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Moved notes into appendices #62

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6815af6
moved notes into appendices
miguelmaso Oct 31, 2025
6719026
clean
miguelmaso Oct 31, 2025
6061514
moved mindmap
miguelmaso Oct 31, 2025
be12490
better layout of appendices
miguelmaso Oct 31, 2025
b3a151d
draft for beams
miguelmaso Oct 31, 2025
8d1a80c
minnor
miguelmaso Oct 31, 2025
79c51a4
minor
miguelmaso Oct 31, 2025
157a8b4
two more beams
miguelmaso Nov 3, 2025
ec716ae
better syntax
miguelmaso Nov 19, 2025
3c2e1b3
added mass damper html
miguelmaso Nov 19, 2025
5343cca
mass damper html adjust
miguelmaso Nov 19, 2025
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1 change: 1 addition & 0 deletions .gitignore
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Original file line number Diff line number Diff line change
Expand Up @@ -5,6 +5,7 @@

# Pdf output
*.pdf
!img/py/*.pdf

# recovery files
*~
Expand Down
0 notes/mindmap/mindmap.tex → img/mindmap.tex
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100 changes: 100 additions & 0 deletions img/pictures.tex
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Expand Up @@ -497,3 +497,103 @@
\end{scope}
}
}

\pgfplotsset{width=7cm}
\pgfplotsset{colormap={white}{color=(gray!20) color=(white)},
colormap/white/.style={colormap name=white}}
\pgfplotsset{colormap={gray}{color=(gray) color=(gray!20)},
colormap/gray/.style={colormap name=gray}}
\pgfplotsset{empty axes/.style={
xtick=\empty,
ytick=\empty,
ztick=\empty,
draw=gray}}
\pgfplotsset{white style/.style={
surf,
colormap/gray,
mesh/interior colormap name=white,
mesh/interior colormap thresh=0.05,
samples=25, % 5 for fast rendering, 25 for accurate rendering
domain=0:1}}

\tikzset{
slab/.pic = {
\draw [fill=gray!20] coordinate (O) rectangle (3,2) coordinate (P);
\path (O) -- node [midway,below] {$l_x$} (O -| P);
\path (O -| P) -- node [midway,right] {$l_y$} (P);
},
slab1/.pic = {
\begin{axis}[empty axes,title={$1^\text{st}$ mode ($n=1$, $m=1$)}]
\addplot3 [white style] {sin(180*x) * sin(180*y)};
\end{axis}
},
slab2/.pic = {
\begin{axis}[empty axes,title={$2^\text{nd}$ mode ($n=2$, $m=1$)}]
\addplot3 [white style] {sin(360*x) * sin(180*y)};
\end{axis}
},
slab3/.pic = {
\begin{axis}[empty axes,title={$3^\text{rd}$ mode ($n=1$, $m=2$)}]
\addplot3 [white style] {sin(180*x) * sin(360*y)};
\end{axis}
},
slab4/.pic = {
\begin{axis}[empty axes,title={$4^\text{th}$ mode ($n=2$, $m=2$)}]
\addplot3 [white style] {sin(360*x) * sin(360*y)};
\end{axis}
}
}

