This chapter presents results and observation on the reference test series obtained from the performed test for the smallest forcing amplitude applied $a_y=5$ mm in direction y for forcing frequencies near/at $f=0.89$ Hz. The results for test series using the same 1 DoF harmonic base excitation signal but at different forcing amplitude, water depth in the model and in hazard condition removing the three steam generators from the experimental setup are presented and are compared each other.\\
Time histories, phase plane plots and Fast Fourier Transform (FFT) are the tools for studying the investigation of sloshing processes in the ALFRED's model. Captures of the video taken with the \textit{Dantec} camera are used to confirm the post-processed information. \texttt{Python} codes for post processing are used as well as spreadsheets for the data collection.\\
The fundamental parameter for the investigation of sloshing is the dimensionless forcing acceleration $\kappa_y$, equal to:
\begin{equation}\label{eq:kappa_y}
    \kappa_y=\frac{a_y\omega_y^2}{g}
\end{equation}
where $g=9.81\,\mathrm{m/s^2}$ is the gravitational acceleration, $a_y$ the external forcing amplitude and $\omega_y$ is the forcing pulsation.
%================================================================================
\section{Basic reference test series}\label{parag:near_linear_test_series}
The horizontal harmonic base excitation with a forcing amplitude equal to $a_y=5$ mm is selected as the test series of reference. The selected free-liquid surface height in the nuclear pool reproduction is equal to the nominal one $h=8.84$ cm, according to the similarity with the actual demonstrator (\S\ref{parag:similarity}). It is the starting point of the experimental investigation of the ALFRED's model, because it's the reference to which compares all the other horizontal harmonic series in 1 DoF. It's also a reference for all the other harmonic series and the earthquakes' simulation events studied in this thesis.\\
The investigation starts with the plots for the maximum amplitude $\zeta_{max}$ reached from the free liquid surface during the test at different forcing frequencies, which are listed in Table \ref{tab:horizontal_harmonic_base_excitation} in \S \ref{parag:test_matrix}. The plots are built taking in account the total uncertainty on the measurement, due to the single probe and to the shaking table displacement (\S\ref{parag:capacitance_probes}), and the capacitance laws obtained after the Monte Carlo method. Taking in account the frequency that gives back the maximum amplitude response equal to $f_1=0.98$ Hz, then $\zeta_{max}$ is plotted as a function of the frequency ratio $f_y/f_1$ in Figure \ref{fig:Harmonic_base_excitation_1DOF_h884_3SG_ay0013}.\\
Data from the P1 gives back relevant results about the real maximum amplitude reached during the sloshing phenomena because it takes the actual peak of the waves. Data from the P2 gives back a hint of when the swirling regime modes starts to initiates, so when another natural frequency is going to be experimentally reached. Maximum wave amplitude is reached for $f_y/f_1=1.$ where $f_y=0.98$ Hz, and doesn't occur for $f=0.89$ Hz, as it would be expected by the numerical studies of reference. The right-shifting of the maximum elevation in function of the frequency ratio is an indication of starting nonlinear phenomena.\\
The test series is then analyzed selecting the following investigation points: point \textbf{AR1} at $f_y/f_1=0.796$ and $f_y=0.78$ Hz; points \textbf{NR} from 1 to 3 at $f_y/f_1=$0.898, 0.918, 0.959 and $f_y$=0.88 Hz, 0.90Hz, 0.94Hz approaching the greater maximum amplitude; point \textbf{R} at $f_y/f_1=1.000$ and $f_y=0.98$ Hz at the higher amplitude; point \textbf{AR2} at $f_y/f_1=1.061$ and $f_y=1.040$ Hz.
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.75]{immagini/harmonic_base_excitation_1DoF/Harmonic_base_excitation_1DOF_h884_3SG_ay0013_MOD.png}
    \caption{Maximum amplitude plot for horizontal harmonic base excitation in $Y$ direction in function of the frequency ratio $f_y/f_1$ when $a_y=5$ mm ($a_y/D=0.013$); point AR1 for $f_y/f_1=0.796$, NR1 for $f_y/f_1=0.898$, NR2 for $f_y/f_1=0.918$, NR3 for $f_y/f_1=0.959$, R for $f_y/f_1=1.000$ and AR2 for $f_y/f_1=1.061$.}
    \label{fig:Harmonic_base_excitation_1DOF_h884_3SG_ay0013}
\end{figure}
%\clearpage
\subsection{Forcing frequency 0.78 Hz}
In the point AR1 the forcing frequency is equal to $f_y=0.78$ Hz, so from Eq. \ref{eq:kappa_y} the dimensionless  forcing acceleration is equal to $\kappa_y=0.0122$. The time evolution of the free-liquid surface motion is shown for ten seconds in Figure \ref{fig:ay5mm_f078Hz_1DOF_time_history}, not taking into account the starting ramp effect. Time histories plots are made dimensionless multiplying the time variable per the pulsation $\omega_1=2\pi f_1$, where $f_1=0.98$ Hz is the frequency at higher wave amplitude. For the current forcing frequency, time evolution confirms a semi-periodic motion, where the P1 presents peaks when the P2 presents troughs because of the position of the probes. 
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f0780Hz_subplot_time_history.png}
    \caption{Dimensionless time histories for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=0.796$ and $f_y=0.78$ Hz; blue is assigned to the P1 signal, while green the P2 signal}
    \label{fig:ay5mm_f078Hz_1DOF_time_history}
\end{figure}
In Figure \ref{fig:ay5mm_f078Hz_1DOF_cine} the video captures for an analogue test series performed in the x-direction are shown. The video had been taken before noticing the faulting of the P0 sensor, but are useful to highlight the modal forms. At the current forcing frequency $f=0.78$ Hz and for $a_y=5$ mm a semilinear behaviour is experienced and the modal form is similar to the first sloshing mode, despite the small amplitudes.
\begin{figure}[H]
    \centering
    \includegraphics[width=0.90\textwidth]{immagini/harmonic_base_excitation_1DoF/078Hz_x.png}
    \caption{Video captures of the model's pool from the $f=0.78$ Hz and $a_y=5$ mm test series but performed in the x-direction to see modal forms: semi-linear first sloshing mode is outlined, but the amplitudes are very small}
    \label{fig:ay5mm_f078Hz_1DOF_cine}
\end{figure}
The studies are conducted by means also of \textit{Fast Fourier Transform} (\textit{FFT}), so by \textit{Power Spectral Density} (\textit{PSD}). In Figure \ref{fig:ay5mm_f078Hz_1DOF_PSD} the PSD associated to the regime signal is shown. The primary peak is at the forcing amplitude frequency ratio of 0.796, while the secondary ones are multiples of the forcing frequency. The low resolution (frequency step $\approx$ 0.02) is not enough to formally identify the nonlinear frequency components, despite secondary peaks at low energy are attended since the phenomenon is nonlinear.
