# Study on Local Vibration Control of the 100 m X-BOW Polar Exploration Cruise Ship
## 2 Method
### Local Mode Equivalent Mass Solution Method  
A discontinuous discrete multi-degree of freedom system is established, as shown in Figure 2.  
After decoupling the multi-DOF system, let the mode mass of the \(i\)-th mode be \(M_{i}\) and the vibration velocity be \(v_{i}\), then the kinetic energy of the mode is  
\[T_{i}=\frac{1}{2}M_{i}v_{i}{}^{2} \tag{6}\]  
Figure 2: Schematic diagram of a discontinuous multi-DOF system.  
The kinetic energy of the mode \(i\)-th is a constant value, but the values of the mode mass \(M_{i}\) and vibration velocity \(v_{i}\) are not unique and are related to the scaling factor of the mode shape vector, assuming the shape vector of the \(i\)-th mode is  
\[X_{i}=\{x_{1i},x_{2i},\cdots,x_{ni}\}^{T} \tag{7}\]  
In this mode, the vibration velocity of each point is proportional to the mode shape, such that  
\[V_{i}=\{\rho x_{1i},\rho x_{2i},\cdots,\rho x_{ni}\}^{T}=\rho\{x_{1i},x_{2i}, \cdots,x_{ni}\}^{T}(m/s) \tag{8}\]  
Then, the total kinetic energy of the system in this mode can be expressed as  
\[T_{i}=\frac{1}{2}\rho^{2}\left(m_{1}x_{1i}^{2}+m_{2}x_{2i}^{2}+\cdots+m_{n}x_{ ni}^{2}\right) \tag{9}\]  
The mode amplitude of the \(j\)-th particle is taken as the scaling factor, such that  
\[\begin{array}{l}T_{i}=\frac{1}{2}\left(m_{1}\Big{(}\frac{x_{1i}}{x_{ij}} \Big{)}^{2}+m_{2}\Big{(}\frac{x_{2i}}{x_{ji}}\Big{)}^{2}+\cdots+m_{j}+\cdots+m _{n}\Big{(}\frac{x_{ni}}{x_{ji}}\Big{)}^{2}\right)\left(\rho x_{ji}\right)^{2} \\ =\frac{1}{2}M_{i-j}\left(\rho x_{ji}\right)^{2}\end{array} \tag{10}\]  
This equation is compared with (9); it can be known that \(M_{i-j}\) is the overall modal mass of the \(i\)-th mode with \(x_{j}\) as the scaling factor, and point \(j\) is the control target point of the \(i\)-th mode; \(M_{i-j}\) can be used as the structural reference quality when TMD controls the \(i\)-th mode; and \(m_{a}\Big{(}\frac{x_{ni}}{x_{ji}}\Big{)}^{2}\) is the mass distribution of modal \(M_{i-j}\) at each point in space, also called the contribution of \(m_{a}\) to the overall modal mass \(M_{i-j}\).  
When the \(k\)-th mode of the multi-DOF system is assumed to be the local mode, two local modes simultaneously occur; as shown in Figure 3, only \(m_{j-2},\ m_{j-1},\ m_{j},\ m_{j+1},\ m_{j+2}\) and \(m_{l-1},\ m_{l},\ m_{l+1}\) vibrate locally.  
So, the total kinetic energy is  
\[\begin{array}{l}T_{k}=\frac{1}{2}M_{k-j}\left(\rho x_{j}\right)^{2}\\ =\frac{1}{2}\left(m_{j-2}\Big{(}\frac{x_{(j-2)k}}{x_{jk}}\Big{)}^{2}+m_{j-1} \Big{(}\frac{x_{(j-1)k}}{x_{jk}}\Big{)}^{2}+m_{j}+m_{j+1}\Big{(}\frac{x_{(j+1 )k}}{x_{jk}}\Big{)}^{2}+m_{j+2}\Big{(}\frac{x_{(j+2)k}}{x_{jk}}\Big{)}^{2}+ \\ m_{l-1}\Big{(}\frac{x_{(l-1)k}}{x_{jk}}\Big{)}^{2}+m_{l}\Big{(}\frac{x_{lk}}{ x_{jk}}\Big{)}^{2}+m_{l+1}\Big{(}\frac{x_{(l+1)k}}{x_{jk}}\Big{)}^{2}\right)\left( \rho x_{jk}\right)^{2}\end{array} \tag{11}\]  
Assuming  
\[M_{local1,k-j}=m_{j-2}\Bigg{(}\frac{x_{(j-2)k}}{x_{jk}}\Bigg{)}^{2}+m_{j-1} \Bigg{(}\frac{x_{(j-1)k}}{x_{jk}}\Bigg{)}^{2}+m_{j}+m_{j+1}\Bigg{(}\frac{x_{( j+1)k}}{x_{jk}}\Bigg{)}^{2}+m_{j+2}\Bigg{(}\frac{x_{(j+2)k}}{x_{jk}}\Bigg{)}^{2} \tag{12}\]  
\[M_{local2,k-j}=m_{l-1}\Bigg{(}\frac{x_{(l-1)k}}{x_{jk}}\Bigg{)}^{2}+m_{l} \Bigg{(}\frac{x_{lk}}{x_{jk}}\Bigg{)}^{2}+m_{l+1}\Bigg{(}\frac{x_{(l+1)k}}{x_{ jk}}\Bigg{)}^{2} \tag{13}\]  
Then  
\[T_{k}=\frac{1}{2}\Big{(}M_{local1,k-j}+M_{local2,k-j}\Big{)}\left(\rho x_{jk} \right)^{2} \tag{14}\]  
Figure 3: Local modal diagram of multi-DOF system.