\documentclass[tikz]{standalone}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[french]{babel}
\usepackage{pgfplots}

\pgfplotsset{compat=1.14}

\begin{document}

\begin{tikzpicture}
\def\r{1}
\def\g{9.81}
    \begin{axis}[xmin=-1.1*pi, xmax=1.1*pi, ticks=none, axis x line=middle, axis y line=middle, ylabel={$\dot\theta/\omega_0$}, xlabel={$\theta$}, width=12cm, enlarge x limits=0.1, enlarge y limits=0.1]
        \foreach \dthetai in {3,...,8,5.5} {
        \addplot [domain=-1.1*pi:1.1*pi, samples=600, smooth] {sqrt(\dthetai^2+2*\g/\r * (-cos(deg(pi/3))+cos(deg(x))))*\r/\g};
        \addplot [domain=-1.1*pi:1.1*pi, samples=600, smooth] {-sqrt(\dthetai^2+2*\g/\r * (-cos(deg(pi/3))+cos(deg(x))))*\r/\g};
    }
    \end{axis}
\end{tikzpicture}


\end{document}
