# Diagnosing and Balancing Approaches of Bowed Rotating Systems: A Review
## 2 Materials and Methods
### Diagnosing
#### 2.1.1 Time Domain  
Signals captured by transducers, e.g., displacement, velocity, and acceleration sensors from a system in industrial applications, are mostly in the time domain, and time-domain analysis is the study of these raw signals without any transformation into the frequency domain. Although in the current computerized era there is more tendency to use automatic fault diagnosing, time-domain analysis can be performed both in manual and smart manners [14]. Two different operating conditions, \"steady-state\" and \"transient\", can be considered when vibration-based methods are used to assess the health condition of a damaged rotor system. The transient signal is only measurable when the rotating acceleration is changing, i.e., when the system is starting up or shutting down. In contrast, the former signal can be measured throughout the system's operation after it starts up and before it shuts down [15]. A bowed rotor system's vibration signals, including transient and steady-state segments, are shown in Figure 3. The related areas for transient (startup), steady-state operation prior to the first critical speed, during the first critical speed, and steady-state operation following the first critical speed are regions 1, 2, 3, and 4. Manual time-domain analysis can be divided into two subfields based on visual and feature-based inspections. Visual inspection means comparing raw vibration signals of a faulty system with a healthy one in the same frequency. Because real signals are usually low-amplitude and polluted with background noise, visual inspection is not considered a reliable procedure to distinguish differences (damages), but this procedure is low-cost and not complicated. Furthermore, this approach can be effective if a specific segment of the response signal is aimed to be examined, for instance, around the system's critical speeds; in the case of real-world signals, firstly, a suitable noise cancellation method should be performed [17]. On the other hand, feature-based inspection in time domain can be employed both in manual and automatic manners. To extract features in time-domain analysis, there are two main approaches, e.g., statistical functions and advanced techniques. As illustrations of the former, peak amplitude, root mean square amplitude, and variance can be mentioned. Time-synchronous averaging and filter-based methods are examples of the latter [17]. Throughout the history of diagnosing bowed rotors, many researchers have used time-domain analysis both in terms of visual and feature-based inspection. Probable effects of different amounts of the residual bow, phase angle, rotational speeds, and system damping ratio on the behavior of the bowed rotor systems have been evaluated by a number of researchers. Although vibrations caused by pure unbalance and a bowed shaft can both produce synchronous excitation, there are some differences between the two situations, particularly in the phase angle during resonance. Figure 3: Different segments of the transient and steady-state signal of a bowed rotor system [16]. [PERSON], [PERSON], and [PERSON] [6] compared the peak responses of unbalanced and bowed rotors for various amounts of bow amplitude, phase angle, and damping ratio. It was demonstrated that the vibration amplitude can be zero if two conditions are satisfied: first, the residual bow and unbalance must be perfectly out of phase (the phase difference must be 180\({}^{\circ}\)); and second, the rotation velocity should be equal to the square root of the ratio of the residual bow amplitude to imbalance eccentricity. A technique for calculating the ratio between the residual bow and unbalance eccentricity was also presented. The residual bow is smaller than the unbalance eccentricity when the response amplitude tends to be minimum below the critical speed. In contrast, it may indicate that the residual bow is larger than the unbalance eccentricity if the response amplitude decreases to a minimal level in the region above the critical speed.