Assessment for the quality of rolling bearing parts based on fuzzy theory

There are many factors leading to rolling bearing failure, and the issue of parts quality like roller error and inner ring error is one of the main influence factors causing premature failure of the bearing. Assessment for bearing parts quality can obviously eliminate or minimize the chance of potential failure, and can improve the service life of bearing, wear-resisting property and reliability. For the sake of the assessment for the rolling bearing parts quality, a new fuzzy comprehensive evaluation method of parts quality was brought forward to account for it. Based on fuzzy theory and mathematical statistics, bearing parts quality evaluation model was established. Fuzzy comprehensive evaluation can objectively and truly reflect the quality of the bearing parts. That we synthetically evaluate the parts quality by the aid of the fuzzy comprehensive evaluation model has great value in theory and practice, and lays a new foundation for the quality evaluation of bearing parts. Then, with the condition of poor information, we take type 30204 tapered roller bearing roller quality evaluation for example to illustrate the application of the model, and the results show that using this model to assess the quality of parts is feasible.


INTRODUCTION
Rolling bearing is an important support for shafts and other rotating components, and it plays an important role in the normal operation of the equipment (Bana et al., 2007;Rho et al., 2005;Ju et al., 2013).It is hoped that information on the roller bearing performance analysis will be useful for the early detection of hidden danger of degradation and the failure of the whole product performance.Thus, greater attention has been paid to evaluate the performance of rolling bearings, with the help of many new findings.For example, gray chaos evaluation model constructed by Xia et al. (2010) for prediction of rolling bearing friction torque; Kovarskii et al. (2010) assessed effectiveness of using rolling bearings in a low-noise electrical engineering; using fuzzy set theory and chaos theory, Sun et al. (2012) evaluate the rolling bearing vibration; Sochting et al. (2006) gave an evaluation of the effect of simulated launch vibration on the friction performance and lubrication of ball bearings; Zaretsky and Branzai (2005) analyzed the effect of rolling bearing refurbishment and restoration on bearing life and reliability.Yu and Yang (2011) made the fatigue failure analysis for a grease-lubricated roller bearing from an electric motor; and a fuzzy chaos method is proposed by Xia and Chen (2013) to evaluate the nonlinearly evolutionary process of rolling bearing performance.
However, so far comprehensive evaluation of the bearing parts quality still in a puzzle, for the fact that the fuzziness and uncertain relation between the quality and its influence factors.For this reason, a new method, viz., fuzzy comprehensive evaluation is proposed to evaluate the bearing parts quality and to lay a new foundation for the quality evaluation of bearing parts.Assessment for bearing parts quality can obviously eliminate or minimize the chance of potential failure, and can hence improve the service life of bearing, wear-resisting property etc. Usually, for quality evaluation of bearing parts, a certain factors index, viz., surface quality, dimensional accuracy or shape and position errors is detected.This single factor evaluation method cannot truly reflect the quality of the bearing parts.The quality of a part should be the comprehensive reflection of three output precision, viz., surface quality, dimensional accuracy or shape and position errors.Based on fuzzy theory (Liu et al., 2004;Lu and Sun (2007), this paper analyzes the parts quality influence factors and creates a comprehensive assessment model of rolling bearing parts quality.An experimental investigation on the roller quality of roller bearing is conducted to illustrate the application of the proposed model.

FUZZY EVALUATION MODEL
Fuzzy comprehensive evaluation (Liu et al., 2004) is an effective multifactor decision-making evaluation method for comprehensive evaluation of the product quality affected by many factors, which has been widely used in the product evaluation.Here, we develop the characteristics of influence factors and the establishment of the model.

Factors set
Factors set are the collection of all the factors affecting the quality of products.Fuzzy evaluation for the quality of rolling bearing parts needs to determine the evaluation factors of quality, and all of the quality evaluation factors constitute the set.The factors set of bearing parts quality can be defined as where u i is the ith influence of product quality, m i , , 2 , 1   ; Usually, they are roundness, convexity, roughness, base surface roughness, etc.

