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/dokumenty/skolni/PRA1/2-pruznost/plot.gp |
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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/skolni/PRA1/2-pruznost/pruhyb.pdf |
---|
Cannot display: file marked as a binary type. |
svn:mime-type = application/octet-stream |
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/dokumenty/skolni/PRA1/2-pruznost/pruhyb.ps |
---|
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\documentclass[12pt,a4paper,oneside]{article} |
\usepackage[colorlinks=true]{hyperref} |
\usepackage[utf8]{inputenc} |
\usepackage[czech]{babel} |
\usepackage{graphicx} |
\textwidth 16cm \textheight 24.6cm |
\topmargin -1.3cm |
\oddsidemargin 0cm |
\pagestyle{empty} |
\begin{document} |
\title{Měření modulu pružnosti ve smyku a tahu} |
\author{Jakub Kákona, kaklik@mlab.cz} |
\date{19.11.2009} |
\maketitle |
\thispagestyle{empty} |
\begin{abstract} |
Pružné vlastnosti homogenního izotropního tělesa při malých deformacích plně určují dvě nezávislé materiálové konstanty, za které mohou být zvoleny např. modul pružnosti v tahu (Youngův modul) $E$ a Poissonovo číslo $\mu $ nebo modul pružnosti v tahu $E$ a modul pružnosti ve smyku $G$. Jejich význam si vysvětlíme na dvou základních experimentech. |
\end{abstract} |
\section{Úvod} |
\begin{enumerate} |
\item Změřte závislost relativního délkového prodloužení $\Delta $l/l ocelového drátu na napětí při zatěžování a odlehčování drátu a sestrojte graf této závislosti. Vypočítejte metodou nejmenších čtverců modul pružnosti v tahu ocelového drátu. |
\item Změřte závislost průhybu $z$ na velikosti síly $F$ při zatěžování i odlehčování ocelového nosníku a narýsujte graf této závislosti. Metodou nejmenších čtverců vypočítejte modul pružnosti v tahu. |
\item V přípravě odvoďte vzorec pro plošný moment setrvačnosti obdélníkového průřezu šířky $a$ a výšky $b.$ |
\item Změřte závislost úhlu zkroucení $\varphi $ ocelového drátu na velikosti kroutícího momentu při postupném zvětšování a postupném zmenšování tohoto momentu. Výsledky měření vyneste do grafu. Metodou nejmenších čtverců vypočtěte modul pružnosti ve smyku $G$ drátu. |
\item Na torzním kyvadle změřte moment setrvačnosti základního systému $I_{0}$ a modul pružnosti ve smyku $G$ ocelového drátu. Dobu torzních kmitů změřte postupnou metodou. |
\item V přípravě odvoďte vzorce pro výpočet modulu pružnosti ve smyku $G$ a momentu setrvačnosti základního systému torzního kyvadla $I_{0}$. |
\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} |
\end{document} |
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hpt neg vpt -1.62 mul V closepath stroke |
Pnt} def |
/TriDF {stroke [] 0 setdash vpt 1.12 mul sub M |
hpt neg vpt 1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt -1.62 mul V closepath fill} def |
/DiaF {stroke [] 0 setdash vpt add M |
hpt neg vpt neg V hpt vpt neg V |
hpt vpt V hpt neg vpt V closepath fill} def |
/Pent {stroke [] 0 setdash 2 copy gsave |
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat |
closepath stroke grestore Pnt} def |
/PentF {stroke [] 0 setdash gsave |
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat |
closepath fill grestore} def |
/Circle {stroke [] 0 setdash 2 copy |
hpt 0 360 arc stroke Pnt} def |
/CircleF {stroke [] 0 setdash hpt 0 360 arc fill} def |
/C0 {BL [] 0 setdash 2 copy moveto vpt 90 450 arc} bind def |
/C1 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 90 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C2 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 90 180 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C3 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 180 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C4 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 180 270 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C5 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 90 arc |
2 copy moveto |
2 copy vpt 180 270 arc closepath fill |
vpt 0 360 arc} bind def |
/C6 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 90 270 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C7 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 270 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C8 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 270 360 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C9 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 270 450 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C10 {BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill |
2 copy moveto |
2 copy vpt 90 180 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C11 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 180 arc closepath fill |
