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Machine Design Under UncertaintyOutlineUncertainty in mechanical components Why consider uncertaintyBasics of uncertaintyUncertainty analysis for machine designExamplesConclusions2Uncertainty in Mechanical Components3

Where Does Uncertainty Come From? Manufacturing impressionDimensions of a componentMaterial propertiesEnvironmentLoadingTemperatureDifferent users

4Why Consider Uncertainty?We know the true solution.We know the effect of uncertainty.We can make more reliable decisions.

5We use probability distributions to model parameters with uncertainty.How Do We Model Uncertainty?6

Probability DistributionWith more samples, we can draw a histogram.

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Normal Distribution8

It indicates how data spread around the mean.It is always non-negative.High std meansHigh dispersionHigh uncertaintyHigh risk

9More Than One Random Variables10Reliability11First Order Second Moment Method (FOSM)12Monte Carlo Simulation (MCS)*A sampling-based simulation method13*This topic is optional.Step 3: Statistic Analysis on model outputExtracting probabilistic informationStep 2: Numerical ExperimentationEvaluating performance functionStep 1: Sampling of random variablesGenerating samples of random variablesAnalysis ModelSamples of input variablesSamples of output variablesProbabilistic characteristics of output variablesDistributions of input variables

Step 1: Sampling on random variablesStep 2: Obtain Samples of Output15Step 3: Statistic Analysis on outputFORM vs MCSFORM is more efficientFORM may not be accurate when a limit-state function is highly nonlinearMCS is very accurate if the sample size is sufficiently largeMCS is not efficient17Example - FOSM18

Example - FOSM19Example - FOSM20Example - MCS21

100 and 1000 Simulations

1e5 SimulationsMore simulations, More accurate result

Reliability Based Design (RBD)Design without considering uncertainty: Low reliability

Design with considering uncertainty: high reliability

24x2x1Failure RegionSafe Regionx2x1Failure RegionSafe RegionNominal design pointActual design pointsRBDRBD ensures that a design has the probability of failure less than an acceptable level, andtherefore ensures that events that lead to catastrophe are extremely unlikely. RBD is achieved by maximizing cost and maintaining reliability at a required level.25ConclusionsFor important mechanical components in important applications,a factor of safety may not be sufficient to account for uncertainties;it is imperative to consider reliability.Uncertainty can be modeled probabilistically.Reliability can be estimated by FOSM and MCS.26

Analysis Model

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