## 1 Introduction
Friction causes mechanical deformation of the rotor at the point of contact in this state. More intense contact, on the other hand, causes the drill string to heat up and lose stiffness over time, which can lead to catastrophic machine failure. As a result, condition monitoring of such a system is required, which is often focused on feature extraction for fault identification in the time or frequency domain. [PERSON] et al., studied the quasistatic stability of an elongated drill string while subjected to torque, axial load, rotational inertial force, and internal mud flow [6]. The inherent frequencies and mode forms of the elongated drill string were presented in their model. Additionally, these features are often load and speed sensitive, and they cannot reliably provide any information about the component problem that has occurred. To address vibration problems in vertical rotors, numerical simulation is essential as it pinpoints the root cause of problems in complex machinery and helps identify parts that need special attention. Therefore, minimizing downtime and maximizing equipment life can be monitored. However, since the majority of signals in mechanical engineering are transient and nonstationary, it also helps in discovering suitable solutions for a variety of nonlinear parameter applications by influencing structural resonance and achieving the best possible results. Methods such as the finite element analysis model of a rotating drill string [7], the Euler finite difference technique [8], the method of disturbance [9], [PERSON]'s fourth-to-fifth-order adaptive Runge-Kutta method [10; 11], and the synchrosqueezing technique [12; 13] were applied to reduce and solve the resulting algebraic equations of the drill pipe model. However, a comprehensive analysis of these issues remains a challenge due to their multidisciplinary nature. Despite the challenges of drill string model formulation, modeling is still considered a powerful method to study vibration propagation along the drill string and to analyze suspected vibration-related failures by decreasing the unwanted string vibration of stems. However, the drill string is not simply a bundle limited to vibrating in the air with mere external forces, but it is capable of operating in a fluid medium with recurrent drill string-casing contact. Many other factors such as fluid pressure, lateral torsional vibration of the drill string, axial vibration of the drill string, the eccentricity of the drill rods, and friction due to the flexibility of the drill string affect the process of drilling. The extraction of drill string characteristics under such operating conditions is usually not measurable in real time, which makes their study one of the difficult and emerging problems in engineering fields. Several works have been conducted on drill string vibrations, and various contributions and simulation techniques have been made regarding the efficient operation of drilling under bit-rock interactions. Since the casing-drill string interaction term is highly nonlinear and unpredictable, mathematical models have been proposed in the literature for the rock-bit interaction, but they do not reflect the dynamics of the drill string under vibration because the lateral/torsional and axial vibrations are strongly coupled. Thus, the objective of this study is to functionally evaluate the effect of inviscid fluid on the friction of a rotating drill string system during casing and bit-rock interactions. To assess these effects, the model of [14] is adopted to compare the dynamic response with and without the fluid-structure interaction mode. The drill string model is also improved based on [15] to include the effects of higher borehole wall friction and fluid propagation around the lateral/torsional/axial deflection of the train drilling. This work's contribution is considering a multi-degree-of-freedom forced excitation parametric drill string with dry and wet friction-induced vibrations in coupled mechanical oscillators.