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f0780Hzstarting_ramp_PSD_probes_plot.png}
    \caption{Power Spectral Density for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=0.796$ and $f_y=0.78$ Hz; while the x-axis are shown until $f_y/f_1=5$; peaks are at the forcing amplitude multiples.}
    \label{fig:ay5mm_f078Hz_1DOF_PSD}
\end{figure}
\textit{Phase plane plots} are also used in the formulation proposed in \cite{Experimental_investigation_sloshing}: for each time instant $t_i$, $\zeta(t_i)=\zeta(t_i+\tau)$ by calculating the mutual information of the attractor, where the $\tau=0.05$ value has been chosen to have a good resolution of the curves. The phase plots typically show circular, clear and regular orbit in case of linear motion, and asymmetric ones in case of nonlinear free-liquid surface motions. 
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f0780Hz_Phase_subplot.png}
    \caption{Dimensionless phase plane plots for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=0.796$ and $f_y=0.78$ Hz; skewness already in the lower forcing amplitude test series for low forcing frequency}
    \label{fig:ay5mm_f078Hz_1DOF_phase_plots}
\end{figure}
\subsection{Forcing frequencies 0.88 Hz, 0.90 Hz, 0.94 Hz}
Near the frequency that gives the maximum amplitude at $f_y=0.98$ Hz (points NR1, NR2, NR3), the amplitude of the wave increases and some significant irregularities start to show in the time histories. Figure \ref{fig:ay5mm_f094Hz_1DOF_time_history} exhibits the time history when $f_y=0.94$Hz and $\kappa_y=0.0177$, associated to video captures in Figure \ref{fig:ay5mm_f094Hz_1DOF_cine}. The maximum amplitude $\zeta_{max}$ for each of the those proposed forcing frequency ratio is increasing. It can be seen that the slope in the evolution of the free-surface amplitude is more steep than the $f_y=0.78$Hz test series, and a nonlinear first sloshing mode is experienced on the free surface, as can be seen in Figure \ref{fig:ay5mm_f094Hz_1DOF_cine}.
\begin{figure}[H]
    \centering
    \includegraphics[width=0.90\textwidth]{immagini/harmonic_base_excitation_1DoF/094Hz_x.png}
    \caption{Video captures of the model's pool from the $f=0.94$ Hz and $a_y=5$ mm test series but performed in the x-direction to see modal forms: semi-linear first sloshing mode is outlined.}
    \label{fig:ay5mm_f094Hz_1DOF_cine}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f0940Hz_subplot_time_history.png}
    \caption{Dimensionless time histories for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=0.959$ and $f_y=0.94$ Hz; P1 time evolution shows up peaks and secondary peaks due to nonlinear sloshing phenomenon.}
    \label{fig:ay5mm_f094Hz_1DOF_time_history}
\end{figure}
The PSD plots in Figure \ref{fig:ay5mm_f094Hz_1DOF_PSD} show that also in this case the primary peaks are at the forcing frequency ratio, but there's a hint of nonlinear component attached to the primary peak, which is instead representative of the external harmonic load. The resolution is too low (frequency step $\approx$ 0.02) to catch exactly the value of the nonlinear component, which in $f_y$[0.94,0.98] Hz.\\
The trend observed in the time history signal is shown in the phase plane plots, in Figure \ref{fig:ay5mm_f094Hz_1DOF_phase_plots}, showing skewed elliptical orbits. The wave-structure interaction is experienced even for a small forcing amplitude, because of the not circular and not even elliptical orbits' shape. The phase plane plot for P1 indicates that as the forcing frequencies increase, then $\zeta(t)$ increases within the positive quarter delineated by $\zeta(t)/D=[0.00,0.04]$ on the x-axis and $\zeta(t+\tau)/D={[}0.00,0.04{]}$ on the y-axis.
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f0940Hzstarting_ramp_PSD_probes_plot.png}
    \caption{Power Spectral Density for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=0.959$ and $f_y=0.94$ Hz; while the x-axis are until $f_y/f_1=5$; peaks at the forcing amplitude multiples and at $f=1$ Hz, but low resolution affects the plots}
    \label{fig:ay5mm_f094Hz_1DOF_PSD}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f0940Hz_Phase_subplot.png}
    \caption{Dimensionless phase plane plots for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=0.959$ and $f_y=0.94$ Hz; P1 phase plane plot is greater than the one for P2 and shows up higher nonlinearity in the positive quarter; a confrontation is made with the phase plane plots for $f=0.78$ Hz and $a_y=5$ mm}
    \label{fig:ay5mm_f094Hz_1DOF_phase_plots}
\end{figure}
\subsection{Forcing frequency 0.98Hz}
The forcing frequency $f_y=0.98$ Hz (in point R) is when there is the maximum free-liquid surface wave amplitude. Dimensionless acceleration is equal to $\kappa_y=0.0194$. For the same adopted methodology in the previous cases, video captures for the same test series but performed in the x-direction, in Figure \ref{fig:ay5mm_f098Hz_1DOF_cine}, are useful to exhibits nonlinear free-surface motion.\\
Time evolution of the free-liquid surface in Figure \ref{fig:ay5mm_f098Hz_1DOF_time_history} shows up higher peaks than the troughs for P1 probe, while secondary peaks are experienced in the P2 time evolution. It's a sign of nonlinearity alongside a non-negligible wave-structure interaction. The maximum amplitude is higher 2.67 times the maximum one for the $f_y=0.78$ Hz test series.
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/harmonic_base_excitation_1DoF/098Hz_x.png}
    \caption{Video captures of the model's pool from the $f=0.98$ Hz and $a_y=5$ mm test series but performed in the x-direction to see modal forms: nonlinear sloshing mode is outlined, which is similar to the first mode.}
    \label{fig:ay5mm_f098Hz_1DOF_cine}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f0980Hz_subplot_time_history.png}
    \caption{Dimensionless time histories for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=1.000$ and $f_y=0.98$ Hz; slopes of the curves are greater than the lower frequency cases and they increase approaching the peaks of the time evolution for P1.}
    \label{fig:ay5mm_f098Hz_1DOF_time_history}
\end{figure}
The wave-structure interaction is the reason way the orbits in phase plane in Figure \ref{fig:ay5mm_f098Hz_1DOF_phase_plots} has a so strong skewness in correspondence of the peaks, and are not even elliptical. The PSD plots for the regime signal in Figure \ref{fig:ay5mm_f098Hz_1DOF_PSD} still indicates the forcing frequency as the major contribution to the motion. The low resolution doesn't allow being sure about the real value of a nonlinear frequency component, which is in range $f=$[0.91,0.98] Hz. This could be the first natural sloshing frequency, that from the numerical studies is about to be $f_1=0.90$ Hz.
