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    Design Tools Shafts & Bearings


  • A: Shafts

  • B: Porous-Metal Bearings

  • C: Plastic and Non-Metallic Bearings

  • D: Ball Bearings

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A: Shafts



(1.0) Introduction




Shafts are used to transmit motion, torque and/or power in any combination. They are also subject to lateral loads. These can be constant or fluctuating. The sizing of shafts, therefore, is usually determined as a function of torsionally induced stresses (shear stresses), bending stresses (tensile or compressive stresses) and the nature of the load (constant or fluctuating).






(2.0) Determination of Stresses for Solid Cylindrical Shafts





(a) Nomenclature:



Let   d = shaft diameter, in.
        H.P. = horsepower
        Km = shock factor for bending loads
        Kt = shock factor for torsional loads
        M = bending moment, in-lbs.
        N = revolutions per minute (RPM)
        S = shear stress, lbs/in2
        Smax = maximum allowable shear stress, lbs/in2
        T = torque, in-lbs.
        Tmax = maximum allowable torque, in-lbs.






(b) Relation Between Torque and Horsepower:





(c) Torsional Loading:


For shafts under steady torsional loads (and no bending loads),

In the particular case of a shaft in which Smax = 12,000 psi (for example in the case of #303 stainless steel and a gradually applied load), equations (1) and (2) can be combined to yield:

Tmax = 2352 d3

and

(H.P.)max = 0.037 d3N


Table 1 shows maximum safe torsional loads based on #303 stainless steel with Smax= 12,000 psi and a gradually applied load.



The maximum torques in this table can be converted to horsepower for a given value of N(RPM) by the power Nomogram given in the Designer Data Section of our Design and Application of Small Standardized Components book.





(d) Combined Torsional and Bending Loads:


Combined bending and torsion arises as a result of component weight, belt tension, gear-tooth forces, etc. In that case shaft size can be determined from the equation:

Where the values of the shock factors (Km, Kt) are given in the following table:


If we arbitrarily consider only minor shock loads and assume that Km = Kt = √ 2 ≅ 1.14 and (as before) Smax = 12,000 psi, the combined allowable bending and torsional moments can be related to shaft size as shown in the following table:


If AISI 1213 steel is used, Smax = 8000 psi and the diameters given in the tables have to be increased by 15%.
Example: A 1/50th H.P. motor drives a shaft rotating at 315 RPM and is subject to a bending moment of 6 in-lbs, the load application being sudden. Determine shaft size for a #303 stainless steel shaft.
From equation (1);

Referring to Table 3 (Combined Loading), we find that the combination of an allowable torsional moment in excess of 4 in-lbs (actual 4.44) together with an allowable bending moment in excess of 6 in-lbs (actual 10.3) first occurs for a 3/16" shaft diameter.

If there were no bending moment and the load were to be applied gradually (Km = Kt 1) Table 1 shows that a 1/8" shaft diameter would be sufficient.

Precision ground stainless steel shafts in undersize, nominal and oversize dimensions, as well as cold-drawn precision low-carbon steel shafts are available from SDP/SI, as featured in our catalog.





(e) Combined Torsional and Bending Loads for Fluctuating Loads with Stress Reversal:


In such cases fatigue failure governs the design. Various failure theories have been proposed (e.g. Soderberg, Mises-Hencky etc.). All of these result in formulas involving both the average loading and the reversing component of the loading. Agreement with experiment varies depending on the nature of the material and failure mode. The equations resulting from these several theories also involve yield-point stresses and endurance limits and the various considerations governing the applicability of a particular set of equations extends beyond the scope of the discussion. For a fuller treatment the reader is referred to the following literature:

(i) R.M. Phelan: "Fundamentals of Mechanical Design", Third Edition, McGraw-Hill, New York, N.Y., 1970, Chapter 6.
(ii) J.E. Shigley: "Mechanical Engineering Design", Third Edition, McGraw-Hill, New York, N.Y., 1977, Chapter 13.
(iii) M.F. Spotts: "Design of Machine Elements", Third Edition, McGraw-Hill, Englewood Cliffs, New Jersey, 1961, Chapter 3.










(3.0) Hollow Cylindrical Shafts




In the case of hollow cylindrical shafts under torsion and/or bending, we can proceed as follows. Suppose a hollow shaft has an outside diameter, do, and an inside diameter, di. To size such a shaft for torsional and/or bending loads, we can convert the design calculation to that for an equivalent solid shaft of the same material by nothing that the stress is inversely proportional to the ratio of moment of inertia (of the shaft cross-section) to shaft radius. Equating this ratio for both the hollow shaft and the equivalent solid shaft of diameter deq. we have:


Solving for deq., we have:







B: Porous-Metal Bearings



(1.0) General Properties



Sintered-metal self-lubricating bearings are based on powder-metallurgy technology. They are economical, suitable for high production rates and can be manufactured to precision tolerances.

