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Fundamentals of Reliability Fundamentals of Reliability Engineering and Applications Part 1 of 3 E. A. Elsayed ©2011 ASQ & Presentation Elsayed Presented live on Nov 30th, 2010http://reliabilitycalendar.org/The_Reliability_Calendar/Short_Courses/Shliability Calendar/Short Courses/Short_Courses.html

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ASQ Reliability Division ASQ Reliability Division Short Course Series Short Course Series The ASQ Reliability Division is pleased to present a regular series of short courses featuring leading international practitioners, academics, and consultants. academics and consultants The goal is to provide a forum for the basic and The goal is to provide a forum for the basic and continuing education of reliability professionals.http://reliabilitycalendar.org/The_Reliability_Calendar/Short_Courses/Shliability Calendar/Short Courses/Short_Courses.html

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Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Nov 30, 2010 1

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Outline Part 1. Reliability Definitions1. Reliability---Time dependent characteristic2. Failure rate3. Mean Time to Failure4. Availability5. Mean residual life 2

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Outline Part 2. Reliability Calculations1. Use of failure data2. Density functions3. Reliability function4. Hazard and failure rates 3

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OutlinePart 3. Failure Time Distributions1. Constant failure rate distributions2. Increasing failure rate distributions3. Decreasing failure rate distributions4. Weibull Analysis – Why use Weibull? 4

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Part 1. Reliability Definitions Some Initial Thoughts Personal Experience …Time Dependent Characteristics More on Reliability Economics Reliability Definition Availability Mean residual life Conclusions 5

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Outline Part 1 Some Initial Thoughts Personal Experience More on Reliability Economics Reliability Definition Reliability Estimation System Reliability Calculations Role of New Technologies Conclusions 6

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Reliability Importance One of the most important characteristics of a product, it is a measure of its performance with time (Transatlantic and Transpacific cables) Products’ recalls are common (only after time elapses). In October 2006, Sony Corporation recalled up to 9.6 million of its personal computer batteries Products are discontinued because of fatal accidents (Pinto, Concord) Medical devices and organs (reliability of artificial organs) 7

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Failure: Types and Prediction Two types of failures:1. Sudden failure (no indicators): Stress exceeds strength ….2. Degradation (gradual wear out): degradation indicator such as crack growth, change of resistance, corrosion, … This is ideal for Condition-Based Maintenance Failure prediction:1. Analysis of field data at normal conditions2. Accelerated life testing3. Accelerated degradation testing other testing 8

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Examples: Component Wear outNov 23 2009, Consumer Product SafetyCommission recalls 2.1 million Stork Craft drop-down-side cribs because at least four infants havedied in them. the drop-down sides of the cribsbecame detached, which resulted in dozens ofbabies either becoming entrapped between theside and the crib frame, or falling out of the cribaltogether. Latch wear out. 9

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Reliability Economics Auto Warranty CostIn 2006, Hyundai chose to woo buyers in America bypromising quality and reliability. It issued an ambitiousnew warranty, good for five years (ten on the engineand transmission), then challenged its engineers toback that up with flaw-proof cars. The early sign arethey have delivered. Hyundai has trimmed itswarranty provision from 5.7% to just 1.8% of itsrevenue… Sales and profits are up. 10

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Prediction of Failure: Auto RecallFeb 13 2010Toyota recalls 2.3 million vehicles, three majorweaknesses in the company’s quality monitoring include:-- Lack of thoroughness in testing new cars and car parts under varying weather conditions, gas-pedal mechanism tended to stick more as humidity increased.-- Failures in gathering information from customer complaints, especially in the United States.-- Inability to analyze and act quickly on complaints that have been received. 11

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Personal Experience: Design of Reliable SystemA new fiber-optic cable to carry 40,000 simultaneouscall (data, voice). The ultrathin glass fibers in thecable carry information on laser beams of light.The glass-fiber line is suited to video transmission. Itprovides a security advantage for banks. Unlikesatellite transmissions, which can be intercepted, aglass-fiber line is almost impossible to tap.Cost $700 M and 8600 miles for transpacificCost $350 M and 3600 miles for transatlanticGoal : No failures in 80 years of service 12

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Global Impact: Oceanic Airspace Data Link Communication Reliability Engine Monitoring AOC or AircraftPosition OperationalMonitoring Communication Ground Earth Stations ARINC SITA Uplink Oakland ARTCC 13

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Reliability Economics Oil Pipeline Shutdown (Hardware Failure)BP shuts oilfield August 8, 2006 Damaged pipeline in Alaska affects 8% of U.S. oil production; crude surges; record gas prices seen. The price of crude jumped $2.22 a barrel on the shutdown news to over $76. Gasoline futures rose 3.35 cents to $2.2650 a gallon. The threat of a stoppage also endangers Alaskas budget: Oil taxes account for more than 90 percent of its revenues. BP officials have acknowledged they did not test the pipes adequately using a so-called pig device which is runs through a pipe to gauge corrosion (utilizes ultrasound to detect corrosion). Lack of proper inspection. Lucent 14

