## 3 Parameter Analyses
### Kinematic Relationship and Water Surface Constraint  
The intersecting surface of the cylindrical paraboloid reflector is simplified into a parabola located in 2-dimensional Cartesian coordinate system XOY of Figure 2, the function of which is defined as \(pz=x^{2}+q\). It is assumed that the origin O is set up at the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Parameters** & **Gear** & **Internal Ring Gear** & **Unit** \\ \hline Module \(m\) & 2 & 2 & mm \\ Number of teeth \(z\) & 20 & 240 & \(\backslash\) \\ Pitch diameter \(d\) & 40 & 480 & mm \\ Addendum circle diameter \(d_{a}\) & 44 & 476 & mm \\ Dedendum circle diameter \(d_{f}\) & 35 & 485 & mm \\ Base circle diameter \(d_{b}\) & 44 & 451 & mm \\ Tooth breadth \(b\) & 20 & 150 & mm \\ Pressure angle \(\alpha\) & 20 & 20 & deg \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters of gear and ring. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Design Parameter** & **Value** & **Unit** \\ \hline Reflector aperture \(D\) & 500 & mm \\ Reflector radius \(R\) & 250 & mm \\ Longitudinal length \(L\) & 150 & mm \\ Shell thickness \(t\) & 5 & mm \\ Track width \(L_{1}\) & 150 & mm \\ Track thickness \(t_{1}\) & 10 & mm \\ Back frame density \(\rho\) & \(1.2\times 10^{3}\) & kg/m\({}^{3}\) \\ Water density \(\rho_{w}\) & \(1.0\times 10^{3}\) & kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial values of design parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Control Variable** & **Initial Value** & **Range** \\ \hline \(a=t/R\) & 0.02 & [0.02, 0.10] \\ \(b=L/R\) & 0.6 & [0.4, 2.0] \\ \(c=m_{s}/m\) & 0.3 & [0.3, 1.0) \\ \(n=\rho/\rho_{w}\) & 1.2 & [1, 10] \\ \(a_{1}=t_{1}/R\) & 0.02 & [0.02, 0.10] \\ \(b_{1}=L_{1}/R\) & 0.6 & [0.4, 2.0] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Ranges of non-dimensional control variables. circle center of the semi-cylindrical shell. The Z axis is perpendicular to this plane by the right-hand rule. The trolley is simplified as a moving mass. Since the FCRA is placed directly on the water, the immersion depth \(h\) into the water's surface is expressed by the central angle \(\theta\) of the arc under the water's surface (Figure 2) as  
\[h=R\bigg{(}1-\cos\frac{\theta}{2}\bigg{)} \tag{1}\]  
As simplified in the XOY plane, the reflector motion is indicated by the elevation angle \(\beta\) between the focal diameter O'F and axis X, where points O' and F are the vertex and focus of the reflect parabola, respectively, in Figure 2. The trolley is located at the center of the track while the focal diameter O'F of the reflector in the upright state coincides with axis Y of the global coordinate system in Figure 2.  
To declare the relationship between the moving mass position and the reflector elevation angle, the FCRA motion process only considering the rigid body kinematics is divided into two stages as shown in Figure 3, namely moving mass motion and structure rotation. Assuming that the back frame structure is immobile at the first stage in Figure 3a, only the motion of moving mass relative to the track is taken into account. The relative motion angle between the moving mass and the track is defined as \(\gamma\). At the second stage, the back frame structure is released to rotate around the central axis until there is an equilibrium for the whole structure. The elevation angle \(\beta\) of the reflector and the position of the moving mass are shown in Figure 3b. There is a torque equilibrium equation as \(m_{c}R_{c}\mathrm{sin}\varphi=m_{s}R_{s}\mathrm{sin}\alpha\). Substituting the non-dimensional variables, it is expressed as Appendix A Equation (A1), where \(\varphi\) is the angle between axis Y and the focus diameter O'F in the equilibrium. \(\alpha\) is the angle between the equilibrium position and initial position of the moving mass. The left and the right terms are the torque generated by the gravity of the back frame and the moving mass, respectively.