General properties of porous-metal bearing materials have been described in Machine Design magazine (Vol. 54, #14 June 17, 1982, pp. 131-132), with whose permission the following material is reprinted:

Sintered-metal self-lubricating bearings “are widely used in home applications, small motors, machine tools, aircraft and automotive accessories, business machines, instruments, and farm and construction equipment.

Most porous-metal bearings consist of either bronze or iron which has interconnecting pores. These voids take up 10% to 35% of the total volume. In operation, lubricating oil is stored in these voids and feeds through the interconnected pores to the bearing surface. Any oil which is forced from the loaded zone of the bearing is reabsorbed by capillary action. Since these bearings can operate for long periods of time without additional supply of lubricant, they can be used in inaccessible or inconvenient places where relubrication would be difficult.

Many variations are possible to meet specific requirements. From 1% to 3.5% graphite is frequently added to enhance self-lubricating properties. High porosity with a maximum amount of lubricating oil is used for high-speed light-load applications, such as fractional-horsepower motor bearings. A low oil-content low-porosity material with a high graphite content is more satisfactory for oscillating and reciprocating motions where it is hard to build up an oil film.

Powder producers can control powder characteristics such as purity, hydrogen loss, particle size and distribution, and particle shape. Each of these properties in some way affects performance. In the bronze system, for example, shrinkage increases as particle size of tin or copper powder in the mix decreases. Graphite additions result in growth but always lower the strength of the bearings. Lubricants used in the mix have only a slight influence on dimensional change, but a more pronounced effect on the apparent density and flow rate.

After sintering, the bearing must be sized to the specified dimensions. Sizing reduces interconnected porosity and produces greater strength, lower ductility and a smooth finish.

   Bronze: The most common porous bearing material. It contains 90% copper and 10% tin. These bearings are wear resistant, ductile, conformable, and corrosion resistant. Their lubricity, imbeddability, and low cost give them a wide range of applications from home appliances to farm machinery.

   Leaded Bronzes: Have a 20% reduction of the tin content of the usual 90-10 bronze and 4% reduction in copper. Lead content is 14% to 16% of total composition and results in a lower coefficient of friction and good resistance to galling in case the lubricant supply is interrupted. These alloys also have higher conformability than the 90-10 bronzes.

   Copper-Iron: The inclusion of iron in the composition boosts compressive strength although the speed limit drops accordingly. These materials are useful in applications involving shock and heavy loads and should be used with hardened shafts.

   Hardenable Copper-Iron: The addition of 1 ½ % free carbon to copper-iron materials allows them to be heat treated to a particle hardness of Rockwell C65. They provide high impact resistance and should be used with hardened-and-ground shafts.

   Iron: Combine low cost with good bearing qualities, widely used in automotive applications, toys, farm equipment, and machine tools. Powered-iron is frequently blended with up to 10% copper for improved strength. These materials have relatively low limiting value of PV (on the V side) but have high oil-volume capacity because of high porosity. They have good resistance to wear but should be used with hardened-and-ground steel shafts.

   Leaded-Iron: Provide improved speed capability but are still low-cost bearing materials.

   Aluminum: In some applications they provide cooler operation, greater tolerance for misalignment, lower weight, and longer oil life than porous bronze or iron. The limiting PV value is 50,000, the same as for porous bronze and porous iron. “






(2.0) Sizing Porous-Metal Bearings




The load-carrying capacity of porous-metal bearings can be measured by a friction/wear criterion, which is a measure of the heat generated by the bearing. It is called the PV factor. The PV factor, as its name implies, is the product of the bearing load, P, expressed in pounds per square inch of projected bearing area, and the surface velocity of the shaft expressed in feet per minute.



Most engineering data relating to the PV factor lists an upper limit of the factor, i.e. a value which should not be exceeded for satisfactory bearing operation. The workng value of the PV factor, however, is often less than this upper limit, for example if the sliding velocity is not sufficiently high to maintain an adequate lubricating film. In addition, the PV limit is affected by the static load-carrying capacity of the material, which should not be exceeded. The latter is a function of environmental factors, bearing clearances and geometry and the nature of the load(continuous, intermittent or shock loading). Detailed information on these considerations is usually furnished by the metal manufacturer. General guidelines are summerized in Table 1.




