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/dokumenty/PRA1/2-pruznost/data.ods
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/dokumenty/PRA1/2-pruznost/hook.txt
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/dokumenty/PRA1/2-pruznost/hooko.pdf
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/dokumenty/PRA1/2-pruznost/hookz.pdf
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%%Trailer
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%%Pages: 1
/dokumenty/PRA1/2-pruznost/plot.gp
0,0 → 1,37
set terminal postscript
 
set key off
set xlabel "hmotnost [g]"
set ylabel "prodlouzeni [10x um]"
 
#hookuv zakon a korekce na spravne jednotky.
 
f(x)=((4*x/1000*9.81*1.15)/(E*pi*0.000195**2))*100000
 
set output "hookz.ps"
 
fit f(x) "hook.txt" using 1:2 via E
plot f(x), "hook.txt" using 1:2
 
set output "hooko.ps"
fit f(x) "hook.txt" using 1:3 via E
plot f(x), "hook.txt" using 1:3
 
!ps2pdf hookz.ps
!ps2pdf hooko.ps
 
 
set output "pruhyb.ps"
set ylabel "pruhyb [mm]"
#f(x)=(-(x/1000*9.81*0.498**3))/(4*pi*0.000195**2))*100000
plot "hook.txt" using 1:2, "hook.txt" using 1:3
!ps2pdf pruhyb.ps
 
set output "torze.ps"
set ylabel "zkrut [°]"
plot "torze.txt" using 1:2, "torze.txt" using 1:3
!ps2pdf torze.ps
 
 
 
 
/dokumenty/PRA1/2-pruznost/pruhyb.pdf
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/dokumenty/PRA1/2-pruznost/pruhyb.txt
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/dokumenty/PRA1/2-pruznost/pruznost.pdf
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/dokumenty/PRA1/2-pruznost/pruznost.tex
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\oddsidemargin 0cm
\pagestyle{empty}
\begin{document}
\title{Cavendishův experiment}
\title{Měření modulu pružnosti ve smyku a tahu}
\author{Jakub Kákona, kaklik@mlab.cz}
\date{19.11.2009}
\maketitle
28,16 → 28,191
\end{enumerate}
 
\section{Úvod}
\subsection{Modul pružnosti v tahu}
Při působení tažné síly na pružné homogenní těleso se jeho rozměr v ose síly prodlužuje podle vztahu.
 
\begin{displaymath} \frac{F}{S} = E \frac{\Delta l}{l}, \end{displaymath}
 
Který nazýváme Hookův zákon. Konstanta $E$ je určena jen vlastnostmi materiálu a nazývá se modul pružnosti v tahu nebo Youngův modul.
Při protahování tělesa se však jeho rozměry kolmé k ose jeho prodloužení zkracují podle vztahu.
 
\begin{displaymath} \frac{\Delta a}{a} = \frac{\Delta b}{b} = \mu \frac{\Delta l}{l} , resp. \frac{\Delta r}{r} = \mu \frac{\Delta l}{l} , \end{displaymath}
 
kde $\mu $ je Poissonovo číslo (nezávislé na $E)$. Poissonovo číslo $\mu $ je v intervalu $\langle $0, 1/2$\rangle $; hodnotu 1/2 nabývá pro nestlačitelné materiály. Protože v našem případě platí
 
\begin{displaymath} S = \pi d^2 \end{displaymath}
a
\begin{displaymath} F = mg \end{displaymath}
 
kde $d$ je průměr drátu a $m$ hmotnost závaží dostaneme po úpravě vztah
 
\begin{equation}
E = \frac{4mgl}{\pi d^{2} \Delta l}
\label{hook_rovnice}
\end{equation}
 
 
\subsection{Modul pružnosti v tahu měřený z ohybu nosníku}
Nosník známé délky $L$ podepřený na obou koncích se při zátěži uprostřed prohýbá podle vztahu.
 
