# Nonlinear Analysis and Reliability Analysis of Multilink Mechanism Considering Mixed Clearance
## 6 Nonlinear Characterizations of Rigid Multilink Mechanism with Clearances
### Driving Speed Effect on Mixed Clearance Rigid Multilink Mechanism Nonlinear Characteristics  
Assuming that clearance values of the rigid multilink mechanism with mixed backlash are all 0.2 mm, drive speed of crank 1 is chosen to be 30 rpm, 60 rpm, and 150 rpm to investigate the impact of driving speed on nonlinear characteristics of multilink mechanism. Figures 21 and 22 show phase diagrams of rotating joint A\({}_{1}\) and rotating joint A\({}_{2}\) in the \(X\) and \(Y\) directions, respectively, and the phase diagram of prismatic joint F. Figures 23 and 24 show the Poincare plots of rotating joint A\({}_{1}\) and rotating joint A\({}_{2}\) in the \(X\) and \(Y\) directions, respectively, and Poincare plots of prismatic joint F.  
Figure 21: Phase diagram at the joint clearance of revolute pair: (**a**) phase diagram of clearance A\({}_{1}\) in the \(X\)-direction; (**b**) phase diagram of clearance A\({}_{1}\) in the \(Y\)-direction; (**c**) phase diagram of clearance A\({}_{2}\) in the \(X\)-direction; (**d**) phase diagram of clearance A\({}_{2}\) in the \(Y\)-direction. Based on Figures 21 and 22, it is inferred that driving speed has a material impact on multilink mechanism nonlinear characteristics with mixed clearance. With increasing drive speed of crank 1, the area of the phase portrait increases. The phase trajectories of rotation pair A\({}_{1}\), rotation pair A\({}_{2}\), and prismatic pair F across horizontal and vertical regions in the X and Y direction phase diagrams at different driving speeds are shown in Table 6.  
Figure 23: Poincaré map at the clearance of revolute pair: (**a**) Poincaré map of clearance A\({}_{1}\) in the \(X\)-direction; (**b**) Poincaré map of clearance A\({}_{1}\) in the \(Y\)-direction; (**c**) Poincaré map of clearance A\({}_{2}\) in the \(X\)-direction; (**d**) Poincaré map of clearance A\({}_{2}\) in the \(Y\)-direction. Figure 22: Phase diagram at the clearance of prismatic joint: (**a**) phase diagram of clearance F in the \(X\)-direction; (**b**) phase diagram of clearance F in the \(Y\)-direction. From Figure 21, which depicts the phase portraits at revolute joints A\({}_{1}\) and A\({}_{2}\), it is seen that as the drive velocity rise from 30 rpm to 60 rpm and further to 150 rpm, the degree of chaos increases, and the chaotic regions expand accordingly. The impact caused by rebound becomes more frequent and intense, and the expanding phase space region means it is becoming increasingly unpredictable, where small disturbances can cause trajectory divergence. Figure 23, showing the Poincare maps of revolute joints A\({}_{1}\) and A\({}_{2}\), reveals that with continuous rise in driving speed, a range of mapped points at these revolute joints expands, indicating an intensification of chaotic phenomena. Based on the observations from these two figures, it is concluded that damage at the revolute joints becomes increasingly severe. Higher speeds inject kinetic energy, causing components to bounce unpredictably in the clearance area. This will excite higher-order vibration modes, leading to complex non periodic motion. Figure 22 presents the phase portrait at prismatic joint F, which indicates that as driving speed increases, a proportion of mapped points