(3.0) Clearances




As in all bearings, satisfactory operation of porous-metal bearings require suitable clearances between shaft and housing. While guidelines depend on the materials used and the nature of the application, a representative chart showing reccomended bearing clearances for porous-bronze and porous-iron bearings is given in Figure 1:

We carry a full line of both thick and thin wall bushings. Please consult the Designer's Data section of this handbook for information on recommended shaft size and bore diameter to be used with various bushing sizes.





The upper curve (maximum) and all allowances above the mean are suggested for iron-based bearings only. The chart is representative of average conditions and each application needs to be evaluated individually.




(4.0) Conclusion




Porous-metal bearings are used widely in instruments and general machinery, in which their self-lubricating characteristics and load-carrying ability is very desirable. When properly designed they can be both economical and highly functional. Calculations of the bearing loads below are also applicable to porous-metal bearings.




C: Plastic & Non-Metallic Bearings



(1.0) General Characteristics




Among the significant characteristics of plastic bearings, the following are noteworthy:
  • Low wear rates
  • Relatively high performance rating (PV) among sleeve bearing materials
  • Bearing O.D.'s compatible with standard sintered bronze sizes for upgrading existing equipment
  • Kinetic and static coefficient of friction as shown in Figure 1
  • Light weight
  • Ability to conform under load

The design characteristics of plastic and non-metallic bearings bear both similarities and differences relative to those of porous-metal bearings. This will now be described in greater detail.




(2.0) Properties of Plastic & Non-Metallic Bearing Materials





Plastics (such as acetal, nylon, PTFE), carbon graphite and other non-metallic materials have been increasingly used as self-lubricating bearings. Their composition has been refined over many years so as to obtain favorable bearing characteristics. These include low friction, corrosion resistance, ability to conform under load (plastic bearings), ability to function over substantial temperature ranges and substantial load-carrying capabilities. Although temperature ranges, dimensional stability and load limitations of plastic gears are in general less than for metallic bearings, plastic bearings are remarkably versatile and economical.

A summary of characteristics of representative plastic and non-metallic has been given by Machine Design Magazine (Vol. 54, #14, June 17,1982, p.132) with whose permission the following material is reprinted:


Phenolics:



Composite materials consisting of cotton fabric, asbestos, or other fillers bonded with phenolic resin. The good compatibility of the phenolics makes them easily lubricated by various fluids.

They have replaced wood bearings and metals in such applications as propeller and rubber-shaft bearings in ships, and electrical switch-gear, rolling-mill, and water-turbine bearings. In small instruments and clock motors, laminated phenolics serve as structural members as well as bearing material. They have excellent strength and shock resistance, coupled with resistance to water, acid and alkali solutions.

Some precautions must be observed with phenolic bearings. Thermal conductivity is low, so heat generated by bearing friction cannot readily be transmitted through the bearing liner. Consequently, larger, heavily loaded bearings must have a generous feed of water or lubricating oil carry away heat. Some swelling and warping of these bearings occurs in the larger sizes, so larger-than-normal shaft clearances are required.





Nylon:



Although the phenolics have predominated in heavy-duty applications, they are frequently replaced by nylon, which has the wildest use in bearings. Nylon bushings exhibit low friction and require no lubrication. Nylon is quiet in operation, resists abrasion, wears at a low rate, and is easily molded, cast, or machined to close tolerances. Possible problems with cold flow at high loads can be minimized by using a thin liner of the material in a well-supported metal sleeve.

Improvement in mechanical properties, rigidity, and wear resistance is obtained by adding fillers such as graphite and molybdenum disulfide to nylon. While the maximum recommended continuous service temperature for ordinary nylon is 170°F, and 250°F for heat-stabilized compositions, filled-nylon parts resist distortion at temperatures up to 300°F.





PTFE:



Has an exceptionally low coefficient of friction and high self-lubricating characteristics, resistance to attack by almost any chemical, and an ability to operate under a wide temperature range. High cost combined with low load capacity has frequently caused PTFE resin to be selected only in some modified form. PTFE is used as a bearing material in automotive knuckle and ball joints, chemical and food processing equipment, aircraft accessories, textile machinery, and business machines.

Although unmodified PTFE can be used to a PV value of only 1,000, PTFE filled with glass fiber, graphite, or other inert materials, can be used at PV values up to 10,000 or more. In general, higher PV values can be used with PTFE bearings at low speeds where its coefficient of friction may be as low as 0.05 to 0.1.

One bearing material combines the low friction and good wear resistance of lead-filled PTFE with the strength and thermal conductivity of a bronze and steel supporting structure. A plated steel backing is covered with a thin layer of sintered, spherical, bronze particles. The porous bronze is then impregnated with a mixture of PTFE and lead to provide a thin surface layer. Service temperatures of -330 to +536°F are possible.

