1 Basic concepts of mechanical reliability design

Reliability is defined as: the ability of a product to perform a specified function under specified conditions and within a specified time. Reliability metrics generally include reliability, failure rate, and mean time between failures. The most commonly used reliability among them has the following characteristics: Reliability is a function of time, and the reliability decreases with time; Reliability is used to express the statistical characteristics of a large number of products, rather than the reliability of a single or a few products ; For the reliability of individual products, it can be expressed in terms of probability.

Mechanical reliability can generally be divided into structural reliability and mechanism reliability. Structural reliability mainly considers mechanical structural strength and failures such as fatigue, wear and tear caused by load; mechanism reliability mainly considers self-locking, crawling, jamming and other faults caused by kinematic problems during the action process of the mechanism .

Mechanical reliability design is a modern design method to ensure that machinery and its components meet the given reliability index, which is divided into qualitative reliability design and quantitative reliability design. Due to differences in products and compositions, common methods of mechanical reliability design include failure prevention design, simplified design, derating design, redundancy design, environmental resistance design, ergonomic design, robust design, probabilistic design, trade-off design, Analog design, etc. The main feature of the commonly used quantitative reliability design is to regard the design parameters (such as load, stress, strength, life, size, etc.) involved in the conventional design as random variables that conform to a certain distribution law, and then according to the reliability of the product. According to the requirements of the degree index, the main parameters and dimensions of the products and parts are designed by the probability method.

2 Features of the reliability design of rolling bearings

Rolling bearings are key or important parts in machinery and have always been in an important position in the field of reliability application technology. They are typical products of mechanical reliability design. Its main features are:

(1) The reliability design of rolling bearings mainly focuses on the structural reliability design, usually quantitative reliability design is adopted, and the method used is the probabilistic design method. The design parameters (or variables) are based on the fatigue life, and the reliability metrics generally only involve reliability .

(2) Rolling bearings are one of the earliest mechanical products to be designed for reliability. For example [1]: Stribeck proposed the calculation method of bearing load capacity and the concept of infinite life as early as 1901; Palmgren combined the load capacity with the total bearing speed in 1924, which directly included the concept of bearing life; Lundberg and Palmgren In 1947, the rating life (90% reliability life) and the basic dynamic load rating calculation method were given, which were incorporated into the ISO international standard in 1962 and are still in use today.

(3) The conventional design of rolling bearings is reliability design.

In the national standards of many countries, especially in the ISO international standard, the definition and calculation method of the rated dynamic load and rated life of the bearing are determined, and they are standardized and become a general criterion. Special design, advanced design, new design, modern advanced design, etc., the conventional design and traditional design of bearings are reliability design. It is an indispensable and important content in bearing design to calculate and check the rated life of specific reliability or the corrected life of higher reliability according to the method specified in the standard.

4 Reliability design method of rolling bearing

1 Distribution of bearing fatigue life

The probabilistic design method used in the reliability design of rolling bearings is based on the theory of stress-strength interference. In the stress-strength interference theory, the generalized stress refers to any factor that causes failure, and the strength refers to any factor that prevents the failure from occurring. According to the working characteristics of the bearing, “stress” refers to the contact stress or bearing load between the rolling element and the raceway, and “strength” refers to the fatigue strength or bearing load capacity (load capacity) of the rolling element or raceway material.

Bearing fatigue is a typical damage accumulation failure. Due to the complex accidental factors affecting failure, the dispersion of bearing fatigue life is extremely large. Under the same working conditions of the same type and batch of bearings, the longest life and the shortest life may be different. several times or even dozens of times. Therefore, the bearing fatigue life cannot be determined by a fixed value method, but must be processed by probability theory and mathematical statistics theory.

A large number of experimental studies and related theoretical analysis have proved that the bearing fatigue life obeys the Weibull distribution, and it is also one of the typical application objects of the recognized Weibull distribution.

2 Calculation method of bearing rating life

The distribution probability curve of bearing fatigue life is shown in Figure 1. It can be seen that there is no significant set of bearing life anywhere.

In order to facilitate the characterization of characteristic quantities, it is necessary to select some life at a certain reliability level (called reliable life) to describe the life characteristics of the bearing.

