## 1 Introduction  
Mechanical driveline systems are employed widely throughout the industrialised world for power transmission. Their applications can be found in automobiles, petrochemical facilities and marine propulsion systems, etc. When it comes to coupling parallel shafts in a fluid medium, high vibrations, mechanical damages and contact friction of specific components are very common, and recurrently lead to premature wear of the operating structures. In practice, a perfect Cardan shaft alignment condition is always present in the maritime and automobile industry. The Cardan shafts (Hooke's joint) constructed to connect two machines that are offset from each other are often challenging for monitoring due to their size, function and accessibility; this makes their alignment study one of the most commonly misunderstood issues a ship's operator can experience. In offset shaft systems, disturbances are often produced by a rub-impact between two components in the system or an unbalanced rotating shaft mechanism. Such frictional vibrations begin spontaneously from the reduction of the clearance gap of an unbalanced shaft, which leads to an increase in amplitude until any nonlinear contact effect limit is reached. Thereare many types of frictional contact in machinery and, in most cases, the impact of the vibration contact poses a hazard to the entire driveline system. The forces acting on the vibrating mechanism are frequently external to the system and independent of the various motions within the system. The authors in [1] considered the dynamics of a Hooke's joint and a crack-induced parametric excitation model using the wavelet technique to analyse the torsional vibration of a ground vehicle propeller shaft system. Furthermore, a crack-induced excitation was identified in a two-axis model for a propeller shaft. The identification considered the torsional and elasticity of an unbalanced and misaligned shaft system using numerical and experimental techniques [2]. Study of the nonlinear dynamic response of a rotor system with multi-faults including unbalance, asymmetric shaft and angular misalignments were numerically investigated by [PERSON] et al., and the influences of the stochastic fault parameters were analysed using a polynomial chaos expansion method for different kinds and levels of uncertainties. The authors demonstrated that variations in faults such as unbalance, bow and parallel misalignment may affect all of the harmonic components of the rotor system [3]. Following the same idea, [PERSON] et al. studied the dynamic response of a misaligned rotor using a nonintrusive analysis method based on Legendre collocation to consider the uncertainties. The results of the multi uncertainties demonstrated significant influences on all harmonic components of the dynamic responses of the nonlinear vibrations in misaligned rotor systems with interval variables [4]. [PERSON] et al. used a nonlinear dynamic model for a four-wheel-drive vehicle driveline connected by a Cardan joint. In this regard, the authors performed the Lagrange formalism to analyse the vibration of the driveline coupling in the torsional and lateral directions. Their study showed that the second-order vibration and noise are generated by the Cardan joint, and the noise can be decreased effectively by reducing the intersection angle of the Cardan joint [5]. Several studies of flexible Cardan shaft instabilities have been conducted to shed more light on the dynamic behaviour of the lateral and torsional vibration caused by the Cardan joint on a vehicle's driveline, except for fault diagnosis in the fluid domain. In the available literature, few research works have addressed the study of specific faults in flexible rotor systems operating in a fluid medium. [PERSON] and [PERSON] [6] studied the dynamic response of a cracked elastic shaft under the laminar flow excitations. Their analysis demonstrated that the critical speed and the whirling radius of a rotor decrease with an increase in liquid viscosity between the shaft and the cylindrical wall.