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Exper_Uncertainty_Analysis

# Exper_Uncertainty_Analysis - Experimental Uncertainty...

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Experimental Uncertainty Analysis Author: John M. Cimbala, Penn State University Latest revision: 07 February 2007 Experimental Uncertainty Analysis Now that the principles of measurement uncertainty and confidence level have been established, we can predict the uncertainty of a calculated quantity as well. Experimental uncertainty analysis provides a method for predicting the uncertainty of a variable based on its component uncertainties. Some authors call this analysis the propagation of uncertainty . Suppose one measures N physical quantities (or variables, like voltage, resistance, power, torque, temperature, etc.), x 1 , x 2 , . .., x N . Also suppose that each of these quantities has a known experimental uncertainty associated with it, which shall be denoted by i x w , i.e., i ii x x xw = ± . Furthermore, unless otherwise specified, each of these uncertainties has a confidence level of 95%. Since the x i variables are components of the calculated quantity, we call the uncertainties component uncertainties . Suppose now that some new variable, R , is a function of these measured quantities, i.e., = R ( x 1 , x 2 , . .., x N ). The goal in experimental uncertainty analysis is to estimate the uncertainty in R to the same confidence level as that of the component uncertainties , i.e., we want to report R as R R Rw = ± , where R w is the predicted uncertainty on variable R . There are two types of uncertainty on variable R : o Maximum uncertainty – We define the maximum uncertainty on variable R as ,max 1 i iN Rx i i R ww x = = = . ± Because of the absolute value signs, this expression assumes that all the errors in the component variable x i measurements are such that the error in R is always the same sign . ± Such a case would be highly unlikely, especially for a large number of variables (large N ), because some of the errors would be positive and some negative, and the errors would cancel each other out somewhat. In other words, this is a worst case scenario . o Expected uncertainty – We define the expected uncertainty on variable R as 2 ,RSS 1 i i i R x = = ⎛⎞ = ⎜⎟ ⎝⎠ . ± Expected uncertainty is also called the root of the sum of the squares uncertainty , or RSS uncertainty , because of the above equation – the square root of the sum of squared quantities. ± This uncertainty estimate is more realistic than the maximum uncertainty since it is unlikely that the maximum error will occur on all component variables simultaneously. ± RSS uncertainty is the engineering standard , and the usual notation is to set w R equal to the RSS uncertainty, i.e., R = w R ,RSS . ± It turns out that we can write R = calculated R ± w R , to the same confidence level as that of each of the x i measurements (the same confidence level as that of the individual component measurements) .

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