## 4 Direct Time Integration Tailored to Systems with Friction Nonlinearities
### Adaptive Time Step Selection for Direct Time Integration Methods  
As stated in [44], stability, convergence, and accuracy all depend on the size of the time step \(\Delta\tau\). Specifically, the chosen Newmark-\(\beta\) method has the \(\Delta\tau/\mathbb{T}<0.318\) limitation to ensure convergence. The quantity \(\mathbb{T}\) is defined as the shortest natural period of the structural system, which is computed through a standard eigenvalue analysis. **Remark 1**. : _When computing \(\mathbb{T}\), one should not consider the \"structural\" stiffness of the system but rather the complete effective stiffness, contact springs included (see Section 3). In other words, the value of \(\mathbb{T}\) changes with the contact state._  
This aspect is not stressed in the few publications where the DTI of friction-damped turbo-machinery systems is carried out. One way to define a value of \(\Delta\tau\) valid for all instants \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Contact State** & **Condition** & \(\mathbf{K_{i}}\) & \(\mathbf{K_{Si}}\) \\ \hline \multirow{2}{*}{Lift-off} & \(\mathrm{n_{IP}}-\mathrm{n_{ID}}<0\) & \(\begin{bmatrix}0&0\\ 0&0\end{bmatrix}\) & \(\begin{bmatrix}0\\ 0\end{bmatrix}\) \\ \hline \multirow{2}{*}{Stick} & \(\mathrm{n_{IP}}-\mathrm{n_{ID}}>0\) & \(\begin{bmatrix}\mathrm{k_{ii}}&0\\ 0&\mathrm{k_{ni}}\end{bmatrix}\) & \(\begin{bmatrix}-\mathrm{k_{ii}}\\ 0\end{bmatrix}\) \\ \hline \multirow{2}{*}{Slip} & \(\mathrm{n_{IP}}-\mathrm{n_{ID}}>0\) & \(\begin{bmatrix}0&\pm\mu\cdot\mathrm{k_{ni}}\\ 0&\mathrm{k_{ni}}\end{bmatrix}\) & \(\begin{bmatrix}0\\ 0\end{bmatrix}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Expression of the contact stiffness matrix depending on the contact state. of the period of vibration is to use the highest effective stiffness that might be encountered, i.e., a full-stick state. With reference to Equation (2), the eigenvalue problem from which \(\mathbb{T}\) is defined is  
\[\left(\sum_{i=\mathrm{L1}}^{\mathrm{R}}\left(\Gamma_{i\mathbf{D}}^{\prime} \cdot\mathbf{K_{i}}\cdot\Gamma_{i\mathbf{D}}\right)\right)\cdot\mathbf{x}= \Lambda\cdot\mathbf{M_{D}}\cdot\mathbf{x} \tag{8}\]  
where all \(\mathbf{K_{i}}\) (\(\forall i\)) matrices have the \"stick\" configuration shown in Table 1. In this case, the structural stiffness matrix \(\mathbf{K_{D}}\) is not present because the damper is modeled as a rigid body. If it were, the overall stiffness matrix would be \(\mathbf{K_{D}}+\sum_{i=\mathrm{L1}}^{\mathrm{R}}\left(\Gamma_{i\mathbf{D}}^{ \prime}\cdot\mathbf{K_{i}}\cdot\Gamma_{i\mathbf{D}}\right)\). Typical values of the contact springs, i.e., \(\approx 10-10^{2}\) N/\(\mu\)m, unfortunately produce \(\Delta\tau\approx 1\)\(\mu\)s. This makes DTI cumbersome, especially at low frequencies. The reader will have noticed that the \"effective stiffness\" of the system changes with time (i.e., with the contact state). Therefore, the limit on \(\Delta\tau\) depends on the specific contact state as well. For this reason, it is here proposed, as shown in Figure 4, that the time step is adapted to the contact state. The contact state is initially assumed to be equal to the previous one, i.e., \(\Delta\tau=\tau_{n}-\tau_{n-1}\): this value is fed into the integration scheme and produces the tentative value of the vector of displacements \(\widetilde{\mathbf{q_{D}}}\). The contact model run at instant \(\tau_{n}\) with \(\widetilde{\mathbf{q_{D}}}\) produces a given contact condition and, consequently, the time step \(\widetilde{\Delta\tau}\). If \(\widetilde{\Delta\tau}=\Delta\tau\), the algorithm goes on; otherwise, the time step is updated; if \(\widetilde{\Delta\tau}<\Delta\tau\), the Newmark algorithm is re-executed with the updated value of \(\widetilde{\Delta\tau}\). There are typically a few transitions per period of vibration (i.e., 4-7). The time savings are quite consistent: the signal shown in Figure 4b is in full slip condition for 54% of the total period of vibration.