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 1set up N_surv nominal survival curve 20 40 60 80 100 time - as "rate" > 0 consistent with time− 1 population decline units are: Nt() ⎞⎟ ⎮⌠t λτdτ () = ⋅ ⎜⎛ ⎮⌠t λτdτ⎟⎞ N0() ⎠ = −⌡0 () Nt NI exp⎜−⌡0 () ⎟ ⎝ ⎠ calculate for modest δt = 0.01 and t =10, 40, 60, 120 100 80 60 40 20 0 −1d ⋅ Nt()Nt() dt ⎛ln ⎜ ⎝ Nt( + δt)− Nt() a typical survival curve might look like this: define the failure rate at time t as proportion_failing_in_δt −δNt() 1λ = = ⋅ = number surviving fail_rate t ( ,δt):= − Nt()⋅δt δt d Nt()dt−1⋅ = λ Nt() to make some estimates based on this sample: fail_rate 40 0.01( , ) = 0.01 Nt() δt dN t()−1⋅ = λ⋅dt Nt() fail_rate 10 0.01( , ) = 0.01 ( , ) = 0.01 ( , ) = 0.01fail_rate 60 0.01 fail_rate 120 0.01looks like λ = failure rate is a constant, not unusual d Nt()dt define ... NI = Nt( = 0) −1⋅ = λ Nt() ⎛⌠t ⎞ set ... or ... Nt() = NI⋅exp ⎜⎜−⎮ λ dτ⎟⎟ or ... N()t⎝⌡0 ⎠ −1dλ = ⋅ Nt() = constant Nt() dt d dtNt() integrate from ⎛ Nt() ⎞⌠t −1⋅ Nt() = λ 0 to t ln⎜ () ⎟⎠ = −⎮⌡0 λ dτ ⎝ N0Iλ := 0.01 N := 100 := NI⋅exp(−λ⋅t) 100 Nt() 50 0 0 50 100 t 12/13/2005 20 1 2 failure rate 100 800 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 three regions are evident: 0 - 100 early failure period = infant mortality rate 100 - 800 intrinsic failure period aka stable failure period => intrinsic failure rate > 800 wearout failure period - materials wear out and degradation failures occur at an ever 0 200 400 600 800 1000 increasing rate for most systems, the failure rate is relatively constant except for wer in and wear out. If the failure rate is constant, the component is said to have random failure. time 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: Nt1Nt1⌠t1 Nt1⎛⌠t1 ⎞() dN t() ⎛()⎞ ()⎜ ⎟Rt1= since ... −1⋅ = λ⋅dt ln = −⎮ λdt Rt1= = exp −⎮ λdt() NI Nt() ⎜⎝ NI ⎟⎠ ⌡0 () NI ⎜⎝⌡0 ⎟⎠ with λ= constant () exp(− t1) and expanding () 1 (λ⋅t1)2 (λ⋅t1)3 Rt1= λ⋅ in a series ... Rt1= − λ⋅t1 + 2!− 3!+ .. and if ... λ*t1 << 1, Rt1= − ⋅ e.g. λ := 0.05 1 − λ = 0.95 exp λ = 0.951() 1 λ t1 t1t1 (− t1) 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 accumulated_lifeMBT OM( )F = number_of_failures number_of_failures 1 For random failures, the failure rate λ = accumulated_life = MBT OM) F ( if ... t1 () 1 t1 < 1 Rt1= − λ⋅t1 = 1 − MBT OM)F ( ⋅ ( MBT OM) F12/13/2005 3Probability of Failure (Q or F) t1 if ... λ*t1since probability of success + failure = 1 R + Q = 1 Q1 − R = 1 − exp(−λ⋅ ) = λ⋅t1 = << 1 = t1 MTBF now consider separate components C1 and C2 having R1 and R2 and Q1 and Q2. then ... (R1 + Q1)⋅(R2 + Q2) = 1 (R1 + Q1)⋅(R2 + Q2) expand → R1⋅R2 + R1⋅Q2 + Q1⋅R2 +