## 3 Experimental and Simulation Analysis of Normal Contact Stiffness in Bolted Flat Plate Structures
### Modeling and Analysis of Contact Stiffness for Different States of the Flat Plate Structure
Therefore, we used the same combination of normal contact stiffness as the first resonance peak frequency band in the low-order frequency bands, focusing our research on the high-order frequency bands where the resonance peaks occur. First, if the change in vibration mode caused by the non-uniform distribution of normal contact stiffness is not considered, set the normal contact stiffness in areas B and C, as shown in the figure, to \(0.338\left(\mathrm{N\cdot mm^{-1}}\right)/\mathrm{mm^{2}}\). At this time, the fundamental frequency of the plate reaches 811 Hz, and the vibration mode remains the same as the first order mode shape, as shown in Table 3. Record the normal contact stiffness value of \(0.338\left(\mathrm{N\cdot mm^{-1}}\right)/\mathrm{mm^{2}}\) currently as the right endpoint of the range for normal stiffness values. Figure 11: The experimental frequency response curves of point 19 and point 22. Figure 10: The schematic diagram of the positions of point 19 and point 22 on the plate. Next, by changing the value of normal contact stiffness in area B, it was found that the normal contact stiffness in area B could increase the fundamental frequency by up to 200 Hz. Therefore, set the normal contact stiffness in areas B and C, as shown in the figure, to \(0.179\left(\mathrm{N\cdot mm^{-1}}\right)/\mathrm{mm^{2}}\); at this time, the fundamental frequency of the plate reaches 631 Hz. Then, by setting the normal contact stiffness on the surface non-uniformly, increasing the normal contact stiffness in the central area, the fundamental frequency of the plate can be increased to 811.9 Hz, and the vibration mode changes to the form shown in Table 2 under contact conditions. The value of \(0.179\left(\mathrm{N\cdot mm^{-1}}\right)/\mathrm{mm^{2}}\) is taken as the left endpoint of the range for normal stiffness values. Then, within the value range of (0.17  
Based on the experimental and simulation analysis of the flat plate structure, it can be concluded that, in the simulation, the normal contact stiffness distribution across the interface was determined based on the mode shape under non-contact conditions. Then, the normal contact stiffness for different areas of the interface and different frequency bands was determined based on the frequency response curve under contact conditions. This method can effectively reproduce the frequency response and modal vibration mode of the plate when it is in contact with the fixture. In summary, by considering the non-uniform distribution and frequency-dependent normal contact stiffness across the interface, it is possible to replicate the experimental results within the 0-1000 Hz range. 2.2 The Simulation Results Obtained by Traditional Modeling Methods and Considering Tangential Contact Stiffness  
This section refers to the traditional joint interface contact stiffness modeling ideas, simulating the flat plate structure under the contact condition of this study. Initially, following the methods and ideas from most of the literature on bolted joint structures, the contact stiffness of the flat plate structure's interface was handled in two different ways: (1) the entire interface's contact stiffness was equivalent to the area under bolt connection pressure; (2) the entire interface's contact stiffness was evenly distributed. As shown in Table 5, neither of these simulation approaches could achieve the change in modal mode of the flat plate under contact conditions and equating the entire interface's contact stiffness to the bolt connection pressure area also failed to realize the fundamental frequency change in the flat plate under contact conditions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Experimental Mode** & **Simulated Mode** & **Frequency Error** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of experimental and simulation results under contact condition. \begin{table}
\begin{tabular}{c c} \hline \hline
**Modeling Method** & **Modal Mode** \\ \hline \hline \end{tabular}
\end{table}