# Simulative and Experimental Investigation of Vibration Transfer Path at Gearboxes
## 5 Results and Discussion
### Experimental Results  
The fast Fourier transform is used to discuss the experimental results in the frequency domain. As described in Section 4, there are two rotating accelerometers mounted on a ring that is attached to the pinion. With rotating sensors, the signals are amplitude-modulated due to the rotational frequency and gravity. The amplitude modulation can be presented through Equation (11):  
\[F(x(t)\cdot\sin(2\pi f_{s}t))=\frac{1}{2\cdot\mathbf{i}}(X(f-f_{s})-X(f+f_{s})), \tag{11}\]  
where \(F\) is the Fourier transform, \(x(t)\) is the original signal, \(X(f)\) is the Fourier transform of \(x(t)\), \(f_{s}\) is the rotational frequency of the shaft, and \(f\) is the frequency. Due to the amplitude modulation by the carrier signal with the frequency \(f_{s}\), the frequency spectrum of the original signal shifts left and right by \(f_{s}\). The effect of the amplitude modulation on the real signal from the rotating accelerometer is shown in Figure 8. In this work, the first three high harmonics of the gear mesh frequencies are discussed. Hence, the upper limit of the frequency spectrum is set to 800 Hz. The high-frequency part of the spectrum will be discussed in the future. These lower frequencies are exclusively investigated, as former works showed that mainly the frequencies around the GMF and between the GMF and the first one higher are affected by gear damage. The investigations at multiple torques and speeds show that vibration levels below \(8\times 10^{-3}\) mm/s\({}^{2}\) can be considered as noise. Several peaks can be observed in this frequency spectrum. At 500 rpm, the rotational frequency is 8.3 Hz. The Outer Race Defect frequency can be calculated in Table 2 and equals 49.8 Hz. The gear mesh frequency is 184 Hz. Additionally, the higher harmonics of these frequencies can also be observed. Due to the amplitude modulation, the influence of the gravitation can be observed at 8.3 Hz. Additionally, due to the amplitude modulation, the peaks at the Outer Race Defect frequency, the gear mesh frequency, and their high harmonics shift left and right by the  
Figure 7: Positions and measurement directions of accelerometers at pinion and housing: (**a**) two rotating sensors at pinion—front view; (**b**) two housing sensors—rear view. rotational frequency, and these shifted peaks are called sidebands [41]. For example, the two peaks beside the peak at the gear mesh frequency (184 Hz) are located at 175.6 Hz and 192.4 Hz. The difference between these two frequencies and 184 Hz is exactly the rotational frequency (8.3 Hz). Also, the difference in the amplitude of the main excitation frequency and that of the sidebands is small because of the amplitude modulation. The post-processed signal, in which the effect of the amplitude modulation is eliminated, is shown in Figure 9. The peak in Figure 9 at the main excitation frequency is much larger than the sideband. Hence, post-processing of the signal is necessary. Figure 8: Frequency spectrum of signal from Channel 2 at 1000 Nm and 500 rpm. Figure 9: Frequency spectrum of signal after post-processing at 1000 Nm and 500 rpm. To avoid the effect of the amplitude modulation, the signal from Channels 1 and 2 should be added [40]. The signal after this post-processing presents only the tangential acceleration. As shown in Figure 10, the measurement directions of these two accelerometers are opposites. The theoretically measured signals can be calculated using Equations (12) and (13). \[a_{1}=\ddot{y}\cdot cos(\alpha)+\ddot{z}\cdot sin(\alpha)-\ddot{\alpha}\cdot r \tag{12}\]  
\[a_{2}=-\ddot{y}\cdot cos(\alpha)-\ddot{z}\cdot sin(\alpha) \tag{13}\]  
where \(a_{1}\) and \(a_{2}\) are the measured accelerations from these two sensors, \(\ddot{y}\) and \(\ddot{z}\) are the accelerations of the center point of the pinion in the y and z directions, and \(\ddot{\alpha}\) is the angular acceleration of the pinion shaft. The sum of \(a_{1}\) and \(a_{2}\) is \(-\ddot{\alpha}\cdot r\), which is the tangential acceleration.