## 2 The Physical Model and Its Dynamic Equations  
The working principle of TM-AFM is illustrated in Figure 1. The physical model of the microcantilever beam system is commonly represented as a spring-mass-damping system, as depicted in Figure 1b. Meanwhile, its equation of motion can be expressed as [21; 22]  
\[m\ddot{x}+c_{1}\dot{x}+c_{s}\dot{x}+k_{1}x+k_{2}x^{3}=F_{vdM}(z)+F_{Cas}(z)+f \cos\omega t, \tag{1}\]  
where \(x\), \(\dot{x}\), and \(\ddot{x}\) denote the displacement, velocity, and acceleration of the microcantilever beam in units of m, m/s, and m/s\({}^{2}\), respectively. During motion, apart from external excitation \(f\cos\omega t\), the microcantilever beam system is also affected by van der Waals force \(F_{vdM}\), linear spring force, nonlinear spring force (compensating for the linear spring force, equivalent stiffness is expressed in \(k_{2}\)), linear damping force, extruded film damping force (compensating for the linear damping force), and Casimir force \(F_{Cas}\)[21; 23; 24; 25]. \(z\) denotes the distance from the tip of the microcantilever beam to the surface of the sample to be measured, assuming that the distance between the tip samples when the microcantilever beam is at rest is \(l_{0}\), then \(z=l_{0}+x(t)\). To facilitate the study of the dynamic characteristics of the cantilever beam, we introduce dimensionless state space Equation (2) for the cantilever beam system. \[\left\{\dot{y}=-ry-bx-hx^{3}+\frac{e}{(a+x)^{3}}-\frac{\dot{x}=y}{(a+x)^{2}}- \frac{p}{(a+x)^{3}}y+\frac{\beta}{(a+x)^{4}}+g\cos(\Omega\tau).\right. \tag{2}\]  
The dimensionless displacement and velocity are represented by \(x\) and \(y\), respectively. The dimensionless linear damping coefficient is denoted as \(r\), while \(b\) and \(h\) represent dimensionless linear and nonlinear stiffness, respectively. The strength of the dimensionless Hamaker repulsion potential is represented by \(e\), and the strength of the dimensionless Hamaker attraction potential is represented by \(d\). The dimensionless Casimir force strength is represented by \(\beta\). The damping coefficient of the extruded film is represented by \(p\) in a dimensionless form. The amplitude of external excitation is represented by \(g\) in a dimensionless form. The frequency ratio is represented by \(\Omega\) (\(\Omega\) (\(\Omega\) = \(\omega/\omega_{n}\)), where \(\Omega\) is the external excitation frequency and \(\omega_{n}\) is the natural vibration frequency of the microcantilever beam system. The equilibrium distance between the tip of the microcantilever beam and the surface of the measured sample under a static state is represented by \(a\). Since, in this paper, we mainly study the effect of damping (including equivalent linear damping and extruded air-film damping) on the motion stability of the TM-AFM microcantilever beam system, we only explain the derivation process of the motion parameters \(r\) and \(p\) here. In general, the linear damping coefficient c of the spring oscillator can be obtained by the measured method or the finite element integration method, but due to the tiny size of the microcantilever beam, it is difficult to obtain the linear damping coefficient c of the microcantilever beam by the measured method. Therefore, the finite element integration method is used to obtain the linear damping coefficient c of the microcantilever beam of the continuum microcantilever beam of the first-order mode of the formation function as follows [7; 18]. \[\varphi(y_{1})=\cos\beta_{1}-\cosh\beta_{1}y_{1}+r_{1}(\sin\beta_{1}y_{1}-\sinh \beta_{1}y_{1}), \tag{3}\]  
where \(y_{1}\) denotes the distance from the reference point on the axial beam to the fixed end of the left end of the microcantilever beam, \(y_{1\max}\)=\(l,l\) denotes the length of the microcantilever beam, \(\beta_{1}=\frac{\pi}{l}\), \(r_{1}=\frac{\sin\beta_{1}l-\sinh\beta_{1}l}{\cos\beta_{1}l+\cosh\beta_{1}l}\).