## 3 Experimental and Simulation Analysis of Normal Contact Stiffness in Bolted Flat Plate Structures  
### Experimental Modal Testing of Different Contact States in Bolted Flat Plate Structures  
For the bolted flat plate structure (300 mm \(\times\) 260 mm \(\times\) 4 mm) shown in Figure 5, four M4 bolts (with thread diameter 4 mm and thread length 25 mm) are used to connect it to an aluminum alloy fixture made of the same material. The fixture is fixed on the shaker table. To investigate the influence of the normal contact stiffness of the contact interfaces on the natural frequencies and mode shapes of the bolted flat plate, modal tests were conducted under two different contact boundary conditions, as shown in Figure 6. Firstly, 0.8 mm thick flat washers were inserted between the plate and the fixture at the bolted connections to ensure that there was no contact between the plate and the fixture, and the bolts were tightened using a torque wrench. Subsequently, the flat washers were removed and the bolts were tightened with the same torque as before, allowing contact between the  
Figure 4: The equivalent dynamic model of normal contact stiffness. plate and the fixture. In this case, the flat washers were used to isolate the test plate from the fixture, preventing any contact between them. The flat plate and fixture used in the experiment are both made of aluminum alloy material. The M4 bolts and flat washers are made of stainless-steel material. The elastic modulus, Poisson's ratio, and density of each material are shown in Table 1.  
This research employed a feature-level swept frequency experiment, applying a vibration acceleration of 1 g in the normal direction of the flat plate. The plate had 36 sampling points, arranged in a transverse sequence on the board, as illustrated in Figure 3 (6 \(\times\) 6 distribution). Points 1 and 19 served as modal reference points, with fixed accelerometers placed at these two points to collect data from each experiment. The remaining thirty-four sampling points were sampled by dividing them into seventeen groups, each \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Elastic Modulus/MPa** & **Poisson’s Ratio** & **Density/(kg\(\cdot\)m\({}^{-3}\))** \\ \hline aluminum alloy & 70,000 & 0.33 & 2770 \\ stainless steel & 200,000 & 0.3 & 7980 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters of plate test. Figure 5: Schematic diagram of flat plate bolt structure. Figure 6: Two boundary conditions of bolted flat plate structure. using two accelerometers. The time-domain data from each sampling point were imported into the software DASP to form a Hankel matrix. The data were then globally fitted using the Stochastic Subspace Identification Method, thus obtaining the modal frequencies and corresponding modal shapes of the flat plate. Through experimental data processing, the experimental results of first-order modal frequency and shape are shown in Table 2. In the mode shape diagram, the darker the color, the greater the modal relative displacement (The color of the modal shapes presented in the subsequent figures is consistent with the previous explanations and will not be elaborated further.). The first-order mode shape of the plate appeared at 116.9 Hz under non-contact conditions, and the first-order mode shape of the plate appeared at 811.9 Hz and 964.5 Hz under contact conditions. The repetition of the plate's first-order mode shape within a certain frequency range is caused by the nonlinearity of contact. This occurrence also confirms the frequency-dependent characteristic of contact stiffness, that is, when the excitation frequency is between 811.9 Hz and 964.5 Hz, the value of normal contact stiffness on the interface changes. All modal frequencies below 1000 Hz were considered in this study. And, the fundamental frequency of the fixture is higher than that of 1200 Hz, so the influence of the fixture on the plate within the 1000 Hz can be ignored.