# Dynamics Analysis and Chaos Identification of Compound Pendulum Jaw Crusher with Joint Clearance
## 5 Analysis of Dynamic Performance
### Influence of Clearance Size  
Given the driving speed of 30 rad/s and the clearance size of 0.03, 0.06, 0.09, 0.12 mm, Figures 11-13 represent the characteristic images of the axial trajectory of the shaft, the vibration force applied to the foundation, and the Poincare mapping at different driving speeds, respectively.  
Analysis of Figure 11 shows that when the clearance size is 0.03 mm or 0.06 mm, the collision point distribution is chaotic. After increasing to 0.09 mm, the axial trajectory is distributed in an elliptical band, and the clearance joint elements contact stably. After increasing to 0.12 mm, the state remains stable and the embedding depth increases. It visually demonstrates the change process of the shock phenomenon that first enhances and then disappears as the clearance size increases. This is because the limited clearance sizes reduce the eccentric shaft's range of motion and increase the li As the clearance size increases, the collision forces increase accordingly and the motion becomes chaotic. When the clearance size is increased to a certain level, the movable jaw is able to maintain contact with the eccentric shaft due to the inertial force and make a stable motion. This is consistent with the scenario of the \"momentum exchange method\" and the \"massless linkage method\" proposed by [PERSON] [26].  
Analysis of Figure 12 shows that the vibration force on the foundation is influenced by changes in the shock phenomenon and exhibits the same dynamic characteristics. When the clearance size is 0.03 mm and 0.06 mm, the peak vibration force is 6 \(\times\) 10\({}^{4}\) N and 7.4 \(\times\) 10\({}^{4}\) N, respectively, when the clearance size increases to 0.09 mm and 0.12 mm, the change curve smooths out and the peak vibration force is 1.1 \(\times\) 10\({}^{6}\) N and 1.7 \(\times\) 10\({}^{6}\) N, respectively. However, at 0.12 mm, the mean vibration force with clearance is 8 \(\times\) 10\({}^{5}\) N and the mean vibration force without clearance is 1.8 \(\times\) 10\({}^{4}\) N, showing an increase of nearly 45 times in comparison, far exceeding the amplitude of oscillation at 0.06 mm. It is shown that the smaller the clearance size, the less the influence on the vibration force on the foundation and the less the influence on the service life of the foundation.  
Analysis of Figure 13 shows that for the clearance size of 0.03 mm, there is no correlation between adjacent Poincare mapping points. After increasing to 0.06 mm, the distribution of mapping points expands. After increasing to 0.09 mm and 0.12 mm, the mapping points form a closed curve. It is shown that as the clearance size increases, the chaotic motion of the system gradually strengthens and then enters the state of quasi-periodic motion, where the stability of the mechanism operation first weakens and then strengthens.  
In summary, for a driving speed of 30 rad/s and a clearance size range of 0.03 mm to 0.12 mm, the clearance size of 0.03 mm is chosen as the best considering the magnitude of the vibration force. Therefore, in the case of a specific driving speed, the minimum value within the clearance size selection range should be selected, which can weaken the chaos and shock phenomenon in the mechanism, enhance the stability of the mechanism operation, weaken the influence of the clearance on the vibration force applied to the foundation, and prolong the service life of the foundation.