From 5ff4419561747c1251aa9bf275e485e4a4ddc8d6 Mon Sep 17 00:00:00 2001 From: victor Date: Sun, 15 Oct 2023 20:41:09 +0300 Subject: [PATCH 01/42] add 1.2.4 preview --- 1.2.4/Baldin_V/1.2.4/1_2_4.fdb_latexmk | 112 +++++ 1.2.4/Baldin_V/1.2.4/1_2_4.fls | 479 ++++++++++++++++++++ 1.2.4/Baldin_V/1.2.4/1_2_4.pdf | Bin 0 -> 272028 bytes 1.2.4/Baldin_V/1.2.4/1_2_4.tex | 139 ++++++ 1.2.4/Baldin_V/1.2.4/pictures/ellipsoid.png | Bin 0 -> 45004 bytes 1.2.4/Baldin_V/1.2.4/pictures/kyb.png | Bin 0 -> 46079 bytes 1.2.4/Baldin_V/1.2.4/pictures/stand.png | Bin 0 -> 19013 bytes 7 files changed, 730 insertions(+) create mode 100644 1.2.4/Baldin_V/1.2.4/1_2_4.fdb_latexmk create mode 100644 1.2.4/Baldin_V/1.2.4/1_2_4.fls create mode 100644 1.2.4/Baldin_V/1.2.4/1_2_4.pdf create mode 100644 1.2.4/Baldin_V/1.2.4/1_2_4.tex create mode 100644 1.2.4/Baldin_V/1.2.4/pictures/ellipsoid.png create mode 100644 1.2.4/Baldin_V/1.2.4/pictures/kyb.png create mode 100644 1.2.4/Baldin_V/1.2.4/pictures/stand.png diff --git 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+\begin{document} + + \maketitle + \section{Аннотация} + \textbf{Цель работы:} измерить периоды крутильных колебаний рамки при различных положениях закрепленного + в ней тела, проверить теоретическую зависимость между периодами крутильных колебаний тела + относительно различных осей, определить моменты инерции относительно нескольких осей для каждого тела, + по ним найти главные моменты инерции тела и посторить эллипсоид инерции. + \bigskip\\ + \textbf{Оборудование:} установка для получения крутильных колебаний, набор исследуемых твердых тел, секундомер. + + \section{Теоретические сведения} + Инерционные свойства твердого тела при вращении определяется + пространственным распределением. Оно характеризуется тензором инерции тела. Тензор инерции твердого тела + является симметричным тензором 2-ого ранга $J\in T_{2}^{0}(V)$ и имеет 6 независимых компонент, + которые в прямоугольной декартовой системе координат выражаются как: + \begin{equation} + I_{ij}=\int (\delta _{ij}r^{2}-r_{i}r_{j}) \ dm =I_{ji}, + \end{equation} + где $r$ — расстояния от точек до центра, относительно которого вычисляется тензор инерции, + а $r_{i}$ — координатные компоненты соответствующих отрезков, $i$ и $j$ — номера координат (от 1 до 3).\\ + Если для какой либо системы координат все 6 компонент известны, то момент инерции тела относительно + произвольной оси $l$, проходящей через начало координат может быть вычислен по формуле: + \begin{equation} + I_{l}=n^{j}n^{i}I_{ij}=\vec{n}^{T} I \vec{n} + \end{equation} + где $\vec{n}$ - единичный вектор-столбец который задает направление оси, $I$ - тензор инерции.\\ + А момент импульса $\vec {L}$ и вращательная энергия тела $E_{\text{вращ}}$ тогда будут выражаться как: + \begin{equation} + E_{\text{вращ}}={1 \over 2}\ {\vec {\omega }}^{\,T} I {\vec {\omega }} ={1 \over 2}\sum _{{ij}}\omega^{i}J_{{ij}}\omega^{j} + \end{equation} + \begin{equation} + {\vec {L}}=I {\vec {\omega }}, \ \ \ \ L_{i}=\sum _{j}I_{{ij}}\omega^{j} + \end{equation} + Отложим вдоль оси $l$ из начала координат радиус-вектор $r$ + равный по длине $1/\sqrt{I_{l}}$. Проведем множество таких отрезков, соответствующих различным направлениям оси $l$. + Геометрическое место концов указанных отрезков, является поверхность второго порядка - эллипсоид. Этот