# Study on Dominant Frequency Attenuation of Blasting Vibration for Ultra-Small-Spacing Tunnel
## 1 Introduction  
In blasting engineering, a part of the explosion energy is used to break a rock and the remainder is transmitted to the surrounding rocks by the blasting vibration waves [1, 2], which may be a significant threat to the safety of the adjacent structure [3, 4]. In small-spacing tunnels, after the primary support and secondary lining of the preceding tunnel are completed, its structure and the middle rock pillar may be affected by the blasting vibrations of the succeeding tunnel, which are manifested as seismic cumulative effects [5]. For example, the repeated blast loading during tunnelling works causes damages on jointed basalt rock masses [6]; and the cyclic blast excavations cause the critical damage PPV for the middle rock pillar drops significantly [7]. Meanwhile, the blast vibration waves may cause a decline in the bearing capacity of the rock [8, 9] and some increases in the stress ofthe anchors [10], concurrently, the excavation damaged zone and deformation modulus of the surrounding rock is also influenced [11]. Recently, owing to the limited space in mountainous areas, small-spacing tunnels are becoming common [12], where the clearance between two tunnels is much shorter than the safe distance to alleviate vibration waves, such as Eurasia Tunnel [13], El-Azhar roadway twin tunnels [14], Liuyuetian bifurcation tunnel [15], Heshang three-lane tunnel [16], and closely spaced triple tunnels in Badaling [17]. Periodic non-uniform natural seismic wave excitation causes shear deformation in underground constructions [18]. And one of the biggest differences between blasting vibration waves and natural seismic waves is the difference in frequency characteristics, where the seismic waves are in low frequency ranges and the blast vibration waves frequency is higher, generally greater than 5 Hz. Therefore, it is necessary to analyze the characteristics of blast vibration waves propagating in the area very close to the tunnels so as to help protect the tunnel structure. The non-uniform seismic excitation, similar to blast vibration waves, has a great influence on the acceleration amplification in soil and the strain and displacement of the long deep buried pipeline [19]. In a tunnel affected by blast vibration waves, the dynamic stress of the structure is directly related to the peak vibration velocity (PPV) [20]. [PERSON] [21] and [PERSON] [22] analyzed the effect of blast vibration waves on the safety of tunnel structures without considering the vibration frequency. However, the adverse effect of blast vibration waves is not only related to the vibration amplitude but also to the relationship between the dominant frequency of a blast vibration and the structural natural frequency. Evidently, [PERSON] [23] found that the PPV at a tunnel wall is amplified compared to that of the incident vibration wave, and the amplification coefficient is positively correlated to the frequency of the blast vibration waves. Therefore, it is necessary to study the frequency distribution range of blast vibration waves. In the tunnel construction, [PERSON] [24] found that the frequency range of blast vibrations is within 0-175 Hz, and the minimum distance between the blast source and the ground measurements point in the study was 30 m. However, the distance has an effect on the signal characteristics of the blast vibration wave. [PERSON] [25] found that the frequency range is closely correlated with the distance values. Additionally, [PERSON] [26] established a mathematical relationship between the blast dominant frequency and its principal influencing factors. [PERSON] [27] found significant differences in the amplitude and spectrum of blast vibration signals collected separately at the surface and underground mines. Nevertheless, these studies on blast vibration frequencies in tunnels do not compare the characteristics of the frequency distribution on the primary support and the lining.