\usetikzlibrary{decorations.pathmorphing, decorations.markings}

\tikzset{
mass/.style = {fill=gray!50, draw, minimum width=2cm, minimum height=1cm
},
smallmass/.style = {mass, minimum width=1cm, minimum height=.5cm
},
wheel/.style = {circle, fill=gray!25, draw, yshift=-0.25cm, minimum size=0.5cm
},
spring/.style = {thick, decorate, decoration={zigzag, pre length=0.3cm, post length=0.3cm, segment length=8}
},
damper/.style = {
thick, decorate, decoration={
markings, mark connection node=dmp, mark=at position 0.5 with {
\node (dmp) [thick, inner sep=0pt, transform shape, minimum width=10pt, minimum height=10pt, draw=none, fill=green!20] {};
\draw [thick] (dmp.south east) -- (dmp.south west) -- (dmp.north west) -- (dmp.north east);
\draw [thick] (dmp |- dmp.35) -- (dmp |- dmp.-35);
\draw [thick] (dmp.center) -- (dmp.east);
}
}
},
springmassdamper/.pic = {
\coordinate (O);
\coordinate (U) at (0,-.3);
\node [mass] (m) at (3,1) {$m$};
\node [wheel] at (m.220) {};
\node [wheel] at (m.320) {};
\draw [spring] (m.165 -| O) -- (m.165) node [midway,above] {$k$};
\draw [damper] (m.195 -| O) -- (m.195) node [midway,below=3pt] {$\xi$};
\draw [very thick,-latex] (m.north east) -- ++(1,0) node [right] {$F=F_0\sin(\Omega t + \phi)$};
\draw (O) -- (0,2);
\draw (O) -- (6,0);
\draw [|->,thick] (m |- U) -- ++(.8,0) node [right] {$u$};
},
springmassdampermass/.pic = {
\coordinate (O);
\coordinate (U) at (0,-.3);
\node [mass] (m) at (3,1) {$m$};
\node [wheel] at (m.220) {};
\node [wheel] at (m.320) {};
\draw [spring] (m.165 -| O) -- (m.165) node [midway,above] {$k$};
\draw [damper] (m.195 -| O) -- (m.195) node [midway,below=3pt] {$\xi$};
\draw [very thick,-latex] (m.north east) -- ++(1,0) node [right] {$F=F_0\sin(\Omega t + \phi)$};
\node [smallmass] (md) at (6,.75) {$m_d$};
\node [wheel] at (md.south) {};
\draw [spring] (md.160 -| m.east) -- (md.160) node [midway,above] {$k_d$};
\draw [damper] (md.200 -| m.east) -- (md.200) node [midway,below=2pt] {$\xi_d$};
\draw (O) -- (0,2);
\draw (O) -- (8,0);
\draw [|->,thick] (m |- U) -- ++(.8,0) node [right] {$u$};
\draw [|->,thick] (md |- U) -- ++(1,0) node [right] {$u+u_d$};
}
}
158 changes: 158 additions & 0 deletions img/py/mass_damper.html
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@@ -0,0 +1,158 @@
<!DOCTYPE html>
<html>
<head>
<meta charset="UTF-8">
<script src="https://cdn.plot.ly/plotly-latest.min.js"></script>
</head>

<body>

<h3>
<math display="inline">
<mi>H</mi><mo>=</mo>
<msqrt>
<mfrac>
<!-- Numerador -->
<mrow>
<msup>
<mrow>
<mo>(</mo>
<msup><mi>&eta;</mi><mn>2</mn></msup>
<mo>−</mo>
<msup><mi>&gamma;</mi><mn>2</mn></msup>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>4</mn>
<msup><msub><mi>&xi;</mi><mi>d</mi></msub><mn>2</mn></msup>
<msup><mi>&gamma;</mi><mn>2</mn></msup>
<msup><mi>&eta;</mi><mn>2</mn></msup>
</mrow>

<!-- Denominador -->
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mo>(</mo>
<msup><mi>&eta;</mi><mn>2</mn></msup>
<mo>−</mo>
<msup><mi>&gamma;</mi><mn>2</mn></msup>
<mo>)</mo>
</mrow>
<mo>(</mo>
<mn>1</mn><mo>−</mo><msup><mi>&gamma;</mi><mn>2</mn></msup>
<mo>)</mo>
<mo>−</mo>
<mi>&rho;</mi>
<msup><mi>&gamma;</mi><mn>2</mn></msup>
<msup><mi>&eta;</mi><mn>2</mn></msup>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>

<mo>+</mo>

<msup>
<mrow>
<mo>(</mo>
<mn>2</mn>
<msub><mi>&xi;</mi><mi>d</mi></msub>
<mi>&gamma;</mi>
<mi>&eta;</mi>
<mo>(</mo>
<mn>1</mn><mo>−</mo>
<msup><mi>&gamma;</mi><mn>2</mn></msup>
<mo>(</mo><mn>1</mn><mo>+</mo><mi>&rho;</mi><mo>)</mo>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
</msqrt>

