## 2 The Specification of a Gearbox Vibration Signal  
A fault in the gear results in a change in the stiffness. This stiffness can be observed in the vibration spectrum at specific characteristic frequencies. These characteristic frequencies represent the tooth meshing stiffness. The meshing frequency in the vibration spectrum of the gearbox represents the symptoms of a defect in the gearbox, as the meshing frequency changes whenever an MDTF occurs in the gearbox [44]. Considering a gearbox operating under normal conditions, the vibration signature obtained from the gearbox is a stationary signal with tooth meshing frequency; this signal can be formulated as follows [45]:  
\[x_{g}(t)\ =\sum_{p=\mathrm{i}}^{p}X_{p}\cos\bigl{(}2\pi pf_{m}\mathrm{t}+ \varepsilon_{p}\bigr{)}, \tag{1}\]where \(x_{g}(t)\) represents the vibration signal of a gear operating under normal conditions, \(X_{p}\) and \(\varepsilon_{p}\) stand for amplitude and phase of \(p\)-th harmonic of a meshing frequency, \(P\) denotes harmonics of the meshing frequency, and \(f_{m}\) denotes meshing frequency, which can be computed using the parameter of pinion wheel (\(f_{m}\) = number of pinion teeth \(\times\) rotational frequency of a pinion wheel) or using a gear wheel (\(f_{m}\) = number of gear teeth \(\times\) rotational frequency of a gear wheel). The meshing frequency and its harmonics are considered useful components for the fault diagnosis process. Figure 1a shows an example of the frequency spectrum of a vibration signal in the perfect condition.  
A fault in the gearbox makes the vibration signal non-stationary, resulting in a complex frequency spectrum. During the gearbox operation, transmission occurs between the motion source (e.g., three-phase motor and a drive shaft) and a load (a non-drive-shaft and a load) through a pair of gears (pinion wheel and gear wheel). The non-stationary impulses start appearing in the vibration signal when there is an impulsive change in the angular velocity. The angular velocity changes impulsively when the two wheels rotate across a faulty tooth (e.g., missing tooth, cracked tooth, chipped tooth, or worn tooth) [46]. Therefore, the vibration signals obtained from a faulty gearbox exhibit non-stationary behavior, for which the frequency spectrum contains harmonics of tooth meshing frequency, sidebands (the frequency tones are distributed in the two sides of harmonics of a meshing frequency), and other oscillation components. The vibration signal can be presented as a combination of phase and amplitude modulation signal [47], as follows:  
\[x_{m}(t)\ =\ \sum_{p=0}^{P}X_{p}\big{(}1+\beta_{p}(t)\big{)}\ \cos\big{(}2\pi pf _{m}\text{t}+\varphi_{p}+\phi_{p}(\text{t})\big{)}. \tag{2}\]  
Here, \(\beta_{p}(t)\ =\sum_{q=0}^{Q}\text{B}_{pq}\cos\big{(}2\pi qf_{b}t+\gamma_{pq} \big{)}\) and \(\phi_{p}(t)\ =\sum_{q=0}^{Q}\Phi_{pq}\cos\big{(}2\pi qf_{b}t+\varepsilon_{pq}\big{)}\) represent the amplitude and phase modulation functions of the defective vibration signal. \(f_{b}\) is the sideband frequency, \(Q\) stands for the total number of sideband tones around the \(p\)-th harmonics, \(\text{B}_{pq}\), \(\Phi_{pq}\) represents the amplitudes, and \(\gamma_{pq}\) and \(\varepsilon_{pq}\) denote phases of the \(q\)-th sideband in the amplitude and phase modulation functions, respectively. Figure 1b shows the frequency spectrum of the vibration signal obtained from the gearbox under defective conditions; the fault signatures or fault-related components are the harmonics of meshing frequency and sideband frequencies.