# Underwater Radiated Noise Prediction Method of Cabin Structures under Hybrid Excitation
## 3 Underwater Radiation Noise Prediction Method for Cabin Structures
Therefore, the \"acoustic-vibration\" transfer function \(H_{P-V}(\omega)\) between the monitoring points inside the cabin can first be constructed by conducting numerical condition tests, and then the acoustic information is used as a reference to solve the coupling part triggered by the sound source in the vibration information, as shown in Equation (10):  
\[A_{0}(\omega)=P(\omega)H_{P-V}(\omega) \tag{10}\]  
where \(A_{0}(\omega)\) is the coupling part triggered by the sound source in the vibration information and \(P(\omega)\) is the acoustic information. The vibration monitoring information is decoupled as shown in Equation (11):  
\[A(\omega)=A_{q}(\omega)-A_{0}(\omega) \tag{11}\]  
where \(A_{q}(\omega)\) is the vibration information and \(A(\omega)\) is the vibration information after decoupling. Based on the assumption that the system is linear and time-invariant, the response function of the response point can be expressed as follows:  
\[Y(\omega)=X(\omega)H_{PV-P}(\omega) \tag{12}\]  
where \(Y(\omega)\) is the acoustic output vector of the target point, \(X(\omega)\) is the vibration and acoustic input vector of the monitoring point, and \(H_{PV-P}(\omega)\) is the \"acoustic/vibration-acoustic\" transmission network inside and outside the cabin structure. The input vector is the combination of the sound pressure signal and vibration signal, and the sound output vector is the sound pressure signal. By measuring the physical quantities of the vibration and acoustic monitoring points and target points under the numerical operation conditionsof the cabin structure system, the sample matrix of the monitored excitation input and target response output can be constructed:  
\[ \begin{pmatrix}A_{11}&\cdots&A_{1k}&P_{11}&\cdots&P_{1m}\\ A_{21}&\cdots&A_{2k}&P_{21}&\cdots&P_{2m}\\ A_{31}&\cdots&A_{3k}&P_{31}&\cdots&P_{3m}\\ A_{41}&\cdots&A_{4k}&P_{41}&\cdots&P_{4m}\\ \vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\ A_{n1}&\cdots&A_{nk}&P_{n1}&\cdots&P_{nm}\\ \end{pmatrix}\begin{pmatrix}H_{a1}\\ \vdots\\ H_{ak}\\ H_{p1}\\ \vdots\\ H_{pmn}\\ \end{pmatrix} = \begin{pmatrix}Y_{1}\\ Y_{2}\\ Y_{3}\\ Y_{4}\\ Y_{n}\\ \end{pmatrix}  \tag{13}\]  
where \(n\) represents the number of test conditions, \(k\) and \(m\) represent the numbers of vibration and harmonic reference points, respectively, \(Y_{n}\) represents the target reference point response of the NTH working condition, \(a_{nk}\) represents the response of the KTH vibration reference point under the NTH working condition, \(p_{nm}\) represents the response of the MTH acoustic reference point under the NTH working condition, \(H_{ak}\) represents the transfer rate of the KTH vibration reference point to the target point, and \(H_{pm}\) represents the transfer rate of the MTH acoustic reference point to the target point. According to the above formula, the numerical working condition transfer matrix of the cabin structural system can be solved:  
\[ \begin{pmatrix}H_{a1}\\ \vdots\\ H_{ak}\\ H_{p1}\\ \vdots\\ H_{pm}\end{pmatrix} = \begin{pmatrix}A_{11}&\cdots&A_{1k}&P_{11}&\cdots&P_{1m} \\ A_{21}&\cdots&A_{2k}&P_{21}&\cdots&P_{2m}\\ A_{31}&\cdots&A_{3k}&P_{31}&\cdots&P_{3m}\\ A_{41}&\cdots&A_{4k}&P_{41}&\cdots&P_{4m}\\ \vdots&\cdots&\vdots&\cdots&\vdots\\ A_{n1}&\cdots&A_{nk}&P_{n1}&\cdots&P_{nm}\\ \end{pmatrix} ^{-1} \begin{pmatrix}Y_{1}\\ Y_{2}\\ Y_{3}\\ Y_{4}\\ Y_{n}\\ \end{pmatrix}  \tag{14}\]  
To accurately obtain the transfer matrix of the system, it is necessary to select a reasonable working condition to test the system. To ensure the uniqueness of the matrix inversion in Equation (14), the principle of \(n>k+m\) should be satisfied. Formula (14) is expressed as follows:  
\[H_{PV-P}=X^{+}Y \tag{15}\]  
where \(X^{+}\) is the generalized inverse matrix of \(X\). Because there is usually correlation between each condition when testing the numerical condition, the direct inversion of \(X\) will lead to incorrect results. In order to avoid the above situation, the singular value decomposition of the transfer matrix is carried out.