## 2 Modeling of the Unified Analysis Model of Vibro-acoustic Characteristics  
### Model Description  
This paper establishes a unified analysis model of the vibro-acoustic characteristics of the coupled system between a composite laminated rotationally stiffened plate and the double-cavity structure of cylindrical-cylindrical, spherical-cylindrical, and conical-cylindrical configurations. These systems represent several examples of the coupled system. Because the coupling principles of different structural-acoustic systems are similar, this chapter employs this unified analysis model to analyze the acoustic-vibration characteristics of the coupled system. Rotational cylinders, spheres, and conical cavities can be represented using a special case of a double-curvature cavity unit. As shown in Figure 1, the unit's bottom surface (\(z=0\)) is taken as a reference plane for establishing an orthogonal coordinate system. Here, \(R_{\alpha}\) and \(R_{\beta}\) represent the base curvature radii along the \(\alpha\) and \(\beta\) axes, while \(L_{\alpha}\) and \(L_{\beta}\) denote the length dimensions along the \(\alpha\) and \(\beta\) axes. Additionally, \(L_{z}\) represents the height dimension of the cavity along the \(z\)-coordinate direction. In order to facilitate the establishment of a unified model of rotational cavity coupling systems, it is necessary to transform the parallelogram cross-sections in the system into  
Figure 1: Geometric parameters and the coordinate system of a double-curvature cavity unit cross-section. composite laminated rotationally stiffened plate and the conical-cylindrical double-cavity configuration includes two local systems of coordination (\(o\)-\(s\), \(\theta\), \(r\)) and (\(o_{n}\)-\(z_{n}\), \(\theta_{n}\), \(x_{n}\)). The conical cavity 1 has small- and large-end radii \(R_{1}\) and \(R_{2}\), with a cone apex angle \(\alpha\) and a generatrix length \(L_{1}\). The inner and outer radii of the composite laminated rotationally stiffened plate and the two cavities are denoted as \(R_{2}\) and \(R_{3}\), respectively. The laminated plate has a thickness of \(h_{\rm p}\), and the curvature radius of the laminated beams is \(R_{\rm bn}\), with a width of \(b_{n}\) and a thickness of \(h_{n}\). The height of the cylindrical cavity is \(L_{2}\), and the overall thickness of the coupled system cavities is \(H\). To research the displacement and acoustic response characteristics of the coupled system, a single sound source \(Q\) and a harmonic point force \(F\), which can be placed at any position within the coupled system, have been added. According to the artificial virtual spring technology [16; 17; 18], as shown in Figure 3(a), for the edge of the laminated, rotationally stiffened plate, the general boundary conditions can be expressed by inserting three sets of linear springs \(k_{u}\), \(k_{v}\), and \(k_{w}\) along the \(u\), \(v\), and \(w\) directions and two sets of torsional springs, \(K_{r}\) and \(K_{\theta}\). These spring groups are consistently distributed along the boundary. \(k_{\theta 0}^{u}\), \(k_{\theta 0}^{v}\), \(k_{\theta 0}^{u}\), \(K_{\theta 0}^{v}\), and \(K_{\theta 0}^{\theta}\) represent five sets of spring of the boundary at the \(\theta=0^{\circ}\) boundary [19]; similarly, the spring of the boundary at \(\theta=\theta\), \(r=0\), and \(r=R_{\rm p}\) can be represented using this approach. For the sector-shaped composite stiffened plate, the spring of the boundary at \(r=0\) has a stiffness of 0. For the ring-shaped composite stiffened plate, the boundary springs at \(\theta=0^{\circ}\) and \(\theta=360^{\circ}\) have a stiffness of 0. For the circular composite stiffened plate, the boundary springs at \(r=0\), \(\theta=0^{\circ}\), and \(\theta=360^{\circ}\) all have a stiffness value of 0.