## 2 Materials and Methods
### Modeling in ABAQUS/CAE  
ABAQUS allows modeling a fault in a system whether by variations in the geometry resulting from the damage such as a crack or by employing the expected force due to the defect such as a bearing defect in a rotary system by means of periodic loads. In this article, the first approach, i.e., change in the geometry of the system, is utilized to model faulty rotor systems. As previously mentioned, an unbalancing occurs when the volume (geometry) and mass (mass) centers of a rotating object--in this case, a disk--are at different points. Figure 2 illustrates a schematic of an unbalanced rotor-disk-bearing system. The disk's volume and mass centers are indicated in the figure by the letters \(C\) and \(M\), respectively. On the other hand, a bowed shaft refers to a situation where the rod is not straight along its main axis. A schematic of this assumption is shown in Figure 2 (with exaggeration). The modeled rotating system in ABAQUS in the current work consisted of a shaft, a central disk, and two couplings at the ends. The couplings are modeled as the kinematic type; as the boundary condition, the ends of the shaft are constrained in all directions except rotation around the fundamental axis of the shaft (\(x\)-axis). It should be mentioned that the shaft and disk are considered integrated parts. On the assumption of the bowed rotor, the shaft is drawn along a curved line, where the maximum curvature is in the center of the shaft. C3D8R solid elements (characterized by 8 nodes with 3 degrees of freedom per node) from the Abaqus Library were used to model both the shaft and the disk, using an average element size of 10 mm. For the bowed rotor, a total of 33,880 elements and 39,096 nodes were created. To model unbalancing in the rotor system, a number of holes have been drilled in the disk. By changing their diameters as well as their distances from the disk's center, the mass center of the disk changed (eccentricity has occurred). The FE model counts a total of 33,353 elements and 39,328 nodes which were created with the same solid elements as the bowed system. Graphs in Figure 3 are the meshed models of the unbalanced and bowed rotor systems, subsequently. Figure 1: Workflow of the article. Figure 2: Schematic of a rotor-disk-bearing system: (**a**) unbalanced and (**b**) bowed. To determine the optimal mesh size, a mesh convergence test was conducted to analyze the L2 norm of transverse displacement, i.e., the \(y\)-axis in Figure 3 at the node of the left side journal bearing across different mesh sizes. The distinctions between the various mesh grid sizes were not stark. So, the most suitable sizes were chosen based on the aforementioned values. The explicit dynamic procedure is performed to analyze the systems' motion while the constant rotating velocity is modeled using a predefined field in the load module; vibration signals are captured in the transverse direction, e.g., \(y\)-axis. To compare vibration signals of the unbalanced and bowed rotor systems in the stationary performance, the time domain waves and frequency spectrums (single-side) for four operational speeds, i.e., 50, 75, 100, and 150 rad/s, are plotted in Figures 4 and 5. Mechanical parameters of these four systems are selected the same as are listed in Table 1. To have a reasonable comparison between these two faults, amounts of the disk eccentricity and the initial bow are selected the same--5 mm. It is worthwhile to emphasize that to avoid resonance or semi-resonance behavior that can result in transient changes in the signal, these velocities are selected well below the first torsional natural frequency (subcritical speed range), e.g., 68.29 Hz (-429 rad/s).