эллипсоид принято называть + эллипсоидом инерции. Он жестко связан с телом для которого он построен. Знание эллипсоида инерции позволяет найти момент инерции тела + относительно любой оси, проходящей через центр эллипсоида. Длина отрезка $r$ будет определять момент инерции тела относительно оси $l$: + \begin{equation} + I_{l} = \frac{1}{r^2} + \end{equation} + \begin{figure}[H] + \centering + \includegraphics[scale = 0.5]{pictures/ellipsoid.png} + \caption{Эллипсоиды вращения для разных тел} + \end{figure} + + Как и всякий симметричный тензор второго ранга может быть диагонализован некоторой заменой координат. + Пусть система координат, в которой он диагонализован имеет оси $Ox,Oy,Oz$, тогда эти оси совпадают с главными осями тела. + Полученные диагональные элементы $I_{x}, I_{y}, I_{z}$ называются главными моментами инерции тела, а уравнение эллипсоида + инерции в этих координатах примет вид: + \begin{equation} + I_{x}r^{2}_{x}+I_{y}r^{2}_{y}+I_{z}r^{2}_{z} = 1 + \end{equation} + Крутильные колебания рамки с телом описываются уравнением: + \begin{equation} + (I + I_p)\ddot{\varphi} + f \varphi = 0 + \end{equation} + Здесь $I$ и $I_{p}$ - моменты инерции тела и рамки относительно + оси вращения, $\varphi$ - угол поворота рамки, меняющийся со + временем $t$, $f$ - модуль кручения проволоки. Отсюда период этих колебаний: + \begin{equation} + T = 2\pi\sqrt{\frac{I+I_{p}}{f}} + \end{equation} + На рисунке показано, как проходят оси вращения в параллелепипеде. + Оси $AA'$, $BB'$ и $CC'$ являются главными. Моменты инерции относительно + этих осей обозначим соответственно $I_{x}, I_{y}, I_{z}$.\\ + + \begin{figure}[H] + \begin{center} + \includegraphics[scale=0.5]{pictures/kyb.png} + \caption{Оси вращения прямоугольного параллелепипеда} + \label{graphic1} + \end{center} + \end{figure} + + Момент инерции $I_{D}$ при вращении относительно диагонали $DD'$ выражается + через главные моменты с помощью формулы: + \begin{equation} + I_{d}=I_{x}\frac{a^2}{d^2}+I_{y}\frac{b^2}{d^2}+I_{z}\frac{c^2}{d^2} + \end{equation} + Используя связь момента инерции с периодом крутильных колебаний + получаем соотношение между периодами колебаний относительно осей $DD'$, $EE'$, + $MM'$ и $PP'$ с периодами крутильных колебаний относительно главных осей. + \begin{equation} + \begin{cases} + (b^2+c^2)T^2_{E}=b^2 T^2_{y}+c^2 T^2_{z} \\ + (a^2+c^2)T^2_{P}=a^2 T^2_{x}+c^2 T^2_{z} \\ + (a^2+b^2)T^2_{M}=a^2 T^2_{x}+b^2 T^2_{y} + \end{cases} + \end{equation} + + Эти соотношения также необходимо проверить экспериментально. + + \section{Методика измерений} + В данной работе используется установка для измерения крутильных колебаний, приведенная + на рисунке 3. Рамка 1 жестко соединена с проволокой 2, закрепленной вертикально в специальных + зажимах 3, позволяющих сообщить начальное закручивание для возбуждения крутильных колебаний + вокруг вертикальной оси. + \begin{figure}[H] + \centering + \includegraphics[scale = 0.5]{pictures/stand.png} + \caption{Схема установки} + \end{figure} + +% +% + \section{Используемое оборудование} + Установка для получения крутильных колебаний, набор исследуемых твердых тел, секундомер. + +% +% \section{Результаты измерений и обработка данных} +% +% \section{Обсуждение результатов} +% \section{Вывод} +\end{document} diff --git a/1.2.4/Baldin_V/1.2.4/pictures/ellipsoid.png b/1.2.4/Baldin_V/1.2.4/pictures/ellipsoid.png new file mode 100644 index 0000000000000000000000000000000000000000..c5f4584a5123cb24e33269e29bf9b6dff17eb31a GIT binary patch literal 45004 zcmbTec{rBs`#nmdgeEBpjpjKbWy&mME@Mb#%n&kDLKG4*WuBshkTO=1ka;E~$sC!