The paper offers a synthesis of some important applications in the aria of analysis and processing of the reliability data of the production equipments documented by computer assisted specific statistical methods and regression model

Adrian Stere PARIS

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 Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012  Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     401  STATISTICAL METHODS AND THE RELIABILITY OF PRODUCTION EQUIPMENTS  Dr.ing. Adrian Stere PARIS, Univ. Politehnica Bucharest, email: s_paris@clicknet.ro   Abstract The paper offers a synthesis of some important applications in the aria of analysis and processing of the reliability data of the production equipments documented by computer assisted specific statistical methods and regression models.   Key words: reliability, statistical methods, production equipments  1. Introduction  Engineering education is traditionally concerned with teaching how manufactured products work: the ways in which products fail, the effects of failure and aspects of design, manufacture, maintenance and use which affect the likelihood of failure are not usually taught. Engineering education is basically deterministic, and does not usually pay sufficient attention to variation. Yet variability and chance play a vital role in determining the reliability of most products. Understanding the laws of chance and the causes and effects of variability is therefore necessary for the creation of reliable products and for the solution of  problems of unreliability [3]. Reliability of manufacturing equipment has a large impact on throughput and productivity.  Hitherto, attempts at modeling manufacturing equipment reliability has concentrated on the use of analytical models or high-level simulation models. At the same time it is important to have available different stastical methods for practical use. The expansion of software applications, many of them free, render possible their use at large scale in engineering fields, and particularly in manufacturing. Focus must be on reliability and not cost, because if reliability starts to improve the cost will definitely go down and it cannot be the other way around. There will be times that focusing on cost will hurt reliability, a lesson that we all should reflect upon.   1. New statistical applications  One important problem in the study of different caharacteristics of production equipments is to focus on the most importants. Discriminant analysis offers a possible way in separation of groups: description of group separation, in which linear functions of the variables (discriminatory functions) are used to describe or elucidate the differences between two or more groups. The goals of descriptive discriminate analysis include identifying the relative contribution of the p variables to the separation of the groups and the possibility to find  out the optimal plane on which the points can be projected to best illustrate the configuration of the groups [6]. The Principal Component Analysis is a standard technique to reduce multivariate data sets in a subspace of small dimension, frequently a tri-, respectively bivariate one. The number of observable attributes gives the dimension of the initial representation space of the objects.    Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012  Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     402 As illustration - an application from the agricultural machinery, concerning selecting usual dehusking equipment from Romanian market, with seven important mechanical and technological properties: 4 agricultural equipments (objects), each of them having 7 attributes(Table1) [7].   Table 1.  Bran finishers characteristics Characteristics FTO FT 30/60  FT 40/80 BRAN BRUSH C1 Mean yeld capacity, kg/h 687 275 550 650 C2 Necessary area for the equipment, m2 0.93 1.064 1.73 1.322 C3 Installed power, kW 4 2.2 5.5 4.4 C4  Equipment mass, kg 285 320 650 530 C5 Dependability  coefficient 0.92 0.85 0.88 0.83 C6 Air flow for aspiration, m3/min 5 3.5 4.5 5.5 C7 Specific loading 85 24.5 27.5 175                      Figure 1. Correlation circle   Table 2. The eigenvalues of the model     Characteristics/  Factors F1 F2 F3 Eigenvalue 3.463 2.131 1.406 Variability (%) 49.471 30.446 20.083 Cumulative (%) 49.471 79.917 100.000  Variables (axes F1 and F3: 69,55 %)Var7Var6Var5Var4Var3Var2Var1-1-0.75-0.5-0.2500.250.50.751-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1F1 (49,47 %)F3 (20,08 %) Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012  Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     403 The results of the analysis are presented in tab. 2 and fig.1 [7]. In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence.Correlation, Covariance and their coefficients are frequently applied in the study of field quality and reliability data. It is important to study the dependence between deviations of working accuracy, namely deviations from cylindricity of the work piece in a normal lathe, and time. In this respect, measurements were performed [2] on 47 pieces of standard form, processed on the same type of normal lathes, recording the actual working time for each machine, and the time since they were put into service on the considered experiment. The evaluation of the correlation coefficient rendered possible the calculation of the technological reliability [2].    3. Regression models   3.1. Regression of reliability data    The reliability community has become well experienced in fitting of survival distributions, the use of design of experiments (DOE) and the associated general linear model (linear regression and analysis of variance methods) approach to analysis. An opportunity to calculate the regression curve for the reliability field data is the software CurveExpert, a comprehensive curve fitting system for Windows. It employs a large number of regression models (both linear and nonlinear) as well as various interpolation schemes to represent data.    