# Prediction of Vibration-Mode-Induced Noise of Structure-Acoustic Coupled Systems
## 2 Induction Characteristics of Acoustic Modes  
### Structure-Acoustic Coupling System  
The acoustic domain coupled with a structure can be expressed as a structure-acoustic coupled system by using an FEM model involving sound boundary conditions [37; 38; 39]. \[ \begin{split} m\ddot{z}+c\dot{z}+kz+W^{\dagger}p=E\delta(t)\\ -\rho_{a}W\ddot{z}+M_{a}\ddot{p}+C_{a}\dot{p}+K_{a}p=0\end{split}  \tag{1}\]  
where \(z\) and \(p\) represent the structural displacement and sound pressure, respectively; \(m\), \(c\), and \(k\) are the structural mass, damping, and stiffness matrices, respectively; \(W\) is the boundary matrix of the structure and acoustic domain; and \(E\) is the location matrix pertaining to the impact load (\(\delta(t)\)). In addition, \(\rho_{a},M_{a},C_{a}\), and \(K_{a}\) represent the air density and mass, damping, and stiffness matrices of the acoustic domain, respectively. Using a modal matrix and modal responses with the structure and acoustic domain in an independent state, the structural displacement and sound pressure can be expressed as follows:  
\[ \begin{split} z&=\Phi\eta\\ p&=\Psi q\end{split}  \tag{2}\]  
where \(\eta\) and \(q\) are the structure and acoustic modes, respectively, and \(\Phi\) and \(\Psi\) are the corresponding modal matrices. By substituting Equation (2) into Equation (1) and using the component mode synthesis technique, the system combining the vibration and acoustic modes can be expressed as follows:  
\[ \begin{split}\ddot{\eta}+\underset{s}{\sum}\ddot{\eta}+\Omega_{s }\eta+\Phi^{T}W^{T}\Psi q=\Phi^{T}E\delta(t)\\ -\rho_{a}\Psi^{T}W\Phi\ddot{\eta}+\ddot{q}+\underset{a}{\sum} \ddot{q}+\Omega_{a}q=0\end{split}  \tag{3}\]  
The mass matrix of the structure and acoustic domain is normalized as a unit matrix (\(I\)) for each of the mode shapes. \[ \begin{split}\Phi^{T}m\Phi=I\\ \Psi^{T}M_{a}\Psi=I\end{split}  \tag{4}\]  
The variables used in Equation (3) can be defined as follows:  
\[ \begin{split}\underset{s}{\sum}&=diag[2\xi_{s} \omega_{s}],\ \Omega_{s}=diag\big{[}\omega_{s}^{2}\big{]}\\ \underset{s}{\sum}&=diag[2\xi_{a}\omega_{a}],\ \Omega_{a}=diag\big{[} \omega_{a}^{2}\big{]}\end{split} \]where _diag_[] represents a diagonal matrix; subscripts \(s\) and \(a\) represent the structural vibration and acoustic modes, respectively; and \(\omega_{a},\omega_{s},\) and \(\xi_{a},\xi_{s}\) denote the natural frequency and damping ratio, respectively. Both sides of \(W\), which represents the boundary between the structure and acoustic domains, are multiplied by the mode shape to represent the coupling effect of the vibration and acoustic modes. The coupling matrix is defined as follows:  
\[\gamma=\Psi^{T}W\Phi \tag{5}\]  
The size of the coupling matrix \(\gamma\) depends on the disparity of the acoustic domain and structural model in the FEM analysis. Although the acoustic mode is influenced by all vibration modes, the influence is minimal when the natural frequencies of the two modes are significantly different. Therefore, only the nearby vibration mode is considered. The second-order coupling motion equation, formulated considering only the vibration and acoustic modes near those in Equation (3), is  
\[ \begin{split}\ddot{\eta_{i}}+2\xi_{s}\omega_{s}\dot{\eta_{i}}+ \omega_{s}^{2}\eta_{i}+\gamma_{ij}\dot{q_{j}}=\Phi_{i}^{T}\mathcal{E}\delta(t) \\ -\rho_{a}\gamma_{ji}\ddot{\eta_{i}}+\ddot{q_{j}}+2\xi_{a}^{*} \omega_{a}\dot{q_{j}}+\omega_{a}^{2}q_{j}=0\end{split}  \tag{6}\]  
where \(\Phi_{i}^{T}\) is the mode shape of the _i_-th vibration mode (\(\eta_{i}\)), which is coupled with the _j_-th acoustic mode (\(q_{j}\)) by the elements of the coupling matrix (\(\gamma_{ij}\)). Only these two modes are assumed to be strongly coupled, as their natural frequencies are similar, and in the subsequent descriptions, the subscripts denoting specific modes are omitted to facilitate equation expansion. The coupling matrix element (\(\gamma_{ij}\)) is defined as a new variable representing the coupling effect:  
\[ \begin{split} g_{s}&=\gamma_{ij}\\ g_{a}&=\gamma_{ji}\end{split}