Crank Bearings

The purpose of the crank bearing is to support loads. The primary loads are radial load from pedal pressure and chain tension. There are smaller axial loads including components of pedal pressure and chain load; also accidents, etc.

Traditional Cup-And-Cone

Traditional cup-and-cone crank bearings are ``backward'', thus increasing bearing loads and shortening the service life of the bearing assembly. This section describes how the loads are ``backwards''.

The following diagram shows a simple crank bearing.

If a force F acts on the end of the spindle, the load creates a radial force at each of the bearing contacts. The bearings are loaded at an angle, not radially. Thus, the radial load on the bearing causes an axial load. Using a shallower bearing contact angle causes a larger axial load.

Bearing loads can be modeled as follows. First, the load F acts at distance d. It causes a torque about the right bearing of F * d. The left bearing, a distance b away opposes that torque with a torque Lr * b. However, loading the bearings causes an axial force La which causes a torque La * 2s in the same direction as the original load. Thus:

(F * d) + (La * 2s) = (Lr * b)

The radial forces must add to zero, else the spindle would be in motion. Thus, the forces pushing up in the diagram must equal the forces pushing down:

F + Lr = Rr

As long as F is the only external force, then the force La pushing to the right is equal Ra, the force pushing left. Thus, the term La * 2s could also be written (La * s) + (Lr * s), or in several other forms. At any rate:

La = Ra

At each bearing, the axial and radial forces are related by the angle of contact across the bearing. If the angle of contact is near radial, then the axial force will be small for a given radial load. If the angle of contact is near axial, then the axial force will be large for a given radial load. Thus:

La = Lr * tan(B)
Ra = Rr * tan(A)

Note that the bearings are shown in the diagram as both having a 45-degree angle. In operation, ball bearings run at different angles depending on the applied load. If the load is largely axial, the bearing will try to move to a position where the contact is more axial and less radial, as shown below in (a). Conversely, if the load is mostly radial, the bearing will try to move as shown in (b).

Motion of the bearings requires slop or elasticity in the overall assembly, but the motions are small and, in practice, the bearings, races, spindle, etc., are all made of slightly flexible materials. Further, note that if the spindle of the earlier figure moves slightly to the left, it both causes the left bearing to assume the position (a) and at the same time causes the right bearing to assume the position (b).

We do not know the exact bearing angles A and B, but we do know the radial forces (F + Lr = Rr) and do know the left and right axial forces are equal (La = Ra). Finally, examining wear on the spindle and cups of a bottom bracket shows the bearings run near a 45 degree angle. Thus, we can tabulate a few results.

Consider a bottom bracket with spindle contact points b 50mm apart and points 2s 15mm apart. With a pedal load of 100 kgf applied at 100mm. The load on the right bearing will be larger than the load on the left, so we are interested in the peak bearing load R, which is Rr/cos(A).

Thus,

Rr = F + Lr

Lr = [(F * d) + (La * 2s)]/b

La = Ra = Rr * tan(A)

Substituting,

Rr = F + [(F * d) + ([Rr * tan(A)] * 2s)]/b

Simplifying,

(Rr * b) = (F * b) + (F * d) + ([Rr * tan(A)] * 2s)

(Rr * b) - ([Rr * tan(A)] * 2s) = (F * b) + (F * d)

Rr * (b - [2s * tan(A)]) = F * (b + d)

Rr = F * (b + d) / (b - [2s * tan(A)])

Tabulating for some values of A:

