## 5 Vibration Response Analysis of Blade with Under-Platform Damper
### Time Domain Signals of Co-Directional and Reverse Excitation Force  
Taking the contact pair of UPD slices in Figure 6a as an example, in a complete vibration cycle, the influence of codirectional vibration and reverse vibration on the contact state was obtained under different levels of roughness. As shown in Figures 7 and 8, under certain calculation conditions, different contact states of stick, slip, or separation at different times appeared on the contact surface. Therefore, different contact conditions can have different nonlinear phenomena.  
The co-excitation force and the reverse excitation force had different effects on the vibration of the UPD, as shown in Figure 6b. The symmetrical nodes on both sides of the UPD were selected as the analysis objects to calculate the time-domain vibration signals. As shown in Figure 9, in the case of co-excitation and reverse excitation, there were vibrations of the UPD nodes on both sides in three directions. As can be seen, the movement of the UPD was translational when the excitation forces moved in the same direction, whereas the UPD produced torsional motion when the excitation forces moved in opposite directions.  
Influence of the Contact Surface Roughness on the Contact State of the Contact Surface of the Under-Platform Damper  
In this study, \(110\) contact pairs were established on the two contact surfaces of the blade-flange plate to describe the vibration of the entire contact surface (see Figures 10 and 11). The  
Figure 8: Time domain signals with the fractal roughness parameter; (**a**) excitation forces in the same direction; (**b**) excitation force in the opposite direction.  
Figure 9: Time domain signals; (**a**) excitation forces in the same direction; (**b**) excitation forces in the opposite directions.  
motion of each contact pair under three excitation forces and different levels of roughness can be observed. The dots of different colors represent the nodes. To be specific, the blue indicates that the node was in a complete stick state, the green indicates that the node was in a stick-slip state, and the red indicates that the node was separated during the motion period but was still in a stick-slip state during contact. Additionally, a dry friction damping effect can be produced in the stick-slip state and the separation-stick-slip state.  
Figure 10: Influence of the roughness fractal dimension \(D\) on the contact state: (**a,d**) single exciting force; (**b,e**) excitation forces in the same direction; (**c,f**) excitation forces in the opposite directions.  
It can be observed in Figures 8 and 11 that in a complete vibration cycle, with the increase in the roughness of the contact surface, the roughness fractal dimension \(D\) increased, the fractal roughness \(G\) decreased, the nodes of the contact surface in the complete stick state declined, the stick-slip state and stick-slip-separation state nodes increased, and the nonlinearity of the system and the dry friction damping effect were strengthened. This  
Figure 11: Influence of the fractal roughness \(G\) on the contact state; (**a,d**) single exciting force; (**b,e**) excitation forces in the same direction; (**c,f**) excitation forces in the opposite directions.  
study also compared the contact state of the contact surface under excitation forces of 5 N and 2 N. When the excitation force was large, there were fewer nodes in the complete stick state, and most of the nodes entered the slip state. At the same time, there were more nodes in the separation state. At this time, the system had obvious nonlinear characteristics, and the dry friction damping effect was also strong.  
Within a certain range, as the contact surface became rough (\(D\) increased and \(G\) decreased), the number of nodes entering the slip state gradually increased, and the dry friction effect of the contact interface gradually strengthened, causing the vibration of the simplified blade model with the under-platform damper to have obvious nonlinear characteristics.