UNCERTAINTY ANALYSIS Significant Figures: The number of significant figures used in expressing the results of a measurement is an indication of the accuracy of that measurement. For example, if a mass is specified as having a value of 12 kg as m = 12 kg then its true value is estimated to be closer to 12 kg than to 11 kg or 13 kg. If this information were given as m = 12.0 kg then its true value would be estimated to be closer to 12.0 kg than to 12.1 kg or 11.9 kg. In 12 kg there are 2 “significant figures” whereas in 12.0 kg there are 3. Note that the number of significant figures can be considered to be infinite for whole numbers (exact numbers); e.g., the number “2” in the denominator of the following formula: m v2 E = 2 Page 29

ME 410 MECHANICAL ENGINEERING SYSTEMS LABORATORY On the other hand, the “total number of digits” may not always correspond to “number of significant figures”, either. This is encountered in two distinct cases: 1. There exist trailing zeros with whole numbers; e.g., x = 600 cm (total number of digits is 3) may mean 550 ≤ x < 650 or 595 ≤ x < 605 or even 599.5 ≤ x < 600.5 In order to clarify this ambiguity, the so-called “scientific notation” with the powers of 10 is employed as: for 1 significant figure 6 102 2 6.0 10 for 2 significant figures 2 6.00 10 for 3 significant figures 2. There exist leading zeros with decimal fractions; e.g., x = 0.032 mm

(total number of digits is 4)

has only 2 significant figures. If the doubt (uncertainty) in values is not equal to the value corresponding to the “least significant digit”, then either • absolute, or • relative value of this uncertainty has to be indicated as: T = 186 ± 8 oC T = 186 ± 4.3 % oC Page 30

ME 410 MECHANICAL ENGINEERING SYSTEMS LABORATORY In computations of measurement data, often it is dealt with quantities having unequal numbers of significant figures. Then the following “rules of thumb” must be used in order to express the results in a meaningful manner: Addition and Subtraction: The result is only as accurate as the least accurate measurement (with the least number of significant fractional digits); hence, it should be given with the same decimal place (significant fractional digits) as the least accurate number; e.g., p1 = 18.7 kPa (3 significant figures but less accurate) p2 = 0.121 kPa (3 significant figures but more accurate) give p1+p2 = 18.821 kPa = 18.8 kPa (1 significant fractional digit)

Multiplication and Division: The total number of digits increase rapidly in multiplication and may even become infinite in division. However, the result should be given with the same number of significant figures as with the number of least significant figures; e.g., F = 172.8 N (4 significant figures & less accurate) v = 0.383 m/s (3 significant figures & more accurate) give P = F.v = 66.1824 W = 66.2 W (3 significant figures)

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ME 410 MECHANICAL ENGINEERING SYSTEMS LABORATORY

Propagation of Uncertainty (Combined Error): When an experimental result is computed through a functional relationship as y = f(x1, x2, … , xn) where x1, x2, … , xn represent measurements, it is often required to determine both the individual and overall contributions of uncertainties ∆xi in xi (i=1,2,…,n) on y. If one defines the uncertainty in y due to ∆xi (i=1,2,…n) as: ∆y = f(x1+∆x1, x2+∆x2, … , xn+∆xn) - f(x1, x2, … , xn) the Taylor series expansion of the first term on the right hand side gives: ∆y ≅

or simply

∂f ∂f ∂f ∆x1 + ∆x 2 + ... + ∆x n ∂x1 ∂x 2 ∂x n n

∆y ≅

∑ i =1

∂f ∆x i ∂x i

assuming that uncertainties ∆xi are small so that quantities of order (∆xi)2 and higher are much smaller than those of (∆xi) hence can be neglected.

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ME 410 MECHANICAL ENGINEERING SYSTEMS LABORATORY

Maximum Uncertainty: It is the “ultimate upper bound of the total error” and can be found as: n

( ∆y)max =

∑ i =1

∂f ∆x i ∂x i

Expected Uncertainty: If the quantities ∆xi’s are considered not as absolute limits of error but rather as “statistical bounds of errors in independent measurements”, then probable errors or uncertainties do not provide odds favoring simultaneous additive contributions. In such a case, a more realistic and mathematically proven expression for the “expected uncertainty” in y is given in “root mean square (rms)” manner as: n

( ∆y)exp =

∑ i =1

(

∂f ∆x i ∂x i

)

2

Note that ( ∆y)exp < (∆y)max always.

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ME 410 MECHANICAL ENGINEERING SYSTEMS LABORATORY

Example: x1 = 7 ± 0.040 cm

z

x2 = 5 ± 0.020 cm

x3

x3 = 5 ± 0.030 cm

x1

x2

The nominal value of z is found as z = f(x1, x2, x2) = [(x1+x2)2+x32]½ = 13 cm The respective partial derivatives are calculated as: x + x 2 12 ∂f = 1 = ∂x1 z 13 x + x 2 12 ∂f = 1 = 13 ∂x 2 z ∂f x 5 = 3 = ∂x 3 z 13

The maximum uncertainty is calculated as: (∆z)max=±[(12/13)*0.040+(12/13)*0.020+(5/13)*0.030)]= ± 0.067 cm

Hence

z = 13 ± 0.067 (max) cm

The expected uncertainty is calculated as: (∆z)exp = ± {[(12/13)*0.040] 2 + [(12/13)*0.020]2 + [(5/13)*0.030)]2}½ = ± 0.043 cm

Hence

z = 13 ± 0.043 (exp) cm

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ME 410 MECHANICAL ENGINEERING SYSTEMS LABORATORY

Example: The maximum stress at the root is given by 6FL σ= t bt2

F L b

When dealing with product functions, it is easier to work with the “relative uncertainty”. Therefore, let the relative error constituents be known as: ∆F/F=±0.10;

∆L/L=±0.05;

∆b/b=±0.05;

∆t/t = ±0.08

Then, an expression for the relative uncertainty in the root stress can be obtained by taking the “ln” first and then “differential” of its equation given above as: ln σ = ln 6 + ln F + ln L - ln b - 2 ln t (dσ/σ) = (dF/F) + (dL/L) - (db/b) - 2 (dt/t) Then, the maximum relative uncertainty in σ is calculated as: (∆σ/σ)max = ± (0.10+0.05+0.05+2*0.08) = ± 0.36

and the expected relative uncertainty in σ is calculated as: (∆σ/σ)exp = ± [(0.10) 2 + (0.05)2 + (0.05)2 + (2*0.08)2]½ = ± 0.20

Notice the unexpectedly increased uncertainty level in the root stress as compared to uncertainty levels given for the measurements in F, L, b, and t. Page 35