ES3D9: Applied Control Revision Guide
Part 1: Basics of Measurement System
Metrological terms and definitions
Error analysis in a measurement
Characteristics of a measurement system
Terms and Definitions
Measurand: quantity subjected to measurement
Range/Span – extent over which measuring system can reliably function
Accuracy – ‘closeness’ of measured value to true value
Precision – ability to stick to same results
Repeatability (measurement conditions = CONSTANT) – ability of instrument to give
identical responses when input is applied repetitively over short period of time, with
same instrument, same observer and same measurement conditions
Reproducibility (measurement conditions = VARY) – closeness in output readings
when changes in environmental settings
Calibrations – applying KNOWN value to a measurement system to establish
relationship between input and output.
o Known values are standards
o Standards = reproducible and stable for long time period
Traceability – chain-like structure in which every instrument in chain is calibrated
against a more accurate instrument immediately above it in the chain
o E.g
Measurement Errors
Types
Systematic Errors – can be corrected
o Zero drifted
o Improper calibration
o Assumptions on linear response and no deformation on contact
o Estimated constants used in calculation
Random Errors – not possible to correct
o Vibrations
o EM interference
o Electrical and electronic noise
Systematic Disturbances – when measuring process
itself is a disturbance to the variable being measured.
In electrical system: To make Em close to Eo,
Rm>>Rab
Signal Processing
For a set of data can apply statistical method to analyse and minimise random errors
o Mean values
o Standard Deviation
For set of data made repeatedly on same work piece, mean standard
𝜎
deviation is: 𝜎𝑚 = √𝑛
o Individual Error, 𝑒𝑖 = 𝑥𝑖 − 𝑥0
𝑛
1
𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 𝑒̅ = ∑ 𝑒𝑖 = 𝑋̅ − 𝑥0
𝑛
𝑖=1
o Precision = repeatability = ± 3𝜎
Error Sources
Alignment Errors
Cosine Errors
Parallax Errors – misalignment between eye and the indicating device
Abbe’s Offset – distance between line of measurement and line of the dimension.
o E.g. Vernier callipers BUT NOT micrometer
Elastic Deformation
Thermal Expansion Errors – all materials have thermal expansion coefficient
Support Points
Bars sag under their own weight
𝐿
=𝑠
√𝑛2 − 1
Standard Devices
Length
Line Standards – length defined by distance between engraved lines
o Rulers, callipers, micrometres
o Contact force needs to be controlled
Ratchet Stop
Friction thimble – more delicate than ratchet stop
End Standards – length defined by nominal distance between opposing faces
o (block gauges, length bars, roller gauges, limit gauges)
o Contains blocks of known range and increments. Assembled together into a
unit by ‘wringing’
o Gauging surfaces are very flat and parallel to each other
o Applications of gauge blocks
Lower grades have smaller tolerance
Compound/Total Error Calculation
Arithmetic Sum – if final measurement M is a function of a number of individual
measurements a,b, c which have individual errors delta_a, b and c then compound
error can be calculated by:
𝜕𝑀 𝜕𝑀 𝜕𝑀
𝛿𝑀 = ±( 𝛿𝑎 + 𝛿𝑏 + 𝛿𝑐)
𝜕𝑎 𝜕𝑏 𝜕𝑐
Quadratic Sum (using example in slides)
∆𝑉 = ± √(∆𝑉ℎ )2 + (∆𝑉ℎ )2 + (∆𝑉ℎ )2 = ± √(𝑤𝑙𝛿ℎ)2 + (ℎ𝑙𝛿𝑤)2 + (𝑤ℎ𝛿𝑙)2
Uncertainty Estimation – G.U.M
Guide to expression of Uncertainty in Measurement
Ensures consistency among research labs and manufacturers
Measurement model
o Define measurand
o Determine mathematical model with input quantities and an output quantity
𝑅 = 𝑓(𝑎, 𝑏, 𝑐)
o Value for output is called output estimate denoted by R
o Derive uncertainty model by total differential theorem to give weightings for
different uncertainties
𝛿𝑓 𝛿𝑓 𝛿𝑓
𝛿𝑅 = 𝛿𝑎 + 𝛿𝑏 + 𝛿𝑐
𝛿𝑎 𝛿𝑏 𝛿𝑐
Uncertainty Classification
Type A – evaluation of uncertainty by statistical analysis
o Modelled by normal distribution (mean, std dev)
o Come from set of observations
Type B – evaluation of uncertainty by means other than stat analysis
o Finite range of (+/- a) equally likely values
o Modelled by rectangular distribution 𝜎 = 𝑎/√3
o Manufacturers specification
o Other reports
Combined uncertainty
o 𝑈𝑐 = √∑𝑈 2 (𝑥𝑖 )
o 𝑈𝑐 = 𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦, 𝑥𝑖 = 𝑠𝑡𝑑 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 𝑜𝑓 𝑖𝑛𝑝𝑢𝑡 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒
Expanded Uncertainty kU_c
o Coverage factor k
K=1 68% confidence
K=2 95% confidence = normal practise
K=3 99.7% confidence
o 𝑌 = 𝑦 ± 𝑘𝑈𝑐 (𝑦 = 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑜𝑓 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑟𝑒𝑠𝑢𝑙𝑡)
Coordinate Measuring Machines (CMM)
Measures physical geometry of an object
Touch-trigger probe to locate coordinates of a line, a
plane or a circle
o Trigger signal is generated on contact with
component and is used to stop the machine
Has three moveable axes
All movements are imperfect against their design
goals
Error Sources
o Alignment errors
o Abbe-offset errors
o Thermal expansion error
o Elastic contact error
o Support points
To calibrate a CMM there are 21 error parameters to be evaluated
