## 3 Results and Discussion
### Effect of Liquid Nitrogen Pulsation on DCM Stability
#### 3.2.1 Frequency Impact  
Under the pulsating liquid nitrogen excitation, the vibration of the DCM is considered as forced vibration. Figures 10a and 11a illustrate the time-domain curves of the DCM angular vibration under pulsating liquid nitrogen excitation. It can be observed that the crystal angular displacement exhibits harmonic response characteristics under the  
Figure 9: Root mean square of pressure amplitude at the monitoring point. influence of pulsating excitation. This harmonic response is jointly induced by the structural nonlinearity within the system and the harmonic excitation generated by liquid nitrogen. The internal system includes elastic elements such as flexible hinges, springs for balancing gravity, and external pipelines. Since the DCM has the minimum stiffness in the pitch direction, its vibration mainly manifests as the swinging of the crystal around the hinge center, with the trajectories of various points approximating a circular arc in the plane. Usually, the restoring force of elastic elements satisfies a proportional relationship with displacement only along the designed force line. However, this linear relationship is easily disrupted when the action point moves in space. Furthermore, the amplitude of pressure pulsation along the fluid path constantly changes, causing pulsating pressure differences at the inlet and outlet of the U-shaped pipe at the second crystal, triggering z direction vibrations of the second crystal. The vibration induces distortion in the flexible hinge. The continuously changing distortion angle causes the hinge stiffness to vary, thereby leading to changes in the elastic coefficient of the flexible hinge in the pitch direction. Figure 11: The angular vibration response of the DCM under the excitation of frequency f2 (the abscissa is the time in units of pulsation periods of liquid nitrogen). (**a**) Time domain feature. (**b**) Frequency domain feature. Figure 10: The angular vibration response of the DCM under the influence of the excitation of frequency f1 (the abscissa is the time in units of pulsation periods of liquid nitrogen, and all short horizontal lines in the figure represent “minus sign”). (**a**) Time domain feature. (**b**) Frequency domain feature. Performing a Fourier transform on the time-domain curves of the crystal vibration response yields Figures 10b and 11b. From these two spectra, it can be observed that the response frequency of the DCM forced vibration is composed of the natural frequency of the DCM and the coolant excitation frequency. Considering the precision mechanical characteristics of the DCM, the amplitudes are generally small, ranging from micrometers to even nanometers. Therefore, linearizing the analysis of such ultra-low amplitude nonlinear vibrations in a specific direction is often possible. In the z direction, by converting the crystal's displacement in the heading direction into an angle, this research can obtain the angular variation of the second crystal. The following linear harmonic excitation vibration response equation [22] exists:  
\[x(t)=XA\cos(\omega t-\phi) \tag{15}\]  
\[X=\frac{1}{\sqrt{\left[1-\left(\frac{\omega}{\omega_{n}}\right)^{2}\right]^{2} +\left(2\xi\frac{\omega}{\omega_{n}}\right)^{2}}} \tag{16}\]  
\[\phi=\tan^{-1}\frac{2\xi\frac{\omega}{\omega_{n}}}{1-\left(\frac{\omega}{ \omega_{n}}\right)^{2}} \tag{17}\]  
\(x(t)\) is the crystal displacement at the time, \(X\) is the amplification factor of the harmonic response, \(A\) is the excitation amplitude, \(\omega\) is the excitation frequency, \(\phi\) is the phase difference of the vibration response, \(\omega_{n}\) is the natural frequency of the structure, and \(\xi\) is the viscous damping coefficient. As shown in Figures 10 and 11, in this forced vibration system, the pulsation frequency of the coolant will significantly excite the nearby natural frequency of the DCM. In the low-frequency range, the first natural and excitation frequencies are the DCM vibration's master frequencies.