## 2 Numerical Model  
Blades and strip are separate structures that interact with each other only through the contact friction forces. From a dynamic point of view, the blades with a strip damper can be considered as a unique system [35; 36] described by the following dynamic balance equations:  
\[M\ddot{\mathbf{x}}(t)+C\dot{\mathbf{x}}(t)+K\mathbf{x}(t)=\mathbf{F_{e}}(t)- \mathbf{F_{nl}}(t) \tag{1}\]where:  
* \(M\), \(C\), and \(K\) are the mass, damping stiffness matrices of the system respectively. They are obtained by combining the mass, damping and stiffness matrices of both strip and blades. Usually these matrices are extracted from the FE models of strip and blades after a Craig Bampton-Component Mode Synthesis (CB-CMS) reduction;
* \(\mathbf{x}\) is the vector of the DOFs of strip and blades. Due to the CB-CMS reduction, this vector includes both modal DOFs and the physical DOFs which are the displacements of the master nodes retained in the reduction. * \(\mathbf{F_{e}}\) is the vector of external forces acting on the system, i.e., the sum of the external excitation vector \(\mathbf{F_{ex}}\), acting on the blade airfoil, and the centrifugal force vector \(\mathbf{F_{c}}\), pushing the strip against the blade platform as shown in Figure 1.
* \(\mathbf{F_{nl}}\) is the vector of friction nonlinear forces generated between the blade-strip coupled nodes by their relative displacements. Equation (1) is nonlinear due to the presence of \(\mathbf{F_{nl}}\) that depends on the relative displacement between strip and blades. Equation (1) is written in the time domain and can be converted in frequency domain and solved by the well-known multi-harmonic balance method (MHBM) [26; 27; 28]. In detail forces and displacements of Equation (1) are approximated by Fourier series as expressed below. \[\mathbf{F_{e}}(t)=\mathbf{F_{e}^{(0)}}+\Re\left(\sum_{n=1}^{N\hbar}\mathbf{F_ {e}^{(n)}}e^{inuct}\right) \tag{2}\]  
\[\mathbf{x}(t)=\mathbf{x}^{(0)}+\Re\left(\sum_{n=1}^{N\hbar}\mathbf{x}^{(n)}e^ {inuct}\right) \tag{3}\]  
Figure 1: Finite Elements (FE) model of blades and strip damper, full disk obtained through cyclic symmetry and relevant external forces applied on blades and strip dampers. The cyan symbols represent the boundary conditions imposed to the disk sector: each sector is tangentially linked to the neighbouring ones and is rigidly connected to the disk hub, not shown in the picture. \[\mathbf{F_{nl}}(t)=\mathbf{F_{nl}^{(0)}}+\Re\left(\sum_{n=1}^{Nh}\mathbf{F_{nl}^{(n )}}e^{inact}\right)\] (4) where:
* \(\omega\) is the fundamental frequency of the excitation force. * \(Nh\) is the number of harmonics that approximate the functions. More the harmonics, better the approximation, but larger the computational time. * The subscript \(n\) refers to the \(n^{th}\) harmonic Fourier coefficient. * \(\Re\) indicates the real part of forces and displacements. Note the composition of the external forces \(\mathbf{F_{e}}=\mathbf{F_{c}}+\mathbf{F_{ex}}\):
* The centrifugal forces \(\mathbf{F_{c}}\), acting on the strip nodes, are static forces and therefore they appear only in the zero-order harmonic \(\mathbf{F_{e}^{(0)}}\). * The external excitations \(\mathbf{F_{ex}}\) are approximated from the first harmonic onwards as \(\mathbf{F_{e}^{(n)}}\), for \(1\leq n\leq Nh\). By substituting Equations (2)-(4) into the set of balance Equation (1) a new set of algebraic complex equations is deduced:  
\[D(\omega)^{(n)}\mathbf{x}^{(n)}=\mathbf{F_{e}^{(n)}}-\mathbf{F_{nl}^{(0)}} \quad with\quad 0\leq n\leq Nh \tag{5}\]  
where:  
* \(D^{(n)}=-(n\omega)^{2}M+in\omega C+K\) is the \(n^{th}\) dynamic stiffness matrix of the system, corresponding to the \(n^{th}\) harmonic. The \(0^{th}\) order represents the static balance equation, where only the centrifugal forces are considered. * The vectors \(\mathbf{x}^{(n)}\), \(\mathbf{F_{e}^{(n)}}\) and \(\mathbf{F_{nl}^{(n)}}\) represent the Fourier coefficients of the \(n^{th}\) harmonic.