2 copy moveto |
2 copy vpt 270 360 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C12 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 180 360 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C13 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 0 90 arc closepath fill |
2 copy moveto |
2 copy vpt 180 360 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/C14 {BL [] 0 setdash 2 copy moveto |
2 copy vpt 90 360 arc closepath fill |
vpt 0 360 arc} bind def |
/C15 {BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill |
vpt 0 360 arc closepath} bind def |
/Rec {newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto |
neg 0 rlineto closepath} bind def |
/Square {dup Rec} bind def |
/Bsquare {vpt sub exch vpt sub exch vpt2 Square} bind def |
/S0 {BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare} bind def |
/S1 {BL [] 0 setdash 2 copy vpt Square fill Bsquare} bind def |
/S2 {BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def |
/S3 {BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare} bind def |
/S4 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def |
/S5 {BL [] 0 setdash 2 copy 2 copy vpt Square fill |
exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def |
/S6 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare} bind def |
/S7 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill |
2 copy vpt Square fill Bsquare} bind def |
/S8 {BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare} bind def |
/S9 {BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare} bind def |
/S10 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill |
Bsquare} bind def |
/S11 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill |
Bsquare} bind def |
/S12 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare} bind def |
/S13 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill |
2 copy vpt Square fill Bsquare} bind def |
/S14 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill |
2 copy exch vpt sub exch vpt Square fill Bsquare} bind def |
/S15 {BL [] 0 setdash 2 copy Bsquare fill Bsquare} bind def |
/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def |
/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def |
/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def |
/D3 {gsave translate 45 rotate 0 0 S3 stroke grestore} bind def |
/D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def |
/D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def |
/D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def |
/D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def |
/D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def |
/D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def |
/D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def |
/D11 {gsave translate 45 rotate 0 0 S11 stroke grestore} bind def |
/D12 {gsave translate 45 rotate 0 0 S12 stroke grestore} bind def |
/D13 {gsave translate 45 rotate 0 0 S13 stroke grestore} bind def |
/D14 {gsave translate 45 rotate 0 0 S14 stroke grestore} bind def |
/D15 {gsave translate 45 rotate 0 0 S15 stroke grestore} bind def |
/DiaE {stroke [] 0 setdash vpt add M |
hpt neg vpt neg V hpt vpt neg V |
hpt vpt V hpt neg vpt V closepath stroke} def |
/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M |
0 vpt2 neg V hpt2 0 V 0 vpt2 V |
hpt2 neg 0 V closepath stroke} def |
/TriUE {stroke [] 0 setdash vpt 1.12 mul add M |
hpt neg vpt -1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt 1.62 mul V closepath stroke} def |
/TriDE {stroke [] 0 setdash vpt 1.12 mul sub M |
hpt neg vpt 1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt -1.62 mul V closepath stroke} def |
/PentE {stroke [] 0 setdash gsave |
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat |
closepath stroke grestore} def |
/CircE {stroke [] 0 setdash |
hpt 0 360 arc stroke} def |
/Opaque {gsave closepath 1 setgray fill grestore 0 setgray closepath} def |
/DiaW {stroke [] 0 setdash vpt add M |
hpt neg vpt neg V hpt vpt neg V |
hpt vpt V hpt neg vpt V Opaque stroke} def |
/BoxW {stroke [] 0 setdash exch hpt sub exch vpt add M |
0 vpt2 neg V hpt2 0 V 0 vpt2 V |
hpt2 neg 0 V Opaque stroke} def |
/TriUW {stroke [] 0 setdash vpt 1.12 mul add M |
hpt neg vpt -1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt 1.62 mul V Opaque stroke} def |
/TriDW {stroke [] 0 setdash vpt 1.12 mul sub M |
hpt neg vpt 1.62 mul V |
hpt 2 mul 0 V |
hpt neg vpt -1.62 mul V Opaque stroke} def |
/PentW {stroke [] 0 setdash gsave |
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat |
Opaque stroke grestore} def |
/CircW {stroke [] 0 setdash |
hpt 0 360 arc Opaque stroke} def |
/BoxFill {gsave Rec 1 setgray fill grestore} def |
/Density { |
