# Dynamics Analysis and Chaos Identification of Compound Pendulum Jaw Crusher with Joint Clearance
## 5 Analysis of Dynamic Performance  
### Influence of Driving Speed  
Given the clearance size of 0.06 mm and the driving speed of 25, 30, 35, and 40 rad/s, Figures 8-10 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. Figure 7: Actual results of influence parameters: (**a**) Displacement of revolute joint B at different clearance sizes; (**b**) Relationship between eccentric shaft speed and production capacity. Figure 8: Axis trajectory of shaft at different driving speeds. (**a**) \(\omega=25\). (**b**) \(\omega=30\). (**c**) \(\omega=35\). (**d**) \(\omega=40\). In Figure 8, the solid circle, which has the origin as its center and the clearance size as its radius, stands for the axial motion boundary of the shaft. The distribution of the collision locations is formed by the contact between the motion trajectory and the solid circle. According to an analysis of Figure 8, the collision points are concentrated in the upper right corner of the boundary at 25 rad/s, chaotically distributed at 30 rad/s and 35 rad/s, and steadily embedded at 40 rad/s. It shows that as the driving speed increases, the shock phenomenon in the mechanism first increases and then decreases. The reason is that when the eccentric shaft is rotating at low speed, the moveable jaw is affected by gravity and fixed load, and the collision occurs mostly on the upper right side of the bearing and shaft, which is easy to cause irregular wear of the revolute joint. As the driving speed increases, the collision forces increase accordingly and the motion becomes chaotic. If the driving speed is further increased, it is difficult for the bearing to make a large movement in a very short period due to the inertia force, and it can only make a stable movement along the axis motion boundary. According to an analysis of Figure 9, compared to the case when there is no clearance, the existence of clearance causes nonlinear oscillations in the vibration forces applied to the foundation. The peak vibration forces at 25 rad/s, 30 rad/s, 35 rad/s and 40 rad/s are \(2.3\times 10^{4}\) N, \(7.4\times 10^{4}\) N, \(4.1\times 10^{4}\) N and \(4\times 10^{4}\) N, which are consistent with the variation in the shock phenomenon. The amplitude of the oscillation is greatest at a driving speed of 30 rad/s, with a peak increase of approximately 180%. It shows that as the driving speed increases, the influence on the vibration force applied to the foundation and the foundation service life first increases and then decreases. The Poincare mapping technique [25] is a qualitative approach to identifying the chaos phenomenon. When the discrete points show one or fewer, closed curves, hierarchical and banded distributions, it means that the system is in a state of periodic, quasi-periodic, and chaotic motion. Figure 10 shows that the Poincare mapping points display a band distribution and no association between neighboring mapping points as the driving speed increases, but the distribution range widens. At the driving speed of 40 rad/s, the distribution range of the Poincare mapping points shrinks and moves in the direction of a point-like trend. It shows that as the driving speed increases, the chaos phenomenon decreases and the stability of the mechanism increases. It is shown that as the driving speed increases, the chaos phenomenon, like the shock phenomenon, first increases and then decreases, i.e., the operating stability of the mechanism first decreases and then increases. In summary, for a clearance size of 0.06 mm and a driving speed range of 25 rad/s to 40 rad/s, the driving speed of 40 rad/s is chosen as the best considering the wear problem.