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f0980Hzstarting_ramp_PSD_probes_plot.png}
    \caption{Power Spectral Density for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=1.000$ and $f_y=0.98$ Hz; low resolution affects the PSD, but a nonlinear component can be spotted before the forcing frequency peak.}
    \label{fig:ay5mm_f098Hz_1DOF_PSD}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f0980Hz_Phase_subplot.png}
    \caption{Dimensionless phase plane plots for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=1.000$ and $f_y=0.98$ Hz P1 phase plane plot is greater than the one for P2 and shows up higher nonlinearity in the positive quarter; a confrontation is made with the phase plane plots for $f=0.78$ Hz and $a_y=5$ mm and the difference is up to 2.67 time the maximum amplitude within the two series.}
    \label{fig:ay5mm_f098Hz_1DOF_phase_plots}
\end{figure}
\clearpage
\subsection{Forcing frequency 1.04 Hz}
The test series at frequency ratio $f_y/f_1=1.061$ at $f_y=1.04$ Hz ($\kappa_y=0.0219$),  (point AR2) away from the greater amplitude is lastly studied. Time history in Figure \ref{fig:ay5mm_f104Hz_1DOF_time_history} indicates that the wave amplitude $\zeta(t)$ for P1 is lower than the previous case for $f_y=0.98$ Hz, while $\zeta(t)$ for P2 is comparable with P1 for the first time in basic reference test series. Phase plane plots in Figure \ref{fig:ay5mm_f104Hz_1DOF_phase_plots} don't show chaotic motion but still strong skewness, in particular from P2. It seems that wave structure interaction is stronger for the P2 signal close to the inner cylinder. 
%Looking at those phase plots for P2 and at the $\zeta_{max}/D$ as a function of $f_y/f_1$ in Figure \ref{fig:Harmonic_base_excitation_1DOF_h884_3SG_ay0013}, a starting swirling mode can be suggested: for the P2 amplitude increases as the frequency ratio is higher than $f_y/f_1=1.000$. Further consideration on the swirling mode cannot be done for lack of numerical results and experimental data.
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/harmonic_base_excitation_1DoF/104Hz_x.png}
    \caption{Video captures of the model's pool from the $f=0.98$ Hz and $a_y=5$ mm test series, but performed in the x-direction to see modal forms: nonlinear sloshing mode is outlined.}
    \label{fig:ay5mm_f104Hz_1DOF_cine}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f1040Hz_subplot_time_history.png}
    \caption{Dimensionless time histories for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=1.061$ and $f_y=1.04$ Hz; P1 peaks are not greater than the ones for P2 as in the previous forcing frequencies.}
    \label{fig:ay5mm_f104Hz_1DOF_time_history}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f1040Hz_Phase_subplot.png}
    \caption{Dimensionless phase plane plots for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=1.061$ and $f_y=1.04$ Hz; phase plane plots for the first forcing frequency $f_y=0.78$ Hz is smaller than the one for $f_y=1.04$ Hz.}
    \label{fig:ay5mm_f104Hz_1DOF_phase_plots}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\textwidth]{immagini/harmonic_base_excitation_1DoF/Sloshing231026_Sine_y_A5mm_f1040Hzstarting_ramp_PSD_probes_plot.png}
    \caption{Power Spectral Density for the horizontal harmonic base excitation $a_y=5$ mm, $f_y/f_1=1.061$ and $f_y=1.04$ Hz; low resolution affects the PSD, but a nonlinear component can be spotted after the forcing frequency at $f=1.11$ Hz}
    \label{fig:ay5mm_f104Hz_1DOF_PSD}
\end{figure}
\clearpage
\begin{sidewaysfigure}
    \centering
    \includegraphics[scale=0.80]{immagini/harmonic_base_excitation_1DoF/video_recap_5mm.png}
    \caption{Evolution of maximum wave amplitude in the pool's model from the video captures; the comparison is associated to the maximum amplitude plot for horizontal harmonic base excitation in $Y$ direction in function of the frequency ratio $f_y/f_1$ when $a_y=5$ mm ($a_y/D=0.013$).}
    \label{fig:capture_video_summary}
\end{sidewaysfigure}
%===============================================================================
\clearpage
\section{Effect of forcing amplitude}\label{parag:effect_of_forcing_amplitude}
The first comparison is about the forcing amplitude for the horizontal harmonic base excitation in one DoF. In addition to reference test series for a maximum forcing amplitude equal to $a_y=5$ mm, $a_y=15$ mm and $a_y=30$ mm are considered (\S \ref{parag:creation_input_signal}). The water height is still the nominal one equal to $h=8.84$ cm. The analysis method used to pursue this investigation is the same used for the reference test series (\S \ref{parag:near_linear_test_series}). For a proper comparison with the test series at forcing amplitude $a_y=5$ mm, $f_1=0.98$ Hz has been selected as reference for the dimensionless frequency ratio. The selected forcing frequency is when the reference test series experiences the greater wave amplitude.\\
Five investigation points have been selected to investigate the effect of the forcing amplitude on the free-liquid surface motion during the sloshing process: point \textbf{A} at $f_y/f_1=0.796$ ($f_y=0.78$ Hz); point \textbf{B} at $f_y/f_1=0.898$ ($f_y=0.88$ Hz); point \textbf{C} at $f_y/f_1=1.000$ ($f_y=0.98$ Hz); point \textbf{D} at $f_y/f_1=1.408$ ($f_y=1.38$ Hz).\\
Table \ref{tab:k_y_forcing_amplitude_horizontal_harmonic_base_excitation} presents the dimensionless forcing acceleration $\kappa_y$ that varies with different forcing amplitude. Dimensionless acceleration is a good indicator for finding patterns in the effect of the forcing amplitudes comparison.
\begin{table}[H]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
$f_y$[Hz]       & 0.78  & 0.88  & 0.98   & 1.38  \\ \hline
$\kappa_y$ for $a_y=5$ mm  & 0.0122 & 0.0156 & 0.0193 & 0.0383 \\ \hline
$\kappa_y$ for $a_y=15$ mm & 0.0367 & 0.0467 & 0.0580 & 0.0115 \\ \hline
$\kappa_y$ for $a_y=30$ mm & 0.0735 & 0.0935 & 0.1159 & 0.2299 \\ \hline
\end{tabular}
\caption{$\kappa_y$ dimensionless acceleration for the horizontal harmonic base excitation in one DoF for different forcing amplitude $a_y$}
\label{tab:k_y_forcing_amplitude_horizontal_harmonic_base_excitation}
\end{table}
The harmonic test series at $a_y=5$ mm presents the maximum wave amplitude for P1 capacitance probe when $f_y=0.98$ Hz. Since the horizontal harmonic base excitation in y direction have the forcing frequencies around the first natural mode, P1 capture the most valuable information about the wave amplitude. Figure \ref{fig:Harmonic_base_excitation_1DOF_h884_3SG_every_ay} presents the complete plot for maximum wave amplitude as a function of the frequency ratio, both for P1 than P2. Taking into account the information about P1, a plateau of the maximum amplitude can be seen for $a_y=15$ mm in correspondence to a forcing frequency range equal to $f_y=$[0.98,1.08] Hz, while for $a_y=30$ mm for $f_y=$[0.98,1.12] Hz. This behavior is not observed for the reference test series $a_y=5$ mm, which reaches the maximum wave amplitude only at $f_y=0.98$ Hz.\\
In the same plot, wave amplitude for P2 is shown. It is always minor than the one detected from P1, but this trend is inverted at higher frequency ratio. A possible explanation could be the starting of a swirling motion of the free-liquid surface. A deeper look at this behavior is provided in the investigation of the investigation point E at $f_y=1.38$ Hz.
