Note: Reading AssignmentTurning ProcessObservations from ExperimentsCNC DataBrake Bending of SheetBending ProcessObservations from Bending ProcessObservations from Injection MoldingObservations from DataObservations from DataHow Model to Distinguish these Effects?Random ProcessesThe WhyBackground NeededHow to Describe Randomness?Example: Thermoforming Histogram (2000 data)How to Describe Continuous RandomnessCumulative FrequencyContinuous EquivalentsProcess Outputs as a Random VariableAnalysis of HistogramsHistogram for CNC TurningHistogram for Bending (MIT 2002 data)Histogram for Bending (MIT 2002 data)Consider: No Intentional Changes (Du = 0)Consider: No Effective Changes (¶Y/¶u= 0)Injection Molding (S’2003)Conclusion?Underlying or “Parent” ProbabilityContinuous Probability FunctionsUse of the pdf : ExpectationStationary ProcessesStationary ProcessesThe Uniform DistributionThe Normal DistributionCumulative DistributionProperties of the Normal pdfSuperposition of Random VariablesUse of the PDF: Confidence IntervalsConfidence IntervalsConfidence IntervalsIs the Process “Normal” ? Kurtosis: Deviation from NormalThe Central Limit TheoremExample: Uniformly Distributed DataSampling: Using Measurements (Data) to Model the Random ProcessSample StatisticsSample Mean Uncertainty ConclusionsMIT OpenCourseWare _________http://ocw.mit.edu___ 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)Spring 2008For information about citing these materials or our Terms of Use, visit: ________________http://ocw.mit.edu/terms.1ManufacturingControl of Manufacturing ProcessesSubject 2.830/6.780/ESD.63Spring 2008Lecture #4Probability Models of Manufacturing Processes February 14, 20082ManufacturingNote: Reading Assignment• May & Spanos– Read Chapter 4• Montgomery– Skim/consult Chapters 2 & 3 if need additional explanations or examples beyond May & Spanos3ManufacturingTurning Process4ManufacturingCNC Turning0.69700.69800.69900.70000.70100.7020Run NumberShift changes• Randomness + Deterministic Changesrandom or unknownΔαObservations from Experiments5ManufacturingCNC Data0.62350.62370.62390.62410.62430.62450.62470.62490.62510.6253123456789101112131415161718192021222324OuterMiddleInnerOperating Points:High Feed=1Low Feed = 26ManufacturingBrake Bending of SheetMMεσOutput:Angle7ManufacturingBending ProcessSpringback8ManufacturingAngle changes with depthΔY ⇒ Δu• Clear Input-Output Effects (Deterministic)• Also Randomness as wellObservations from Bending Process2030405060708090Aluminum 0.3 InSteel 0.3 InAluminum 0.6 InSteel 0.6 In9ManufacturingObservations from Injection MoldingRun Chart for Injection Molded Part40.6040.6540.7040.7540.8040.8540.9040.9541.000 102030405060Number of RunWidth (mm)AverageHolding Time = 5 secInjection Press = 40%Holding Time = 10 secInjection Press = 40%Holding Time = 5 secInjection Press = 60%Holding Time = 10 secInjection Press = 60%2/3/05 50QuickTime™ and aGraphics decompressorare needed to see this picture.10ManufacturingObservations from Data• Clearly some measurement “noise”?2030405060708090shift 1shift 2shift 311ManufacturingObservations from Data• Systematic/traceable “operator error”Sheet Shearing0.850.870.890.910.930.950.970.991.01shift changeshift changealuminumsteel steel12ManufacturingInputsA Random Process + A Deterministic ProcessHow Model to Distinguish these Effects?ProcessDisturbances (Reducible)Irreducible DisturbancesOutputs + "Noise"13Manufacturing• Consider the Output-only, “Black Box” view of the Run Chart• How do we characterize the process?– Using Y(t) only• WHY do we characterize the process– Using Y(t) only?ProcessY(t)Random Processes14ManufacturingThe Why• Did output really change?• Did the input cause the change?• If not, why did the output vary?• How confident are we of these answers?• Can we model the randomness?Process15ManufacturingBackground Needed• Theory of Random Processes and Random Variables• Use of Sample Statistics Based on Measurements– SPC basis– DOE: use of experimental I/O data– Feedback control with random disturbances16ManufacturingHow to Describe Randomness?• Look at a Frequency Histogram of the Data• Estimates likelihood of certain ranges occurring:–Pr(y1< Y <y2)– “Probability that a random variable Y falls between the limits y1and y2”17ManufacturingExample: Thermoforming Histogram (2000 data)0246810121.48 1.482 1.484 1.486 1.488 1.49 1.492 1.494 1.496 1.498 1.5 1.502Thermoforming Run Chart1.4751.481.4851.491.4951.51.505Run NumberPart DiameterShift Changes18ManufacturingHow to Describe Continuous Randomness• Process outputs Y are continuous variables• The Probability of Y(t) taking on any specific value for a continuumProb(Y(t) = y*) = 0• Must use instead a Cumulative Probability FunctionPr(Y(t)<y*)– Look at Cumulative Frequency19ManufacturingCumulative Frequency01020304050601.48 1.482 1.484 1.486 1.488 1.49 1.492 1.494 1.496 1.498 1.5Pr(Y<1.49)=14/50= 0.280246810121.48 1.482 1.484 1.486 1.488 1.49 1.492 1.494 1.496 1.498 1.5 1.502A Continuous equivalent?20Manufacturing• Probability Function: (P(x))• Probability Density Function pdf(x) = dP/dxxPxp(x)Continuous Equivalents01020304050601.48 1.482 1.484 1.486 1.488 1.49 1.492 1.494 1.496 1.498 1.50246810121.48 1.482 1.484 1.486 1.488 1.49 1.492 1.494 1.496 1.498 1.5 1.50221ManufacturingProcess Outputs as a Random Variable• The Histogram suggests a pdf– Parent or underlying behavior “sampled” by the process• Standard Forms (There are Many)– e.g. The Uniform and Normal pdf’s0.10.20.30.4-4-3 -2 -1 0 1 2 34Normal Probability Density Function for σ2 = 1, μ=0xp(x)22ManufacturingAnalysis of Histograms• Is there a consistent pattern?• Is an underlying “parent” distribution suggested?23ManufacturingHistogram for CNC Turning05101520250.697 0.698 0.699 0.7 0.701 0.702Bin Dimension (in)CNC Turning0.69700.69800.69900.70000.70100.70201357911131517192123252729313335373941434547Run NumberShift changes24Manufacturing02468101214134 134.5 135 135.5 136 136.5 137 137.5 138 138.5 139 139.5 140 140.5 141 141.5 1420.3 in depth onlyHistogram for Bending(MIT 2002 data)Bending Data Sorted by Depth809010011012013014015015913172125293337414549535761656973778185899397101105109113117121125129133Angle (Deg.)Steel0.6 in depthAluminum0.6 in depthSteel0.3 in depthAluminum0.3in depthDashed Lines are at group changes25Manufacturing02468101214161898 98.5 99 99.5 100
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