# Dual-Indicator Micro-Electro-Mechanical System Monitoring Method for Rock Instability Early Warning
## 2 Fundamental Theory and Methodology
### Quantification Metrics for MEMS-Acquired Stability Indicators  
Through MEMS sensing, the vibration time history curve of bedrock is obtained as the vibration input of the spring-mass model, and the input curve of unstable rock mass is serves as the response output. The frequency response function (FRF) \(H(f)\) representing the vibration response of unstable rock mass under ambient vibration can be obtained, and its calculation formula is shown in Equation (6). Under the excitation of a sinusoidal signal, the theoretical magnification factor of the spring-mass model system \(D(s)\) is shown in Equation (7). The NF and damping ratio of unstable rock mass can be obtained by  
Figure 3: SDOF spring–mass oscillator of an unstable rock.  
analyzing the relationship between the theoretical amplitudes and the actual amplitudes under the multi-frequency excitation [21].  
\[H(f)=\cancel{P_{xy}(f)}\cancel{P_{x}(f)} \tag{6}\]  
\[D(s)=1/\sqrt{\left(1-s^{2}\right)^{2}+\left(2\xi s\right)^{2}} \tag{7}\]  
where \(P_{xy}(f)\) is the cross-correlation power spectral density function between bedrock and unstable rock mass; \(P_{x}(f)\) is the matrix autocorrelation power spectral density function; and \(f\) is the excitation frequency of the input unstable rock mass under constant fretting.  
When the frequency ratio reaches the unity (\(f=f_{0}\)), the amplification factor \(D(s)\) attains its maximum value. This indicates that the peak value in the actual obtained FRF is the magnification of the response amplitude caused by the resonance of the unstable rock mass, and the horizontal coordinate frequency corresponding to the peak value is the NF of the unstable rock mass.  
The amplitude-amplification factor \(D(s)\) is the ratio of the vibration amplitude of unstable rock mass to that of bedrock under different excitation frequencies. As the NF \(f_{0}\) of unstable rock mass decreases, the amplitude-amplification factor of the system under the same excitation frequency increases. Therefore, the ratio of the vibration amplitude of unstable rock mass to that of bedrock also increases. In order to reduce the error caused by the random interference of the MEMS sensor in the acquisition process of the vibration amplitude of the unstable rock mass and bedrock, the ratio of mean square velocity to the amplitude of the unstable rock mass and bedrock is adopted as the monitoring index of unstable rock mass collapse. The equation is as follows:  
\[\eta=\sqrt{\int\limits_{t_{1}}^{t_{2}}(1/(t_{2}-t_{1}))y^{2}(t)}\Bigg{/}\int \limits_{t_{1}}^{t_{2}}(1/(t_{2}-t_{1}))x^{2}(t) \tag{8}\]  
where \(x(t)\) is the velocity time history curve of the bedrock; and \(y(t)\) is the velocity time history curve of unstable rock mass.  
The stability state of an unstable rock mass evolves dynamically. MEMS-derived NF and RMS-VAR parameters effectively characterize the progressive destabilization of unstable rock masses. Using MEMS sensing technology, triaxial vibration time-history curves of unstable rock masses and bedrock are collected in real time, with a principal component analysis (PCA) employed to identify dominant vibration directions. Through time-domain window optimization, the FRF of unstable rock masses is calculated, and their natural frequencies are determined through fitting of amplitude amplification factors. The RMS-VAR between unstable rock masses and bedrock is derived by converting detrended acceleration curves to velocity curves. Furthermore, in engineering practice, unstable rock masses are typically formed through prolonged geological processes and often exhibit precursor phenomena such as localized rockfall events and minor deformations prior to collapse [22; 23]. MEMS sensors are utilized to monitor instantaneous impact acceleration and subtle tilt angles caused by surrounding rockfalls, providing critical data support for early-warning systems of rock mass collapse.