## 2 Blade-Damper Model and Theoretical Development  
The structural model of the blades with under-platform dry friction dampers is shown in Figure 2: the \(X\)-\(Y\)-\(Z\) coordinate system (the global cylindrical coordinate system) is defined in accordance with the convention of tangential (\(X\)), radial (\(\gamma\)), and axial (\(Z\)) directions; the \(x_{1}\)-\(y_{1}\) coordinate system which is a plane Cartesian coordinate frame (called the left damper coordinate system) is defined with the local coordinate origin being at the left initial contact point, the \(x_{2}\)-\(y_{2}\) coordinate system which is a plane Cartesian coordinate frame (called the right damper coordinate system) is defined similarly. In this paper, the blade-platform structure is simplified and regarded as two rigidly connected Euler-Bernoulli beams, as illustrated in Figure 3. Unlike a conventional under-platform damper which is a floating mass actuated by centrifugal force, the under-platform damper studied here is mounted on the free end of a small vertical cantilever beam. The mechanical models of the blades and under-platform dampers are shown in Figures 3 and 4. In Figures 3 and 4, \(F\) is the external excitation force acting on a blade, \(N_{1}\) and \(N_{2}\) are the normal contact forces acting on the blade platform from the dampers, \(f_{1}\) and \(f_{2}\) are the friction forces at the contact interface, \(F_{1}\) and \(F_{2}\) are the centrifugal forces of the under-platform dampers, and \(k\) is a spring constant (the equivalent lateral stiffness of the tip of the small vertical cantilever beam). \(F_{1}=m_{1}r\Omega^{2}\), \(F_{2}=m_{2}r\Omega^{2}\), where \(m_{1}\) and \(m_{2}\) are the mass of the dampers, \(r\) is the distance between the platform and the centre of the disk, and \(\Omega\) is the rotating speed of the disk. The weights of the dampers are very small in comparison with the centrifugal forces and thus can be ignored. In this investigation, the normal contact forces and the tangential friction forces at the contact interface are modeled as moving loads (which are a distinct feature of this paper and have not been modeled as such in previous research of blade under-platform friction dampers), and the two  
Figure 4: The mechanical model of the under-platform dampers. Figure 3: The mechanical model of the simplified blade-platform system. Figure 2: The structural model of the blades with under-platform dry friction dampers. dampers are assumed to undergo horizontal and vertical vibrations, but no rotation. They are in point contact with the platform and do not separate with the platform during vibration excited by excitation force \(F\). Therefore, displacements \(y_{1}\) and \(y_{2}\) of the dampers are equal to the local transverse deflections of the platform; but they can slide along the underside of the platform in the global \(X\) direction. The horizontal vibration and the vertical vibration of the two dampers, and the horizontal and transverse platform vibrations are coupled by friction at the contact interface. A classical Coulomb friction model [39] is adopted in this paper: before sliding occurs, the friction force equals the shear force at the contact interface due to the applied external force and acts to resist the initiation of sliding; during sliding the friction force is proportional to the normal force at the contact interface and acts in the direction opposite to sliding. A further assumption is that the static coefficient of friction \(\mu_{\text{s}}\) is greater than the kinetic coefficient of friction \(\mu_{\text{k}}\) (so that stick-slip vibration of the damper is possible) and both are constant. Figure 5 describes the Coulomb friction model. Accordingly, the relationship between the friction force \(f(v_{\text{rel}})\) and the relative velocity \(v_{\text{rel}}\) and the local normal force \(N\) is  
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