## 2 Blade-Damper Model and Theoretical Development
\begin{array}{c}f(v_{\text{rel}})=\mu_{\text{k}}N(v_{\text{rel}}>0)\\ f(v_{\text{rel}})=-\mu_{\text{k}}N(v_{\text{rel}}<0)\\ -\mu_{\text{s}}N\leq f(v_{\text{rel}})\leq\mu_{\text{s}}N(v_{\text{rel}}=0) \end{array} \right. \tag{1}\]  
The equations of free vibration in the local transverse and longitudinal directions of the component of the blade modeled as an Euler-Bernoulli beam [40] (denoted by 1) in its local coordinate \(\varepsilon\) are  
\[\left\{ \begin{array}{c}\rho_{1}A_{1}\frac{\partial^{2}w_{1}}{\partial t^{2} }(\varepsilon,t)+E_{1}I_{1}\frac{\partial^{4}w_{1}}{\partial x^{4}}( \varepsilon,t)=0\\ \rho_{1}A_{1}\frac{\partial^{2}w_{1}}{\partial t^{2}}(\varepsilon,t)-E_{1}A_ {1}\frac{\partial^{2}w_{1}}{\partial x^{2}}(\varepsilon,t)=0\end{array} \right. \tag{2}\]  
In the same approach, the equations of free vibration of the platform component modeled as an Euler-Bernoulli beam (denoted by 2) are  
\[\left\{ \begin{array}{c}\rho_{2}A_{2}\frac{\partial^{2}w_{2}}{\partial t^{2} }(\varepsilon,t)+E_{2}I_{2}\frac{\partial^{4}w_{2}}{\partial t^{4}}( \varepsilon,t)=0\\ \rho_{2}A_{2}\frac{\partial^{2}w_{1}}{\partial t^{2}}(\varepsilon,t)-E_{2}A_ {2}\frac{\partial^{4}w_{2}}{\partial x^{2}}(\varepsilon,t)=0\end{array} \right. \tag{3}\]  
where \(w_{1}\) and \(w_{2}\) are the transverse displacements, and \(u_{1}\) and \(u_{2}\) are the longitudinal displacements, all in their local coordinates; \(E_{1}\) and \(E_{2}\) are [PERSON]'s moduli of these two components; \(\rho_{1}\) and \(\rho_{2}\) are the mass densities; \(A_{1}\) and \(A_{2}\) are the cross-sectional areas; and \(I_{1}\) and \(I_{2}\) are the second moment of the areas. The \(w\)-component and \(u\)-component of the \(k\)-th beam's \(n\)-th analytical mode can be denoted as \(\psi_{w\text{k}n}(\varepsilon)\) and \(\psi_{w\text{k}n}(\varepsilon)\), \((k=1,\ 2)\) in its local coordinate \(\varepsilon\), which consist of base functions of sine, cosine,  
Figure 5: The classical Coulomb friction model. hyperbolic sine and hyperbolic cosine functions when expressed exactly [41]. Equations (2) and (3) are solved to get the natural frequencies and modes of the cross structure, which will be used in the mode superposition method for subsequent analysis. However, when the configuration of the beam structure is complex, analytical modes are difficult to obtain accurately. In general, the FE method provides a simple and effective approach to obtain the frequencies and modes of a structure of arbitrary configuration and hence is used. The blade-platform system is discretized with a number of two-node Euler-Bernoulli Beam elements. Each node has three degrees-of-freedom: one translation along the longitudinal axis of the beam element, denoted by \(u\); one translation lateral to the beam axis, denoted by \(w\); and a rotation around the axis normal to this plane, denoted by \(\theta\)[42]. Denote the \(k\)-th beam's \(n\)-th numerical mode as \(\Psi_{kn}^{\text{FE}}\)  
\[\Psi_{kn}^{\text{FE}}=\big{\{}u_{k1},w_{k1},\theta_{k1},u_{k2},w_{k2},\theta_{k 2},\ldots u_{ki},w_{ki},\theta_{ki},u_{k(i+1)},w_{k(i+1)},\theta_{k(i+1)}, \ldots\big{\}}_{n}^{\text{T}}\]  
\[\Psi_{ukn}^{\text{FE}}=\big{\{}u_{k1},u_{k2},\ldots u_{ki}\,,u_{k(i+1)},\ldots \big{\}}_{n}^{\text{T}},\,\Psi_{ukn}^{\text{FE}}=\big{\{}w_{k1},w_{k2},\ldots w _{ki}\,,w_{k(i+1)},\ldots\big{\}}_{n}^{\text{T}},\]  
\[\Psi_{\theta kn}^{\text{FE}}=\big{\{}\theta_{k1},\theta_{k2},\ldots\theta_{ki} \,,\theta_{k(i+1)},\ldots\big{\}}_{n}^{\text{T}}\]  
\(k=1,\ 2\), where subscript \(i\) is the left node of the \(i\)-th beam element and the right node of the \((i-1)\)-th beam element, and superscript T stands for matrix transpose. When the number of beam elements is sufficient, the numerical modes and frequencies will be very close to the analytical modes and frequencies of the whole structure. Denote the \(i\)-th element shape function matrix of the \(k\)-th beam for \(w\)-displacement and \(u\)-displacement by \(\mathbf{N}_{uki}(\varepsilon)\) and \(\mathbf{N}_{uki}(\varepsilon)\)\((k=1,\ 2)\).