Fdb2sBLrLARrR
100 kgf100 mm50 mm15 mm0 deg200 kgf200 kgf0 deg300 kgf300 kgf
100 kgf100 mm50 mm15 mm7 deg208 kgf210 kgf5 deg308 kgf309 kgf
100 kgf100 mm50 mm15 mm14 deg217 kgf224 kgf10 deg317 kgf322 kgf
100 kgf100 mm50 mm15 mm20 deg226 kgf241 kgf15 deg326 kgf338 kgf
100 kgf100 mm50 mm15 mm26 deg237 kgf263 kgf20 deg337 kgf358 kgf
100 kgf100 mm50 mm15 mm31 deg249 kgf289 kgf25 deg349 kgf385 kgf
100 kgf100 mm50 mm15 mm35 deg263 kgf319 kgf30 deg363 kgf419 kgf
100 kgf100 mm50 mm15 mm38 deg280 kgf355 kgf35 deg380 kgf464 kgf
100 kgf100 mm50 mm15 mm41 deg301 kgf396 kgf40 deg401 kgf523 kgf
100 kgf100 mm50 mm15 mm43 deg329 kgf447 kgf45 deg429 kgf606 kgf
100 kgf100 mm50 mm15 mm44 deg367 kgf512 kgf50 deg467 kgf726 kgf
100 kgf100 mm50 mm15 mm45 deg425 kgf604 kgf55 deg525 kgf915 kgf
100 kgf100 mm50 mm15 mm46 deg524 kgf753 kgf60 deg624 kgf1249 kgf
100 kgf100 mm50 mm15 mm46 deg741 kgf1063 kgf65 deg841 kgf1990 kgf
100 kgf100 mm50 mm15 mm45 deg1607 kgf2270 kgf70 deg1707 kgf4991 kgf
100 kgf100 mm50 mm15 mm43 deg-2608 kgf3560 kgf75 deg-2508 kgf-9690 kgf
100 kgf100 mm50 mm15 mm39 deg-528 kgf675 kgf80 deg-428 kgf-2463 kgf
100 kgf100 mm50 mm15 mm29 deg-224 kgf255 kgf85 deg-124 kgf-1417 kgf

Note first that a radial bearing -- where the angle A is near zero -- has a very small bearing load. Radial bearings are discussed below.

Note also that as the angle A gets past about 50 degrees, the bearing force R skyrockets. By 75 degrees the equation shows bearing forces have gone negative. In practice, bearing forces cannot go negative. What happens is a small applied radial load causes a large axial force, and the large axial force ``aids and abets'' the original force to such a degree the bearing is no longer able to support the applied load.

As noted above, as the radial load increases, the bearing configuration tends to change so the right bearing is ``more radial'' (smaller angels of A).

The bearing angle and the amount by which the bearing moves depends in part on how tight the bearing is to start. If the bearing is adjusted tight, the bearing angles will remain high. If the bearing is adjusted loose, the bearing angles will go low, so most of the load is carried radially.

Unfortunately, this presents us with a conundrum: both theory and practice of bearings shows that, all else equal, a bearing works better if it is initially tight [1]. Here, ``works better'' means either that the bearing can carry more load, or it lasts longer (is more durable) when carrying a given load. The reason is that bearing preload helps to ensure the load is carried by more balls. Although preload increases the total bearing forces, it decreases the load carried by any one ball. Again, that is ``all else equal''.

In the case of a bottom bracket bearing, decreasing preload causes more load on each bearing because of worse load sharing, but increasing the preload causes total bearing loads to rise, so even though preload makes sharing better, it also causes total loads by enough to make preload have only a small benefit.

The upshot of this discussion is that conventional bottom bracket bearings are loaded poorly due to the bearing orientation. Although they show fair durability, failures are common, and overall durability is limited by the design.

Radial Bearings

The above analysis suggests that radial bearings would fare much better. In practice they do not last well, this section discusses some reasons why.

Two standard radial bearings are ``offset fill'' and ``slot fill''. Offset fill cartridge bearings are used commonly in industry and are the predominant type for bicycle bottom brackets.

The offset fill bearings use inner and outer races with a ``U'' shape which holds both sides of the bearing. The bearing is filled by placing the inner race inside the outer race, but off axis. Bearings are then fed in the resulting gap, then spread around the bearing and a retainer is installed to keep the bearings at an even spacing.

A problem with the offset-fill bearings is that it is not possible to get a full complement of bearings. That is, the available space might have enough space for 11 bearings, but the way the filling is done limits the bearing assembly to at most 9 balls. The reduced number of bearings increases the load on each bearing and hurts bearing durability.

The slot-fill bearing has races which have an extra groove to allow bearings to be slid in place when the grooves are aligned. Slot-fill bearings can have a full complement, but are relatively poor at carrying side loads.

An issue with both types of bearings is that common radial bearings are designed to have a high radial load capacity but are relatively poor at tolerating bearing misalignment. However, bicycles are of relatively crude construction so initial bearing alignment is often poor, and this is exaggerated by spindle flexibility. When a high-radial-load bearing is run misaligned, it suffers high internal loads which reduce its life.