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Reliability Economics Oil Pipeline No-Shutdown (Hardware Failure)BP Fails to Shutdown Oil Pipeline (April 10, 2010) A “blowout” on an oil rig occurs when some combination of pressurized natural gas, oil, mud, and water escapes from a well, shoots up the drill pipe to the surface, expands and ignites. Wells are equipped with structures called blowout preventers that sit on the wellhead and are supposed to shut off that flow and tamp the well. Deepwater Horizon’s blowout preventer failed. Two switches — one manual and an automatic backup — failed to start it (System Design). 15

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Reliability Engineering Air Traffic Delays (Software Failure) Nov 19 2009: A computer glitch caused flight cancellations and delays across the U.S. The problem involved the FAA computer systems in Salt Lake City and Atlanta that handle automated flight plans, forcing air traffic controllers to revert to the much more time- consuming approach of entering flight plans by hand. Software failure (7000 flights) 16

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Reliability Definitions “Measurements”► When you a buy a product or service… you request “high quality” and “high reliability” How do you measure it? What is “high”? How long? Reliability: 0.99 at year 5, 0.999 at year 4…► Time dependent quality…reliability► How do companies predict reliability and estimate warranty?► Reliability of cold standby units …New tires and old tires… 17

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Some Initial Thoughts Repairable and Non-RepairableAnother measure of reliability is availability (probabilitythat the system provides its functions when needed). Maximum Reliability level With R epairs Reliability No Rep airs Time 18

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Some Initial Thoughts Warranty • Will you buy additional warranty? • Burn in and removal of early failures. (Lemon Law). Early Failures Increasing Constant Failure Failure Rate Rate Failure Rate19 Time

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Reliability DefinitionsReliability is a time dependent characteristic. It can only be determined after an elapsed time but can be predicted at any time. It is the probability that a product or service will operate properly for a specified period of time (design life) under the design operating conditions without failure. 20

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Other Measures of ReliabilityAvailability is used for repairable systems It is the probability that the system is operational at any random time t. It can also be specified as a proportion of time that the system is available for use in a given interval (0,T). 21

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Other Measures of ReliabilityMean Time To Failure (MTTF): It is the averagetime that elapses until a failure occurs.It does not provide information about the distributionof the TTF, hence we need to estimate the varianceof the TTF.Mean Time Between Failure (MTBF): It is theaverage time between successive failures.It is used for repairable systems. 22

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Mean Time to Failure: MTTF ∞ ∞ MTTF = ∫ tf (t )dt = ∫ R(t )dt 0 0 1 n MTTF = ∑ ti n i =11 2 is better than 1? 2R(t) 1 Time t 0 23

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Mean Time Between Failure: MTBF 24

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Other Measures of ReliabilityMean Residual Life (MRL): It is the expected remaining life, T-t, given that the product, component, or a system has survived to time t. 1 ∞ L(t ) = E[T − t | T ≥ t ] = ∫ τ f (τ )dτ − t R(t ) tFailure Rate (FITs failures in 109 hours): The failure rate in a time interval [ t1 −t2 ] is the probability that a failure per unit time occurs in the interval given that no failure has occurred prior to the beginning of the interval.Hazard Function: It is the limit of the failure rate as the length of the interval approaches zero. 25

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Strengthmean =20 kg/mm2, Sigma=5 kg/mm2 26

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Stressmean =10 kg/mm2, Sigma=5 kg/mm2 27

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Stress-Strength 28

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Safety Factors and ReliabilityLet S denote the strength random variable and sthe stress random variable. The random variable y= (S – s) is then related to the reliability of thecomponent by R = P( y ≥ 0) (1)When strength and stress random variables havenormal density functions, y is normally distributedand the reliability R is given by 29

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Safety Factors and Reliability − z2 1 ∞ (2) R= 2π ∫ ( µ −µ ) − S s 2 2 e 2 dz σ S +σ s µS − µs z0 =Where σ S + σ s2 2This integral can be estimated using numericalintegration orhttp://www.fourmilab.ch/rpkp/experiments/analysis/zCalc.html 30

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Calculations of Factor of SafetyUsing the mean values of the stress and strength weobtain factor of safety = 20/10=2The probability of failure is calculated as µS − µs z0 = =1.4142 σ +σ 2 S 2 sThe corresponding probability of failure =0.078652 orAbout 1 in 13. The reliability is 0.921348 31

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Example 32

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Calculations of Factor of Safetywhere z is the standard normal random variable, µS isthe mean value of the strength, µ s is the mean value ofthe stress, and σ S and σ s are the standarddeviations of strength and stress, respectively. Thereliability clearly depends on the lower limit of theintegral in Equation 2. A higher value of reliability canbe obtained by lowering the lower limit.Table 1 gives the reader an idea about the variabilityin reliability related to different magnitudes ofvariability in strength and stress random variables.The factor of safety is given by µ S / µ s . 33

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Solution using Matlab%Load excel sheet file[A,B]=xlsread(P1A1S2009.xls);>> S1=A(:,1);>> S2=A(:,2);>> mu1=mean(S1)mu1 = 9.8616>> mu2=mean(S2)mu2 =19.7842>> sigma1=std(S1)sigma1 = 0.9476>> sigma2=std(S2)sigma2 =1.9592% Calculate z>> z=(mu2-mu1)/(sqrt(sigma1^2+sigma2^2))z = 4.5593Probability corresponding to z is 0.0003 34