# The Vibration Response Characteristics of Neighboring Tunnels Induced by Shield Construction
## 2 Basic Theory  
### Vibration Energy Density Decay Theory  
Vibration wave power density, derived from wave acceleration, represents the energy distribution of squared acceleration over a unit frequency range. It is widely used in vibration analysis to characterize the frequency properties and energy distribution of acceleration signals, with applications in mechanical vibration, civil engineering, seismic wave propagation, and related fields. Factors such as frequency, propagation path, and structural damping significantly influence power density attenuation [26; 27].  
The power density represents the energy distribution of the square value of the acceleration of the signal in the unit frequency range. In vibration analysis, the power density _S(f)_ of the acceleration _a(t)_ of a time-domain signal is defined as follows:  
\[S_{X}(f_{k})=\frac{P|k|}{T}=\frac{A|k|\cdot A^{*}|k|}{N\Delta t} \tag{1}\]  
where _A(k)_ is the Fourier transform of the acceleration signal _a(t)_ and \(T\) is the observation time of the signal. The unit is (mm/s\({}^{2}\))/Hz.  
The energy distribution of the vibration signal on different frequency components can be analyzed by the power density, which provides a basis for the energy change in the signal propagation process.  
When a vibration wave passes through a damped structure, its acceleration power density decay can be expressed as follows:  
\[S_{B}(f)=S_{A}(f)\cdot|H(f)|^{2}\cdot e^{-2\xi\omega(f)d} \tag{2}\]  
where _S\({}_{A}\)(f)_ and _S\({}_{B}\)(f)_ are the vibration acceleration power density before and after the structure, respectively; _H(f)_ is the frequency response function of the structure; \(e^{-2\xi\omega(f)d}\) is the exponential decay due to structural damping; \(\xi\) is the damping ratio of the structure; \(\omega\)(f) = 2\(\pi\)f is the angular frequency of the signal; and \(d\) is the propagation distance of the vibration signal.  
The equation describes in the frequency domain how the vibration acceleration power density decays with the propagation process, focusing on the role of structural damping and frequency.  
The frequency response function _H(f)_ describes the transmission properties of a structure for signals of different frequencies. For structures with complex geometrical and material properties, _H(f)_ exhibits nonlinear variations due to resonance, material absorption, and geometrical reflection. In the vicinity of the resonance frequency, the response of the structure to vibration is enhanced and the power density _S\({}_{B}\)(f)_ may show a localized increase. For the frequency response function of a structure, the solution can be based on a dynamic model of the structure. For example, for the vibration analysis of a simply supported beam, the frequency response function can be calculated from the stiffness matrix \(K\) and the mass matrix \(M\) of the structure. The dynamic equations of this system are as follows:  
\[M\ddot{u}(t)+Ci\dot{u}(t)+Ku(t)=F(t) \tag{3}\]where \(u(t)\) is the displacement and \(F(t)\) is the excitation force. By analyzing this equation in the frequency domain, the frequency response function is obtained, as follows:  
\[H(f)=\frac{1}{\left(M(2\pi f)^{2}+j2\pi fC+k\right)} \tag{4}\]  
where \(j\) is an imaginary unit.  
The damping effect reflects the ability of the structure to dissipate vibration energy. The larger the damping ratio \(\xi\) is, the faster the energy decays during signal propagation. The exponential term shows that higher frequency signals (larger \(f\)) decay more significantly than lower frequency signals and that, the farther the signal travels from \(d\), the more the power density \(S_{B}(f)\) decreases.