Precise pressure core motion, including translation and rotation, is the basis and core part of the Analysis and Transfer System of Natural Gas Hydrate Pressure Core, which is crucial to nondestructive analyses, core cutting, and transfer. This paper mainly proposes a driving device, whereby a pressure core, up to 3 m long, can be transferred from pressure core drilling tools to proceed to nondestructive analyses and transferring the cores into other chambers. The lead screw is one of the most important components of this driving device. Therefore, the modal analyses of the lead screw are performed, which can help researchers to analyze the stability of this device. The analyzed data shows that the different positions of the slider have a great impact on the natural frequency of the lead screw. Furthermore, the lead screw with a support slider has a larger natural frequency than that without a support slider. According to data analysis, we can derive that the device with the support slider has a much larger rigidity, which can contribute to the stability of the device. To verify the feasibility of this device, the deformation of the lead screw was tested by the Micro-Electro-Mechanical Systems (MEMS) accelerometer array. Experimental results show that the deformation of the lead screw with the support slider is much less than that without the support slider.