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\title[RANM 2019]{{\small Rome, September 15-19 2019} \\
\medskip {\small AIMETA 2019} \\
\medskip PERIODIC ORBITS OF A BOUNCING MASS ON A FLEXIBLE HINGED-HINGED BEAM}
\author[L. Demeio - S. Lenci]{Lucio Demeio$^1$ - Stefano Lenci$^2$}
\institute{
$^1$Dipartimento di Ingegneria Industriale e Scienze Matematiche \\
$^2$Dipartimento di Ingegneria Civile, Edile e Architettura \\
Universit\`{a} Politecnica delle Marche}
\date{}
\input{macro_lucio}
\newcommand{\bul}{\color{green}$\bullet$}
\newcommand{\fa}{\varphi_{1,\Omega}}
\newcommand{\fb}{\varphi_{2,\Omega}}
\newcommand{\tw}{\widetilde{w}}


\begin{document}
\setbeamertemplate{items}[ball]
%\setbeamertemplate{footline}[frame number]
%\setbeamertemplate{footline}[author]
 
\frame{\titlepage}

%\section[]{}
%\frame{\tableofcontents}


\begin{frame}[label=intro]
\frametitle{\bf Introduction}
\begin{block}<1->{Summary}
%\small
\bit
\item<1-> In this work we present a collection of numerical simulations of a ball repeatedly bouncing on a flexible beam with hinged-hinged boundary conditions with $x \in [0,1]$;
\item<2-> we select one particular periodic orbit in the case of a static beam and then investigate numerically the orbits generated by perturbing the initial conditions; 
\item<3-> we then study the effect of the beam's vibration on the periodic orbits;
\item<4-> impacts are modelled in a very simple way, by using momentum conservation;
\item<5-> this is an entirely numerical work, so all results must be understood ``within numerical errors''.
\eit
\end{block}
\end{frame}


\frame
{
\begin{block}<1->{The impact model}
\begin{center}
\includegraphics[height=4cm]{impact.eps} 
\end{center}
\end{block}
\begin{block}<2->{}
%\small
\bit
\item<2-> The velocity of the ball relative to the beam immediately before the impact ${\bf u'}_k$ is projected into the parallel and perpendicular components $u'_{\parallel}$ and $u'_{\perp}$ (primes denote relative velocities); 
\item<3-> the post-impact relative velocity ${\bf v}'_k$ is then obtained by $v'_{\parallel}=u'_{\parallel}$ and $v'_{\perp}=-r \, u'_{\perp}$, with $r$ the restitution coefficient ($r=1$ in all simulations);
\item<4-> the relative velocity is then transformed into an absolute velocity ${\bf v}_k$.
\eit
\end{block}
}

\frame
{
\frametitle{\bf Return map}
\begin{block}<1->{}
\bit
\item<1-> The sequence of impacts generates a return map $f:[0,1] \times \mathbb{R}^2 \to [0,1] \times \mathbb{R}^2$ such that 
$$(x_{k+1},{\bf u}_{k+1})=f(x_k,{\bf u}_k)$$
and given by
\item<2-> $(x_k,{\bf u}_k) \to (x_k,{\bf v}_k) \to$ ball's trajectory to next impact (determined numerically as root of a nonlinear equation) $\to (x_{k+1},{\bf u}_{k+1})$;
\item<3-> a periodic orbit is a fixed point of the map or of an iterate:
$$(x_{k+n},{\bf u}_{k+n})=f^n(x_k,{\bf u}_k)=(x_k,{\bf u}_k)$$
\eit
\end{block}
}

\frame
{
  \frametitle{\bf Periodic orbits with static beam}
\begin{block}<1->{}
\bit
\item<1-> A class of peridic orbits is generated by dropping the ball vertically from a suitable height $h=h_0$ with $v_y=0$ impacting at $x=x_0$;
\item<1-> the height $h_0$ is chosen so that the ball bounces exactly on the point symmmetric w.r. to the midplane of the beam;
\item<1-> then the trajectory consists od a parabolic arc and two straight lines, all travelled twice with opposite velocities.
\eit 
\end{block}

\begin{block}<1->{}
\begin{center}
\includegraphics[height=3.8cm]{Traiettoria_03_h0.eps} 
\includegraphics[height=3.8cm]{x_03_h0.eps} 
$x_0=0.3$, $h_p\approx 2.10564$, $\omega_p\approx 0.382$, $T_p\approx 16.45$
\end{center}
\end{block}
}

\frame
{
\frametitle{\bf Periodic orbits with static beam}
\begin{block}<1->{}
\bit
\item<1-> These peridic orbits exist for every $x_0$;
\item<1-> the height $h_0$ can be calculated analytically (for each $x_0$):
$$
h= \frac{2 \, x_0-1}{4 \, \sin (2\, \theta_0) \, \cos (2\, \theta_0)}
$$
\eit 
\end{block}
\begin{block}<1->{}
\begin{center}
\includegraphics[height=4.2cm]{curva_x0h} 
\end{center}
\end{block}
}

