to the type of bearing service, a factor of safety may be applied to the basic load rating.
Interruptions in the rolling path such as those caused by permanent deformations result in increased friction, noise, and vibration. Chapter 14 discusses the noise and vibration phenomenon in substantial detail. In this chapter, the discussion centered on bearing static load ratings, which, if not exceeded while the bearing was not rotating, would preclude permanent deformations of significant magnitude.
The ratings were based on a maximum allowable permanent deformation of 0. Subsequently, it was determined that for various types of ball and roller bearings, this deformation could be related to a value of rolling element—raceway contact stress. In accordance with this stress, basic static load ratings are developed for each rolling bearing type and size. Generally, a load of magnitude equal to the basic static load rating cannot be continuously applied to the bearing with the expectation of obtaining satisfactory endurance characteristics.
Rather, the basic static load rating is based on a sudden overload or, at most, one of short duration compared with the normal loading during a continuous operation. Exceptions to this rule are bearings that undergo infrequent operations of short durations, for example, bearings on doors of missile silos or dam gate bearings.
For these and simpler applications, the bearing design may be based on basic static load rating rather than on endurance of fatigue. Whereas the current static load ratings are based on damage during nonrotation, during operation under heavy load and slow speed, rolling contact, components experience significant microstructural alterations. Because of the relatively slow speeds of rotation and infrequent operation, neither vibration nor surface fatigue may be as significant in such applications as excessive plastic flow of subsurface material.
The bearings could thus be sized to eliminate or minimize such plastic flow and ultimately bearing failure. Units mm2 in. In this book, treatment has been restricted to shaft or outer-ring rotation or oscillation.
Unlike hydrodynamic or hydrostatic bearings, motions occurring in rolling bearings are not restricted to simple movements. For instance, in a rolling bearing mounted on a shaft that rotates at n rpm, the rolling elements orbit the bearing axis at a speed of nm rpm, and they simultaneously revolve about their own axes at speeds of nR rpm. In most applications, particularly those operating at relatively slow shaft or outer-ring speeds, these internal speeds can be calculated with sufficient accuracy using simple kinematical relationships; that is, the balls or rollers are assumed to roll on the raceways without sliding.
This condition will be considered in this chapter. Resisting the rotary motion of the bearing is a friction torque that, in conjunction with shaft or outer-ring speed, can be used to estimate bearing power loss.
On the basis of laboratory testing of rolling bearings, empirical equations have been developed to enable the estimation of this friction torque in applications where speeds are relatively slow, that is, where inertial forces and contact friction forces are not significantly influenced by contact deformations and speed.
These empirical equations are presented in this chapter. This slow-speed behavior is called kinematical behavior. As a general case, it will be initially assumed that both inner and outer rings rotate in a bearing having a common contact angle a as indicated in Figure Notwithstanding the fact that the motions of the contacting elements in rolling bearings are more complex than indicated by pure rolling, rolling bearings exhibit much less friction than most fluid-film or sleeve bearings of comparable size, speed, and load-carrying ability.
A notable exception to this generalization is the hydrostatic gas bearing; however, such a bearing is not self-acting, as is a rolling bearing, and it requires a complex and expensive gas supply system.
Friction of any magnitude retards motion and results in energy loss. In an operating rolling bearing, friction causes temperature increase and may be measured as a motionresisting torque.
It will be shown in a later chapter that, even considering the name rolling bearing, the principal causes of friction in moderately-to-heavily loaded ball and roller bearings are sliding motions in the deformed rolling element—raceway contacts.
In addition, in tapered roller bearings, the major source of friction is the sliding motion between the roller ends and the large end flange on the inner ring or cone.
In cylindrical roller bearings, the sliding between the roller ends and roller guide flanges on the inner, outer, or both rings is a major source of friction. Rolling bearings with cages experience sliding between the rolling elements and cage pockets; if the cage is piloted on the inner or outer ring, sliding friction occurs between the cage rail and the piloting surface.
In all of the above conditions, the amount of friction occurring depends considerably on the type of lubricant used. This frictional resistance is a function of the lubricant properties, the amount of lubricant in the free space, and the orbital speed of the rolling elements.
In the Second Volume of this handbook, it will be demonstrated that this friction component influences the rolling element speeds. These tests were conducted under loads ranging from light to heavy, at slow-to-moderate shaft speeds, and using a variety of lubricants and lubrication techniques.
In evaluating the test results, Palmgren  separated the measured friction torque into a component due to applied load and a component due to the viscous property of the lubricant type, the amount of the lubricant employed, and bearing speed. Actually, as suggested in Section For the purposes of the simplified analytical methods that follow, however, it is sufficient to attribute this friction torque component to applied loading.
This is particularly true, for example, when providing comparison between rolling bearings and fluid-film bearings. These terms were explained in Chapter 9. Table Fb in Equation It may be expressed in equation form as follows for radial ball bearings: In Equation Equation Palmgren  gave a more complete formula for oils of different densities.
For grease-lubricated bearings, no refers to the oil within the grease, and the equation is valid shortly after the addition of the lubricant. TABLE Lower values pertain to light series bearings; higher values to heavy series bearings. See Example In this case, the rollers are loaded against one flange on each ring. The bearing friction torque due to the roller end motions against properly designed and manufactured flanges is given by TABLE Therefore, Equation For thrust spherical roller bearings, Table Values of bearing load torque as calculated from Equation Harris  used these data successfully in the thermal evaluation of a submarine propeller shaft radial and thrust bearing assembly.
The typical roller length is at least three to four times larger than the diameter. Longer rollers lead to more sliding at the roller ends with small amounts of roller—raceway misalignment or skewing that is typically not significant in other bearing types. This is especially true for thrust needle roller bearings, where the raceway surface velocity is dependent on the contact diameter, while the roller surface velocity is constant along its length, necessitating a sliding motion at the ends of the roller—raceway contacts.
Also, needle bearings are frequently mounted directly in contact with the shaft or housing manufactured by the bearing user. This results in surface roughness and textures slightly different from those found on the raceways fabricated onto bearing rings by bearing manufacturers. Both of these situations cause different friction conditions to exist in the operation of needle roller bearings as compared with other bearing types.
In this catalog, the friction torque calculation method has been modified. Nevertheless, Equation Later chapters will cover direct methods for estimating contact friction and hence bearing running friction torque; however, empirical friction torque equations for radial and thrust needle bearings developed by Chiu and Myers  will be presented here.