Woven PTFE fabrics are often readily handled and applied. With their resistance to cold flow, they are used as bearings in a wide variety of high-load applications as automotive thrust washers, ball-and-socket joints, aircraft controls and accessories, bridge bearings, and electrical switch gear. To provide a strong bond to either steel or other rigid backing material, a secondary fiber such as polyester, cotton, or glass is commonly interwoven with the PTFE. The woven fabric then is bonded to a steel backing.

Improved versions of this type of bearing have woven or braided "socks" (of PTFE and a bondable material). The bearing sleeve is then filament wound with a fiberglass-epoxy shell. These bearings have been reported to carry dynamic loads as high as 50,000 psi.





Acetal:



Has been used for inexpensive bearings in a wide variety of automotive, appliance, and industrial applications. It is particularly useful in wet enviornmentals because of its stability and resistance to wet abraison.





Polyimide, Polysulfone, Polyphenylene Sulfide:



High-temperature materials with excellent resistance to both chemical attack and burning. With suitable fillers, these moldable plastics are useful for PV factors to 20,000 and 30,000. Polyimide molding compounds employing graphite as a self-lubricating filler show promise in bearing, seal, and piston ring applications at temperatures to 500°F. Polyphenylene sulfide can be applied as a coating through use of a slurry spray, dry powder, or fluidized bed. These coating techniques require a final bake at about 700°F.





Ultrahigh-Molecular-Weight Polyethylene:



Resists abrasion and has a smooth, low-friction surface. Often an ideal material for parts commonly made from acetal, nylon, or PTFE materials.





Carbon-Graphite:



The self-lubricating properties of carbon bearings, their stability at temperatures up to 750°F, and their resistance to attack by chemicals and solvents, give them important advantages in fields where other bearing materials are unsatisfactory. Carbon-graphite bearings are used where contamination by oil or grease is undesirable, as in textile machinery, food handling machinery, and pharmaceutical processing equipment. They are used as bearings in and around ovens, furnaces boilers, and jet engines where temperatures are too high for conventional lubricants. They are also used with low-viscosity and corrosive liquids in such applications as metering devices or pumps for gasoline, kerosene, hot and cold-water, sea water, chemical processes streams, acids, alkalis, and solvents.

The composition and processing used with carbon bearings can be varied to provide characteristics required for particular applications. Carbon-graphite has from 5% to 20% porosity. These pores can be filled with a phenolic or epoxy resin for improved strength and hardness, or with oil or metals (such as silver, copper, bronze, cadmium, or babbitt) to improve compatibility properties.

A PV limit of 15,000 ordinarily can be used for dry operation of carbon bearings. This should be reduced for continuous running with a steady load over a long period of time to avoid excessive wear. When operating with liquids which permit the developement of a supporting fluid film, much higher PV values can be used.

A hard, rust-resistant shaft with at least a 10-uin. finish should be used. Hardened tool steel or chrome plate is recommended for heavy loads and high-speed applications. Steel having a hardness over Rockwell C50, bronzes, 18-8 stainless steels, and various carbides and ceramics also can be used.

Certain precautions should be observed in applying carbon-graphite. Since this material is brittle, it is chipped or cracked easily if struck on an edge or a corner, or if subjected to high thermal, tensile, or bending stresses. Edges should be relieved with a chamfer. Sharp corners, thin sections, keyways, and blind holes should be avoided wherever possible. Because of brittleness and low coefficient of expansion (about 1/4 that of steel), carbon-graphite bearings are often shrunk into a steel sleeve. This minimizes changes in shaft clearance with temperature variations and provides mechanical support for the carbon-graphite elements."





The comparative properties of three proprietary materials are summarized in Table 1.

Data reprinted with the permission of the following manufacturers:
(i) "Graphitar" Wickes, 1621 Holland Ave., Saginaw, Mich. 48601;
(ii) "Oilon PV® -80 DEsign Guide:", TFE Industries, 148 Parkway Kalamazoo, Mich. 49006;
(iii) "Rulon® Standard Stock Bearings, Engineering Manual, Cat. 75", Dixon Corp., Div. of Dixon Industries, Bristol, R.I., 02809.



