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### MEASUREMENT AND INSTRUMENTATIONBMCC 3743 :

MEASUREMENT AND INSTRUMENTATIONBMCC 3743 LECTURE 4: EXPERIMENTAL UNCERTAINTY ANALYSIS Mochamad Safarudin Faculty of Mechanical Engineering, UTeM 2010

### Contents :

andims - FKM - UTeM 2 Contents Propagation of uncertainties Consideration of systematic and random components of uncertainty Sources of elemental error Uncertainty of the final result Design-stage uncertainty analysis Applying uncertainty-analysis in digital data acquisition system

### Propagation of uncertainties :

andims - FKM - UTeM 3 Propagation of uncertainties Uncertainty analysis is important to identify corrective measures while validating and performing experiments. Propagation of uncertainties => ‘total uncertainties’, e.g. P = VI = n Two important factors in uncertainty: Random uncertainty (or precision uncertainty) : imprecision in measurements Systematic uncertainty (or bias uncertainty): estimated maximum fixed error

### General consideration :

andims - FKM - UTeM 4 General consideration If R is a function of n measured variables x1, x2, …. xn, i.e. Then a small change in is due to small changes in s in xi’s via the differential equations: Sensitivity coefficient (1) (2)

### General consideration :

andims - FKM - UTeM 5 General consideration For calculated result based on measured xi’s, Eq. (2) can be rewritten as where | | is to make sure we don’t get zero uncertainty in R. However, this can produce high estimate for wR. (3) Uncertainty in result Uncertainty in variables

### General consideration :

andims - FKM - UTeM 6 General consideration Hence Eq. (3) is better represented by =>root of the sum of the squares (RSS) In this case, the confidence level must be the same for all uncertainties (typically 95%). Assumption is made that each measured variables (hence, error) are independent of each other. (4)

### Exercise :

andims - FKM - UTeM 7 Exercise To calculate the power consumption of an electric circuit, we have P = VI where V = 100 2 V and I = 10 0.2 A Calculate the maximum possible error (uncertainty) and best-estimate uncertainty (RSS). Hint: Use Eq. (3) and Eq. (4) respectively.

### Answer to Exercise :

andims - FKM - UTeM 8 Answer to Exercise Because P=VI dP/dV=I=10.0 A , dP/di=V=100.V then

### Contents :

andims - FKM - UTeM 9 Contents Propagation of uncertainties Consideration of systematic and random components of uncertainty Sources of elemental error Uncertainty of the final result Design-stage uncertainty analysis Applying uncertainty-analysis in digital data acquisition system

### Consideration of systematic and random components of uncertainty :

andims - FKM - UTeM 10 Consideration of systematic and random components of uncertainty Random uncertainty depends on sample size (usually large, n>30) Systematic uncertainty is independent of sample size & does not vary during repeated reading Need to separate for detailed uncertainty analysis

### Random uncertainty :

andims - FKM - UTeM 11 Random uncertainty Using t-distribution, the random uncertainty for all measurements is given by where Sx is the standard deviation of the sample For a single measurement (also for each individual measurement), the random uncertainty is (5) (6)

### Systematic uncertainty :

andims - FKM - UTeM 12 Systematic uncertainty Sometimes assumed as level of accuracy Depends on manufacturer’s specification, calibration tests, mathematical modelling, considerable judgement as well as comparisons between independent measurements.

### Systematic uncertainty – some examples :

andims - FKM - UTeM 13 Systematic uncertainty – some examples Radiation heat transfer => lower measured value Instrument location => spatial error, e.g. a single thermometer measures temperature in a box oven Dynamic errors

### Combining random & systematic uncertainties :

andims - FKM - UTeM 14 Combining random & systematic uncertainties Total uncertainty is obtained, using RSS (Eq. 4) for all measurements, is given by For a single measurement of x, (7) (8)

### Contents :

andims - FKM - UTeM 15 Contents Propagation of uncertainties Consideration of systematic and random components of uncertainty Sources of elemental error Uncertainty of the final result Design-stage uncertainty analysis Applying uncertainty-analysis in digital data acquisition system

### Sources of elemental error :

andims - FKM - UTeM 16 Sources of elemental error ‘Chain of uncertainties’, e.g. A/D converter would have quantisation errors, sensitivity errors and linearity errors. Each of these components contribute to further errors. Can be random or systematic error.

