## 2 Vibration Isolation System of the DCM  
The working principle of a DCM involves using Bragg diffraction to extract monochromatic light from the continuous spectrum of synchrotron radiation that is tailored for experimental needs. With the increasing brightness of third- and fourth-generation synchrotron light sources, the power density on the crystal surface of monochromators can reach several tens of watts per square millimeter. Thermal loads can cause local protrusions on the crystal, overall warping, and changes in lattice constants, leading to decreased surface accuracy of the crystal. Therefore, it is essential to incorporate a cooling system to ensure the thermal stability of the crystals. Figure 2: (**a**) The compression buckling process commonly used in other similar studies; (**b,c**) the compression buckling process of the structure studied in this paper. The DCM cooling system creates vibrational excitation of the crystal; the root mean square angular vibration amplitude of the crystals can reach several hundred nrad. However, fourth-generation synchrotron light sources require the angular stability of crystals to be about 15-30 nrad, i.e., the relative angular displacement between the first and second crystals, as shown in Figure 3, should not exceed 15-30 nrad. The traditional DCM to solve the problem of crystal angular stability is to obtain a higher angular displacement resolution to ensure the precision of the Bragg angle through the high rigidity mechanical shaft and stepping motor or piezoelectric drive. Therefore, the traditional DCM mainly through the optimization of the mechanical structure to reduce the crystal amplitude to ensure the angular stability; however, through the optimization of the mechanical structure, it is difficult to meet the stability requirements of the fourth generation of light sources [38]. In order to meet the requirements of the fourth-generation light source, scholars proposed a new design method: using a laser interferometer to measure the crystal angular displacement and completing the active control by driving the crystal rotation to keep the crystal angle stable through the voice coil motor, as show in Figure 4. Active control methods enable the DCM to meet the 15-30 nrad angular stability requirement. The frame also needs to have stronger vibration isolation to prevent vibration excitation (external vibration) from the cooling system and peripheral facilities from being transmitted to the active regulating mechanism (core element) to affect its regulating accuracy. Key parameters affecting vibration isolation performance include the damping ratio and natural frequency. When the external excitation amplitude \(A\) and the frequency ratio \(\omega/\omega_{n}\) (where \(\omega\) is excitation frequency and \(\omega_{n}\) is natural frequency) exceed \(\sqrt{2}\), increasing the damping ratio \(\xi\) increases the steady-state response amplitude \(X\) of the system, but the transmissibility \(X/A\) remains less than one. Conversely, when the frequency ratio is less than \(\sqrt{2}\), increasing the damping ratio reduces the steady-state response amplitude, while  
Figure 4: DCM structure, the first crystal is mounted on the basic frame, the second crystal active adjustment mechanism is mounted on the peripheral frame, and the basic frame and the periphery frame are connected in the middle using a vibration isolation device. Figure 3: (**a**) Bragg diffraction principle; (**b**) principle of DCM operation. the transmissibility remains greater than one, as show in Figure 5. The natural frequency of the isolation system is primarily influenced by stiffness \(k\) and mass \(m\), as shown in Equation (1).