## 1 Introduction
In the same order of ideas, works relating to fluid-shaft interaction under the condition of breathing crack and rubs of a flexible shaft under the boundary layer excitation have been reported in [7; 8; 9; 10; 11]. Researches on the dynamic response of the misalignment rotor with specific faults exhibiting nonlinearity, such as breathing crack, rub, etc., on the driveline is limited. More importantly, there are discrepancies in the techniques of identifications and findings from different investigators. More so, excitation to the flexible structure and fluid forces are provided using a mathematical model of a rotating shaft in a fluid medium. In these works, excitation to the flexible structure and fluid forces are provided using a mathematical model of a rotating shaft in the fluid medium. Despite the effects of torsional and lateral shaft deformation being considered, the dynamic fluid forces imposed by the fluid on the shaft allowed the identification of the characteristics of the crack. The authors of [12; 13] carried out numerical and experimental investigations on flow-induced vibration in a low Reynolds number and low Mach number flow stream to an external exciting flexible plate. The nonlinear Navier-Stokes equations considered bidirectional fluid-structure interaction aided to calculate the flow field. They found that flow-induced and external excitations greatly influence the dynamic performance of the coupled system in turbulent and unstable flow. For a particular combination of frequency and amplitude of the external excitation, an experimental study of the vibration induced by the flow of an externally actuated flexible structure is reported in [13]. They investigated, experimentally under a wind tunnel, the flow-induced vibration of a harmonically actuated flexible plate in the wake of an upstream bluff body. They observed that with the help of a laser vibrometer, a pressure microphone and a high-speed camera, for a certain range of excitation frequency, the vibration of the plate decreases first, reaches a minimum value and then increases with a rise in external excitation. Similarly, the repeated failure of a flexible rubber coupling in a similar vessel with a \(Z\)-drive propeller was investigated in [14] using torsional vibration theory and global torsional vibration. In a power transmission system using a Cardan shaft, such as the propulsion system, the authors concluded that an elastic coupling should be used because it offers more radial flexibility for inhibiting the self-excited torsional vibrations effects. To ensure safe design and reliable operation, numerous studies investigated the dynamic characteristics of misalignment shafts connected to a universal joint in the transmission system. However, nonlinear analysis of the misaligned rotor-stator coupled with fluid forces is still limited due to the difficulties in obtaining the analytical nonlinear model of the misaligned shafts and hydrodynamic forces from the complicated fluid dynamics. The current trend towards higher speeds performance and driveshafts subjected to torsional, bending and recurrent contact in the fluid medium with any component of the system reinforces the importance of the accurate modelling of rotor-bearing systems. To overcome that difficulty, the present work contributes to the dynamic analysis of a misaligned flexible rotor-stator system subjected to rub effect in a viscous medium. The system is modelled based on the assumptions that the driveshaft during operation must transmit its maximum torque developed by the engine, and must also operate at constant varied speeds and angles between the transmission and the driven shaft. The model consists of a system of five-degree freedoms, four of them associated with the bending of the shafts, and the other linked to the rotating driveshaft. The use of two-dimensional Navier-Stokes equations based on the change in flow field due to the oscillating driveshaft was to establish the hydrodynamic forces.