For most applications, 90% reliability is sufficient. Therefore, setting L10 as the rating life can be easily used to evaluate the bearing life as the basis for selection. In the early days, the median life L50 was also used to characterize bearing life, but the reliability of 50% was obviously too low, so it was gradually abandoned. Equation (9) not only gives the relationship between bearing life and load with an explicit function, but also gives the bearing life and the main parameters of the bearing internal structure design, as well as the part geometry. The relationship between manufacturing accuracy, material and other related parameters. Taking the calculation formula of radial dynamic load rating Cr of radial ball bearings as an example, we can understand the relationship between them. Dw≤25． At 4 mm, it is the rated factor of high-quality hardened bearing steel commonly used in contemporary times and good machining methods, and the value varies with bearing type and design; fc is the factor related to the bearing part geometry, manufacturing accuracy and material; i is Number of rows of balls in the bearing; α is the nominal contact angle of the bearing, (°); Z is the number of balls in the bearing; Dw is the diameter of the ball, mm. Using the bearing rated life calculation formula, for a given bearing design and load conditions, it can be checked and calculated whether the bearing meets the life requirements under the reliability. or according to the set rating life

and load conditions, design or adjust the main parameters of the bearing internal structure design.

3 Calculation method of bearing corrected rating life

For applications with a reliability higher than 90%, L10 is generally still used as the basis, and the bearing life at the corresponding reliability is obtained through correction calculation, that is, Ln = a1 L10, (13) where: Ln is the bearing correction rating life , × 106 r; a1 is the reliability life correction factor. In the international standard ISO 281:1990 [5], a1 adopts a 2-parameter Weibull distribution.

4 Infinite life design of rolling bearings

For general machinery, the life design criterion is [9]: when it belongs to a high-cycle fatigue problem, that is, when the stress level of the component or system is low and the number of stress cycles is high (such as drive shafts, vibration components, etc.), it can be carried out infinitely. Design for life, whose safety is controlled by stress; when low cycle fatigue dominates, i.e. when the component or system is operating at high stress levels and the number of stress cycles is low (eg aircraft structures, heavy machinery components, etc.), Then a limited life design should be carried out, and its safety is controlled by the life.

Due to the point and line contact between the rolling element and the raceway in the rolling bearing, the contact stress level is relatively high, usually around 2 000 MPa, and it can reach more than 3 000 MPa under severe load conditions, and the number of stress cycles is also high. Therefore, the bearing fatigue life is mainly designed according to the limited life requirements. In the early stage of the development of the bearing industry, the concept of infinite bearing life was proposed and applied. For example, literature [1] believes that if the load on the bearing is less than its load capacity (the bearing load when the maximum contact stress between the rolling element and the raceway is equal to the specified strength of the material), the bearing may be used permanently. But later, more theories believed that even if the bearing is installed correctly, lubricated well, and used properly, it will eventually fail due to fatigue due to repeated alternating stress, and it cannot run forever. Therefore, bearing life is only possible to a limited life.

The new life theory published by the Swedish SKF company in 1984 reintroduced the concept of bearing with infinite life: under ideal conditions of lubrication, cleanliness and other operating conditions, if the load on the bearing is lower than the fatigue load limit Pu, it will not be affected. Fatigue damage occurs, i.e. bearing life is infinite. For conventional bearing steels, the Pu-based contact stress is about 1 500 MPa. When designing the bearing with infinite life, the specific Pu can refer to SKF’s bearing product catalog [10]

, can also be estimated from the bearing static load rating C0.

The infinite life and the minimum life are the life with 100% reliability, but the preconditions for determining the two are different: under normal load and operating conditions, the minimum life of the bearing is about 0. 05L10 ; If the contact stress is very low (less than 1 500 MPa) and the operating conditions are ideal, the bearing may reach infinite life.

5 Examples of Reliability Design of Rolling Bearings

Example: Deep groove ball bearing 6204, radial basic dynamic load rating Cr = 12 000 N, radial static load rating C0 = 6 500 N, radial equivalent dynamic load Pr = 2 000 N, speed is 1 500 r/min. Try calculation:

(1) Rated life;

(2) The reliability when the bearing life is 2 000 h;

(3) Bearing life with a reliability of 99%;

(4) The minimum life is the life of 100% reliability;

(5) At what level the equivalent radial load Pr is, the bearing has infinite life.

6 Conclusion

The distribution law of the fatigue life of rolling bearing determines that it must adopt probabilistic design and reliability design. Although bearing reliability design has long been a traditional design and conventional design, many concepts and contents are still developing, such as high reliability life, infinite life, etc. are the latest achievements. The factors affecting bearing life are very complex. Although reliability design can better reveal the essence of the problem and bearing life theory is relatively mature, the current bearing reliability design is still only an “estimation” due to the randomness of many factors. Therefore, it is still a long-term task to improve the bearing reliability design level to accumulate more data through reliability tests and establish a more complete mathematical model in order to predict and evaluate the bearing life more accurately.

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