## 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  
Modeling and Analysis of Normal Contact Stiffness Considering Distribution Non-Uniformity and Frequency-Dependent Characteristics  
In this paper's research, we constructed a finite element model using HyperMesh 2023 and solved it using OptiStruct 2022, both software tools are products of Altair Engineering Inc. (Troy, MI, USA). First, a finite element model of the bolted connection structure between the flat plate and the fixture was established without contact between them, as shown in Figure 7. The finite element models for each component were built using hexahedral solid elements, where the plate comprised 15,876 elements, the fixture included 64,550 elements, the bolts contained 576 elements, and the flat washers had 192 elements. The boundary conditions were defined by constraining the six degrees of freedom at the nodes of the eight screw holes on the fixture model. Additionally, preload forces were incorporated into the bolt model, corresponding to the torque applied during the experimental setup. \begin{table}
\begin{tabular}{c c c} \hline
**Non-Contact Condition** & **Contact Condition** \\ \hline
116.9 Hz & 811.9 Hz & 964.5 Hz \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of experimental modal shapes under non-contact and contact conditions. The comparison of experimental mode and simulation mode results for the flat plate without contact with the fixture is presented in Table 3. The table clearly demonstrates that the first six mode shapes of the flat plate structure exhibit a high level of consistency. Moreover, the absolute errors in the natural frequencies for each mode shape are generally around 5%, indicating the accuracy of the finite element model. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Modal Order** & **Experimental Mode** & **Simulated Mode** & **Frequency Error** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of experimental mode and simulation mode results for the flat plate without contact with the fixture. Figure 7: Finite element model under non-contact condition of plate and fixture. According to the findings presented in Table 3, the first mode shape of the model is associated with bending deformation along the normal direction of the flat plate. In the deformed region, the mode displacement gradually decreases from the center of the plate towards the periphery. The magnitude of the mode displacement serves as an indicator of the extent of the response to external excitation. A larger mode displacement implies a higher contact pressure on the contact surface in cases where contact occurs. Contact stiffness is influenced by factors such as material properties, surface roughness, and contact pressure in the study of mechanical contact stiffness [19; 20; 21; 22]. As increased contact pressure leads to an increase in contact stiffness, the regions with larger mode displacements are associated with higher contact stiffness. Conversely, regions with smaller mode displacements correspond to lower contact pressures and contact stiffness. Based on the mode displacement of the first mode shape when there is no contact between the flat plate and the fixture, it is assumed that regions with larger mode displacements will experience higher contact pressures and contact stiffness when considering the contact between the flat plate and the fixture. First, the normal contact stiffness distribution is determined according to the mode shape. In this paper, taking the mode shape corresponding to the first mode frequency in Table 3 as an example, the contact interface between the flat plate and the fixture is divided into three different regions based on the magnitude of the mode displacement in the simulation analysis under the non-contact condition, as shown in Figure 8.  
Region A represents the bolted connection area, which is tightly connected and has a large contact stiffness.