Today

1. Today • MonteCarlo Method • Combining uncertainties

2. Uncertainty: GUM approach SIMPLIFIED FORM: • To be used only if: • Data reduction equation is a closed form equation • Result function is linearly approximable near its estimated value • Parameters involved aren’t interdependent But what if some of the constrains aren’t met?

3. Uncertainty: GUM approach PROBLEM: • Sometimes solving the equation in closed form is not possible at all for instance when determining the temperature procedure of a refrigeration wing: how to get mean h coefficient? t w L

4. Uncertainty: GUM approach • Measuring, in stationary conditions, the surface temperature T0=37.5°C at the junction, and at the same time, temperature Tx at an x distance x=30mm, I need to estimate the convective thermal exchange coefficient h of a cantilever in stale air at Tinf=23°C. • The cantilever is aluminium made (k=204W/mK) and long 414mm, having a rectangular section of 20mmx5.5mm • Measurement of Tx is repeted 10 times obtaining the following results, in °C:35.2 35.434.8 35.035.2 34.935.0 35.135.3 34.8

5. GUM Approach review ADVANTAGES: • Easiness of use, simple calculus required • Gives access to useful indicators (UPC,UMF) • Allows for a global vision of the measurement DISADVANTAGE: • Approximates to the first order • Requires a closed form data reduction equation • Can hide the degrees of freedom

6. a σ x Uncertainty: Montecarlo method 1) Make the system replicable automatically • Identify the input variables and their uncertainty PDFs [as for GUM approach, for A uncertainty use Student]

7. σ x Uncertainty: Montecarlo method 3) Generate numerically M casual samples for each of the N input variables following their PDF, obtaining an MxN matrix(GUM rule M>30 000, best situation M=100 000) • Please check that the casual samples follow the desired PDF after random generation K1….KM k

8. a Uncertainty: Montecarlo method How can I extract from a PDF a random number? Using a pseudorandom generator that follows uniform distribution AND the cumulate PDF. For cumulate PDF is usually possible to use built-in function (eg INV.T)of spreadsheet and calculus programs. Symmetry of most distributions simplifies their implementation.

9. Uncertainty: Montecarlo method 4) Apply the reduction system automatically to each row of the NxM matrix, thus obtaining M samples of the measurement result. Usually this means replicating M times the solving algorithm.

10. Uncertainty: Montecarlo method • Plot an histogram of the M “simulated” results. • Extract from the histogram:mean value (average) => estimatestandard deviation => standard uncertaintypercentile (1-a/2)-P(a/2) => extended uncertainty

11. Montecarlo method: PRACTICE EXERCISE 4: simple statistics • Simulate the rolling of one eight faces dice and check if it conforms to the uniform PDF using an histogram • Simulate the rolling of 3d8 and plot the result of 10000 simulations using an histogram in order to describe the PDF of the result EXERCISE 5: uncertainty estimation • Look at exercise 1 of GUM review (Shear Modulus) and check whether the results are consistent with the MonteCarlo approach

12. Exercise 1: Shear Modulus Given the picture below, I want to measure the G elastic shear modulus of the steel beam shown, applying a T torque and measuring the ϑ angular displacement produced. θ a 2R F L

13. Exercise 1: Shear Modulus I collected the following informations about the parameters involved: 2R 16 mm (1/20 caliper) L 1 m (production tolerance ±10mm) 2a 240 mm (ruler - 1 mm stepped) ϑ 0.81 rad (optical encoder with 360 units) F is measured using repeted measurements with a digital dynamometer, giving the following results [in N]:

14. Exercise 6: Pin On Disk We were asked to measure the load applied in a PIN-DISK contact during friction tests. The load is given by an hydraulic actuator using a pressure multiplier as shown. Knowing the diameter shown was measured using 1/20 calliper, and considering the working pressures shown measured using a transducer with 1% overall uncertainty declared and 3MPa full scale estimate the frictioncoefficientf at the pin/disk contact. Suppose lateralforcemeasuredusing a loadcellseveraltimes, obtaining the followingmeasurements: 59.996 N 60.041 N 60.012 N 59.983 N 58.044 N p1=2.5MPa d0=10mm FT LoadCell

15. Uncertainty: Montecarlo method ADVANTAGES • Suitable for strongly non linear systems • Does not require approximations • Do not need a closed form equation (solving algorithms are allowed and welcomed) DISADVANTAGES • “heavy” numerical implementation • Getting UMF and UPC is quite difficult • Result stability depends on samples number

16. Uncertainty: Weighted Average Repeated measurements with different uncertainty It’s a particular application of the least square theory: each measurement result is weighted on the inverse of its uncertainty. Choose a tentative value for σ0 then check its consistency with the chi-square test

17. Uncertainty: Weighted Average EXERCISE 7: Temperature using different transducers Using different measurement techniques the same temperature is measured at the same time, obtaining: • Tc1:25.2±1.0°C (P=95%) • Tc2:25.2±0.5°C (P=95%) • Tc3:25.2±0.5°C (P=99%) Estimate the temperature and its uncertainty as a weighted average of the three inputs. Before using a set of measurement check whether they are compatible or not.