## 5 Analyses on Dynamic Behavior of Test FPSE  
### Stiffness and Damping Coefficients of FPSE  
The dynamics of FPSEs are highly associated with the nature of the pressure generation mechanism during the thermodynamic cycle generated by piston motions. In particular, the pressure stiffness relating the pressure generation with the piston motion, defined from Equation (7), has a high influence on the dynamic characteristics of FPSEs.  
Figure 5a shows the predicted system stiffness coefficients per mass with an increasing charge pressure of 0-50 bar for the reference case 3--The PP mass of 3.6 kg and PP stiffness of 84 kN/m. The stiffness coefficients are divided by the PP and DP masses, respectively. The DP mass is 1.64 kg and DP stiffness is 84 kN/m for all the cases. The first and second subscripts of the stiffness coefficients mean the forced piston and the displaced piston, respectively.  
Basically, the pressure force with the positive stiffness acts in the opposite direction to the displacement and generates a restoring force toward an equilibrium position, while the negative stiffness is the opposite and causes a diverging force.  
First of all, the stiffness element is composed of the flexure spring stiffness with a constant value and the pressure stiffness variable depending on pressure and temperature conditions; particularly, the cross-coupling coefficients have only the pressure stiffness.  
Figure 5: Predicted (a) system stiffness and (b) damping coefficients with charge pressure increasing from 0 to 50 bar for case 3.  
Under the increases in charge pressures, \(K_{pp}\) increases greatly with positive pressure stiffness from the PP spring stiffness, which is the value at zero charge pressure; while \(K_{dd}\) decreases slowly due to the negative pressure stiffness from the DP spring stiffness, and maintains positive. The greater slope of \(K_{pp}\) results from the larger cross-sectional area of the PP than the DP rod, which are the pressure-acting areas. These increasing and decreasing tendencies are also observed in \(K_{dp}\) and \(K_{pd}\), respectively. In particular, \(K_{pd}\) has negative values over the charge pressures.  
In detail, when the PP moves up, the working space of the engine is compressed to a higher pressure, so that the pressure force pushes both the PP and DP downwards, acting as a restoring force thus having a positive stiffness, as in \(K_{pp}\) and \(K_{dp}\). On the other hand, when the DP moves upwards, the working gas moves from the expansion space to the compression space via the heat exchanger part, and is cooled, so that the working space has a lower pressure. Therefore, the bounce space pressure pushes up both the DP and PP, which then acts as a diverging force with a negative stiffness, shown as the decreasing tendencies in \(K_{dd}\) and \(K_{pd}\).  
Eventually, the \(K_{pd}\) with the negative values leads to the divergent motion of PP due to the DP-induced thermodynamic processes. The \(K_{pp}\) and \(K_{dd}\) are still positive owing to the flexure spring stiffness, keep providing restoring forces and maintaining the piston vibrations. The effect of the pressure stiffness strengthens with increases in the charge pressure.  
On the other hand, Figure 5b shows the system damping coefficients per mass, where the damping coefficients are divided by the PP and DP mass, respectively. They are assumed constant over the charge pressures, and only the direct terms are considered. The electromagnetic load damping of linear alternator is approximately 2.5 times that of the heat exchanger damping.  
Note that the damping load on the PP by the alternator limits the piston amplitudes by extracting the dynamic power to the load device, while the damping load on DP dissipates energy as a loss in the heat exchangers.