This work investigates the free vibrations of a flexible inner cylinder immersed within a concentric rigid outer cylinder, separated by a viscous Newtonian fluid. Motivated by applications such as the Jules Horowitz Reactor, this study develops a novel theoretical framework to predict the vibration frequencies and decay rates. Based on a modal expansion of the cylinder displacement and fluid forces, the framework accommodates arbitrary aspect ratios and boundary conditions, without assuming a narrow gap. To validate the theoretical predictions, numerical simulations of the Navier–Stokes equations were performed using the open-source code TrioCFD. This Computational Fluid Dynamics code is designed for incompressible or quasi-incompressible fluids, coupled with an Arbitrary Lagrange-Euler module. The equations were time discretized using a first-order backward Euler scheme, and a hybrid Finite Element Volume method was used for spatial discretization on unstructured grids. A 3D mesh with 15 million elements, generated by the MGCADSurf-MGTetra mesh generator from the SALOME platform, was employed for the simulations. Nonlinear convective terms were approximated with a second-order MUSCL scheme upwind, while iterative solvers from the PETSc library were used to solve the discrete linear systems. For additional details on TrioCFD. The results reveal that vibration characteristics are strongly influenced by the aspect ratio of the cylinders, the separation distance, the Stokes number, and the density ratio. For the pinned-pinned boundary condition, the theoretical predictions closely align with numerical results for specific Stokes numbers and density ratios, validating the robustness of the model. This research provides a comprehensive framework for understanding fluid-structure interactions in confined geometries, setting the stage for future investigations into other configurations, such as clamped-free systems, and the effect of varying fluid properties.