# Nonlinear Analysis and Reliability Analysis of Multilink Mechanism Considering Mixed Clearance
## 6 Nonlinear Characterizations of Rigid Multilink Mechanism with Clearances  
In this section, the influence of various clearance values and driving velocities on the nonlinear characteristics of a stiff multilink structure with mixed clearances, phase plane analysis, and Poincare plot analysis are used for the discussion, respectively. Clearance value and drive speed are essential to influence the mechanism's nonlinear characteristics. Phase plane analysis method is a direct observation technique that allows a geometrical  
Figure 16: Center trajectories at clearances with different driving speeds over time: **(a)** revolute joint clearance A\({}_{1}\); **(b)** revolute joint clearance A\({}_{2}\); **(c)** prismatic joint clearance F.  
representation of the reciprocal acyclic kinematic properties of chaotic motions in terms of phase plane maps. The Poincare map analysis method provides an effective method to studying chaotic motion, which enables a more precise determination of the chaotic phenomena at the clearance. Effect of Clearance Values on Nonlinearity Characteristics of Rigourous Multilink Structures with Mixed Clearances  
Assuming equal clearance values at the composite revolute pair clearance A (consisting of rotating pair clearance A\({}_{1}\) and rotating pair clearance A\({}_{2}\)) and the prismatic pair clearance F, a comparative analysis is conducted to investigate the effect of clearance values of 0.1 mm, 0.2 mm, and 0.3 mm on the motion states of the revolute pairs and sliders with mixed clearances. The driving speed of crank 1 is uniformly set at 60 rpm. Phase portraits and Poincare maps for rotating pair clearance A\({}_{1}\), rotating pair clearance A\({}_{2}\), and prismatic pair clearance F are illustrated in Figures 17-20. As shown in Figures 17-20, phase portraits and Poincare maps at rotating pair clearance A\({}_{1}\), rotating pair clearance A\({}_{2}\), and prismatic pair clearance F reveal that with the rise in clearance values, the scope of the phase portraits and the Poincare maps expands, the scattering degree of the mapped points rises, resulting in a more pronounced chaotic  
Figure 17: 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. phenomenon. Under different clearance values, the phase trajectories in the X and Y direction phase diagrams of revolute pair A\({}_{1}\), revolute pair A\({}_{2}\), and prismatic pair F, spanning horizontal and vertical regions, are presented in Table 5.  
Figure 19: Poincaré map at the clearance of revolute pair: **(a)** [PERSON] map of clearance A\({}_{1}\) in the \(X\)-direction; **(b)** Poincaré map of clearance A\({}_{1}\) in the \(Y\)-direction; **(c)** [PERSON] map of clearance A\({}_{2}\) in the \(X\)-direction; **(d)** [PERSON] map of clearance A\({}_{2}\) in the \(Y\)-direction. Figure 18: 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 the data shown in Figures 17 and 18 and Table 5, it is evident that, as the clearance size rises, the degree of disorder in the phase diagrams at revolute clearance A\({}_{1}\), revolute clearance A\({}_{2}\), and prismatic clearance F intensifies, accompanied by an expansion of the chaotic regions. This chaos originates from intermittent collisions and nonlinear contact forces caused by clearances, leading to instability and energy dissipation of the motion trajectory, indicating a decrease in motion accuracy. The expansion of the phase diagram range is related to higher energy dissipation and chaotic effects between joint surfaces.