### Estimation of uncertainty :

andims - FKM - UTeM 17 Estimation of uncertainty Systematic uncertainty: just combine all elemental uncertainties Random uncertainty: 3 approaches to determine Sx Run entire test in a sufficient number of times Run auxiliary tests for each measured variable x. Combine elemental random uncertainties => Based on experiment requirement.

### 5 categories of elemental errors :

andims - FKM - UTeM 18 5 categories of elemental errors Calibration Uncertainties: residual systematic errors due to; uncertainty in standards, uncertainty in calibration process, randomness in the process Data-Acquisition Uncertainties: during measurement due to; random variation of measurand, A/D conversion uncertainties, uncertainties in recording devices Data-Reduction Uncertainties: due to interpolation, curve fitting and differentiating data curves Uncertainties Due to Methods: due to assumptions/constant in calculation, spatial effects and uncertainties due to hysterisis, instability, etc. Other Uncertainties

### Combining elemental systematic & random uncertainties (RSS) :

andims - FKM - UTeM 19 Combining elemental systematic & random uncertainties (RSS) Calibration Uncertainties Data-Acquisition Uncertainties Data-Reduction Uncertainties Uncertainties Due to Methods Other Uncertainties Reproduced from Wheeler’s book: ASME 1998

### Degrees of freedom, vx :

andims - FKM - UTeM 20 Degrees of freedom, vx When sample size is large, vx is simply number of sample, n, minus 1. When sample size is small, then vx is given by => Welch-Satterthwaite formulation (ASME 1998) (9) Degrees of freedom of individual elemental error

### Contents :

andims - FKM - UTeM 21 Contents Propagation of uncertainties Consideration of systematic and random components of uncertainty Sources of elemental error Uncertainty of the final result Design-stage uncertainty analysis Applying uncertainty-analysis in digital data acquisition system

### Uncertainty of the final result (Multiple measurement) :

andims - FKM - UTeM 22 Uncertainty of the final result (Multiple measurement) Referring to Eq. 1, then for multiple measurements, M, the mean results is given by Little exercise: Derive the standard deviation (SR) and random uncertainty ( ) of R. (10)

### Uncertainty of the final result (Multiple measurement) :

andims - FKM - UTeM 23 Uncertainty of the final result (Multiple measurement) Rearranging Eq. 4 (RSS), we get the systematic uncertainty in terms of the combination of elemental systematic uncertainties, given by (11)

### Uncertainty of the final result (Multiple measurement) :

andims - FKM - UTeM 24 Uncertainty of the final result (Multiple measurement) Therefore, the total uncertainty estimate of the mean value of R is To estimate random uncertainty for multiple measurements, results are more reliable using the test results themselves, compared to auxiliary tests or combination of elemental uncertainties. Practical applications: The life of a light bulb, the life span of a certain brand of tyre or car engine (12)

### Uncertainty of the final result (Single measurement) :

andims - FKM - UTeM 25 Uncertainty of the final result (Single measurement) To deal with uncertainty of a single test result only Practical applications: measuring blood pressure/ heartbeat, speed of car, etc To estimate random uncertainty of the result, must use or combine auxiliary tests and elemental random uncertainties.

### Uncertainty of the final result (Single measurement) :

andims - FKM - UTeM 26 Uncertainty of the final result (Single measurement) Similar to Eq. 11, standard deviation of the result is given by Hence, the total uncertainty in the final result is given by (13) (14)

### Uncertainty of the final result (Single measurement) :

andims - FKM - UTeM 27 Uncertainty of the final result (Single measurement) For a large n, then t is independent of v, the degree of freedom, (and has a value of 2.0 for a 95% confidence level). For a small n, again using Welch-Satterthwaite formulation, we get (15)

### Slide 28:

andims - FKM - UTeM 28 Example The manufacturer of plastic pipes uses a scale with an Accuracy of 1.5% of its range of 5 kg to measure the Mass of each pipe the company produces in order to Calculate the uncertainty in mass of the pipe. In one batch Of 10 parts, the measurements are as follows: 1.93, 1.95, 1.96, 1.93, 1.95, 1.94, 1.96, 1.97, 1.92, 1.93 (kg) Calculate The mean mass of the sample The standar deviation of the sample and the standar deviation of the mean c. The total uncertainty of the mass of a single product at a 95% confidence level The total uncertainty of the average mass of the product at a 95% confidence level

