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| % Default to the notebook output style | |
| % Inherit from the specified cell style. | |
| \documentclass[11pt]{article} | |
| \usepackage[T1]{fontenc} | |
| % Nicer default font (+ math font) than Computer Modern for most use cases | |
| \usepackage{mathpazo} | |
| % Basic figure setup, for now with no caption control since it's done | |
| % automatically by Pandoc (which extracts  syntax from Markdown). | |
| \usepackage{graphicx} | |
| % We will generate all images so they have a width \maxwidth. This means | |
| % that they will get their normal width if they fit onto the page, but | |
| % are scaled down if they would overflow the margins. | |
| \makeatletter | |
| \def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth | |
| \else\Gin@nat@width\fi} | |
| \makeatother | |
| \let\Oldincludegraphics\includegraphics | |
| % Set max figure width to be 80% of text width, for now hardcoded. | |
| \renewcommand{\includegraphics}[1]{\Oldincludegraphics[width=.8\maxwidth]{#1}} | |
| % Ensure that by default, figures have no caption (until we provide a | |
| % proper Figure object with a Caption API and a way to capture that | |
| % in the conversion process - todo). | |
| \usepackage{caption} | |
| \DeclareCaptionLabelFormat{nolabel}{} | |
| \captionsetup{labelformat=nolabel} | |
| \usepackage{adjustbox} % Used to constrain images to a maximum size | |
| \usepackage{xcolor} % Allow colors to be defined | |
| \usepackage{enumerate} % Needed for markdown enumerations to work | |
| \usepackage{geometry} % Used to adjust the document margins | |
| \usepackage{amsmath} % Equations | |
| \usepackage{amssymb} % Equations | |
| \usepackage{textcomp} % defines textquotesingle | |
| % Hack from http://tex.stackexchange.com/a/47451/13684: | |
| \AtBeginDocument{% | |
| \def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code | |
| } | |
| \usepackage{upquote} % Upright quotes for verbatim code | |
| \usepackage{eurosym} % defines \euro | |
| \usepackage[mathletters]{ucs} % Extended unicode (utf-8) support | |
| \usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document | |
| \usepackage{fancyvrb} % verbatim replacement that allows latex | |
| \usepackage{grffile} % extends the file name processing of package graphics | |
| % to support a larger range | |
| % The hyperref package gives us a pdf with properly built | |
| % internal navigation ('pdf bookmarks' for the table of contents, | |
| % internal cross-reference links, web links for URLs, etc.) | |
| \usepackage{hyperref} | |
| \usepackage{longtable} % longtable support required by pandoc >1.10 | |
| \usepackage{booktabs} % table support for pandoc > 1.12.2 | |
| \usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment) | |
| \usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout) | |
| % normalem makes italics be italics, not underlines | |
| % Colors for the hyperref package | |
| \definecolor{urlcolor}{rgb}{0,.145,.698} | |
| \definecolor{linkcolor}{rgb}{.71,0.21,0.01} | |
| \definecolor{citecolor}{rgb}{.12,.54,.11} | |
| % ANSI colors | |
| \definecolor{ansi-black}{HTML}{3E424D} | |
| \definecolor{ansi-black-intense}{HTML}{282C36} | |
| \definecolor{ansi-red}{HTML}{E75C58} | |
| \definecolor{ansi-red-intense}{HTML}{B22B31} | |
| \definecolor{ansi-green}{HTML}{00A250} | |
| \definecolor{ansi-green-intense}{HTML}{007427} | |
| \definecolor{ansi-yellow}{HTML}{DDB62B} | |
| \definecolor{ansi-yellow-intense}{HTML}{B27D12} | |
| \definecolor{ansi-blue}{HTML}{208FFB} | |
| \definecolor{ansi-blue-intense}{HTML}{0065CA} | |
| \definecolor{ansi-magenta}{HTML}{D160C4} | |
| \definecolor{ansi-magenta-intense}{HTML}{A03196} | |
| \definecolor{ansi-cyan}{HTML}{60C6C8} | |
| \definecolor{ansi-cyan-intense}{HTML}{258F8F} | |
| \definecolor{ansi-white}{HTML}{C5C1B4} | |
| \definecolor{ansi-white-intense}{HTML}{A1A6B2} | |
| % commands and environments needed by pandoc snippets | |
| % extracted from the output of `pandoc -s` | |
| \providecommand{\tightlist}{% | |
| \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} | |
| \DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}} | |
| % Add ',fontsize=\small' for more characters per line | |
| \newenvironment{Shaded}{}{} | |
| \newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}} | |
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| \newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} | |
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| \newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}} | |
| \newcommand{\NormalTok}[1]{{#1}} | |
| % Additional commands for more recent versions of Pandoc | |
| \newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}} | |
| \newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} | |
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| \newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} | |
| % Define a nice break command that doesn't care if a line doesn't already | |
| % exist. | |
| \def\br{\hspace*{\fill} \\* } | |
| % Math Jax compatability definitions | |
| \def\gt{>} | |
| \def\lt{<} | |
| % Document parameters | |
| \title{ME3263\_Lab-02} | |
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| \def\PY@tok#1{\csname PY@tok@#1\endcsname} | |
| \def\PY@toks#1+{\ifx\relax#1\empty\else% | |
| \PY@tok{#1}\expandafter\PY@toks\fi} | |
| \def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{% | |
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| % for compatibility with earlier versions | |
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| \begin{document} | |
| \maketitle | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}87}]:} \PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np} | |
| \PY{k+kn}{from} \PY{n+nn}{scipy}\PY{n+nn}{.}\PY{n+nn}{stats} \PY{k}{import} \PY{n}{norm}\PY{p}{,} \PY{n}{t} | |
| \PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{pyplot} \PY{k}{as} \PY{n+nn}{plt} | |
| \PY{k+kn}{import} \PY{n+nn}{pandas} \PY{k}{as} \PY{n+nn}{pd} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{rcParams}\PY{o}{.}\PY{n}{update}\PY{p}{(}\PY{p}{\PYZob{}}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{font.size}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{l+m+mi}{22}\PY{p}{\PYZcb{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{rcParams}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{lines.linewidth}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} \PY{o}{=} \PY{l+m+mi}{3} | |
| \PY{k+kn}{import} \PY{n+nn}{check\PYZus{}lab02} \PY{k}{as} \PY{n+nn}{p} | |
| \end{Verbatim} | |
| \hypertarget{me-3263-introduction-to-sensors-and-data-analysis-fall-2018}{% | |
| \section{ME 3263 Introduction to Sensors and Data Analysis (Fall | |
| 2018)}\label{me-3263-introduction-to-sensors-and-data-analysis-fall-2018}} | |
| \hypertarget{lab-2---static-beam-deflections-with-strain-gage}{% | |
| \section{Lab \#2 - Static beam deflections with strain | |
| gage}\label{lab-2---static-beam-deflections-with-strain-gage}} | |
| \hypertarget{what-is-a-strain-gage}{% | |
| \section{What is a Strain Gage?}\label{what-is-a-strain-gage}} | |
| A strain gage consists of a looped wire that is embedded in a thin | |
| backing. Two copper coated tabs serve as solder points for the leads. | |
| See Figure 1a. The strain gage is mounted to the structure, whose | |
| deformation is to be measured. As the structure deforms, the wire | |
| stretches (increasing its net length ) and its electrical resistance | |
| changes: \(R=\rho L/A\), where \(\rho\) is the material resistivity, | |
| \(L\) is the total length of the wire, and \(A\) is the cross sectional | |
| area of the wire. Note that as \(L\) increases, the cross sectional area | |
| changes as well due to the Poisson contraction; the resistivity also | |
| changes. | |
| \begin{figure} | |
| \centering | |
| \includegraphics{./figure_01.png} | |
| \caption{Figure 1: a) A typical strain gage. b) One common setup: the | |
| gage is mounted to measure the x-direction strain on the top surface. | |
| It's engaged in a quarter bridge configuration of the Wheatstone bridge | |
| circuit.} | |
| \end{figure} | |
| \emph{Figure 1: a) A typical strain gage. b) One common setup: the gage | |
| is mounted to measure the x-direction strain on the top surface. It's | |
| engaged in a quarter bridge configuration of the Wheatstone bridge | |
| circuit.} | |
| \hypertarget{digital-image-correlation}{% | |
| \section{Digital Image Correlation}\label{digital-image-correlation}} | |
| \url{https://github.com/dicengine/dice} | |
| \begin{figure} | |
| \centering | |
| \includegraphics{https://cloud.githubusercontent.com/assets/15202746/26112065/167c12b0-3a14-11e7-8361-a4978dc54bf0.png} | |
| \caption{DICe Digital Image Correlation engine} | |
| \end{figure} | |
| Digital image correlation (DIC) is a method to track movement of groups | |
| of pixels with sub-pixel accuracy used in solid mechanics {[}0{]}. A | |
| region of interest (ROI)~is chosen, shown in green above, and groups of | |
| pixels are tracked in subsequent ``deformed'' images. | |
| In this lab, we will use DIC to determine the curvature \(\kappa\) in | |
| 1/pixels and multiply by the thickness in pixels to determine the strain | |
| on the surface of the beam. The kinematics is detailed in the next | |
| section ``Validate strain gage measurements'' | |
| \hypertarget{validate-static-strain-gage-measurements}{% | |
| \section{Validate static strain gage | |
| measurements}\label{validate-static-strain-gage-measurements}} | |
| In this lab we will calibrate strain measurements using Euler-Bernouli | |
| beam theory kinematics {[}1{]}. The axial strain in a beam is directly | |
| proportional to the distance from the neutral axis and curvature as such | |
| \(\epsilon_x=-\kappa z\). (1) | |
| The curvature of the beam is approximated as such | |
| \(\kappa=\frac{d^2 w}{dx^2}\). (2) | |
| Equations 1 and 2 relate beam deflection to axial strain. These are | |
| called kinematic equations because they define the geometry of the beam | |
| deflection. | |
| The Euler-Bernouli beam theory uses Newton's second law (kinetics) and | |
| Hooke's law (constitutive) to derive the linear equations of motion for | |
| one dimensional deformable objects {[}1,4{]}. The relation between a | |
| static applied force q(x) and static deflection w(x) is as such | |
| \(\frac{\partial^2}{\partial x^2}\left(EI\frac{\partial^2 w}{\partial x^2}\right)=q(x)\) | |
| (3) | |
| where \(E\) is the Young's modulus, \(x\) is the distance along the | |
| neutral axis, and \(I\) is the second moment of area of the beam's | |
| cross-section. For a rectangular cross-section \(b \times t\), width by | |
| thickness the second moment of area is | |
| \(I=\frac{bt^3}{12}\). (4) | |
| We will design our strain gage validation such that \(E\) and \(I\) are | |
| not necessary. The boundary conditions for a clamped beam that is | |
| deflected by a distance \(\delta\) are \(w(0)=0\), \(w'(0)=0\), | |
| \(w(L)=\delta\), and \(w''(L)=0\). Therefore, the functions \(w(x)\) and | |
| \(w''(x)\) can be determined by integrating equation 3 four times and | |
| using the four boundary conditions as such | |
| \(w(x)=-\frac{1}{2}\left(\frac{\delta}{L^3}x^3-3\frac{\delta}{L^2}x^2\right)\) | |
| (5a) | |
| and the curvature | |
| \(w''(x)=-\left(3\frac{\delta}{L^3}x-3\frac{\delta}{L^2}\right)\) (5b) | |
| Using equations 5a-b, the only quantities needed to determine strain at | |
| a given location on a linear, homogeneous beam are \(z\), \(\delta\), | |
| \(w(L)\), and \(L\). | |
| \begin{figure} | |
| \centering | |
| \includegraphics{./figure_02.png} | |
| \caption{Figure 2: Diagram of the validation process. The strain gage is | |
| placed at a distance \(x_{SG}\) from the cantilever support. A | |
| linear-elastic beam of length \(L\) is deflected by distance, | |
| \(\delta\).} | |
| \end{figure} | |
| \emph{Figure 2: Diagram of the validation process. The strain gage is | |
| placed at a distance \(x_{SG}\) from the cantilever support. A | |
| linear-elastic beam of length \(L\) is deflected by distance, | |
| \(\delta\).} | |
| \begin{itemize} | |
| \item | |
| take a picture using a cross-hair to get the beam as close to | |
| horizontal as possible | |
| \item | |
| Apply a known tip displacement by applying a load on the beam at | |
| multiple locations as seen in Figure 2. Measure and record the strain | |
| at your strain gage location, the displacement, \(\delta\) and length | |
| where load was applied, \(L\). | |
| \item | |
| take a picture of the deformed beam | |
| \item | |
| Calculate the strain at the location of the strain gage using | |
| equations 1,2, and 5a-b. | |
| \end{itemize} | |
| Below are two functions that calculate displacement and curvature given | |
| a bar of length \texttt{L}, that is deflected by a distance, | |
| \texttt{delta}. | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}10}]:} \PY{k}{def} \PY{n+nf}{disp\PYZus{}at\PYZus{}x}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{L}\PY{p}{,}\PY{n}{delta}\PY{p}{)}\PY{p}{:} | |
| \PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}returns the displacement w(x) given the position, x, } | |
| \PY{l+s+sd}{ length of bar, L, and displacement at L, delta\PYZsq{}\PYZsq{}\PYZsq{}} | |
| \PY{n}{wx}\PY{o}{=}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{*}\PY{p}{(}\PY{n}{delta}\PY{o}{/}\PY{n}{L}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{3}\PY{o}{*}\PY{n}{x}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{3}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{o}{*}\PY{n}{delta}\PY{o}{/}\PY{n}{L}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{x}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{)} | |
| \PY{k}{return} \PY{n}{wx} | |
| \PY{k}{def} \PY{n+nf}{k\PYZus{}at\PYZus{}x}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{L}\PY{p}{,}\PY{n}{delta}\PY{p}{)}\PY{p}{:} | |
| \PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}returns the curvature w\PYZsq{}\PYZsq{}(x) given the position, x, } | |
| \PY{l+s+sd}{ length of bar, L, and displacement at L, delta\PYZsq{}\PYZsq{}\PYZsq{}} | |
| \PY{n}{wx}\PY{o}{=}\PY{o}{\PYZhy{}}\PY{p}{(}\PY{l+m+mi}{3}\PY{o}{*}\PY{n}{delta}\PY{o}{/}\PY{n}{L}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{3}\PY{o}{*}\PY{n}{x}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{o}{*}\PY{n}{delta}\PY{o}{/}\PY{n}{L}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{)} | |
| \PY{k}{return} \PY{n}{wx} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}7}]:} \PY{n}{w20mm}\PY{o}{=}\PY{n}{disp\PYZus{}at\PYZus{}x}\PY{p}{(}\PY{l+m+mi}{20}\PY{p}{,}\PY{l+m+mi}{400}\PY{p}{,}\PY{l+m+mi}{10}\PY{p}{)} | |
| \PY{n}{k20mm}\PY{o}{=}\PY{n}{k\PYZus{}at\PYZus{}x}\PY{p}{(}\PY{l+m+mi}{20}\PY{p}{,}\PY{l+m+mi}{400}\PY{p}{,}\PY{l+m+mi}{10}\PY{p}{)} | |
| \PY{n+nb}{print}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{displacement of 400 mm bar deflected 10 mm, 20 mm from support =}\PY{l+s+si}{\PYZpc{}1.3f}\PY{l+s+s1}{ mm}\PY{l+s+s1}{\PYZsq{}}\PY{o}{\PYZpc{}}\PY{k}{w20mm}) | |
| \PY{n+nb}{print}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{ curvature of 400 mm bar deflected 10 mm, 20 mm from support =}\PY{l+s+si}{\PYZpc{}1.3f}\PY{l+s+s1}{ 1/m}\PY{l+s+s1}{\PYZsq{}}\PY{o}{\PYZpc{}}\PY{p}{(}\PY{n}{k20mm}\PY{o}{*}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{)} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| displacement of 400 mm bar deflected 10 mm, 20 mm from support =0.037 mm | |
| curvature of 400 mm bar deflected 10 mm, 20 mm from support =0.178 1/m | |
| \end{Verbatim} | |
| \hypertarget{problem-1}{% | |
| \section{Problem 1}\label{problem-1}} | |
| If your beam is 6000 pixels long by 500 pixels thick and it deflects 700 | |
| pixels at the tip, determine the following: | |
| \begin{enumerate} | |
| \def\labelenumi{\alph{enumi}.} | |
| \item | |
| displacement \texttt{w300} of the beam 3000 pixels from the base | |
| \item | |
| the curvature \texttt{k300} of the beam 3000 pixels from the base | |
| \item | |
| the strain \texttt{e300} of the beam 3000 pixels from the base | |
| \end{enumerate} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor} }]:} \PY{c+c1}{\PYZsh{} your work here} | |
| \PY{n}{p}\PY{o}{.}\PY{n}{check\PYZus{}p01}\PY{p}{(}\PY{n}{w300}\PY{p}{,}\PY{n}{k300}\PY{p}{,}\PY{n}{e300}\PY{p}{)} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}90}]:} \PY{n}{x}\PY{o}{=}\PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{400}\PY{p}{)} | |
| \PY{n}{wx}\PY{o}{=}\PY{n}{disp\PYZus{}at\PYZus{}x}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{l+m+mi}{400}\PY{p}{,}\PY{l+m+mi}{10}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{wx}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{position along beam (mm)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+sa}{r}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{deflection of }\PY{l+s+se}{\PYZbs{}\PYZbs{}}\PY{l+s+s1}{neutral axis w(x) (mm)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Shape of deflected cantilever beam}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{outcolor}Out[{\color{outcolor}90}]:} Text(0.5, 1.0, 'Shape of deflected cantilever beam') | |
| \end{Verbatim} | |
| \begin{center} | |
| \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_12_1.png} | |
| \end{center} | |
| { \hspace*{\fill} \\} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}91}]:} \PY{n}{x}\PY{o}{=}\PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{400}\PY{p}{)} | |
| \PY{n}{kx}\PY{o}{=}\PY{n}{k\PYZus{}at\PYZus{}x}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{l+m+mi}{400}\PY{p}{,}\PY{l+m+mi}{10}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{kx}\PY{o}{*}\PY{l+m+mi}{1000}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{position along beam (mm)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+sa}{r}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{curvature }\PY{l+s+se}{\PYZbs{}\PYZbs{}}\PY{l+s+s1}{of neutral axis }\PY{l+s+se}{\PYZbs{}\PYZbs{}}\PY{l+s+s1}{w(x) (1/m)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Curvature of deflected cantilever beam}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{outcolor}Out[{\color{outcolor}91}]:} Text(0.5, 1.0, 'Curvature of deflected cantilever beam') | |
| \end{Verbatim} | |
| \begin{center} | |
| \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_13_1.png} | |
| \end{center} | |
| { \hspace*{\fill} \\} | |
| \hypertarget{using-dice-results-files}{% | |
| \section{Using DICe results files}\label{using-dice-results-files}} | |
| You can load the displacement calculated in DICe into your Python | |
| workspace for analysis. Included here is a folder ``results-dice'' with | |
| 2 images, the reference and the deformed image. The following creates | |
| two arrays from the two image analyses (reference = 0 and deformed = 1) | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}92}]:} \PY{n}{data0}\PY{o}{=}\PY{n}{pd}\PY{o}{.}\PY{n}{read\PYZus{}csv}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{./results\PYZhy{}dice/DICe\PYZus{}solution\PYZus{}00.txt}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{data1}\PY{o}{=}\PY{n}{pd}\PY{o}{.}\PY{n}{read\PYZus{}csv}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{./results\PYZhy{}dice/DICe\PYZus{}solution\PYZus{}01.txt}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{data0}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{COORDINATE\PYZus{}X}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{+}\PY{n}{data0}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{DISPLACEMENT\PYZus{}X}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,}\PY{n}{data0}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{COORDINATE\PYZus{}Y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{+}\PY{n}{data0}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{DISPLACEMENT\PYZus{}Y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{o}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{reference}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{data1}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{COORDINATE\PYZus{}X}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{+}\PY{n}{data1}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{DISPLACEMENT\PYZus{}X}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,}\PY{n}{data1}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{COORDINATE\PYZus{}Y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{+}\PY{n}{data1}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{DISPLACEMENT\PYZus{}Y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{o}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{deformed}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{camera x\PYZhy{}axis}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{camera y\PYZhy{}axis}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{n}{bbox\PYZus{}to\PYZus{}anchor}\PY{o}{=}\PY{p}{(}\PY{l+m+mf}{1.05}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{n}{loc}\PY{o}{=}\PY{l+m+mi}{2}\PY{p}{,} \PY{n}{borderaxespad}\PY{o}{=}\PY{l+m+mf}{0.}\PY{p}{)} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{outcolor}Out[{\color{outcolor}92}]:} <matplotlib.legend.Legend at 0x7f652bbe8358> | |
| \end{Verbatim} | |
| \begin{center} | |
| \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_15_1.png} | |
| \end{center} | |
| { \hspace*{\fill} \\} | |
| The raw data has two problems: | |
| \begin{enumerate} | |
| \def\labelenumi{\arabic{enumi}.} | |
| \item | |
| The deformed image has poorly tracked regions | |
| \item | |
| The reference image is not a flat line (the camera is tilted) | |
| \end{enumerate} | |
| We fix these issues with two solutions: | |
| \begin{enumerate} | |
| \def\labelenumi{\arabic{enumi}.} | |
| \item | |
| Use a measure of how well the regions were tracked be the image | |
| correlation method | |
| \item | |
| Use best-fit lines to determine the shape of the | |
| \end{enumerate} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}93}]:} \PY{n}{data0}\PY{o}{=}\PY{n}{data0}\PY{p}{[}\PY{n}{data1}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{UNCERTAINTY}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{\PYZgt{}}\PY{l+m+mf}{0.05}\PY{p}{]} | |
| \PY{n}{data1}\PY{o}{=}\PY{n}{data1}\PY{p}{[}\PY{n}{data1}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{UNCERTAINTY}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{\PYZgt{}}\PY{l+m+mf}{0.05}\PY{p}{]} | |
| \PY{n}{X0}\PY{o}{=}\PY{n}{data0}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{COORDINATE\PYZus{}X}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{+}\PY{n}{data0}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{DISPLACEMENT\PYZus{}X}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} | |
| \PY{n}{Y0}\PY{o}{=}\PY{n}{data0}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{COORDINATE\PYZus{}Y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{+}\PY{n}{data0}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{DISPLACEMENT\PYZus{}Y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} | |
| \PY{n}{X1}\PY{o}{=}\PY{n}{data1}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{COORDINATE\PYZus{}X}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{+}\PY{n}{data1}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{DISPLACEMENT\PYZus{}X}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} | |
| \PY{n}{Y1}\PY{o}{=}\PY{n}{data1}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{COORDINATE\PYZus{}Y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{+}\PY{n}{data1}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{DISPLACEMENT\PYZus{}Y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{X0}\PY{p}{,}\PY{n}{Y0}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{o}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{reference}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{X1}\PY{p}{,}\PY{n}{Y1}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{o}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{deformed}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{camera x\PYZhy{}axis}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{camera y\PYZhy{}axis}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{n}{bbox\PYZus{}to\PYZus{}anchor}\PY{o}{=}\PY{p}{(}\PY{l+m+mf}{1.05}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{n}{loc}\PY{o}{=}\PY{l+m+mi}{2}\PY{p}{,} \PY{n}{borderaxespad}\PY{o}{=}\PY{l+m+mf}{0.}\PY{p}{)} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{outcolor}Out[{\color{outcolor}93}]:} <matplotlib.legend.Legend at 0x7f652acbb828> | |
| \end{Verbatim} | |
| \begin{center} | |
| \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_17_1.png} | |
| \end{center} | |
| { \hspace*{\fill} \\} | |
| In the graph above all the points with `UNCERTAINTY'\textless{}0.05 were | |
| removed. This is a measure made during the cross-correlation method | |
| {[}0{]}. | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}94}]:} \PY{k+kn}{from} \PY{n+nn}{scipy}\PY{n+nn}{.}\PY{n+nn}{optimize} \PY{k}{import} \PY{n}{curve\PYZus{}fit} | |
| \PY{k}{def} \PY{n+nf}{reference\PYZus{}line}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{A}\PY{p}{,}\PY{n}{B}\PY{p}{)}\PY{p}{:} | |
| \PY{k}{return} \PY{n}{A}\PY{o}{*}\PY{n}{x}\PY{o}{+}\PY{n}{B} | |
| \PY{k}{def} \PY{n+nf}{deformed\PYZus{}line}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{A}\PY{p}{,}\PY{n}{B}\PY{p}{,}\PY{n}{C}\PY{p}{,}\PY{n}{D}\PY{p}{,}\PY{n}{E}\PY{p}{)}\PY{p}{:} | |
| \PY{k}{return} \PY{n}{A}\PY{o}{*}\PY{n}{x}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{4}\PY{o}{+}\PY{n}{B}\PY{o}{*}\PY{n}{x}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{3}\PY{o}{+}\PY{n}{C}\PY{o}{*}\PY{n}{x}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{o}{+}\PY{n}{D}\PY{o}{*}\PY{n}{x}\PY{o}{+}\PY{n}{E} | |
| \PY{n}{Al}\PY{p}{,}\PY{n}{pcov}\PY{o}{=}\PY{n}{curve\PYZus{}fit}\PY{p}{(}\PY{n}{reference\PYZus{}line}\PY{p}{,} \PY{n}{X0}\PY{p}{,} \PY{n}{Y0}\PY{p}{)} | |
| \PY{n}{A0}\PY{p}{,}\PY{n}{B0}\PY{o}{=}\PY{n}{Al} | |
| \PY{n}{Aq}\PY{p}{,}\PY{n}{pcov}\PY{o}{=}\PY{n}{curve\PYZus{}fit}\PY{p}{(}\PY{n}{deformed\PYZus{}line}\PY{p}{,} \PY{n}{X1}\PY{p}{,} \PY{n}{Y1}\PY{p}{)} | |
| \PY{n}{A1}\PY{p}{,}\PY{n}{B1}\PY{p}{,}\PY{n}{C1}\PY{p}{,}\PY{n}{D1}\PY{p}{,}\PY{n}{E1}\PY{o}{=}\PY{n}{Aq}\PY{p}{[}\PY{p}{:}\PY{p}{]} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}95}]:} \PY{n}{Y0\PYZus{}fit}\PY{o}{=}\PY{n}{reference\PYZus{}line}\PY{p}{(}\PY{n}{X0}\PY{p}{,}\PY{n}{A0}\PY{p}{,}\PY{n}{B0}\PY{p}{)} | |
| \PY{n}{Y1\PYZus{}fit}\PY{o}{=}\PY{n}{deformed\PYZus{}line}\PY{p}{(}\PY{n}{X1}\PY{p}{,}\PY{n}{A1}\PY{p}{,}\PY{n}{B1}\PY{p}{,}\PY{n}{C1}\PY{p}{,}\PY{n}{D1}\PY{p}{,}\PY{n}{E1}\PY{p}{)} | |
| \PY{n}{x\PYZus{}fit}\PY{o}{=}\PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{min}\PY{p}{(}\PY{n}{X0}\PY{p}{)}\PY{p}{,}\PY{n}{np}\PY{o}{.}\PY{n}{max}\PY{p}{(}\PY{n}{X0}\PY{p}{)}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{X0}\PY{p}{,}\PY{n}{Y0}\PY{o}{\PYZhy{}}\PY{n}{Y0\PYZus{}fit}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{o}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{reference}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{X1}\PY{p}{,}\PY{n}{Y1}\PY{o}{\PYZhy{}}\PY{n}{Y0\PYZus{}fit}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{o}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{deformed}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{X0}\PY{p}{,}\PY{n}{Y0\PYZus{}fit}\PY{o}{\PYZhy{}}\PY{n}{Y0\PYZus{}fit}\PY{p}{,}\PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{ref fit}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{x\PYZus{}fit}\PY{p}{,}\PY{n}{deformed\PYZus{}line}\PY{p}{(}\PY{n}{x\PYZus{}fit}\PY{p}{,}\PY{n}{A1}\PY{p}{,}\PY{n}{B1}\PY{p}{,}\PY{n}{C1}\PY{p}{,}\PY{n}{D1}\PY{p}{,}\PY{n}{E1}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{reference\PYZus{}line}\PY{p}{(}\PY{n}{x\PYZus{}fit}\PY{p}{,}\PY{n}{A0}\PY{p}{,}\PY{n}{B0}\PY{p}{)}\PY{p}{,}\PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{def fit}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x\PYZhy{}axis}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y\PYZhy{}axis}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{n}{bbox\PYZus{}to\PYZus{}anchor}\PY{o}{=}\PY{p}{(}\PY{l+m+mf}{1.05}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{n}{loc}\PY{o}{=}\PY{l+m+mi}{2}\PY{p}{,} \PY{n}{borderaxespad}\PY{o}{=}\PY{l+m+mf}{0.}\PY{p}{)} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{outcolor}Out[{\color{outcolor}95}]:} <matplotlib.legend.Legend at 0x7f652ae3b240> | |
| \end{Verbatim} | |
| \begin{center} | |
| \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_20_1.png} | |
| \end{center} | |
| { \hspace*{\fill} \\} | |
| In the figure above, we have done 3 things: | |
| \begin{enumerate} | |
| \def\labelenumi{\arabic{enumi}.} | |
| \item | |
| removed the untracked sections of the beam | |
| \item | |
| subtracted a line of best fit for the reference and deformed images | |
| \item | |
| plotted the displacement of the beam | |
| \end{enumerate} | |
| \hypertarget{problem-2}{% | |
| \section{Problem 2}\label{problem-2}} | |
| Based upon the deformation from the reference to the deformed | |
| configuration, write a function that returns the strain on the surface | |
| of the deformed beam if the beam is 7 pixels thick. *Hint: you'll have | |
| to calculate the second derivative of the displacement and use | |
| \$\epsilon = -z\kappa | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}1}]:} \PY{k}{def} \PY{n+nf}{strain\PYZus{}in\PYZus{}beam}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{A}\PY{p}{,}\PY{n}{B}\PY{p}{,}\PY{n}{C}\PY{p}{,}\PY{n}{D}\PY{p}{,}\PY{n}{E}\PY{p}{)}\PY{p}{:} | |
| \PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}function to return the strain in a beam that has the shape:} | |
| \PY{l+s+sd}{ w(x) = A*x**4+B*x**3+C*x**2+D*x+E} | |
| \PY{l+s+sd}{ the thickness of the beam is t=7 px in this case\PYZsq{}\PYZsq{}\PYZsq{}} | |
| \PY{n}{t}\PY{o}{=}\PY{l+m+mi}{6} | |
| \PY{c+c1}{\PYZsh{} your work here e=...} | |
| \PY{k}{return} \PY{n}{e} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor} }]:} \PY{k+kn}{import} \PY{n+nn}{check\PYZus{}lab02} \PY{k}{as} \PY{n+nn}{p} | |
| \PY{n}{p}\PY{o}{.}\PY{n}{check\PYZus{}p02}\PY{p}{(}\PY{n}{strain\PYZus{}in\PYZus{}beam}\PY{p}{)} | |
| \end{Verbatim} | |
| \hypertarget{measure-constitutive-properties}{% | |
| \section{Measure constitutive | |
| properties}\label{measure-constitutive-properties}} | |
| \hypertarget{constitutive-model}{% | |
| \subsection{Constitutive Model}\label{constitutive-model}} | |
| Now the strain gage should be calibrated. We need to determine the | |
| Young's modulus, \(E\) and second moment of area, \(I\). This | |
| measurement requires three components: kinematic (which we solved in the | |
| validation), kinetic (\(\sum{F}=0\)), and a constitutive model (Hooke's | |
| Law). The constitutive equation for a linear-elastic beam subject to a | |
| moment is as such | |
| \(M=EI\kappa\). (6) | |
| In equation 6, \(M\) is the applied moment, \(E\) is the Young's | |
| modulus, \(I\) is the second moment of inertia of the area of the beam, | |
| and \(\kappa\) is the curvature of the beam. The constant \(I\) is based | |
| upon the geometry from equation 4. The material constant \(E\) is | |
| unknown. | |
| Use the weights to apply forces at different distances, \(r\), from the | |
| support as seen in Figure 3. Increasing the weight or the distance the | |
| weight is applied will increase the applied moment. Use Table 1 to | |
| record the trial no., strain, force, distance, and moment. Use at least | |
| two trials per measurement. | |
| \begin{figure} | |
| \centering | |
| \includegraphics{./figure_03.png} | |
| \caption{Figure 3: A linear-elastic beam of length \(L\) has a force | |
| applied at distance, \(r\). The strain gage is placed at a distance | |
| \(x_{SG}\) from the cantilever support.} | |
| \end{figure} | |
| \emph{Figure 3: A linear-elastic beam of length \(L\) has a force | |
| applied at distance, \(r\). The strain gage is placed at a distance | |
| \(x\) from the cantilever support.} | |
| \hypertarget{fitting-your-data-to-your-model}{% | |
| \subsection{Fitting your data to your | |
| model}\label{fitting-your-data-to-your-model}} | |
| Once you have filled in a number of data points for \(\kappa\) and | |
| \(M\), you can use a linear regression to determine the slope of the | |
| data. The constitutive model in equation (6) predicts that the moment | |
| and curvature will be related by a proportional constant, \(EI\). If we | |
| know \(EI\), the total squared error is as such | |
| \(SSE=\sum_i^N{(M_i-EI\kappa_i)^2}\) (7) | |
| where SSE is the sum of squares error between the predicted moment and | |
| measured moment for the \(i^{th}\) measurement with \(N\) total | |
| measurements | |
| \href{https://www.amazon.com/Numerical-Methods-Engineers-Steven-Chapra/dp/0073401064}{{[}2{]}}. | |
| We can choose a of \(EI\) that minimizes \(SSE\), but it will never be | |
| zero. Below is an example calculation for a linear least squares | |
| regression in python for a beam with cross-section \(12\times3\) mm. | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}7}]:} \PY{n}{k}\PY{o}{=}\PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{6.95685737e\PYZhy{}07}\PY{p}{,} \PY{l+m+mf}{9.93992373e\PYZhy{}06}\PY{p}{,} \PY{l+m+mf}{3.25200211e\PYZhy{}05}\PY{p}{,} \PY{l+m+mf}{3.55750721e\PYZhy{}05}\PY{p}{,} | |
| \PY{l+m+mf}{5.32023782e\PYZhy{}05}\PY{p}{,} \PY{l+m+mf}{7.48128585e\PYZhy{}05}\PY{p}{,} \PY{l+m+mf}{7.61625461e\PYZhy{}05}\PY{p}{,} \PY{l+m+mf}{8.54476229e\PYZhy{}05}\PY{p}{,} | |
| \PY{l+m+mf}{1.02089509e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{1.02841452e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{1.32351731e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{1.43996022e\PYZhy{}04}\PY{p}{,} | |
| \PY{l+m+mf}{1.45204793e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{1.56867759e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{1.73435915e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{1.95625232e\PYZhy{}04}\PY{p}{,} | |
| \PY{l+m+mf}{2.00670618e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{2.12900332e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{2.24582886e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{2.41141396e\PYZhy{}04}\PY{p}{,} | |
| \PY{l+m+mf}{2.45991618e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{2.55608426e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{2.76117673e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{2.96128593e\PYZhy{}04}\PY{p}{,} | |
| \PY{l+m+mf}{3.07157389e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{3.20509718e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{3.20363196e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{3.35535814e\PYZhy{}04}\PY{p}{,} | |
| \PY{l+m+mf}{3.53706984e\PYZhy{}04}\PY{p}{,} \PY{l+m+mf}{3.64130433e\PYZhy{}04}\PY{p}{]}\PY{p}{)} | |
| \PY{n}{M}\PY{o}{=}\PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[} \PY{l+m+mf}{0.} \PY{p}{,} \PY{l+m+mf}{23.82866379}\PY{p}{,} \PY{l+m+mf}{47.65732759}\PY{p}{,} \PY{l+m+mf}{71.48599138}\PY{p}{,} | |
| \PY{l+m+mf}{95.31465517}\PY{p}{,} \PY{l+m+mf}{119.14331897}\PY{p}{,} \PY{l+m+mf}{142.97198276}\PY{p}{,} \PY{l+m+mf}{166.80064655}\PY{p}{,} | |
| \PY{l+m+mf}{190.62931034}\PY{p}{,} \PY{l+m+mf}{214.45797414}\PY{p}{,} \PY{l+m+mf}{238.28663793}\PY{p}{,} \PY{l+m+mf}{262.11530172}\PY{p}{,} | |
| \PY{l+m+mf}{285.94396552}\PY{p}{,} \PY{l+m+mf}{309.77262931}\PY{p}{,} \PY{l+m+mf}{333.6012931} \PY{p}{,} \PY{l+m+mf}{357.4299569} \PY{p}{,} | |
| \PY{l+m+mf}{381.25862069}\PY{p}{,} \PY{l+m+mf}{405.08728448}\PY{p}{,} \PY{l+m+mf}{428.91594828}\PY{p}{,} \PY{l+m+mf}{452.74461207}\PY{p}{,} | |
| \PY{l+m+mf}{476.57327586}\PY{p}{,} \PY{l+m+mf}{500.40193966}\PY{p}{,} \PY{l+m+mf}{524.23060345}\PY{p}{,} \PY{l+m+mf}{548.05926724}\PY{p}{,} | |
| \PY{l+m+mf}{571.88793103}\PY{p}{,} \PY{l+m+mf}{595.71659483}\PY{p}{,} \PY{l+m+mf}{619.54525862}\PY{p}{,} \PY{l+m+mf}{643.37392241}\PY{p}{,} | |
| \PY{l+m+mf}{667.20258621}\PY{p}{,} \PY{l+m+mf}{691.03125} \PY{p}{]}\PY{p}{)} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}20}]:} \PY{k+kn}{from} \PY{n+nn}{scipy}\PY{n+nn}{.}\PY{n+nn}{optimize} \PY{k}{import} \PY{n}{curve\PYZus{}fit} | |
| \PY{k}{def} \PY{n+nf}{func}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{EI}\PY{p}{)}\PY{p}{:} | |
| \PY{k}{return} \PY{n}{EI}\PY{o}{*}\PY{n}{x} | |
| \PY{n}{EI}\PY{p}{,}\PY{n}{pcov}\PY{o}{=}\PY{n}{curve\PYZus{}fit}\PY{p}{(}\PY{n}{func}\PY{p}{,} \PY{n}{k}\PY{p}{,} \PY{n}{M}\PY{p}{)} | |
| \PY{n}{EI\PYZus{}error}\PY{o}{=}\PY{n}{np}\PY{o}{.}\PY{n}{sqrt}\PY{p}{(}\PY{n}{pcov}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{)} | |
| \PY{n}{I}\PY{o}{=}\PY{l+m+mi}{12}\PY{o}{*}\PY{l+m+mi}{3}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{3}\PY{o}{/}\PY{l+m+mf}{12.0} | |
