# Study on Vibration Characteristics of Multi-Beam Structures with Stick and Slip at Joints
## 1 Introduction  
Beam structures are fundamental components in modern machinery, and the dynamic analysis of such structures has always been a significant research focus for scholars [1]. Theoretical models, including those of Euler-Bernoulli [2], [PERSON] [3], [PERSON] [4], and the shear deformation beam [5], have been proposed and extensively investigated. Various numerical methods such as the semi-analytical method [6], [PERSON] method [7], [PERSON] method [8], dynamic stiffness method [9], differential quadrature method [10], and finite element method [11] have demonstrated remarkable capabilities in accurately predicting the dynamic behavior of beam structures. Engineering systems commonly involve beam structures that have diverse types of connections. To accurately describe their mechanical properties, various mechanical models have been developed based on different connection conditions [12]. For instance, theoretical models employing stiffness damping [13], the Iwan model utilizing the density function integral method [14, 15, 16], the Valanis model [17], the Bouc-Wen model [18, 19],and diverse finite element models [20; 21] have been constructed. Analyzing structural vibrations essentially involves solving boundary value problems consisting of governing differential equations and their corresponding boundary conditions. This collective set of connections and conditions can be referred to as the boundary value problem. A research approach that incorporates spring stiffness to simulate boundaries and connections [22; 23; 24; 25; 26; 27; 28] enables a more comprehensive analysis of vibration characteristics within the framework of linear theory. [PERSON] et al. [29] used an improved Fourier series and Rayleigh-Ritz method to establish an analysis model for the self-vibration characteristics of continuous multi-span beams with concentrated mass. [PERSON] et al. [30] used an improved Fourier-Ritz method to study the free vibration problem of the double-beam system under general boundary conditions. [PERSON] et al. [31] provided a general method for analyzing the dynamic characteristics of multi-span beam systems under arbitrary boundary conditions, where the coupling between adjacent beams is described by linear and rotational springs, and each beam is independently elastically constrained/coupled. Based on the Fourier series, the displacement basis function is established, and continuity is not required at the connection, which has strong scalability and can be used for modal analysis research into periodic beam structures. Numerical examples show that under general boundary and coupling conditions, the modal characteristics of the multi-beam system can be accurately calculated. In the linear theoretical framework, the idealized description of the linear stiffness in the connection transitions from 'free' to 'clamped'. However, when considering beam structures connected by bolts or rivets, it is important to account for the relative displacement of the assembly joint surface, caused by transverse vibration, which can lead to nonlinear behavior due to frictional viscous effects [32]. The study conducted by [PERSON] et al. [33] utilized nonlinear stiffness and damping to characterize the axial connection of the rotor system and established a dynamic model of a rotor-bearing system incorporating bolted joints. The impact of connection looseness faults and other factors on the system's dynamic and mechanical behavior was investigated by analyzing the vibration waveform and frequency spectra. The analysis of nonlinear behavior [34], which incorporates the influence of friction on damping [35], is essential for accurately predicting the dynamic characteristics of structures and facilitating structural design. [PERSON] [36] investigated the nonlinear mixing of Timoshenko beams with breathing cracks by considering the crack interface as a bilinear stiffness-connected structure and incorporating two springs to simulate the local bending and shear deformations induced by the crack.