\frame
{
\frametitle{\bf Perturbed periodic orbits with static beam}
\begin{block}<1->{}
\bit
\item<1-> We choose a slightly different value for $h$;
\item<1-> the orbits remain close to the periodic one, suggesting stability;
\item<1-> the $x-$coordinate shows an oscillation at high frequency with an oscillating amplitude with smaller frequency.
\eit
\end{block}
\begin{block}<1->{}
\begin{center}
\includegraphics[width=5cm]{Traiettoria_03_h212.eps} 
\includegraphics[width=5cm]{x_03_h212.eps} \\
$x_0=0.3$ and $h=2.12 > h_0$
\end{center}
\end{block}
}


\frame
{
\frametitle{\bf Perturbed periodic orbits with static beam}
\begin{block}<1->{}
\bit
\item<1-> The periodic orbit is characterized by four points on the return map (or on a phase-space projection);
\item<1-> with a slightly different value of $h$ (larger or smaller than $h_0$) small closed loops form in the vicinity of the periodic points.
\eit
\end{block}
\begin{block}<1->{}
In the following pictures, $h=h_0=2.10564$ (black dots), $h=2.055$ (red line) and $2.08$ (blue line).
\begin{center}
\includegraphics[width=5cm]{ret_x_x_03_hhh.eps} 
\includegraphics[width=5cm]{x_ux_03_hhh.eps} 
\end{center}
\end{block}
}

\frame
{
\frametitle{\bf Merging of loops}
\begin{block}<1->{}
\begin{center}
As $h$ decreases further from $h_0$ (black dots) to $2.054$ (blue lines) and $h=2.04$ (red line)  the loops  merge and form a single loop. \\
%\vspace*{0.1cm}
\includegraphics[width=4.cm]{ret_x_x_03_hhhh.eps} 
\includegraphics[width=4.cm]{x_ux_03_hhhh.eps} 
\end{center}
\end{block}
\begin{block}<2->{}
\begin{center}
Trajectories in physical space. \\
%\vspace*{0.1cm}
\includegraphics[width=4.cm]{x_y_03_h2055.eps} 
\includegraphics[width=4.cm]{x_y_03_h204.eps} 
\end{center}
\end{block}
}

\frame
{
\frametitle{\bf Higher-period case}
\begin{block}<1->{}
At $h=2.31$ there is evidence of period-7 orbit 
\begin{center}
%\vspace*{0.1cm}
\includegraphics[width=4.cm]{ret_x_x_h231_p7.eps} 
\includegraphics[width=4.cm]{x_y_03_h231_p7.eps} 
\end{center}
\end{block}
\begin{block}<2->{}
$h=2.35$: route to chaos ?
\begin{center}
\includegraphics[width=4.cm]{ret_x_x_03_h235.eps} 
\includegraphics[width=4.cm]{x_y_03_h235.eps} 
\end{center}
\end{block}
}

\frame
{
\frametitle{\bf Vibrating beam}
\begin{block}<1->{}
\bit
\item<1-> A hinged-hinged beam is considered with only the first mode excited: $w(x,t)= (a \, \cos(\omega \, t) + b \, \sin(\omega \, t)) \, \varphi_1(x)$;
\item<1-> with $\varphi_n(x) = \sqrt{2} \, \sin(n \, \pi \, x)$ the beam eigenfunctions;
\item<1-> we vary $\omega$ and $a$ parametrically, while $b=0$;
\item<1-> we initialize the system at $h=h_0$ (i.e. at the periodic orbit).
\eit
\end{block}
\begin{block}<2->{Low frequency}
\begin{center}
\includegraphics[width=4.2cm]{Trave_omega01/a0002.eps} \hspace*{1 cm}
\includegraphics[width=4.2cm]{Trave_omega01/a0001.eps} \\
$\omega=0.1$, $a=0.0002$ and $a=0.0001$
\end{center}
\end{block}
}
\frame
{
\frametitle{\bf Vibrating beam}
\begin{block}<1->{}
\begin{center}
\includegraphics[width=4.2cm]{Trave_omega01/a0005.eps} \hspace*{1 cm}
\includegraphics[width=4.2cm]{Trave_omega01/a0018.eps} \\
$\omega=0.1$, $a=0.0005$ and $a=0.00018$
\end{center}
\end{block}
\begin{block}<2->{}
\begin{center}
\includegraphics[width=4.2cm]{Trave_omega01/a002.eps} \hspace*{1 cm}
\includegraphics[width=4.2cm]{Trave_omega01/ret_x_x_03_a0027.eps} \\
$\omega=0.1$, $a=0.002$ and $a=0.0027$
\end{center}
\end{block}
}

