# Dynamic Characteristics Analysis of a Rod Fastening Rotor System Considering Contact Roughness
## 4 Results and Discussion  
### Normal Contact Stiffness of Joint's Interface  
The relationship between normal contact stiffness and contact load for the rough surface is conducted in this part. The main parameters of the rough surface are listed in Table 1.  
The contact stiffness of the joint's interface is the sum of the normal contact stiffness of all grades of asperities, namely  
\[K_{r}=K_{re}+K_{rep1}+K_{rep2}+K_{rp} \tag{31}\]  
Meanwhile, the nominal contact area of the joint's interface is defined as  
\[A_{a}=l_{r}^{2}=(l/\gamma^{n})^{2} \tag{32}\]  
Correspondingly, the dimensionless contact load and the dimensionless normal contact stiffness can, respectively, be expressed as  
\[F_{r}^{*}=\frac{F_{r}}{EA_{a}} \tag{33}\]  
and  
\[K_{r}^{*}=\frac{K_{r}}{E\sqrt{A_{a}}} \tag{34}\]  
According to Table 1, the elastic, first elastoplastic and second elastoplastic critical frequency indexes can be calculated as \(n_{cc}=28\), \(n_{epc}=34\) and \(n_{pc}=45\), respectively. Then the variation of dimensionless normal contact stiffness is depicted in Figure 5. It is clear that with the increase in the dimensionless contact load, the dimensionless normal contact  
\begin{table}
\begin{tabular}{c c} \hline \hline Fractal dimension \(D\) & 1.34 \\ \hline Characteristic length scale \(G\) (m) & \(7.342\times 10^{-11}\) \\ \hline Grade of asperity \(n(a)\) & 20 \(\sim\) 50 \\ \hline Equivalent elastic modulus \(E\) (GPa) & 115.4 \\ \hline Hardness \(H\) (GPa) & 6.1 \\ \hline Poisson’s ratio \(\
u\) & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Model parameters of the rough surface.  
stiffness rises rapidly at first and then tends to stabilize. This change is consistent with the test results in reference [28]. Because \(n_{\text{min}}\) is smaller than \(n_{cc}\), the contact stiffness of the joint's interface is mainly determined by the asperity, which corresponds to the frequency index of this range. As a result, the relation between the dimensionless normal contact stiffness and the contact load is approximately linear.  
In addition, the effect of the frequency index on the dimensionless normal contact stiffness is also depicted in Figure 5. As the frequency index mounts up, the dimensionless normal contact stiffness becomes larger.  
When the characteristic length scale is \(G=7.342\times 10^{-11}\) m and the fractal dimension changes from \(D=1.34\) to \(D=1.7\), the influences of the fractal dimension on the dimensionless normal contact stiffness change, as shown in Figure 6. With the increase in the fractal dimension \(D\), the dimensionless normal contact stiffness \(K_{r}^{*}\) increases synchronously. The main reason for this phenomenon is that the large fractal dimension can lead to the obvious refinement of the surface concave-convex and raise the distribution of the asperity.  
Figure 7 shows the influence of the characteristic length scale \(G\) on the dimensionless normal contact stiffness. When the fractal dimension is \(D=1.34\) and the level range of the asperity is \(n=[20,\ 50]\), the dimensionless normal contact stiffness decreases with an increase in the characteristic length scale. This is because the larger the characteristic length scale \(G\), the rougher the contact surface becomes.  
Figure 5: Variation of dimensionless normal contact stiffness due to different parameters: (a) dimensionless contact force \(F_{r}^{*}\) and (b) frequency index \(n\).  
Figure 6: Relation between dimensionless normal stiffness \(K_{r}^{*}\) and fractal dimension \(D\).