# Diagnosing and Balancing Approaches of Bowed Rotating Systems: A Review
## 2 Materials and Methods
### Smart Inspection
#### 2.1.2 Frequency Domain  
The most common tool in industrial applications for condition monitoring of rotary machines is the Fourier transform (FT). This method provides indications of the amount of a distinct frequency band in a signal [17]. It should be emphasized that methods such as Bode plot, Nyquist plot, and order analysis, which are based on the frequency response or tracking the higher frequency components, are placed in this category. Experimental observations in several comprehensive books and handbooks expressed that the incidence of higher-frequency harmonics (further than the 1X) is a sign of a bowed rotor system. In [48], [PERSON] stated that according to the graphs plotted using the fast Fourier transform (FFT), the 1X frequency component is the typical symptom of a bowed, and also a misaligned, rotor system. The 2X component, as well as the 3X and 4X components, may appear frequently depending on the severity of the fault and its location. Radial and axial vibrations are appropriate for the detection of these two faults. [PERSON] [49] claimed that the 1X frequency component is visible in the frequency spectrum due to faults such as unbalance, bowed shaft, misalignment, bearing clearance, bearing looseness, and so on. Furthermore, the second frequency harmonics, 2X, can be produced by misalignment, loose internal components, and sometimes a thermally bowed shaft. [PERSON] [50] declared that, in addition to analyzing vibration signals in the time domain, where both the radial and axial amplitudes increase and a 180\({}^{\circ}\) phase difference exists in these two directions, the frequency domain should be considered as symptoms of a bowed rotor system. Because of a shaft bend in this domain, 1X and 2X frequency harmonics can be seen; the magnitude of the 2X component can lead us to the approximate location of the bow. Furthermore, some authors evaluated the effects of a bowed shaft in the frequency domain theoretically and experimentally. [PERSON], [PERSON], and [PERSON] [51] investigated the stability and unbalance response of a bowed rotor system with three different types of supports. A bow in the shaft was revealed by the presence of 2X and 3X frequency harmonics. Other small peaks appeared at various speeds although they were ascribed to resonances (structural ones) from the driving motor and noise, as well as coupling. Oil whip occurred at approximately twice the first rotor critical speed. Axial groove bearings were better for low-speed use and near the critical speed. At the first critical speed, the pressure dam bearing exhibited controlled vibration, but a whip occurred at twice and three times the critical speed. Tilting pad bearings, on the other hand, showed increased vibration near the critical speed but did not whip at the greater speeds. [PERSON] et al. [52] compared the unbalance response in the presence of mechanical/electrical runout, and shaft bow for a Jeffcott rotor model experimentally and theoretically. A compensation vector was extracted from the response at a low speed where the dynamic effects were negligible. The low-speed compensation was subtracted from the responses in other speeds to distinguish bowed shaft response. To study the responses, Bode (for the synchronous responses) and Nyquist (for the complex responses) plots were created. The maximum magnitude of vibration for a bowed rotor was dramatically dependent on the bow phase angle (angle between the bow and mass unbalance), whereas the response of the rotor with a runout had a poor dependency on the angle between the rotor runout vector and timing mark. In contrast with the bowed rotor, the minimum response in the case of the runout rotor was not zero. In the Nyquist plot, the shape of the response for the bowed rotor was close to a circle. In the case of the bowed rotor, the size of the response circles was a function of the bow phase angle, whereas the size of the runout system was not related to its phase angle.