## 4 Finite Element Analysis
### Vibration Isolation Performance Analysis  
The primary purpose of applying the quasi-zero stiffness system to the DCM is to isolate vibrations from the crystal cooling system, preventing vibration interference with the core components of the DCM. In evaluating vibration isolation performance, key metrics include not only transmissibility but also 3\(\sigma\) displacement. In various practical application scenarios, vibration excitations exhibit randomness, with their amplitudes generally following a Gaussian distribution where higher amplitudes occur less frequently. Figure 16: (**a**) Finite element simulation results for the 2D stress distribution of an LS with a \(l_{0}\) of 60, \(b\) of 20, \(h\) of 0.3, \(u_{x}\) of 0.3, and \(\delta_{1}=1\); (**b**) comparison between 1D stress distribution calculated for different \(l_{0}\) and the finite element simulation results for an LS with a \(h\) of 0.1, \(u_{x}\) of 0.1, and \(\delta_{1}=1\); (**c**) comparison between 1D stress distribution calculated for different \(l_{0}\) and the finite element simulation results for a leaf spring with a \(h\) of 0.3, \(u_{x}\) of 0.3, and \(\delta_{1}=1\). The 3\(\sigma\) displacement refers to the maximum amplitude within the 3\(\sigma\) confidence interval of random vibration statistical data, with a confidence level of 99.737%. This implies that the probability of random vibration amplitudes being lower than the 3\(\sigma\) displacement is 99.737%. In this section, using ANSYS Fluent, the vibration excitation of the cooling system under different flow pulsations is computed as the input to analyze the vibration isolation performance of the quasi-zero stiffness system. The finite element model shown in Figure 13 utilizes Static Structure, Modal, and Random Vibration solvers. Slightly differently from the previous chapters, firstly, the displacement boundary condition of the peripheral frame is removed from the simulation here because there is no need to examine the negative stiffness value for random vibration; secondly, spring 1 is removed, and the stiffness coefficient of spring 2 is modified to 25.3, which is more in line with the practical application scenarios. Lastly, to achieve quasi-zero stiffness, the dimensions of the LS are modified based on Equation (30) to a \(h=0.3\), \(l_{0}=70\), \(b=81\), and \(u_{x}=0.4\). The finite element simulation steps include compressing the LS through slider displacement, solving for the first 10 modal frequencies, and applying PSD excitation along the y-axis direction on the crystal surface for final solution completion. The calculation formula for the PSD response (34) is as follows, where \(S_{o}\) represents the PSD response, \(S_{i}\) represents the PSD excitation, and \(H(\omega)\) represents the frequency response function, which here equals the transmissibility function calculated by Equation (2). \[S_{o}=(H(\omega))^{2}S_{i} \tag{34}\]  
As shown in Figure 17a, the overall trend of the FEM and analytical curves is relatively consistent, significant vibration isolation at an excitation frequency of 1 Hz, and the PSD curve of the decibel is 10 lg. In the random vibration response, it can be seen that the peak of the natural frequency of the analytical and FEM curves at the very-low frequency has a small offset, which is caused by the computational error. The results at 100 Hz after the high-frequency region of the FEM are different from the analytical peak of the PSD. The reason for this is that the analytical calculated response PSD does not consider the frame models, while the finite element simulation uses the modal superposition method to comprehensively calculate the effects of each order of modals, so the analytical value is bound to have a small error in the integral calculation of the 3\(\sigma\) displacement. The figure shows the results of the integral calculation using the PSD response output from the solver and the theoretically calculated PSD response; the displacements are very close and both are controlled on the nanometer scale.