# Diagnosing and Balancing Approaches of Bowed Rotating Systems: A Review
## 2 Materials and Methods
### Diagnosing
#### 2.1.1 Time Domain
Parallel with an increase in the initial shaft bow, the magnitude of the lateral response, contact ratio, and the gear pressure angle raised. If a higher external force will be applied to the system, the initial shaft bow can increase the gear pressure angle, and also the contact ratio. Conversely, if the phase angle of the initial shaft bow increases, the gear contact ratio diminishes. [PERSON] [30] investigated the translational movement of a two-stage geared rotor-bearing system under the effects of deformation in a spinning shaft. The contact ratio of the gear and pressure angle were considered time-dependent variables. The residual shaft bow increased the magnitudes of lateral vibrations, the phase angle of gear pairs, and the contact ratio in the geared system. This indicated that the presence of residual shaft bow had a significant impact on the system's lateral response, increasing amplitudes and thus destabilizing the system. Multiple scholars have studied the impacts of a residual shaft in rotating machines equipped with fluid film bearings. [PERSON] and [PERSON] [31] utilized the transfer matrix method for a multi-mass rotor system with fluid film supports suffering unbalance, shaft bow, and disc skew. The effects of these three faults in a water pump were illustrated. It was shown that because of the shaft bow and disc skew, the unbalance response at the critical speeds changed dramatically. The synchronous responses in the first three critical speeds were evaluated. It was claimed that the shaft bow was dominant (increased the amplitude) at the first critical speed, while the effects of both shaft bow and disc skew were visible at the second whirling speed. The effects of disc skew, on the other hand, were dominant at the third critical speed. Theoretically and experimentally, [PERSON] and [PERSON] [32] investigated the synchronous responses of an unbalanced and bowed rotor system. Effects of four different types of fluid film bearings, i.e., pressure dam, four-lobe, tilting pad, and two kinds of axial groove, on the response of the malfunctioned system were researched; they used the transfer matrix procedure to extract the equation of motion. When rigid supports are used and the shaft bow and unbalance have the same amplitude and a 180\({}^{\circ}\) phase difference, the response reaches zero at the critical speed. The vibration amplitude of fluid film bearings, on the other hand, is not exactly zero in the same situations but is nearly constant. [PERSON], [PERSON], and [PERSON] [33] solved the equation of motion for a bowed rotor system consisting of fluid film bearing, then compared the responses in midplane and bearing points experimentally and theoretically. It was stated that there is no self-balancing speed in the case of fluid film bearing, but the system response is at its minimum magnitude. Although the peak amplitudes decreased when the disc was closer to the fluid bearing, moving the disc along the shaft length had no effect on the self-balancing speed. When measuring at the mid-plane, the low-speed shutdown was introduced as a good tool for recognizing the shaft bow, but measurements at the fluid bearing did not yield a reliable result. In contrast, phase measurements in mid-plane and fluid bearing at low speeds accurately represented bow angle. The effects of a residual shaft bow on the onset of rubbing, as well as the contact force between the rotor and stator in rotating systems, have been examined too. [PERSON], [PERSON], and [PERSON] [16] modeled the rotor-stator contact force in two ways, linearly and nonlinearly, and for various compositions of mass unbalancing and shaft bent. The resulting responses of the permanent contact were shown to be independent of the modeling assumption for the force in question, while for the first contact, the nonlinear contact hypothesis sowed greater response. The numerical results were tuned with the experimental ones.