# Study on Local Vibration Control of the 100 m X-BOW Polar Exploration Cruise Ship
## 2 Method
### Conversion Method of Concentration Quality Matrix  
By the finite element method, the internal displacement of the cell is generally expressed by interpolation, so that the displacement of each node of the cell is \(\left\{\delta\right\}^{\epsilon}\), then the internal displacement of the cell is  
\[\left\{u\right\}=[N]\{\delta\}^{\epsilon} \tag{15}\]  
where \([N]\) is the shape function matrix; the in-cell acceleration is  
\[\left\{\ddot{u}\right\}=[N]\left\{\ddot{\delta}\right\}^{\epsilon} \tag{16}\]  
where \(\left\{\ddot{\delta}\right\}^{\epsilon}\) is the acceleration matrix of the cell nodes.  
Using the inertial force \(\left\{\ddot{\delta}\right\}^{\epsilon}\) in the cell as a volume-distributed load and distributing it to the nodes of the cell according to the distribution principle, we obtain  
\[\left\{Q\right\}_{\rho}^{\epsilon}=-\int[N]^{T}\rho\{\ddot{u}\}dv \tag{17}\]  
Substituting Equation (15) into Equation (16) yields  
\[\left\{Q\right\}_{\rho}^{\epsilon} =-\int[N]^{T}\rho[N]\left\{\ddot{\delta}\right\}^{\epsilon}dV \tag{18}\] \[=-\int[N]^{T}\rho[N]dV\left\{\ddot{\delta}\right\}^{\epsilon}=-[M] ^{\epsilon}\left\{\ddot{\delta}\right\}^{\epsilon}\]  
where \([M]^{\epsilon}\) is a consistent mass matrix, which is not diagonal; there is some inertial coupling between cells.  
The consistent mass matrix can be equated to a diagonal matrix, and the equivalent matrix is the concentrated mass matrix.  
Let the total kinetic energy of the structure be \(T\), and the total kinetic energy of the sum of the kinetic energies of the cells, such that  
\[T=\sum T^{\epsilon} \tag{19}\]  
The kinetic energy of the unit is  
\[\begin{split} T^{\epsilon}=\frac{1}{2}\int_{V^{\epsilon}}\left\{ \dot{u}\right\}^{T}\rho\left\{\dot{u}\right\}dV=\frac{1}{2}\int_{V^{\epsilon} }\left([N]\left\{\dot{\delta}\right\}^{\epsilon}\right)^{T}\rho[N]\left\{ \dot{\delta}\right\}^{\epsilon}dV\\ =\frac{1}{2}\left\{\dot{\delta}^{\epsilon}\right\}^{T}\int_{V^{ \epsilon}}[N]^{T}\rho[N]dV\left\{\dot{\delta}\right\}^{\epsilon}=\frac{1}{2} \left\{\dot{\delta}^{\epsilon}\right\}^{T}[M]^{\epsilon}\left\{\dot{\delta} \right\}^{\epsilon}\end{split} \tag{20}\]  
When the overall mass is equally distributed among all cells, the kinetic energy of each cell is  
\[T^{\epsilon}=\sum_{k=1}^{n}\frac{1}{2}m_{k}\dot{\delta}_{k}^{2} \tag{21}\]  
where \(m_{k}\) is the nodal mass corresponding to the displacement component of node \(k\), and \(n\) is the number of cell degrees of freedom, at which point the cell mass matrix is diagonal, such that  
\[[M]^{\epsilon}=\begin{bmatrix}m_{1}&0&0&\cdots&0\\ 0&m_{2}&0&\cdots&0\\ 0&0&m_{3}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&m_{n}\end{bmatrix} \tag{22}\]For thin beam and thin plate and shell units, the kinetic energy of rotation of the nodes can be neglected when using the concentrated mass matrix to represent the structural mass. Assume a two-node straight beam cell, \(i\) and \(j\) being the two nodes, and \(M\) being the mass of the cell. However, when the unit masses are equally distributed by the two nodes, the nodal displacements can be expressed as  
\[\{\delta\}^{e}=\begin{bmatrix}\omega_{i}&\theta_{i}&\omega_{j}&\theta_{j}\end{bmatrix} \tag{23}\]  
The kinetic energy is  
\[T^{e}=\frac{1}{2}\frac{M}{2}\omega_{i}^{2}+\frac{1}{2}\frac{M}{2}\omega_{j}^{ 2} \tag{24}\]  
The matrix of concentrated masses is  
\[\left[M\right]^{e}=\begin{bmatrix}\frac{M}{2}&0&0&0\\ 0&0&0&0\\ 0&0&\frac{M}{2}&0\\ 0&0&0&0\end{bmatrix} \tag{25}\]