add presentation files

4 files changed, 775 insertions, 1 deletions

diff --git a/buch/papers/kra/images/simple_mass_spring.tex b/buch/papers/kra/images/simple_mass_spring.tex
index 207f1e0..e0e869a 100644
--- a/buch/papers/kra/images/simple_mass_spring.tex
+++ b/buch/papers/kra/images/simple_mass_spring.tex
@@ -14,7 +14,7 @@
     }
 
     \tikzmath{
-        \hWall = 1.5;
+        \hWall = 1.2;
         \wWall = 0.3;
         \lWall = 3.5;
         \hMass = 0.6;
diff --git a/buch/papers/kra/presentation/presentation.tex b/buch/papers/kra/presentation/presentation.tex
new file mode 100644
index 0000000..eb6541b
--- /dev/null
+++ b/buch/papers/kra/presentation/presentation.tex
@@ -0,0 +1,491 @@
+\documentclass[ngerman, aspectratio=169, xcolor={rgb}]{beamer}
+
+% style
+\mode<presentation>{
+	\usetheme{Frankfurt}
+}
+%packages
+\usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-}
+\usepackage[english]{babel}
+\usepackage{graphicx}
+\usepackage{array}
+
+\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
+\usepackage{ragged2e}
+
+\usepackage{bm} % bold math
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{mathtools}
+\usepackage{amsmath}
+\usepackage{multirow} % multi row in tables
+\usepackage{booktabs} %toprule midrule bottomrue in tables
+\usepackage{scrextend}
+\usepackage{textgreek}
+\usepackage[rgb]{xcolor}
+
+\usepackage[normalem]{ulem} % \sout
+
+\usepackage{ marvosym } % \Lightning
+
+\usepackage{multimedia} % embedded videos
+
+\usepackage{tikz}
+\usepackage{pgf}
+\usepackage{pgfplots}
+
+\usepackage{algorithmic}
+
+%citations
+\usepackage[style=verbose,backend=biber]{biblatex}
+\addbibresource{references.bib}
+
+
+%math font
+\usefonttheme[onlymath]{serif}
+
+%Beamer Template modifications
+%\definecolor{mainColor}{HTML}{0065A3} % HSR blue
+\definecolor{mainColor}{HTML}{D72864} % OST pink
+\definecolor{invColor}{HTML}{28d79b} % OST pink
+\definecolor{dgreen}{HTML}{38ad36} % Dark green
+
+%\definecolor{mainColor}{HTML}{000000} % HSR blue
+\setbeamercolor{palette primary}{bg=white,fg=mainColor}
+\setbeamercolor{palette secondary}{bg=orange,fg=mainColor}
+\setbeamercolor{palette tertiary}{bg=yellow,fg=red}
+\setbeamercolor{palette quaternary}{bg=mainColor,fg=white} %bg = Top bar, fg = active top bar topic
+\setbeamercolor{structure}{fg=black} % itemize, enumerate, etc (bullet points)
+\setbeamercolor{section in toc}{fg=black} % TOC sections
+\setbeamertemplate{section in toc}[sections numbered]
+\setbeamertemplate{subsection in toc}{%
+	\hspace{1.2em}{$\bullet$}~\inserttocsubsection\par}
+
+\setbeamertemplate{itemize items}[circle]
+\setbeamertemplate{description item}[circle]
+\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true]
+\beamertemplatenavigationsymbolsempty
+
+\setbeamercolor{footline}{fg=gray}
+\setbeamertemplate{footline}{%
+	\hfill\usebeamertemplate***{navigation symbols}
+	\hspace{0.5cm}
+	\insertframenumber{}\hspace{0.2cm}\vspace{0.2cm}
+}
+
+\usepackage{caption}
+\captionsetup{labelformat=empty}
+
+%Title Page
+\title{KRA}
+\subtitle{Kalman Riccati Abel}
+\author{Samuel Niederer}
+% \institute{OST Ostschweizer Fachhochschule}
+% \institute{\includegraphics[scale=0.3]{../img/ost_logo.png}}
