Statistics of Failure and Weibull Analysis - Assignment Example

MECHANICAL SYSTEM RELIABILITY: Student Name: Student No. Course: Lecturer: Sponsor: Director of Engineering Widgets plc. Subject. : Mechanical Systems Reliability. Assignment: Statistics of Failure and Weibull Analysis. Date of Submission: Introduction. Background information. In the analysis of life data, also known as (Weibull analysis) , the practitioner main objective is to predict life of all products or samples in a given population. This is usually done by fixing a statistical distribution of life sample data, in our case the data from the 20 machines that were treated. This distribution set of data is used to approximate life behaviour of products, for example their probability of failure after a given time period and their reliability, the failure rate and their mean life. When conducting this experiment, you are required to: Collect sample life data for the product. Have a lifetime distribution that fits the sample data together with the model of life of the product. Approximate the parameters Plot graphs and the results that give estimate of the life behaviour of the product, i.e. mean life or reliability. Life data is the product life measurement. Many parameters can be used to measure product life and some of them include: cycles miles, hours and other metric. Since time parameter is commonly used to measure data life , its mostly called ‘times-to-failure’ and the description of the product life will be in reference to time . 1.1 Life data Models An engineering example Take for example that we have tested some gears and found that the percentage of tooth failures after specific load cycles is given in the table below. Cycles (100’s) Number of failures Percentage failures Sum of failures (%) < 240 0 0 240 – 269 1 9 9 270 – 279 4 36 45 280 – 309 3 27 73 310 – 339 2 18 91 340 – 369 1 9 100 370 + 0 0 Total 11 100 When we increase the number of samples, we also increase the number of the cells, and what we eventually get is a continuous curve for the failure distribution function F(t) and the pdf. F(t) is simply the area under f(t) as we increase the cycles . Therefore F(T), the value of F(t) at t=T is obtained from - (t is the x-coordinate): F(t) is known as probability failure or the failure distribution function. It is normally interpreted as the probability that a sample of components will fail by a given time frame. Of key interest is, the probability that a product will survive. This is called the Reliability Function, R (t), and it is defined as follows: i.e. the area marked in the diagram below If the area under the pdf is unity then we can say that Another interest we have is to know what is the rate of failure is at any time T. This explains that in the proportion of those still surviving will fail at time T. This is known as the instantaneous failure rate (also called the hazard rate) λ, and is defined as follows: The numerator is nf(T) and denominator is nR(T), therefore So we have: f(t) – Probability Density Function is the (the frequency distribution of failures) F(T) – Failure Distribution Function – is defined the probability that failure occurs by T R(T) – Reliability Function – is the probability of survival beyond time T λ (T) – Instantaneous failure rate, or hazard rate – rate of failure of at time T 1.2 Common Probability Density Functions of Failure Exponential or Infant Mortality This is seen when each device or machine has an equal Probability of failure after a given time span. Bathtub This shape has an exponential characteristic due to high initial failures, then the devices are highly reliable for a period. When they come towards the end of their lives, they start to fail because of ‘wear-out’, which is evident in the increased probability of individual failure. Gaussian with delay Here there are negligible failures for a period of time, and then at t=to , they begin to fail, most failing around t=tm Clearly the probability of an individual device failing is almost virtual zero until to , after which it starts to rise as indicated. Fitting models to failure data When we have performed some tests on certain components and have collected a sample failure data, then it can be used to try to fit a mathematical expression to the data. This enables us t to predict a number of important parameters. It’s a standard practise not to test the whole population of products or their components; only a sample is tested and we try to draw some conclusions about the whole population. (Clearly, it goes without saying that the smaller the sample, the less accurate our predictions for the population are likely to be). The most commonly used approach is to try to fit a Weibull distribution to the failure data. This distribution is given by: t is a statistical variable of e.g. no of cycles, time etc T is a specific value of t e.g. a specific number of cycles or a particular time β is the shape parameter; γ is the period from the start of the experiment during which there are absolutely no failures; and η is the scale parameter (=T- γ) Mechanical Systems Reliability Widgets plc. Engineering Company. Test Report Sponsor: Director of Engineering. Remit: We have been asked to test 20 of the new design of washing machine, and make recommendations on whether the 5-year guarantee currently offered can be sustained. Method 20 