# Study on Dominant Frequency Attenuation of Blasting Vibration for Ultra-Small-Spacing Tunnel
## 5 Data Analysis
### Difference in Dominant Frequency Distributions on Primary Support and Secondary Lining

Table 4: Fitting analysis of vibration data at primary support in blasting experiments. Figure 6: Relationships between frequencies and SD. (**a**) ZCF and (**b**) DF. dominant frequencies on the lining are concentrated in an interval when the RD is less than a certain value. As the blast face in the succeeding tunnel advances gradually, the blast-affected area in the preceding tunnel changes from the secondary lining to the primary support. As shown in Figure 7, when the lining is located at different RD, the dominant frequency distributed on it is variant. For example, the dominant frequencies do not have a specific range when the blasting face is close to the lining (RD = 0). When the RD is less than a certain value (RD = \(-\)16 m), the dominant frequencies on the lining are concentrated in an interval, either ZCFs or DFs. In contrast, in the range of RD > 16 m, the dominant frequencies on the primary support are still relatively discrete. To a clear illustration, Figure 7a,b are divided into zones I, II, and III, respectively. The measurement points on the lining are distributed in zones I and II, while the measurement points on the primary support are distributed in zone II and III. The dominant frequencies of the primary support are obeyed with the characteristics of high-frequency attenuation, while the data on the lining are not submitted to it. Another phenomenon is that the propagation of dominant frequencies on the lining does not decay regularly but gradually concentrates in an interval besides a certain distance. For example, in Figure 7a, the dominant frequencies on the lining in zone II are widely distributed within 200 Hz, while only distributed in a range of 50-100 Hz in zone I. The specific range in Figure 7b is from 50 Hz to 100 Hz. Each type of rock mass and each terrain has a propagated dominant frequency, which is conducive to the propagation of a wave having this frequency [35]. The test data of this study show that the advantageous propagation of the dominant frequency of the lining is in the range of 50-150 Hz, which favours wave propagation for that frequency. Figure 7: Relationship between dominant frequencies and RD. (**a**) ZCF and (**b**) DF. The difference of the dominant frequency distributed on the primary support and lining is because of the changes that have occurred during the propagation of the vibration wave. First, the distribution of the dominant frequencies on the primary support is wider than that on the lining, which is evidently in zone II. For example, in Figure 7a, in the range higher than 150 Hz of zone II, more data are distributed on the primary support than on the lining; moreover, in Figure 7b, the range higher than 150 Hz of zone II is distributed including the data of the primary support without any data of the lining. A portion of the frequency of the vibration wave has been wholly attenuated and is related to the gap between the primary support and the secondary lining. The primary support is closely bonded with the surrounding rock mass and directly receives the full vibration waves from the inside of the surrounding rock. Due to space between the lining and the primary support for installing a waterproof board and a geotextile (as shown in Figure 3) and inevitable cavities caused by concrete casting in situ, there is a notifiable interface for seismic wave propagation. Second, blasting seismic wave is usually composed of body waves (P wave and S wave) and Rayleigh surface waves (R wave) [36]. The surface wave propagates on the free surface, and the body waves propagate inside the surrounding rock; body waves at a non-vertical incidence angle on a free surface will produce reflected and mode-converted body waves as well as surface waves [37]. Therefore, when the vibration waves are transmitted from the primary support to the lining, part of the blasting vibration waves (mainly body waves) did not pass through from an abrupt change in the medium.