< zu-EPVe)sYFXCM36`#CR$lBiE(yA#~{I+(4s)C0aUr$0}rq@im`3DyA%FVyY 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OfWZJ>Uf6RLz<&V(NuLP- literal 0 HcmV?d00001 diff --git a/1.2.5/Baldin_V/1.2.5.tex b/1.2.5/Baldin_V/1.2.5.tex new file mode 100644 index 00000000..6155bd03 --- /dev/null +++ b/1.2.5/Baldin_V/1.2.5.tex @@ -0,0 +1,77 @@ + \documentclass[a4paper,12pt]{article} +\usepackage[a4paper,top=1.3cm,bottom=2cm,left=1.5cm,right=1.5cm,marginparwidth=0.75cm]{geometry} +\usepackage{setspace} +\usepackage{cmap} +\usepackage{mathtext} +\usepackage[T2A]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[english,russian]{babel} +\usepackage{multirow} +\usepackage{graphicx} +\usepackage{wrapfig} +\usepackage{tabularx} +\usepackage{float} +\usepackage{longtable} +\usepackage{hyperref} +\hypersetup{colorlinks=true,urlcolor=blue} +\usepackage[rgb]{xcolor} +\usepackage{amsmath,amsfonts,amssymb,amsthm,mathtools} +\usepackage{icomma} +\mathtoolsset{showonlyrefs=true} +\usepackage{euscript} +\usepackage{mathrsfs} + +\DeclareMathOperator{\sgn}{\mathop{sgn}} +\newcommand*{\hm}[1]{#1\nobreak\discretionary{} + {\hbox{$\mathsurround=0pt #1$}}{}} + +\title{\textbf{Исследование прецессии уравновешенного гироскопа (1.2.5)}} +\author{Балдин Виктор} +\date{6 ноября 2023} + +\begin{document} + + \maketitle + + \section{Введение} + + \textbf{Цель работы:} исследовать вынужденную прецессию гироскопа, установить зависимость скорости вынужденной прецессии от величины момента сил, действующий на ось гироскопа и сравнить ее со скоростью, рассчитанной по скорости прецессии.\\ + \textbf{Оборудование:} гироскоп в кардановом подвесе, секундомер, набор грузов, отдельный ротор гироскопа, цилиндр известной массы, крутильный маятник, штангенсциркуль, линейка. + + \section{Теоретические сведения} + + В этой работе исследуется зависимость скорости прецессии гироскопа от момента силы, приложенной к его оси. Для этого к оси гироскопа подвешиваются грузы. Скорость прецессии определяется по числу оборотов рычага вокруг вертикальной оси и времни, которое на это ушло, определяемоу секундомером. В процессе измерений рычаг не только поворачивается в результате прецессии гироскопа, но и опускается. Поэтому его в начале опыта следует преподнять на 5-6 градусов. Опять надо закончить, когда рычаг опустится на такой же угол.\\ + \begin{center}$ + \begin{array}{cc} + \includegraphics[width=0.40\textwidth]{img1.png}& + \includegraphics[width=0.40\textwidth]{img2.png}\\ + \end{array}$ + \end{center} + + Измерение скорости прецессии гироскопа позволяет вычислить угловую скорость вращения его ротора. Расчет производится по формуле: + + \begin{equation} + \Omega = \frac{mgl}{I_z\omega_0}, + \end{equation} + + где $m$ -- масса груза, $l$ -- расстояние от центра карданова подвеса до точки крепления груза на оси гироскопа, $I_z$ -- момент инерции гироскопа по его главной оси вращения. $\omega_0$ -- частота его вращения относительно главной оси, $\Omega$ -- частота прецессии.