A second issue with both types of bearings is that they are typically used as separate cartridges slid over the spindle and slid in to bearing ``cups''. This arrangement limits the amount of metal which supports the bearing compared to a bearing which is integral with the cups and the spindle. In addition, bearings are often pressed in or pressed on to the bottom bracket parts at relatively low loads in order to aid assembly, disassembly, and bearing service. The looser the press fit, the worse the bearing races are supported, and the less durable the bearing at a given load.

In practice, cartridge bearing bottom brackets show similar durability to conventional cup-and-cone units.

There are only two bottom brackets I know of which use slot-fill bearings:

Reversed Bearings

A third design reverses the position of the cups and cones so that axial forces help to carry (oppose) the load, rather than ganging up with it to overload the bearings.

[0.5-cup-cone-reverse.jpg]

Note a serious problem with this design: The overlap x makes assembly impossible. It may be possible to reduce x so that assembly is possible, though doing so may compromise the bearing function. Even if there is no overlap, it may be difficult or impossible to load the bearings, provide suitable seals, and so on. The remainder of this discussion assumes a usable design is possible, but that is an assumption, not a given.

The derivation of equations is similar to the conventional bottom bracket, but note that the axial load now helps to resist the load F.

(F * d) = (Lr * b) + (La * 2s)

As above:

F + Lr = Rr

La = Ra

La = Lr * tan(B)
Ra = Rr * tan(A)

Thus,

Rr = F + Lr

Lr = [(F * d) - (La * 2s)]/b

La = Ra = Rr * tan(A)

Substituting,

Rr = F + [(F * d) - ([Rr * tan(A)] * 2s)]/b

Simplifying,

(Rr * b) = (F * b) - (F * d) + ([Rr * tan(A)] * 2s)

(Rr * b) + ([Rr * tan(A)] * 2s) = (F * b) + (F * d)

Rr * (b + [2s * tan(A)]) = F * (b + d)

Rr = F * (b + d) / (b + [2s * tan(A)])

Tabulating for some values of A:

Fdb2sBLrLARrR
100 kgf100 mm50 mm15 mm0 deg200 kgf200 kgf0 deg300 kgf300 kgf
100 kgf100 mm50 mm15 mm8 deg192 kgf194 kgf5 deg292 kgf293 kgf
100 kgf100 mm50 mm15 mm15 deg185 kgf191 kgf10 deg285 kgf289 kgf
100 kgf100 mm50 mm15 mm22 deg178 kgf192 kgf15 deg278 kgf287 kgf
100 kgf100 mm50 mm15 mm28 deg170 kgf194 kgf20 deg270 kgf288 kgf
100 kgf100 mm50 mm15 mm34 deg163 kgf197 kgf25 deg263 kgf290 kgf
100 kgf100 mm50 mm15 mm39 deg156 kgf201 kgf30 deg256 kgf295 kgf
100 kgf100 mm50 mm15 mm44 deg148 kgf205 kgf35 deg248 kgf303 kgf
100 kgf100 mm50 mm15 mm48 deg140 kgf208 kgf40 deg240 kgf313 kgf
100 kgf100 mm50 mm15 mm51 deg131 kgf209 kgf45 deg231 kgf326 kgf
100 kgf100 mm50 mm15 mm54 deg121 kgf208 kgf50 deg221 kgf344 kgf
100 kgf100 mm50 mm15 mm57 deg110 kgf204 kgf55 deg210 kgf366 kgf
100 kgf100 mm50 mm15 mm60 deg97 kgf197 kgf60 deg197 kgf395 kgf
100 kgf100 mm50 mm15 mm63 deg83 kgf185 kgf65 deg183 kgf432 kgf
100 kgf100 mm50 mm15 mm67 deg64 kgf167 kgf70 deg164 kgf481 kgf
100 kgf100 mm50 mm15 mm73 deg42 kgf143 kgf75 deg142 kgf547 kgf
100 kgf100 mm50 mm15 mm84 deg11 kgf110 kgf80 deg111 kgf640 kgf
100 kgf100 mm50 mm15 mm-64 deg-32 kgf75 kgf85 deg68 kgf777 kgf

Note the above table shows that at 85 degrees, the left bearing radial load is negative. Again, negative bearings loads are not possible. The equation shows a negative value; as with a conventional bottom bracket, this is not a stable configuration.

