METROLOGIA

Wykład 2

Literatura:

1. „Podstawy metrologii – wybrane zagadnienia” Mirosław Wereszko, Radosław Wereszko. ISBN 978-83-61234-20-3.

Niepewność złożona, to inaczej niepewność typu C.

Niepewność typu A zaokrąglamy do tylu miejsc po przecinku, jaka jest dokładność przyrządu służącego do pomiaru badanego obiektu.

OBLICZENIA

• Kula:

$$V = \frac{4}{3}\pi r^{3}$$

$$r = \frac{1}{2}S$$

$$V = \frac{4}{3}\pi\left( \left. \ \frac{S}{2} \right) \right.\ \ ^{3}$$

$$V = \frac{4}{3}\pi\frac{S^{3}}{8}$$

$$V = \frac{1}{6}\pi S^{3}$$

$$U_{\mathbf{c}}\left( V \right) = \ \frac{\partial V}{\partial S} \bullet U_{A}\left( S \right)$$

$$\frac{\partial V}{\partial S} = \left( \left. \ \frac{1}{6}\pi\ \bullet S^{3} \right)\ _{S}^{'} = \left( \left. \ \frac{\pi}{6} \right)^{'} \bullet S^{3} + \left( S^{3} \right)^{'} \right.\ \bullet \frac{\pi}{6} \right.\ $$

$$\left( \text{XV} \right)^{'} = X^{'}V + V^{'}\text{X\ \ \ \ \ \ \ }\overset{\Rightarrow}{\ }\text{\ \ \ \ \ \ \ \ \ \ \ }\left( \text{CX} \right)^{'} = C^{'}X + X^{'}C = C$$

(xⁿ)^′ = nx^n − 1

(S³)^′ = 3S^3 − 1 = 3S²

$$3S^{2} \bullet \frac{\pi}{6} = \frac{1}{2}\pi S^{2}\text{\ \ \ \ \ }\overset{\rightarrow}{\ }\text{\ \ Wynik\ obliczenia\ po}chodnej\ czastkowej.$$

$$U_{c}\left( V \right) = \frac{1}{2}\pi S^{2} \bullet U_{A}\left( S \right)$$

Gdy mamy jeden pomiar i przyrząd jest mianowany:

$$U_{c}\left( V \right) = \frac{1}{2}\pi S^{2} \bullet U_{B}\left( S \right)$$

• Walec:

V = πr² • H

$$r = \frac{1}{2}S$$

$$V = \pi\left( \left. \ \frac{S}{2} \right)\ ^{2} \bullet H \right.\ $$

$$V = \pi\frac{S^{2}}{4} \bullet H$$

$$U_{c}\left( V \right) = \sqrt{\left( \left. \ \frac{\partial V}{\partial S} \bullet U_{A}(S) \right)\ ^{2} \right.\ } + \left( \left. \ \frac{\partial V}{\partial H} \bullet U_{A}(H) \right)\ ^{2} \right.\ $$

$$\frac{\partial V}{\partial S} = \left( \left. \ \frac{1}{4}\pi H \bullet S^{2} \right)\ _{S}^{'}\text{\ \ \ \ } \right.\ $$

Bardzo wazne,  by pamietac o (  ) _S^′.  Jedyna zmienna jest S, H jest traktowana jako liczba 

(VX)^′ = V^′X + X′V

(CV)^′ = C^′V + V′C

$$\frac{\partial V}{\partial S} = \left( \left. \ \frac{1}{4}\pi H \bullet S^{2} \right)\ _{S}^{'} = \left( \left. \ \frac{1}{4}\text{πH} \right)^{'} \bullet S^{2} + \left( \left. \ S^{2} \right)' \bullet \right.\ \right.\ \frac{1}{4}\pi H = \left( \left. \ S^{2} \right)' \bullet \right.\ \ \frac{1}{4}\pi H = 2S \bullet \ \frac{1}{4}\pi H = \ \frac{1}{2}\text{SπH} \right.\ $$

