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authorsamuel niederer <samuel.niederer@ost.ch>2022-07-24 12:03:25 +0200
committersamuel niederer <samuel.niederer@ost.ch>2022-07-24 12:03:25 +0200
commit5e25727877da020b0b23132fec8c0ea70288a18b (patch)
treec827e261e5a3d94ad2f48f8fea98cecb8722a112 /buch
parentadd phase space plot (diff)
downloadSeminarSpezielleFunktionen-5e25727877da020b0b23132fec8c0ea70288a18b.tar.gz
SeminarSpezielleFunktionen-5e25727877da020b0b23132fec8c0ea70288a18b.zip
add presentation files
Diffstat (limited to 'buch')
-rw-r--r--buch/papers/kra/images/simple_mass_spring.tex2
-rw-r--r--buch/papers/kra/presentation/presentation.tex491
-rw-r--r--buch/papers/kra/scripts/animation.py243
-rw-r--r--buch/papers/kra/scripts/simulation.py40
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()