\\ + Момент инерции ротора относительно оси симметрии $I_0$ измеряется по крутильным колебаниям точной копии ротора, подвешиваемой вдоль оси симметрии на десткой проволоке. Период крутильных колебаний $T_0$ зависит от момента инерции $I_0$ и модуля кручения проволоки $f$: + + \begin{equation} + T_0 = 2\pi\sqrt{\frac{I_0}{f}}. + \end{equation} + + Чтобы исключить модуль кручения проволоки, вместо ротора гироскопа к той же проволоке подвешивают цилиндр правильной формы с известными размерами и массой, для которого легко можно вычислить момент инерции $I_\text{ц}$. Для определения момента инерции ротора гироскопа имеем: + + \begin{equation} + I_0 = I_\text{ц}\frac{T_0^2}{T_\text{ц}^2}, + \label{moment} + \end{equation} + Здесь $T_\text{ц}$ -- период крутильных колебаний цилиндра.\\ + \section{Методика измерений} + \begin{center} + \includegraphics[width=0.8\textwidth]{img3.png} + \end{center} + + Скорость вращения ротора гироскопа можно определить и не прибегая к исследованию прецессии. У используемых в работе гироскопов статор имеет две обмотки, необходимые для быстрой раскрутки гироскопа. В данной работе одну обмотку искользубт для раскрутки гироскопа, а вторую -- для измерения числа оборотов ротора. Ротор электромотора всегда немного намагничен. Вращаясь, он наводит во второй обмотке переменную ЭДС индукции, частота которой равна частоте врещения ротора. Частоту этой ЭДС можно, в частности, измерить по фигурам Лиссажу, получаемым на экране осциллографа, если на один вход подать исследуемую ЭДС, а на другой -- переменное напряжение с хорошо прокалиброванного генератора. При совпадении частот на эеране получаем эллипс. +\end{document} diff --git a/1.2.5/Baldin_V/img1.png b/1.2.5/Baldin_V/img1.png new file mode 100644 index 0000000000000000000000000000000000000000..5300e0ec639c590231f9af4b157f0e07ff450610 GIT binary patch literal 21053 zcmZ6y1yCGq7d1#o0t9#W46Y$qaE8I%-2%a#!97@j5ZnUi=%mcG zyhYsNBA^o@RB*w4{f&oj^b2LxdAZx(Ii@<~YjSmqwXF~E=V9B}XT6xMIZ{i)dr`Y( zZDe)3O4L0WkB+(Me*Gl!B}1drmjJf)Ea9!4Cn9_AL0~V-G0m~{{T`GD0|sUlkyHk_ zAw?J%p-dPUOyKT7fGH%f|F`K2dG~)eVIpZ^{@a8Jf&K63f1CgN6tMZUKycbq%kzkg3cLc*DHa+=%W=Vv4?*yeB$ zEmeSwh`4#-T4#%sdcTQOH1l|P=L6&QQ3E}Gj<#^Zj8QEtZ+5)AJpJVv5A9@4@OE&Q z>48C6sjI6iBMSzp=aNu#Noz?A`ZSDL3Q6Qgh^b!Yurx$NU0t1g;=WXlh&59QHhoQPuKP%%(-GU(0u}meG#C(%k47AqeM` zc5-=+RvRCJyO=hlNeBrotjzd4Zi|%)mVs}16~FBYqFH}gr_rC+kwOUU9MR@)>$I>9iBq92iaEp!z6Q-IubnQX^8<0Nx z{9=gx5jPH^`;N>AQ}?o{i8AUJIHNrr9yc74>VnTQN>!}Xpo3aq@7DuTlV-0K5Ez@Sg;a31EHaB(aNOVDymcnE}_BPU#;Vy}h{!BtrZD zJ4B2W^ud@F)>o8Hah~KOOj-Q@>m>glulWDDRy(*HTM8y+#KgaYH9o5U07{R8$VWPM 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У используемых в работе гироскопов статор имеет две обмотки, необходимые для быстрой раскрутки гироскопа. В данной работе одну обмотку искользубт для раскрутки гироскопа, а вторую -- для измерения числа оборотов ротора. Ротор электромотора всегда немного намагничен. Вращаясь, он наводит во второй обмотке переменную ЭДС индукции, частота которой равна частоте врещения ротора. Частоту этой ЭДС можно, в частности, измерить по фигурам Лиссажу, получаемым на экране осциллографа, если на один вход подать исследуемую ЭДС, а на другой -- переменное напряжение с хорошо прокалиброванного генератора. При совпадении частот на эеране получаем эллипс. + + \section{Оборудование} + + Гироскоп в кардановом подвесе, секундомер, набор грузов, отдельный ротор гироскопа, цилиндр известной массы, крутильный