+\date{\today}
+
+\input{../packages.tex}
+
+\newcommand*{\QED}{\hfill\ensuremath{\blacksquare}}%
+
+\newcommand*{\HL}{\textcolor{mainColor}}
+\newcommand*{\RD}{\textcolor{red}}
+\newcommand*{\BL}{\textcolor{blue}}
+\newcommand*{\GN}{\textcolor{dgreen}}
+\newcommand{\dt}[0]{\frac{d}{dt}}
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+
+\makeatletter
+\newcount\my@repeat@count
+\newcommand{\myrepeat}[2]{%
+	\begingroup
+	\my@repeat@count=\z@
+	\@whilenum\my@repeat@count<#1\do{#2\advance\my@repeat@count\@ne}%
+	\endgroup
+}
+\makeatother
+
+\usetikzlibrary{automata,arrows,positioning,calc,shapes.geometric, fadings}
+
+\begin{document}
+
+\begin{frame}
+	\titlepage
+\end{frame}
+
+\begin{frame}
+	\frametitle{Content}
+	\tableofcontents
+\end{frame}
+
+\section{Einführung}
+
+\begin{frame}
+	\begin{itemize}
+		\item<1|only@1> \textbf{K}alman
+		\item<1|only@1> \textbf{R}iccati
+		\item<1|only@1> \textbf{A}bel
+
+		\item<2|only@2> \textcolor{red}{\sout{\textbf{K}alman}}
+		\item<2|only@2> \textbf{R}iccati
+		\item<2|only@2> \textbf{A}bel
+
+		\item<3|only@3> \textcolor{red}{\sout{\textbf{K}alman}} \textcolor{green}{Federmassesytem}
+		\item<3|only@3> \textbf{R}iccati
+		\item<3|only@3> \textbf{A}bel
+
+		\item<4|only@4> \textcolor{red}{\sout{\textbf{K}alman}} \textcolor{green}{Federmassesytem}
+		\item<4|only@4> \textbf{R}iccati
+		\item<4|only@4> \uwave{\textbf{A}bel}
+	\end{itemize}
+\end{frame}
+
+\section{Riccati}
+
+\begin{frame}
+	\frametitle{Riccatische Differentialgleichung}
+	\begin{equation*}
+		% y'(x) = f(x)y^2(x) + g(x)y(x) + h(x)
+		x'(t) = f(t)x^2(t) + g(t)x(t) + h(t)
+	\end{equation*}
+
+	\pause
+
+	\begin{equation*}
+		\dot{X}(t) = - X(t)BX(t) - X(t)A + DX(t) + C
+	\end{equation*}
+
+	% \pause
+	% Anwendungen
+	% \begin{itemize}
+	% 	\item Zeitkontinuierlicher Kalmanfilter
+	% 	\item Regelungstechnik LQ-Regler
+	% 	\item Federmassesyteme
+	% \end{itemize}
+\end{frame}
+
+\begin{frame}
+	\frametitle{Auftreten der Gleichung}
+	\begin{columns}
+		\column{0.4 \textwidth}
+		\begin{equation*}
+			\dt
+			\begin{pmatrix}
+				X \\
+				Y
+			\end{pmatrix}
+			=
+			\underbrace{
+				\begin{pmatrix}
+					A & B \\
+					C & D
+				\end{pmatrix}
+			}_{H}
+			\begin{pmatrix}
+				X \\
+				Y
+			\end{pmatrix}
+		\end{equation*}
+
+		\pause
+
+		\column{0.4 \textwidth}
+		\begin{equation*}
+			U = YX^{-1} \qquad \dt U = ?
+		\end{equation*}
+	\end{columns}
+
+	\pause
+
+	\begin{align*}
+		\dt U         & = \dot{Y} X^{-1} + Y \dt X^{-1}                                                                                                    \\
+		\uncover<4->{ & = (CX + DY) X^{-1} - Y (X^{-1} \dot{X} X^{-1})\\}
+		\uncover<5->{ & = C\underbrace{XX^{-1}}_\text{I} + D\underbrace{YX^{-1}}_\text{U} - Y(X^{-1} (AX + BY) X^{-1})\\}
+		\uncover<6->{ & = C + DU - \underbrace{YX^{-1}}_\text{U}(A\underbrace{XX^{-1}}_\text{I} + B\underbrace{YX^{-1}}_\text{U})\\}
+		\uncover<7->{ & = C  + DU - UA - UBU}
+	\end{align*}
+\end{frame}
+
+\begin{frame}
+	\frametitle{Lösen der Gleichung}
+	\begin{equation*}
+		\begin{pmatrix}
+			X(t) \\
+			Y(t)
+		\end{pmatrix}
+		=
+		\Phi(t_0, t)
+		\begin{pmatrix}
