## 2 Blade-Damper Model and Theoretical Development
The nodal displacement vector of the whole structure can be expressed as  
\[\left\{ \begin{array}{c}\mathbf{u}_{1}(t)\\ \mathbf{u}_{2}(t)\\ \mathbf{w}_{1}(t)\\ \mathbf{w}_{2}(t)\\ \Theta_{1}(t)\\ \Theta_{2}(t)\end{array} \right\}=\sum_{i=1}^{n}\left\{ \begin{array}{c}\Psi_{u 1i}^{\text{FE}}\\ \Psi_{u2i}^{\text{FE}}\\ \Psi_{u2i}^{\text{FE}}\\ \Psi_{w2i}^{\text{FE}}\\ \Psi_{w1i}^{\text{FE}}\\ \Psi_{\theta 2i}^{\text{FE}}\end{array} \right\}q_{i}(t)=\left\{ \begin{array}{c} \mathbf{\Psi}_{u1}^{\text{FE}}\\ \mathbf{\[PERSON]}_{u2}^{\text{FE}}\\ \mathbf{\[PERSON]}_{u1}^{\text{FE}}\\ \mathbf{\[PERSON]}_{w2i}^{\text{FE}}\\ \mathbf{\[PERSON]}_{w2i}^{\text{FE}}\\ \mathbf{\[PERSON]}_{\theta 2i}^{\text{FE}}\end{array} \right\}\mathbf{q}(t) \tag{9}\]  
where \(q_{i}(t)\) is the \(i\)-th modal coordinate of the blade-platform structure. \[\mathbf{q}(t)=\left\{q_{1}(t),q_{2}(t),q_{3}(t),\ldots q_{n}(t)\right\}^{ \text{T}}\]  
\[\mathbf{\Psi}_{u1}^{\text{FE}}=\left[\Psi_{u11}^{\text{FE}},\Psi_{u12}^{\text {FE}},\Psi_{u13}^{\text{FE}},\ldots\ \Psi_{u1n}^{\text{FE}}\right]\mathbf{\Psi}_{u2}^{\text{FE}}=\left[\Psi_{u21}^ {\text{FE}},\Psi_{u22}^{\text{FE}},\Psi_{u23}^{\text{FE}},\ldots\ \left[\Psi_{u2n}^{\text{FE}}\right]\]  
\[\mathbf{\Psi}_{w1}^{\text{FE}}=\left[\Psi_{w11}^{\text{FE}},\[PERSON]_{w12}^{\text {FE}},\[PERSON]_{w13}^{\text{FE}},\ldots\ \[PERSON]_{w1n}^{\text{FE}}\right]\mathbf{\[PERSON]}_{w2}^{\text{FE}}=\left[\[PERSON]_{w21}^ {\text{FE}},\[PERSON]_{w22}^{\text{FE}},\[PERSON]_{w23}^{\text{FE}},\ldots\ \[PERSON]_{w2n}^{\text{FE}}\right]\]  
\[\mathbf{\[PERSON]}_{\theta 1}^{\text{FE}}=\left[\[PERSON]_{\theta 11}^{\text{FE}},\[PERSON]_{ \theta 12}^{\text{FE}},\[PERSON]_{\theta 13}^{\text{FE}},\ldots\ \[PERSON]_{\theta 1n}^{\text{FE}}\right]\mathbf{\[PERSON]}_{\theta 2}^{\text{FE}}=\left[\[PERSON]_{\theta 21}^ {\text{FE}},\[PERSON]_{\theta 22}^{\text{FE}},\[PERSON]_{\theta 23}^{\text{FE}},\ldots\ \[PERSON]_{\theta 2n}^{\text{FE}}\right]\]  
\(\[PERSON]_{wiki}^{\text{FE}},\[PERSON]_{wiki}^{\text{FE}}\) and \(\[PERSON]_{\theta 3i}^{\text{FE}}\) (\(k=1,\ 2\)) form the \(i\)-th mass-normalized FE mode of the \(k\)-th beam (in the order of ascending frequencies).  
It follows from Equation (9) that the horizontal and the vertical displacements of the platform can be expressed as  
\[[u_{2}(\varepsilon,t),w_{2}(\varepsilon,t)]=\mathbf{q}^{\text{T}}(t)[\Psi_{u2 }(\varepsilon),\Psi_{w2}(\varepsilon)] \tag{10}\]  
where \(\Psi_{u2}(\varepsilon)\) and \(\Psi_{w2}(\varepsilon)\) are approximate analytical \(u\)-component and \(w\)-component mode vectors converted from \(\mathbf{\Psi}_{u2}^{\text{FE}}\), \(\mathbf{\Psi}_{w2}^{\text{FE}}\) and \(\mathbf{\Psi}_{\theta 2}^{\text{FE}}\) through element shape functions. Very similarly, the two displacements of the blade can also be expressed like Equation (10). Analytical modes (even though approximate), instead of numerical modes, are particular useful in dealing with moving loads [44]. Regardless of relative sticking or slipping of the dampers to the platform, friction forces \(f_{1}\) and \(f_{2}\) and normal forces \(N_{1}\) and \(N_{2}\) at the contact points always satisfy  
\[\frac{1}{2}m_{1}\ddot{x}_{1}+\frac{1}{2}kx_{1}=f_{1} \tag{11}\]  
\[\frac{1}{2}m_{2}\ddot{x}_{2}+\frac{1}{2}kx_{2}=f_{2} \tag{12}\]  
\[\frac{1}{2}F_{1}-\frac{1}{2}m_{1}\ddot{y}_{1}=N_{1} \tag{13}\]  
\[\frac{1}{2}F_{2}-\frac{1}{2}m_{2}\ddot{y}_{2}=N_{2} \tag{14}\]  
where \(x_{1}\) and \(x_{2}\), and \(y_{1}\) and \(y_{2}\) are the local horizontal displacements and the vertical displacements of the two dampers, respectively; and the \(\frac{1}{2}\) in the above equations is due to the equal share of a damper by two adjacent blades (see Figure 2). Platform horizontal displacements \(u_{2}\) at the contact points can be found from Equation (10). Denote \(X_{1}=u_{2}-x_{1}+x_{10}\), \(X_{2}=u_{2}-x_{2}+x_{20}\), \(\frac{1}{2}F_{1}=\frac{1}{2}F_{2}=N_{0}\).