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.65]{immagini/effect_of_forcing_amplitude/Harmonic_base_excitation_1DOF_h884_3SG_every_ay.png}
    \caption{Maximum amplitude plot for horizontal harmonic base excitation in $Y$ direction in function of the frequency ratio $f_y/f_1$ when $a_y=5,15,30$ mm ($a_y/D=0.013,0.038,0.076$); investigation point are outlined.}
    \label{fig:Harmonic_base_excitation_1DOF_h884_3SG_every_ay}
\end{figure}
\subsection{Forcing frequency 0.78 Hz}
The first investigation point A is at $f_y/f_1=0.796$ for $f_y=0.78$ Hz. Video captures of the free-liquid surface approaching and reaching the maximum amplitude are presented in Figure \ref{fig:a30_f0780Hz_cine} for the $a_y=30$ mm test series. Motion is strongly nonlinear and some bubbles start to show up already for a low forcing frequency. Maximum amplitude is reached when a wave coming from the right edge merges with another one from the left edge. The merge takes place not in the center line of the model because the internal structure damping.
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/effect_of_forcing_amplitude/a30_f0780Hz.png}
    \caption{Cine file capture near, at and beyond the maximum amplitude when $f=0.78$ Hz: the maximum amplitude is reached in proximity of the P1 sensor after a merge within two waves from the right edge and the left edge; centreline of the tank is outlined in red.}
    \label{fig:a30_f0780Hz_cine}
\end{figure}
Phase plane plots in Figure \ref{fig:effect_of_ay_078Hz_Phase_plot} for showing immediately that the amplitude $\zeta(t)$ is increasing when the forcing amplitude increase. The orbits for each of the studied cases are different, highlighting different wave-structure interaction. Nonlinearity is stronger as the dimensionless acceleration $\kappa_y$ is increasing for increasing forcing amplitude.
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.35]{immagini/effect_of_forcing_amplitude/CONFRONTATION_f0780Hz_Phase_plot.png}
    \caption{Phase plane plots for $a_y=5,15,30$ mm, $f_y/f_1=0.796$ at $f_y=0.78$ Hz: skewness makes assuming different orbits due to different wave structure interaction.}
    \label{fig:effect_of_ay_078Hz_Phase_plot}
\end{figure}
The corresponding PSD plots in Figure \ref{fig:effect_of_ay_078Hz_PSD} gives back a hint on the nonlinear frequency associated to the motion. Frequency x-axis have been cut up to 5 Hz because the predominant motion is still periodic each 15 or 20 seconds, so the most energetic peaks are about the forcing frequency. Despite the low resolution in the PSD plots doesn't allow being sure about the value of the secondary peak indicated, an estimation can be done and is about $f=0.74$ Hz, at the left of $f_y=0.78$ Hz peak of the external excitation.
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{immagini/effect_of_forcing_amplitude/CONFRONTATION_f0780Hz_PSD_plot_MOD.png}
    \caption{Power Spectral Density for the horizontal harmonic base excitation for $a_y=5,15,30$ mm, $f_y/f_1=0.796$ at $f_y=0.78$ Hz: a possible nonlinear component is outlined at $f=0.74$ Hz for the $a_y=30$ mm test series.}
    \label{fig:effect_of_ay_078Hz_PSD}
\end{figure}
\subsection{Forcing frequency 0.88 Hz}
The second investigation point B is taken fo $f_y/f_1=0.898$ at $f_y=0.88$ Hz as an example of approaching to the maximum wave amplitude. Figure \ref{fig:a30_f0880Hz_cine} presents the video capture near/at the maximum wave amplitude. The complex geometry, the inner cylinder and the damping internal element are responsible for this behavior highly nonlinear behavior of the free-liquid surface.
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/effect_of_forcing_amplitude/a30_f0880Hz.png}
    \caption{Cine file capture near, at and beyond the maximum amplitude when $f=0.88$ Hz: highly nonlinear modes are occurring in the model's pool.}
    \label{fig:a30_f0880Hz_cine}
\end{figure}
Figure \ref{fig:effect_of_ay_088Hz_Phase_plot} shows increasing amplitude $\zeta(t)$. If the maximum amplitude reached is compared in taking as reference the $a_y=5$ mm test series, $\zeta_{max}$ of P1 for 15 mm series is 2.31 times and 3.29 time for 30 mm series. Wave structure interaction is particular strong at the troughs for the medium and the larger forcing amplitude test series: phase plane presents a small lobe in the negative quarter for negative $\zeta(t)/D$ and negative $\zeta(t+\tau)/D$.\\
The PSD in Figure \ref{fig:effect_of_ay_088Hz_PSD} presents a secondary peak at $f=0.85$ Hz despite the more energetic ones are about the forcing frequency $f_y=0.88$ Hz. Despite the resolution, this peak can be outlined in particular in the $a_y=5$ mm test series, while in the other two the low resolution affects too much the plots.
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.35]{immagini/effect_of_forcing_amplitude/CONFRONTATION_f0880Hz_Phase_plot.png}
    \caption{Phase plane plots for $a_y=5,15,30$ mm, $f_y/f_1=0.898$ at $f_y=0.88$ Hz: skewness makes assuming similar orbit for the troughs of the medium and the larger forcing amplitude series.}
    \label{fig:effect_of_ay_088Hz_Phase_plot}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{immagini/effect_of_forcing_amplitude/CONFRONTATION_f0880Hz_PSD_plot_MOD.png}
    \caption{Power Spectral Density for the horizontal harmonic base excitation for $a_y=5,15,30$ mm, $f_y/f_1=0.898$ at $f_y=0.88$ Hz: a possible nonlinear component is outlined at $f=0.85$ Hz for the $a_y=30$ mm test series.}
    \label{fig:effect_of_ay_088Hz_PSD}
\end{figure}
\subsection{Forcing frequency 0.98 Hz}
The third investigation point C for $_y/f_1=1.000$ at $f_y=0.98$ Hz. From \S \ref{parag:near_linear_test_series} this forcing frequency is the one that excites more the free-liquid surface motion for the $a_y=5$ mm test series in terms of maximum amplitude detected from the P1 capacitance probe. This forcing frequency is the first of a range that goes until $f_y=1.08$ Hz for the medium forcing amplitude series and $f_y=1.12$ Hz for the greater forcing amplitude series. comparing with the $a_y=5$ mm test series, maximum wave amplitude for the $a_y=15$ mm series is 2.01 times higher and 3.32 times for the $a_y=30$ mm series.\\
The video captures in Figure \ref{fig:a30_f0980Hz_cine} near/at the maximum wave amplitude leads to a similar conclusion from the smaller forcing frequency: the greater amplitude measured from P1 is after the merge of two waves from the right and the left edge of the ALFRED's model. The violence increases for higher forcing frequencies, so at higher $\kappa_y$ value in terms of dimensionless external acceleration.