+			I(t) \\
+			U_0(t)
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			\Phi_{11}(t_0, t) & \Phi_{12}(t_0, t) \\
+			\Phi_{21}(t_0, t) & \Phi_{22}(t_0, t)
+		\end{pmatrix}
+		\begin{pmatrix}
+			I(t) \\
+			U_0(t)
+		\end{pmatrix}
+	\end{equation*}
+
+	\pause
+
+	\begin{equation*}
+		U(t) =
+		\begin{pmatrix}
+			\Phi_{21}(t_0, t) + \Phi_{22}(t_0, t) U_0(t)
+		\end{pmatrix}
+		\begin{pmatrix}
+			\Phi_{11}(t_0, t) + \Phi_{12}(t_0, t) U_0(t)
+		\end{pmatrix}
+		^{-1}
+	\end{equation*}
+
+	\pause
+
+	% wobei $\Phi(t, t_0)$ die sogennante Zustandsübergangsmatrix ist.
+
+	\begin{equation*}
+		\Phi(t_0, t) = e^{H(t - t_0)}
+	\end{equation*}
+\end{frame}
+
+\section{Federmassystem}
+\begin{frame}
+	\frametitle{Federmassesystem}
+	\begin{columns}
+		\column{0.5 \textwidth}
+		\input{../images/simple_mass_spring.tex}
+
+		\column{0.5 \textwidth}
+		\begin{align*}
+			\uncover<2->{F_R      & = k \Delta_x  \\}
+			\uncover<3->{F_a      & = am = \ddot{x} m \\}
+			\uncover<4->{F_R      & = F_a \Leftrightarrow k \Delta_x = \ddot{x} m\\}
+			\uncover<5->{\ddot{x} & = \frac{k \Delta_x}{m} \\}
+			\uncover<6->{x(t)     & = A \cos(\omega_0 + \Phi), \quad \omega_0 = \sqrt{\frac{k}{m}}}
+		\end{align*}
+	\end{columns}
+\end{frame}
+
+\begin{frame}
+	\frametitle{Phasenraum}
+	\begin{columns}
+		\column{0.3 \textwidth}
+		\begin{tikzpicture}[scale=3]
+			\draw[->, thick] (0, 0) -- (1,0) node[right] {$q$};
+			\draw[->, thick] (0.5, -0.5) -- (0.5,0.5) node[above]{$p$};
+		\end{tikzpicture}
+		\column{0.7 \textwidth}
+		Impulskoordinaten $p = (p_{1}, p_{2}, ..., p_{n}), \quad p=mv$ \\
+		Ortskoordinaten $q = (q_{1}, q_{2}, ..., q_{n})$ \\
+
+
+
+		\begin{align*}
+			\uncover<2->{\mathcal{H}(q, p) & = \underbrace{T(p)}_{E_{kin}} + \underbrace{V(q)}_{E_{pot}} = E_{tot} \\}
+			\uncover<3->{                  & = \frac{p^2}{2m}+ \frac{k q^2}{2}}
+		\end{align*}
+
+
+
+		\begin{equation*}
+			\uncover<4->{
+				\dot{q_{k}} = \frac{\partial \mathcal{H}}{\partial p_k}
+				\qquad
+				\dot{p_{k}} = -\frac{\partial \mathcal{H}}{\partial q_k}
+			}
+		\end{equation*}
+
+		\pause
+
+		\begin{equation*}
+			\uncover<5->{
+				\begin{pmatrix}
+					\dot{q} \\
+					\dot{p}
+				\end{pmatrix}
+				=
+				\begin{pmatrix}
+					0  & \frac{1}{m} \\
+					-k & 0
+				\end{pmatrix}
+				\begin{pmatrix}
+					q \\
+					p
+				\end{pmatrix}
+			}
+		\end{equation*}
+
+	\end{columns}
+\end{frame}
+
+\begin{frame}
+	\frametitle{Phasenraum}
+	\input{../images/phase_space.tex}
+\end{frame}
+
+\begin{frame}
+	\frametitle{Federmassesystem}
+	\begin{columns}
+		\column{0.6 \textwidth}
+		\scalebox{0.8}{\input{../images/multi_mass_spring.tex}}
+		\begin{align*}
+			\uncover<2->{\mathcal{H} & = T + V \\}
+			\uncover<7->{            & = \frac{p_1^2}{2m_1} + \frac{p_2^2}{2m_2} + \frac{k_1 q_1^2}{2} + \frac{k_c (q_2 - q_1)^2}{2} + \frac{k_2 q_2^2}{2}}
+		\end{align*}
+
+		\column{0.4 \textwidth}
+		\begin{align*}
+			\uncover<3->{T & = T_1 + T_2}                                                               \\
+			\uncover<5->{  & = \frac{p_1^2}{2m_1} + \frac{p_2^2}{2m_2}   }                              \\
