## 6 Conclusions  
Based on the 3D dry friction model and fractal geometry theory, a friction mechanical model was established to represent the contact behavior of the rough contact surface of an under-platform damper. On this basis, a finite element dynamic model of a blade with an under-platform damper was established, and the finite element model was reduced to decrease the calculation time. Furthermore, the harmonic balance method was used to analyze the influence of the roughness level of the contact surface on the vibration reduction performance of the under-platform damper under different centrifugal forces, excitation amplitudes, and phases (in the same direction and opposite directions). The numerical analysis proves that:  
1. Under excitation forces in the same direction, the UPD appeared to undergo translational motion, whereas under the opposite excitation force, rotation appeared. Under excitation forces in different directions, the contact state of the UPD between the blades  
Figure 14: Influence of the fractal roughness \(G\) on vibration reduction characteristics. Figure 15: Dissipative energy of different levels of fractal roughness \(G\). appeared to have viscous slip separation at different times, which led to different nonlinear phenomena in the vibration process;
2. When the centrifugal force of the under-platform damper was small, there were more nodes on the contact surface of the under-platform damper, which led to a strong nonlinear phenomenon. With the increase in the centrifugal force, the contact nodes that were in a complete stick state gradually increased, while the nonlinear phenomena and vibration suppression performance weakened;
3. When the two blades were subjected to the same excitation force, there were more nodes in the slip state of the contact surface of the under-platform damper, which led to a strong nonlinear phenomenon, and the contact surface produced more dissipated energy. When the two blades were subjected to a single excitation force or a reverse excitation force, the contact nodes in a complete stick state increased, the nonlinear phenomenon weakened, the contact surface produced less dissipation energy, and the vibration suppression performance weakened. 4. The fractal dimension \(D\) determined the contribution of the high- and low-frequency components to the surface profile. Additionally, the fractal roughness \(G\) refers to a height-scaling parameter independent of the frequency. As the contact surface of the shock absorber under the platform became rougher, the \(D\) increased, whereas the \(G\) decreased. The number of nodes in the sliding state of the contact surface of the under-platform damper increased, which presents a strong nonlinear phenomenon. Moreover, the dissipation energy generated by the contact surface increased, the maximum response corresponding to the resonance frequency was reduced, and the vibration suppression performance was enhanced. In order to study the influence of roughness on a UPD, a more general simplified model was adopted because the simplified model had the problem of a small structural angle. In a real engine, the damper may not roll or have liftoff at such a shallow vertex angle. Only one UPD angle was considered in this paper. The influence of different angles will be studied later. Therefore, it is essential to further study the selection of the UPD angle. During the rotation of the blade, there is wear between the blade and the under-platform damper. According to the wear evolution law of the contact surface, this characterization model can not only obtain the interface contact stiffness at different wear stages by changing the contact surface morphology and the pressure distribution of the contact surface, but it can also determine the contact force characterization of the complex dry friction contact interface considering the wear of the contact surface. Therefore, it is concluded that the wear and roughness characterization model can be combined for further study. Conceptualization, H.S. and S.H. ; methodology, S.H.