The history is cornerstone of the
present time as well as challenge to the future.
The history is cornerstone of the
present time as well as challenge to the future.
5.1 General information
5.2 Roller bearing reliability
5.3 Dynamic Load Capacity
5.4 Durability
5.5 Basic durability equation
5.6 Modified durability equation
5.7 Durability according ZKL
5.8 Equivalent dynamic load
5.9 Effect of temperature
5.10 Static Load Rating
A properly installed and lubricated roller-contact bearing will operate under normal conditions, i.e. absent extreme speeds and temperatures, until it fails due to fatigue of materials at acting surfaces. Repeated stress on the contact surfaces between roller-contact surfaces and rings will manifest after a certain period depending on the magnitude of load as a stress fracture. This will expand until a part of the bearing ring material or roller element material breaks off (pitting) and causes failure. Many bearings are also discarded for other reasons than material fatigue, but these failures can be avoided if the bearing is treated properly, if it is properly installed, lubricated, and overloading is avoided.
When a certain number of identical bearings are tested for fatigue under specified operating conditions (load and rpm), there is a large variance of durability between individual bearings. In a group of 30 or more bearings, the ratio between the shortest and longest durability can be 20-fold or more. A durability variance curve can be drawn for each tested group of bearings that illustrates the relationship between the durability and the number of bearings, which were discarded.
The required bearing size is determined on the basis of externally acting forces and based on the durability and reliability demands of the seated bearing. The size, direction, purpose, and nature of the bearing load as well as the revolution operating speed are determinant when selecting the bearing type and size. Meanwhile, other special or important conditions of each individual case must be considered, e.g. operating temperature, spatial allowances, ease of installation, lubrication requirements, packing, etc., which can affect the selection of the most suitable bearing. Various types of bearings may, in many cases, be suitable for the given specific conditions.
In terms of the action of external forces and the function of the bearing in the respective node or unit, we distinguish two types of roller bearing loads in bearing technology:
The durability limited by failure of a particular bearing component (bearing rings, roller elements, cage, lubricant and seal) is, in the first case, decisive for calculating bearing safety. In the second case, permanent deformities of functional surfaces at contact points between rolling elements and orbits is decisive.
The reliability of a group of apparently identical roller bearings, operating under identical conditions, is the percentage of the group, expected to achieve or exceed the specified durability.
The reliability of an individual roller bearing is the probability that the bearing will achieve or exceed the specified durability.
The equation for calculating durability includes the effect of stress induced by external loads, lubrication, and surface kinematics at the site of rolling contact. Including the impact of the comprehensive system of stress on bearing durability makes it possible to better anticipate the actual manner, in which a bearing behaves within a specific housing. International standards, such as e.g. ISO 281, are based on the theory of material fatigue at the site of rolling contact. One must keep in mind that a complete bearing can be considered as a system, the individual components of which (bearing rings, rolling elements, cage, lubricant and seal) have the same effect on durability and, in certain cases, are even a decisive factor in determining the bearing durability during operation. The optimal operating durability is theoretically achieved when all of the components achieve the same durability. In other words, the calculated durability corresponds to the actual operating durability if the operating durability of related components is at least as long as the calculated bearing durability. Related components in such case are the cage, seal and lubricant. The most important factor in practise is metal fatigue.
Dynamic load capacity is, according to ISO 281:1990, a constant invariable load that a bearing can theoretically carry at a basic durability of one million revolutions.
The dynamic load capacity Cr for radial bearings relates to constant, invariable, entirely radial loads. For thrust bearings, the dynamic load capacity Ca relates to the invariable, purely axial load acting in the bearing’s axis.
The dynamic load capacity Cr and Ca, whose magnitude depends on the bearing dimensions, the number of rolling elements, the bearing material and design, is provided in the table for each bearing. The dynamic load capacity values were determined in accordance with ISO standard 280. These values are verified on testing equipment and confirmed in operating results.
The numeric values specified in this catalogue apply for chrome steel bearings, heat treated to a minimal hardness of 58 HRC and normal operating conditions. NEW FORCE bearings display, among others, improved material properties and advanced manufacturing processes. To determine the dynamic load capacity in these bearings, thus requires the use of correction factors according to ISO 281. More information about these bearings is available in separate chapter 7.7.
It is the number of revolutions that a bearing lasts, before fatigue of one of its components occurs, which manifests as flaking of material. It is expressed either as the total number of revolutions or operating hours, or in vehicles, by the distance travelled (number of driven km).