+			\uncover<4->{V & = V_1 + V_c + V_2 }                                                        \\
+			\uncover<6->{  & = \frac{k_1 q_1^2}{2} + \frac{k_c (q_2 - q_1)^2}{2} + \frac{k_2 q_2^2}{2}}
+		\end{align*}
+	\end{columns}
+\end{frame}
+
+\begin{frame}
+	\frametitle{Federmassesystem}
+	\begin{equation*}
+		\begin{pmatrix}
+			\dot{q_1} \\
+			\dot{q_2} \\
+			\dot{p_1} \\
+			\dot{p_2} \\
+		\end{pmatrix}
+		=
+		\begin{pmatrix}
+			0            & 0            & \frac{1}{2m_1} & 0              \\
+			0            & 0            & 0              & \frac{1}{2m_2} \\
+			-(k_1 + k_c) & k_c          & 0              & 0              \\
+			k_c          & -(k_c + k_2) & 0              & 0              \\
+		\end{pmatrix}
+		\begin{pmatrix}
+			q_1 \\
+			q_2 \\
+			p_1 \\
+			p_2 \\
+		\end{pmatrix}
+		\Leftrightarrow
+		\dt
+		\begin{pmatrix}
+			Q \\
+			P \\
+		\end{pmatrix}
+		\underbrace{
+			\begin{pmatrix}
+				0 & M \\
+				K & 0
+			\end{pmatrix}
+		}_{H}
+		\begin{pmatrix}
+			Q \\
+			P \\
+		\end{pmatrix}
+	\end{equation*}
+
+	\pause
+
+	$U = PQ^{-1} \qquad \dt U = ?$
+
+	\pause
+
+	\begin{align*}
+		\dt U & = C  + DU - UA - UBU \\
+		      & = K - UMU
+	\end{align*}
+
+\end{frame}
+
+\begin{frame}
+	\frametitle{Einfluss der Anfangsbedingung:}
+	\begin{columns}
+		\column{0.4 \textwidth}
+		\begin{equation*}
+			\uncover<2->{q_0 =
+				\begin{pmatrix}
+					q_{10} \\
+					q_{20}
+				\end{pmatrix}
+				=
+				\begin{pmatrix}
+					3 \\
+					1
+				\end{pmatrix}
+			}
+		\end{equation*}
+		\begin{equation*}
+			\uncover<3->{q_0 =
+				\begin{pmatrix}
+					q_{10} \\
+					q_{20}
+				\end{pmatrix}
+				=
+				\begin{pmatrix}
+					3 \\
+					3
+				\end{pmatrix}
+			}
+		\end{equation*}
+		\begin{equation*}
+			\uncover<4->{q_0 =
+				\begin{pmatrix}
+					q_{10} \\
+					q_{20}
+				\end{pmatrix}
+				=
+				\begin{pmatrix}
+					2 \\
+					-2
+				\end{pmatrix}
+			}
+		\end{equation*}
+		\column{0.6 \textwidth}
+		\scalebox{0.8}{\input{../images/multi_mass_spring.tex}}
+	\end{columns}
+\end{frame}
+
+\section{Schlussteil}
+\begin{frame}
+	\frametitle{Zusammenfassung}
+	\begin{itemize}
+		\pause
+		\item{Riccatische Differentialgleichung}
+		      \pause
+		      \begin{itemize}
+			      \item{Ausgansgleichung}
+			            \pause
+			      \item{Lösung}
+		      \end{itemize}
+		      \pause
+		\item{Harmonischer Ozillator}
+		      \pause
+		      \begin{itemize}
+			      \item{Hamiltonfunktion}
+			            \pause
+			      \item{Phasenraum}
+		      \end{itemize}
+		      \pause
+		\item{Gekoppelter harmonischer Ozillator}
+		      \pause
+		      \begin{itemize}
+			      \item{Riccatische Differentialgleichung}
+			            \pause
+			      \item{Einfluss der Anfangsbedingungen}
+		      \end{itemize}
+		      \pause
+		\item{\uwave{Abel}}
+		      \begin{itemize}
+			      \pause
+			      \item{Nichtlineare Federkonstante}
+		      \end{itemize}
+
+	\end{itemize}
+\end{frame}
+
+\end{document}
diff --git a/buch/papers/kra/scripts/animation.py b/buch/papers/kra/scripts/animation.py
new file mode 100644
index 0000000..5e805ae
--- /dev/null
+++ b/buch/papers/kra/scripts/animation.py
@@ -0,0 +1,243 @@
+import numpy as np