Chain Tension Loads

The above discussion focused on loads caused by a force F on the left pedal. The spindle is also under chain loads. Pedaling loads are typically down, while chain loads are typically at about 90 degrees, pulling back.

A chainring one-third the diameter of the crank will have three times the chain force but it will be exerted more nearly in line with the right bearing, perhaps offset only 20mm. Here are similar computations but showing only the chain tension force.

This table shows the bearing loads for a conventional bottom bracket, with 300 kgf chain tension at 20mm offset.

Fdb2sBLrLARrR
300 kgf20 mm50 mm15 mm0 deg120 kgf120 kgf0 deg420 kgf420 kgf
300 kgf20 mm50 mm15 mm16 deg131 kgf137 kgf5 deg431 kgf433 kgf
300 kgf20 mm50 mm15 mm28 deg143 kgf163 kgf10 deg443 kgf450 kgf
300 kgf20 mm50 mm15 mm37 deg157 kgf196 kgf15 deg457 kgf473 kgf
300 kgf20 mm50 mm15 mm43 deg171 kgf235 kgf20 deg471 kgf502 kgf
300 kgf20 mm50 mm15 mm48 deg188 kgf279 kgf25 deg488 kgf539 kgf
300 kgf20 mm50 mm15 mm51 deg208 kgf328 kgf30 deg508 kgf587 kgf
300 kgf20 mm50 mm15 mm53 deg232 kgf383 kgf35 deg532 kgf649 kgf
300 kgf20 mm50 mm15 mm54 deg261 kgf445 kgf40 deg561 kgf733 kgf
300 kgf20 mm50 mm15 mm55 deg300 kgf520 kgf45 deg600 kgf849 kgf
300 kgf20 mm50 mm15 mm55 deg354 kgf613 kgf50 deg654 kgf1017 kgf
300 kgf20 mm50 mm15 mm54 deg435 kgf743 kgf55 deg735 kgf1281 kgf
300 kgf20 mm50 mm15 mm53 deg574 kgf950 kgf60 deg874 kgf1749 kgf
300 kgf20 mm50 mm15 mm51 deg878 kgf1382 kgf65 deg1178 kgf2787 kgf
300 kgf20 mm50 mm15 mm47 deg2090 kgf3067 kgf70 deg2390 kgf6987 kgf
300 kgf20 mm50 mm15 mm42 deg-3811 kgf5102 kgf75 deg-3511 kgf-13566 kgf
300 kgf20 mm50 mm15 mm33 deg-899 kgf1075 kgf80 deg-599 kgf-3448 kgf
300 kgf20 mm50 mm15 mm20 deg-473 kgf503 kgf85 deg-173 kgf-1984 kgf

This table shows the bearing forces for a similar load applied to a reversed bottom bracket.

Fdb2sBLrLARrR
300 kgf20 mm50 mm15 mm0 deg120 kgf120 kgf0 deg420 kgf420 kgf
300 kgf20 mm50 mm15 mm18 deg109 kgf115 kgf5 deg409 kgf411 kgf
300 kgf20 mm50 mm15 mm35 deg99 kgf121 kgf10 deg399 kgf405 kgf
300 kgf20 mm50 mm15 mm49 deg89 kgf134 kgf15 deg389 kgf402 kgf
300 kgf20 mm50 mm15 mm59 deg79 kgf152 kgf20 deg379 kgf403 kgf
300 kgf20 mm50 mm15 mm66 deg68 kgf170 kgf25 deg368 kgf407 kgf
300 kgf20 mm50 mm15 mm72 deg58 kgf188 kgf30 deg358 kgf413 kgf
300 kgf20 mm50 mm15 mm77 deg47 kgf205 kgf35 deg347 kgf424 kgf
300 kgf20 mm50 mm15 mm81 deg36 kgf219 kgf40 deg336 kgf438 kgf
300 kgf20 mm50 mm15 mm84 deg23 kgf230 kgf45 deg323 kgf457 kgf
300 kgf20 mm50 mm15 mm88 deg9 kgf237 kgf50 deg309 kgf481 kgf
300 kgf20 mm50 mm15 mm-89 deg-6 kgf241 kgf55 deg294 kgf513 kgf
300 kgf20 mm50 mm15 mm-84 deg-24 kgf241 kgf60 deg276 kgf553 kgf
300 kgf20 mm50 mm15 mm-79 deg-44 kgf236 kgf65 deg256 kgf605 kgf
300 kgf20 mm50 mm15 mm-72 deg-70 kgf227 kgf70 deg230 kgf673 kgf
300 kgf20 mm50 mm15 mm-62 deg-102 kgf217 kgf75 deg198 kgf766 kgf
300 kgf20 mm50 mm15 mm-47 deg-145 kgf211 kgf80 deg155 kgf895 kgf
300 kgf20 mm50 mm15 mm-25 deg-205 kgf226 kgf85 deg95 kgf1088 kgf

