# Diagnosing and Balancing Approaches of Bowed Rotating Systems: A Review
## 2 Materials and Methods
### Diagnosing
#### 2.1.1 Time Domain
Influences of three faults, i.e., unbalancing, initial bow, and disc skew, were investigated by [PERSON] and [PERSON] [4] for a simply supported rotor system. Furthermore, the possible impacts of changing the disc position alongside the shaft were investigated. In the case of a thick disc, the square root of the ratio of initial bend magnitude to lumped imbalance eccentricity was found to be equal to the first self-balancing speed, and in the case of a thin disc, it was also shown to be equal to the self-balancing speed. The imparted moments acted as restoring moments in the thin disc case, pitching the disc in the opposite direction of the initial skew. The applied moments, on the other hand, tended to raise the disc pitch in the same direction as the original skew in the thick disc casing. The first critical speed shifted somewhat as the moment of inertia difference decreased, but the second critical speed varied dramatically as the moment of inertia difference decreased. Moreover, the critical speed was discovered to be only a function of the disc location between bearings, also the moment of inertia differential. The magnitude of the vibration was observed to reach its maximum level when both the disc and the maximum initial bow of the shaft were placed in the center of the shaft. Both the translational and rotational vibration magnitudes decreased as the spinning speed raised when the damping was further than 30%. In [18], [PERSON] stated that the speed at which the rotating system operates can change the effects of the shaft bow. If the rotor system works below the critical speed, the effects of the unbalancing are smaller than the shaft bow. The effects of these two defects, i.e., shaft bow and unbalance, are the same at the critical speed, although at speeds more than the critical one, the influence of the shaft bow is masked by the effects of unbalance. [PERSON], [PERSON], and [PERSON] [19] proposed a novel process to identify parameters of the excitation forces and the foundations of a bowed rotating system in a single rundown or runup. Moreover, the shaft's bent profile was estimated with reasonable accuracy. Using this procedure, the magnitude of unwanted vibrations resulting from unbalancing and bowed shaft neutralized by about 92%. It was stated that the vibrations in the bent shaft have a 180\({}^{\circ}\) phase angle difference at resonance speed. In [20], [PERSON] studied the influences of various conditions of an initial bend on the behavior of rotor systems. It was stated that at resonance, the response phase shift is independent of the damping ratio, but changing the amount of the residual bow can alter the phase angle. The whirl amplitude decreases in the slow runout operation, parallel to a rise in the speed, if the bow and unbalancing are in the opposite location and the amount of bow is smaller than the eccentricity. When the bow amplitude exceeds the eccentricity, two different conditions can occur. If the bow amplitude is at most twice the eccentricity, the rotor whirl amplitude falls to zero after resonance and then rises to a unit value as the speed increases. Because the amplitudes have already fallen to very low values, it may be difficult to notice the trend in the previous note for large values of rotor bow (further than twice) in comparison to eccentricity. There were differences in the observed trend between low-speed and high-speed operations. Because the unbalance force is proportional to the rotor system's angular velocity, the bowed force was dominant at low speeds; however, the unbalance force could mask the bowed force effects at high speeds. The phase angle in the slow shutting down (transient signal) presents the angle of the bow position to the imbalance eccentricity. If the phase angle is zero or close to it, the rotor may not have a bow, or it may have a bow in the same direction as the unbalance.