\begin{displaymath} z(0) = - \frac{F L^3}{48 E I}. \end{displaymath}
 
Kde $I$ je plošný moment setrvačnosti určený profilem nosníku. Pro nosník obdélníkového průřezu je roven.
 
\begin{displaymath} I = \frac{ab^3}{12} \end{displaymath}
 
Po sloučení obou výrazů a vyjádření $E$ dostáváme
 
\begin{displaymath} E = - \frac{mgL^3}{4ab^3} \end{displaymath}
 
\subsection{Modul pružnosti ve smyku měřený statickou torzí}
Při zatížení drátu momentem síly
\begin{displaymath} M = 2mgr \end{displaymath}
 
Se jeho konec bude stáčet o úhel $\Phi$ Tento úhel bude záviset na momentu síly a poloměru drátu. Tím je určen modul pružnosti ve smyku $G$
 
\begin{displaymath} G = \frac{2mL}{\pi R^4 \Phi} \end{displaymath}
 
Po dosazení za moment síly dostáváme výraz
\begin{displaymath} G = \frac{2rgmL}{\pi R^4 \Phi} \end{displaymath}
 
\subsection{Modul pružnosti ve smyku měřený torzním kyvadlem}
Stočením drátu o úhel $\Phi$ bude kyvadlo působit momentem síly o velikosti
 
\begin{displaymath} M = K \Phi \end{displaymath}
 
kde $K$ je direkční moment kyvadla, pro který platí
 
\begin{displaymath} K = \frac{G \pi R^4}{2L} \end{displaymath}
 
Kde $R$ je poloměr drátu a $L$ jeho délka. Tento moment bude stáčet drát zpět do rovnovážné polohy a tím vznikne kmitavý pohyb s periodou
 
\begin{displaymath} T = 2 \pi \sqrt{\frac{I}{K}} \end{displaymath}
Kde $I$ je moment setrvačnosti kyvadla. Z vlastností $K$ plyne vztah
 
\begin{displaymath} \frac{I_1}{T_1^2} = \frac{I_2}{T_2^2} \end{displaymath}
 
Kde $I_1$ a $I_1$ jsou momenty setrvačnosti kyvadla pro dvě různé vzdálenosti závaží od osy.
 
\section{Postup měření}
 
Měření modulu pružnosti v tahu jsme měřili napínáním svisle pověšeného drátu závažími o hmotnosti 101g průtah drátu jsme zjistili pomocí měřících hodinek. Naměřené hodnoty ukazuje tabulka \ref{hook}.
 
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
počet zavaží & hmotnost[g] & zatezovani[10um] & odlehcovani[10um] \\ \hline
1 & 101 & 51 & 59 \\ \hline
2 & 202 & 77 & 81 \\ \hline
3 & 303 & 96 & 100 \\ \hline
4 & 404 & 114 & 118 \\ \hline
5 & 505 & 133 & 135 \\ \hline
6 & 606 & 151 & 153 \\ \hline
7 & 707 & 169,5 & 170 \\ \hline
8 & 808 & 183,5 & 187 \\ \hline
9 & 909 & 202 & 202 \\ \hline
10 & 1010 & 221 & 221 \\ \hline
\end{tabular}
\end{center}
\label{hook}
\caption{Prodlužování a zkracování drátu }
\end{table}
 
Proložením naměřených dat funkcí \ref{hook_rovnice} jsme dostali hodnotu $E = (1.59619e+11 \pm 6.539e+09)$ Pa.
 
Proložení ukazuje graf \ref{Graf_hookz} a \ref{Graf_hooko}.
 
\begin{figure}
\begin{center}
\includegraphics[width=150mm]{hookz.pdf}
\end{center}
\caption{prodloužení drátu délky 1,15m v závislosti na zátěži}
\label{Graf_hookz}
\end{figure}
 
\begin{figure}
\begin{center}
\includegraphics[width=150mm]{hooko.pdf}
\end{center}
\caption{prodloužení drátu délky 1,15m v závislosti na zátěži během odlehčování}
\label{Graf_hooko}
\end{figure}
 
Průhyb nosníku jsme měřili zatěžováním hranolu o rozměrech 498x10,1x4,1 mm. Hodnotu jsme měřili mikroskopem v prostřední části. Naměřené hodnoty uvádí tabulka \ref{pruhyb}.
 
