# Optimization of the Housing Shape Design for
## 4 Radiated Noise Reduction Using Topology Optimization  
### Equivalent Radiated Power  
To optimize the shape design of the gearbox housing to reduce radiated noise, the weak areas of the structure against noise and vibration should be identified, and the design areas should be modeled. To analyze the weak areas of the structure, the sound power according to the gear mesh excitation was predicted using a validated FE model. The design areas were modeled based on the operating deflection shape (ODS) and elastic strain energy (ESE) in the relevant mode under operating conditions for high radiated noise.  
As sound power represents the properties of a sound source, it does not depend on distance and direction, unlike sound pressure or sound intensity, which represents the properties of a sound wave at an arbitrary point in space. In a vibroacoustic problem, the surface velocity of the vibrating structure is the same as the velocity of the air particles at the interface owing to the speed continuity. Accordingly, the sound power can be expressed as shown in Equation (2), as a function of the surface velocity of the structure:  
\[P=\frac{1}{2}\sigma\rho c\int\overline{v}^{2}\;d\mathrm{S}\;. \tag{2}\]Here, \(P\) is the sound power, W; \(\sigma\) is the sound radiation efficiency; \(\rho\) is the density of air, kg/m\({}^{3}\); \(c\) is the speed of sound, m/s; S is the vibrating surface area, m\({}^{2}\); \(\overline{v}\) is the root mean square value of surface velocity at the vibrating structure, m/s.  
In the sound power analysis based on the FEM, Equation (2) can be calculated by discretizing in finite elements, and the prediction can be made by assuming the sound radiation efficiency as a unit value. This is called equivalent radiated power (ERP) and can be calculated as shown in Equation (3) [17]. Since the radiation efficiency is assumed to be 1, the ERP can be calculated to be larger than the actual sound power; however, the upper bound of the sound power radiated by structural vibration can be approximately estimated:  
\[P_{ERP}=\frac{1}{2}\rho c\sum_{e=1}^{N_{e}}S_{e}\overline{v}_{e}^{2}. \tag{3}\]  
Here, \(P_{ERP}\) is the equivalent radiated power, W; \(S_{e}\) is the surface area of the finite element at the vibrating structure, m\({}^{2}\); \(\overline{v}_{e}\) is the RMS surface velocity of the finite element at the vibrating structure, m/s; \(N_{e}\) is the number of finite elements.  
Depending on the rotational direction of the input shaft, the gear mesh force transmitted to the housing is changed. In this study, to consider the forward and reverse motions of an agricultural electric vehicle, ERP values were calculated when the input shaft was rotated clockwise (CW) and counterclockwise (CCW), respectively. Figure 9 shows the direction of the bearing reaction force along the rotational direction of the input shaft.  
A modal frequency response analysis was performed for the ERP analysis. Figure 10 shows the ERP results when the first and second harmonic components of the first stage of the gear train were applied as the excitation force. When the input shaft was rotated in the CW direction, the ERP value was high at approximately 3500 rpm in the first harmonic component and approximately 2636 rpm in the second harmonic component. When the input shaft was rotated in the CCW direction, the ERP value was high at approximately 3788 rpm in the first harmonic component and approximately 2724 rpm in the second harmonic component. The rotational speed range at which the peak value appears depending on the gear mesh excitation was similar to the experimental noise measurement result in Figure 8, and this could confirm the reliability of the FEM-based numerical analysis results.  
Figure 9: Bearing reaction force according to the rotational direction of the input shaft. (**a**): Clockwise rotation of the input shaft; (**b**): Counterclockwise rotation of the input shaft.