# Diagnosing and Balancing Approaches of Bowed Rotating Systems: A Review
## 2 Materials and Methods
### Smart Inspection
#### 2.1.2 Frequency Domain
Theoretically, an instantaneous phase change (180\({}^{\circ}\)) should occur when the bow phase angle is 180\({}^{\circ}\), whereas in the experimental response, this did not occur. It was claimed that for an unbowed rotor with no runout, the system damping can be determined by measuring the slope of the phase angle in a Bode plot at the critical speed. Without compensation, such a procedure for a bowed rotor frequently results in severe inaccuracies. However, it is projected that adopting such compensation will reduce mistakes to 5% or less. To identify multiplex defects, including the shaft bow, unbalancing, and coupling misalignment in a rotating machine, a model-based procedure was developed by [PERSON], [PERSON], and [PERSON] [39]. They employed a least-squares spectral analysis in identifying miscellaneous defects in the rotor system. The system was modeled by means of the model-based approach. For this purpose, the residual multidimensional vibrations (residual map) between measuring planes and the calculated vibrations resulting from the acting malfunctions were minimized. This procedure could identify the position, module, and phase of the damages. [PERSON], [PERSON], and [PERSON] [53] modeled the equation of motion of a cracked rotor system that suffered a bowed rotor, too. They analyzed the steady-state and transient responses of the defected system. The goal of this research was to observe how the residual bow affects the stiffness of a rotating cracked shaft and what changes, if any, a bow could make to the dynamics of a cracked rotor. The stiffness variation was not significantly influenced by the usual level of the bow, and the nonlinear nature of the crack response was not significantly altered. When the phase angle between bow and crack was zero, the bow tended to keep the crack open. The bow tended to close the crack when the bow phase was 180\({}^{\circ}\) opposite the crack direction. However, the above results showed that if a bow existed out of phase with the crack, the presence of a crack might not cause any changes in the orbit of the asymmetric rotor. The bow, on the other hand, completely obscured the sensitivity of the orbital response of a cracked rotor to unbalance phase at half the critical speed, and the influence of the unbalance phase on the orbital response of a cracked rotor at half the critical speed could not be used for detection. Higher harmonics' directional nature and amplitude, such as the 3X frequency component, did not change significantly. The zero response at the self-balancing speed of the bowed rotor for a specific phase of unbalance was not observed due to the presence of a crack and the influence of gravity. [PERSON] [5], in an experimental work, discovered that 1X and 2X vibration components appeared as a result of a bow in the shaft of a rotating system; a high level of the first axial vibration, i.e., 1X was observed as a sign of the fault in the time domain. Because these harmonic components, i.e., 1X and 2X, were visible at various rotating speeds, the presence of these signs was interpreted to be independent of the system's speed, but by increasing the amount of the system's speed, the vibration magnitude experienced a rise. A vibration analysis technique based on order analysis was proposed for bent shaft diagnosis by [PERSON] and [PERSON] [54]. Both phase and amplitude were obtained through order analysis. The fault type (whether unbalancing or shaft bow) and location were determined based on phase and amplitude. The amplitude of the bent shaft's order spectra was higher at the first order, 1X. The bent shaft was confirmed by the phase difference, which was approximately 180\({}^{\circ}\) between the axial directions at the driver and driven end bearings. [PERSON] and [PERSON] [55] evaluated a cracked-bowed-unbalanced rotating machine that was equipped with an auxiliary active magnetic bearing in a model-based manner (using the least-square fitting procedure). To achieve this, the full-spectrum harmonics, Nyquist, and Bode diagrams were considered.