# Dynamic Characteristics Analysis of a Rod Fastening Rotor System Considering Contact Roughness
## 3 Characterization of Contact Stiffness of Joint's Interface
### Contact of Joint's Interface  
In this paper, the W-M function is adopted to describe the two-dimensional rough joint's interface. According to reference [25], the specific expression is given by  
\[Z(x)=G^{D-1}\sum_{n=n_{\text{min}}}^{\infty}\frac{\cos(2\pi\gamma^{n}x)}{ \gamma^{(2-D)n}},\;(1<D<2) \tag{4}\]  
where \(Z(x)\) denotes the height of the surface profile, \(D\) denotes the fractal dimension of a surface profile, \(G\) denotes the length scale of a surface, \(n_{\text{min}}\) corresponds to the low cut-off frequency of the profile, \(\gamma^{n}\) denotes the frequency spectrum of the surface roughness and \(n\) denotes the frequency index of the surface profile, which represents the level of each asperity. According to Figure 4, for the \(n\)th level asperity surface, the surface profile curve can be obtained according to Equation (4), namely  
\[Z_{n}(x)=G^{D-1}l^{2-D}\cos\Bigl{(}\frac{\pi Tx}{l}\Bigr{)} \tag{5}\]  
where \(l\) denotes the length scale of the asperity, which obeys \(l=1/\gamma^{n}\). Figure 4: Schematic diagram of contact between the rigid plane and single asperity. Figure 3: Schematic diagram of an equivalent model. As shown in Figure 4, the contact between a single asperity and a rigid plane happens. According to Equation (5), for the two-dimensional rough surface, the radius of the asperity can be written as  
\[R_{n}=\left|\frac{\left(1+\left(\
icefrac{{dz}}{{dx}}\right)^{2}\right)^{3/2}}{ \left(\
icefrac{{dz}}{{dx}^{2}}\right)}\right|_{x=0}=\frac{1}{\pi^{2}G^{D-1} \gamma^{nD}} \tag{6}\]  
The height of the \(n\)-th level asperity can be derived as  
\[\delta_{n}=Z_{n}(0)=G^{D-1}\gamma^{-n(2-D)} \tag{7}\]  
As introduced in the theory of contact mechanics, there are three steps in the process of contact between the rigid plane and the \(n\)-th level asperity surface, including the elastic part, elastic-plastic part and plastic part, respectively. According to reference [35], combined with the above different steps, the relations between the deformation of single asperity and the contact area can be derived, respectively, as  
\[ \begin{cases}a_{ne}=\pi R_{n}w_{n}&(w_{n}\leq w_{nec})\\ a_{nep1}=a_{nec}0.93\left(\frac{w_{n}}{w_{nec}}\right)^{1.136}&(w_{nec}<w_{n} \leq 6w_{nec})\\ a_{nep2}=a_{nec}0.94\left(\frac{w_{n}}{w_{nec}}\right)^{1.146}&(6w_{nec}<w_{n} \leq 110w_{nec})\\ a_{np}=2\pi R_{n}w_{n}&(110w_{nec}<w_{n})\end{cases}  \tag{8}\]  
Correspondingly, the contact force of the single asperity at the different stages can be expressed as  
\[\left\{ \begin{array}{l}F_{ne}=\frac{2}{3}E\pi^{-1}R_{n}^{-0.5}a_{nec}^{1.5} \\ (w_{n}\leq w_{nec})\\ F_{nep1}=\frac{2}{3}\psi H\times 1.1282a_{nec}^{-0.2544}a_{nep1}^{1.2544}\\ (w_{nec}<w_{n}\leq 6w_{nec})\\ F_{nep2}=\frac{2}{3}\psi H\times 1.4988a_{nec}^{-0.1021}a_{nep2}^{1.1021}\\ (6w_{nec}<w_{n}\leq 110w_{nec})\\ F_{np}=Ha_{np}\\ (110w_{nec}<w_{n})\end{array} \right. \tag{9}\]  
where the subscripts _ne_, _nep1_, _nep2_ and _np_ represent the elastic stage, the first elastic-plastic stage, the second elastic-plastic stage and the plastic stage, respectively. The subscript _nec_ represents the critical elastic deformation of the asperity. \(H\) denotes the material hardness. According to reference [35], critical elastic deformation can be expressed as  
\[w_{nec}=\left(\frac{\pi\psi H}{2E}\right)^{2}R_{n} \tag{10}\]  
where the hardness coefficient \(\psi\) and the equivalent elastic modulus \(E\), respectively, obey  
\[\psi{=0.454+0.41\
u_{2}} \tag{11}\]  
\[E=\frac{1}{\frac{1-\
u_{1}^{2}}{E_{1}}-\frac{1-\
u_{2}^{2}}{E_{2}}} \tag{12}\]  
where \(E_{1}\) and \(E_{2}\) denote the elastic modulus of hard and soft materials, and \(\
u_{1}\) and \(\
u_{2}\) denote the Poisson's ratio of hard and soft materials, respectively.