The material is primarily responsible for significant variance in durability in a broader range of identical bearings tested under the same conditions. No material or bearing steel is entirely homogenous and contains certain weak points. If a weak point is located on the orbit, where large load (stress) is generated, then the durability of the bearing will be small. The durability is higher where the load is decreased. Poor material has a large amount of weak points and, in all likeliness, some of them lie in areas of greatest load. The variance of durability will thus be less in poor material and larger in first-class material.
Variance of durability is also affected by manufacturing tolerances of individual components. The tolerances of roller diameters and radiuses of raceways significantly affect loads on roller surfaces. For manufacturing reasons, the radial clearance in a bearing varies within a specific tolerance, and as such, it also affects the distribution of pressure on individual roller elements. The distribution of forces within the bearing in the same manner cause expansion and decrease the orbit diameter due to the prescribed placement of rings on the shaft and within the housing.
Adherence to the prescribed material composition, its purity, and heat treatment is also an important indicator of bearing quality. Large variances in durability of large quantities of identical roller bearings, testing under identical conditions is but a natural consequence of the specified individual influences. Current research shows that even the quality of lubrication, its purity, and quantity may significantly impact bearing durability. Lubrication is taken into account in the modified durability calculation, see further.
The results of performed durability tests and practical operating experiences indicate that identical bearings, operating under identical conditions, do not achieve the same durability. The term “durability” must thus be correctly defined.
The basic durability of a bearing is mathematically defined by the durability equation, which applies for all types of bearings.
L_{10} = [C / P]^{P} or C / P = [L_{10}]^{1/P}
L_{10} . . . . . basic durability | [10^{6} ot] |
---|---|
C . . . . . . . dynamic load capacity (the C_{r} a C_{a} values are specified in the product section of the catalogue) | [kN] |
P . . . . . . . bearing equivalent dynamic load capacity (the equations for calculating P_{r} and P_{a} are provided in the chapter Equivalent Dynamic Load Capacity and for each structural group of bearings) | [kN] |
p . . . . . . . ball bearing exponent | p = 3 |
. . . . . . . For cylindrical roller, needle roller, spherical-roller, and tapered-roller bearings | p = 10 / 3 |
The basic durability of a bearing is thus understood to mean the durability that 90 % of bearings achieve or exceed from a set of identical bearings, working under the same operating conditions. All standard durability calculations are performed for this reliability level. Mean durability Ls is the durability that 50 % of bearing from the same set achieve; it is about 5 times higher than the basic durability. In contrast, the durability achieved by 99 % of bearings is about one fifth when compared with the basic durability. The impact of the degree of reliability on the durability calculation is specified in chapter 5.6.
Table 5.1 lists the relationship of durability L10 in millions of revolutions and the corresponding C/P ratio. If the revolution speed is unchanged, then the durability can be calculated using the modified equation, which expresses the basic durability in terms of operating hours:
L_{10h} = [C / P]^{P} × [10^{6} / P = [60_{n}]
L_{10h} . . . . . basic durability | [h] |
---|---|
n . . . . . . . revolution speed | [min^{-1}] |
The relationship of the C/P ratio on basic durability L_{10h} and on the revolution speed for ball bearings is specified in table 5.2 and in table 5.3 for cylindrical roller, needle, spherical-roller, and tapered-roller bearings.
In road and rail vehicle axle supports, we can express the basic durability using the modified relationship in terms of kilometres driven.
L_{10km} = [C / P]^{P} × [π × D / 1000]
L_{10km} . . . . . basic durability | [10^{6}km] |
---|---|
D . . . . . . . wheel diameter | [m] |
In cases, when the required durability for the given housing is not provided in advance, we can appropriately use the values provided in tables 5.4 and 5.5.
The operating durability, as previously described, depends on many factors. Research and operating results demonstrated that greater durability can be achieved through thorough lubrication, when the roller elements are entirely separated by a layer of grease. It was further demonstrated that greater resistance against stress damage of materials is provided using advanced manufacturing processes. This technical advance was incorporated into standard ISO 281 as a modified durability calculation, which includes reliability a_{1}, material a_{2}, and operating condition a_{3} factors. Additional test results concluded that the impact of materials on operating conditions, in particular, lubrication, are in close correlation. This led to the merger of both factors into one a_{23}.
The modified durability is thus the modified basic durability which, aside from taking into account load, also considers the impact of bearing material components, the physical and chemical properties of the lubricant, and the temperature regime of the bearing operating environment.
L_{na} = a_{1} + a_{23} + L_{10}
L_{na} . . . . . . modified durability for reliability (100 - n) % and other than normal operating conditions [10^{6} ot]
a_{1} . . . . . . eliability coefficient for other than 90 % reliability, see table 5.6
a_{23} . . . . . material, lubricant, manufacturing technology, and operating condition coefficient, see fig. 5.3
L_{10} . . . . . basic durability [10^{6} ot]
The diagram in fig. 5.4 is used to determine the basic values of coefficient a_{23}.
