## 4 Structural Energy Response Spectrum of the Tunnel Blasting Vibration
### Influence of Blasting Vibration Characteristics on the Input Energy Spectrum  
The acceleration signal in Figure 5a is selected as the input data of the structural dynamic calculation. The natural period and damping ratio of the structure are selected as 0.1 s and 0.05, respectively. The acceleration, velocity, and displacement time history curves of the tunnel structure under blasting vibration, as shown in Figures 6-8, are obtained by the Newmark-\(\beta\) step-by-step method. Then, the time history curves of input energy \(E_{I}\), kinetic energy \(E_{K}\), elastic strain energy \(E_{E}\), damping energy \(E_{D}\) and the ratio of different forms of energy to total energy are calculated by the method in [27], as shown in Figures 9 and 10, respectively. The natural period of the structure is defined within 0-1.0 s, and the time interval is selected as 0.01 s. The input energy of the structure under different periods is calculated, which is shown in Figure 11.  
Figure 5: Acceleration signals and its Hilbert spectra. (**a**) With a dominant frequency of 28.6 Hz; (**b**) With a dominant frequency of 37.1 Hz; (**c**) With a dominant frequency of 50.0 Hz. Figure 6: Time history curve of the elastic response acceleration. Figure 8: Time history curve of the elastic response displacement. Figure 7: Time history curve of the elastic response velocity. Figure 9: Time history curve of the elastic response energy. Figure 9 and 10 show that the input energy basically increases with the vibration time, and the input energy generally reaches the peak when the vibration ends. However, parts of the input energy curve are declining, which is partly due to the hysteresis of the structural dynamic response caused by damping, and also due to the integral definition of the input energy causing the blasting vibration force to be negative. The kinetic energy and elastic strain energy will increase with time in the elastic SDOF system and also transform each other in the process. Finally, all the energy is converted into damping energy and dissipated; thus, the damping energy dissipation reflects the cumulative effect over time. To compare the influence of the acceleration peak on the input energy spectrum, the peak acceleration signal in Figure 5a is adjusted to 0.2 m/s\({}^{2}\), 0.5 m/s\({}^{2}\), and 0.8 m/s\({}^{2}\), and its input energy spectrum with a damping ratio of 0.05 is shown in Figure 12. Results show that the change in the acceleration peak does not affect the shape of the input energy spectrum. The peak input energy increases with increasing peak acceleration in the square. The peak acceleration signal is normalised to analyse the influence of the dominant frequency on the input energy spectrum. The input energy spectrum of the acceleration signal at three dominant frequencies with a damping ratio of 0.05 is shown in Figure 13, which shows that the input energy spectrum varies markedly with the dominant frequency of the signal. The peak value of the input energy spectrum and its corresponding structural natural period decrease as the dominant frequency increases. The number of peak input energies with a dominant frequency of 50 Hz is more than that of other waves. The larger the dominant frequency is, the richer the frequency component of the vibration waveform is. The dominant frequency of other blasting waveforms is near that of the full-time waveform; thus, there are many peaks. Figure 11: Input energy spectrum. Figure 10: Ratio of each energy to the input energy. The acceleration signal in Figure 5a is selected to analyse the influence of the blasting vibration duration on the input energy spectrum. The duration is adjusted by doubling or tripling the acceleration signal, and the peak acceleration of each segment signal is normalised. The damping ratio is 0.05. Figure 14 shows that the change in duration has no effect on the shape of the input energy spectrum, but the smoothness of the curve is markedly deteriorated.