# Underwater Radiated Noise Prediction Method of Cabin Structures under Hybrid Excitation
## 2 Theoretical Basis of Acoustic Information Monitoring for Cabin Structures  
### Analysis of the Vibration and Sound Characteristics of Cabin Structures under Hybrid Excitation  
In this section, the sound field characteristic in the cabin is expressed by the mean square sound pressure near the inner surface of the cabin, the vibration characteristic of the cabin structure is expressed by the mean square vibration velocity of the cabin's inner surface, and the sound radiation characteristic of the cabin structure is expressed by the radiated sound power of the exterior surface of the cabin. The mean square sound pressure inside the cabin is calculated as follows:  
\[P=\frac{\int\limits_{s}P_{\text{in}}(z,\theta,R)\ast\overline{P}_{\text{in}}(z,\theta,R)ds}{S} \tag{1}\]  
where \(P_{\text{in}}\) is the sound pressure near the interior surface of the cabin and \(S\) is the internal surface area. The mean square vibration velocity of the cabin surface is calculated as follows:  
\[V=\frac{\int\limits_{s}V_{\text{in}}(z,\theta,R)\ast\overline{V}_{\text{in}}(z,\theta,R)ds}{S} \tag{2}\]where \(V_{\text{in}}\) is the radial vibration velocity of the surface inside the cabin and \(S\) is the internal surface area. The calculation expression of radiated sound power is shown as follows:  
\[W=\frac{1}{2}\text{Re}\left\{\int\limits_{s}P_{out}(M)V_{r}^{*}(M)ds\right\} \tag{3}\]  
In order to simplify the calculation model, a single-layer cylindrical shell was selected as the research object. A cylindrical coordinate system was established with the center of the left end face of the cabin as the origin of the coordinates, as shown in Figure 1. The length of the cylindrical shell \(L\) = 2.3 m, the radius \(R\) = 0.8 m, and the thickness of the cylindrical shell \(h\)s = 0.008 m. The Young's modulus of the shell \(E\) = 2.1 \(\times\) 10\({}^{11}\) N/m\({}^{2}\), the Poisson's ratio \(\sigma\) = 0.3, and the density of the shell material \(\rho_{\text{s}}\) = 7800 kg/m\({}^{3}\). The interior of the shell is air, the density \(\rho_{0}\) = 1.29 kg/m\({}^{3}\), the sound speed \(c_{0}\) = 340 m/s, the outer part of the shell is water, the density \(\rho_{\text{e}}\) = 1000 kg/m\({}^{3}\), and the sound speed \(c_{e}\) = 1500 m/s. Two excitation sources are arranged inside the cabin. The coordinate of the sound source is \((0.5R_{0},0,0.5L_{0})\), the sound pressure amplitude at a distance of 1 m from the sound source in the free field is 1 Pa, the coordinate of the force source is \((R_{0},0,0.5L_{0})\), and the excitation strength is 1 N. The initial condition is 0 state, and the boundary conditions at both ends of the single-layer cylindrical shell are simply supported. For the structure shown in Figure 1, the vibration equation of its shell is as follows:  
\[[L]\{u\}=\frac{R^{2}\big{(}1-v^{2}\big{)}}{Eh}\{-\{F\}\} \tag{4}\]  
The constitutive equation of the sound cavity is as follows:  
\[p(r,\omega)\equiv\int_{V}G_{0}(r,r^{\prime},\omega)\widetilde{s}(r^{\prime}, \omega)dr^{\prime} \tag{5}\]  
Firstly, the structural vibration characteristics under the excitation of the sound source and force source were analyzed, as shown in Figure 2. When the sound source excitation is 1 Pa and the force source excitation intensity is 1 N, the sound field in the cabin is mainly determined by the sound source excitation, and the sound field monitoring in the cabin mainly reflects the sound source excitation information. The vibration characteristics of the cabin surface are mainly determined by the force source in the middle- and low-frequency bands, and the vibration of the shell caused by the sound source gradually increases with the increase in the frequency. Figure 1: Calculation model. Force sources with different amplitudes were used for the excitation, so that the radiated sound power of the cabin was the same in each one-third octave frequency band under the excitation of the sound sources.