## 1 Introduction  
In mechanical structures, the surfaces that come into contact with each other during the assembly of different components are commonly referred to as \"bonding interfaces\" or \"contact interfaces.\" For example, bolted connections are widely used in industries such as aerospace, automobile, and shipbuilding due to their reliability, high strength, and ease of disassembly [1, 2, 3, 4]. The contact stiffness of the mechanical interface is an important component of the overall stiffness of the mechanical structure. The contact stiffness significantly influences the dynamic response of the mechanical structure. However, due to the strong nonlinearity of the contact stiffness in the mechanical interface, modeling methods for contact stiffness have been a focal point of research. When modeling and simulating bolted joint interfaces in finite element analysis, researchers commonly employ virtual materials or generalized spring elements to simulate the contact conditions of the interface. [PERSON] et al. [5] assumed the micro-contact portion between two contacting surfaces of a machine tool as a virtual isotropic material and derived analytical solutions for the elastic modulus, shear modulus, Poisson's ratio, and density of the virtual material for application in finite element analysis, and the comparison between the theoretical and experimental vibration modes showed good agreement. [PERSON] et al. [6] utilized thin layer elements in finite element analysis to simulate contact stiffness. By varying the stiffness of the thin layer elements, they were able to replicate experimental phenomena. [PERSON] et al. [7] proposed a gradient virtual material model based on the uneven pressure distribution on the bolt connection interface. The modeling method mentioned above can simulate the stiffness and damping behavior of bolt connections to a certain extent. [PERSON] et al. [8] proposed modeling the rough mating surfaces of bolted connections using a virtual material layer and validated the effectiveness of this method through numerical analysis and experiments. [PERSON] et al. [9] utilized a nonlinear virtual material simulation based on surface contact stress to analyze the dynamic performance of bolted components. [PERSON] [10] used experimental methods to calculate the normal and tangential characteristics of the contact area in a flange connection. They then simulated the contact stiffness of the contact layer using spring elements. [PERSON] et al. [11] utilized beam elements and spring elements to simulate the elastic characteristics in the analysis of multi-bolt connections. This approach allows for an effective reproduction of the experimental results. [PERSON] et al. [12] addressed the uneven pressure distribution on the bolt connection interface in thin plates by employing displacement-correlated, nonlinear elastic spring elements with non-uniform distribution parameters for finite element analysis. The simulation results successfully reproduced the soft nonlinear vibration phenomenon of bolt-connected thin plates. [PERSON] et al. [13] developed mathematical and finite element models for multi-plate structures connected via flange geometry bolts. In the finite element model, they incorporated spring elements to simulate the changes in connection stiffness under different conditions. [PERSON] et al. [14] simulated the inconsistent preloading conditions of bolts by assigning different parameters to the spring elements. They validated the accuracy of the model through modal experiments. In the engineering field, some scholars have also conducted research on the phenomenon of variable stiffness in actual structures; [PERSON] et al. [15] noted that the varying mesh stiffness in gear systems can cause the system's natural frequency to be significantly different from that of a system with constant stiffness. [PERSON] et al.