# Dynamics Analysis and Chaos Identification of Compound Pendulum Jaw Crusher with Joint Clearance
## 3 Dynamic Modeling  
Figure 2 depicts the working mechanism of the jaw crusher with clearance in the revolute joint, which includes frame 0, eccentric shaft 1, moveable jaw 2, and elbow plate 3. The driving part is the eccentric shaft 1, which provides power to the moveable jaw 2 via the revolute joint A to accomplish the crushing operation. The components are all hard and difficult to bend, so the influence of revolute joint A on the mechanism is greater and the existence of its clearance must be taken into account [17], which has been represented in Figure 2a in an exaggerated manner. The system generalized coordinates are \(\mathbf{q}=[x_{1},y_{1},\theta_{1},x_{2},y_{2},\theta_{2},x_{3},y_{3},\theta_{3}]^{T}\). Ideally, eight constraint equations can be introduced for the four moving subsets; however, the existence of the clearance causes the constraint to fail at revolute joint A. The constraint equation can be shown as follows:  
\[\mathbf{\Phi}(\mathbf{q},t)=\left[ \begin{array}{c}x_{1}-l_{0}\text{sin}\theta_{0}-l_ {11}\text{cos}\theta_{11}\\ y_{1}-l_{0}\text{cos}\theta_{0}-l_{11}\text{sin}\theta_{11}\\ x_{2}-l_{3}\text{cos}\theta_{3}+l_{2}\text{sin}\theta_{2}-l_{21}\text{sin} \theta_{21}\\ y_{2}+l_{3}\text{sin}\theta_{3}-l_{2}\text{cos}\theta_{2}+l_{21}\text{cos} \theta_{21}\\ x_{3}-(l_{3}/2)\text{cos}\theta_{3}\\ y_{3}+(l_{3}/2)\text{sin}\theta_{3}\end{array} \right]=0 \tag{13}\]  
where \(x_{1},y_{1},x_{2},y_{2},x_{3},y_{3}\) represent the horizontal and vertical coordinates of the centers of mass \(S_{1},S_{2},S_{3}\), respectively. \(\theta_{0},\theta_{1},\theta_{2},\theta_{3}\) represent the angle between each linkage and the coordinate axis, respectively. \(l_{0},l_{1},l_{2},l_{3}\) represent the length of each linkage, respectively. \(l_{11},l_{21}\) represent the length of the center of mass \(S_{1},S_{2}\) to their left endpoints, respectively. \(\theta_{11},\theta_{21}\) represent the angle between the line of the center of mass and the left endpoint and the coordinate axis, respectively, \(\theta_{11}=\theta_{1}+30^{\circ},\theta_{21}=\theta_{2}+15^{\circ}\). By taking the first-order derivative of Equation (13) with respect to time, the velocity constraint equation is stated as follows:  
\[\mathbf{\Phi}_{q}\ddot{\mathbf{q}}=-\mathbf{\Phi}_{t}\equiv v \tag{14}\]  
where \(\mathbf{\Phi}_{q}\) is the Jacobian matrix of the constraint equation. \(\mathbf{\Phi}_{t}\) is the first-order derivative of the constraint equation with respect to time. By taking the second-order derivative of Equation (13) with respect to time, the acceleration constraint equation is stated as follows:  
\[\mathbf{\Phi}_{q}\ddot{\mathbf{q}}=-\big{(}\mathbf{\Phi}_{q}\ddot{\mathbf{q}}\big{)}_{q}\dot{ \mathbf{q}}-2\mathbf{\Phi}_{qt}\ddot{\mathbf{q}}-\mathbf{\Phi}_{tt}\equiv\mathbf{\gamma} \tag{15}\]  
where \(\mathbf{\Phi}_{qt}\) is the time partial derivative of the Jacobian matrix. \(\mathbf{\Phi}_{tt}\) is the second-order partial derivative of the constraint equation with respect to time. The dynamic equation based on the Lagrange multiplier method (LMD) [18] can be defined as the following:  
\[M\ddot{\mathbf{q}}+\mathbf{\Phi}_{q}^{\text{T}}\lambda=\mathbf{Q}-\mathbf{F} \tag{16}\]  
Figure 2: Working mechanism of the jaw crusher with clearance. (**a**) Demonstration of mass, rotational inertia and load; (**b**) Demonstration of length and angle. where \(M\) is the generalized quality matrix, which can be shown as follows:  
\[\mathbf{M}=diag[m_{1},m_{1},J_{1},m_{2},m_{2},J_{2},m_{3},m_{3},J_{3}] \tag{17}\]  
where \(\mathbf{Q}\) is the system generalized external forces, which can be shown as follows:  
\[\mathbf{Q}=[0,-m_{1}g,M_{0},F_{load}\text{sin}\theta_{2},-m_{2}g-F_{load}\text{cos} \theta_{2},0,0,-m_{3}g,0]^{T} \tag{18}\]  
where \(\mathbf{F}\) is the collision force matrix, which can be shown as follows:  
\[F=\left\{