Generation & Propagation of Uncertainty Analysis P M V Subbarao Professor Mechanical Engineering Department A Measure of Confidence Level in compound Experiments…..

Generation of Uncertainty in Independent Measurements

Propagation of Uncertainty into Derived Variable Uncertainty in the measurement of individual variables, X i, propagate through the Measurement Equation, resulting in an uncertainty in measurand, y. It is possible to understand how a small error in one of the measured variables propagates into the measurand, using Taylor’s expansion. Taylor’s expansion of single variable function: Y=f(X)

Propagation of Error

Simplified Taylor’s Expansion of Y w.r.t to X true But error is random and hence only uncertainty is estimated

Where Y k is the vaule of Y at k th measured value of X Verification

Combining uncertainty An estimate of the measurand or output quantity Y, denoted by y y = f(x 1, x 2,..., x N ).

The combined standard uncertainty of the measurement result y, designated by u c (y) and taken to represent the estimated standard deviation of the result, is the positive square root of the estimated variance u c 2 (y) obtained from

where u(x i,x j ) is the covariance between x i and x j The covariance is related to the correlation coefficient r ij by where -1 ≤ r ij ≤ 1.

If x 1, x 2,..., x N are independent variables, then r ij =0 and is partial derivative of y w.r.t x i and called sensitivity coefficient