+import matplotlib.pyplot as plt

+import matplotlib.patches

+import matplotlib.transforms

+import matplotlib.text

+from matplotlib.animation import FuncAnimation

+import imageio

+

+from simulation import Simulation

+

+

+class Mass:

+    def __init__(self, x_0, width, height, **kwargs):

+        self._x_0 = x_0

+        xy = (x_0, 0)

+        self._rect = matplotlib.patches.Rectangle(xy, width, height, **kwargs)

+

+    @property

+    def patch(self):

+        return self._rect

+

+    @property

+    def x(self):

+        return self._rect.get_x()

+

+    @property

+    def width(self):

+        return self._rect.get_width()

+

+    def move(self, x):

+        self._rect.set_x(self._x_0 + x)

+

+

+class Spring:

+    def __init__(self, n, height, ax, resolution=1000, **kwargs):

+        self._n = n

+        self._height = height

+        self._N = resolution

+        (self._line,) = ax.plot([], [], "-", **kwargs)

+

+    def set(self, x_0, x_1):

+        T = (x_1 - x_0) / self._n

+        x = np.linspace(x_0, x_1, self._N, endpoint=True)

+        t = np.linspace(0, x_1 - x_0, self._N)

+        y = (np.sin(2 * np.pi * t / T) + 1.5) * self._height / 2

+        self.line.set_data(x, y)

+

+    @property

+    def line(self):

+        return self._line

+

+

+class LinePlot:

+    def __init__(self, ax, **kwargs):

+        (self._line,) = ax.plot([], [], "-", **kwargs)

+        self._x = []

+        self._y = []

+

+    @property

+    def line(self):

+        return self._line

+

+    def update(self, x, y):

+        self._x.append(x)

+        self._y.append(y)

+        self._line.set_data(self._x, self._y)

+

+

+class ScatterPlot:

+    def __init__(self, ax, **kwargs):

+        self._color = kwargs.get("color", "tab:green")

+        self._line = ax.scatter([], [], **kwargs)

+        self._ax = ax

+        self._x = []

+        self._y = []

+

+    @property

+    def line(self):

+        return self._line

+

+    def update(self, x, y, **kwargs):

+        self._x.append(x)

+        self._y.append(y)

+        self._line.remove()

+        self._line = self._ax.scatter(self._x, self._y, color=self._color, **kwargs)

+

+

+class QuiverPlot:

+    def __init__(self, ax, **kwargs):

+        self.x = []

+        self.y = []

+        self.u = []

+        self.v = []

+        self.ax = ax

+        self.ln = self.ax.quiver([], [], [], [])

+

+    def update(self, x, y, u, v):

+        self.x.append(x)

+        self.y.append(y)

+        self.u.append(u)

+        self.v.append(v)

+        self.ln.remove()

+        self.ln = self.ax.quiver(self.x, self.y, self.u, self.v)

+

+    @property

+    def line(self):

+        return self.ln

+

+

+anim_folder = "anim_0"

+img_counter = 0

+

+sim = Simulation()

+params = {

+    "x_0": [2, -2],

+    "k_1": 1,

+    "k_c": 2,

+    "k_2": 1,

+    "m_1": 0.5,

+    "m_2": 0.5,

+}

+

+time = 2.1

+

+

+# create axis

+fig = plt.figure(figsize=(20, 15), constrained_layout=True)

+fig.suptitle(

+    " ,".join([f"${key} = {val}$" for (key, val) in params.items()]), fontsize=20

+)

+spec = fig.add_gridspec(3, 4)

+ax0 = fig.add_subplot(spec[-1, :])

+ax1 = fig.add_subplot(spec[:-1, :2])

+ax2 = fig.add_subplot(spec[:-1, 2:])

+

+ax0.set_yticks([])

+

+mass_height = 0.5

+spring_height = 0.6 * mass_height

+x_max = 21

+y_max = 2 * mass_height

+

+mass_1 = Mass(

+    7,

+    2,

+    mass_height,

+    color="tab:red",

+)

+mass_2 = Mass(14, 2, mass_height, color="tab:blue")

+masses = [mass_1, mass_2]

+patches = [mass.patch for mass in masses]

+

+spring_1 = Spring(4, spring_height, ax0, color="tab:red", linewidth=10)

+spring_2 = Spring(4, spring_height, ax0, color="tab:gray", linewidth=10)

+spring_3 = Spring(4, spring_height, ax0, color="tab:blue", linewidth=10)

+springs = [spring_1, spring_2, spring_3]

+

+linePlot_1 = LinePlot(ax1, color="tab:red", label="$m_1$", alpha=1)

+linePlot_2 = LinePlot(ax1, color="tab:blue", label="$m_2$", alpha=1)

+linePlots = [linePlot_1, linePlot_2]

+

+# quiverPlot = QuiverPlot(ax2)

+scatterPlot = ScatterPlot(ax2)

+

+lines = [spring.line for spring in springs]

+lines.extend([plot.line for plot in linePlots])

+# lines.append(quiverPlot.line)

+lines.append(scatterPlot.line)

+

+objects = lines + patches

+

+ax0.plot(

+    np.repeat(mass_1.x, 2),

+    [0, y_max],

+    "--",

+    color="tab:red",

+    label="Ruhezustand $m_1$",

+)

