## 6 A Critical Comparison of the Proposed Methods  
This section is concerned with offering the user practical guidelines and an indication of expected results when one of the three methods proposed in this paper is applied. In order to achieve this goal, a test case has been defined. The test case at hand sees an underplatform damper between a set of blade platforms (as shown in Figure 2b) vibrating at 200 Hz for one period of vibration. The blade platform kinematics is large enough to cause the damper to reach the bilateral gross slip condition (both the right and left interfaces are sliding). This results in six changes in contact states during one period of vibration. The resulting hysteresis cycle at the left (flat-on-flat) contact interface is shown in Figure 6.  
The test case defined above has been solved using the following four methods:  
1. a **standard** Newmark-\(\beta\) algorithm, as described in Appendix A.3, to be used as a reference case, which applies standard techniques and no particular adaptation to friction damped systems;
2. a standard Newmark-\(\beta\) algorithm with an **adaptive time step** (see Section 4.2);
3. a **piecewise-linear** implementation of the Newmark-\(\beta\) algorithm, as described in Section 4.3 and Appendix A.4;
4. a **quasi-static** solution method, as described in Section 5 and Appendix B with a user-defined 300 time steps per period.  
All methods yield the same set of results; forces and displacements are perfectly superposed. Accuracy is not impaired in the slightest by the proposed methods. Computational time, on the other hand, varies from case to case. The computational time needed for the case described in Method 1 is used as a reference to normalize other values.  
Figure 7: DTI vs. quasi-static results comparison. (**a**) Tangential displacement of the right damper contact point. (**b**) Vertical component of the right contact force.  
As shown in Figure 8 all proposed methods lead to a non-negligible decrease in the computational time. The adaptive time step method (Method 2) introduces a 20% reduction in the computational time and can be applied to ALL existing DTI algorithms. It enables the user to automatically set the largest possible time step that still ensures convergence. Method 3 is instead applicable to all DTI methods and no restrictions on the test case are observed. It is especially useful for implicit algorithms (in this case, Newmark-\(\beta\)) since it enables the user to make the best possible assumption when proceeding to the following time step (i.e., unchanged contact state) without unnecessary iterations. This leads to halved computational times. Methods 2 and 3 can be applied together, thus leading to an overall reduction in the computational time of \(\approx\)60%. The reader will notice that the quasi-static method (Method 4) reduces the overall computational time by more than one order of magnitude. This solution method, which is an alternative to DTI algorithms (and therefore to Methods 2 and 3) is, however, applicable only in those cases where inertial effects are negligible. When applicable, it is a very competitive alternative, especially in all those cases where displacements are large (leading to highly nonlinear systems) and the HBM needs large harmonic support, resulting in convergence issues.