\frame
{
\frametitle{\bf Vibrating beam}
\begin{block}<1->{Frequency moderately close to resonance}
\begin{center}
\includegraphics[width=3.9cm]{Trave_omega03/a002.eps} \hspace*{1 cm}
\includegraphics[width=3.9cm]{Trave_omega03/a003.eps} \\
{\small $\omega=0.3$, $a=0.002$ and $a=0.003$}
\end{center}
\end{block}
\begin{block}<2->{}
\begin{center}
\includegraphics[width=3.9cm]{Trave_omega03/a004.eps} \hspace*{1 cm}
\includegraphics[width=3.9cm]{Trave_omega03/a005.eps} \\
{\small $\omega=0.3$, $a=0.004$ and $a=0.005$}
\end{center}
\end{block}
}

\frame
{
\frametitle{\bf Vibrating beam}
\begin{block}<1->{Frequency very close to resonance}
\begin{center}
\includegraphics[width=3.9cm]{Trave_omega0382/a001.eps} \hspace*{1 cm}
\includegraphics[width=3.9cm]{Trave_omega0382/a0015.eps} \\
{\small $\omega=0.382$, $a=0.001$ and $a=0.0015$}
\end{center}
\end{block}
\begin{block}<2->{}
\begin{center}
\includegraphics[width=3.9cm]{Trave_omega0382/a0021.eps} \hspace*{1 cm}
\includegraphics[width=3.9cm]{Trave_omega0382/a00222.eps} \\
{\small $\omega=0.382$, $a=0.0021$ and $a=0.0022$}
\end{center}
\end{block}
}

\begin{frame}
\frametitle{\bf Conclusions}
\begin{block}<1->{}
%\small
\bit
\item<1-> We have generated a class of periodic orbits of a point-mass bouncing on a static beam;
\item<2-> when the initial conditions giving the periodic orbit are changed, the points on the return map (or on the phase space) turn into loops which represent motions of the ball bouncing always within bounded intervals near the periodic points;
\item<3-> as the perturbation becomes larger, the loops increase in size and eventually merge into a single loop comprising the whole space, meaning that the bounces of the ball occur over the whole spatial domain $[0,1]$; 
\item<4-> if the beam is set into vibration the effect is similar, but the loops acquire a much richer structure, possibly leading to chaotic behaviour;
\item<5-> more analysis is needed in order to make stronger statements and to accompany the numerical findings.
\eit
\end{block}
\end{frame}

\frame{
\medskip
\begin{center}
{\bf \huge \color{red} THANK YOU !!}
\end{center}
}
\end{document}

\frame
{
\frametitle{\bf Vibrating beam}
\begin{block}<1->{}
\begin{center}
\includegraphics[width=4.2cm]{a002.eps} \hspace*{1 cm}
\includegraphics[width=4.2cm]{ret_x_x_03_a0027.eps} \\
$\omega=2$ and $\omega=3$
\end{center}
\end{block}
}

\frame
{
\frametitle{\bf Vibrating beam}
\begin{block}<1->{}
\begin{center}
\includegraphics[width=4.2cm]{ret_x_x_03_a002_oooo.eps} \hspace*{1 cm}
\includegraphics[width=4.2cm]{ret_x_x_03_a001_om1.eps} \\
$\omega=0.2$, $\omega=0.3$ and $\omega=0.382$ \hspace*{2 cm} $\omega=1$
\end{center}
\end{block}
\begin{block}<2->{}
\begin{center}
\includegraphics[width=4.2cm]{ret_x_x_03_a001_om2.eps} \hspace*{1 cm}
\includegraphics[width=4.2cm]{ret_x_x_03_a001_om3.eps} \\
$\omega=2$ and $\omega=3$
\end{center}
\end{block}
}

\frame
{
\frametitle{\bf Vibrating beam}
\begin{block}<1->{}
\begin{center}

\includegraphics[width=4.3cm]{ret_x_x_03_a001_om4.eps} \hspace*{1 cm}
\includegraphics[width=4.3cm]{ret_x_x_03_a001_om5.eps} \\
$\omega=4$ and $\omega=5$
\end{center}
\end{block}
\begin{block}<2->{}
\begin{center}

\includegraphics[width=4.3cm]{ret_x_x_03_a001_om6.eps} \hspace*{1 cm}
\includegraphics[width=4.3cm]{ret_x_x_03_a001_om8.eps} \\
$\omega=6$ and $\omega=8$
\end{center}
\end{block}
}

\frame
{
\frametitle{\bf Vibrating beam}
\begin{block}<1->{}
\begin{center}

\includegraphics[width=4.5cm]{ret_x_x_03_a001_om9.eps} \hspace*{1 cm}
\includegraphics[width=4.5cm]{ret_x_x_03_a001_om10.eps} \\
$\omega=9$ and $\omega=10$
\end{center}
\end{block}
}

\frame
{
\frametitle{\bf Conclusions}
\begin{block}<1->{}
\bit
\item<1->
\item<2->
\eit
\end{block}
}