(A) Oilon PV® -80-Class I
(B) Oilon PV® -80-Class II
(C) Acetal -Class I
(D) Nylon MoS2 -Class II
(E) Nylon MoS2 -Class I
(F) PTFE glass filled -Class II
(G) Oil impregnated sintered copper alloy -Class II
(H) White Metal -Class I

Class I -Grease applied externally prior to start-up.
Class II -No grease applied prior to start-up

Test Conditions:
Velocity - 46 ft/min. (350 RPM)
Load - 140 lbs./sq.in., addition applied at 10 min. intervals
Dimensions of Test Specimen - 5/8" O.D. x 3/8" long
Mating Material - Steel 113°F HR-B 90

A comparison of frictional characteristics of various metallic and plastic material is given in Figure 1. In some plastic materials the coefficient of friction decreases with load, thereby greatly reducing or eliminating the stick-slip problem in the start-up of machinery.

In recent years the properties of plastic bearing materials have been materially enhanced by the addition of fillers (such as fiber, powder, graphite, and molybdenum disulfide) and composite (metal or other backings). If the cost is warrented the mechanical properties of such bearings can be dramatically improved.




(3.0) Sizing Plastic And Non-Metallic Bearings




The load-carrying capacity of plastic and non-metallic bearings is determined by means of the PV-factor, as described in the section on porous-metal bearings.

The upper bound or limiting value of the PV factor again depends on operating conditions (speed, temperature, etc.) and a limit on the allowable unit loading. In addition to its use as a design guide for limiting load/speed values the PV factor can also be used to estimate a relative wear factor, K. Table 2 summarizes data for the PV and K factors for typical and non-metallic bearing materials.


A more detailed treatment of the PV-curve for any particular material involves additional data, which can often be obtained from the material manufacturer. For example, in the case of Oilon PV° -80, Figure 2 presents such additional information.

("A") - low wear region, no external lubrication required.
("B") - low wear region, initial external lubrication recommended.
("C") - feasible for Oilon with testing. External lubrication required.




(4.0) Conclusion




Plastic and non-metallic bearings are widely used in applications, toys, general machinery and applications ranging from cameras and toys to office machinery and automobiles. When properly designed their light weight and economy can be highly attractive.

* Reprinted with the permission of TFE Industries, 148 Parkway, Kalamazoo, Michigan 49006; from p. 7 of "Oilon PV® - Design Guide"





D: Ball Bearings



(1.0) Introduction




Ball bearings are used widely in instruments and machines in order to minimize friction and power loss. While the concept of the ball bearing dates back at least to Leonardo da Vinci, their design and manufacture has become remarkably sophisticated. In the following we shall review their basic characteristics.




(2.0) Types of Ball Bearings




Commercially available ball bearings, which are usually made from hardened steel, involve many forms of construction. These have been summarized by A.O. DeHart ("Which Bearings and Why", ASME Paper 59-MD-12, 1959), from which source the following material (including Figures 1 & 2) is hereby reprinted*


"A typical deep-groove ball bearing designed for high-speed operation is shown in Figure 1. In this bearing, the separator serves to keep the balls from rubbing against one another as is piloted on the inner race OD. Alternatively, the separator may be piloted by the rolling elements or by the outer race ID. Where rotative speeds are low, the separator often is omitted. The rolling elements may take many forms — cylinders, balls, tapered rollers, barrels, or very slim rollers known as needles — and the whole bearing name is generally taken from this form.

Ball Bearings:
There are several types of ball bearings that fit specific needs. The deep-groove ball bearing, Figure 2(a), is the most versatile. Radial loads and thrust-load capacities may be approximately equal in this bearing. When it has the proper separator, it is very good for high-speed operation. At low speeds, no bearing separator is required; at intermediate speeds, a ball control separator of steel-ribbon construction is adequate; while the ultimate high-speed performance is obtained with a race controlled (or piloted), fully machined separator.

Since balls are assembled into the bearing by eccentric displacement of the races the number of balls in this type of bearing is limited. More balls can be introduced into the bearing if a notch is machined into one of the races, Figure 2(b). Radial load capacity is higher in this bearing than in the standard deep-groove construction, but high-speed performance and thrust-load capacity is impaired. When large thrust loads in one direction are coupled with radial loads, angular contact ball bearings, Figure 2(c), are usually superior. Most high-speed and precision spindles use axially preloaded pairs of these bearings. Preload is controlled by the length of the spacers, which determine axial location of the races, or by mounting the bearings against one another in a "back-to-back" or "face-to-face" fashion. The double-row, angular-contact bearing, Figure 2(d), is a simpler arrangement from the standpoint of the user. The preload is built into the bearing at the factory.

In contrast th the previously discussed bearings, in which alignment is a very critical item, the self-aligning ball bearing, Figure 2(e) by virtue of the spherically ground outer race can tolerate considerable misalignment of shaft and housing. On the other hand, load-carrying capacity is reduced due to the high contact stresses that result from the large difference in curvature between the balls and the outer race.

