## 5 Fault Diagnosis Using the Proposed Method
### Case Study 2: Fault Diagnosis of Gearbox  
The common faults of gears include local faults (pitting and teeth broken), distribution faults of worn types, and multiple coupled faults. Accurate fault identification is necessary for the safety of a mechanical system. In this section, a gearbox experimental dataset under different speeds was employed to validate the robustness of the proposed method. The test data were gathered on the gearbox platform shown in Figure 16, which consisted of a gearbox, a diesel engine, a bearing seat, a flexible coupling, a base, etc. The speed of the test system was controlled by electrical machinery. The gearbox contained two gears (pinion and wheel gear); their parameters are shown in Table 1. When the diesel engine ran, the Signal-Noise-Ratio was small. Therefore, the dataset validated the robustness to the noise of the proposed algorithm.  
Figure 16: Platform of multi-fault gearbox.  
Figure 15: Row vectors of \(W\): (**a**) Vectors of \(N_{in}\) = 200 in the time domain; (**b**) Vectors of \(N_{in}\) = 200 in the frequency domain.  
The gear dataset contained four kinds of faults: a coupled fault of wheel pit and pinion worn, a single pit of wheel, coupled fault of wheel broken and pinion worn, and a single worn of pinion. For convenience, the above four types of gear faults are named Type-2, Type-3, Type-4 and Type-5, respectively. In addition, the normal state of the gear is referred to as Type-1. Each fault type was tested under three different speeds. Note that the speed fluctuations also existed among different faults and the values of speeds are shown in Table 2. The dataset of gearbox was averagely divided into two non-overlapping parts which were used as the training set and testing set.  
The proposed method achieved excellent performance in its application to the gearbox fault diagnosis. The diagnosis results, compared with the original method by using different \(N_{in}\), are displayed in Figure 17; notably, the paramater setting is the same as the bearing case. As seen in the figure, testing accuracies improved greatly. In addition, the non-montonicity phenomenon was significantly weak. The diagnosis accuracies with different percentages of training samples are shown in Figure 18. The experimental results show that the proposed method can effectively identify the gear health conditions with different fault types and severities, exhibiting a better performance than the original method.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Gear Name** & **Teeth Number** & **Gear Modulus (mm)** & **Gear Pressure Angle (\({}^{\circ}\))** & **Gear Material** \\ \hline Pinion gear & 55 & 2 & 20 & S45C \\ Wheel gear & 75 & 2 & 20 & S45C \\ \hline \hline \end{tabular}
\end{table}
Table 1: Gear parameters of the test gearbox.  
Figure 17: Diagnosis result using various input dimensions of modified and original sparse filtering.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Speed** & **Type-1** & **Type-2** & **Type-3** & **Type-4** & **Type-5** \\ \hline Speed1 (rpm) & 800 & 825 & 834 & 812 & 822 \\ Speed2 (rpm) & 820 & 849 & 850 & 842 & 845 \\ Speed3 (rpm) & 852 & 864 & 866 & 860 & 861 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Speeds of different fault types.  
To demonstrate the diagnosis information, the confusion matrix of the proposed method is displayed in Figure 19. Notably, the proposed method misclassifies 0.01% of testing samples of Type-2 as Type-1 and 0.02% of testing samples of Type-5 as Type-4. It is possible that the concurrent types were similar, making it more difficult to classify the two faults than other types.