## 4 Gear ACTRAN Noise Prediction
### Theory of Radiated Noise and Its Measurement  
Noise in gear systems is primarily caused by vibrations. To obtain the radiated noise characteristics of the arc gear system, an acoustic model of the arc gear system is established using acoustic software to predict the radiated noise of the gear. After the model was built, the field point mesh was defined, and finally, the gear noise was solved and analyzed using acoustic analysis software to obtain the acoustic response of the gear subfield point.  
In acoustic calculations, the most common calculation method is to solve for the sound field by integrating the Helmholtz formula:  
\[\
abla^{2}p+k^{2}p=0 \tag{4}\]  
In Formula (4), \(\
abla^{2}\) is [PERSON]'s algorithm, \(\
abla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y ^{2}}+\frac{\partial^{2}}{\partial z^{2}}\); \(p\) is the sound pressure; \(k\) is wave number, \(k=\frac{\omega}{c}\); and \(\omega\) is the Excitation Circle Frequency (rad/s).  
The Helmholtz equation describes the spatial distribution of the sound pressure in a stable acoustic field, and given a certain acoustic boundary condition, the radiated sound pressure at that boundary can be determined.  
Acoustic quantities such as sound pressure, sound intensity, and sound power can be used to measure the strength of sound. However, owing to the wide range of human hearing and the size of the sound pressure, the sensitivity of the human ear is not a linear  
Figure 8: Displacement frequency curve.  
relationship between the size of the sound pressure; so, the use of the sound pressure size to calibrate the strength of the sound in an actual project will bring a lot of inconveniences. The same applies to sound intensity and power level. For comparison, a logarithmic scale was used to measure the sound pressure and sound intensity.  
Sound pressure level: the ratio of the actual sound pressure value to the reference sound pressure value is taken as a logarithm and then multiplied by 10 with the expression as in Equation [12].  
\[L_{P}=10 lg\frac{P_{e}}{P_{r}} \tag{5}\]  
In Formula (5), \(P_{e}\) is the actual sound pressure value and \(P_{r}\) is the reference sound pressure value (\(P_{r}=2\times 10^{-5}\)Pa).  
The sound intensity level can be expressed as in Equation (6):  
\[L_{I}=10 lg\frac{I}{I_{0}} \tag{6}\]  
In Formula (6), \(I\) is the actual sound intensity value and \(I_{0}\) is the benchmark sound intensity value (\(I_{0}=10^{-12}\) W/m\({}^{2}\)), which is the lowest sound intensity perceived by the human ear.  
The main research objective of this study is to select the sound pressure level to reflect the noise value of the system by understanding the radiation of noise outward through the air from the gearing system, combined with an analysis of the noise measurement acoustic quantities. For comparison, the RMS value of the sound pressure level of the gearing system over the entire scanning frequency band was chosen as the system noise value. As shown in Figure 9, five suitable field points were selected to verify the sound pressure levels at these points, and the corresponding sound pressure level curves were plotted.