The quality of the lubrication process is given by the extent of separation of the roller surfaces. Viscosity is a decisive factor for the formation of lubricant film, which is strongly related to temperature. The viscosity ratio, as follows, decides on the use of lubricant:
ν . . . . . lubricant kinematic viscosity at bearing operating temperature [m^{2} . s^{-1}]
ν_{1} . . . . . kinematic viscosity for the defined revolution speed and the given dimension of the bearing [m^{2} . s^{-1}]
We determine the ν and ν_{1} values based on the diagram found in fig. 5.4 and 5.5. In the diagram on fig. 5.3, line I applies for radial ball bearings that operate in a very clean environment. In all other cases, we select a lower a_{23} coefficient, proportional to the cleanliness of the environment, while a decreasing tendency is dependent on the structural group of the bearing in the following order:
Line II can be used to determine coefficient a_{23} for spherical-roller bearings that operate in a dusty environment. We recommend that these issues be resolved in consultation with the ZKL technical and consultation services department.
The use of the L_{10} calculation of basic durability as bearing performance parameter criteria has demonstrated, over many years, to be satisfactory. This calculation is associated with 90 % reliability in conjunction with the use of superior materials, a superior technological design, and under normal operating conditions.
Notwithstanding, many applications require that the calculation be performed for a different reliability level or for more precise lubrication and contamination conditions. It was determined, with the use of advanced high quality bearing steel, that under favourable operating conditions and when contact stresses fall below the limit values and provided that the bearing steel fatigue stress limit is not exceeded, a higher durability than L_{10} can be achieved. Under unfavourable operating conditions, on the other hand, the bearing durability can in fact be shorter than L_{10}.
A system approach of fatigue-related durability was applied when creating the method of calculating ZKL modified durability. The impact on the durability of the system (bearing) is described in the following text and considers the influence of variance and the interaction of mutually related factors on the overall life. These factors are demonstrated through increased contact stress in the contact area, which leads to decreased service life.
These factors are used in the modified durability equation.
L_{m} = a_{1} × a_{ZKL} × L_{10}
a_{1} . . . . . . reliability coefficient for other than 90 % reliability, see table 5.6
a_{ZKL} . . . . . modified life coefficient
L^{10} . . . . . basic durability [10^{6} ot]
Provided that the lubrication conditions, cleanliness of the environment, and other operation conditions are favourable, an advanced, high-quality bearing can, under a certain load, achieve infinite service life. The fatigue load limit for bearings manufactured from generally high-quality bearing material and workmanship is such a load, that the contact pressure exerted on roller elements in the bearing is approximately 1500 MPa. This stress value takes into account the additional stresses caused by manufacturing tolerances and operating conditions. Decreased product precision and quality of materials leads to a lower fatigue load limit.
The contact stress in many applications is greater than 1500 MPa. Such operating conditions lead to reduced bearing life.
The operating influences can be related to the applied stress and rigidity of the material.
Various influences on bearing durability are mutually dependant. Consequently, a systemic approach to calculating fatigue durability is entirely appropriate.
A theoretical explanation of how to incorporate additional influences, such as the radial clearance during operation and the variable stress on raceways from tilting, is explained in ISO/TS 16281.
The modified durability coefficient a_{ZKL} can be expressed as function
(fatigue load limit divided by the real stress σ, while considering all potential influencing factors).
If the actual stress decreases to fatigue stress limit, then a_{ZKL} asymptotically approaches infinity. Generally, the orthogonal shear stress is used as a fatigue criterion. The diagram on fig. 5.6 is also based on the shear fatigue limit.
Analogous to the C_{or} static load rating, defined in ISO 76, the fatigue load limit is defined as the load, during which the stress fatigue limit is reached at the most burdened point on the orbit.
The ratio σ_{u} / σ can then be estimated according to the ratio C_{or} / P and the modified life coefficient can be expressed as:
The following must be considered when calculating the C_{or} static load rating:
The modified durability coefficient takes into consideration the following:
The effect of bearing clearance and the effect of tilt on bearing durability is described in ISO/TS 16281.
The a_{ZKL} Fatigue life coefficient is derived from the following equation:
Factors e_{C} and κ adjust for contamination and lubrication conditions.
If the grease is contaminated with sold particles, notches may form in the orbit due to rolling. Stress points (concentrations) form later on these notches, which results in decreased bearing life. The given decrease in life caused by the contamination of lubricant is adjusted for in the e_{C} contamination factor.
