## 2 Mechanical Model and Dynamics Equations
### The Dynamics Equations of Blade-Coupled Vibration  
In the _yoz_ plane, the sizes of the damper and the platform are neglected when calculating the convective acceleration. In accordance with Figure 2, the dynamic equation of the system in vector form is established with the theories of compositive motion, which can be written as:  
\[\left\{ \begin{array}{l}m_{1}\Big{(}\overset{\rightarrow}{a}_{1r}+\overset{ \rightarrow}{a}_{1e}+\overset{\rightarrow}{a}_{1k}\Big{)}+c_{1}\overset{ \rightarrow}{v}_{1r}+\overset{\rightarrow}{F}_{K1}=-m_{1}\overset{ \rightarrow}{g}+\overset{\rightarrow}{Q}_{1}+\overset{\rightarrow}{f}_{1} +\overset{\rightarrow}{N}\\ m_{2}\Big{(}\overset{\rightarrow}{a}_{2r}+\overset{\rightarrow}{a}_{2e}+ \overset{\rightarrow}{a}_{2k}\Big{)}+c_{2}\overset{\rightarrow}{v}_{2r}+ \overset{\rightarrow}{F}_{K2}=-m_{2}\overset{\rightarrow}{g}+\overset{ \rightarrow}{f}_{1}\)' \(-\overset{\rightarrow}{N}\end{array} \right. \tag{1}\]  
where \(m_{1}\overset{\rightarrow}{g}\) and \(m_{2}\overset{\rightarrow}{g}\) are gravity vectors, which can be ignored, as they are very small in comparison with the other forces. The relative acceleration vectors of \(m_{1}\) and \(m_{2}\) in the corresponding local coordinate systems are \(\overset{\rightarrow}{a}_{1r}\) and \(\overset{\rightarrow}{a}_{2r}\), respectively (in \(y\) direction). The corresponding convective acceleration vectors are \(\overset{\rightarrow}{a}_{1e}\) and \(\overset{\rightarrow}{a}_{2e}\). (When \(\overset{\rightarrow}{\omega}\) is constant, the convective acceleration is along the \(z\) direction. When \(\overset{\rightarrow}{\omega}\) changes, the tangential component of the convective acceleration is along \(y\) direction, and the normal component of the convective acceleration is along the \(z\) direction.) The corresponding Coriolis acceleration vectors are \(\overset{\rightarrow}{a}_{1k}\) and \(\overset{\rightarrow}{a}_{2k}\). (The direction of \(\overset{\rightarrow}{\omega}\) is perpendicular to the \(yoz\) plane; according to \(\overset{\rightarrow}{a}_{k}=2\overset{\rightarrow}{\omega}\times \overset{\rightarrow}{v}_{r}\), the Coriolis acceleration is along the \(z\) direction.) The corresponding relative velocity vectors are \(\overset{\rightarrow}{v}_{1r}\) and \(\overset{\rightarrow}{v}_{2r}\)(in \(y\) direction). \(\overset{\rightarrow}{F}_{K1}\) and \(\overset{\rightarrow}{F}_{K2}\) are the corresponding stiffness forces vectors in \(y\) direction. \(\overset{\rightarrow}{Q}_{1}\) is the aerodynamic excitation forces' vector in \(y\) direction. \(\overset{\rightarrow}{N}\) is the normal pressure vector between the blade and the damper in \(z\) direction. Ignoring \(m_{1}\overset{\rightarrow}{g}\) and \(m_{2}\overset{\rightarrow}{g}\), the dynamic Equation (1) is projected to the \(y\) direction, and the scalar form of it could be written as  
\[\left\{ \begin{array}{l}m_{1}\ddot{y}_{1}+c_{1}\dot{y}_{1}+k_{1}y_{1}=Q_{1}-f _{1}\\ m_{2}\ddot{y}_{2}+c_{2}\dot{y}_{2}+k_{2}y_{2}=f_{1}\prime=f_{1}\end{array} \right. \tag{2}\]  
Figure 2: Dynamic model of a blade with an under-platform damper. \[\left\{ \begin{array}{l}m_{1}(\ddot{y}_{1}+a_{1\sigma})+c_{1}\dot{y}_{1}+k_{1}y_{ 1}=Q_{1}-f_{1}\\ m_{2}(\ddot{y}_{2}+a_{2\sigma})+c_{2}\dot{y}_{2}+k_{2}y_{2}=f_{1}{}^{\prime}=f_{ 1}\end{array} \right. \tag{3}\]  
The dynamic equation of the system in \(y\) direction is Equation (2) when the rotation speed of the bladed disc is constant, with the rotation speed varying the equation shown as Equation (3). The response characteristics of the blade vibration relative to the bladed disc can be studied at working speed (uniform rotation) and in start-stop condition (non-uniform rotation). In Equation (3), \(a_{1\sigma}\) and \(a_{2\sigma\tau}\) exist with the bladed disc's rotation considered, and the tangential component of the convective acceleration is described as  
\[\left\{ \begin{array}{l}a_{1\sigma\tau}=-l_{1}\alpha=-l_{1}\dot{\omega}\\ a_{2\sigma\tau}=-l_{2}\alpha=-l_{2}\dot{\omega}\end{array} \right.