L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 16 1 MER301: Engineering Reliability LECTURE 17: Measurement System Analysis and Uncertainty Analysis-Part 2

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 16 2 Measurement System Analysis  Total Error in a measurement is defined as the difference between the Actual Value and Observed Value of Y  Two general categories of error – Accuracy or Bias and Precision Accuracy or Bias of Measurement System is defined as the difference between a Standard Reference and the Average Observed Measurement Precision of a Measurement System is defined as the standard deviation of Observed Measurements of a Standard Reference Total Error = Bias Error + Precision Error for independent random variables  Measurement System Error is described by Average Bias Error (Mean Shift)and a statistical estimate of the Precision Error (Variance) Measurement System Analysis is a Fundamental Part of Every Experiment

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 16 3 Measurement System Analysis  Bias or Accuracy error is a constant value and is dealt with by calibrating the measurement system  Variation or Precision error is a random variable which depends on the measurement equipment(the instruments used) and on the measurement system repeatability and reproducibility. Instrument Capability Analysis, Test/retest (repeatability)and Gage R&R studies are used to quantify the size of these errors.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 16 Gage Performance relative to required Tolerance Band  R&R less than 10% - Measurement system is acceptable.  R&R 10% to 30% - Maybe acceptable - make decision based on classification of characteristic, hardware application, customer input, etc.  R&R over 30% - Not typically acceptable. Find the problem using root cause analysis(fishbone), remove root causes GRR is a measure of “noise” in the data 4

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 16 5 Summarizing how it all fits together…..  When a set of measurements are made, the results are always observed values,  If the actual mean and standard deviation are known then the measurement system bias and variance can be calculated If the item being measured is a standard reference  If the measurement system bias and variance are known then the actual mean and actual variance can be calculated

L Berkley Davis Copyright 2009 Engineering Reliability Lecture 16 6 Measurement System Errors Repeatability (precision) Reproducibility Operator B Operator A Stability Time 1 Time 2 Observed Average Accuracy (Bias) True Average True Average Accuracy (Low End) Accuracy (High End) Observed Average (Low End) Observed Average (High End) Linearity

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 16 7 Elements that contribute to Accuracy and Precision Errors  Instrument Capability Resolution Gage Repeatability Linearity  Measurement System - Short Term (ST) Instrument Capability Equipment Calibration(Bias) Test/Re-Test Study(Repeatability)  Measurement System - Long Term (LT) Use Measurement System - Short Term Use Reproducibility Stability First Two are Entitlement….Third is Reality

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 16 8 Elements that contribute to Precision or Variation Errors  Instrument Capability Resolution Gage Repeatability Linearity  Measurement System- Short Term (ST) Use Instrument Capability Equipment Calibration(Bias) Test/Re-Test Study(Repeatability)  Measurement System - Long Term (LT) Use Measurement System - Short Term Use (ST) Reproducibility(Gage R&R) Stability(Gage R&R) First Two are Entitlement….Third is Reality

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 16 9 Measurement System Analysis From pages …

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Updating how variances all fit together  When a set of measurements are made, the results are always observed values,  If the actual mean and standard deviation are known then the measurement system bias and variance can be calculated If the item being measured is a standard reference  If the measurement system bias and variance are known then the actual mean and actual variance can be calculated

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Elements that contribute to Accuracy and Precision Errors  Instrument Capability Resolution Gage Repeatability Linearity Measurement System- Short Term(ST) Use Instrument Capability Equipment Calibration System Repeatability  Measurement System- Long term (LT) Use Measurement System -Short Term(ST) Use Reproducibility Stability

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture  Establish magnitude and sources of measurement system error due to bias and precision errors  Tools Instrument Capability Analysis Test/Re-test – system precision/repeatability Calibration - bias “Continuous Variable” Gage R&R (Gage Reproducibility and Repeatability) Attribute Variable Gage R&R Destructive Gage R&R How Can we Address Accuracy and Precision Errors?

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Measurement System Analysis  Instrument Capability Analysis…..  Resolution-smallest increment that the gage can resolve in the measurement process. Gage should be able to resolve tolerance band into ten or more parts. Resolution Uncertainty =  Instrument Precision- measure of instrument repeatability or instrument “noise”.. Found by repeated measurements of the same test item. Uncertainty =  Linearity- consistency of the measurement system across the entire range of the measurement system. Linearity Uncertainty =  The variations are combined as follows

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Measurement System Analysis  Measurement System Short Term Use Includes Instrument Capability Repeatability - variation when one operator repeatedly makes the same measurement with the same measuring equipment Test/Re-test Study Calibration/Bias  Measurement System-Long Term Use Includes Measurement System –Short Term Use Reproducibility- variation when two or more operators make same measurement with the same measuring equipment Stability-variation when the same operator makes the same measurement with the same equipment over an extended period of time

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Measurement System Analysis  Measurement System-Short Term Use  Repeatability-variation when one operator repeatedly makes the same measurement with the same measuring equipment Test/Re-test Study  Measurement System - Long Term Use  Reproducibility- variation when two or more operators make same measurement with the same measuring equipment  Stability-variation when the same operator makes the same measurement with the same equipment over an extended period of time

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Measurement System Analysis

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Measurement System Analysis  Instrument Capability Resolution Gage Repeatability Linearity  Measurement System - Short Term (ST) Use Instrument Capability Equipment Calibration Test/Re-Test Study  Measurement System - Long Term (LT) Use Measurement System (Short Term Use) Reproducibility(Gage R&R) Stability(Gage R&R) First Two are Entitlement….Third is Reality

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Mathematics of Measurement System Analysis  The Partial Derivative(Propagation of Errors) Method can be used to estimate variation when some X’s are related to actual product variation and other X’s are related to the measurement system (some may relate to both)  Those X’s that represent actual characteristics of the quantity Y contribute to the product variation while those associated with measurements of Y will contribute to measurement variation

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 17.1-Variation Equations Fuel Consumption Example

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture The Uncertainty Variables and  The quantity is a measure of the uncertainty in the value of the variable.It is a band wide that is a 95% confidence interval on the value of Define an Uncertainty Variable for any variable as so that  A dimensionless Relative Uncertainty is defined as

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Measurement System Uncertainty  The quantity is a measure of relative uncertainty in the measurement R and is an uncertainty band wide arising from variation in the x’s. It represents a 95% CI on the size of the variation expected in the reading  The equation for relative uncertainty for a measurement system can be written as  The individual x terms can be written as a relative uncertainty u X

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 17.2: Uncertainty Equations…. Viscometer and Triangle Examples

L Berkley Davis Copyright 2009 Viscometer Example- Lecture =1.0% reproducible Empirical Instrument Constant Y=fn(K,densities, time)

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 17.3  Uncertainty in Liquid Mass Flow Rate The mass flow rate of water through a tube is to be determined by collecting water in a beaker. The mass flow rate is calculated from the net mass of water collected divided by the time interval. Where Error Estimates are:  Mass of full beaker,  Mass of empty beaker,  Collection time interval,

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Lecture 17 Summary  Review of Measurement System Analysis from Lecture 16 Instrument Capability Measurement System in Short Term (ST) Use…  Instrument capability  Repeatability  Calibration/Bias Measurement System in Long Term (LT) Use…  Measurement System in Short term Use…  Reproducibility(Gage R&R)  Stability(Gage R&R)  Mathematics of Measurement System Analysis and Uncertainty Analysis