маятник, штангенсциркуль, линейка. + \end{document} From 9638226914f48df44e5b38e806b9e84eb69201af Mon Sep 17 00:00:00 2001 From: victor Date: Sun, 19 Nov 2023 21:15:29 +0300 Subject: [PATCH 10/42] add 1.2.3 --- 1.2.3/Baldin_V/1.2.3.pdf | Bin 249866 -> 431380 bytes 1.2.3/Baldin_V/1.2.3.tex | 165 ++++++++++++++++++++- 1.2.3/Baldin_V/Grafik.jpg | Bin 0 -> 275181 bytes 1.2.3/Baldin_V/data/data.ods | Bin 0 -> 11777 bytes 1.2.3/Baldin_V/data/disk.csv | 4 + 1.2.3/Baldin_V/data/ring.csv | 14 ++ 1.2.3/Baldin_V/position.jpg | Bin 0 -> 40751 bytes 1.2.3/Varlamov_A/Report_lab_work_1.2.3.pdf | Bin 0 -> 207176 bytes 1.2.3/Varlamov_A/Report_lab_work_1.2.3.tex | 160 ++++++++++---------- 9 files changed, 262 insertions(+), 81 deletions(-) create mode 100644 1.2.3/Baldin_V/Grafik.jpg create mode 100644 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\frac{T_p^2}{(T_z^{\text{цил}})^2 - T_p^2} = 4213\; \text{кг} \cdot \text{мм} ^2 $$ $$ \varepsilon_{I_p} = \frac{\sigma_I}{I} + 4\varepsilon_T = 0.06 $$ + \item Вычислим величины $1/\sqrt{T^2 - T_p^2}$ для осей $x$ и $z$. + $$ \frac{1}{\sqrt{T_x^2 - T_p^2}} = 0.64 \;\text{c} ^{-2} $$ + $$ \frac{2}{\sqrt{T_z^2 - T_p^2}} = 0.54 \;\text{c} ^{-2}$$ + + \item Нарисуем сечения эллипсоида инерции плоскостью $XOZ$ для цилиндра: + \begin{figure} + \centering + \caption{Сечение плоскостью $AA'M$} + \includegraphics[scale = 0.8]{pictures/graph.png} + \end{figure} + + \item Повторим все те же действия теперь для куба: \begin{table}[H] \centering @@ -227,9 +239,6 @@ Так как они примерно одинаковые, можно рассчитать погрешность лишь для одного случая: $$ \varepsilon_I = \varepsilon_{I_p} + 4\varepsilon_{T} = 0.12 $$ - \item Нарисуем сечения эллипсоида инерции главными плоскостями: - - \end{enumerate} diff --git a/1.2.4/Baldin_V/1.2.4/graph1.py b/1.2.4/Baldin_V/1.2.4/graph1.py new file mode 100644 index 00000000..0fbf80c2 --- /dev/null +++ b/1.2.4/Baldin_V/1.2.4/graph1.py @@ -0,0 +1,17 @@ +import numpy as np +from matplotlib import pyplot as plt +from math import pi + +u=0 #x-position of the center +v=0 #y-position of the center +a=0.64 #radius on the x-axis +b=0.54 #radius on the y-axis + +t = np.linspace(0, 2*pi, 200) +plt.plot( u+a*np.cos(t) , v+b*np.sin(t) , label=r'$Апроксимация$') +plt.xlabel(r'$OX$', fontsize=10) +plt.ylabel(r'$OZ$', fontsize=10) +plt.legend(loc='upper right', fontsize=7) +plt.grid(color='lightgray',linestyle='--', linewidth = 0.5) +plt.plot([0.64, -0.64, 0, 0], [0, 0, -0.54, 0.54], 'ro') +plt.show() diff --git a/1.2.4/Baldin_V/1.2.4/pictures/graph.png b/1.2.4/Baldin_V/1.2.4/pictures/graph.png new file mode 100644 index 0000000000000000000000000000000000000000..5809c60aca5c4b867fef274a58e62d6e48ebb626 GIT binary patch literal 37326 zcmeFZWmr|+w>G{20cimVNhPGD1OaKKLFp1{ly2z`0Vxp>Hwp-dl(cj=sDQL|Y>@8W zT5HZR#~g8wd(3cEWjO*|DqIMH2o&U{)gcHy0D{nF 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zlNQoS+K5luVK#U5T>W955`TvqMZ23_u1DCnf8h|_SvPPMXdZ);jhWEt*#U*O6~`2)OEGl!Rq S2L&DrH!