The thrust ball bearing, Figure 2(f), is adaptable to large thrust loads that have almost no radial component. Very large sizes of this bearing are used in gun turrets and large earth moving machinery"






(3.0) Bearing Selection




Bearing selection represents a compromise among many factors including the nature of the application, performance requirements and cost. A useful bearing-selection chart, which summarizes the principal considerations involved, has been given by A.O. DeHart and is reproduced in Table 1.

For more details, which are beyond the scope of this presentation, the reader is referred to the technical literature.




(4.0) Bearing Loads




(a): Radial Shaft Load Between Bearings



The first step in sizing a suitable ball bearing for a given application is the determination of the loads which support. In this section we list some of the most frequently occurring mechanical configurations and the bearing loads imposed by them.





(b): Overhung Radial Load







(c): Flat Belt Drives






The maximum bearing load on either pulley shaft occurs when the belt is transmitting the maximum horsepower (i.e., the belt would slip if the horsepower were increased above this level). Under this condition the maximum bearing load is given by:

Note: In the case of chain drives the bearing load is often approximated by the pull on the tight side of the chain, the slack side being assumed tensionless.




(d): Unbalanced Rotors








(e): Cams




The bearing load on the camshaft bearings due to load, P, can be determined according to Cases (a) or (b), if the camshaft is supported by two bearings.

(ii) Disc cam with translating roller follower


The bearing load on the camshaft bearings due to the load, P, can be determined according to Cases (a) and (b), if the camshaft is supported by two bearings. Note that the force P in the above two cases is equivalent to a radial force, P, together with a torque about the cam axis.




(f): Spur Gears (External)








(g) Helical Gears


We consider here only the case of helical gears on parallel shafts.


Note that:
(i) The helices on mating gears are of opposite hand;
(ii) The direction of the thrust load is determined by the condition (see Figure 12) that the vector sum of the radial force and the thrust load is normal to the helix. This implies that reversal of rotation causes reversal of thrust.

Total radial shaft load

The thrust load in the case of helical gears implies that the bearings be capable of carrying both the radial load and the thrust load.

The calculation of the radial bearing load in the case of shafts with two bearings can be obtained from Cases (a) and (b).

Again we note that since action and reaction are equal and opposite, the three orthogonal force components F, FR and FT act on gears (and shafts), but in opposite directions.





(h): Straight Bevel Gears





Note that the direction of F (in all gear drives) depends on the direction of rotation of the driving gear. The thrust loads FTG and FTP are components of the tooth separating force, which must be taken up by both pinion and gear bearings. The directions acting on the gear and the pinion are opposite. Total bearing force on each gear is the vector sum of three forces: tangential, gear thrust and pinion thrust. These forces are shown in Figure 14.

Total radial shaft load

With the aid of these figures the radial bearing loads for shafts with two bearings can be obtained from Cases (a) and (b). The presence of thrust loads again necessitates axial takeup capabilities in bearings.





(i): Worms and Worm Gears



Total radial shaft load

Note that the direction of F depends on the direction of rotation of the worm. The three force components, F, FR and FTW must be taken up by both worm and gear bearings. The directions acting on the worm gear and worm are opposite. Total bearing force on each member is the vector sum of these three forces. With the worm as driver and the gear rotating as shown in Figure 15, the direction of these forces on each member are shown in Figures 16a and b.

With the aid of these figures the radial bearing loads for shafts with two bearings can be obtained from Cases (a) and (b). Once again both thrust and radial forces need to be taken up by the bearings.





(j) Compound Spur-Gear Train



As an example of the bearing-reaction calculations for an entire gear train we consider the spur-gear train shown in Figure 17.

The gear train shown in Figure 17 transmits 1/20 horsepower. Shaft S-1 is the driver. If shaft S-2 rotates at 100 rpm CW as shown, what are the bearing reaction forces on Shaft S-2?

The free body diagram of S-2 is shown in Figure 18a, and component forces are shown in Figure 18b. From the horsepower the tramsitted forces are obtained as follows:

Horsepower forces

These transmitted forces are generated from contact tooth forces given by Equation 2:

Equation 2 Contact Force

Where the double subscripts designate transmission of forces between members. For example, F12 means the force of gear 1 on gear 2.

The above contact tooth forces plus the bearing reaction forces hold the shaft in equilibrium as pictured in Figure 18a. Resolving all forces into X and Y components, as shown in Figure 19, the equilibrium equations can be applied.

