# Dual-Indicator Micro-Electro-Mechanical System Monitoring Method for Rock Instability Early Warning
## 2 Fundamental Theory and Methodology
### Dynamic Interdependence: NF Versus Safety Factor  
According to vibration mechanics theory, unstable rock masses exhibit characteristic vibration under external excitation, which inherently reflect their mechanical properties [20]. To establish quantitative correlations between rock mass stability and vibrational signatures, we systematically analyzed dominant stability control factors through mechanical characterization. Figure 2 illustrates the force equilibrium of typical toppling-prone rock masses, where two critical moments coexist: (1) toppling moment induced by gravitational forces; (2) resisting moment is generated through structural plane bonding. Figure 1: Working principle of MEMS sensor. (**a**) Application method of MEMS sensors. (**b**) MEMS sensing mechanisms. Where \(l\) denotes the bonding length of the principal structural plane. When \(l\) decreases below critical thresholds, the resisting moment becomes insufficient to counteract gravitational toppling forces, ultimately triggering collapse. The instability failure characteristics of this unstable rock mass are primarily manifested as a rotational failure mode around a specific pivot point, caused by unbalanced moments resulting from either load increase or cohesion decrease. This demonstrates that structural plane bonding length constitutes the pivotal control parameter for rock mass stability. According to limit equilibrium principles, the safety factor \(K\) for toppling-prone rock mass is defined as the ratio of the resisting moment to the toppling moment, as expressed in Equation (2):  
\[K=\left(F\times l/2\right)\raisebox{-1.29 pt}{\includegraphics[width=14.29 pt ]{_R_}}\times e=\delta_{s}Tl^{2}s\raisebox{-1.29 pt}{\includegraphics[width=14.29 pt]{_R_}} \tag{2}\]  
where \(F\) is the resisting force; \(e\) is the distance between the center of gravity of rock mass and the structural plane; \(\delta_{s}\) is the tensile strength of rock mass; \(T\) is the thickness of rock mass; and \(s\) is the width of rock mass. \(G\) represents the gravitational force of the unstable rock mass. The square of the bonding length \(l^{2}\) in unstable rock mass structural planes exhibits a positive correlation with the safety factor \(K\). When \(l\) diminishes to the critical stability threshold \(l_{c}\), the safety factor reaches the unity (\(K=1\)) [2]. Rock mass failure initiates when \(K<1\), indicating a loss of structural equilibrium. This early-warning methodology employs MEMS sensors to track vibrational frequency modulations that correlate with the reduction in the bonding length of the principal structural plane. The NF constitutes an intrinsic parameter of rock mass vibrational dynamics. As shown in Figure 3, the unstable rock mass can be simplified into a single-degrees-of-freedom (SDOF) spring-mass oscillator. The motion balance shown in Equation (3) is established according to Newton's Second Law, and NF expression is shown in Equation (4) [20]. Figure 2: Schematic diagram of overthrown unstable rock mass. Where the stiffness decreases proportionally with bonding length reduction. This stiffness degradation induces measurable frequency attenuation, enabling the quantitative assessment of structural integrity through frequency-domain analysis. \[me\bar{\beta}(t)+\left( \begin{matrix}I^{3}ETs&\\ &\sqrt{3d}\end{matrix} \right)\beta(t)=0 \tag{3}\]  
\[f_{0}=(1/2\pi)\sqrt{(ET^{3}sg)/(3 dGe^{2})} \tag{4}\]  
where \(\beta(t)\) is the change in tilt angle in the vibration process; \(m\) is the quality of unstable rock mass; and \(f_{0}\) is the NF of unstable rock mass. According to Equations (2) and (4), the mathematical expression of NF and safety factor is obtained, as shown in Equation (5):  
\[K=\sqrt[3]{18\pi^{4}eTd^{2}\delta_{sf}f_{0}^{4}/GE^{2}g^{2}} \tag{5}\]  
Equation (5) established a bijective relationship between the NF of unstable rock masses and their safety. The NF of toppling-type unstable rocks exhibits a quantitative proportional relationship of 4:3 with the safety factor.