## 3 Kinetic Model of Trigger Mechanism
### Contact Mechanics Modelling of Contact Pins  
The impact-function and compensation methods were used to calculate contact force. For contact modelling under ADAMS, we adopted the collision-function method. The shock-function method calculates contact force on the basis of Hertzian elastic contact theory, which is essentially modelled as a nonlinear spring damper [34]. Through simulation, ADAMS/Solver can produce a continuous stream of responses, including the acceleration, velocity, position, and force of all elements and points of contact [35]. Contact force is divided into two parts, elastic force and damping force [36]. Taking the collision contact between a small ball and a plane as an example, as shown in Figure 4, the normal force of the contact is given by the following relation:  
\[F_{N}=\left\{ \begin{array}{c}k\delta^{e}+step(\delta,0,0,d_{max},c_{max}) \dot{\delta},\;\delta>0\\ 0,\;\delta\leq 0\end{array} \right. \tag{4}\]  
where \(F_{N}\) is the normal contact force, and \(k\) is the contact stiffness coefficient, which depends on the material and the radius of curvature of the contacting solid surface, and can be expressed by the following formula:  
\[k=\frac{4}{3\pi\big{(}h_{i}+h_{j}\big{)}}R^{\frac{1}{2}},\;R=\frac{R_{i}R_{j} }{R_{i}+R_{j}},\;h_{k}=\frac{1-v_{k}^{2}}{\pi E_{k}},\;k=i,j \tag{5}\]  
where \(R_{i}\), \(v_{i}\), and \(E_{i}\) represent the curvature radius, Poisson's ratio, and the elastic modulus of element \(i\) (the 304 stainless steel material designed in this paper had an elastic modulus of 220 GPa), respectively; \(\delta\) is the penetration depth of the contact point; \(e\) is the shape index, which determines the shape of the force-displacement curve. For the material defined in this model, \(e=1.5\); step is the step function; \(d_{max}\) is the maximal allowable mutual penetration depth; \(c_{max}\) is the maximal value of the damping coefficient; and \(\dot{\delta}\) is the penetration velocity of the contact point. In ADAMS, the instantaneous damping coefficient is a cubic step function of the penetration:  
\[step(\delta,0,0,d_{max},c_{max})=\left\{ \begin{array}{c}0,\;\delta\leq 0\\ c_{max}\Big{(}\frac{\delta}{d_{max}}\Big{)}^{2}\Big{(}3-2\frac{\delta}{d_{max}} \Big{)},\;0<\delta<d_{max}\\ c_{max}\;\delta>d_{max}\end{array} \right. \tag{6}\]  
Friction between contacting surfaces is defined as the Coulomb friction:  
\[f=\mu_{s}F_{N} \tag{7}\]  
Figure 4: Example of contact mechanics model based on impact function. Figure 5 shows the movement state and force diagram of the contact pin in the trigger mechanism during the whole process from starting to locking and unlocking. The three pictures correspond to the actions in Figure 2b, and describe the force state of the locking and unlocking actions. If the simulation experiment of the mechanism could reach the motion state of the third picture, the design could meet the functional requirements, that is, the sleeve could achieve the rebound giving the initial pressure to the valve cover. Figure 6 is the 3-dimensional design diagram of the trigger mechanism installed in the core that shows the specific shape of the trigger mechanism. The interaction between contact pin and core tube, tube shoe, and the inner wall of the outer tube is defined as the contact force and Coulomb friction force. To simplify the force model and reduce the calculation amount of the numerical simulation, the constraint relationship between sleeve and contact pin is defined as a hinge pair. The groove depth of the outer tube is \(a\), the thickness of the tube shoe is \(b\), the radius difference between core barrel and outer tube is \(B\), and angles between the inclined plane of the tube shoe and the groove of the outer tube, and the vertical direction are \(a\) and \(\beta\), respectively. The radii of the two circular contact surfaces are \(r_{1}\) and \(r_{2}\), and distances from the center of the circle to the center of rotation of the contact pin were \(R_{1}\) and \(R_{2}\), respectively.