(Rm3MC~)PeuyYaW?z_ diff --git a/1.2.4/Baldin_V/1.2.4/1_2_4.tex b/1.2.4/Baldin_V/1.2.4/1_2_4.tex index 562d259d..5490333f 100644 --- a/1.2.4/Baldin_V/1.2.4/1_2_4.tex +++ b/1.2.4/Baldin_V/1.2.4/1_2_4.tex @@ -239,9 +239,18 @@ Так как они примерно одинаковые, можно рассчитать погрешность лишь для одного случая: $$ \varepsilon_I = \varepsilon_{I_p} + 4\varepsilon_{T} = 0.12 $$ + \end{enumerate} -% \section{Обсуждение результатов} -% \section{Вывод} + \section{Обсуждение результатов} + В результате этой работы мы убедились в выполнении теоретических + соотношений. Построили эллипсоид инерции для цилиндра, а также убедились в + равенстве главных моментов инерции для куба, то есть его эллипсоидом инерции + действительно является сфера. + \section{Вывод} + Использование установки дает некоторую погрешность, но ее + точности вполне достаточно для получения результатов. По виду эллипсоида инерции были вычислены соотношения между моментами + инерции тела относительно главных осей, + которые совпадают с теоретическим расчетом в пределах погрешности. \end{document} From 0c417465dd06f153b67f2020d80e48477771c371 Mon Sep 17 00:00:00 2001 From: victor Date: Sat, 28 Oct 2023 21:21:03 +0300 Subject: [PATCH 07/42] add 1.2.3 --- 1.2.3/Baldin_V/1.2.3 graph.png | Bin 0 -> 105602 bytes 1.2.3/Baldin_V/1.2.3 ustan.png.jpg | Bin 0 -> 92573 bytes 1.2.3/Baldin_V/1.2.3.pdf | Bin 0 -> 249866 bytes 1.2.3/Baldin_V/1.2.3.tex | 121 +++++++++++++++++++++++++++++ 4 files changed, 121 insertions(+) create mode 100644 1.2.3/Baldin_V/1.2.3 graph.png create mode 100644 1.2.3/Baldin_V/1.2.3 ustan.png.jpg create mode 100644 1.2.3/Baldin_V/1.2.3.pdf create mode 100644 1.2.3/Baldin_V/1.2.3.tex diff --git a/1.2.3/Baldin_V/1.2.3 graph.png b/1.2.3/Baldin_V/1.2.3 graph.png new file mode 100644 index 0000000000000000000000000000000000000000..e569f5c2094c36f0d80e8330e83d2a9255b5f9f5 GIT 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кодировка +\usepackage{cmap} % поиск в PDF l +\usepackage[T2A]{fontenc} % кодировка \usepackage[utf8]{inputenc} % кодировка исходного текста \usepackage[english,russian]{babel} % локализация и переносы @@ -18,7 +19,7 @@ \graphicspath{pictures} \author{Балдин Виктор} -\title{Работа 1.2.4 \\ Определении главных моментов инерции твердых тел с помощью крутильных колебаний} +\title{Работа 1.2.4 \\ Определение главных моментов инерции твердых тел с помощью крутильных колебаний} \begin{document} @@ -27,7 +28,7 @@ \textbf{Цель работы:} измерить периоды крутильных колебаний рамки при различных положениях закрепленного в ней тела, проверить теоретическую зависимость между периодами крутильных колебаний тела относительно различных осей, определить моменты инерции относительно нескольких осей для каждого тела, - по ним найти главные моменты инерции тела и посторить эллипсоид инерции. + по ним найти главные моменты инерции тела и построить эллипсоид инерции. \bigskip\\ \textbf{Оборудование:} установка для получения крутильных колебаний, набор исследуемых твердых тел, секундомер. @@ -132,8 +133,106 @@ Установка для получения крутильных колебаний, набор исследуемых твердых тел, секундомер. % -% \section{Результаты измерений и обработка данных} -% + \section{Результаты измерений и обработка данных} + \begin{enumerate} + \item Измерим период сначала для ненагруженной рамки (здесь и далее + период измеряем $N = 15$ раз, погрешность секундомера считаем приблизительно + равной $\sigma_t^{\text{ сист }} = 0.20$ с): + \begin{table}[H] + \centering + \caption{Колебания рамки} + \begin{tabular}{|c|c|c|c|c|c|} + \hline + $t$, c & 38.4 & 38.6 & 38.6 & 38.6 & 38.5 \\ \hline + $T$, c & 2.56 & 2.57 & 2.57 & 2.57 & 2.57 \\ \hline + \end{tabular} + \end{table} + + Отсюда