## 4 Conclusions  
This paper presents a study on the forced vibration response of a simplified turbine blade model with a new kind of under-platform dry friction dampers. It establishes a numerical approach which uses numerical modes of the structure and is capable of accommodating the discontinuous classical Coulomb's law of friction and dealing with moving loads efficiently.  
In this study, both the normal contact forces and the resultant friction forces at the damper-platform interfaces are modeled as moving loads, and the normal contact force variation as a result of platform vibration is also considered. Parametric analysis demonstrates that the dampers will experience three motion regimes: gross slip regime, stick-slip regime, and pure stick regime, as the preloads at the contact points between the dampers and blade platform increase. The following conclusions can be drawn:  
1. When the pre-load \(N_{0}\) is very small, the two dampers are all in gross slip regime in the steady state. The motion of the blade tip is harmonic. The normal contact forces fluctuate around the preload and are periodic, the friction forces \(f_{1}\) and \(f_{2}\) undulate between \(\pm\mu_{k}N_{1}\) and \(\pm\mu_{k}N_{2}\), respectively, and are also periodic. The motions of the two dampers are not harmonic but periodic whose periods are the period of the excitation. The phase plane maps of the dampers are smooth closed curves.
2. When the pre-load \(N_{0}\) is bigger and appropriate, the two dampers could undergo stick-slip vibration in the steady state. The motion of the blade tip is periodic rather than harmonic. The normal contact forces continue to fluctuate around the preload and are periodic as in gross slip regime. Friction forces \(f_{1}\) and \(f_{2}\) fluctuate between \(\pm\mu_{s}N_{1}\) and \(\pm\mu_{s}N_{2}\), respectively, and are periodic. At times, \(f_{1}\) equals \(\pm\mu_{k}N_{1}\) and \(f_{2}\) equals \(\pm\mu_{k}N_{2}\); at any other time, friction forces \(f_{1}\) and \(f_{2}\) could reach \(\pm\mu_{s}N_{1}\) and \(\pm\mu_{s}N_{2}\), separately. The motions of the dampers are similar to those in the gross slip regime. The phase plane maps of the dampers are closed curves but not smooth.
3. When the pre-load \(N_{0}\) is high enough, the two dampers are all in pure stick regime in the steady state. During pure stick scheme, the blade-platform structure and the dampers form one new system. The normal contact forces and the friction forces display a quasi-periodic variation, and the motion of the blade tip changes to quasi-periodic vibration as well. The motions of the dampers are non-periodic but quasi-periodic. There are three peaks in the frequency spectrum plots: the frequency of the first peak is the same as the excitation frequency, but the frequencies of the second peak and the third peak are now the first-order bending frequency and the fifth-order bending frequency of the whole new system.
4. There is an optimal pre-load which can make the biggest vibration reduction.  
The authors are grateful to the support by the National Natural Science Foundation of China (Grants No. 51405452 and No. 11672052) and \"the Fundamental Research Funds for the Central Universities\" (DUT16 RC(3)027). Part of this work was carried out during the first author's visit to the University of Liverpool.  
**Author Contributions:** [PERSON] and [PERSON] conceived the study. [PERSON] wrote the paper and clarified the key methods and results. [PERSON] edited the manuscript. [PERSON] and [PERSON] helped in simulation and data analysis.  
**Conflicts of Interest:** The authors declare no conflict of interest.