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
zatez[g] & pruhyb[mm] & pruhyb[mm] \\ \hline
101 & 2,53 & 0,06 \\ \hline
202 & 5,05 & 0,13 \\ \hline
303 & 7,58 & 0,19 \\ \hline
404 & 10,1 & 0,25 \\ \hline
505 & 12,63 & 0,32 \\ \hline
606 & 15,15 & 0,38 \\ \hline
707 & 17,68 & 0,44 \\ \hline
808 & 20,2 & 0,51 \\ \hline
909 & 22,73 & 0,57 \\ \hline
1010 & 25,25 & 0,63 \\ \hline
\end{tabular}
\end{center}
\caption{Průhyb nosníku}
\label{pruhyb}
\end{table}
 
Grafické vyjádření je v grafu \ref{pruhyb_graf}.
 
 
\begin{figure}
\begin{center}
\includegraphics[width=150mm]{pruhyb.pdf}
\end{center}
\caption{Průhyb nosníku}
\label{pruhyb_graf}
\end{figure}
 
Zde nám vyšla hodnota modulu pružnosti $E = 165,66$ GPa.
 
Měření modulu pružnosti ve smyku jsme provedli zkrutem drátu delky 665mm, a průměru 1,99mm naměřené výsledky jsou v tabulce \ref{torze_static} a grafu \ref{Torze_graf}.
 
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
hmotnost & Zatezovani [°] & Odlehcovani [°] \\ \hline
0 & 290 & 291 \\ \hline
101 & 302 & 305 \\ \hline
202 & 312 & 317 \\ \hline
303 & 324 & 329 \\ \hline
404 & 336 & 341 \\ \hline
505 & 350 & 350 \\ \hline
\end{tabular}
\end{center}
\caption{Statická torze drátu}
\label{torze_static}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=150mm]{torze.pdf}
\end{center}
\caption{Torze drátu}
\label{Torze_graf}
\end{figure}
 
Po výpočtu nám vyšel modul pružnosti ve smyku 91 GPa.
 
Posledním měřením bylo požití torzního kyvadla k měření modulu pružnosti ve smyku. Zde jsme zvolili dvě různé vzdálenosti závaží a změřili pro ně periodu kmitů.Pro vzdálenost závaží 31mm nám vyšla perioda 5,91s a pro vzdálenost 222mm 16,8s. Po zjištění potřebných momentů setrvačnosti již bylo možné vypočítat hodnotu modulu pružnosti ve smyku 83,2 GPa.
\section{Diskuse}
Při měření bylo poměrně komplikované určit správné rozměry měřených profilů. zvláště pak výšku nosníku kde bylo třeba mikrometrem měřit přesně kolmo k jeho ose, tuto jsem vyhodnotil jako nepřesnou a výšku se pokusil změřit posuvným měřítkem o kterém mám ale nyní podezření, že mohlo proměřovat až o 0,2mm, což je hodnota kterou bych od měřícího prostředku tohoto typu nečekal. Výsledkem je, že měření jsou zatížena poměrně neznámou chybou určení rozměrů. Na druhou stranu si ale myslím, že výsledné moduly pružnosti vyšly v celku reálné, takže konečná chyba nebude velká.
 