Because of the particular shaft orientation given for this problem the X and Y components of contact force F12 are the tangential and normal components, but this is not true of F43 which is inclined 50° to the X axis.

basic equilibrium equations

Thus, there are 4 unknowns and two equations. However, if the moment equilibrium equations are written, the unknowns can be reduced.




Taking moments about Bearing A, first about the X-axis and then about the Y-axis (using the convention Positive moments are CCW):

Equation 26

Note that the sign of Equation R y/b is negative. This means its direction is actually opposite of that assumed in equilibrium Equation 26. Thus, in Figure 19 component Equation R y/b should be drawn in reversed direction to that shown. Conversely, component Equation R y/bhas a positive sign, so its' direction assumed for the equilibrium and Figure 19 is correct.

To determine the X reaction components moments are taken about the Y-axis at bearing A:

X reaction components moments are taken about the Y axis at Bearing A
The resultant bearing reaction forces and orientations are pictured in Figure 20.










(5.0) Sizing Ball Bearings




(a): Basic Definitions



In the course of many years of experience with ball bearings and extensive testing, it has been found that the prediction of the load capacity of a ball bearing is a statistical event related to the fatique life of the bearing. This makes the sizing of ball bearings more difficult than that of many other machine elements.

A basic phenomenon in ball bearings is that ball bearing life has been found to be inversely proportional to the cube of the bearing load. This means that when the load is doubled the life expectancy of the bearing is reduced by a factor of eight. This phenomenon has been studied extensively and has led to the adoption of an industry-wide national standard for rating ball bearings pioneered by the AFBMA (Anti-Friction Bearing Manufacturers Association, 1235 Jefferson Davis Highway, Arlington, Virginia, 22202). The following represents a summary of the load rating of ball bearings less than one inch in diameter, according to ANSI-AFBME Standard 9, 1978: "Load Rating and Fatique Life for Ball Bearings" - reprinted with the permission of the American National Standards Institute Inc., 1430 Broadway, New York, N.Y., 10018:

"Life Criterion. Even if ball bearings are properly mounted, adequately lubricated, protected from foreign matter, and are not subjected to extreme operating conditions, they can ultimately fatigue. Under ideal conditions, the repeated stresses developed in the contact areas between the balls and the raceways eventually can result in fatigue of the material which manifests itself as spalling of the load carrying surfaces. In most applications the fatigue life is the maximum useful life of a bearing. This fatigue is the criterion of life used as the basis for the first part of this standard."

The material in the standard which follows assumes bearings having non-truncated contact area, hardened good quality steel as the bearing material, adequate lubrication, proper ring support and alignment, nominal internal clearances, and adequate groove radii. In addition certain high-speed effects such as ball centrifugal force and gyroscopic moments are not considered. We now continue with the standard.

"Life. The "life" of an individual ball bearing is the number of revolutions (or hours at some given constant speed) which the bearing runs before the first evidence of fatigue develops in the material of either ring (or washer) or of any of the rolling elements."

"Rating life. The "RATING LIFE", L10, of a group of apparently identical ball bearings is the life in millions of revolutions that 90% of the group will complete or exceed. For a single bearing, L10 also refers to the life associated 90% reliability. As presently determined, the life which 50% of the group of ball bearings will complete or exceed ("median life", L50) is usually not greater than five times the RATING LIFE."

"Basic Load Rating. The "basic load rating", C, for a radial or angular contact ball bearing is that calculated, constant, radial load which a group of apparently identical bearings with stationary outer ring can theoretically endure for a RATING LIFE of one million revolutions of the inner ring. For a thrust ball bearing it is that calculated, constant, centric, thrust load which a group of apparently identical bearings can theoretically endure for a RATING LIFE of one million revolutions of one of the bearing washers. The basic load rating is reference value only, the base value of one million revolutions RATING LIFE having been chosen for ease of calculation. Since applied loading as great as the basic load rating tends to cause local plastic deformation of the rolling surfaces, it is not anticipated that such heavy loading would normally be applied."





(b): Determination of Basic Load Rating



Based on the preceding definitions the standard lists the equations required for the determination of the basic load rating:

"Calculation of Basic Load Rating. The magnitude of the basic load rating, C, for radial and angular contact ball bearings with balls not larger than 25.4 mm (1 inch) in diameter is:

Calculation of Basic Load Rating





(c): Illustration



Consider ball bearing A 7Y55-FS5025 described in the SDP/SI inch catalog. This is an ABEC-3 single-row, radial ball bearing having 10 balls of 1/16" diameter, 0.330" inner race diameter and 0.452" outer race diameter in a single shield configuration.



This means that for a load of P =143 lbs. the rating life of this ball bearing will be one million revolutions and 90% of a group of such ball bearings will be expected to complete or exceed this value.