Fig.2. Adapted beta hazard-rate function   As example for reliability regressions here is used the software LAB Fit [10], with the main application Curve Fitting (nonlinear regression - least squares method, Levenberg-Marquardt algorithm -, almost 500 functions at the library with one and two independent variables, functions finder, option: write fitting function with up to 150 characters, 6 independent variables and 10 parameters). An example for an adapted beta model for the hazard rate function h(x), as representative reliability form, is presented in fig. 2:  ]     Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012  Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     404 3.2. Technological reliability models   To emphasize the problem of accuracy for manufacturing equipments it was introduced and developed the term of technological reliability [2, 5], that is quantitatively defined at t moment as the probability of a manufacturing equipment (namely a machine-tool) to maintain its working accuracy limits by the time t: this means to check the machine-tool accuracy at different time moments and establish the corresponding function of technological reliability, as example for a family of lathes [5]. The results are processed using multivariate data analysis, especially correlation theory and regression analysis.  3.3. Logistic regression application   Logistic regression analysis (LRA) extends the techniques of multiple regression analysis to research situations in which the outcome variable is categorical. Interesting developments in the reliability field are the asset health reliability estimation based on condition data, the prognostic algorithm for machine performance assessment and its application [8], etc. An example (Fig. 3), uses bearing failure simulation data and experiment run-to-failure data[1].    Fig.3 Schematic diagram of machine degradation assessment model [1]  One-dimensional feature, namely kurtosis, is calculated initially and represents the information of bearing from normal to failure condition. Failure degradation is calculated using the LR method for case of simulated data and case of experiment data. The results are regarded as target vectors of failure probability. RVM is used for training the run-to-failure kurtosis data and target vectors of failure probability and predict the component. To evaluate the training performance, root mean square error (RMSE) and correlation (R) are used [1].  4. Reliability and redundancy  Redundancy is a common approach to improve the reliability and availability of a system.  Adding redundancy increases the cost and complexity of a system design and with the high reliability of modern electrical and mechanical components, many applications do not need redundancy in order to be successful. However, if the cost of failure is high enough, redundancy may be an attractive option [12n]. Against the above description, in the case of manufacturing systems it must be considered the technological and production aspect. A new concept, the manufacturing redundancy, should be introduced, to quantify the overlap of manufacturing targets by more complex equipments, with multiple working possibilities.      Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012  Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     405 5. Conclusions  A few statistical methods of analysis and processing the reliability data are presented in the paper. A concise overview of some statistical software with life data analysis is complementary presented. New applications of more complex methods as Principal Component Analysis and Logistic Regression Analysis are detailed. It is highlighting the concepts of technological reliability and manufacturing redundancy, for real interest in production development and reliability.      References  1. Caesarendra, W., Widodo, A., Yang, B., Application of relevance vector machine and logistic regression for machine degradation assessment, Mechanical Systems and Signal Processing 24 (2010) 1161–1171 7. Ionescu, A., Târcolea, C., Paris, A., S. - Metodă de stabilire a criteriului de calcul al fiabilităţii tehnologice la maşinile-unelte. În: Construcţia de maşini, 1976, vol.23, nr.9,  p. 485-487  3. O'Connor,P.,D.,T.,- Practical Reliability Engineering, 2002, ISBN: 978-0-470-84463-2  4n. Paris, A., S. - Models in mechanical reliability data- 3rd Symposyum „DURABILITY AND RELIABILITY OF MECHANICAL SISTEMS” Univ C. Brancusi Tg. Jiu, mai 2010, In: Fiability and Durability, no. 1(5)/2010, Ed.Academica , Tg. Jiu, ISSN 1844 – 640X p.81-86 5. Paris, A,. S., Târcolea, C., - Regression models applied to manufacturing systems, (2010),  Proceedings in Manufacturing Systems, Editura Academiei Române, vol.5, nr.4,  pp. 249-253.  6. Târcolea, C., Paris, A., S.- Discriminant Analysis and Applied Regression The International Conference of Differential Geometry and Dynamical BSG PROCEEDINGS 18, (DGDS-2010) August 2010, Bucharest, Geometry Balkan Press, ISSN 1843-2654 (printed version) pp. 221-226, ISSN 1843-2859 (online version), 2011, pp.95-100 7. Târcolea, C., Paris, A, S., Voicu, P.- Principal Component Analysis Applied to Agricultural Equipments, Tarım Makinaları Bilimi Dergisi (Journal of Agricultural Machinery Science)  2011, 7 (3) pp.305-308  8. Yan, J., Koc, M., Lee, J., A prognostic algorithm for machine performance assessment and its application, (2004), Production Planning & Control, Vol. 15, No. 8, Dec., p796–801  9.*** http://www.ni.com/tutorials/ National Instruments Tutorial, Redundant System Basic Concepts,  Publish Date: Jan 11, 2008  10. *** http://zeus.df.ufcg.edu.br/labfit/    

STATISTICAL METHODS AND THE RELIABILITY OF PRODUCTION EQUIPMENTS

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STATISTICAL METHODS AND THE RELIABILITY OF PRODUCTION EQUIPMENTS

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