###### Abstract
[PERSON] and [PERSON] employed a model-based method in two various procedures, i.e., equivalent loads minimization and vibration minimization to investigate the effects of unbalancing in a rotor system. It has been found that the main symptoms of this impairment in the time and frequency domain are the increased vibration amplitude and the appearance of 1X frequency harmonic, respectively [6]. Rotating machines can experience distributed unbalancing, or shaft bow, for a variety of causes, including creep, thermal distortion, or a significant imbalance force. Shaft bows can occasionally be the result of too much heat, a long length, or a physical bend. This fault can be fixed in its initial phases, somewhat like a concentrated unbalancing [7]. On the assumption that the malfunction is permanent, a limited number of modification procedures from balancing and optimization (to relocate other components) can be carried out. The presence of this malfunction may occasionally be transient and can be resolved on its own [8]. Since, in some circumstances, the temporary shaft bending is removed soon after the device starts, the analysis of start-up and shot-down signals has, thus far, been the primary focus in the study of bent rotor systems. Although both model-based and data-driven approaches have been researched, the quantity of recent studies is not particularly impressive when compared to some defects, such as concentrated unbalanced or cracked systems; most researchers focus on contrasting this defect with imbalance. Debates on the specific signs of this damage go far beyond unbalancing because it has been discovered that the significant indicators of this fault in the time domain are an increased vibration response [9, 10, 11, 12, 13, 14] although the severity of the vibration is essentially constant, and it is not significantly impacted by changes in rotating speed [15]. The self-balancing circumstance refers to a very scarce event in which the unbalancing and bow vectors are completely out of phase, as well as having the same magnitude also was observed [16, 10, 17]. The steady-state analysis also revealed changes in the critical speed and phase difference [18, 19]. In the frequency domain, 1X, 2X, and very rarely 3X frequency components have been announced as possible characteristics [20; 21; 22]. A comprehensive review of the previous investigations can be found in [7]. There is a long history of employing the finite element method (FEM) to model and troubleshoot rotating systems. In fact, many complex systems that were previously impossible or challenging to model using numerical methods have been modeled, thanks to the advancement of computer science and the emergence of cutting-edge software such as ABAQUS using FEM. A rotor system was modeled by [PERSON] and [PERSON] [23] using the FEM, which is valid until now. Characteristic matrices of the system were extracted; the force vector resulting from an unbalancing was presented. Additionally, [PERSON] et al. [16] utilized FEM to model a bowed rotor system. A cosine function was applied to simulate the harmonic nature of a bowed shaft's force by multiplying it by the stiffness matrix and the initial bow vector. [PERSON] and [PERSON] [24] used ABAQUS to simulate the effects of a crack's breathing behavior on a rotor system; for this, a quasi-static approach was applied. Primarily, uncertainties should be taken into account in the design of mechanical systems, especially in dynamic systems to avoid unexpected vibrational responses [25]. On the other side, the effect of uncertainties should be considered when theorizing and designing such systems' fault detection mechanisms as well, since as will be discussed below, fault uncertainty is also one of the sources of uncertainty. Damage symptoms should be distinguishable from those of other defects when the physical and/or operational conditions of a system are changed.