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Today. MonteCarlo Method using Excel using Labview. Montecarlo method in short. Make the problem replicable numerically Identify parameters uncertainty PDFs Generate M sample for each parameter Verify the obtained PDFs Compute M results Analize the results PDF.

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Today

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  1. Today • MonteCarlo Method • using Excel • using Labview

  2. Montecarlo method in short • Make the problem replicable numerically • Identify parameters uncertainty PDFs • Generate M sample for each parameter • Verify the obtained PDFs • Compute M results • Analize the results PDF

  3. Generating a random sample xm Triangularmean=estimatehalf width=a Uniformmean=estimatehalf width=a Normalmean=estimatevariance=σ² Studentmean=estimates.variance=s² a x xm a x xm xm

  4. a Generating a random sample 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.

  5. Basic assumption • We need to have access to a pseudorandom generator able to recreate a random number between 0 and 1with an UNIFORM distribution LABVIEW: C [0..1) EXCEL: =CASUALE() =RANDOM() 0 1

  6. Uniform distribution • Rescale and translate the 0-1 interval or • Use the inverse cumulate distribution xm Uniformmean=estimatehalf width=a a LABVIEW: EXCEL: =CASUALE()*2*a+(xm-a)

  7. Triangular distribution • Use the inverse cumulate distributionuse mode=(min+max)/2 in most cases Triangularmean=estimatehalf width=a LABVIEW: EXCEL: =SE(CASUALE()>0.5; xm-a+a*(2*casuale()/2)^0.5; xm+a-a*(2*casuale()/2)^0.5) We use the distributionsimmetry to simplifygeneration

  8. Normal distribution • Use the inverse cumulate distribution Normalmean=estimatevariance=σ² LABVIEW: EXCEL: =INV.NORM(CASUALE();xm;u) =INV.NORM.ST(CASUALE())*u+xm

  9. Student distribution • Describes the PDF of the AVERAGE of a set of samples, such as in repeated measurements Studentmean=estimates.variance=s² LABVIEW: EXCEL: =SE(CASUALE()>0.5;xm-s/n^0.5*INV.T(CASUALE());xm+s/n^0.5*INV.T(CASUALE())) We use the distributionsimmetry to simplifygeneration

  10. 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

  11. 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]:

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