( S²)^′=2S 

$$\frac{\partial V}{\partial H} = \left( \left. \ H \bullet \frac{1}{4}\pi S^{2} \right)\ _{H}^{'} = H' \right.\ \bullet \frac{1}{4}\pi S^{2} + \left( \left. \ \frac{1}{4}\pi S^{2} \right)' \right.\ \bullet H = \frac{1}{4}\pi S^{2}$$

(H¹)^′ = 1H^1 − 1 = H⁰ = 1

$$U_{c}\left( V \right) = \sqrt{\left( \left. \ \frac{1}{2}S\pi H \bullet U_{A}\left( S \right) \right)\ ^{2} + \left( \left. \ \frac{1}{2}S^{2} \bullet U_{A}\left( H \right) \right)\ ^{2} \right.\ \right.\ }$$

• Prostopadłościan:

V = abH

$$U_{c}\left( V \right) = \sqrt{\left( \left. \ \frac{\partial V}{\partial a} \bullet U_{A}(a) \right)\ ^{2} \right.\ } + \left( \left. \ \frac{\partial V}{\partial b} \bullet U_{A}(b) \right)\ ^{2} + \left( \left. \ \frac{\partial V}{\partial H} \bullet U_{A}(H) \right)\ ^{2} \right.\ \right.\ $$

$$\frac{\partial V}{\partial a} = \left( \text{abH} \right)\ _{a}^{'} = a^{'} \bullet bH + \left( \text{bH} \right)^{'} \bullet a = bH$$

$$\frac{\partial V}{\partial b} = \left( b \bullet aH \right)\ _{b}^{'} = b^{'} \bullet aH + \left( \text{aH} \right)^{'} \bullet b = aH$$

$$\frac{\partial V}{\partial H} = ab$$

$U_{c}\left( V \right) = \sqrt{\left( \left. \ bH \bullet U_{A}(a) \right)\ ^{2} + \left( \left. \ aH \bullet U_{A}(b) \right)\ ^{2} + \left( \left. \ ab \bullet U_{A}(H) \right)\ ^{2} \right.\ \right.\ \right.\ }$

• Walec wydrążony:

V_R = V_WS − V_Wp

V_WS −   Objetosc walca stalowego

V_Wp −   Objetosc walca powietrznego

V = πr² • H

$$V_{R} = \ \pi\left( \left. \ \frac{S_{M}}{2} \right)\ ^{2} \bullet H - \pi\left( \left. \ \frac{S_{p}}{2} \right)\ ^{2} \bullet H \right.\ \right.\ $$

$$V_{R} = \frac{1}{4}\text{πH}\left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ $$

$$U_{c}\left( V \right) = \sqrt{\left( \left. \ \frac{\partial V_{R}}{\partial S_{M}} \bullet U_{A}(S_{M}) \right)\ ^{2} \right.\ } + \left( \left. \ \frac{\partial V_{R}}{\partial S_{p}} \bullet U_{A}(S_{p}) \right)\ ^{2} + \left( \left. \ \frac{\partial V_{R}}{\partial H} \bullet U_{A}(H) \right)\ ^{2} \right.\ \right.\ $$

$$\frac{\partial V_{R}}{\partial S_{M}} = \left( \left. \ \frac{1}{4}\text{πH}\left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ \right) \right.\ \ _{S_{M}}^{'} = \left( \left. \ \frac{1}{4}\text{πH} \right)' \bullet \right.\ \left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ + \left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ ^{'} \bullet \frac{1}{4}\pi H = 2S_{M} \bullet \frac{1}{4}\pi H = \frac{1}{2}S_{M}\text{πH}$$