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/effect_of_forcing_amplitude/a30_f0980Hz.png}
    \caption{Cine file capture near, at and beyond the maximum amplitude when $f=0.98$ Hz: the maximum amplitude is reached in proximity of the P1 sensor after a merge within two waves from the right edge and the left edge; the violence of the motion is increasing.}
    \label{fig:a30_f0980Hz_cine}
\end{figure}
Phase plane plots in Figure \ref{fig:effect_of_ay_098Hz_Phase_plot} present orbits more asymmetric than the previous investigation points at lower forcing frequency. Nonlinearity is higher for the test series at higher dimensionless acceleration parameter $\kappa_y$ (Table \ref{tab:k_y_forcing_amplitude_horizontal_harmonic_base_excitation}). As an example, a selected time history is presented for the higher forcing amplitude at $f_y=0.98$ Hz in Figure \ref{fig:time_history_30mm_f098Hz}: periodicity cannot be emphasized as in the reference test series (\S\ref{parag:near_linear_test_series}, and peaks for P1 reach different amplitudes in the time evolution as well the ones for P2.
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.35]{immagini/effect_of_forcing_amplitude/CONFRONTATION_f0980Hz_Phase_plot.png}
    \caption{Phase plane plots for the horizontal harmonic base excitation for $a_y=5,15,30$ mm, $f_y/f_1=1.000$ at $f_y=0.98$ Hz: phase plane plots make asymmetric but out of focus orbits.}
    \label{fig:effect_of_ay_098Hz_Phase_plot}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.35]{immagini/effect_of_forcing_amplitude/Sloshing231026_Sine_y_A30mm_f0980Hz_subplot_time_history.png}
    \caption{Dimensionless time histories for the horizontal harmonic base excitation $a_y=30$ mm, $f_y/f_1=1.000$ and $f_y=0.98$ Hz; P1 peaks and troughs reach different amplitudes in the time evolution, as well as P2 peaks and troughs.}
    \label{fig:time_history_30mm_f098Hz}
\end{figure}
The PSD plots are affected by low resolution, so the nonlinear component cannot be clearly identified, but studying the $a_y=5$ mm curve a secondary peak is at $f=0.91$ Hz, while the primary one is at the forcing frequency $f_y=0.98$ Hz.
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{immagini/effect_of_forcing_amplitude/CONFRONTATION_f0980Hz_PSD_plot_MOD.png}
    \caption{Power Spectral Density for the horizontal harmonic base excitation for $a_y=5,15,30$ mm, $f_y/f_1=1.000$ at $f_y=0.98$ Hz; a possible nonlinear component is outlined at $f=0.91$ Hz for the $a_y=5$ mm test series.}
    \label{fig:effect_of_ay_098Hz_PSD}
\end{figure}
\subsection{Forcing frequency 1.38 Hz}
The investigation point D for $f_y/f_1=1.408$ at $f_y=1.38$ Hz is here discussed. The first observation is about the wave amplitude of P1 related to the one of P2: the second probe, which is near the inner cylinder of ALFRED's model, captures higher values than the one on the edge of the annular tank. It's the first time that this magnitude of amplitude is reached in the test series at different forcing amplitude. The video captures in Figure \ref{fig:a30_f1380Hz_cine} for $a_y=30$ mm show a violent motion that leads to bubble making events both at the surface and under it. A very complex motion is undergoing in the model pool, which is also not resembling so easily to the first sloshing mode.
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/effect_of_forcing_amplitude/a30_f1380Hz.png}
    \caption{Cine file capture near, at and beyond the maximum amplitude when $f=1.38$ Hz: the maximum amplitude is not more reached in proximity of the P1 sensor and the violence of the motion is increasing by bubbles making events.}
    \label{fig:a30_f1380Hz_cine}
\end{figure}
The phase plane plots in Figure \ref{fig:effect_of_ay_138Hz_Phase_plot} show violent motion for the larger forcing amplitude because at equal resolution [$\tau=0.05$ s] the curves are not smooth. This observation is in continuity with the investigated $f_y=0.98$ Hz forcing frequency in the previous paragraph. P2 contribution becomes greater than the one of P1, so phase plane shows up the strong wave structure interaction with the inner cylinder.\\
The PSD plots are still affects of low resolution and basically show the forcing frequency contribution and its multiples. Despite the difficulties, smaller peaks can be spotted before and after the $2\cdot f_y$ peak in the P2's PSD, that becomes more representative since the magnitude of wave amplitude is higher than the P1 one.
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.35]{immagini/effect_of_forcing_amplitude/CONFRONTATION_f16456Hz_Phase_plot.png}
    \caption{Phase plane plots for the horizontal harmonic base excitation for $a_y=5,15,30$ mm, $f_y/f_1=1.408$ at $f_y=1.38$ Hz: phase plane plots show not smooth orbits due to violent free-liquid surface motion.}
    \label{fig:effect_of_ay_138Hz_Phase_plot}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{immagini/effect_of_forcing_amplitude/CONFRONTATION_f16456Hz_PSD_plot_MOD.png}
    \caption{Power Spectral Density for the horizontal harmonic base excitation for $a_y=5,15,30$ mm, $f_y/f_1=1.408$ at $f_y=1.38$ Hz: smaller peaks can be spotted before and after the $2\cdot f_y$ peak in the P2's PSD.}
    \label{fig:effect_of_ay_138Hz_PSD}
\end{figure}
\clearpage
%===========================================================================
\section{Effect of water depth}\label{parag:effect_of_water_depth_harmonic}
The influence of water depth on ALFRED's model is investigated across five different depths. Particularly, the study focuses on shallow water due to the compact dimensions of the model. The water depth is measured starting from the diaphragm as a convention, because the main sloshing processes take places in the upper part of the model's pool.\\
The reference test series involves horizontal harmonic base excitation in one degree of freedom (DoF), with a force amplitude set to $a_y=15$ mm and the model's free-liquid surface resting at a level of $h_{model}=8.84$ cm. To explore the effect, variations of $\pm 11\%$ are applied to $h_{model}$, along with two other initial conditions smaller than the nominal value. These tests are identified by the following nomenclature: $h_1=5.00$ cm, $h_2=6.50$ cm, $h_3=7.84$ cm, $h_{model}=h_4=8.84$ cm, and $h_5=9.90$ cm.\\
Table \ref{tab:max_amplitude_water_depth} summarizes the forcing frequencies corresponding to the maximum wave amplitudes at each tested water depth. Notably, the frequency required for achieving maximum amplitude increases with greater water depths, indicating a stiffening behavior of the free-liquid surface. Additionally, with increasing $h$, there is a considerable rise in maximum wave amplitude. For instance, an $+11\%$ increase in water depth from $h_{model}=8.84$ cm results in a $+23\%$ increase in wave amplitude. Figure \ref{fig:effect_of_water_depth_P1} visually illustrates this increasing amplitude, consolidating insights from Table \ref{tab:max_amplitude_water_depth}.
