## 8 Conclusions  
This paper presents a linear dynamic analysis of beta-type FPSEs on the operable charge pressure and working frequency along with experimental verifications.  
The equations of motion of a FPSE are formulated as a 2-DOF vibration system of the PP and DP based on the state equation of ideal gas and the isothermal Stirling cycle model. Each piston is modeled to have a flexure spring and a direct damping. The internal working pressure, which couples the PP and DP motions, are linearized and consequently becomes pressure stiffness terms in the stiffness matrix.  
The dynamic models of FPSE that we considered are (1) the 1-DOF simple vibration model of each piston and (2) the 2-DOF root locus model of coupled pistons. The simple vibration model assumes that the engine operates when two natural frequencies of each piston are equal. The root locus method predicts the operating point based on root trajectories in the state space.  
We developed a test FPSE for the verification of the dynamic models, and conducted a series of experiments to measure the dynamics behavior of PP and DP under varying charge pressures for various masses and stiffness of the PP.  
As a result, an operable range of charge pressure existed during operations of the test FPSE under varying charge pressures. The increases in the PP mass led to increases in the operable charge pressure, while the PP stiffness yielded the opposite results. The working frequencies are not affected by those changes.  
Both the prediction models showed good agreements with experimental results. The 1-DOF vibration model was found to be simple and effective for predicting the operating frequency and charge pressure of FPSE. Likewise, the root locus method showed high predictions with the operation criterion of the DP-PP phase angle of 90\({}^{\circ}\), where the root switch occurs. It was found that DP tends to determine the operating frequency and PP does the operable charge pressure.  
In addition, the FPSE was confirmed to operate in resonant oscillations when the DP-PP phase angle is 90\({}^{\circ}\), based on the analysis of the force vector diagram of two pistons. In the resonant operation, the cross-coupling effects of piston motions are maximized, and the pressure force is used solely to overcome damping loads without being used to balance the other inertial and stiffness forces.  
Finally, the dynamic model developed in this paper can be used in terms of FPSE design and testing by predicting the operating frequency and charge pressure with the FPSE design parameters including the piston mass and spring stiffness, so as to satisfy design specifications or to setup testing conditions. In particular, it can be very helpful in testing with variable piston mass and spring stiffness for engine development and optimization.  
Conceptualization, K.S.; methodology, K.S.; software, D.-J.K.; validation, D.-J.K. and K.S.; formal analysis, D.-J.K.; investigation, D.-J.K.; writing--original draft preparation, D.-J.K.; writing--review and editing, K.S.; supervision, K.S. Both authors have read and agreed to the published version of the manuscript.  
This study was supported by the research program funded by SeoulTech (Seoul National University of Science and Technology).  
The authors declare no conflict of interest.