## 2 Theoretical Modeling
### Vibrating Response of the Ribbed Plate with Point Control Force
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Geometrical Parameters** & **Value** & **Material Properties** & **Value** \\ \hline The length and width of the base plate & \(l_{x}=0.8\) m & The material of the base plate and the ribs & aluminum \\  & \(l_{y}=0.6\) m & The density of aluminum & \(\rho=2790\) kg/m\({}^{3}\) \\ \hline The thickness of the base plate & \(h=0.004\) m & The density of aluminum & \(\rho=2790\) kg/m\({}^{3}\) \\ \hline The locations of the vertical and horizontal ribs & \(x_{a}=\)0.6 m & [PERSON]’s modulus of aluminum & \(E=7.2\times 10^{10}\) N/m\({}^{2}\) \\  & \(y_{a}=\)0.45 m & The [PERSON]’s modulus of aluminum & \(\
u=0.34\) \\ \hline The size of the rectangular cross section of the ribs & \(A=0.003\times 0.04\) m\({}^{2}\) & The [PERSON]’s ratio of aluminum & \(\
u=0.34\) \\  & \(P_{0}=1\) Pa & & \\ The primary excitation & \(\theta=\frac{3}{2}\) & The density of air & \(\rho_{0}=1.21\) kg/m\({}^{3}\) \\  & \(\alpha=\frac{3}{4}\) & The sound speed of air & \(c_{0}=344\) m/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the model. According to the modal expansion method, the bending displacement \(w\) of the base plate, the flexural and torsional displacements of two beams \((U,\theta_{r})\) can be decomposed over their mode shape functions as \(w=\sum_{m=1}^{M}\sum_{n=1}^{N}q_{mn}(t)s_{nm}(x,y)\), \(U=\sum_{n=1}^{N}u_{n}(t)\varphi_{n}(y)\), \(\theta_{r}=\sum_{n=1}^{N}\theta_{r,n}(t)\varphi_{n}(y)\). \(s_{mn}(x,y)=\phi_{m}(x)\varphi_{n}(y)\) is the mode shape function of the simply supported base plate. \(q_{mn}(t)\), \(u_{n}(t)\), \(\theta_{r,n}(t)\) are the modal amplitudes with respect to each subsystem. \(M\times N\) is the upper limit number of modes. As for the ribbed plate; by combining Equations (1) to (3), applying the orthogonality of the mode shape functions and two compatibility conditions at the beam/plate interface (the displacement and rotation continuity conditions [14]: \(U_{v}(y)=w(x_{a},y)\),\(U_{h}(x)=w(x,y_{a})\),\(\theta_{r,v}(y)=\frac{\partial w}{\partial x(x_{a},y)}\) and \(\theta_{r,h}(x)=\frac{\partial w}{\partial y(x,y_{a})}\) can yield the equation that the modal amplitude \(q_{mn}(\omega)\) of the base plate satisfies:  
\[ \begin{split}&\big{(}k_{mn}-\omega^{2}\mu_{mn}\big{)}q_{mn}( \omega)+\sum\limits_{p=1}^{M}[\phi_{m}(x_{a})\big{(}k_{Bn}-\omega^{2}\mu_{Bn} \big{)}\phi_{p}(x_{a})-\phi_{m}^{\prime}(x_{a})\big{(}k_{Tn}-\omega^{2}\mu_{Tn }\big{)}\phi_{p}^{\prime}(x_{a})]\cdot q_{pn}(\omega)\\ &+\sum\limits_{q=1}^{N}\Big{[}\varphi_{n}(y_{a})\big{(}k_{Bm}- \omega^{2}\mu_{Bm}\big{)}\varphi_{q}(y_{a})-\phi_{n}^{\prime}(y_{a})\big{(}k _{Tm}-\omega^{2}\mu_{Tm}\big{)}\varphi_{q}^{\prime}(y_{a})\Big{]}q_{mq}\\ &=Q_{p,mm}+F_{s}Q_{s,mm}.\end{split}  \tag{4}\]  
The variables \(k_{mn}\), \(\mu_{mn}\), \(k_{Bn}\), \(\mu_{Bn}\), \(k_{Tn}\),\(\mu_{Tn}\),\(k_{Bm}\), \(\mu_{Bm}\), \(k_{Tm}\) and \(\mu_{Tm}\) can be referenced in [4]. \(Q_{p,mm}\) and \(Q_{s,mm}\) are the generalized primary and secondary modal force, \(Q_{p,mm}=\int_{S}f_{p}(\mathbf{r},\omega)\phi_{m}(x)\varphi_{n}(y)ds\), \(Q_{s,mm}=\phi_{m}(x_{s})\varphi_{n}(y_{s})\). Then, the \(M\times N\) -length modal amplitude vector \(q\) of the base plate satisfies the following matrix equation,  
\[\big{(}\mathbf{K}-\omega^{2}\mathbf{M}\big{)}\mathbf{q}=\mathbf{Q}_{p}+F_{s}\mathbf{Q}_{s} \tag{5}\]  
where \(\mathbf{K}\) and \(\mathbf{M}\) are the \((M\times N)\times(M\times N)\) stiffness and mass matrixes of the base plate. These can be referenced in [4], where the coupling items of the horizontal rib should be added as well into their respective expressions. \(\mathbf{Q}_{p}=\big{[}\mathbf{Q}_{p,11},\mathbf{Q}_{p,12},\ldots,\mathbf{Q}_{p,MN}\big{]}^{T}\) and \(\mathbf{Q}_{s}=\big{[}\mathbf{Q}_{s,11},\mathbf{Q}_{s,12},\ldots,\mathbf{Q}_{s,MN}\big{]}^{T}\) are the generalized modal force vectors. \(\mathbf{q}=\big{[}q_{11},q_{12},\ldots,q_{MN}\big{]}^{T}\) is the modal amplitude vector of the base plate. \(F_{s}\) is the amplitude of the control force.