<!-- definiciones -->
<mspace width="1em"/>
<mo>,</mo>
<mspace width="0.5em"/>
<mi>&rho;</mi><mo>=</mo>
<mfrac><msub><mi>m</mi><mi>d</mi></msub><mi>m</mi></mfrac>
<mo>,</mo>
<mspace width="0.5em"/>
<mi>&eta;</mi><mo>=</mo>
<mfrac><msub><mi>&omega;</mi><mi>d</mi></msub><mi>&omega;</mi></mfrac>
</math>
</h3>


<label>ρ = <span id="rho_tag"></span></label><br>
<input type="range" id="rho" min="0" max="0.1" step="0.001" value="0.05" style="width:300px;"><br>
<label>η = <span id="eta_tag"></span></label><br>
<input type="range" id="eta" min="0.9" max="1.05" step="0.001" value="0.975" style="width:300px;"><br>
<label>ξd = <span id="xi_tag"></span></label><br>
<input type="range" id="xi" min="0" max="0.2" step="0.001" value="0.10" style="width:300px;"><br>

<div id="plot" style="width:800px;height:400px;"></div>

<script>
function updatePlot() {

let rho = parseFloat(document.getElementById("rho").value);
document.getElementById("rho_tag").innerHTML = rho;
let eta = parseFloat(document.getElementById("eta").value);
document.getElementById("eta_tag").innerHTML = eta;
let xi = parseFloat(document.getElementById("xi").value);
document.getElementById("xi_tag").innerHTML = xi;

let x = [];
let y = [];
let z = [];
for (let i=0; i<=3; i+=0.01) {
a = eta**2 - i**2;
b = 2*xi*i*eta;
c = a**2 + b**2;
d = (a*(1-i**2) -rho*i**2*eta**2)**2 + b**2*(1-i**2*(1+rho))**2;

f = Math.abs(1 - i*i);

x.push(i);
y.push(Math.sqrt(c/d));
z.push((f < 5e-2) ? NaN : 1/f);
}

Plotly.newPlot("plot", [
{
x: x,
y: y,
name: "H",
mode: "lines"
},{
x: x,
y: z,
name: "no mass damper",
mode: "none",
fill: "tozeroy",
fillcolor: "rgba(200,50,50,0.2)",
hoverinfo: "none"
}
], {
margin: { t: 10 },
dragmode: false
});
}

document.getElementById("rho").addEventListener("input", updatePlot);
document.getElementById("eta").addEventListener("input", updatePlot);
document.getElementById("xi").addEventListener("input", updatePlot);

updatePlot();
</script>

</body>
</html>
0 notes/mass_damper/mass_damper.py → img/py/mass_damper.py
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11 changes: 11 additions & 0 deletions include/beam_frequencies.tex
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@@ -0,0 +1,11 @@

Analyitical expressions for the frequencies of vibration for beams can be found by means of the Rayleigh's method. Generally, sinusoidal functions are used, since the appropriate combination of them allow to consider multiple boundary conditions and the main modes of vibration.
The expressions shown in the table below coincide with the expressions proposed in most of the codes and application guidelines, see for instance \cite{sci-p354}. All of them are the result of applying the Raileigh's method with appropriate shape functions.

\begin{center}
\begin{tabular}{lcc}
Model & $f$ (Hz) & $M_\Psi$ \\ \hline
simply supp & $\displaystyle \frac{\pi}{2}\sqrt{\frac{EI}{L^3m}}$ & $\frac{m}{2}$ \\
cantilever & $\displaystyle \frac{\pi}{8\sqrt{3-8/\pi}}\sqrt{\frac{EI}{L^3m}}$ & $0.64m$ \\
\end{tabular}
\end{center}
66 changes: 8 additions & 58 deletions notes/floor_frequency/floor_frequency.tex → include/floor_frequency.tex
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@@ -1,30 +1,3 @@
\documentclass{article}

\usepackage{../sty/notes}
\notenum{2}

\pgfplotsset{width=7cm}
\pgfplotsset{colormap={white}{color=(gray!20) color=(white)},
colormap/white/.style={colormap name=white}}
\pgfplotsset{colormap={gray}{color=(gray) color=(gray!20)},
colormap/gray/.style={colormap name=gray}}
\pgfplotsset{empty axes/.style={
xtick=\empty,
ytick=\empty,
ztick=\empty,
draw=gray}}
\pgfplotsset{white style/.style={
surf,
colormap/gray,
mesh/interior colormap name=white,
mesh/interior colormap thresh=0.05,
samples=5, % 5 for fast rendering, 25 for accurate rendering
domain=0:1}}


\begin{document}

\section*{Natural frequencies of floor slabs}

The natural stiffness of a floor slab can be approximated making use of the Rayleigh's method using an appropriate shape function satisfying the boundary conditions. From the static equilibrium equation of a slab, the modal mass and modal stiffness are rewritten according to the follwong expressions:
\begin{equation*}
Expand All @@ -39,12 +12,8 @@ \section*{Natural frequencies of floor slabs}
$$

\begin{wrapfigure}{l}{4cm}
\vspace{-1em}
\begin{tikzpicture}
\draw [fill=gray!20] coordinate (O) rectangle (3,2) coordinate (P);
\path (O) -- node [midway,below] {$l_x$} (O -| P);
\path (O -| P) -- node [midway,right] {$l_y$} (P);
\end{tikzpicture}
\vspace{-1em}
\tikz \pic {slab};
\end{wrapfigure}

For a rectangular floor \emph{simply supported at the four sides}, the natural frequencies can be approximated using the following shape function, where $n$ and $m$ stand for the $i^\text{th}$ mode of vibration at the $x$ and $y$ directions,
Expand Down Expand Up @@ -75,8 +44,8 @@ \section*{Natural frequencies of floor slabs}
\begin{align*}
\omega^2 &= \frac{k_\psi}{m_\psi} = \frac{\pi^4D}{4}\frac{4}{\rho t}
\left(\frac{n^4}{l_x^4} + \frac{m^4}{l_y^4} + 2\frac{n^2m^2}{l_x^2l_y^2}\right)
= \frac{\pi^4D}{\rho t}\left(\frac{n^2}{l_x^2} + \frac{m^2}{l_m^2}\right)^2 \\
\omega &= \pi^2\sqrt{\frac{D}{\rho t}} \left(\frac{n^2}{l_x^2} + \frac{m^2}{l_y^2}\right)^2
= \frac{\pi^4D}{\rho t}\left(\frac{n^2}{l_x^2} + \frac{m^2}{l_m^2}\right)^2, \\
\omega &= \pi^2\sqrt{\frac{D}{\rho t}} \left(\frac{n^2}{l_x^2} + \frac{m^2}{l_y^2}\right),
\end{align*}
and the first natural frequency is
\begin{equation*}
Expand All @@ -85,30 +54,11 @@ \section*{Natural frequencies of floor slabs}


\begin{center}
\begin{tikzpicture}
\begin{axis}[empty axes,title={$1^\text{st}$ mode ($n=1$, $m=1$)}]
\addplot3 [white style] {sin(180*x) * sin(180*y)};
\end{axis}
\end{tikzpicture}
\tikz \pic {slab1};
\hspace{1em}
\begin{tikzpicture}
\begin{axis}[empty axes,title={$2^\text{nd}$ mode ($n=2$, $m=1$)}]
\addplot3 [white style] {sin(360*x) * sin(180*y)};
\end{axis}
\end{tikzpicture}
\tikz \pic {slab2};

\begin{tikzpicture}
\begin{axis}[empty axes,title={$3^\text{rd}$ mode ($n=1$, $m=2$)}]
\addplot3 [white style] {sin(180*x) * sin(360*y)};
\end{axis}
\end{tikzpicture}
\tikz \pic {slab3};
\hspace{1em}
\begin{tikzpicture}
\begin{axis}[empty axes,title={$4^\text{th}$ mode ($n=2$, $m=2$)}]
\addplot3 [white style] {sin(360*x) * sin(360*y)};
\end{axis}
\end{tikzpicture}
\tikz \pic {slab4};
\end{center}


\end{document}
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