+ax0.plot(

+    np.repeat(mass_2.x, 2),

+    [0, y_max],

+    "--",

+    color="tab:blue",

+    label="Ruhezustand $m_2$",

+)

+

+

+def init():

+    ax0.set_xlim(0, x_max)

+    ax0.set_ylim(0, y_max)

+

+    ax1.set_xlim(0, time)

+    ax1.set_ylim(-4, 4)

+    ax1.set_xlabel("time", fontsize=20)

+    ax1.set_ylabel("$q$", fontsize=20)

+

+    ax2.set_xlim(-4, 4)

+    ax2.set_ylim(-4, 4)

+    ax2.set_xlabel("$q_1$", fontsize=20)

+    ax2.set_ylabel("$q_2$", fontsize=20)

+

+    for patch in patches:

+        ax0.add_patch(patch)

+

+    spring_1.set(0, mass_1.x)

+    spring_2.set(mass_1.x + mass_1.width, mass_2.x)

+    spring_2.set(mass_2.x + mass_2.width, x_max)

+

+    return objects

+

+

+def update(frame):

+    global img_counter

+    x_1, x_2 = sim(frame, **params)

+

+    mass_1.move(x_1)

+    mass_2.move(x_2)

+

+    spring_1.set(0, mass_1.x)

+    spring_2.set(mass_1.x + mass_1.width, mass_2.x)

+    spring_3.set(mass_2.x + mass_2.width, x_max)

+

+    linePlot_1.update(frame, x_1)

+    linePlot_2.update(frame, x_2)

+

+    scatterPlot.update(x_1, x_2, alpha=0.25)

+

+    img_counter += 1

+    return objects

+

+

+anim = FuncAnimation(

+    fig,

+    update,

+    frames=np.linspace(0, time, int(time * 30)),

+    init_func=init,

+    blit=False,

+)

+

+ax0.legend(fontsize=20)

+ax1.legend(fontsize=20)

+FFwriter = matplotlib.animation.FFMpegWriter(fps=30)

+anim.save("animation.mp4", writer=FFwriter)

diff --git a/buch/papers/kra/scripts/simulation.py b/buch/papers/kra/scripts/simulation.py
new file mode 100644
index 0000000..8bccb6a
--- /dev/null
+++ b/buch/papers/kra/scripts/simulation.py
@@ -0,0 +1,40 @@
+import sympy as sp

+

+

+class Simulation:

+    def __init__(self):

+        self.k_1, self.k_2, self.k_c = sp.symbols("k_1 k_2 k_c")

+        self.m_1, self.m_2 = sp.symbols("m_1 m_2")

+        self.t = sp.symbols("t")

+        K = sp.Matrix(

+            [[-(self.k_1 + self.k_c), self.k_c], [self.k_c, -(self.k_2 + self.k_c)]]

+        )

+        M = sp.Matrix([[1 / self.m_1, 0], [0, 1 / self.m_2]])

+        A = M * K

+

+        self.eigenvecs = []

+        self.eigenvals = []

+        for ev, mult, vecs in A.eigenvects():

+            self.eigenvecs.append(sp.Matrix(vecs))

+            self.eigenvals.extend([ev] * mult)

+

+    def __call__(self, t, x_0, k_1, k_c, k_2, m_1, m_2):

+        params = {

+            self.k_1: k_1,

+            self.k_c: k_c,

+            self.k_2: k_2,

+            self.m_1: m_1,

+            self.m_2: m_2,

+        }

+        x_0 = sp.Matrix(x_0)

+        eig_mat = sp.Matrix.hstack(*self.eigenvecs).subs(params)

+        g = eig_mat.inv() * x_0

+        L = sp.Matrix(

+            [

+                g[0] * sp.cos(self.eigenvals[0].subs(params) * self.t),

+                g[1] * sp.cos(self.eigenvals[1].subs(params) * self.t),

+            ]

+        )

+        x = eig_mat * L

+        f = sp.lambdify(self.t, x, "numpy")

+        return f(t).squeeze()