## 2 Blade-Damper Model and Theoretical Development
Then, one acquires an approximate expression of the \(n\)-th analytical mode of the blade-platform structure as  
\[\Psi_{w1n}(\varepsilon)=\left\{\begin{array}{c}\mathbf{N}_{w11}(\varepsilon _{1})\{u_{11},\theta_{11},u_{12},\,\theta_{12}\}_{n  
\[\psi_{w2n}(\varepsilon)=\left\{\begin{array}{c}\mathbf{N}_{w21}(\varepsilon_{1}) \{w_{21},\theta_{21},w_{22},\theta_{22}\}_{n}^{\mathrm{T}}\varepsilon\subset \varepsilon_{1}\\ \mathbf{N}_{w22}(\varepsilon_{2})\{w_{22},\theta_{22},w_{23},\theta_{23}\}_{ n}^{\mathrm{T}}\varepsilon\subset\varepsilon_{2}\\ \cdot\\ \cdot\\ \cdot\\ \mathbf{N}_{w2i}(\varepsilon_{i})\Big{\{}w_{2i},\theta_{2i},w_{2(i+1)}, \theta_{2(i+1)}\Big{\}}_{n}^{\mathrm{T}}\varepsilon\subset\varepsilon_{i}\\ \cdot\\ \cdot\\ \cdot\\ \cdot\\ \psi_{u2n}(\varepsilon)=\left\{ \begin{array}{c}\mathbf{N}_{u21}(\varepsilon_ {1})\{u_{21},\,u_{22}\}_{n}^{\mathrm{T}}\varepsilon\subset\varepsilon_{1}\\ \mathbf{N}_{u22}(\varepsilon_{2})\{u_{22},\,u_{23}\}_{n}^{\mathrm{T}}\varepsilon \subset\varepsilon_{2}\\ \cdot\\ \cdot\\ \cdot\\ \mathbf{N}_{u2i}(\varepsilon_{i})\Big{\{}u_{2i},u_{2(i+1)}\Big{\}}_{n}^{ \mathrm{T}}\varepsilon\subset\varepsilon_{i}\\ \cdot\\ \cdot\\ \cdot\\ \cdot\\ \cdot\end{array} \right. \tag{7}\]  
where \(\varepsilon_{i}\) is the local longitudinal coordinate within the \(i\)-th element of the platform; \(N_{wki}(\varepsilon_{i})=\frac{1}{L_{ki}}^{3}[\left(L_{ki}^{3}-3L_{ki}\varepsilon _{i}^{2}+2\varepsilon_{i}^{3}\right)\), \(\left(L_{ki}^{2}\varepsilon_{i}-2L_{ki}\varepsilon_{i}^{2}+\varepsilon_{i}^{3 }\right)\), \(\left(3L_{ki}\varepsilon_{i}^{2}-2\varepsilon_{i}^{3}\right)\), \(\left(\varepsilon_{i}^{3}-L_{ki}\varepsilon_{i}^{2}\right)]\); and \(N_{uki}(\varepsilon_{i})=\left[1-\frac{\varepsilon_{i}}{L_{ki}},\frac{ \varepsilon_{i}}{L_{ki}}\right]\); \(\left(k=1,\,2\right)\); and \(L_{ki}\) is the length of the \(i\)-th element of the \(k\)-th beam. The reason for converting the FE modes to approximate analytical modes is to accommodate the tracking of horizontal positions of the friction dampers relative to the platform, which is a common formulation of moving-load problems [43]. Such a numerical-analytical combined formulation allows dynamic responses of complicated structures excited by moving loads to be determined conveniently [44]. The combined equations of motion of the blade-platform structure under external excitation can be written as  
\[\mathbf{M}\left\{ \begin{array}{c}\ddot{u}_{1}\\ \ddot{u}_{2}\\ \ddot{w}_{1}\\ \ddot{w}_{2}\end{array} \right\}+\mathbf{K}\left\{ \begin{array}{c}u_{1}\\ u_{2}\\ w_{1}\\ w_{2}\end{array} \right\}=\mathbf{f} \tag{8}\]  
\[\mathbf{M}=\left[ \begin{array}{cccc}\rho_{1}A_{1}&0&0&0\\ 0&\rho_{2}A_{2}&0&0\\ 0&0&\rho_{1}A_{1}&0\\ 0&0&0&\rho_{2}A_{2}\end{array} \right]\mathbf{K}=\left[ \begin{array}{cccc}-E_ {1}A_{1}\frac{\partial^{2}}{\partial\varepsilon^{2}}&0&0&0\\ 0&-E_{2}A_{2}\frac{\partial^{2}}{\partial\varepsilon^{2}}&0&0\\ 0&0&E_{1}I_{1}\frac{\partial^{4}}{\partial\varepsilon^{4}}&0\\ 0&0&0&E_{2}I_{2}\frac{\partial^{4}}{\partial\varepsilon^{4}}\end{array} \right]\]  
\[\mathbf{f}=\left\{ \begin{array}{c}0\\ f_{1}\delta(\varepsilon-X_{1})+f_{2}\delta(\varepsilon-X_{2})\\ F\delta(\varepsilon-Y_{0})\\ N_{1}\delta(\varepsilon-X_{1})+N_{2}\delta(\varepsilon-X_{2})\end{array} \right\}\]  
where \(\mathbf{M}\) and \(\mathbf{K}\) are the \"mass\" and \"stiffness\" operator matrices of the blade-platform structure; and \(\{u_{1},\,u_{2},\,w_{1},\,w_{2}\}^{\mathrm{T}}\) and \(\mathbf{f}\) are, respectively, the nodal displacement vector and force vector of the structure. The dot over a symbol represents differentiation with respect to time \(t\), \(Y_{0}\) is the location of the external excitation force \(F\), \(X_{1}\) and \(X_{2}\) are the location of the contact points which are unknown a priori and vary with time. Please note that the first element of \(\mathbf{f}\) is zero because there is no external force acting in the \(u_{1}\) direction for beam 1 (the blade) while the second element denotes the friction forces at the two damper-platform interfaces.