( S_M ²−S_p ²) ^′ = (S_M)^′ − (S_p)^′ = 2S_M

$$\frac{\partial V_{R}}{\partial S_{p}} = \left( \left. \ \frac{1}{4}\text{πH}\left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ \right) \right.\ \ _{S_{p}}^{'} = \left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ ' \bullet \left( \left. \ \frac{1}{4}\text{πH} \right) + \left( \left. \ \frac{1}{4}\text{πH} \right)^{'} \bullet \left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ = \left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ ' \bullet \left( \left. \ \frac{1}{4}\text{πH} \right) \right.\ = (0 - 2S_{p})\left( \left. \ \frac{1}{4}\text{πH} \right) = - 2S_{p} \bullet \frac{1}{4}\pi H = - \frac{1}{2}S_{p}\text{πH} \right.\ \right.\ \right.\ $$

$$\frac{\partial V_{R}}{\partial H} = \left( \left. \ H \bullet \frac{1}{4}\text{πH}\left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ \right)\ _{H}^{'} \right.\ = \left( H \right)^{'} \bullet \frac{1}{4}\text{πH}\left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ + \left( \left. \ \frac{1}{4}\text{πH}\left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ \right)^{'} \bullet H = 1 \bullet \frac{1}{4}\text{πH}\left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ + 0 = \frac{1}{4}\text{πH}\left( \left. \ S_{M}\ ^{2} - S_{p}\ ^{2} \right) \right.\ \right.\ $$

POCHODNA CZĄSTKOWA

$$S = \frac{\alpha \bullet \beta}{\gamma \bullet}$$

$$\frac{\partial S}{\partial\alpha} = \left( \left. \ \frac{\alpha \bullet \beta}{\gamma \bullet} \right)\ _{\alpha}^{'} \right.\ = \left( \left. \ \alpha \bullet \frac{\beta}{\gamma} \right)\ _{\alpha}^{'} = \right.\ \frac{\beta}{\gamma}$$

$$\frac{\partial S}{\partial\alpha} = \left( \left. \ \frac{\alpha \bullet \beta}{\gamma \bullet} \right)\ _{\alpha}^{'} \right.\ = \frac{\left( \left. \ \alpha \bullet \beta \right)'\left( \left. \ \gamma \bullet \right) + \left( \left. \ \gamma \bullet \right) \right.\ '\left( \left. \ \alpha \bullet \beta \right) \right.\ \right.\ \right.\ }{\left( \left. \ \gamma \bullet \right) \right.\ \ ^{2}} = \frac{\beta}{\gamma}$$

( α•β) ^′ = α^′β + β^′α = β

$$\frac{\partial S}{\partial\gamma} = \left( \left. \ \frac{1}{\gamma} \bullet \frac{\alpha \bullet \beta}{} \right)\ _{\gamma}^{'} = \left( \left. \ \gamma^{- 1} \bullet \frac{\alpha \bullet \beta}{} \right)\ _{\gamma}^{'} = - x^{- 2} \bullet \frac{\alpha \bullet \beta}{} = - \frac{\alpha \bullet \beta}{x^{2}} \right.\ \right.\ $$

(x⁻¹) = −1x^−1 + 1 = −x⁻²

NIEPEWNOŚĆ ZŁOŻONA WIELKOŚCI SKORELOWANYCH

$$r_{U_{i}}l = \frac{\sum_{i = 1}^{N}\left( \left. \ U_{i} - \overset{\overline{}}{U} \right) \bullet \left( l_{i} - \overset{\overline{}}{l} \right) \right.\ }{\sqrt{\sum_{i = 1}^{N}{\left( \left. \ U_{i} - \overset{\overline{}}{U} \right) \right.\ \ ^{2} \bullet \sum_{i = 1}^{N}{\left( l_{i} - \overset{\overline{}}{l} \right)\ ^{2}}}}}$$

$$U_{c}\left( R \right) = \sqrt{\left( \left. \ \frac{\partial R}{\partial U} \bullet U_{A}(U) \right)\ ^{2} \right.\ } + \left( \left. \ \frac{\partial R}{\partial I} \bullet U_{A}(l) \right)\ ^{2} + 2 \bullet \frac{\partial R}{\partial U} \bullet \right.\ \frac{\partial R}{\partial I} \bullet U_{A}(U) \bullet U_{A}(l) \bullet r_{U_{i}}l$$

WYKRES: