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% Packages
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\usepackage{hhline}             % Border under tables
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\usepackage{seqsplit}           % Splits long words.
%\usepackage{opensans}          % Can't make this work so far. Shame. Would be lovely.
\usepackage[normalem]{ulem}     % For underlining links
% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
\usepackage{amsmath}            % Symbols
\usepackage{MnSymbol}           % Symbols
\usepackage{wasysym}            % Symbols
%\usepackage[english,german,french,spanish,italian]{babel}              % Languages

% Document Info
\author{MostAncientDream}
\pdfinfo{
  /Title (alvl-p2-shm-ch11.pdf)
  /Creator (Cheatography)
  /Author (MostAncientDream)
  /Subject (Alvl P2: SHM (ch11) Cheat Sheet)
}

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% This two commands together give roughly
% the right line height in the tables
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\onehalfspacing

% Commands
\newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour
\newcommand{\mymulticolumn}[3]{\multicolumn{#1}{>{\columncolor{\RowColorName}}#2}{#3}} % For coloured multi-cols
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\color{TextColor}

% Header and Footer
\pagestyle{fancy}
\fancyhead{} % Set header to blank
\fancyfoot{} % Set footer to blank
\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
    \SetRowColor{DarkBackground}
    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
    }
\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Alvl P2: SHM (ch11) Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{MostAncientDream} via \textcolor{DarkBackground}{\uline{cheatography.com/168994/cs/42401/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}MostAncientDream \\
  \uline{cheatography.com/mostancientdream} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 15th February, 2024.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
  \SetRowColor{white}
  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{2}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{basics}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{conditions: \newline % Row Count 1 (+ 1)
- force/acceleration is proportional to and in the opposite direction to the displacement  \newline % Row Count 3 (+ 2)
a = w\textasciicircum{}2\textasciicircum{}x \newline % Row Count 4 (+ 1)
\textgreater{} amplitude: max displacement from equilibrium \newline % Row Count 5 (+ 1)
a max = w\textasciicircum{}2\textasciicircum{}A \newline % Row Count 6 (+ 1)
\textgreater{} displacement at any point: \newline % Row Count 7 (+ 1)
x = Acos(2pi.ft) (rads)  \newline % Row Count 8 (+ 1)
\textgreater{} velocity at any point: \newline % Row Count 9 (+ 1)
v = +- 2pi.f(A\textasciicircum{}2\textasciicircum{}-x\textasciicircum{}2\textasciicircum{})\textasciicircum{}1/2\textasciicircum{} \newline % Row Count 10 (+ 1)
v max = 2pi.fA = wA (at equilibrium)% Row Count 11 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{resonance and vibrations}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{when frequency of external driving force F matches natural frequency \newline % Row Count 2 (+ 2)
it is 90' out of phase with each other \newline % Row Count 3 (+ 1)
{\bf{free vibrations}}- frequency a system tends to vibrate at in a vibration is called the natural frequency  \newline % Row Count 6 (+ 3)
{\bf{forced vibrations}}- a driving force causes systems to vibrate at a different frequency% Row Count 8 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{damping}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{{\bf{when external force opposes motion/opposite direction of v}} \newline % Row Count 2 (+ 2)
- this occurs when energy is transferred out of the system adn the total energy is no longer being constant  \newline % Row Count 5 (+ 3)
main types of damping: \newline % Row Count 6 (+ 1)
1. {\bf{light dampening}} where the oscillations are damped slowly eg. air resistance/friction \newline % Row Count 8 (+ 2)
2. {\bf{heavy damping}} where the oscillations still continue but are brought to a stop more quickly \newline % Row Count 10 (+ 2)
3. {\bf{critical damping}} involves stopping oscillations in the quickest time possible \newline % Row Count 12 (+ 2)
4. {\bf{overdamping}} caused when the force is too great and stops the oscilations but takes longer to return to equilibrium position% Row Count 15 (+ 3)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{resonance and vibrations}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{when frequency of external driving force F matches natural frequency \newline % Row Count 2 (+ 2)
it is 90' out of phase with each other \newline % Row Count 3 (+ 1)
{\bf{free vibrations}}- frequency a system tends to vibrate at in a vibration is called the natural frequency  \newline % Row Count 6 (+ 3)
{\bf{forced vibrations}}- a driving force causes systems to vibrate at a different frequency% Row Count 8 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}