# 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
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Speed (rpm)** & **Direction** & **Horizontal Range (mm)** & **Vertical Range (mm)** \\ \hline \multirow{6}{*}{rotation pair A\({}_{1}\)} & 30 & X & \([-1.39\times 10^{-4}\), \(-7.13\times 10^{-5}]\) & \([-9.64\times 10^{-4}\), \(9.19\times 10^{-4}]\) \\  & & Y & \([-2.12\times 10^{-4}\), \(-1.70\times 10^{-4}]\) & \([7.14\times 10^{-4}\), \(7.10\times 10^{-4}]\) \\  & & X & \([-2.13\times 10^{-4}\), \(2.23\times 10^{-4}]\) & \([-1.01\times 10^{-1}\), \(7.09\times 10^{-2}]\) \\  & & Y & \([-2.27\times 10^{-4}\), \(2.15\times 10^{-4}]\) & \([-9.11\times 10^{-2}\), \(6.336\times 10^{-2}]\) \\  & & X & \([-2.29\times 10^{-4}\), \(2.34\times 10^{-4}]\) & \([-2.08\times 10^{-1}\), \(1.35\times 10^{-1}]\) \\  & & Y & \([-2.72\times 10^{-4}\), \(2.67\times 10^{-4}]\) & \([-2.10\times 10^{-1}\), \(1.69\times 10^{-1}]\) \\  & & X & \([-1.57\times 10^{-4}\), \(-7.97\times 10^{-5}]\) & \([-5.13\times 10^{-4}\), \(5.22\times 10^{-4}]\) \\  & & Y & \([-1.93\times 10^{-4}\), \(-1.33\times 10^{-4}]\) & \([-3.57\times 10^{-4}\), \(3.50\times 10^{-4}]\) \\  & & X & \([-2.03\times 10^{-4}\), \(2.07\times 10^{-4}]\) & \([-4.10\times 10^{-2}\), \(1.72\times 10^{-2}]\) \\  & & Y & \([-2.09\times 10^{-4}\), \(5.26\times 10^{-5}]\) & \([-2.95\times 10^{-2}\), \(4.07\times 10^{-2}]\) \\  & & X & \([-2.08\times 10^{-4}\), \(2.31\times 10^{-4}]\) & \([-3.06\times 10^{-2}\), \(4.32\times 10^{-2}]\) \\  & & Y & \([-2.11\times 10^{-4}\), \(2.06\times 10^{-4}]\) & \([-6.38\times 10^{-2}\), \(8.48\times 10^{-2}]\) \\  & & X & \([-2.00\times 10^{-4}\), \(2.00\times 10^{-4}]\) & \([-1.12\times 10^{-7}\), \(8.53\times 10^{-8}]\) \\  & & Y & \([1.74\times 10^{-2}\), \(1.42]\) & \([-2.23\), \(2.42]\) \\ prismatic & & X & \([-2.00\times 10^{-4}\), \(2.00\times 10^{-4}]\) & \([-1.13\times 10^{-2}\), \(1.22\times 10^{-1}]\) \\ joint F & 60 & Y & \([1.74\times 10^{-2}\), \(1.42]\) & \([-4.54\), \(4.90]\) \\  & & X & \([-2.00\times 10^{-4}\), \(2.01\times 10^{-4}]\) & \([-3.18\times 10^{-1}\), \(2.35\times 10^{-1}]\) \\  & & Y & \([1.76\times 10^{-2}\), \(1.43]\) & \([-11.16\), \(12.16]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The phase range of rotation pair A\({}_{1}\), rotation pair A\({}_{2}\), and prismatic joint F varies at different driving speeds. Figure 24: Poincaré map at the clearance of prismatic joint: (**a**) Poincaré map of clearance F in the \(X\)-direction; (**b**) Poincaré map of clearance F in the \(Y\)-direction. in both \(X\) and \(Y\) directions expands, with a phase portrait in \(Y\) direction resembling a circular ring. The Poincare map at prismatic joint F, shown in Figure 24, demonstrates that the mapped points in the \(Y\) direction are relatively concentrated, with many overlapping, especially at driving speeds of 30 rpm and 150 rpm, where the Poincare maps are solid small circles. Combined with the trajectory of the prismatic joint's center, it can be inferred that slider undergoes periodic motion as driving speed continuously increases, without an occurrence of chaotic phenomena. The Influence of the Prismatic Joint Clearance on the Nonlinear Characteristics at the Revolute Joint Clearance  
Assuming that the clearance values of the rigid multilink mechanism with clearance are all 0.2 mm, and the driving speeds of crank 1 are selected as 60 rpm, the influence of the clearance of the moving pair on the nonlinear characteristics at the clearance of the compound rotating pair of the multilink mechanism is studied. Figure 25 shows the phase diagrams of revolute joint A\({}_{1}\) and revolute joint A\({}_{2}\) in the X and Y directions. Figure 26 shows the Poincare maps of revolute joint A\({}_{1}\) and revolute joint A\({}_{2}\) in the X and Y directions.