machines were put on accelerated test in our laboratories. The machines were tested over a period of four weeks, and all failed within this period. The failure data for 20 of the new design of washing machines is presented below, where the time to failure of each machine has been scaled up to estimate the actual life. Thus, for example, machine 3 actually lasted for 28 days, but because of the duty cycle employed, we can say that this represents 200 months of life in a typical household. Analysis: The advantage of Weibull analysis is that it useful when working with inadequacies in the data. Even poor Weibull plots are usually informative to practitioners trained to read them. Methods for Weibull can be described for: • Identification of mixtures failure modes, • Problems with the origin not located at zero, • Investigation of an aging parameter • Handling of data when some part ages are not known. • Construction of a Weibull curve even when no failures have occurred, The Weibull distribution gives the best fit of life data. Some distributions are included in the Weibull family either approximately or exactly, including the exponential, the normal, the Rayleigh, and at times even the Binomial and Poisson . Other distributions should be given consideration. 1.9 Engineering Frequency distribution table Other Methods and Problems Weibull analysis is remains main theme of this text, but there are some other types of data and some other types of problems that can be analyzed better using other techniques a part from math models. For e.g. the Gumbel distribution and both the maximum and minimum forms may have useful applications as well. An example is gust loads on aero plane that are modeled with a Gumbel maximum distribution. Certain organizations have the feeling that the Crow-AMSAA (C-A) model is more useful application than Weibull analysis. . It is more robust compared to Weibull, that is, it provides more reasonable and accurate results when the data has serious conflicting deficiencies. It works well even at extreme cases with mixtures of failure modes and missing parts of a data sample proportion. Weibull plots can only allow one failure mode at a given time period. It tracks major changes in the reliability which Weibull is unable to do. C-A remains the best tool in trending vital events for management, such as interruptions, outages, accidents in-flight loss of power and production cutbacks. The original C-A original objective was to track the reliability growth in R&D testing. It still stood the test of time and remains best practice for the application. It is also best procedure to be used in predicting claims of warranty by using annual calendar time and also in tracking systems in-service Engineering Change Test Substantiation In the event that redesigning is done in order to correct am failure mode in existence, tests are done in order to prove the improvement of the new design. The tests are necessary because improvements cannot be guaranteed to apply for all designs. What is the number of units that must be tested without failure, duration, to clear any doubts that the new design is significantly a better one than the one in existence? Also, the objective can be to explain a design requirement expressed as a probability or reliability of failure at some life design. The test gives a success data that can be of use in determination of lower confidence bound for the Weibull line. In the newer design called a "Weibayes" line. The criteria for test design may give zero failures or one or zero failure, to be alternatives. "Sudden Death" testing is one other technique which is useful. Failure test plans for Zero gives the absolute minima test time. In order to find the optimum plan of the test which can minimize cost and duration trade studies are used. 1.10 Maintenance Planning The Weibull plot is most important and useful for maintenance planning, especially the maintenance reliability centered. Beta, (β) tells the analyst if there’s need for overhauls and scheduled inspections. If β is less than or equal to one, they are not cost effective. With greater than one, its period or inspection interval is obtained directly from the plot within acceptable probability of failure. In wear out failure modes, where cost of an unplanned failure happens to be much greater than the initial cost of a planned overhaul, then optimum replacement duration for minimum cost comes into play. With Weibull failure forecasting, Quantitative trade can be done between, scheduled and unscheduled maintenance as well as forced ones. Conclusion My conclusion is that this new design for the washing machines cannot be used to guarantee the 5- year warranty that you are currently offering because only five machines out of the tested 20 machines will have life after 60 months, this shows that in the event you proceed with their mass production, you risk facing legal challenges and compensations due to losses from your customers, reputation and market. References. 1. Heinz, Bloch P. Shortcuts to Machinery Reliability and Improvement, Houston, Texas 1997. 2. Geitner, K.Fr. Machinery Failure Analysis .Volume 2, Third Edition, Publishing Company of Gulf, Houston, Texas, 1996    3. Robert, A Dovich,. Reliability Statistics, ASKC Quality Press, wilwaukee, WK, 1998. (Paperback) 4. Laboratory Manual 5.Lecturer Notes   Read More