ME 363 Elementary Instrumentation LAB 6: Propagation of Uncertainty INTRODUCTION In this lab you will measure the density of a rubber stopped and determine the uncertainty of the measurement. OBJECTIVES • Improved understanding of propagation of uncertainty • Improved understanding of statistics OVERVIEW For this lab 4 measurements will be taken of a rubber stopper: 3 of its dimensions and one of its weight. These will be used to calculate its density. More importantly, the accuracy of the measurement will also be determined. Note: Some rubber stoppers have a hole in the middle. For purposes of this lab, neglect this hole. The instruments used to take measurement in this lab will be: 1.
2.
3.
4.
A ruler in inches for the top radius A ruler in mm for the bottom radius Calipers in inches for the height The weight in lb on a digital scale The formula for the density of a truncated cone is: !=
ℎ!
!!!
3!
+ !! !! + !!!
In this lab there is both systematic and random uncertainty. The formula for combining the systematic and random uncertainty is: !
!! = ± !!! + !!,!" !! (95%) Where PR is the random uncertainty, which comes from the variation inherent in taking multiple measurements. The systematic uncertainty, BR, comes from errors in the measurement system, including the instrument accuracy. When a final value is calculated from other measured values, the uncertainty in the final value is found from the uncertainty in the measured values using the expressions: !
!! = ±
!! !!!
!!!
!
!
!! = ±
!! !!!
!
!!!
Where L is the number of measured values needed to calculate the final value. !!! and !!! are the individual uncertainties associated with each of the measured variables, and !! is the linearized sensitivity of the ith measurement. !! is determined by the derivative of the governing equation which defines the variable’s relationship, in respect to each variable. In this case !! is equal to and !"
!!!
!"
!"
!"
!"
!!
!!!
, , , . The random uncertainty, !!! , is equal to the standard deviation of the means of a measured value. On the other hand, the systematic uncertainty, !!! , is the root-­‐sum-­‐square of the zero-­‐order uncertainty and instrument uncertainty. In this lab the instrument uncertainty is neglected, so the systematic uncertainty reduces simply to the zero-­‐order uncertainty. The zero order uncertainty (!! ) is defined as half the instrument resolution with an assumed 95% probability. PROCEDURE Phase I: Design Stage Uncertainty First take all four measurements only once and round them to the nearest logical value. Do not round the weight. In the absence of reported values from the manufacturer these will be our assumed nominal values for our design stage uncertainty. Convert all your measurements to the metric units of meters and kilograms. Determine the zero-­‐order uncertainty for each instrument (see Analysis Question 2). Then convert this uncertainty to metric units also. Calculate the linear sensitivity of each variable using the equation above. Use the nominal values you determined earlier in this calculation. Take the root-­‐sum-­‐square of the product of the zero-­‐order uncertainty and the sensitivity to determine !! as indicated in the equation above. Phase II: Random Uncertainty First, take 6 measurements of each dimension and record them in Excel. Take all measurements in the units specified above. Afterward, convert them all to standard metric units of meters and kilograms. Using Excel, compute the mean and standard deviation of the means of your collected data. Use the AVERAGE() command for the mean and STDEV() for the sample standard deviation. Remember that the standard deviation of the means is equal to the sample standard deviation divided by the square root of the sample size. Use your average values as your nominal values when calculating the sensitivity this time. Finally, calculate the random uncertainty as the root-­‐sum-­‐square of the product of the sensitivity and standard deviation of the means. Phase III: Combine Finally, put everything together. First calculate the t-­‐value for the probability function of your sample using Excel’s TINV() function. Be sure to use the correct number of degrees of freedom, and 5% as the probability (See Analysis Question 1). Then calculate uR as per the equation at the start of this lab. Finally, using your mean values, calculate the density of the rubber stopper and express it alongside the uncertainty in the form ! ! = ! ± !! . Phase IV: Correct Finally, change out one of your instruments that you believe was causing a large error and substitute it for another. Redo the needed calculations and show how your uncertainty changes. Turn in a printout of screenshot of your Excel sheet from the steps above. Show all your work! Ideas for above and beyond 1. How low can you get your uncertainty? Would taking measurements another way help? 2. Determine the uncertainty associated with taking other measurements. Analysis: 1. How many degrees of freedom does your sample have? 2. What is the zero-­‐order uncertainty for each of the instruments listed above? (Remember to convert all units to metric) Discussion: 1. Where does most of the error come from? Is this what you expected? 2. What are some practical ways error could be reduced? 3. Is the uncertainty you measured acceptable for practical applications? Think about the cost of shipping by weight, for example. Report: 1. Please write your report in the form of a technical report to your lab TA. Use the template from the undergraduate ME guide on writing technical reports. Use this as an opportunity to discuss some of the uncertainty in the measurements you have been performing with your project this semester. The hope here is that you will be able to use a portion of what you write in your term paper.