/Fillden exch def |
currentrgbcolor |
/ColB exch def /ColG exch def /ColR exch def |
/ColR ColR Fillden mul Fillden sub 1 add def |
/ColG ColG Fillden mul Fillden sub 1 add def |
/ColB ColB Fillden mul Fillden sub 1 add def |
ColR ColG ColB setrgbcolor} def |
/BoxColFill {gsave Rec PolyFill} def |
/PolyFill {gsave Density fill grestore grestore} def |
/h {rlineto rlineto rlineto gsave closepath fill grestore} bind def |
% |
% PostScript Level 1 Pattern Fill routine for rectangles |
% Usage: x y w h s a XX PatternFill |
% x,y = lower left corner of box to be filled |
% w,h = width and height of box |
% a = angle in degrees between lines and x-axis |
% XX = 0/1 for no/yes cross-hatch |
% |
/PatternFill {gsave /PFa [ 9 2 roll ] def |
PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add translate |
PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec |
gsave 1 setgray fill grestore clip |
currentlinewidth 0.5 mul setlinewidth |
/PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def |
0 0 M PFa 5 get rotate PFs -2 div dup translate |
0 1 PFs PFa 4 get div 1 add floor cvi |
{PFa 4 get mul 0 M 0 PFs V} for |
0 PFa 6 get ne { |
0 1 PFs PFa 4 get div 1 add floor cvi |
{PFa 4 get mul 0 2 1 roll M PFs 0 V} for |
} if |
stroke grestore} def |
% |
/languagelevel where |
{pop languagelevel} {1} ifelse |
2 lt |
{/InterpretLevel1 true def} |
{/InterpretLevel1 Level1 def} |
ifelse |
% |
% PostScript level 2 pattern fill definitions |
% |
/Level2PatternFill { |
/Tile8x8 {/PaintType 2 /PatternType 1 /TilingType 1 /BBox [0 0 8 8] /XStep 8 /YStep 8} |
bind def |
/KeepColor {currentrgbcolor [/Pattern /DeviceRGB] setcolorspace} bind def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke} |
>> matrix makepattern |
/Pat1 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke |
0 4 M 4 8 L 8 4 L 4 0 L 0 4 L stroke} |
>> matrix makepattern |
/Pat2 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop 0 0 M 0 8 L |
8 8 L 8 0 L 0 0 L fill} |
>> matrix makepattern |
/Pat3 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop -4 8 M 8 -4 L |
0 12 M 12 0 L stroke} |
>> matrix makepattern |
/Pat4 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop -4 0 M 8 12 L |
0 -4 M 12 8 L stroke} |
>> matrix makepattern |
/Pat5 exch def |
<< Tile8x8 |
/PaintProc {0.5 setlinewidth pop -2 8 M 4 -4 L |
0 12 M 8 -4 L 4 12 M 10 0 L stroke} |
>> matrix makepattern |
/Pat6 exch def |
<< Tile8x8 |
/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 |
gnudict begin |
gsave |
50 50 translate |
0.100 0.100 scale |
90 rotate |
0 -5040 translate |
0 setgray |
newpath |
(Helvetica) findfont 140 scalefont setfont |
1.000 UL |
LTb |
854 448 M |
63 0 V |
6073 0 R |
-63 0 V |
770 448 M |
( 290) Rshow |
1.000 UL |
LTb |
854 1185 M |
63 0 V |
6073 0 R |
-63 0 V |
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( 300) Rshow |
1.000 UL |
LTb |
854 1923 M |
63 0 V |
6073 0 R |
-63 0 V |
-6157 0 R |
( 310) Rshow |
1.000 UL |
LTb |
854 2660 M |
63 0 V |
6073 0 R |
-63 0 V |
-6157 0 R |
( 320) Rshow |
1.000 UL |
LTb |
854 3397 M |
63 0 V |
6073 0 R |
-63 0 V |
-6157 0 R |
( 330) Rshow |
1.000 UL |
LTb |
854 4135 M |
63 0 V |
6073 0 R |
-63 0 V |
-6157 0 R |
( 340) Rshow |
1.000 UL |
LTb |
854 4872 M |
63 0 V |
6073 0 R |
-63 0 V |
-6157 0 R |
( 350) Rshow |
1.000 UL |
LTb |
854 448 M |
0 63 V |
0 4361 R |
0 -63 V |
854 308 M |
( 0) Cshow |
1.000 UL |
LTb |
1877 448 M |
0 63 V |
0 4361 R |
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0 -4501 R |
( 100) Cshow |
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0 63 V |
0 4361 R |
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0 63 V |
0 4361 R |
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( 400) Cshow |
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0 -63 V |
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( 500) Cshow |
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0 63 V |
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( 600) Cshow |
1.000 UL |
LTb |
1.000 UL |
LTb |
854 4872 N |
854 448 L |
6136 0 V |
0 4424 V |
-6136 0 V |
Z stroke |
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 |
1.000 UL |
LTb |
1.000 UP |
1.000 UL |
LT0 |
/Helvetica findfont 140 scalefont setfont |
854 448 Pls |
1887 1333 Pls |
2920 2070 Pls |
3953 2955 Pls |
4986 3840 Pls |
6018 4872 Pls |
1.000 UP |
1.000 UL |
LT1 |
/Helvetica findfont 140 scalefont setfont |
854 522 Crs |
1887 1554 Crs |
2920 2439 Crs |
3953 3324 Crs |
4986 4208 Crs |
6018 4872 Crs |
1.000 UL |
LTb |
854 4872 N |
854 448 L |
6136 0 V |
0 4424 V |
-6136 0 V |
Z stroke |
1.000 UP |
1.000 UL |
LTb |
stroke |
grestore |
end |
showpage |
/dokumenty/skolni/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 |