# Nonlinear Analysis and Reliability Analysis of Multilink Mechanism Considering Mixed Clearance
## 6 Nonlinear Characterizations of Rigid Multilink Mechanism with Clearances
In the Poincare maps at revolute clearance A\({}_{1}\) and revolute clearance A\({}_{2}\), as illustrated in Figure 19, it can be seen that as the clearance value increases, a range of mapped points at revolute clearance A\({}_{1}\) and revolute clearance A\({}_{2}\) expands, indicating enhanced chaotic behavior and increasingly severe damage to the revolute joints with clearances. The Poincare mapping transforms from clustered points to dispersed points, \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **Clearance (mm)** & **Direction** & **Horizontal Range (mm)** & **Vertical Range (mm)** \\ \hline \multirow{4}{*}{rotation pair A\({}_{1}\)} & \multirow{2}{*}{0.1} & X & \([-1.19\times 10^{-4},1.24\times 10^{-4}]\) & \([-6.88\times 10^{-2},4.82\times 10^{-2}]\) \\  & & Y & \([-1.27\times 10^{-4},1.19\times 10^{-4}]\) & \([-5.98\times 10^{-2},4.62\times 10^{-2}]\) \\  & & X & \([-2.14\times 10^{-4},2.23\times 10^{-4}]\) & \([-1.01\times 10^{-1},7.09\times 10^{-2}]\) \\  & & Y & \([-2.27\times 10^{-4},2.16\times 10^{-4}]\) & \([-9.11\times 10^{-2},6.36\times 10^{-2}]\) \\  & & X & \([-3.16\times 10^{-4},3.24\times 10^{-4}]\) & \([-1.60\times 10^{-1},9.71\times 10^{-2}]\) \\  & & Y & \([-3.26\times 10^{-4},3.18\times 10^{-4}]\) & \([-1.22\times 10^{-1},8.73\times 10^{-2}]\) \\ \hline \multirow{4}{*}{rotation pair A\({}_{2}\)} & \multirow{2}{*}{0.2} & X & \([-1.06\times 10^{-4},1.10\times 10^{-4}]\) & \([-2.38\times 10^{-2},1.43\times 10^{-2}]\) \\  & & Y & \([-1.11\times 10^{-4},2.59\times 10^{-5}]\) & \([-1.56\times 10^{-2},2.29\times 10^{-2}]\) \\  & & X & \([-2.03\times 10^{-4},2.08\times 10^{-4}]\) & \([-4.10\times 10^{-2},1.72\times 10^{-2}]\) \\  & & Y & \([-2.29\times 10^{-4},5.26\times 10^{-5}]\) & \([-2.96\times 10^{-2},4.06\times 10^{-2}]\) \\  & & X & \([-3.05\times 10^{-4},3.10\times 10^{-4}]\) & \([-5.78\times 10^{-2},4.11\times 10^{-2}]\) \\  & & Y & \([-3.11\times 10^{-4},1.46\times 10^{-4}]\) & \([-4.75\times 10^{-2},7.08\times 10^{-2}]\) \\ \hline \multirow{4}{*}{prismatic joint F} & \multirow{2}{*}{0.1} & X & \([-1.00\times 10^{-4},1.01\times 10^{-4}]\) & \([-1.10\times 10^{-1},1.24\times 10^{-1}]\) \\  & & Y & \([1.81\times 10^{-2},1.42]\) & \([-4.52,4.88]\) \\ \cline{1-1}  & & X & \([-2.00\times 10^{-4},2.00\times 10^{-4}]\) & \([-1.13\times 10^{-1},1.22\times 10^{-1}]\) \\ \cline{1-1}  & & Y & \([1.74\times 10^{-2},1.42]\) & \([-4.54,4.90]\) \\ \cline{1-1}  & & X & \([-3.00\times 10^{-4},3.01\times 10^{-4}]\) & \([-1.50\times 10^{-1},1.51\times 10^{-1}]\) \\ \cline{1-1}  & & Y & \([1.71\times 10^{-2},1.43]\) & \([-4.58,4.94]\) \\ \hline \end{tabular}
\end{table} 
Table 5: The phase portrait range of rotation pair A\({}_{1}\), rotation pair A\({}_{2}\) and prismatic pair F under different clearance values. Figure 20: 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. confirming the bifurcation into chaos at larger clearances. The Poincare plots of the prism clearance F in Figure 20 shows that the proportion of area occupied by the mapped points increases as the clearance value increases, and all the points are approximately circular in shape. Although the phase diagram of the moving pair F shows a large range in the Y direction, its Poincare mapping points are approximately circularly distributed; it can be inferred that the slider exhibits a periodic motion without confusion.