# Dynamics Analysis and Chaos Identification of Compound Pendulum Jaw Crusher with Joint Clearance
## 5 Analysis of Dynamic Performance
### Influence of Driving Speed and Clearance Size  
Figure 14a shows the root-mean-square values of the vibration force on the foundation for a driving speed range of [25; 40] and a clearance size range of [0.03, 0.12]. The three-dimensional surface is able to show the magnitude of \(F_{rms}\) corresponding to different \((r,\omega)\). The contour plots of \(F_{rms}\) are shown in Figure 14b, which can graphically express the effects of the driving speed and the clearance size together on the vibration force applied to the foundation.  
Analysis of Figure 14a,b shows that the contours are densely distributed on the right side, and the value of each contour increases tenfold, from 1 \(\times\) 10\({}^{4}\) N all the way to 9 \(\times\) 10\({}^{4}\) N. It indicates that the vibration force increases sharply after the clearance size increases to a certain degree, which verifies the analysis of the vibration force in Section 5.2. The combination of driving speed and clearance size should be chosen from the darkest color part since its vibration force is much lower than that of the other parts and its influence on the service life of the foundation is likewise much smaller.  
According to the previous analysis, the peak is able to reflect the shock phenomenon and the operational stability of the mechanism. Based on the conclusion in Section 5.2 on the selection of clearance size, the local surface and contour maps of the peak vibration force are plotted based on the darkest color part in Figure 14b with \(r\) = 0.06 mm as the dividing line. Figure 15a shows the peak vibration forces on the foundation for a driving speed range of [25, 40] and a clearance size range of [0.03, 0.06]. The three-dimensional surface is able to show the magnitude of \(F_{max}\) corresponding to different \((r,\omega)\). The contour plots of \(F_{max}\) are shown in Figure 15b.  
Analysis of Figure 15a,b shows that the contour distribution is complex, increasing from \(2\times 10^{4}\) N all the way to \(2\times 10^{5}\) N, where the darkest color part is distributed near the driving speed of 25 rad/s. Considering that a smaller driving speed can lead to irregular wear of the revolute joint, the combination solution is chosen from the part with the second darkest color. Combining the conclusions on the driving speed and clearance size selection in Sections 5.1 and 5.2, the optimal combination solution is 39 rad/s and 0.03 mm, with a mean vibration force of 2.19 \(\times\) 10\({}^{4}\) N and a peak vibration force of 3.8 \(\times\) 10\({}^{4}\) N.  
the Poincare mapping of the crusher with clearance when the optimal driving speed and clearance size suggested in this study are utilized.  
Comparing Figure 8b with Figure 16a, Figure 9b with Figure 16b, and Figure 10b with Figure 16c, it is clear that the optimal combination solution significantly reduces the shock and chaos phenomenon in the mechanism, maintains the stable contact between the revolute joint's elements, and reduces the peak vibration force applied to the foundation from 7.4 \(\times\) 10\({}^{4}\) N to 3.8 \(\times\) 10\({}^{4}\) N, or about 48.6%, with a smooth change curve.