Reliability and Availablity This set of notes is a combination of material from Prof Doug Carmichael s notes for 13 21 and Chapter 8 of Engineering Statistics Handbook NIST SEMATECH e Handbook of Statistical Methods http www itl nist gov div898 handbook 2005 available free from see NIST SEMATECH e Handbook of Statistical Methods on CD Including and improving reliability of propulsion and other systems is a challenging goal for system designers An approach has developed to tackle this challenge 1 a design and development philosophy 2 a test procedure for components and total systems 3 a modelling procedure based on test results field tests and probability statistics Design and development philosophy recognition that reliability is a product is essentiall the abscence of failures or substandard performance of all critical systems in the design followed by an examination of the factors leading to failure Causes of failure a loading inaccurate estimates of thermal mechanical or electriacl including vibrations b strength inaccurate estimates of the load carrying capacity of the components c environment presence of dirt high temperature shock corrosion moisture etc d human factors heavy handed operators sailor proof wrong decisions operator error criminal activities sabatoge poor design tools left in critical components use of incorect replacements e quality control or lack thereof loose control of materials and manufacture lack of inspection loose specifications f accident act of God freak accidents collisions g acts of war terorism war damage designer should recognize these potential causes for failure and try to design devices that will resist failure Detailed Design Features a try to account for all possible situations in the design stage and eliminate possible failures Delivering maximumloads and minimum strengths b assume that every component can fail examine the outcome of the failure and try to reduce the risk of damage Failure Modes and Effects Analysis FEMA c institute strict quality control in manufacture and maintenance d have cleaarly defined specifications including material specifications and methods of testing e develop technology to meet new challenges conduct development testing f consider possible war damage and ship collision g carry out development testing in arduous conditions System Design Features a calculate probability of failures reliability and availability analysis b improve system design by standby or redundant systems c analyze failures note trends d specify clearly all operating procedures good operating manuals e require inspection maintenance and replacement procedures trend analysis Failure testing and analysis from field or laboratory tests on components or systems determine number of operating units as a function of time life 12 13 2005 1 set up N surv nominal survival curve 100 a typical survival curve might look like this number surviving 80 60 40 20 0 20 40 60 80 100 time define the failure rate at time t as 1 proportion failing in t t d N t dt N t N t 1 1 d N t N t t N t dt as rate 0 consistent with population decline units are 1 to make some estimates based on this sample d N t N t N t ln d N 0 0 dt fail rate t t N t t N t N t t fail rate 40 0 01 0 01 fail rate 10 0 01 0 01 1 N N t 0 define I N t NI exp d t 0 d N t dt N t set or calculate for modest t 0 01 and t 10 40 60 120 fail rate 60 0 01 0 01 looks like failure rate is a constant not unusual or t N t NI exp d 0 t 1 1 time fail rate 120 0 01 0 01 1 d N t constant N t dt d N t dt N t t integrate from N t ln d 0 to t N 0 0 0 01 N t NI exp t NI 100 100 N t 50 0 0 50 t 12 13 2005 2 100 N B failure rate is not necessarily the same as but can be related to in this case it is the probability of failure see Engineering Statistics Handbook an actual failure rate curve might look like this set up bath tub nominal failure rate 2 100 three regions are evident 800 failure rate 0 100 early failure period infant mortality rate 1 0 100 800 intrinsic failure period aka stable failure period intrinsic failure rate 0 200 400 600 800 800 wearout failure period materials wear out and degradation failures occur at an ever increasing rate 1000 for most systems the failure ratetime is relatively constant except for wer in and wear out If the failure rate is constant the component is said to have random failure Reliability applies to a particular mission with a defined duration defined as the probability of operating without degraded performance during a specific time period At time t 1 the number operating is N t1 and NI is the initial number The reliability is R t1 1 N t1 ln dt 0 NI t N t1 1 since NI with constant d N t N t dt R t1 exp t1 and if t1 1 and expanding in a series R t1 1 t1 e g t1 R t 1 exp dt NI 0 R t1 1 t1 t1 0 05 N t1 2 3 t1 t1 1 t1 0 95 2 3 exp t1 0 951 Mean Time Between Operational Mission Failure MTB OM F with field testing data is collected in the form of operating time failures and repair time During the field operation of a component or a system there is a total number of operating hours and a total number of failures MTB OM F is defined MBT OM F accumulated life number of failures For random failures the failure rate if t1 MBT OM F 12 13 2005 1 number of failures accumulated life 1 MBT OM F t1 R t1 1 t1 1 MBT OM F 3 Probability of Failure Q or F since probability of success failure 1 t1 if t1 Q 1 R 1 exp t1 t1 MTBF 1 R Q 1 now consider separate components C1 and C2 having R 1 and R2 and Q1 and Q2 then R1 Q1 R2 Q2 1 R1 Q1 R2 Q2 expand R1 R2 R1 Q2 Q1 R2 Q1 Q2 R1 R2 probability both C1 and C2 operating R1 Q2 probability C1 operating and C2 failed R2 Q1 probability C2 operating and C1 failed Q1 Q2 probability C1 and C2 failed Series Systems If it is necessary for all systems to operate then this termed a series system and is represented as a circuit as Rseries R1 R2 From above the probability that both are operating is e g i t1 Ri Rseries R1 R2 R3 Rn exp more generally R1 0 9 R2 0 9 n R3 0 9 n 2 components R1 R2 0 81 Rn 0 9 6 components 6 Rn 0 531 Parallel Systems If there is redundancy and either C1 or C2 is required for operation then this is a parallel scheme Rparallel R1 R2 R1 Q2 Q1 R2 1 Q1 …