# Interval Analysis of Vibro-Acoustic Systems by the Enclosing Interval Finite-Element Method
## 2 Formulation of Vibro-Acoustic Coupling System with Uncertain-But-Bounded Parameters  
### Interval Vibro-Acoustic Coupling Equation  
For a deterministic interior coupled vibro-acoustic system, as shown in Figure 1, the structure, fluid, and coupling are, respectively, described with the following governing equations [31]. \[\
abla^{\top}\sigma_{s}+\mathbf{F}_{s}=\rho_{s}\frac{\partial^{2}w_{s}}{ \partial t^{2}}\quad\mathbf{\Omega}_{s},\mathbf{\Gamma}_{s} \tag{1}\]  
\[\frac{\partial^{2}p}{\partial^{2}t}-c_{0}^{2}\
abla^{2}p=0\quad\mathbf{\Omega }_{f},\mathbf{\Gamma}_{f} \tag{2}\]  
\[\left. \begin{array}{cc}w_{s}\big{|}_{n}=\left.w_{f}\right|_{n}&\mathbf{\Gamma }_{fs}\\ \sigma_{s}\big{|}_{n}=-p&\mathbf{\Gamma}_{fs}\end{array} \right. \tag{3}\]  
where \(\sigma_{s},\mathbf{F}_{s}\) represent stresses and external force respectively, \(w_{s},p\) denote the deflection and sound pressure, the subscript \(n\) denotes the outward normal direction, and \(\rho_{s},c_{0}\) represent the density of structures and sound speed of acoustic fluid, respectively. The boundary of fluid subsystem can be divided into two categories, one is the original acoustic boundary denoted as \(\mathbf{\Gamma}_{f}\), which includes three types in detail, the pressure boundary (Dirichlet boundary) \(\mathbf{\Gamma}_{p}\), the velocity boundary (Neumann boundary) \(\mathbf{\Gamma}_{v}\), and the impedance boundary (Robin boundary) \(\mathbf{\Gamma}_{Z}\). As a matter of fact, the interaction between structure and acoustic can be simplified as the additional boundary conditions, respectively [32]. One of the effects from structural bending vibration can be regarded as the velocity boundary for acoustic, while the other is acoustic influence to structure, regarded as the stress boundary condition. If neglecting the damping of structure and the impedance boundary of acoustic domain, after the finite-element discretization, the coupling process would be described with the following formulations [32]. \[\left(\mathbf{K}_{s}-\omega^{2}\cdot\mathbf{M}_{s}\right)\cdot\mathbf{w}_{s}- \mathbf{R}_{cp}\cdot\mathbf{p}_{a}=\mathbf{F}_{s} \tag{4}\]  
\[\left(\mathbf{K}_{a}-\omega^{2}\cdot\mathbf{M}_{a}\right)\cdot\{p_{j}\}-\rho_ {a}\omega^{2}\cdot\mathbf{R}_{cp}^{\top}\{w_{i}\}=\mathbf{F}_{a}^{v}+\mathbf{ F}_{a}^{p} \tag{5}\]  
\[\mathbf{R}_{cp}=\int_{\mathbf{F}_{s}}\mathbf{N}_{s}^{\top}\mathbf{n}_{f} \mathbf{N}_{a}d\Omega \tag{6}\]  
\[\mathbf{F}_{a}^{v}=\int_{\mathbf{F}_{s}}\left(-\mathrm{i}p_{a}\omega\{N_{i} \}_{a}v_{n}\right)\cdot d\Omega \tag{7}\]  
Figure 1: The interior vibro-acoustic coupling system. \[\mathbf{F}_{a}^{p}=\int_{\mathbf{F}_{p}}\left(-\mathrm{i}\rho_{a}\omega\{N_{i}\}_{ a}\mathbf{v}\cdot\mathbf{n}_{f}\right)\cdot d\Omega \tag{8}\]  
where \(\mathbf{K}_{s},\mathbf{M}_{s},\mathbf{K}_{a},\mathbf{M}_{a}\) represent stiffness and mass matrix for structure and acoustic, respectively, and \(\rho_{a}\) is the density of acoustic fluid inside the cavity. When considering the velocity and pressure boundaries, the corresponding loads are defined with input velocity load \(v_{n}\) on \(\mathbf{F}_{a}^{v}\) and input pressure load \(\mathbf{F}_{a}^{p}\). By employing the coupling matrix \(\mathbf{R}_{cp}\) for two-way coupling, Equations (4)-(8) can be rewritten into a non-symmetrical form. \[\left\{\left[ \begin{array}{cc}\mathbf{K}_{s}&-\mathbf{R}_{cp}\\ \mathbf{0}&\mathbf{K}_{a}\end{array} \right]-\omega^{2}\left[ \begin{array}{ cc}\mathbf{M}_{s}&\mathbf{0}\\ \rho_{a}\mathbf{R}_{cp}^{T}&\mathbf{M}_{a}\end{array} \right]\right\}\left\{ \begin{array}{c}\mathbf{w}_{s}\\ \mathbf{p}_{a}\end{array} \right\}=\left\{ \begin{array}{c}\mathbf{F}_{s}\\ \mathbf{F}_{a}\end{array} \right\} \tag{9}\]  
If the density of fluid is light as air, it is apparently that the effect by structure plays the leading role to the interaction.