In this work, we propose a fixed mesh finite element formulation to solve the fluid dynamic on an Eulerian mesh dealing with immersed bodies in motion. The study is focused on the computation of the fluid dynamic forces acting on immersed bodies which strongly depend on the evolution of the vortex shedding. The frequency of vortex detachment for flow past cylinder problems can be modified when the cylinder moves, promoting the modification of the wake of vortices. Synchronization phenomena appear when the frequencies of the resulting flow pattern coincide with the frequency of the imposed body motion. To study this problem, we propose to describe the immersed body surface by a collection of markers that moves according to the imposed body motion. The markers are updated using a Lagrangian scheme. In this framework, a distinct aspect of the present work is the imposition of the body velocity as an internal immersed boundary condition for the fluid dynamic analysis. To transfer the body velocity to the fluid along the fluid–solid interface, a restriction on the flow velocity is added into the weak form of the Navier–Stokes equations by means of a penalty technique. This work encompasses the study of flows past a crossflow, streamwise, and rotational oscillating cylinders. The results are satisfactorily compared with numerical data reported in the literature, showing a proper behavior for the analysis of long-term vibrating systems at low Reynolds numbers.

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