# Flow-Induced Vibration of Cantilever Type Elastic Material in Straight Tricylinder
## 1 Introduction  
The fluid-structure interaction of slender flexible cylinders in a cross-flow is pivotal in various engineering applications, particularly in nuclear reactors, low-frequency energy harvesting, and offshore engineering involving cantilever slender flexible structures [1, 2]. The elongated offshore structures are prone to VIV when exposed to ocean currents. [PERSON] [3] highlights that VIV is characterized by nonlinearity, self-excitation, self-limitation, and multiple degrees of freedom. Even the simplest form of VIV, such as the single-cylinder vortex-induced vibration in two-dimensional uniform flow, remains incompletely understood. In the case of slender and flexible cylinders, it has been observed that the dynamics of VIV response are chaotic, complex, and consist of unpredictable motions [4]. A vibrating flexible cylinder subjected to cross-flow generates a complex vortex wake that significantly influences the trajectory of response motion, phase synchronization between fluid force structural motion, as well as the interaction between the in-line (IL) and cross-flow (CF) responses. For cantilevered flexible cylinders with fewer constraints imposed on them, predicting the characteristics of wake flow becomes more challenging. At present, the study of vortex-induced vibration of flexible cylinders has attracted the attention of many scholars [5, 6]. [PERSON] et al. [7] checked various early empirical models (IVVA, VIVANA, VI-Co Mo, SHEAR7, and ABAIV) according to the experimental results of flexible cylinder vortex-induced vibration and found that there were some errors between the model prediction results and the experiment. [PERSON] et al. [8] studied threerelatively new models, SST-PANS, SST-SAS, and SST-IDDES, and the results showed that SST-IDDES had the best simulation results. The numerical results from SST-IDDES show its comparable capabilities for the simulation of massively separated hydrodynamic flows and its potential application in the prediction of industrial turbulent flows for vortex-induced motions (VIM). [PERSON] [9] studied the coupling mechanism of the cross-flow direction and forward-flow direction of flexible cylindrical VIV and found that the fluid force coefficient was affected by the phase between CF and IL displacement. In addition, when the rigid cylinder VIV occurs, the structure does not deform, and the fluid force at different section locations is roughly the same, while the flexible cylinder has a large length-diameter ratio (the ratio of length to diameter), and the form of vortex shedding and the fluid force change irregularly along the axis. VIV has typical three-dimensional characteristics, and the spatial distribution of fluid force needs further study. In research on three cylinders, [PERSON] [10] studied the flow past three circular cylinders in equilateral triangular arrangements at Re = 50-200 from the viewpoint of vortex-shedding suppression. Their results showed that for one upstream cylinder and the other two side-by-side downstream, the vortex shedding from the upstream cylinder is suppressed for Re = 100-200. However, for two upstream cylinders side-by-side and one downstream cylinder, the suppression occurs on the downstream cylinder at Re \(\leq\) 175. At a high Re = 1.1 \(\times\) 10\({}^{4}\)-6 \(\times\) 10\({}^{4}\), [PERSON] [11] studied the vortex-induced vibration of a cylinder with two degrees of freedom via experiments and found that the amplitude of the lateral displacement response could reach a maximum of 1.35 times the diameter. [PERSON] et al. [12] simulated the flow past three circular cylinders with the incidence angle of 30\({}^{\circ}\) for 200 \(\leq\) Re \(\leq\) 3900 and 1.25 \(\leq\) D/L \(\leq\) 4.0, and five flow patterns were identified. The boundary spacing ratios between the five regimes are affected by the Reynolds number. With the increase in the spacing ratio or the Reynolds number, the three-dimensionality of the wake flow becomes stronger.