## 3 Modification of the Normal Stiffness Fractal Model for Joint  
[PERSON] and [PERSON] proposed a contact fractal model (M-B fractal model) based on fractal geometry. The M-B fractal model simplifies the contact between rough surfaces to the contact between a rough surface and the ideal rigid plane, as shown in Figure 6, and this rough surface has fractal properties [3]. In the schematic diagram, \(P\) is the load between the contact points, \(R\) is the radius of curvature of the tip of asperity, \(\delta\) is the deformation, and \(r\) is the radius of the contact area. [PERSON] found experimentally that the deformation of asperity changed from elasticity to plasticity when the deformation \(\delta\) is greater than the critical deformation \(\delta_{c}\), in which case the critical deformation is given as [2]  
\[\delta_{c}=\frac{(\pi k\phi)^{2}}{2}R \tag{7}\]  
\(k\) and \(\phi\) satisfy the following relations, respectively:  
\[k=\frac{H}{\sigma_{y}} \tag{8}\]  
\[\phi=\frac{\sigma_{y}}{E} \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **First Order** & **Second Order** & **Third Order** \\  & **(Hz)** & **(Hz)** & **(Hz)** \\ \hline Experiment & 48.7 & 166.5 & 310.7 \\ Considering joint & 48.8 & 165.8 & 313.6 \\ Without considering & 402.1 & 842.5 & 1068.9 \\ joint & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Natural frequency comparison. Figure 6: Joint contact model. where \(H\) denotes the hardness of the softer material in Brinell units, \(\sigma_{y}\) denotes the yield strength of the softer material, and \(E\) denotes the composite Young's modulus of the contact material, which is defined by the elastic modulus and Poisson's ratio of the two contacting materials, as shown in Equation (10):  
\[E=\frac{1-{\
u_{1}}^{2}}{E_{1}}+\frac{1-{\
u_{2}}^{2}}{E_{2}} \tag{10}\]  
The surface contour line of the joint can be determined by the W-M function, as shown in Equation (11) [3]:  
\[Z(x)=G^{(D-1)}\cdot l^{(2-D)}\cdot\cos\frac{\pi x}{l} \tag{11}\]  
where \(D\) is the fractal dimension (1 < \(D\) < 2), \(G\) is the fractal roughness parameter, and \(l\) denotes the length scale. From Equation (11), the radius of curvature of the tip of asperity can be derived as  
\[R=\left|\frac{1}{\left|\frac{\partial^{2}_{x}}{\partial x^{2}}\right|_{x=0}} \right|=\frac{a^{\frac{D}{2}}}{\pi^{2}G^{(D-1)}} \tag{12}\]  
\(a\) is the contact area, as shown in Equation (13). \[a=\pi r^{2} \tag{13}\]  
The relationship between the contact area and the contact length is given as  
\[l=a^{\frac{1}{2}} \tag{14}\]  
From Equations (11) and (14), the deformation of asperity can be derived as  
\[\delta=G^{(D-1)}\cdot l^{(2-D)}=G^{(D-1)}\cdot a^{\frac{2-D}{2}} \tag{15}\]  
When the deformation \(\delta\) of asperity is smaller than the critical deformation \(\delta_{c}\), it is in the elastic deformation stage. According to the Hertz contact theory, the relationship between load and deformation of a single asperity in the elastic deformation stage is given as follows (in this paper, \(P\) denotes the load of the individual asperity, and \(P^{\prime}\) denotes the total load of the joint):  
\[P=\frac{4}{3}ER^{\frac{1}{2}}\delta^{\frac{4}{3}} \tag{16}\]  
It was found that asperity interaction significantly affected the mean surface spacing and asperity load redistribution, and the deformation of asperity had to take into account the combined effect of load and asperity interaction. The actual deformation of asperity is given as [7]  
\[\delta=\delta^{\prime}-\delta^{\prime\prime}=\left(\frac{9P^{2}}{16E^{2}R} \right)^{\frac{1}{3}} \tag{17}\]  
where \(\delta^{\prime}\) is the deformation due to normal loads, and \(\delta^{\prime\prime}\) is the additional deformation generated by asperity interaction.