## 4 Modal Criterion for Random Vibration Suppression  
In theory the response of all the modes in the bandwidth of the excitation must be considered and controlled. In themindependent modal resonant shunt method, the number of transducers equals the number of modes to be controlled, so a modal criterion is needed to indicate which modes are worthy to control with the given input-output condition of the structure in order to optimize the number of transducers. The state-space representation of the uncontrolled mechanical structure is:  
\[ \begin{array}{l}\left\{\dot{x}\right\}=\left[A\right]\{x\}+\left[B\right] \{w\}\\ \left\{y\right\}=\left[C\right]\{x\}\end{array}  \tag{16}\]The state vector \(\{x\}\) is as follows:  
\[\{x\}= \begin{Bmatrix}\{x_{\text{s}}\}\\ \{\dot{x}_{\text{s}}\}\end{Bmatrix}  \tag{17}\]  
while system matrix \([A]\) is:  
\[[A]=\left[ \begin{array}{cc}0&I\\ -[M_{\text{s}}]^{-1}[K_{\text{s}}]&-[M_{\text{s}}]^{-1}[D_{\text{s}}]\end{array} \right] \tag{18}\]  
The input matrix \([B]\) consisting of 0 and 1 represents locations of the disturbances, \(\{w\}\) is the disturbance vector, and the output matrix \([C]\) consisting of 0 and 1. The frequency-response function \([G]\) is defined as the complex gain between \(\{w\}\) and \(\{y\}\):  
\[[G(\omega)]=[C](j\omega[I]-[A])^{-1}[B] \tag{19}\]  
The relation between the power spectral density (PSD) of \(\{w\}\) and \(\{y\}\) is:  
\[\big{[}S_{y}(\omega)\big{]}=[G(\omega)][S_{w}(\omega)][G(\omega)]^{*} \tag{20}\]  
where the asterisk represents conjugate transpose. When the disturbances are uncorrelated white noise, e.g., \([S_{w}(\omega)]=[I]\), the RMS response of the output of system (16) is:  
\[ \begin{array}{rl}\sigma_{y}^{2}&=\frac{1}{2\pi}\int_{0}^{\infty}trace\big{(} \big{[}S_{y}(\omega)\big{]}\big{)}d\omega\\ &=\frac{1}{2\pi}\int_{0}^{\infty}trace\big{(}\big{[}G(\omega)\big{]}^{*}[G( \omega)]\big{)}d\omega\\ &=\|G\|_{2}^{2}\end{array}  \tag{21}\]  
This is indicated that the RMS power response equals the \(H_{2}\) norm \(\|G\|_{2}\) of the system. For a lightly damped structure, the \(H_{2}\) norm can be approximated as [23]:  
\[\|G\|_{2}\cong\sqrt{\sum_{i=1}^{N}\|G_{i}\|_{2}^{2}} \tag{22}\]  
where \(N\) is the total number of modes in the bandwidth, and  
\[\|G_{i}\|_{2}\cong\frac{\|B_{mi}\|_{2}\|C_{mi}\|_{2}}{2\sqrt{\xi_{i}\omega_{i}}} \tag{23}\]  
is \(H_{2}\) norm of the \(i\)th mode. Quantity of \(H_{2}\) norm of a mode represents the contribution of this mode to the response. A natural idea of modal criterion is using \(H_{2}\) norm of the modes, say, controlling the modes with highest \(H_{2}\) norm. However, in an independent modal resonant shunt, the added damping ratios cannot be an arbitrary set, as it is decided by the quantity \(\lambda_{i}\) in (15). In fact, \(\lambda_{i}\) is linked to the MSE of the replaced rod. In [21] it is indicated that the amount of the added damping ratio is directly proportional to square root of the fraction of MSE \(\delta_{i}\) of the replaced rod, which can be calculated as:  
\[\delta_{i}=\frac{\frac{1}{2}K_{\text{a}}\lambda_{i}^{2}}{\frac{1}{2}\omega_{ \text{si}}^{2}} \tag{24}\]  
where \(\omega_{\text{si}}\) is the short-electrode frequency of the \(i\)th mode with short-electrode transducer stiffness \(K_{\text{a}}\). Note that because the transducers are free of bending and torsion, only axial MSE is considered. Thus, a modal criterion is proposed here using decrement of \(H_{2}\) norm of a mode when controlled by independent modal resonant shunt.