### Vibration Analysis  
Vibration analysis can be considered a fundamental approach in respect of CM [20]. It has been extensively used over the past years and it is considered the most common process for CM, which is broadly applied in this field. Since vibrations can be found in all rotating machines, a modification or damage to any system component can modify a vibrational signature. Most of the kinds of damage that a rotating machine is subject to have vibration patterns which can be specified in relation to the structure, development, construction, and health state of the machine components. All machines that have moving components can substantially generate sound and vibration from specific characteristics, and these characteristics can change according to their behavioral process if the machine condition changes. Indeed, such change can allow early damage detection before the occurrence of a severe problem [21]. Moreover, vibration analysis is able to detect abnormal vibration events which can enable an evaluation of the overall condition of the object and the identification of variations in its spectra. Then, CM can be used as a pattern recognition process to compare spectra over time and identify these variations. Since vibration analysis is a process that monitors the levels and patterns of vibration signals within a structure, excessive vibration in a machine (e.g., gearbox transmission system) can indicate damage. Indeed, by analyzing spectra, it is possible to identify the frequencies and relate the differences to the baseline (signature) with the mechanical characteristics of the transmission system and thus identify potential defects. With respect to this issue, previous works have reported that the frequencies of a system can shift given the occurrence of damage. Ref. [22] observed differences in the amplitudes of gear harmonics which gave indications of gear faults. Ref. [23] showed that a noise-like distortion in the vibrational response of a structure, for SHM, can be nonstationary and thus requires a time-frequency analysis. Ref. [24] analyzed the relationships among the natural frequencies of the system studied and compared the least affected to the most affected modes of vibration, allowing the method to be more sensitive to damage. Ref. [25] used an SHM methodology to predict the edge frequencies of a wind turbine blade to check the health state of the system. The proposed methodology was able to identify when the blades started behaving differently over time. Ref. [26] researched a frequency band identification framework for gearbox fault diagnosis under time-varying operating conditions. Ref. [27] studied a technique for cases in which the monitoring of vibrations excited by machine operations with accelerometers is required. According to this technique, mechanical coupling between sensors and measurement objects influences high-frequency vibrational responses. In addition, excessive vibration can reflect the characteristics of the damage problem, which is generally related to the ratio of the forces that can act on the gear. Indeed, time-varying mesh stiffness and related errors can affect the dynamic variations of meshing forces [28]. The procedures used for gear vibration analysis can be subdivided into three categories: time domain, frequency domain, and time-frequency domain. Time domain procedures use tools to determine the condition of a rotating machine and methods to collect and interpret data to investigate patterns and trends in terms of time. Frequency domain refers to an analytic environment wherein signals are displayed in terms of frequency. The time-frequency domain covers two techniques, whose investigation comprehends the study of signals in both time and frequency domains. Moreover, the harmonics are considered important characteristics to be analyzed with respect to amplitude and range of frequency. In addition, analysis of the modulation phenomena associated with sidebands may indicate a fault condition in the system depending on the amplitude of the spectra.