## 7 Discussions on FPSE Resonance with Phase Angle  
The resonant operation of FPSEs is discussed with the DP-PP phase angle using the force vector diagram and the root locus analysis. Figure 12 shows the force vector diagrams of the DP and PP, which are drawn from the linearized EOMs in Equation (7). The force vector diagram is composed of the inertial force, damping force, stiffness force, and pressure force, which is frequently used in the FPSE dynamics analyses [26, 27, 28]. In order to maintain a finite amplitude vibration, called also as limit cycle, the pressure force overcomes the damping force: the flow resistance of heat exchanger part for DP and the external load damping of linear alternator for PP. In addition, the inertial and stiffness forces must balance for the mechanical kinetic and potential energy conservation. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{**Experiment**} & \multicolumn{2}{c}{**Simple Vibration**} & \multicolumn{2}{c}{**Root Locus (Root Switch)**} & \multicolumn{2}{c}{**Root Locus (Real = 0)**} \\ \cline{2-9}  & **Pressure** & **Pressure** & **Error** & **Pressure** & **Error** & **Pressure** & **Error** \\  & **(bar)** & **(bar)** & **(bar)** & **(bar)** & **(bar)** & **(bar)** & **(bar)** & **(bar)** \\ \hline
1 & 22.5 & 19.8 & 12.1\% & 19.5 & 13.4\% & 12.1 & 46.3\% \\
2 & 28.0 & 24.3 & 5.8\% & 23.4 & 16.5\% & 15.0 & 46.5\% \\
3 & 15.0 & 15.8 & 5.6\% & 14.4 & 3.8\% & 10.9 & 27.2\% \\
4 & 7.1 & 7.2 & 2.0\% & 6.8 & 3.6\% & 12.3 & 74.3\% \\
5 & 10.9 & 11.5 & 5.1\% & 12.0 & 9.6\% & 10.0 & 8.6\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of experiments with two dynamic model predictions for all the cases 1–5: mid-point operable charge pressure. Figure 11: Comparison of experiments with two dynamic model predictions for all the cases 1–5: (**a**) mid-point operable charge pressure and (**b**) operating frequency. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{**Experiment**} & \multicolumn{2}{c}{**Simple Vibration**} & \multicolumn{2}{c}{**Root Locus (Root Switch)**} & \multicolumn{2}{c}{**Root Locus (Real = 0)**} \\ \cline{2-7}  & **Frequency** & **Frequency** & **Error** & **Frequency** & **Error** & **Frequency** & **Error** \\  & **(Hz)** & **(Hz)** & **(Hz)** & **(Hz)** & **(Hz)** & **(Hz)** & **(Hz)** \\ \hline
1 & 31.3 & 34.9 & 11.3\% & 35.0 & 11.7\% & 34.8 & 11.2\% \\
2 & 30.8 & 34.7 & 12.5\% & 34.7 & 12.6\% & 34.2 & 11.0\% \\
3 Interestingly, the pressure force of each piston is represented by the cross-coupling pressure stiffness force caused by the other piston displacement. Therefore, the phase angle between the excitation force and piston displacement becomes identical to the DP-PP phase angle. For example, in the case of DP with a phase angle greater than 90\({}^{\circ}\), the pressure force can be divided into the vertical component in the opposite direction to the damping force and the horizontal component in the same direction of the inertial force. Therefore, the vertical component is used to overcome the damping force, while the horizontal component assists the inertial force to balance the stiffness force. If the phase angle is less than 90\({}^{\circ}\), the horizontal component assists the stiffness force. In particular, when the phase angle is 90\({}^{\circ}\), the pressure force is used solely to overcome damping load without being used for the balance of inertial and stiffness forces. In the same way, as the charge pressure increases in the PP vector diagram, the direct stiffness force of PP increases (see Figure 5) and thus the phase angle decreases. Consequently, the DP-PP phase angle becomes 90\({}^{\circ}\), resulting in a resonant vibration condition. Note that the \(K_{pd}\) has a negative value and the DP motion is in the opposite direction to the pressure force. Therefore, the DP motion is ahead of the PP by the phase angle, which is the same as that of the DP vector diagram.