средний период + $$ \overline {T}_{p} = \frac{1}{N}\sum_{i} T_i = 2.57\; \text{с}, $$ + Среднеквадратичное отклонение + $$ \sigma_T^{\text{ случ }} = \sqrt{\frac{ \sum_{i} (T_i - \overline {T})^2}{N} } = 0.01\; \text{с} $$ + % + Общая погрешность таким образом равна: + $$ \sigma_T = \frac{\sigma_t^{\text{ сист }}}{N} + \sigma_T^{\text{ случ }} = 0.03\; \text{с}$$ + + \item Измерим размеры и массу цилиндра: + \begin{table}[H] + \centering + \caption{Измерения размеров цилиндра} + \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} + \hline + № & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & Среднее & $\sigma$ \\ \hline + $h$, мм & 49.1 & 49.5 & 49.3 & 49.3 & 49.2 & 49.2 & 49.2 & 49.4 & 49.2 & 49.2 & 49.3 & 0.2 \\ \hline + $d$, мм & 88.3 & 88.1 & 88.1 & 88.1 & 88.1 & 88.1 & 88.1 & 88.1 & 88.1 & 88.0 & 88.1 & 0.2 \\ \hline + \end{tabular} + \end{table} + + Масса $m_{\text{цил}} = 2.264\pm0.001$ кг. + Отсюда момент инерции + $$I_{\text{цил}} = \frac{1}{8}md^2 = 2197\; \text{кг} \cdot \text{мм} ^2, $$ + $$ \sigma_I = I\sqrt{\left( \frac{\sigma_m}{m} \right) ^ 2 + 2\left( \frac{\sigma_d} + {\overline{d}} \right) ^ 2} = 8\; \text{кг} \cdot \text{мм} ^2 $$ + + \item Измерим период колебаний цилиндра + \begin{table}[H] + \centering + \caption{Колебания цилиндра} + \begin{tabular}{|cccccc|} + \hline + \multicolumn{6}{|c|}{Главная ось} \\ \hline + \multicolumn{1}{|c|}{$t$, c} & \multicolumn{1}{c|}{47.9} & \multicolumn{1}{c|}{47.6} & \multicolumn{1}{c|}{47.2} & \multicolumn{1}{c|}{47.4} & 47.6 \\ \hline + \multicolumn{1}{|c|}{$T$, c} & \multicolumn{1}{c|}{3.19} & \multicolumn{1}{c|}{3.17} & \multicolumn{1}{c|}{3.15} & \multicolumn{1}{c|}{3.16} & 3.17 \\ \hline + \multicolumn{6}{|c|}{Боковая ось} \\ \hline + \multicolumn{1}{|c|}{$t$, c} & \multicolumn{1}{c|}{45.1} & \multicolumn{1}{c|}{45.0} & \multicolumn{1}{c|}{45.3} & \multicolumn{1}{c|}{45.0} & 44.9 \\ \hline + \multicolumn{1}{|c|}{$T$, c} & \multicolumn{1}{c|}{3.01} & \multicolumn{1}{c|}{3.00} & \multicolumn{1}{c|}{3.02} & \multicolumn{1}{c|}{3.00} & 2.99 \\ \hline + \end{tabular} + \end{table} + + $$ \overline{T}_{z}^{\text{цил}} = 3.17 \; \text{с}, \\\ \overline{T}_{x}^{\text{цил}} = 3.00\; \text{с} $$ + + \item Найдем момент инерции рамки через формулу (8): + $$ \frac{T_p}{T_{z}^{\text{цил}}} = \sqrt{\frac{I_p}{I_p + I_{\text{цил}}}}, $$ + $$ I_p = I_{\text{цил}} \frac{T_p^2}{(T_z^{\text{цил}})^2 - T_p^2} = 4213\; \text{кг} \cdot \text{мм} ^2 $$ + $$ \varepsilon_{I_p} = \frac{\sigma_I}{I} + 4\varepsilon_T = 0.06 $$ + + \item Повторим все те же действия теперь для куба: + \begin{table}[H] + \centering + \caption{Измерения для куба} + \begin{tabular}{|ccccccccccccc|} + \hline + \multicolumn{13}{|c|}{Измерения длины} \\ \hline + \multicolumn{1}{|c|}{№} & \multicolumn{1}{c|}{1} & \multicolumn{1}{c|}{2} & \multicolumn{1}{c|}{3} & \multicolumn{1}{c|}{4} & \multicolumn{1}{c|}{5} & \multicolumn{1}{c|}{6} & \multicolumn{1}{c|}{7} & \multicolumn{1}{c|}{8} & \multicolumn{1}{c|}{9} & \multicolumn{1}{c|}{10} & \multicolumn{1}{c|}{$\overline{a}$} & $\sigma_{a}$ \\ \hline + \multicolumn{1}{|c|}{$a$, мм} & \multicolumn{1}{c|}{92.7} & \multicolumn{1}{c|}{92.8} & \multicolumn{1}{c|}{92.8} & \multicolumn{1}{c|}{92.6} & \multicolumn{1}{c|}{92.6} & \multicolumn{1}{c|}{92.6} & \multicolumn{1}{c|}{92.6} & \multicolumn{1}{c|}{92.6} & \multicolumn{1}{c|}{92.7} & \multicolumn{1}{c|}{92.8} & \multicolumn{1}{c|}{92.7} & 