Plan for the Session ? Guest lecture by Eric Feron (1 hour) ? Quiz on Design of Dynamic Systems (15 minutes) ? Review of Reliability Improvement Case Study -- Router Bit Life (35 minutes) 16.881 MIT Learning Objectives ? Introduce some basics of reliability engineering ? Relate reliability to robust design ? Practice some advanced construction techniques for orthogonal arrays ? Introduce analysis of ordered categorical data ? Practice interpreting data from robust design case studies MIT16.881 Reliability Terminology ? Reliability function R(t) -- The probability that a product will continue to meet its specifications over a time interval ? Mean Time to Failure MTTF -- The average ∞ time T before a unit fails MTTF = ∫ R(t)dt ? Instantaneous failure rate λ(t) 0 λ(t) = Pr(System survives to t + dt System survives to t) t R(t) = e ? ∫ 0 λ(ξ )dξ 16.881 MIT Typical Behavior ? Early failure period often removed by “burn-in” ? Wear out period sometimes avoided by retirement ? What will the reliability curve R(t) look like if early failure and wear out are avoided? Early Wear λ(t) failure out Useful life 16.881 t MIT ?? Weibull Distributions ? Common in component failure probabilities (e.g., ultimate strength of a test specimen) ? Limit of the minimum of a set of independent random variables s ?? ? t ?t o ? ? R(t) = e ? η ? 16.881 MIT System Reliability ? Components in series (system fails when any subsystem fails) R SYST = ∏ R i i ? Components in parallel (system fails only when all subsystems fail) F SYST = ∏ F i i 16.881 MIT Router Bit Life Case Study ? Printed wiring boards cut to size by a routing operation ? Router bit rotated by a spindle ? Machine feeds spindle in X-Y plane 16.881 MIT Taylor Tool Life Equation VT n = C ? V - cutting speed ? T - time to develop a specified amount of flank wear ? C and n - experimentally determined constants for cutter material / workpiece combinations n=0.08-0.2 High speed steel n=0.2-0.5 Carbide 16.881 MIT Noise factors ? Out of center rotation of spindle ? Router bit properties ? PWB material properties ? Spindle speed 16.881 MIT Noise Factor in the Inner Array ? When can this be done? ? Why can this work? ? What are the advantages? 16.881 MIT Quality Characteristic ? We want to maximize useful life of the bit ? Phadke chose to measure inches cut prior to excessive dust production ? Does this meet the guidelines for additivity? ? What other choices would you suggest? 16.881 MIT Control Factors ? Suction (1 in Hg, 2 in Hg) ? XY feed rate (60”/min, 80”/min) ? In-feed (10”/min, 50” in/min) ? Bit type (four types) ? Suction foot (solid ring, bristle brush) ? Stacking height (3/16”, 1/4”) ? Backup slot depth (0.06”, 0.1”) ? Spindle speed (30K rpm, 40K rpm) Do these meet the definition of a control factor? 16.881 MIT Reqts of the Matrix Experiment ? Two 4 level factors ? Seven 2 level factors ? Interactions – XY feed X speed – In-feed X speed – Stack height X speed – XY feed X stack height ? Compute required DOF 16.881 MIT Linear Graphs and Experimental Design ?Draw the required linear graph ? Compare with standard linear graphs ? Modify the standard linear graph to suit using modification rules – Breaking a line a c b a c b a c b a c b – Forming a line 16.881 MIT Speed Assigned as an Outer Factor ? Guarantees that all interactions with all other factors can be computed ? Doubles the experimental effort (for a two level outer factor) 16.881 MIT S/N for Fraction Defective ? Average quality L(y) loss in passing products Q = k p 1? p k p 1-p ? Signal to noise m-? ο m m+? ο y η= ?10log 10 ? ? ? 1? p p ? ? ? "goal post" loss function 16.881 MIT Ordered Categorical Data ? The response of the system is summarized by membership in discrete categories ? The categories have an unambiguous ordering ? For example, survey data often takes the form: – Poor –Fair – Good – Excellent 16.881 MIT Router Bit Life Conclusions ? Two fold increase in bit life realized while maintaining throughput ? Robust design can be used to improve life – Life as larger the better – Probability of failure as ordered categorical ? Noise can be induced in inner array ? A control factor can be used in an outer array to study interactions 16.881 MIT Next Steps ? Homework #8 due 8 July ? Next session Wednesday 8 July 1:05-3:55 – Quiz on reliability and robust design – Mid deck active control experiment case study ? Last on-campus session 9 July 1:05-3:55 - Review session for final exam ? Each student may resubmit up to three quizzes and/or home works by Monday 13 July (grades will be averaged with the original grades) 16.881 MIT