## 3 Results and Discussion
### Effect of Liquid Nitrogen Pulsation on DCM Stability
#### 3.2.1 Frequency Impact
Under high-frequency excitation, the first mode mainly dominates the DCM vibration. When the excitation frequency is 40 Hz, this frequency is close to the first mode of the structure, resulting in a significant amplification factor of the vibration, and the first mode is significantly excited. Currently, the response frequency of the DCM is the frequency of the excitation and the first mode. When the excitation frequency is 60 Hz, the DCM vibration amplitude is minimal, and this excitation frequency is almost equidistant from the first mode and the second mode frequencies. In this case, the response of the DCM is still dominated by the first mode and the excitation frequency. However, the amplitude of the first mode is lower than the operating condition at 40 Hz, since when \(\omega/\omega_{n}>X_{max}\), the amplification factor X decreases faster than \(\omega/\omega_{n}<X_{max}\) (\(X_{max}\) is the amplification factor at resonance, the maximum gain of amplitude). When the excitation frequency is 80 Hz, the DCM frequency is still the first mode and the excitation frequency. This is mainly due to the significant increase in excitation amplitude caused by the increase in excitation frequency. Although the amplification factor X decreases slightly at this time, the first mode frequency remains the master vibration frequency due to increased excitation amplitude. The second mode of the system is manifested as the U-shaped pipe outlet section swinging in the z direction, with a small contribution to the DCM's vibration in the pitch direction. In addition, at the end of the outlet pipe, the pressure pulsation amplitude is minimal. Although the amplification factor of the second mode is significant at this frequency, the peak at 88.2 Hz in the frequency amplitude diagram is not significant. When the excitation frequency is 100 Hz, it is close to the third mode frequency (99.4 Hz) of the DCM, making the system resonant. At the same time, with the increase in the amplitude of the pipeline pressure pulsation, the third mode of the DCM is significantly excited. At this time, the pipe at the entrance of the U-shaped circuit undergoes significant swinging, triggering a significant vibration of the DCM in the pitch direction. In the high-frequency range of excitation, the first mode is the dominant manifestation of the DCM vibration response. The DCM has minor stiffness in the pitch direction, and it only exhibits rotation of the crystal around the flexible hinge in the first mode. This modal vibration contributes maximally to the DCM's pitch-directional motion. In addition, the system's natural frequency does not match the high-frequency excitation (200 Hz, 300 Hz, 400 Hz). With the significant increase in the excitation amplitude, the amplitude of the first mode response of the DCM gradually increases. Specifically, the increase in the excitation frequency causes a larger amplitude of the DCM response, indicating that the impact of the pressure amplitude caused by the rapid conversion of energy is stronger than the impact of the amplification factor in the high-frequency excitation region. The above analysis shows that the DCM's vibration response is mainly influenced by the pressure pulsation caused by the velocity pulsation of liquid nitrogen, and the pressure pulsation is also influenced by the pipeline resistance. The vibration response amplitude of the DCM is positively correlated with the amplitude of pressure pulsation. In the high-frequency range, the impact of excitation amplitude on vibration is more pronounced than the amplification factor changes caused by frequency, suggesting that the pressure amplitude variation caused by fluid frequency has a more significant impact on vibration in the high-frequency region.