Suppose now it is desired to determine the "L10" life of this bearing when operating at 200 RPM and a load of 50 lbs, the life being evaluated in hours of operation.

Let the life in hours be denoted by l10 and let N denote the RPM of the bearing. We then have

Equation 29

Substituting N = 200, P = 50 and C = 143 into Equation (29), we obtain l10 = 1949 hours.

NOTE: L10 is bearing life in millions of revolutions l10 is bearing life in hours.

A chart showing required life at constant operating speed has been given by N. Chironis ("Today's Ball Bearings", Product Engineering, December 12, 1960, pp. 63-77, chart on p. 68). This chart is hereby reproduced with the permission of McGraw-Hill Book Company, New York, N.Y.





(d): Combined Axial and Radial Loads



Such cases can be evaluated according to the methods previously outlined by combining the axial and radial loads into an equivalent radial load. This is defined in ANSI/AFBMA Standard 9, 1978 as follows:

"Calculation of Equivalent Radial Load. The magnitude of the equivalent radial load P, for radial and angular contact bearings, under combined constant radial and constant thrust loads, is:

P = XFr + YFa

Values of X and Y are given in Table 4.

The rating and sizing of ball bearings involve many considerations, many of which are beyond the scope of this introductory presentation. For further information the reader is reffered to the technical literature.









(6.0) Tolerances and Clearances




For satisfactory operation of a ball bearing, suitable shaft and housing tolerances are extremely important. Standard tolerance ranges have been established by the industry and Tables 5 and 6 show recommended deviations of shaft diameters and housing bores from nominal.

For normal conditions the recommendations of many manufacturers for rotating shafts and stationary housings, as given by Wilcock and Booser*, recommend fits in the approximate range K5 and J6 for shaft fits, and J6 and H7 for housing fits.

A fuller discussion of tolerances and their relation to bearing applications, installation and design is a complex subject beyond the scope of this presentation. This would include considerations involving temperature effects, high-speed operation, shock loading, lubrication, environmental conditions etc. For a discussion of such topics the reader is referred to the technical literature.

* "Bearing Design and Application" by D.F. Wilcock and E.R. Booser, McGraw Hill, New York, N.Y., 1st Ed., 1957. p.69



    1. When calculating the basic load rating for a unit consisting of two similar, single row, radial contact ball bearings, in a duplex mounting, the pair is considered as one, double row, radial contact ball bearing.

    2. When calculating the basic load rating for a unit consisting of two, similar, single row, angular contact ball bearings in a duplex mounting, "Face-to-Face" or "Back-to-Back", the pair is considered as one, double row, angular contact ball bearing.

    3. When calculating the basic load rating for a unit consisting of two or more similar, single angular contact ball bearings mounted "in Tandem", properly manufactured and mounted for equal load distribution, the rating of the combination is the number of bearings to the 0.7 power times the rating of a single row ball bearing. If the unit may be treated as a number of individually interchangeable single row bearings, this footnote 1c does not apply.

  1. Use to obtain C in newtons when D is given in mm.
  2. Use to obtain C in pounds when D is given in inches.

* Reprinted by permission of the American National Standards Institute, 1430 Broadway, New York, N.Y. 10018 (from ANSI-AFBMA Std 9-1978)


* Reproduced from "Today's Ball Bearings" by N. Chironis, Product Engineering, December 12, 1960, pp. 68 with the permission of McGraw-Hill Book Co. Inc., New York, N.Y.


  1. Two similar, single row, angular contact ball bearings mounted "Face-to-Face" or "Back-to-Back" are considered as one, double row, angular contact bearing.
  2. Values of X,Y and e for a load or contact angle other than shown in Table 2 are obtained by linear interpolation.
  3. Values of X,Y and e shown in Table 2 do not apply to filling slot bearings for applications in which ball-raceway contact areas project substantially into the filling slot under load.
  4. For single row bearings when Fa / Fr ≤ e, use X = 1, Y = 0.

* Reprinted by permission of the American National Standards Institute, 1430 Broadway, New York, N.Y., 10018 (force ANSI-AFBMA Std. 9-1978).


* After SKF
** Reprinted by permission of McGraw-Hill Book Co. Inc., New York, N.Y. from "Bearing Design and Application" by D.F. Wilcock and E.R. Booser, 1st Ed., 1957, pp. 64-65


* After SKF
** Reprinted by permission of McGraw-Hill Book Co. Inc., New York, N.Y. from "Bearing Design and Application" by D.F. Wilcock and E.R. Booser, 1st Ed., 1957, p. 67





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