## 3 Theoretical Analysis of VP  
The structural cross-section of the cylindrical VP is depicted in Figure 2, where point \(O\) is the suppression pivot, \(l_{1}\) is the effective length of the flat elastic spring, \(l_{2}\) and \(R_{2}\) are the length and the diameter of the upper solid pole that has a column shape, respectively, and \(l_{3}\) and \(R_{3}\) are the length and the diameter of the solid cylinder below the pole, respectively. The dynamic equation of the VP's response to the tilt vibration \(\theta\) is expressed by [14]  
\[l_{O}\ddot{\psi}=-\eta\dot{\psi}-M_{O}+k_{S}(1+j\kappa)\phi \tag{1}\]  
Figure 1: The schematic diagram of the tilt AVI. (1) is the supporting frame, (2) is the flat elastic spring, (3) is the pendulum body, (4) is the capacitance plates, (5) is the adjustable supporting plot, (6) is the piezoelectric actuator, (7) is the plate of the tiltmeter’s base. Figure 2: The structural cross-section of the vertical pendulum (VP). where \(I_{O}\), expressed by Equation (2), is the moment of inertia of the VP, \(\eta\) is the angular velocity dissipation coefficient caused by the viscous damping resulting from the surrounding air, \(\kappa\) refers to the elastic material's inertial friction, \(j=\sqrt{-1}\) is an imaginary unit, \(\psi\) and \(\phi=\theta-\psi\) are the angles at which the vertical pendulum deviates from the vertical section and the frame, respectively, \(M_{O}\), expressed by Equation (3), is the gravity torque, and \(k_{S}\) is the elastic coefficient of the flat spring, caused by flexural vibration. \[I_{O}=\frac{1}{3}m_{2}\left(l_{1}^{2}+3l_{1}l_{2}+3l_{2}^{2}\right)+\frac{1}{3 }m_{3}\left[l_{3}^{2}+3(l_{1}+l_{2})l_{3}+3(l_{1}+l_{2})^{2}\right] \tag{2}\]  
\[M_{O}=0.5\{m_{2}(l_{2}+2l_{1})+m_{3}[l_{3}+2(l_{1}+l_{2})]\}g\psi \tag{3}\]  
where \(g=9.8\) m/s\({}^{2}\) is the gravitational acceleration, and \(m_{1}=0.0246\) kg and \(m_{2}=0.2953\) kg are, respectively, the masses of the upper and lower solid cylinders of the pendulum's body, \(l_{1}=5\) mm, \(l_{2}=116\) mm and \(l_{3}=40\) mm. Following Equation (1), the function of amplitude transfer, \(T_{\phi\theta}(f)\), of the vertical pendulum to ground tilt vibration is expressed by  
\[T_{\phi\theta}(f)=\left|\frac{\phi(f)}{\theta(f)}\right|=\frac{\sqrt{\left( \beta_{0}-\mu^{2}\right)^{2}+\left(\frac{\mu}{Q_{1}}\right)^{2}}}{\sqrt{ \left(1-\mu^{2}\right)^{2}+\left(\frac{\mu}{Q_{1}}+\frac{\alpha_{0}}{Q_{2}} \right)^{2}}} \tag{4}\]  
where \(\beta_{0}=c/(k_{S}+c)<1\) is a scale factor representing the mechanical response of the vertical pendulum to low-frequency tilt vibration, \(c=0.5\{m_{2}(l_{2}+2l_{1})+m_{3}[l_{3}+2(l_{1}+l_{2})]\}g\) is a constant with the unit of energy \(\alpha_{0}=1-\beta_{0}\), \(Q_{1}=\sqrt{I_{O}(c+k_{S})/\eta}\) and \(Q_{2}=1/\kappa\) are, respectively, the quality factor of the mechanical oscillator caused by the viscous damping resulting from the surrounding air and the internal damping within the materials, and \(f_{0}=\frac{1}{2\pi}\sqrt{(c+k_{S})/I_{O}}\) is the natural frequency, \(\mu=f/f_{0}\). When the vibration frequency satisfies \(f\ll f_{0}=\sqrt{(c+k_{S})/I_{O}}/(2\pi),\ \mu\to 0\). As \(k_{S}\ll c\) and \(\alpha_{0}/Q_{2}\ll 1\), in general, for the designed VP, \(T_{\phi\theta}(\mu\to 0)\to 1\), and this means that the VP undergoes hardly any displacement relative to the vertical direction. A damped harmonic oscillator undergoes dissipation fluctuation caused by the thermal noise motion of the oscillator at a certain temperature. Thermal noise establishes a significant noise floor within these precise measurements. The mechanical thermal noise of the tilt-sensitive probe is dependent on the loss of the mechanical oscillator, which is due to the viscous damping caused by the surrounding air or the internal damping within the materials. With a vacuum encapsulating the resonating structure, the main limitation is the internal damping within the materials.