# Dynamics Analysis and Chaos Identification of Compound Pendulum Jaw Crusher with Joint Clearance
## 4 Validation Analysis
### Validation of Influence Parameters  
Crusher production capacity is defined as the volume of material that a crusher can crush per unit of time, which is an important indicator of crusher performance [22]. According to the lamination crushing theory (LCT), the movable jaw follows the eccentric shaft to move reciprocally once, and the ore will be crushed and moved down layer by layer until they are discharged from the crushing chamber. The specific discharge volume per unit of time is obtained by calculating the discharge volume of a single layer. The cross-section diagram of the broken layer is shown in Figure 6. \(B\) and \(b\) are the inlet size and discharge size when the movable jaw moves to the loose edge position, \(a\) is the engagement  
Figure 5: Comparison of results from Adams and MATLAB: (**a**) Moveable jaw displacement; (**b**) Moveable jaw velocity; (**c,d**) Collision force at the revolute joint A.  
angle of the crushing chamber, \(H\) is the height of the crushing chamber, \(s\) is the horizontal travel of the lower end point of the movable jaw, AA\({}_{1}\)BB\({}_{1}\) is the trapezoidal cross-section of the discharge prism during one movement cycle, and \(h\) is the height of the cross-section. The crusher production capacity can be expressed as follows:  
\[Q=60 nV\mu \tag{21}\]  
where \(V\) is the volume of material discharged once by the reciprocating motion of the moveable jaw, which can be defined as the following:  
\[V=S\cdot L \tag{22}\]  
where \(S\) is the area of the trapezoidal section of the prism, which can be defined as follows:  
\[S=\frac{2b-s}{2}h \tag{23}\]  
when the material breaks away from the moveable jaw, it does free fall motion, so the height of the cross-section can be shown as follows:  
\[h=\frac{1}{2}gt^{2} \tag{24}\]  
In a motion cycle, the discharge time occupies half of the time, so the discharge time can be expressed as follows:  
\[t=\frac{30}{n} \tag{25}\]  
where \(n\) is the eccentric shaft speed, \(g\) is the acceleration of gravity, \(L\) is the feed opening length, and \(\mu\) is the loosening factor of the mineral. Combined with the above equations, it can be seen that the production capacity of the crusher is determined by the eccentric shaft speed, the horizontal travel of the lower end point of the movable jaw and the discharge size. While the driving speed and the clearance size are important influencing factors of the shock and chaos phenomenon in the mechanism [23], if these two parameters are analyzed, their influence on the production capacity also needs to be considered. The driving speed corresponds to the eccentric shaft speed, and the clearance size corresponds to the horizontal travel of the lower end point of the movable jaw and the discharge size, i.e.,  
Figure 6: Cross-section diagram of the broken layer. Given a driving speed of 30 rad/s and common fit tolerances of 0.06 and 0.12 mm for the revolute joint A, a comparison of the displacement of the revolute joint B under different clearance size conditions is shown in Figure 7a. It can be seen that the displacement curves basically overlap, indicating that the change in clearance size will not affect the horizontal travel of the lower end point of the movable jaw and the discharge size, which will not affect the production capacity. Taking granite ore crushing as an example, the given feed opening length is 0.4 m, the crushing chamber height is 0.6 m and the loosening factor is 0.7. According to Equation (21), the relationship between driving speed and production capacity of a PE250 \(\times\) 400 compound pendulum jaw crusher can be obtained, as shown in Figure 7b. As can be seen from the figure, the production capacity decreases as the driving speed increases. The single-layer material fall height \(h\) should be less than the crushing chamber height \(H\). Combining with Equations (24) and (25), the driving speed should be greater than 25 rad/s. Meanwhile, the maximum production capacity of the crusher is calculated as 9.41 m\({}^{3}\)/h.