| \PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Best fit for Young}\PY{l+s+s2}{\PYZsq{}}\PY{l+s+s2}{s Modulus is }\PY{l+s+si}{\PYZpc{}1.1f}\PY{l+s+s2}{ +/\PYZhy{} }\PY{l+s+si}{\PYZpc{}1.1f}\PY{l+s+s2}{ GPa}\PY{l+s+s2}{\PYZdq{}}\PY{o}{\PYZpc{}}\PY{p}{(}\PY{n}{EI}\PY{o}{/}\PY{n}{I}\PY{o}{*}\PY{l+m+mf}{1e\PYZhy{}3}\PY{p}{,}\PY{n}{EI\PYZus{}error}\PY{o}{/}\PY{n}{I}\PY{o}{*}\PY{l+m+mf}{1e\PYZhy{}3}\PY{p}{)}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{k}\PY{o}{*}\PY{l+m+mf}{1e3}\PY{p}{,}\PY{n}{M}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{o}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{experiment}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{k}\PY{o}{*}\PY{l+m+mf}{1e3}\PY{p}{,}\PY{n}{func}\PY{p}{(}\PY{n}{k}\PY{p}{,}\PY{n}{EI}\PY{p}{)}\PY{p}{,}\PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{model}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{curvature (1/m)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Moment (N\PYZhy{}mm)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| Best fit for Young's Modulus is 70.2 +/- 0.3 GPa | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{outcolor}Out[{\color{outcolor}20}]:} Text(0,0.5,'Moment (N-mm)') | |
| \end{Verbatim} | |
| \begin{center} | |
| \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_28_2.png} | |
| \end{center} | |
| { \hspace*{\fill} \\} | |
| \emph{Note:} | |
| The least-squares method used above can be used to fit any function to a | |
| data set. You would just need to update the \texttt{func} definition to | |
| return the desired function based upon your unkown fitting constants. | |
| The outout \texttt{popt} is the covariance matrix | |
| \href{./least_squares-error_with_covariance.pdf}{{[}3{]}}. In practice, | |
| we can use the square root of the diagonal terms to estimate the error | |
| in our least-squares fit. We make a few assumptions when performing this | |
| best-fit: | |
| \begin{enumerate} | |
| \def\labelenumi{\arabic{enumi}.} | |
| \item | |
| There is a random error in the measured dependent variable (here the | |
| moment \(M\)). | |
| \item | |
| There is no error in the reported independent variable (here the | |
| curvature \(\kappa\)). | |
| \item | |
| The measured dependent variables are uncorrelated with the measured | |
| error | |
| \item | |
| The random error has a mean of zero | |
| \end{enumerate} | |
| We can test assumption 4 by plotting the ``residuals'' of the fit | |
| i.e.~the error. The plot below demonstrates that our data has a mean | |
| error of 0 and is uncorrelated with the | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{incolor}In [{\color{incolor}25}]:} \PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{k}\PY{o}{*}\PY{l+m+mf}{1e3}\PY{p}{,}\PY{n}{M}\PY{o}{\PYZhy{}}\PY{n}{func}\PY{p}{(}\PY{n}{k}\PY{p}{,}\PY{n}{EI}\PY{p}{)}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{o}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{curvature (1/m)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+sa}{r}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Error=\PYZdl{}M\PYZhy{}EI}\PY{l+s+s1}{\PYZbs{}}\PY{l+s+s1}{kappa\PYZdl{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} | |
| \end{Verbatim} | |
| \begin{Verbatim}[commandchars=\\\{\}] | |
| {\color{outcolor}Out[{\color{outcolor}25}]:} Text(0,0.5,'Error=\$M-EI\textbackslash{}\textbackslash{}kappa\$') | |
| \end{Verbatim} | |
| \begin{center} | |
| \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_1.png} | |
| \end{center} | |
| { \hspace*{\fill} \\} | |
| \emph{Table 1: Fill in the measured and calculated values.} | |
| \begin{longtable}[]{@{}llllll@{}} | |
| \toprule | |
| trial & distance, \(r\) & force= \(mg\) & strain (mm/mm) & curvature | |
| (1/mm) & Moment=\(r\times F\)\tabularnewline | |
| \midrule | |
| \endhead | |
| 1 & & & & &\tabularnewline | |
| 1 & & & & &\tabularnewline | |
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| \bottomrule | |
| \end{longtable} | |
| \hypertarget{your-report}{% | |
| \subsubsection{Your Report}\label{your-report}} | |
| \begin{enumerate} | |
| \def\labelenumi{\arabic{enumi}.} | |
| \item | |
| Introduction | |
| \item | |
| Procedure | |
| \item | |
| Results and Discussion | |
| \item | |
| Conclusion | |
| \end{enumerate} | |
| \hypertarget{references}{% | |
| \subsubsection{References}\label{references}} | |
| References | |
| \begin{enumerate} | |
| \def\labelenumi{\arabic{enumi}.} | |
| \setcounter{enumi}{-1} | |
| \item | |
| Sutton, M. A., Orteu, J. J., \& Schreier, H. (2009). Image correlation | |
| for shape, motion and deformation measurements: basic concepts, theory | |
| and applications. Springer Science \& Business Media. | |
| \item | |
| F.P. Beer and E.R. Johnson, Mechanics of Materials, 2nd Edition, | |
| McGraw-Hill, 1992. | |
| \item | |
| S. Chapra, Numerical Methods for Engineers, ch.~14-15, 6th Edition, | |
| McGraw-Hill, 2009. | |
| \item | |
| \href{./least_squares-error_with_covariance.pdf}{C. Salter, Error | |
| Analysis Using the Variance--Covariance Matrix, J. of Chem. Ed., | |
| 2000.} | |
| \item | |
| \href{https://en.wikipedia.org/wiki/Euler\%E2\%80\%93Bernoulli_beam_theory}{Euler-Bernoulli | |
| Beam Theory - Wikipedia} | |
| \item | |
| \href{https://courses.washington.edu/phys431/uncertainty_notes.pdf}{Uncertainty | |
| Notes courses.washington.edu} | |
| \end{enumerate} | |
| % Add a bibliography block to the postdoc | |
| \end{document} |