\begin{table}[H]
\centering
\begin{tabular}{|l|l|l|l|l|l|}
\hline
& $h_1$ & $h_2$ & $h_3$ & $h_{model}$ & $h_5$ \\ \hline
$f_1[\mathrm{Hz}]$ & 0.85       & 0.98       & 0.98       & 1.04       & 1.08       \\ \hline
$\%\zeta_{max,h-model}$ & $-43.1\%$       & $-20.3\%$       & $-6.6\%$       & $\times$       & $+23.0\%$       \\ \hline
\end{tabular}
\caption{Forcing frequencies and maximum wave amplitudes when $a_y=15$ mm for the five water depths: $h_1=5.00$ cm, $h_2=6.50$ cm, $h_3=7.84$ cm, $h_4=8.84$ cm, $h_5=9.90$ cm.}
\label{tab:max_amplitude_water_depth}
\end{table}
Figure \ref{fig:effect_of_water_depth_cine} shows the maximum amplitude reached for each test series. Video captures confirm the previous observation on the wave amplitude. It is visually confirmed that higher water depths, in addition to the small model dimensions, may lead to sloshing processes that could hit the top plate.\\
Time histories in Figure \ref{fig:effect_of_water_depth_time_history} are taken in a two/three periods of time windows. The different forcing frequencies for each water depth test series are there outlined, as the signal from the probes are not synchronized and the PSD in Figure \ref{fig:effect_of_water_depth_PSD} exhibits peaks in frequency ratio equal to 1, but the $f_1$ value is suited of each of the water depth. Phase plane plots in Figure \ref{fig:effect_of_water_depth_phase_plot} exhibit nonlinear behaviour for all the water depth, but as they increase, the free-liquid surface motion becomes more violent and critical. The violence of the motion is confirmed also in the previous Figure \ref{fig:effect_of_water_depth_cine}.
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/effect_of_water_depth/Harmonic_base_excitation_1dof_ay15mm_P1.png}
    \caption{Maximum amplitude plot of P1 for horizontal harmonic base excitation in $Y$ direction and forcing amplitude $a_y=15$ mm in function of frequency ratio $f_y/f_1$; $f_1(h_1)=0.85$ Hz, $f_1(h_2)=0.98$ Hz, $f_1(h_3)=0.98$ Hz, $f_1(h_4)=1.04$ Hz, $f_1(h_5)=1.08$ Hz; greater wave amplitude are reached for higher water depths.}
    \label{fig:effect_of_water_depth_P1}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/effect_of_water_depth/water _depth_cine.png}
    \caption{Video captures for the $a_y=15$ mm test series at when the maximum amplitude is approached for $h_1$, $h_2$, $h_3$ and $h_4$: sloshing process reaches higher amplitude as the water depth increases; maximum amplitude are exhibited at increasing forcing frequencies.}
    \label{fig:effect_of_water_depth_cine}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.50\textwidth]{immagini/effect_of_water_depth/CONFRONTATION_waterdepth_Time_history.png}
    \caption{Dimensionless time histories (two periods window) at different water depth for the horizontal harmonic base excitation $a_y=15$ mm when maximum amplitudes are reached at forcing frequency: $f_1(h_1)=0.85$ Hz, $f_1(h_2)=0.98$ Hz, $f_1(h_3)=0.98$ Hz, $f_1(h_4)=1.04$ Hz, $f_1(h_5)=1.08$ Hz.}
    \label{fig:effect_of_water_depth_time_history}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/effect_of_water_depth/CONFRONTATION_waterdepth_PSD.png}
    \caption{Dimensionless PSD plots at different water depth for the horizontal harmonic base excitation $a_y=15$ mm when the forcing frequency is: $f_1(h_1)=0.85$ Hz, $f_1(h_2)=0.98$ Hz, $f_1(h_3)=0.98$ Hz, $f_1(h_4)=1.04$ Hz, $f_1(h_5)=1.08$ Hz; forcing frequency peaks are in 1 because the $f_1$ is suited for each of the performed water depth.}
    \label{fig:effect_of_water_depth_PSD}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.50\textwidth]{immagini/effect_of_water_depth/CONFRONTATION_waterdepth_phaseplot.png}
    \caption{Dimensionless phase plane plots at different water depth for the horizontal harmonic base excitation $a_y=15$ mm when the forcing frequency is: $f_1(h_1)=0.85$ Hz, $f_1(h_2)=0.98$ Hz, $f_1(h_3)=0.98$ Hz, $f_1(h_4)=1.04$ Hz, $f_1(h_5)=1.08$ Hz.}
    \label{fig:effect_of_water_depth_phase_plot}
\end{figure}
\clearpage
%===============================================================================
\section{Simulation of hazard in the primary pool}\label{parag:effect_of_hazard_harmonic}
The simulation of a major hazard in the actual demonstrator is performed removing the 3 steam generators from the model, which are the most damping elements. Water depth of similitude $h_{model}=8.84$ cm has been setup. The Figure \ref{fig:effect_of_damage_max_amplitude} represents the maximum amplitude plots $\zeta_{max}$ for the test series at $a_y=15$ mm and $a_y=30$ mm in the two cases of internal structure with and without the 3 steam generators. In the plot the wave amplitude is reported as a function of the dimensionless acceleration $\kappa_y$. The shift of the larger amplitude test series is due to the definition of the parameter.\\
The first observation is about the wave amplitudes. In both the two forcing amplitude cases, at equal $\kappa_y$ an higher maximum amplitude is reached from the liquid surface in the ALFRED's model. The damping role of the three steam generators is confirmed. Maximum amplitude is reached for similar forcing frequencies: for the $a_y=15$ mm test series is for $f_y=1.08$ Hz when the internal structure is complete and for $f_y=1.12$ Hz when the hazard is simulated; for the $a_y=30$ mm test series is for $f_y=1.08$ Hz in both the internal configurations.\\
The second observation is about the frequency range when the maximum wave amplitude is reached. When the internal structure is complete of the three SG, a plateau at the maximum amplitude is reached for a range of forcing frequencies $f_y=$[0.98,1.12] Hz. On the contrary when the configuration without the three SG is set-up, the wave-structure interaction is still present but less significant and smaller plateau are recorded.
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/effect_of_damage/max_amplitude_hazard_P1.png}
    \caption{Maximum amplitude plot as a function of the dimensionless acceleration $\kappa_y$ for the horizontal harmonic test series for the nominal configuration (3SG) and the hazardous one (no3SG): higher wave amplitudes are reached in the hazardous structural configuration at equal $\kappa_y$.}
    \label{fig:effect_of_damage_max_amplitude}
\end{figure}
\subsection{Forcing amplitude 15 mm at forcing frequency 1.12 Hz}
The impact of the internal structural arrangement, both with and without the inclusion of three steam generators, is examined at a frequency of $f_y=1.12$ Hz for the harmonic base excitation test series with an amplitude of $a_y=15$ mm. The chosen frequency corresponds to the maximum wave amplitude during the hazardous configuration.\\
The video footage in Figure \ref{fig:video_capture_3sg_effect_15mm} captures moments before, during, and after reaching peak wave amplitude in both configurations of the model. The attained maximum amplitude significantly differs between the configurations, with the less-damped configuration reaching 1.61 times that of the normal configuration. Additionally, observations note the intensity of sloshing motion, with occurrences of bubble formation and vigorous free-liquid surface movements evident in the hazardous configuration.
