# Study on Vibration Characteristics of Multi-Beam Structures with Stick and Slip at Joints
## 1 Introduction
The intermittent contact between the two crack surfaces was simulated by switching between intact and cracked beams. The wavelet finite element method was employed to explain the iterative process of solving the nonlinear equation system required for interpreting the solution while analyzing the phenomenon of nonlinear mixing in a beam with lateral edge cracks. [PERSON] et al. [37] proposed a dynamic analysis model for a generally constrained double-beam structure with nonlinear stiffness and damping in its connections, and the Galerkin truncated method (GTM) was employed to solve the nonlinear governing equations of the double-beam structure. The study conducted by [PERSON] et al. [38] investigated the modal coupling resulting from joint friction and explored passive vibration reduction techniques involving friction. Under proximity to the conjugate eigenfrequency, the nonlinear characteristics of friction give rise to nonlinear modal interaction. When harmonic excitation is applied in the vicinity of a specific mode's natural frequency, significant contributions to the vibration response can be attributed to exciting another mode through nonlinear energy transfer. Within the resonance range, adjusting the ratio between the first two natural frequencies allows the triggering or suppression of mode interactions. In certain scenarios, vibration levels can decrease due to energy redistribution. [PERSON] et al. [39] introduced dry friction at the contact interface of an imperfectly supported cantilever and proposed an analytical solution for predicting the nonlinear dynamic behavior of a friction-clamped rod under axial and torsional motion. The dynamic behavior of a cantilever beam supported by contact forces with friction was analyzed by solving the control differential equation using a numerical method known as frequency-time harmonic balance, and the influence of contact parameters on the system's behavior was investigated. [PERSON] et al. [40] investigated the soft nonlinearity of the structural response curve of a bolted connection between cylinders by incorporating a viscous friction model into the joint and calculating the resulting connection force. Loose connections [41; 42] are a prevalent type of mechanical structural fault. The literature mentioned above shows that the characteristic representation of structural fault diagnosis can be enhanced by investigating the laws of the influence of nonlinear mechanical state changes on structural vibration characteristics. Specifically, an in-depth understanding of the nonlinear modalities of connection structures [43] and their responses is crucial for implementing damping measures using friction-damping nonlinear characteristics [44; 45]. This can be developed into strategies for designing vibration suppression friction contacts that minimize energy transmission or maximize energy dissipation [46; 47]. In summary, the existing research has primarily focused on developing friction force function expressions and a dynamic analysis framework that incorporates friction force into structural analysis, ranging from discrete system analysis for nonlinear connection conditions with friction function expressions [48; 49; 50; 51] to continuous structure analysis considering the dynamics of friction force [52; 53; 54]. This work will mainly examine the beam structure that often appears in actual projects, and will analyze the influence of changes in connection parameters on the structural connection state and vibration response, which lays the foundation for vibration reduction and complex structure design.