## 2 Modelling
### The Model of Normal Tuning Fork Probes  
At first, the multistage coupling model as shown in Figure 3 is utilized to specify normal tuning fork probes. The feature of the new model is two mechanical coupling stages are added to describe the cantilever (or the tip) and the sample respectively. The couplings are regarded as ideal transformers. All the mechanical parameters will be transformed to electrical parameters in an RLC circuit using the principle of ideal transformers.  
This electromechanical coupling model consists of three parts: a typical electromechanical coupling part describing the tuning fork, a mechanical coupling part representing the connection between the cantilever (or tip) and the tuning fork, and another mechanical coupling part explaining the contact between the cantilever tip and the sample. In the model, the parameters, \(b\), \(k\), \(m\), \(F\), \(v\), with different subscripts, denote damping, spring constant, mass, force, and the velocity of the tuning fork (subscript tf), and cantilever (subscript cl), respectively; \(C_{\mathrm{p}}\) is the parasitic capacitance; \(i\) is the current; \(V_{\mathrm{in}}\) is the excitation voltage; \(p\) is the electromechanical coupling factor; \(n\), \(b_{\mathrm{sa}}\), and \(k_{\mathrm{eff}}\) are the mechanical coupling factor, the damping, and effective force gradient between the tip and the sample. If the normal tuning fork's tip is regarded as a rigid body, the spring constant (\(k_{\mathrm{cl}}\)) of the tip is infinite and the damping (\(b_{\mathrm{cl}}\)) is zero. The coupling factor between the tip and the tuning fork, and the factor between the tip and sample (factor \(n\) in Figure 3) are one because the tip is adhered to the tuning fork directly, \(v_{\mathrm{tf}}=v_{\mathrm{cl}}=v_{\mathrm{tip}}\). So the model can be transformed to:  
With Equation (4), the model in Figure 4 can be transform to a simple RLC circuit as shown in Figure 5.  
Figure 4: The model of rigid tip.  
Figure 5: The equivalent RLC circuit of the tuning fork probe.  
Figure 3: The electromechanical coupling model of normal probes.  
And the parameters in Figure 5 are  
\[\begin{split} R_{\text{probe}}&=p^{2}(b_{\text{tf}}+b_{ \text{sa}})\\ L_{\text{probe}}&=p^{2}(m_{\text{tf}}+m_{\text{cl}})\\ C_{\text{probe}}&=\frac{1}{p^{2}(k_{\text{tf}}+k_{ \text{eff}})}\end{split} \tag{6}\]  
So the frequency shift after contact is  
\[\begin{split}&\Delta f=f^{\prime}-f_{0}=\frac{1}{2\pi\sqrt{l_{ \text{probe}}C_{\text{probe}}^{\prime}}}-\frac{1}{2\pi\sqrt{l_{\text{probe}}C_{ \text{probe}}}}\\ &=\frac{1}{2\pi\sqrt{l_{\text{probe}}C_{\text{probe}}^{\prime}C_{ \text{probe}}}}(\sqrt{C_{\text{probe}}}-\sqrt{C_{\text{probe}}^{\prime}})\\ &=\frac{f_{0}}{\sqrt{C_{\text{probe}}^{\prime}}}(\sqrt{C_{\text{ probe}}}-\sqrt{C_{\text{probe}}^{\prime}})\end{split} \tag{7}\]  
\[\begin{split}&\frac{\Delta f}{f_{0}}=\sqrt{\frac{C_{\text{probe}}}{C_{ \text{probe}}^{\prime}}}-1\\ &=\sqrt{\frac{k_{\text{tf}}+k_{\text{eff}}}{k_{\text{tf}}}}-1 \\ &=\sqrt{1+\frac{k_{\text{eff}}}{k_{\text{tf}}}}-1\\ &\Rightarrow(\frac{\Delta f}{f_{0}})^{2}+2(\frac{\Delta f}{f_{0} })=\frac{k_{\text{eff}}}{k_{\text{tf}}}\end{split} \tag{8}\]  
For \(\Delta f<<f_{0}\),  
\[k_{\text{eff}}\approx 2\frac{\Delta f}{f_{0}}k_{\text{tf}} \tag{9}\]  
where \(k_{\text{tf}}\) is the spring constant of the tuning fork's prong, \(\Delta f\) is the frequency shift, \(f_{0}\) is the resonance frequency without contact. This equation is the same with the conclusion that regards the prong as a cantilever [11; 12]. That means this model is correct to describe the normal tuning fork probes.