## 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
The contact between the flat plate and the fixture in this area is simulated using a shared node approach. Region B, located in the center of the flat plate, is modeled using BUSH elements with high stiffness to simulate the large contact stiffness caused by significant surface pressure. Region C, outside of the aforementioned areas, is modeled using BUSH elements with low stiffness to simulate the small contact stiffness under lower surface pressure. A total of 458 BUSH elements are established in region B, and 4598 BUSH elements are established in region C, using the flat plate node-fixture node approach. The finite element model under the contact conditions between the flat plate and the fixture is established based on the above analysis, as shown in Figure 9, because the plate is in contact with the fixture; the length of the BUSH unit is zero. In the established finite element model, the stiffness values of the BUSH elements need to be determined. Since the flat plate only underwent a sinusoidal sweep frequency test along the normal direction, and it was also confirmed in the simulation that the tangential stiffness of the BUSH elements has minimal influence on the simulation results, the influence of tangential contact stiffness on the normal vibration of the flat plate can be neglected. Therefore, when assigning values to the BUSH elements, only the normal stiffness values need to be considered. In our paper, the approach to calculate the stiffness values of the BUSH elements was based on the commonly used response surface methodology [23, 24, 25] in the relevant field, with some simplifications made according to our needs. The overall approach was to calculate the contact stiffness values based on experimental data, combined with a simplified response surface methodology. To replicate the effect of normal contact stiffness in the simulation, we used the midpoint value between 811.9 Hz and 964.5 Hz, which is 888 Hz, as the dividing point. We divided the vibration response of the flat plate in the 0-1000 Hz frequency range into two phases: firstly, due to the effect of non-uniformly distributed normal contact stiffness, the fundamental frequency of the plate increases to 811.9 Hz in the range of 0-888 Hz, and at the same time, the modal shape changes to the form shown in Table 2 under contact conditions; then, in the range of 888-1000 Hz, because of the frequency-dependent characteristic of the normal contact stiffness, the same modal shape is presented again at 964.5 Hz. Following the analysis above, based on the changes in the modal shape of the plate in the conditions of non-contact and contact with the fixture, it was found that the modal shape performances in the central and edge areas of the plate are the most different. Therefore, points 19 and 22 marked in the experiment were selected as feature points, and the schematic diagram of their locations is shown in Figure 10; the frequency response curves and modal shapes of these points were used as references for determining the values of the normal contact stiffness, and the experimental frequency response curves for these two points are shown in Figure 11.  
Figure 9: Finite element model under contact condition of plate and fixture. The specific approach was as follows: determine the ratio of the normal stiffness of the BUSH units in area B and area C, so that the frequency response analysis shows peaks near 811.9 Hz and 964.5 Hz, respectively, and the ratio of normal stiffness can make the plate vibration mode consistent with the experimental results. Essentially, the approach described above yields the values of normal contact stiffness in the frequency band near the resonance peak. In simulations, we found that the low-order frequency bands far from the resonance peak can be approximated using the same values of normal contact stiffness as those used in the resonance peak frequency band. This is because, when the normal contact stiffness is sufficiently high, the low-order frequencies are still far from causing the plate's resonance response.