Isis Notes

The Isis spindle/crank joint design has several changes which should reduce the rate of catastrophic (sudden and dangerous) failures. However, there are numerous reports of rapid bearing failure. Some riders report bearing failures as often as every few months.

What follows are several guesses (NOT backed by any evidence) of why the failure rates are high and what to do about them.

First, the spindle diameter is larger, about 22mm for Isis vs. 17mm for standard. Although the larger diamter should help reduce failures, it also means there is less space for bearings. For a given bearing outside diameter, a small number of large balls is typically more durable than a large number of small balls [1]. Thus, a large spindle automatically puts the bearings at a disadvantage.

Second, the bearings are often pressed in to aluminum cups. Although light, aluminum cups do not support the bearing as well, leading to larger outer race deflections. That will tend to shorten the outer race life, though it may improve the inner race life.

Third, bearings are often pressed lightly in to the cups and pressed lightly on the spindle. A loose fit makes assembly and disassembly much easier, but for best life, radial bearing assemblies should be under high radial preload [1].

Fourth, radial cartridge bearings fail much more quickly if they are used misaligned. Although the larger Isis spindle is less flexible than the square taper spindle, relatively large crank loads may still produce some flexing. Also, the bottom bracket threads and shell face are likely to be misaligned -- using threads to ensure alignment is always risky. Note also that bottom bracket thread/shell alignment is outside the control of the bottom bracket manufacturer. (I believe, though I am not sure, that small-ball radial bearings of a given cartridge outside dimension are more sentivie to misalignment than large-ball bearings of the same outside dimension.)

Fifth, all bearing problems are exagerated by the introduction of dirt. Most bicycle bearings, if used a while, show dirt contamination.

One solution is to use an oversize shell. There are numerous attempts to introduce new ``standards'', all of which are remarkably close to the existing oversize standard for one-piece cranks. The manufacturers could do us all a favor by minimizing the introduction of gratuitous standards; we can do ourselves a favor by not buying them when they are introduced.

Pressing the bearings in to steel cups should improve outer race durability. It would be useful to examine some failed bearings to determine where the failure usually starts.

Pressing the bearings tightly in to cups and on the spindle should improve bearing life.

Using special bearing cartridges might help with misalignment. Standard cartridges are designed for high radial capacity and limited tolerance for misalignment. A bearing which is more tolerant of misalignement would have a lower radial capacity, but the radial capacity might still be suffcient, and at the same time be more durable in the face of misalignment.

Better seals -- labyrinth seals, most likely -- would help to delay dirt-related problems.

Finally, note that doubled bearings seem to be of dubious benefit in improving bearing life. Doubled bearings have a higher radial load capacity, but are also more sensitive to misalignment and also more sensitive to axial assembly mis-spacing. It seems likely that using a double-row bearing would bring many of the benefits of using a pair of bearings and at the same time would be less sensitive to msialignment.

Other Notes

The spindle (cone) typically fails first before the cup. The spindle shape is convex in one direction, making a saddle shape where it contacts each ball. In contrast, the cup is convex, so the ball fits the shape better. The spindle shape means the spindle has less ball contact area and higher point loads than the cup.

The bearings typically outlast the spindle because they have no features and so can be manufactured in large volume to better tolerances and with more uniform material.

References

[1] Tedric A. Harris, ``Rolling Bearing Analysis'', 4. Edition - March 2001 1086 Pages, Hardcover. ISBN 0-471-35457-0 - John Wiley & Sons.