## 2 Theoretical Modeling  
### Vibrating Response of the Ribbed Plate with Point Control Force  
Figure 1a shows the sketch of the system. For both simplifying the theoretical model and also suitably representing the practical complicated ribbed structure, an orthogonally rib stiffened plate is used in the study. The orthogonally rib stiffened plate is constituted with a simply supported base plate and two simply supported ribs with one rib along the horizontal and another along the vertical directions. An oblique incident plane wave is applied as the primary excitation. The beam/plate interface is considered as a nonslip line connection in which a pair of coupling force \(F\) and moment \(M_{m}\) is considered to simulate the beam/plate coupling effects [27], as shown in Figure 1b. The vertical rib locates at \(x=x_{a}\) and the horizontal rib locates at \(y=y_{a}\). A controllable point force [16]\(f_{\rm s}(\mathbf{r}_{s},t)=F_{\rm s}\delta(x-x_{s},y-y_{s})e^{j\omega t}\) is added on the ribbed plate to control the vibration and noise radiation of the ribbed plate. When the decentralized control system with \(S\) feedback loops is used, the control force can be expressed as \(f_{\rm s}=\sum_{i=1}^{S}f_{\rm s,i}(\mathbf{r}_{\rm s,j},t)\). Under the excitation of an oblique incident plane wave, the base plate is subjected to the coupling force and moment of the ribs, the sound pressure, and the control force. The transverse bending displacement \(w\) satisfies the motion of Equation [17]:  
\[D\
abla^{4}w+\rho h\frac{\partial^{2}w}{\partial t^{2}}=f_{p}(\mathbf{r},t)+F_{v} \delta(x-x_{a})+M_{m,v}\delta^{\prime}(x-x_{a}),+F_{h}\delta(y-y_{a})+M_{m,h} \delta^{\prime}(y-y_{a})+f_{\rm s}(\mathbf{r}_{s},t), \tag{1}\]  
where \(\
abla^{4}\) denotes the Laplacian, \(D\) is the flexural rigidity of the base plates, \(D=\frac{E\hbar^{3}}{12(1-\sigma^{2})}\). \(E_{r}\rho h\) and \(v\) are the Young's modulus, density, thickness and Poisson's ratio of the base plate. \(f_{p}(\mathbf{r},t)\) is the primary excitation of the oblique incident plane wave [28]. The rib is assumed as a uniform beam element with a rectangular cross section. It is assumed as well that the beam is symmetrical with respect to its neutral plane so that the flexural and torsional vibration is uncoupled. The governing equations of the flexural and torsional displacements \((U,\theta_{r})\) of the ribs can be expressed as [29] (using a vertical rib for example),  
\[EI\frac{\partial^{4}U}{\partial y^{A}}+\rho A\frac{\partial^{2}U}{\partial t^{2 }}=-F_{v}, \tag{2}\]  
\[EI_{r,w}\frac{\partial^{4}\theta_{r}}{\partial y^{A}}-GJ\frac{\partial^{2} \theta_{r}}{\partial y^{2}}+\rho l_{0}\frac{\partial^{2}\theta_{r}}{\partial t ^{2}}=-M_{m,v}, \tag{3}\]  
where \(E\) and \(I\) are the [PERSON]'s modulus and sectional moment of inertia of the beam. \(\rho\) and \(A\) are the density and sectional area of the beam. The material of the base plate and the rib is the same in this research (mentioned in Table 1)--hence the symbols of the [PERSON]'s modulus \(E\) and density \(\rho\) for the rib are used as well. \(EI\) is the flexural rigidity and \(\rho A\) is the mass per unit length. \(I_{r,w}\) is the warping moment of inertia related to the torsion and \(EI_{r,w}\) is the warping rigidity. \(G\) is the shear modulus and its relation with the [PERSON]'s modulus \(E\) can be expressed as \(G=\frac{E}{2(1+\alpha)}\). \(J\) is the Saint-Venant torsional constant, and \(GJ\) is the torsional rigidity. \(\rho l_{0}\) is the mass moment of inertia per unit length. The warping rigidity \(EI_{r,w}\) is much smaller than the torsional rigidity \(GJ\), and the differential term \(\frac{\partial^{4}\theta_{r}}{\partial y^{A}}\) in Equation (3) can be neglected for simplicity. Figure 1: Systemic model: (**a**) active orthogonally rib stiffened plate; (**b**) the coupling effects between the base plate and the rib for the vertical rib; (**c**) the coupling effects for the horizontal rib.