# Diagnosing and Balancing Approaches of Bowed Rotating Systems: A Review
## 2 Materials and Methods
### Diagnosing
#### 2.1.1 Time Domain
The phase angle for rotors with a bow magnitude less than the eccentricity diminishes with increasing speed from the slow runout condition up to roughly the self-balancing speed, then increases. In [21], [PERSON] and [PERSON] experimentally confirmed the aforementioned results, too. [PERSON] and [PERSON] [3] developed a mathematical model of a bowed rotating system to investigate the effects of a bowed component (shaft) on the vibrations of the bowed and straight parts of the system. The system was assumed to have several different initial bends, also various operation speeds, then vibration signals were analyzed in the disc position and in the shaft in the time domain. The vibration of the bent part was primarily affected by the bending force, while the vibration of other straight parts was less affected. The intensity of the vibration response was mostly governed by the amount of bend. The vibration mode was generally stationary, with a lateral and vertical difference. If the cooling process in a rotor system is not performed correctly, e.g., the system does not work in turning gears, then restarts immediately, the occurrence of a thermal bow is plausible. In this case, the vibration amplitude can be greatly influenced by the thermal gradient [22]. Several researchers have studied the probable impacts of a thermal bow in rotor systems, as well as methods for diagnosis of this defect. To separate the vibrations resulting from an electrical failure and a bowed shaft, [PERSON] and [PERSON] [23] proposed an approach based on the vibration amplitude in time domain. Observing the motor immediately after power cutting off, a bowed rotor can be determined. Although vibration suspension immediately after disconnecting the power supply can be a sign of an electrical failure in the device, a gradual decrease in vibration can signify a bend. If the motor is tested quickly after starting, it is probable that no defect will be revealed. After operating temperatures and loads have been reached, temperature inspections of the equipment can be informative. In diagnosing a thermal bow in a power unit generator, a model-based identification approach was used by [PERSON] and [PERSON] [24]. During shutdowns and runups, the transient 1X vibrations were measured. A thermal bow was discovered using differences in 1X vibration amplitude during a sudden shutdown versus a planned shutdown. Bending moments were applied to the FE (finite element) model of the system to model a thermal bow on the rotor. Furthermore, by comparing experimental and modeled results, it was determined that the current fault is a bow rather than an unbalance. The accuracy of the fault identification was assessed by statistically analyzing the errors resulting from the experimental and theoretical responses. [PERSON] and [PERSON] [25] found that the thermal bend can be removed naturally if the rotating machine starts working. They also observed that the highest vibration level resulting from a bent shaft is measured when the highest temperature difference is measured. To identify a thermal bow, [PERSON], [PERSON], and [PERSON] [2] dealt with a case study consisting of a combinatorial rotary machine made up of a low-pressure turbine, a high-pressure-intermediate-pressure steam turbine, and a generator. The flexural critical velocities of the first balance resonance were split in two. In the theoretical model, the faults were simulated by various equivalent forces and excitations. The difference between the vibrations measured during runup and rundown and in speeds close to the first balance resonance caused suspicion of a thermal root fault. On the 1X vibration, the effects of unbalancing and shaft bow appeared to be identical. A coherence factor that measured the error between experimental and theoretical vibrations was used to distinguish distributed unbalancing (shaft bow) from local unbalance; if the amplitude of the vibration changes with the temperature, the fault is assumed to be bow rather than unbalancing. [PERSON] et al.