## 6 Experimental Results and Comparison to Analysis
### Comparisons to Analysis Results  
Figure 9 shows a comparison of the experimental results with the simple vibration analyses in terms of working frequency and piston strokes over charge pressures for cases 1-5. Both ends of the three measurements represent the lower and upper limits of operable charge pressures; out of the range, the test engine did not operate. Overall, the changes in PP mass and spring stiffness lead to large variations in the operable charge pressures from 6 bar to 31 bar. Both the experiments and analyses show that a smaller stiffness and higher mass of PP yield higher operable charge pressures. In addition, at higher levels of charge pressures, the operable pressure range tends to widen. The engine mostly operates near the predicted pressure, i.e., the crossing point of natural frequency curves of PP and DP. The experiment frequencies are also approximately 3 Hz lower than the predicted values with an error around 10%. The working frequencies are relatively constant for all the cases since the DP mass and stiffness are fixed. Figure 10 shows a comparison of the experimental results with the simple vibration analyses in terms of working frequency and piston strokes over charge pressures for cases 1-5. The upper and lower parts in each figure compare the measured piston strokes and working frequencies with the real and imaginary part values, respectively. Figure 9: Comparison of experimental and the simple vibration analysis results for cases 1–5. Predicted PP and DP damped natural frequencies with increasing charge pressure of 0–50 bar (prediction) and the operable pressure ranges, the frequency, and the two piston strokes (experiment) are given. Especially, the experiment case 3 shows the measured operable range of charge pressures from 13 bar to 17 bar, and the predicted root-zero pressure of 11 bar and root-switch pressure of 14 bar are near the measured pressure range. The measured working frequencies around 33 Hz are close to the imaginary part line around 36 Hz. In detail, the measured operable ranges of charge pressures locate around or slightly above the root-switch pressures, instead of the root-zero pressures where the real part becomes positive. The piston amplitudes show a tendency to increase with the real part values. The predicted operating frequencies are \(\sim\)3 Hz higher than the measurements with an error less than 10%. The changes in the experiment factors of PP mass and PP spring stiffness lead to large variations in the operable charge pressures from 6 bar to 31 bar. The increases in the PP mass shift the operable charge pressures to the right, as in cases 1 and 2 and cases 4 and 5, while the PP stiffness increases result in the opposite, shown in cases 1 to 4 and 2 to 5. Under the simultaneous decreases in the PP spring stiffness and increases in the PP mass, as in cases of 4, 3, and 2, the real part values increase with higher operable charge pressure. These results imply a proportional relationship of the charge pressure to the strength of resonance in piston motions. Note that higher real part values lead to increase in the speed of divergence of piston motions. Figure 11 shows the comparison of the experiments with the two dynamic model predictions in terms of the mid-point operable charge pressure and the operating frequency for all the cases 1-5. The root locus predictions employ two criterion of real part zero and root switch for the FPSE operability. Regarding the operable charge pressure, both the simple vibration analyses and the root locus predictions based on the root switch criterion agree very well with the experiments with an average error \(\sim\)10%. On the other hand, the criterion of real part zero does not show reasonable and consistent prediction accuracies with an average error \(\sim\)40%. For the working frequencies, all the models show good agreements with the experiments with a consistent error around 10%. Tables 4 and 5 summarize the comparison results. Figure 10: Comparison of root locus analyses with experiments for cases 1–5.