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/effect_of_damage/3sgeffect_a15mm_f1120Hz.png}
    \caption{Video captures for the $a_y=15$ mm test series at $f_y=1.12$ Hz with the 3SG (first row) and without them (second row) when the maximum amplitude is approached, reached and passed: in the hazardous configuration higher wave amplitude are reached and the bubbles suggest a more violent sloshing process.}
    \label{fig:video_capture_3sg_effect_15mm}
\end{figure}
The time history shown in Figure \ref{fig:effect_of_damage_f1120Hz_a15mm_3sgeffect} demonstrates higher peaks during the hazardous configuration setup. The interaction between waves and the structure appears more pronounced in the nominal configuration, as irregularities affect the trough, resulting in less smooth peaks compared to the other configuration. This phenomenon is also evident in the phase plane plots, where the orbits exhibit more prominences when the steam generators are installed. Although the PSD plots suffer from low resolution, a secondary peak at $f_y=2.12$ Hz is discernible alongside the primary peak at $f_y=0.98$ Hz and its multiples.
\begin{figure}[H]
    \centering
    \includegraphics[width=0.68\textwidth]{immagini/effect_of_damage/CONFRONTATION_3sgeffect_15mm_1120Hz_MOD.png}
    \caption{Dimensionless time history, phase plane plots and PSD plot for the $a_y=15$ mm test series st $f_y=1.12$ Hz for the two structural configuration: wave amplitude for the 3SG configuration is 1.61 times than for the other configuration; phase plane plots outline a stronger wave structure interaction in the nominal configuration; secondary peak at $f_y=2.12$ Hz}
    \label{fig:effect_of_damage_f1120Hz_a15mm_3sgeffect}
\end{figure}
\subsection{Forcing amplitude 30 mm at forcing frequency 1.08 Hz}
Test series at the greater forcing amplitude at $a_y=30$ mm enhance the behavior outlined for the previous series at $a_y=15$ mm. The maximum wave amplitude is reached at $f_y=0.98$ Hz, where the free surface in the hazardous configuration reaches 1.50 times the one for the nominal internal structure of ALFRED. Violent motion is experienced for both of the set-up because bubbles making events can be equally seen. In \S\ref{parag:effect_of_forcing_amplitude} is outlined how applying 30mm of external forcing amplitude at $f_y=0.98$ Hz is critical also for the ALFRED's complete internal structure.\\
The dimensionless time history confirms the differences in terms of maximum wave amplitude within the two examined structural configurations. Phase plane plots outline a stronger wave-structure interaction for the ALFRED's complete set-up. Unlike the $a_y=15$ mm test series, the PSD exhibits a secondary peak at $f_y=0.91$ Hz, despite the low resolution.
\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{immagini/effect_of_damage/3sgeffect_a30mm_f0980Hz.png}
    \caption{Video captures for the $a_y=30$ mm test series at $f_y=0.98$ Hz with the 3SG (first row) and without them (second row) when the maximum amplitude is approached, reached and passed: wave amplitude is higher for the hazardous configuration; the bubbles suggest a more violent sloshing process..}
    \label{fig:video_capture_3sg_effect_30mm}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.68\textwidth]{immagini/effect_of_damage/CONFRONTATION_3sgeffect_30mm_1080Hz_MOD.png}
    \caption{Dimensionless time history, phase plane plots and PSD plot for the $a_y=30$ mm test series st $f_y=0.98$ Hz for the two structural configuration: wave amplitude for the 3SG configuration is 1.50 times than for the other configuration; phase plane plots outline a stronger wave structure interaction in the nominal configuration; secondary peak at $f_y=0.91$ Hz}
    \label{fig:effect_of_damage_f1120Hz_a15mm_3sgeffect}
\end{figure}
\clearpage
%=========================================================================
\section{Scale effect assessment}\label{parag:full_scale_discussion_harmonic}
Main objective in the scale effect assessment for the horizontal harmonic base excitation is to have hints of the effect of the previous consideration, which are made on a simplified experimental case study. The observation could be of interest when a more realistic seismic signal is then performed (\S\ref{parag:full_scale_discussion_seismic}). Scale effect assessment is studied by means of five parameters: forcing amplitude, forcing frequency, wave amplitude, water depth and internal configuration.\\
Forcing amplitude and frequency are necessary to compute the external forcing acceleration, which is then compared to the ground acceleration by literature. From the similarity in \S\ref{parag:similarity}, it's know that:
\begin{equation}
    \begin{cases}
        A_{model}=0.054\times A_{ALFRED}\\
        f_{model}=4.29\times f_{ALFRED}
    \end{cases}
\end{equation}
where subscript $m$ stands for model and A for ALFRED. External acceleration for the horizontal harmonic base excitation are then as follows:
\begin{equation}
        \Ddot{a}_t=A_{ALFRED}\times(2\pi f_{ALFRED})^2=\frac{A_{model}}{0.054}\times\left(2\pi \frac{f_{model}}{4.29}\right)^2=\frac{\Ddot{a}_y}{0.95}
\end{equation}
where $\Ddot{a}_t$ stands for the ground acceleration needed to theoretically simulate the harmonic base excitation in a real event scenario.\\
Wave amplitude is computed because the Froude scaling is valid by the similarity studies \cite{ALFRED_Mockup}:
\begin{equation}\label{eq:Fr_ALFRED_model_wave_amplitude}
    Fr_m=Fr_A \to \frac{U^2_m}{gD_m}=\frac{U^2_A}{gD_A} \to U_A=\sqrt{\frac{D_A}{D_m}}U_m \to \left(\frac{d\zeta}{dt}\right)_A=\sqrt{\frac{D_A}{D_m}}\left(\frac{d\zeta}{dt}\right)_m
\end{equation}
where $D_a/D_m=$18.4 by the geometry model scaling \S\ref{parag:similarity}. The total uncertainty on the wave amplitude measured in the model is equal to 1.352 mm for the P1 probe and to 1.302 mm for the probe P2. In first approximation, the calculation of the uncertainty on ALFRED is simply computed as a wave amplitude value, so it's equal to $\simeq$5.80 mm for P1 and $\simeq$5.59 mm for P2 (\S\ref{parag:total_uncertainty}).\\
Table \ref{tab:full_scale_harmonic} presents the scaling-up of the previous parameters on some of the most interesting test series among the harmonic base excitation results.\\
The effect of the forcing amplitude is highlighted in rows of colour light-yellow: the maximum wave amplitude equal to 0.231 m is reached for a forcing amplitude of $A_{ALFRED}=0.03$m$/0.054=0.556$m, with a theoretical ground acceleration equal to $\Ddot{a}_t=1.081\,\mathrm{m/s^2}$.\\
The effect of water depth, starting as a convention from the diaphragm in the ALFRED's model because is the region where the majority of the sloshing processes takes place, is outline in the rows in colour light-green: keeping constants the forcing amplitude, the selected frequencies corresponds to the first sloshing ones, which are equal to the forcing frequencies for those relevant cases. Higher wave amplitude exhibits when the water depth is deeper, but an advantage is that the sloshing frequencies are higher than for the other ones. On the other hand, smaller amplitude are reached for more shallow water conditions, but the maximum amplitude is reached at lower frequencies. In terms of ground acceleration, the maximum value reported is for $h=9.90$ cm, and it's minor than the higher from the previous comparison for $A_{ALFRED}=0.556$ m.\\
The effect of the removal of the three steam generators is the most critical in terms of wave amplitude. Figure \ref{fig:effect_of_damage_max_amplitude} already exhibits higher wave amplitude in the model for equal $k_y$ dimensionless acceleration in y-direction. In the full scale discussion, supposing that a similar-sinusoidal excitation is experienced in real life events, the ground acceleration is greater than all the previous cases when the hazardous configuration is verified. In the worst-case scenario, wave amplitude reaches 0.357 m in a pool filled of molten lead for a deep equal to 1.627 m, which is the 22$\%$ of it.
