Multiple information fusion based on single component feature and diagnosis of gear’s early fault

2.1. Definition of the FRFT

For a signal $x\left(t\right)$, the FRFT can be defined as follow:

The nucleus function of $X_p\left(u\right)$ is:

where $A_{\alpha }=\sqrt{\left(1-j\mathrm{c}\mathrm{o}\mathrm{t}\alpha \right)/2\pi }$, $p$ is FRFT’s important parameter, which can reflect the rotation angle in time-frequency distribution, and the rotating angle can be calculated by $\alpha =p\pi /2$.

2.2. Extraction of LFM signal based on FRFT

A signal having many LFM components is difficult to be separated from each other either in time domain or frequency domain, as shown in Fig. 1. FRFT is able to rotate time-frequency distribution of signal by any angle, so if $\alpha$ and $\beta$ is orthogonal, the component $f=f_0+f_{m}t$ will be filtered from other components, and now FRFT’s best order is $p_0=2\alpha /\pi$.

Fig. 1Extraction of LFM signal based on FRFT

2.3. Multiple information fusion based on single component feature

Diagnosis of gear’s early fault by analyzing only one sensor’s signal is not accurate enough. So, a method of multiple information fusion based on single component feature (MIFSCF) is proposed. Vibration signal is sampled by many sensors installed on different positions of transmission. Meshing frequency component is separated by FRFT as single component signal $F_i$, whose OEDS is calculated next, and the value of OEDS at feature order of gear’s fault is selected as feature parameter $P_i$. $P_i$ is fused by radar graph and MIFSCF is carried out.

4.1. Separation of order components

Rotation frequency and meshing frequency curves of different speeds are shown as Fig. 4. The time-frequency distribution of original vibration signal is shown as Fig. 5, it is difficult to distinguish each component, because different components are mixed together.

Fig. 4Frequency curve of rotate speed and meshing component

Fig. 5Gabor time-frequency spectrum of vibration signal

Fig. 6Gabor time-frequency spectrum of separated components

Fig. 7Order spectrum of separated components

Second speed and fourth speed component are filtered by FRFT separately, the result is added together and Gabor time-frequency distribution is shown as Fig. 6, in which second speed and fourth speed component is clearly separated from other components and noise. In order to test the validity of this method, order spectrum of filtered second speed and fourth speed component is calculated and shown as Fig. 7. Order 12 and 19 is corresponding to theoretical value of second speed and fourth speed, which proves that components separation based on FRFT is valid and correct.

4.2. Extraction of fault feature

OEDS of original vibration signal is calculated as shown in Fig. 8, in which there is no obvious energy amplitude at feature order 0.43, so it is difficult for OEDS to extract the submerged feature of early fault. OEDS of second speed meshing component filtered by FRFT is shown as Fig. 9. Energy of order 0.4375 is evident and close to theoretical order 0.43, which proves that the weak fault feature is effectively extracted by FRFT.

Fig. 8OEDS of original vibration signal

Fig. 9OEDS of second speed signal extracted by FRFT

4.3. Diagnosis based on MIFSCF

8 sensors’ signal of gear in normal state and fault state is separately filtered by FRFT, then OEDS of filtered components are calculated, and feature parameters are listed in Table 1.

Fig. 10Diagnosis based on MIFSCF

Parameters in Table 1 can’t show the diagnosis result of gear early fault directly. So, parameters in Table 1 are fused by radar graph and shown as Fig. 10. Fig. 10 shows that P1, P2, P3 of fault are clearly different from normal state, so normal and fault state are clearly separated. P4, P5, P6 of fault are so close to normal state that difficult to be extinguished from each other. P7 and P8 of normal and fault state are crossed and can’t be used to determine a fault is occurred. So MIFSCF is more effective in diagnosis by fusing multiple information of different sensors rather than only one sensor’s information.