2 Topics Today Review of Error Analysis Theory & Experimentation in EngineeringSome Considerations in Planning ExperimentsReview of Statistical formulas and theoryBegin Statistical Design of experiments (“DOE” or “DOX”)
3 Part 1: Review of Error Analysis Uncertainty or “random error” is inherent in all measurementsStatistical basisUnavoidable- seek to estimate and take into accountCan minimize with better instruments, measurement techniques, etc.
4 Review of Error Analysis Systematic errors (or “method errors”) are mistakes in assumptions, techniques etc. that lead to non-random biasCareful experimental planning and execution can minimizeDifficult to characterize; can only look at evidence after the fact, troubleshoot process to find source and eliminate
5 Graphical Description of Random and Systematic Error
6 Why do we need to estimate uncertainty and include in stated experimental values? Probability of being wrong will influence process and/or financial decisionsCost / benefit of accepting result as “fact”?What would be the effect downstream as the uncertainty propagates through the process?When comparing two values and determining if they are differentOverlap of uncertainty?What is the probability that the difference is significant?
7 Stating Results +/- Uncertainty Rule for Stating UncertaintiesExperimental uncertainties should almost always be rounded to one significant figure.Rule for Stating AnswersThe last significant figure in any stated answer should usually be of the same order of magnitude (in the same decimal position) as the uncertainty.Express Uncertainty as error bars and confidence interval for graphical data and curve fits (regressions) respectively
8 Determining Propagated Error: Non-statistical Method Compute from total differential
9 Propagated errorOR Can do sensitivity analysis in spreadsheet of other software programCompute possible uncertainty in calculated result based on varying values of inputs according to the uncertainty of each inputExample: Use “Solver” optimization tool in Excel to find maximum and minimum values of computed value in a cell by varying the value of each input cellSet constraint that the input values lie in the range of uncertainty of that value
10 Or Can Use repeat measurements to estimate uncertainty in a result using probability and statistics for random errors:meanstandard deviation of each measurementstandard deviation of the mean of the measurementsConfidence intervals on dependant variableConfidence intervals on regression parameters1)Mean all familiar with- average or “best” value2)std. dev. Is a good meas. Of variation in repeat meas.- tells how far apart individual measurements are from each other.3) SDOM gives estimate of how much ind. Measurements vary form the mean value.4) one std. dev defines a 66% CI. . ..if you take more meas., then 66% should lie within 1 std. dev.
22 Statistical Process Control Very Widely UsedUsed for quality control and in conjunction with DOE for process improvementControl Charts provide statistical evidenceThat a process is behaving normally or if something wrongServe as data output (dependant variable )from process in designed statistical experiments
23 Variation from expected behavior in control charts- similar to regression and point statistics Control Limit is the mean of a well behaved process output (i.e. product)Upper and lower Control Limits represent confidence limit on mean of “well behaved” process ouptutExpect random deviations form mean just like in regression
24 Part 2: Theory and Experimentation in Engineering
25 Theory and Experimentation in Engineering Two fundamental approaches to problem solving problems in the discovery of knowledge:Theoretical (physical/mathematical modeling)Experimental measurement(Most often a combination is used)Each application is differentBut in General:-math models: ALWAYS only an approximation of the “real world”Good? Fair? Poor?-Engineering as opposed to pure science: more money & time constraintsTheoretical: apply to whole class of problemsExperimental; more specific to application at hand(Note example of correlations based on theoretical development- ie. Correlated data based on dimensionless groups, etc.)
26 Example of combination of theory and experimentation to get semi-empirical correlation Here, the researchers have correlated data of several experimenters:1) Have used similarity principles / Dimensional Analysis ( Buckingham Pi method) to reduce the number of variables-theory, reason to determine the variables that the phenomena or process depends on- Construct dimensionless “Pi variables2) This process is somewhat complex: heat transfer in a fluidized bed (recall you should have done fluid part- model as many straight tubes, find a type of friction factor from model= Ergun Eqn, Then too the theoretical model needed to be curve fit to data to get coefficeints- LOOK UP)-Did regression analysis of log transformed data to find exponents on dimensionless groups
27 Features of alternative methods Theoretical ModelsSimplifying assumptions neededGeneral resultsLess facilities usually neededCan start study immediatelyExperimental approachStudy the “real world”-no simplifying assumptions neededResults specific to apparatus studiedHigh accuracy measurements need complex instrumentsExtensive lab facilities maybe neededTime delays from building apparatus, debugging
28 Functional Types of Engineering Experiments Determine material propertiesDetermine component or system performance indicesEvaluate/improve theoretical modelsProduct/process improvement by testingExploratory experimentationAcceptance testingTeaching/learning through experimentationMixing a chem E example, or polymer properties, etc.Specifications for product: ie if you buy a stereo: signal/noise, frequency response, total harmonic distortion, etcSelf ex.**VERY IMPORTANT- most applicable to what you might doie, what research faculty, grad students doLarge of Expensive pieces of equipment- before you pay, test it out (do your tests meet performance indices in #2?)What we do in this class. . .
29 Some important classes of Experiments Estimation of parameter mean valueEstimate of parameter variabilityComparison of mean valuesComparison of variabilityModeling the dependence of dependant Variable on several quantitative and/or qualitative variables
30 Practical Experimental Planning Experimental design:Consider goalsConsider what data can be collected.Difficulty of obtaining dataWhat data is most importantWhat measurements can be ignoredType of data: Categorical? Quantitative?Test to make sure that measurements/apparatus are reliableCollect data carefully and document fully in ink using bound notebooks. Make copies and keep separatelyPLANNING SAVES YOU TIME!!!
31 Preview of Uses for DOE Lab experiments for research Industrial process experiments-first is similar to what we have been doing-second is likely more relevant to what you may do in future
32 Four engineering problem classes to which DOE is applied in manufacturing 1. Comparison2. Screening/ characterization3. Modeling4. Optimization-will talk briefly about each and try to present examplesIe two means the same for tow diff. processes?Which factors important: ID the significant few from the many insignificant- complex manufacturing, etc.Model to predict output given values of significant few inputs (response = dependant variable)Make process give best product possible, or least expensive process, combination of the two, etc.
33 ComparisonCompares to see if a change in a single “factor” (variable) has resulted in a process change (ideally an improvement)Ties in with what we discussed about SPC. Often SPC would be used as output data from a process to tell us whether the product or intermediate is better, etc.
34 Screening/Characterization Used when you want to see the effect of a whole range of factors so as to know which one(s) are most important.Two factorial experiments usually usedRecall when we talked about SPC we said that it was important to “understand your process”
35 ModelingUsed when you want to be able to construct a mathematical model that will predict the effect on a process of manipulating a variables or multiple variables-example would be our Kla experiment-continuous variables where in many process applications we are often talking about discreet of even binary factors
36 OptimizationWhen you want to determine the optimal settings for all factors to give an optimal process response.For example in an industrial process, the optimal response might be the best product spec or the least cost to manufacture, or some optimal combination of those 2 responses, etc.