\begin{table}[H]
\centering
{\fontsize{11}{10}\selectfont
\begin{tabular}{|cl|lll|lll|}
\hline
\multicolumn{2}{|l|}{}                                                                                                           & \multicolumn{3}{c|}{Model}                                                                                                                 & \multicolumn{3}{c|}{ALFRED}                                                                                                                \\ \hline
\multicolumn{1}{|l|}{\textbf{Test series}}                                                                        & \textbf{3SG} & \multicolumn{1}{l|}{\textbf{h {[}m{]}}}            & \multicolumn{1}{l|}{\textbf{$\Ddot{a}_y$ {[}$\mathrm{m/s^2}${]}}} & \textbf{$\zeta_m$ {[}m{]}} & \multicolumn{1}{l|}{\textbf{h\_A {[}m{]}}}         & \multicolumn{1}{l|}{\textbf{$\Ddot{a}_t$ {[}$\mathrm{m/s^2}${]}}} & \textbf{$\zeta_A$ {[}m{]}} \\ \hline
\rowcolor[HTML]{F5F5D2} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{F5F5D2}\begin{tabular}[c]{@{}c@{}}$a_y=0.005$m\\ $f_y=0.98$Hz\end{tabular}} & Yes          & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}0.088} & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}0.190}       & 0.015                      & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}1.627} & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}0.180}       & 0.065                      \\ \hline
\rowcolor[HTML]{F5F5D2} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{F5F5D2}\begin{tabular}[c]{@{}c@{}}$a_y=0.015$m\\ $f_y=0.98$Hz\end{tabular}} & Yes          & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}0.088} & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}0.569}       & 0.033                      & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}1.627} & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}0.540}       & 0.143                      \\ \hline
\rowcolor[HTML]{F5F5D2} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{F5F5D2}\begin{tabular}[c]{@{}c@{}}$a_y=0.030$m\\ $f_y=0.98$Hz\end{tabular}} & Yes          & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}0.088} & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}1.137}       & 0.054                      & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}1.627} & \multicolumn{1}{l|}{\cellcolor[HTML]{F5F5D2}1.081}       & 0.231                      \\ \hline
\rowcolor[HTML]{DFF5DF} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{DFF5DF}\begin{tabular}[c]{@{}c@{}}$a_y=0.015$m\\ $f_y=0.85$Hz\end{tabular}} & Yes          & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.050} & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.428}       & 0.020                      & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.920} & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.406}       & 0.087                      \\ \hline
\rowcolor[HTML]{DFF5DF} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{DFF5DF}\begin{tabular}[c]{@{}c@{}}$a_y=0.015$m\\ $f_y=0.98$Hz\end{tabular}} & Yes          & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.065} & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.569}       & 0.028                      & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}1.196} & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.540}       & 0.119                      \\ \hline
\rowcolor[HTML]{DFF5DF} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{DFF5DF}\begin{tabular}[c]{@{}c@{}}$a_y=0.015$m\\ $f_y=0.98$Hz\end{tabular}} & Yes          & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.078} & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.569}       & 0.032                      & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}1.443} & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.540}       & 0.139                      \\ \hline
\rowcolor[HTML]{DFF5DF} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{DFF5DF}\begin{tabular}[c]{@{}c@{}}$a_y=0.015$m\\ $f_y=1.04$Hz\end{tabular}} & Yes          & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.088} & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.640}       & 0.035                      & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}1.627} & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.608}       & 0.149                      \\ \hline
\rowcolor[HTML]{DFF5DF} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{DFF5DF}\begin{tabular}[c]{@{}c@{}}$a_y=0.015$m\\ $f_y=1.08$Hz\end{tabular}} & Yes          & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.099} & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.691}       & 0.043                      & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}1.822} & \multicolumn{1}{l|}{\cellcolor[HTML]{DFF5DF}0.656}       & 0.183                      \\ \hline
\rowcolor[HTML]{D5F7F5} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{D5F7F5}\begin{tabular}[c]{@{}c@{}}$a_y=0.015$m\\ $f_y=1.04$Hz\end{tabular}} & Yes          & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}0.088} & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}0.640}       & 0.035                      & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}1.627} & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}0.608}       & 0.149                      \\ \hline
\rowcolor[HTML]{D5F7F5} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{D5F7F5}\begin{tabular}[c]{@{}c@{}}$a_y=0.015$m\\ $f_y=1.04$Hz\end{tabular}} & No           & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}0.088} & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}0.640}       & 0.053                      & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}1.627} & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}0.608}       & 0.229                      \\ \hline
\rowcolor[HTML]{D5F7F5} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{D5F7F5}\begin{tabular}[c]{@{}c@{}}$a_y=0.030$m\\ $f_y=1.12$Hz\end{tabular}} & Yes          & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}0.088} & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}1.486}       & 0.055                      & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}1.627} & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}1.411}       & 0.238                      \\ \hline
\rowcolor[HTML]{D5F7F5} 
\multicolumn{1}{|c|}{\cellcolor[HTML]{D5F7F5}\begin{tabular}[c]{@{}c@{}}$a_y=0.030$m\\ $f_y=1.12$Hz\end{tabular}} & No           & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}0.088} & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}1.486}       & 0.083                      & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}1.627} & \multicolumn{1}{l|}{\cellcolor[HTML]{D5F7F5}1.411}       & 0.357                      \\ \hline
\end{tabular}
}
\caption{Scale effect on some selected test series in the horizontal harmonic base excitation with different forcing amplitudes, frequencies, water depths and internal configurations; the uncertainty on the displacements for the wave amplitude in model from capacitance probe P1 is equal to $\simeq$0.0058 m (\S\ref{parag:total_uncertainty}).}\label{tab:full_scale_harmonic}
\end{table}
If the theoretical ground acceleration peaks in those horizontal harmonic base excitation are taken into account, the differences in numerical value from a peak in a real scenario are minimum in some cases. The order of magnitude for $\Ddot{a}_t$ is of 1 $\mathrm{m/s^2}$ in Mioveni \cite{ALFRED_Mockup}, Romania, the building site of ALFRED. For example, the conventional peak ground acceleration of Turin is $\Ddot{a}_t\leq1.4$ because is a zone 3 in the \textit{Protezione Civile} updated archive \cite{Protezione_civile}. In the end, the mean of the peak ground acceleration of Belgium \cite{Earthquake_ground_acceleration} is reported and is quite low: $\Ddot{a}_t=0.40\,\mathrm{m/s^2}$.