Judgment set
Fuzzy evaluation for rolling bearing parts quality needs evaluation level, which are parts quality processing requirements by national standards, mechanical standards or industry standards, and the parts quality evaluation set can be constituted by all of quality evaluation rating.The ith evaluation set of the bearing parts quality can be defined as where v ij is the judgment standard of the ith quality influence factors in level j,

Single factor evaluation
Suppose that there are m factors in the quality factor set, through the performance test, a part's quality data sequence of the ith influence factors u i can be described as where i is the serial number of quality influence factors, ; u i (k) is the kth test status value of the quality influence factors.The composite matrix U of the quality affecting factors test data sequence made up of parts quality affecting factors sequence u i can be obtained as: From Equation (2), judgment standard matrix V made up of the evaluation standard value of the parts quality influence factors can be expressed as: Let v ij be level evaluation standard value of the ith bearing parts quality influence factor in grade j (serial number j = 1, 2,...,J), u i (k) is the test status value.If a status value of rolling bearing quality influence factors meets the following condition (6) then the status value u i (k) corresponding to quality grade is level j.According to the above description, fuzzy transformation can be made for each influence factors of parts quality, and the fuzzy mapping relation can be defined as: Fuzzy relations can be induced by the fuzzy mapping f as: where f ij is number of a test value of ith quality influence factors in the scope of level j.Fuzzy matrix R composed of R f can be expressed as: R can be said to be a single factor evaluation matrix.

Weighting set
The selection of the affecting factors weighting A is more complex.We can consider the mathematical expectation (average) and dispersion scope (standard deviation or range) of each output factors in the test results, and also can consider technical requirements of the parts machining process.The ith influence factor weight can be defined as: where u Ri is the ith range or standard deviation of the quality influence factors in the n test data; u mi is the average of the ith quality influence factors u i .
From Equation ( 11), the weight set of components quality affecting factors can be attained as:

Fuzzy comprehensive evaluation model
Bearing parts quality comprehensive evaluation model can be defined as where "  " is the fuzzy operator, with

EXPERIMENTAL INVESTIGATION
Taking the roller (type 30204) as an example to illustrate the application of the model, and the number of the bearing samples is 30, viz., n=30.There are many factors affecting the quality of roller, in this paper mainly considering the processing quality parameters of angle error, crown diameter error parameters, namely, m = 8, and related symbols and their meaning are shown in Table 1.
In the scene of the production, we randomly selected 30 sets of tapered roller bearing, and measured influence factors status values after numbered and disassembled.Main technical parameters and measuring instrument are shown in Table 2. Then we respectively measured the influence factors value of roller after disassembled, and recorded the various influence factors measured value.According to bearing parts processing requirements of the national standard, each evaluation standard of influence factor, evaluation standard score 6, influence factors judgment set V composed of the evaluation standard value can be attained as From Equations to (10), fuzzy transform for the record test data, obtaining fuzzy matrix R  A= (0.0585, 0.1749, 0.0307, 0.1712, 0.3071, 0.0880, 0.1057, 0.0638) From Equation ( 14), the comprehensive evaluation for roller quality can be expressed as B= (0.2000, 0.1712, 0.3000, 0.2000, 0.1057, 0.1333) According to the principle of maximum membership, under the given conditions, the bearing roller belongs to grade 3 for the consideration factors.The experimental investigation on the roller shows that the evaluation value is in very good accordance with the practice value, and the roller can meet the accuracy and quality requirement of the roller bearing.

CONCLUSIONS
Based on fuzzy theory, the paper puts forward a new method of rolling bearing parts quality evaluation, and establishes a fuzzy evaluation model; with the condition of poor information, the experimental investigation on bearing roller quality evaluation shows that the model can be used to evaluate bearing roller quality only with small samples and without any prior information of probability distributions.

R
The weight of all the factors can be calculated by Equation (11), thus getting weighting set A

Table 1 .
symbols and meanings.

Table 2 .
Measuring instrument of parameter.