## 1 Introduction  
Pneumatic down-the-hole (DTH) hammer drilling technology has been widely used in the fields of thermal storage mining and geological exploration [1, 2]. Its principle is that the piston inside the DTH hammer is mainly driven by compressed air to carry out high-speed reciprocating motion to break the rock [3], which has the advantages of high drilling efficiency and small pollution to the formation compared with traditional rotary drilling [4, 5]. In the process of pneumatic DTH hammer drilling, the near-bit measurement while drilling (MWD) is crucial. It is a device that is used to monitor various important drilling parameters relating to the near drill bit in real time. By accurately measuring data, the real-time drilling status can be monitored, and the drilling efficiency can be optimized. However, the vibration at the drill bit during pneumatic DTH hammer drilling is as high as 40 g, which poses a serious challenge to the performance of the near-bit MWD. The existing MWD tools are mainly designed for oil drilling and geological drilling. Its vibration level is only 20 g. Adapting to the high-intensity and low-frequency vibrations in pneumatic DTH hammer drilling is difficult. Therefore, for the special vibration environment of pneumatic DTH hammer drilling, it is important to research and develop a new type of anti-vibration MWD to improve drilling efficiency and ensure operational safety. Since the vibration response of a circuit unit is caused by the vibration of the drill string near the drill bit, studying the vibration characteristics of the drill string is crucial for establishing a vibration model. Many scholars have carried out research on the vibration mechanics and vibration control of drill strings. For example, [PERSON] [6] used the finite difference method to analyze the effect of damping, mass, and other factors on vibration by simplifying the drilling system. [PERSON] [7] and [PERSON] et al. [8] focused on the vibration dissipation process of a drill string by using the linear elasticity theory and experimental methods. They pointed out that damping plays an important role in the vibration of a drill string. [PERSON] et al. [9; 10] investigated the effect of the damper's installation position on the longitudinal vibration of a drill string by establishing a mathematical model and using a computer simulation. They determined the optimal installation position of the damper. [PERSON] [11] calculated the intrinsic frequencies of a drill string's longitudinal, transverse, and torsional vibration by using the differential quadrature method (DQM). [PERSON] [12] investigated the longitudinal forced vibration of a drill string under the action of cyclic impacts of the drilling cycle of a DTH hammer in hard rock. [PERSON] [13] explored vibration and impact-suppression techniques for drill strings. They proposed three control methods: passive control, active control, and semi-active control. Further, they comparatively analyzed the characteristics of the various methods, equipment, and application scopes. [PERSON] [14] analyzed the effects of stiffness and damping of the damper and the installation position on the vibrations of downhole drilling tools. [PERSON] et al. [15] eliminated the severe stick-slip vibrations that appear along the drill string of a rotary drilling system according to the LQG observer-based controller approach. The above studies revealed the vibration mechanisms of a drill pipe and quantified its vibration characteristics. However, there is still a gap in the research related to the forced vibration response of the internal unit of the near-bit MWD during DTH hammer drilling and its damping technology. In other fields, the research and development of vibration-damping methods is more mature. [PERSON] [16] proposed the principle of elastic vibration isolation and the frequency design criterion. She established a dynamic model of an engine's elastic system and determined the dynamic characteristic parameters.