Example Calculations of Uncertainty

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Abigayle James
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1 nertainty Examples. Example alulations of nertainty Instrument Bias nertainty Example # A D voltage of -.65 volts is measured with an Agilent #05A Multimeter. Aording to the data sheet (p.5.) the D auray of this meter in the V range is ±0.05% of the reading ±0.005% of full sale. Therefore the instrument or bias unertainty of this D voltage is V.65V V 0.006V V 0. 00V This value was rounded to the ten-thousandths position 0.00V to math the nominal value s least signifiant digit in the ten-thousandths position.65v. Instrument Bias nertainty Example # A D voltage of -.65 volts is measured with a G Preision #DM-B True M Digital Multimeter. Aording to the data sheet (p.5.) the auray of this meter in the 0-0V range is 0.% digits. The ±0.% value applies to the reading (-.65 volts) and the ± digits apply to the resolution whih is mv = 0.00 V at this setting. Therefore the instrument or bias unertainty of this D voltage is 0. V.65V 0.00V 0.065V V 0. 0V 00 This value was rounded to the thousandths position 0.0V to math the nominal value s least signifiant digit in the thousandths position.65v. Instrument Bias nertainty Example # An A voltage of.6 volts is measured with a lue 5 Digital Multimeter. Aording to the data sheet (p.5.) the auray of this meter in the.- V range is % digits. The ±% value applies to the reading (.6 volts) and the ± digits apply to the resolution whih is 0.0 V at this setting. Therefore the instrument or bias unertainty of this A voltage is V.6V 0.0V 0.5V V 0. 5V 00 This value was rounded to the hundredths position 0.5V to math the nominal value s least signifiant digit in the hundredths position.6v. Instrument Bias nertainty Example # The same A voltage of.6 volts (at 00 Hz) is measured with a G Preision #DM-B True M Digital Multimeter. Aording to the data sheet (p.5.) the auray of this meter in the 0-00 V range is 0.5% 0digits and the resolution is 0 mv = 0.0 V. The instrument or bias unertainty of this A voltage is

2 nertainty Examples. 0.5 V.6V 00.0V 0.085V V 0. V 00 This value was rounded to the hundredths position 0.V to math the nominal value s least signifiant digit in the hundredths position.6v. Instrument Bias nertainty Example #5 A resistane of. is measured with an Agilent #05A Multimeter. Aording to the data sheet (p.5.) the auray of this meter in the range is ±0.05% of the reading ±0.00% of full sale. The instrument or bias unertainty of this resistane measurement is therefore This value was rounded to the thousandths position 0.0 to math the nominal value s least signifiant digit in the thousandths position Instrument Bias nertainty Example #6 A resistane of ohms is measured with a alibeur DT80D Digital Multimeter. Aording to the data sheet (p.5.5) the auray of this meter in the range is 0.8% digits and the resolution is. The instrument or bias unertainty of this resistane measurement is This value was rounded to the ones position to math the nominal value s least signifiant digit in the ones position. Instrument Bias nertainty Example # A apaitane of 8. n is measured with an Agilent #05A Multimeter. Aording to the data sheet (p.5.) the auray of this meter in the n range is ±% of the reading ±0.5% of full sale. The instrument or bias unertainty of this apaitane measurement is therefore This value was rounded to the tenths position signifiant digit in the tenths position 8. n. 8.n 00.0n.n. n.n to math the nominal value s least

3 nertainty Examples. nertainty Example # The hoop stress in a thin-walled ylinder is found by Pd t P d where P is the interior pressure d is the ylinder diameter and t is the wall thiness. If we assume that eah of the three terms on the right ontributes to the overall unertainty in stress then P P t d d t t Nominal values for this problem are P = 0 psi d = inh t = inh. The nominal value for hoop stress is lbf 0.50in in in lbf 50 in ine the equation for stress is a simple polynomial in eah of the three variables the simplified form wors for this problem P d t P d t The numerial value of the unertainty in hoop stress is % lbf 0.08* 0.08* 50 in 580 psi or % Note that the first term in the equation above ( ) was due to the unertainty in the pressure measurement and that this term ontributed the most to the overall unertainty in the hoop stress. Any attempt to redue the overall unertainty in the measurement of stress in this experiment should begin with the pressure measurement.