\section{Závěr}
Z výsledků měření vidíme že hodnota modulu pružnosti v tahu se u ocelových materiálů pohybuje okolo 160 GPa zatímco, modul pružnosti ve smyku má menší hodnoty v oblasti 80 GPa. Tento fakt vysvětluje známou snahu konstruktérů ocelové díly namáhat pouze na tah a vyhnout se namáhání na zkrut a střih.
 
 
\begin{thebibliography}{99}
\bibitem{pruznost}{Zadání úlohy 2 - Měření modulu pružnosti v tahu a modulu pružnosti ve smyku} \href{http://praktika.fjfi.cvut.cz/Pruznost/}{http://praktika.fjfi.cvut.cz/Pruznost/}
\end{thebibliography}
/dokumenty/PRA1/2-pruznost/torze.pdf
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/PaintProc {0.5 setlinewidth pop -2 0 M 4 12 L
0 -4 M 8 12 L 4 -4 M 10 8 L stroke}
>> matrix makepattern
/Pat7 exch def
<< Tile8x8
/PaintProc {0.5 setlinewidth pop 8 -2 M -4 4 L
12 0 M -4 8 L 12 4 M 0 10 L stroke}
>> matrix makepattern
/Pat8 exch def
<< Tile8x8
/PaintProc {0.5 setlinewidth pop 0 -2 M 12 4 L
-4 0 M 12 8 L -4 4 M 8 10 L stroke}
>> matrix makepattern
/Pat9 exch def
/Pattern1 {PatternBgnd KeepColor Pat1 setpattern} bind def
/Pattern2 {PatternBgnd KeepColor Pat2 setpattern} bind def
/Pattern3 {PatternBgnd KeepColor Pat3 setpattern} bind def
/Pattern4 {PatternBgnd KeepColor Landscape {Pat5} {Pat4} ifelse setpattern} bind def
/Pattern5 {PatternBgnd KeepColor Landscape {Pat4} {Pat5} ifelse setpattern} bind def
/Pattern6 {PatternBgnd KeepColor Landscape {Pat9} {Pat6} ifelse setpattern} bind def
/Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def
} def
%
%
%End of PostScript Level 2 code
%
/PatternBgnd {
TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse
} def
%
% Substitute for Level 2 pattern fill codes with
% grayscale if Level 2 support is not selected.
%
/Level1PatternFill {
/Pattern1 {0.250 Density} bind def
/Pattern2 {0.500 Density} bind def
/Pattern3 {0.750 Density} bind def
/Pattern4 {0.125 Density} bind def
/Pattern5 {0.375 Density} bind def
/Pattern6 {0.625 Density} bind def
/Pattern7 {0.875 Density} bind def
} def
%
% Now test for support of Level 2 code
%
Level1 {Level1PatternFill} {Level2PatternFill} ifelse
%
/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont
dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall
currentdict end definefont pop
end
%%EndProlog
%%Page: 1 1
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854 1185 M
63 0 V
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( 300) Rshow
1.000 UL
LTb
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63 0 V
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( 310) Rshow
1.000 UL
LTb
854 2660 M
63 0 V
6073 0 R
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( 320) Rshow
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854 3397 M
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6073 0 R
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854 4135 M
63 0 V
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0 63 V
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3922 448 M
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4945 448 M
0 63 V
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5967 448 M
0 63 V
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1.000 UL
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6990 448 M
0 63 V
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854 4872 N
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LCb setrgbcolor
280 2660 M
currentpoint gsave translate 90 rotate 0 0 M
(zkrut [°]) Cshow
grestore
LTb
LCb setrgbcolor
3922 98 M
(hmotnost [g]) Cshow
LTb
1.000 UP
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/Helvetica findfont 140 scalefont setfont
854 448 Pls
1887 1333 Pls
2920 2070 Pls
3953 2955 Pls
4986 3840 Pls
6018 4872 Pls
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/Helvetica findfont 140 scalefont setfont
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1887 1554 Crs
2920 2439 Crs
3953 3324 Crs
4986 4208 Crs
6018 4872 Crs
1.000 UL
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854 4872 N
854 448 L
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1.000 UL
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stroke
grestore
end
showpage
/dokumenty/PRA1/2-pruznost/torze.txt
0,0 → 1,7
0 290 291
101 302 305
202 312 317
303 324 329
404 336 341
505 350 350