Decreased bearing life caused by the effect of sold particles in the lubricant film depends on:
Approximate contamination factor values can be taken from table 5.7.
Detailed calculation of the contamination factor
Table 5.7 lists the approximate contamination factor values. If the situation requires the use of more detailed calculations, the more precise calculation, provided below, must be used.
A contamination factor may be established for the following types of lubricants:
Definition of the β_{χ} filtration ratio:
β_{χ} . . . . . . filtration ratio for particles of determined size x
n_{1} . . . . . . number of particles per unit of volume (100 ml) larger than x, prior to passage through filter
n_{2} . . . . . . number of particles per unit of volume (100 ml) larger than x, after passage through filter
The filter ratio determined the filter efficiency.
Circulating lubrication with on-line filtration
The β_{χ} filter ratio with particles of size x in μm according to standard ISO 16889 is the most influential factor when choosing the corresponding diagram.
Oil bath lubrication or circulating lubrication with off-line filtration
Mazání tukem
The effectiveness of the lubricant is primarily given by the degree of separation of contact elements. The formation of adequate lubricating film is subject to the given minimal viscosity that the lubricant must possess, when the application achieves its operating temperature. A requirement for the formation of lubricating film is specified by the viscosity ratio κ, which is defined as the ratio between the real (actual) kinematic viscosity ν and the reference kinematic viscosity ν_{1}. The kinematic viscosity ν is the viscosity of the lubricant, when the given lubricant achieves its operating temperature.
In order to create sufficient lubricating film, the lubricant must maintain a certain minimal viscosity at operating temperature. The bearing life may be increased by increasing the operating viscosity ν.
The reference kinematic viscosity can be determined from figure 5.4 or by using the following equations:
ν_{1} = 45 000 × n^{-0,8} × D_{pw}^{-0,5} for n < 1 000 rev/min
ν_{1} = 45 000 × n^{-0,5} × D_{pw}^{-0,5} for n >= 1 000 rev/min
D_{pw} = 0,5 × (d + D) is the bearing mean diameter
The modified durability coefficient a_{ZKL} may be easily determined from the following graphs:
The bearing in the structural node is exposed generally to acting forces of various magnitudes at various revolution speeds and with various periods of action. In terms of the calculation method, the applied forces must be recalculated at constant load, during which the bearing has the same durability as achieved under actual load. This recalculated constant radial or axial load is called equivalent load P, or P_{r} (radial) or P_{a} (axial), resp.
Constant load method
The external forces applied on the bearing do not change in size or in relation to time.
Radial bearings
If constant radial or axial forces simultaneously act on a radial bearing, the following equation for calculating the radial dynamic load applies:
P_{r} = X × F_{r} + Y × F_{a} | [kN] |
---|---|
P_{r} . . . . . . radial equivalent dynamic load | [kN] |
F_{r} . . . . . . . radial force acting on the bearing | [kN] |
F_{a}. . . . . . . axial force acting on the bearing | [kN] |
X . . . . . . . radial load coefficient | |
Y . . . . . . . axial load coefficient |
Coefficients X and Y are dependent on the ratio F_{a} / F_{r}. The values X and Y are provided in the table or in the commentary preceding each structural group, where further information is provided for bearing calculations of the respective structural group.
Thrust bearings
Thrust ball bearings can only transfer forces acting axially and the following equation applied for calculating the axial equivalent dynamic load:
P_{a} = F_{a} | [kN] |
---|---|
P_{a} . . . . . . axial equivalent dynamic load | [kN] |
F_{a} . . . . . . axial bearing load | [kN] |
Spherical-roller thrust bearings can also transfer certain radial loads, however, only when a simultaneous axial load is applied, while observing the following condition:
P_{a} = F_{a} + 1,2 × F_{r} | [kN] |
---|
Variable loading method
A real variable load, whose time course is known, is replaced by a mean intended load to enable calculation. This intended load has the same effect on the bearing as an actual variable load.