4 nertainty Examples. nertainty Example # An aluminum rod of diameter ( 0.00) inh is 5. ( 0.0) inhes long. A onentrated mass of.0 ( 0.0) lbm is attahed to the end of the rod. Estimate the natural frequeny of the system and the unertainty in the estimate. Assume that the modulus of elastiity for aluminum is 0.E6 ( %) psi. The natural frequeny of a massless beam with a onentrated mass M at the end is given by the following formula EI M where E = modulus of elastiity of the beam material (lbf/in) I = area moment of inertia of the beam (in) = antilever length (in) M = onentrated mass (lbm g slugs or lbf-se /ft) = natural frequeny (rad/se or se-). The fator of is developed during the solution to a th order partial differential equation and is a dimensionless quantity. The remaining terms must therefore have units of radians/se sine this is the units of. Assume the beam has a uniform irular ross-setion. The area moment of inertia I is therefore I 6 d ubstituting this value for I into the equation for we have E d 6M 6 / E / d / M / / If we assume that eah of the parameters (the "" and "6" terms are exat onstants) on the right-hand side of the equation ontributed to the overall unertainty in we would have the following equation E E d d M M ine the equation for stress is a simple polynomial in eah of the three variables the simplified form wors for this problem

5 nertainty Examples.5 E E d d M M alulating the Nominal Value Inorret Method: E d (0.E6) (0.500) 6M 6(.0)(5.) rad.96 se alulating the Nominal Value orret Method: lbf ft lbm 0.E in Ed in lbf se 6M 6(.0lbm)(5.in) in ft rad.8 se The unertainty is found from E E d d M M % 00% 0.00in in % lbm.0 lbm 0.0in 5. in The nominal value for frequeny was.8 rad/se therefore the unertainty in frequeny is (.8rad / se) 0.5rad / se

6 nertainty Examples.6 nertainty Example # A first order system onsisting of two resistors in series and two apaitors in parallel is shown in the figure on the right. The time onstant is given by esistor values are = and =. apaitor values are = and = The nominal time onstant is therefore 0.00se se farad volt amp ohm amp volt farad x farad x ohm x ohm x The unertainty in the time onstant is given by This is an example of a problem that annot be wored the easy way due to the additions in the formula. There is a lot of symmetry in the problem whih maes it relatively easy to find the four partial derivatives. Dividing eah term by the time onstant gives ubstituting into the equation for unertainty ubstituting numerial values into the equation.9% se se E o (t) + - E i (t)

7 nertainty Examples. nertainty Example # There is another way to wor the previous problem with the first order system onsisting of two resistors in series and two apaitors in parallel. esistor values are = = and =. apaitor values are = = and = 0.. The unertainty in the time onstant is given by Note the that easy form does apply when we define the etive resistane and apaitane. alulations for the unertainty in the etive resistane and apaitane are relatively easy The alulation for the unertainty in the time onstant is now quite simple % se se.09 0

8 nertainty Examples.8 nertainty Example #5 eries resistane for seven resistors an be alulated from:... The simplified or easy method does not apply to alulating unertainty for series resistane due to the additions. alulations for the unertainty in the series resistane is relatively easy though using the partial derivative method:... The partial derivatives are easily determined from the series resistane formula above... ubstituting the partial derivatives into the partial derivative form of the omputed unertainty gives:...

Torsion. Torsion is a moment that twists/deforms a member about its longitudinal axis

Torsion. Torsion is a moment that twists/deforms a member about its longitudinal axis Mehanis of Solids I Torsion Torsional loads on Cirular Shafts Torsion is a moment that twists/deforms a member about its longitudinal axis 1 Shearing Stresses due to Torque o Net of the internal shearing