## 2 Governing Equations  
All multi-degree of freedom (multi-DOF) systems subjected to an external dynamic forcing and to friction share similar equilibrium equations. In the time domain, they can be written as  
\[\mathbf{M}\cdot\ddot{\mathbf{q}}(\tau)+\mathbf{C}\cdot\dot{\mathbf{q}}(\tau)+ \mathbf{K}\cdot\mathbf{q}(\tau)=\mathbf{f}_{\mathbf{E}}(\tau)+\mathbf{f}_{ \mathbf{C}}(\mathbf{q},\tau) \tag{1}\]  
the dependence of the contact forces vector \(\mathbf{f}_{\mathbf{C}}\) from the vector of displacements \(\mathbf{q}\) is defined by the contact model [18; 22], further discussed below. Equation (1) can be tailored to represent an underplatform damper inserted between two adjacent turbine blades. The dissipative function of underplatform dampers has been discussed in the introduction, while Figure 1 shows a graphical representation of a turbine disk with underplatform dampers. The numerical code concerned with the solution of Equation (1), already applied and validated in [38; 39], can be considered as a \"plug-in\" software used to represent any given underplatform damper between blades modeled using finite elements (see also Figure 2a). The damper is here considered as a rigid body. The rigid body assumption is well justified by the bulkiness of \"solid-bar\" underplatform dampers and has already been validated against specific experimental evidence on underplatform dampers and blades [23; 31; 32]. The same conclusion has been drawn for similar applications in [4]. Furthermore, this choice has two advantages: (1) damper mass, shape, and contact points are easily editable by the user, and (2) the number of physical damper DOFs is limited, regardless of the selected number of contact points. In this paper, the system is further simplified, to be easily used as a demonstrator. A 2D version of the damper between a set of blades is considered, as shown in Figure 2b. This version of the damper system has been effectively used and validated in a series of previous works [19; 39; 40], where the blade platforms undergo bending (i.e., pure in plane) modes of vibration. Furthermore, it should be noted that the techniques described throughout Sections 3-5 can be easily extended to more complex systems. Their formulation is general. With reference to Figure 2b, Equation (1) becomes  
\[\mathbf{M_{D}}\cdot\mathbf{\ddot{q}_{D}}(\tau)=\mathbf{f_{ED}}(\tau)+\mathbf{f_ {CD}}(\mathbf{q_{D}},\mathbf{q_{P}},\tau) \tag{2}\]  
Figure 1: (a) CAD model of a turbine bladed disk with underplatform dampers (not directly visible in the figure). (b) Sketch representing a close up of three blades with interposed underplatform dampers. Figure 2: (a) Contact model with three contact springs. (b) Contact patches on the blade platforms and on the strip damper. where \(\mathbf{M_{D}}=diag(m_{D},m_{D},I_{z})\) is the mass matrix, \(\mathbf{f_{ED}}=(0,-|\mathrm{CF}|,0)^{\prime}\) is the vector of external forces on the damper, \(\mathbf{q_{D}}=(u_{D},w_{D},\mathrm{f_{ID}})^{\prime}\) is the vector of unknowns at the damper center of mass, and \(\mathbf{q_{P}}=(\mathbf{q_{LIP}},\mathbf{q_{LIP}},\mathbf{q_{RF}})^{\prime}\) is the vector of input platform displacements, where \(\mathbf{q_{IP}}=(u_{\mathrm{i}P},w_{\mathrm{i}P})^{\prime}\) and i = L1, L2, R. The vector \(\mathbf{q_{P}}\) is here seen as an input because it comes from the displacement of the nodes on the blade platforms, provided by \"higher-level\" software [38] concerned with the blades' dynamics. The indication of the time instant \(\tau\) has been dropped for the sake of brevity. In this particular set of equations, there are no harmonic \"external forces.\" The input platform displacements act on the contact elements and produce the contact forces that serve as the harmonic external excitation to the system. Further details on the contact model implementation are given in Section 3.