### Slide 29:

andims - FKM - UTeM 29 Solution:

### Slide 30:

andims - FKM - UTeM 30

### Contents :

andims - FKM - UTeM 31 Contents Propagation of uncertainties Consideration of systematic and random components of uncertainty Sources of elemental error Uncertainty of the final result Design-stage uncertainty analysis Applying uncertainty-analysis in digital data acquisition system

### Design-stage uncertainty analysis (Based on ASME 1998) :

andims - FKM - UTeM 32 Design-stage uncertainty analysis (Based on ASME 1998) Define the measurement process State test objectives, identify independent parameters and their nominal values, etc List all elemental error sources To do a complete list of possible error sources for each measured parameter. Estimate the elemental errors Estimate the systematic uncertainties and standard deviations. If error is random in nature and/or data is available to estimate the std dev. of a parameter, then classify it as random uncertainties, which must have the same confidence level. For small samples, to determine degrees of freedom. Refer Table 1.

### Guideline to assign elemental error (Table 1), from Wheeler :

andims - FKM - UTeM 33 Guideline to assign elemental error (Table 1), from Wheeler *assume no. of samples > 30

### Design-stage uncertainty analysis (Based on ASME 1998) :

andims - FKM - UTeM 34 Design-stage uncertainty analysis (Based on ASME 1998) Calculate the systematic and random uncertainty for each measured variable Use the RSS formulation with data & procedure in Step 3. Propagate the systematic uncertainties and standard deviations all the way to the result(s) Use the RSS formulation to find the final test results, with the same confidence level in all calculations. Calculate the total uncertainty of the results Use the RSS formulation to find the total uncertainty of the result(s).

### Contents :

andims - FKM - UTeM 35 Contents Propagation of uncertainties Consideration of systematic and random components of uncertainty Sources of elemental error Uncertainty of the final result Design-stage uncertainty analysis Applying uncertainty-analysis in digital data acquisition system

### Applying uncertainty-analysis in digital data acquisition system :

andims - FKM - UTeM 36 Applying uncertainty-analysis in digital data acquisition system A digital DAS typically consists of sensor, sensor signal conditioner, amplifier, filter, multiplexer, A/D converter, Data reduction and analysis Problem may occur due to sequential components which may have different range from adjacent components. So, adjustment to uncertainty data must be done.

### Slide 37:

andims - FKM - UTeM 37 Another example In using a temperature probe, the following uncertainties were determined: Hysteresis ±0.10C Linearization error ±0.2% of the reading Repeatability ±0.20C Resolution error ±0.050C Zero offset ±0.10C Determine the type of these error (random or systematic) and the total uncertainty due to these effects for a temperature reading of 1200C

### Slide 38:

andims - FKM - UTeM 38 Assuming that the random errors have been determined with samples>30, So total uncertainty hysteresis systematic Lineariz.error systematic Resolution error 0.05C random zero off set 0.1C systematic repeatability random

### Slide 39:

andims - FKM - UTeM 39 Two resistors, R1=100.0 ±0.2 and R2=60.0 ±0.1 are connected (a) in series and (b) in parallel. Calculate the uncertainty in the resistance of the resultants circuits. What is the maximum possible error in each case?

### Slide 40:

andims - FKM - UTeM 40 (a) In series (b) In parallel

### Slide 41:

andims - FKM - UTeM 41 Another example: A mechanical speed control system works on the basis of centrifugal force, which is related to angular velocity through the formula: F=mrw2 where F is the force, m is the mass of the rotating weights, r is the radius of rotation, and w is the angular velocity of the system. The following values are measured to determine w : r=20± 0.02 mm, m=100 ± 0.5 g and F=500 ±0.1%N Find the rotational speed in rpm and its uncertainty. All measured values have a confidence level of 95%.

### Slide 42:

andims - FKM - UTeM 42 Solution

### Next Lecture :

andims - FKM - UTeM 43 Next Lecture Signal Conditioning End of Lecture 4