## 5 A Time-Marching Method for Systems with Negligible Inertial Effects
### The Influence of Damper Dynamics on the Overall Damper Equilibrium  
The equations solved through DTI are dynamic equations; i.e., they take into account inertial and viscous damping effects. However, given the reduced mass of the damper and the lack of a damping matrix \(\mathbf{C_{D}}\), it is worth investigating what role the dynamics inside Equation (2) play and what the consequences of neglecting it are. Furthermore, in many applications, the presence of a viscous damping matrix can be neglected as the damping provided by friction is far larger. To properly answer the questions posed above, two aspects have to be considered:  
1. the magnitude of the damper inertia forces;
2. the damper \"inner resonance\" (i.e., the natural frequencies of the damper oscillating on the contact springs). To tackle the first point, let us consider a damper of mass \(\mathbf{m_{D}}\). Both the centrifugal force (and therefore the contact forces) and the inertial effects are proportional to the mass of the damper; specifically, the centrifugal force is \(\mathrm{CF}=\mathbf{m_{D}}\cdot\mathbf{R_{D}}\cdot\Omega^{2}\), and the inertial effects are \(\propto\mathbf{m_{D}}\cdot\omega^{2}\cdot|\mathbf{q_{D}}|\), where \(\Omega\) is the rotation speed of the bladed disk, and \(\omega\) is the frequency of vibration, i.e., close to the mode under investigation \(\omega_{\mathrm{B}}\). Let us define a ratio between the two quantities:  
\[\tau_{\mathrm{I/CF}}=\frac{\omega_{\mathrm{B}}^{2}\cdot|\mathbf{q_{D}}|}{ \mathrm{R_{D}}\cdot\Omega^{2}}. \tag{9}\]  
It should be noted that this ratio does not depend on the mass of the damper. Let us consider a damper mounted on a turbine for energy production; it is therefore reasonable to set \(\Omega=3600\) rpm and \(\mathrm{R_{D}}\approx 0.5\) m. Based on the authors' experience, the magnitude of the damper displacement is at most \(10^{-5}-10^{-4}\) m. In this specific case, the frequency of vibration \(\omega\approx\omega_{\mathrm{B}}\) (i.e., the mode being excited) would have to be \(>\)30 kHz in order to have the inertial forces at least 5% of the centrifugal force (i.e., \(\tau_{\mathrm{I/CF}}=0.05\)). Consequently, the damper vibrating with the first modes of a bladed system, typically in the range of 150-600 Hz, will not be affected by the inertia forces. This aspect allows for the safe reconstruction of the damper equilibrium starting from experimental data at low frequencies [11; 19; 29] and at higher realistic frequencies (i.e., \(f_{\mathrm{B}}=2\pi\omega_{\mathrm{B}}=200\)-500 Hz) [39; 40]. Regarding the damper inner dynamics, the presence of the contact springs causes high-frequency oscillations of the damper: they are visible as superposed waves in Figure 6. The frequency of oscillation \(\omega_{D}\), given by the mass of the damper (12 g in this example) and the values of contact springs (determined in ), is, in this case, \(\approx\)10 kHz. These oscillations are perfectly explained and predictable from a purely numerical point of view. However, they have not been encountered experimentally at either low [29] or higher frequencies [38; 39; 40]. For this reason, they must be removed \"a posteriori\" in order to make results more legible. This can be done only when the damper internal resonant frequency is far from the mode of vibration of the blades \(\omega_{\rm D}>>\omega_{\rm B}\), i.e., for \(\omega_{\rm B}\leq 1\) kHz. Both \(\omega_{\rm B}\) and \(\omega_{\rm D}\) refer to modes of vibration of the blade/damper coupled system, but at \(\omega=\omega_{\rm B}\) the blade is vibrating (e.g., bending mode) and \"driving\" the damper with it. At \(\omega=\omega_{\rm D}\), the blade itself may be far from its resonance (low mobility), and the damper is vibrating on the contact springs (local mobility mode). If \(\omega_{\rm D}>>\omega_{\rm B}\), the oscillations mentioned above can be filtered out thorough a low-pass filter (see Figure 6).