## 4 Methodology  
Based on the above analysis and evaluation, the main diagnostic symptom of gearbox fault is the detection of sidebands that appear around a gear meshing frequency. Only the meshing frequency will be present when a gearbox is operated in good condition. On the contrary, a gearbox with any fault will result in sidebands associated with the gear meshing frequency. As the intensity of sidebands increases, the level of fault also increases. Assume that the faulty gearbox operated stationary, causing sidebands to be observable in the frequency spectrum. Conversely, with non-stationary conditions, the spectrum's sidebands become blurred. Identification of the time-varying frequency changes requires representation of the signal in the time-frequency domain. This section will outline the mathematical model of a faulty gearbox operated in non-stationary situations via simulation and experimentation. Based on the proposed approach, a suitable method is suggested to identify fault symptoms quickly. A measured signal from faulty gearboxes is a multi-component signal formed by modulation of amplitude and frequency. Assume that the signal \(x(t)\) is a multi-component of a faulty gearbox representation in the following [90]  
\[x(t)=A(t)cos(\varphi(t)) \tag{2}\]  
where the instantaneous amplitude of the signal:  
\[A(t)=A(1+B\cos(2\pi f_{s}(t)t)) \tag{3}\]  
and instantaneous phase of a signal:  
\[\varphi(t)=2\pi Zf_{s}(t)t+\phi(t) \tag{4}\]  
Regarding this situation, \(Z\) is the number of the gear teeth, \(\phi(t)\) presents a frequency modulation function; \(f_{s}(t)\) is shaft rotating frequency. Based on the characteristics of the signal, the instantaneous frequency can be determined as follows:  
\[f(t)=\frac{d\varphi(t)}{2\pi dt} \tag{5}\]  
By putting Equations (3) and (4) into Equation (2), it is obtained:  
\[x(t)=x_{1}(t)+x_{2}(t)+x_{3}(t) \tag{6}\]  
in which the component signals are computed in the following manner:  
\[\left\{ \begin{array}{l}x_{1}(t)=A\cos(2\pi Zf_{s}+\phi(t))\\ x_{2}(t)=\frac{1}{2}AB\cos(2\pi(Z+1)f_{s}+\phi(t))\\ x_{3}(t)=\frac{1}{2}AB\cos(2\pi(Z-1)f_{s}+\phi(t))\end{array} \right. \tag{7}\]  
By utilizing Equation (5), the instantaneous frequencies of the component signals can be calculated, resulting in the following outcomes:  
\[\left\{ \begin{array}{l}f_{1}(t)=Zf_{s}+\phi^{\prime}(t)\\ f_{2}(t)=(Z+1)f_{s}+\phi^{\prime}(t)\\ f_{3}(t)=(Z-1)f_{s}+\phi^{\prime}(t)\end{array} \right. \tag{8}\]  
As was said above, it is easy to extract the sidebands when the frequencies do not overlap. However, in cases of time-varying frequency, there will be a phenomenon of frequency overlap that is difficult to extract. In order to overcome the limitation, this study utilizes the Generalized Fourier Transform (GFT) developed by [PERSON] [91]. The GFT is utilized to transform the varying frequency into a constant frequency, hence preventing any overlap. The GFT transformation is performed by multiplying a mapping function \(e^{-i2\pi x_{0}(t)}\) with an analytical signal \(\widetilde{x}(t)\) of the original signal \(x(t)\). At that time, the time varying frequency is mapped to a constant frequency. Assume a multi-component signal \(x(t)\), the GFT transform is defined as:  
\[X_{g}(f)=\int_{-\infty}^{+\infty}\widetilde{x}(t)e^{-j2\pi[ft+x_{0}(t)]}dt \tag{9}\]  
By employing transformations as described in the document [92], it is possible to extract the constant frequency component \(f_{0}\) from the instantaneous frequency according to the following formula:  
\[f(t)=\frac{dx_{0}(t)}{dt}+f_{0} \tag{10}\]  
The above equation proves that the signal \(x(t)\) with time-varying frequency can be straightened out using the mapping function with phase \(x_{0}(t)\) determined by (10). After straightening the frequency, the phenomenon of frequency overlap will no longer exist. Thus, this work proceeds to bandpass filter around the straightened frequency. Afterward, the original frequency is restored by applying the inverse GFT transformation according to Equation (9).