If a load acts on a bearing in a constant direction, whose size changes in relation to time, while the revolution speed is constant (fig. 5.25), we calculate the mean intended load F_{s} according to the equation
[kN]
P_{r} = X × F_{r} + Y × F_{a} | [kN] |
---|---|
F_{s} . . . . . . intended mean constant load | [kN] |
F_{s} = F_{1}, …, F_{n} . . .constant partial actual load | [kN] |
q_{i} = q_{1}, …, q_{n} . . .proportion of partially acting loads | [%] |
If a variable load acts on a bearing, while the rotation speed meanwhile changes (fig. 5.26), we calculate the mean intended load using the equation
[kN]
Provided that the actual load has a sinusoid shape (fig. 5.27), the mean intended load is given by
[kN]
If a variable load acts on a bearing along with variable rotation speed, the intended mean load is derived from the equation
[kN]
n_{i} = n_{1}, …, n_{n} . . . constant rotation speed during action of partial loads F_{1}, …, F_{n} | [min^{-1}] |
---|---|
q_{i} = q_{1}, …, q_{n} . . . proportion of partially acting loads and frequencies | [%] |
If the rotation speed only changes in relation to time, the intended mean rotation speed is calculated using the equation
[min^{-1}]
n_{s} . . . . . . mean rotation speed | [min^{-1}] |
---|
During oscillating motion with oscillating amplitude γ (fig. 5.28), it is easiest to substitute the oscillating motion by the notion of rotation, provided that the frequency of rotation is equal to the oscillating frequency. For radial bearings, we calculate the mean intended load using the equation
[kN]
kde
F_{s} . . . . . . mean intended load | [kN] |
---|---|
F_{r} . . . . . . . actual radial load | [kN] |
γ . . . . . . . amplitude of oscillation | [°] |
p . . . . . . . ball bearing exponent | p=3 |
For roller, needle roller, spherical-roller, and tapered-roller bearings
The supplied range of bearings is designated for use in environments with a temperature of up to 120 °C. Larger spherical roller bearings are manufactured, by default, for operation in temperatures up to 200 °C. The exception are particular double row spherical roller bearings with polyamide races and single row ball bearings equipped with seals (RS, 2RS, RSR, 2RSR), which may be used short-term in temperatures up to 150 °C. More information about these bearings is available in chapter 12 “Manufacturer data”.
Rolling bearings designed for higher operating temperatures are manufactured to ensure their required physical and mechanical properties and dimensional stability. Housing solutions at higher operating temperatures should be consulted with the supplier.
The and dynamic load rating values C_{r} and C_{a} provided within the tables of the publication must, in the case of higher operating temperatures, be multiplied by the coefficient f_{t}, as specified in table 5.9.
The radial static load rating C_{or} and axial static load rating C_{oa} for each bearing is specified in the table section of the publication. The values C_{or} and C_{oa} were determined by calculation according to international standard ISO 76.
The static load rating is the load that corresponds to the calculated contact stress in the roller element and raceway contact zone, under the greatest load.
This stress permanently deforms the rolling elements and raceways by approximately 0.0001 the diameter of the rolling element. The load is purely radial for radial bearings and purely axial within the bearing axis for thrust bearings.
The static load rating C_{or} is used for calculations, if the bearings
It is equally very important to check the safety in short-acting loads, such as e.g. shock loads and peak loads that act on a rotating bearing (dynamic load) or on a stationary bearing.
The maximum load that can act on a bearing should be used when calculating the equivalent static load of a bearing.
The equivalent static load is the recalculated radial load P_{or} for radial bearings and the axial load P_{oa} for thrust bearings.
P_{or} = X_{0} × F_{r} + Y_{0} × F_{a} | [kN] |
---|---|
P_{oa} = Y_{0} × F_{a} | [kN] |
P_{or} . . . . . . radial equivalent static load | [kN] |
---|---|
P_{oa} . . . . . axial equivalent static load | [kN] |
F_{r} . . . . . . . radial load | [kN] |
F_{a} . . . . . . axial load | [kN] |
X_{0} . . . . . . radial load coefficient | |
Y_{0} . . . . . . axial load coefficient |
Coefficients X_{0} and Y_{0} are specified in the table section of the publication. Detailed information is also provided here for determining the equivalent static load of bearings of a particular structural group.
In practice, the bearing safety under static load is determined from the ratio C_{or}/P_{or} or C_{oa}/P_{oa} and compared with the data in table 5.10, where the smallest permissible coefficient values s_{0} are specified for various operating conditions.
S_{0} = C_{or} / P_{or} and/or C_{oa} / P _{oa}
S_{0} . . . . . . safety coefficient under static load | [kN] |
---|---|
C_{or} . . . . . . radial dynamic load capacity | [kN] |
C_{oa} . . . . . axial dynamic load capacity | [kN] |
P_{or} . . . . . . radial equivalent static load or max. acting force F_{rmax} (fig. 5.29) under significant impact load, resp. | [kN] |
P_{oa} . . . . . axial equivalent static load or max. acting force F_{rmax} (fig. 5.29) under significant impact load, resp. | [kN] |
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ZKL is the biggest manufacturer of large-scale spherical-roller, special and split bearings in Central Europe.