0.1 \\ \hline + \multicolumn{13}{|c|}{Ось $AA’$} \\ \hline + \multicolumn{1}{|c|}{№} & \multicolumn{1}{c|}{1} & \multicolumn{1}{c|}{2} & \multicolumn{1}{c|}{3} & \multicolumn{1}{c|}{4} & \multicolumn{1}{c|}{5} & \multicolumn{1}{c|}{Среднее} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \\ \hline + \multicolumn{1}{|c|}{$t$, с} & \multicolumn{1}{c|}{45.4} & \multicolumn{1}{c|}{45.4} & \multicolumn{1}{c|}{45.3} & \multicolumn{1}{c|}{45.3} & \multicolumn{1}{c|}{45.2} & \multicolumn{1}{c|}{45.3} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \\ \hline + \multicolumn{1}{|c|}{$T$, c} & \multicolumn{1}{c|}{3.03} & \multicolumn{1}{c|}{3.03} & \multicolumn{1}{c|}{3.02} & \multicolumn{1}{c|}{3.02} & \multicolumn{1}{c|}{3.01} & \multicolumn{1}{c|}{3.02} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \\ \hline + \multicolumn{13}{|c|}{Ось $MM’$} \\ \hline + \multicolumn{1}{|c|}{№} & \multicolumn{1}{c|}{1} & \multicolumn{1}{c|}{2} & \multicolumn{1}{c|}{3} & \multicolumn{1}{c|}{4} & \multicolumn{1}{c|}{5} & \multicolumn{1}{c|}{Среднее} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \\ \hline + \multicolumn{1}{|c|}{$t$, c} & \multicolumn{1}{c|}{45.6} & \multicolumn{1}{c|}{45.6} & \multicolumn{1}{c|}{45.5} & \multicolumn{1}{c|}{45.3} & \multicolumn{1}{c|}{45.5} & \multicolumn{1}{c|}{45.5} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \\ \hline + \multicolumn{1}{|c|}{$T$, c} & \multicolumn{1}{c|}{3.04} & \multicolumn{1}{c|}{3.04} & \multicolumn{1}{c|}{3.03} & \multicolumn{1}{c|}{3.02} & \multicolumn{1}{c|}{3.03} & \multicolumn{1}{c|}{3.03} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \\ \hline + \multicolumn{13}{|c|}{Ось $DD’$} \\ \hline + \multicolumn{1}{|c|}{№} & \multicolumn{1}{c|}{1} & \multicolumn{1}{c|}{2} & \multicolumn{1}{c|}{3} & \multicolumn{1}{c|}{4} & \multicolumn{1}{c|}{5} & \multicolumn{1}{c|}{Среднее} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \\ \hline + \multicolumn{1}{|c|}{$t$, c} & \multicolumn{1}{c|}{45.5} & \multicolumn{1}{c|}{45.7} & \multicolumn{1}{c|}{45.4} & \multicolumn{1}{c|}{45.5} & \multicolumn{1}{c|}{45.5} & \multicolumn{1}{c|}{45.5} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \\ \hline + \multicolumn{1}{|c|}{$T$, c} & \multicolumn{1}{c|}{3.03} & \multicolumn{1}{c|}{3.05} & \multicolumn{1}{c|}{3.03} & \multicolumn{1}{c|}{3.03} & \multicolumn{1}{c|}{3.03} & \multicolumn{1}{c|}{3.03} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \\ \hline + \end{tabular} + \end{table} + + Вычислим моменты инерции: + $$ I_{AA'} = I_p \frac{T_{AA'}^2 - T_p^2}{T_p^2} = 1605\; \text{кг} \cdot \text{мм} ^2, $$ + $$ I_{MM'} = I_p \frac{T_{MM'}^2 - T_p^2}{T_p^2} = 1643\; \text{кг} \cdot \text{мм} ^2, $$ + $$ I_{DD'} = I_p \frac{T_{DD'}^2 - T_p^2}{T_p^2} = 1643\; \text{кг} \cdot \text{мм} ^2 $$ + Так как они примерно одинаковые, можно рассчитать погрешность лишь для одного случая: + $$ \varepsilon_I = \varepsilon_{I_p} + 4\varepsilon_{T} = 0.12 $$ + + \item Нарисуем сечения эллипсоида инерции главными плоскостями: + + + \end{enumerate} + + % \section{Обсуждение результатов} % \section{Вывод} \end{document} From 8de2202fb51f0265e4c05b13dbee55208425b9b3 Mon Sep 17 00:00:00 2001 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