## 3 Kinetic Model of Trigger Mechanism
### Contact Mechanics Modelling of Contact Pins
The angle between \(R_{1}\) and \(R_{2}\) was \(\gamma\), and the distance between the centers of the two circles is \(r\). The design principle of the contact pin relies on the contact force between the tube shoe and the contact surface to push the contact pin to rotate and reach the groove of the outer tube. According to the designed geometric relationship, rotation angle \(\theta\) and included angle \(\gamma\) of the contact pin can be calculated as:  
\[\theta=sin^{-1}\frac{b}{B} \tag{8}\]  
\[\gamma=cos^{-1}\frac{R_{1}}{R_{2}} \tag{9}\]  
The annular space of radius difference \(B\) between core barrel and outer tube is the design space of the contact pin, and its relationship with the design size of the contact pin is:  
\[B=r+r_{1}+r_{2} \tag{10}\]  
Figure 5: Geometric design of contact pin and force state during action. The action process of the trigger mechanism is as follows: the core tube is driven by the top to lift at a constant speed of \(v_{1}\), the contact force of the core tube to the contact pin is \(T_{N1}\), the contact pin transmits pressure \(F_{N2}\) to the inner wall of the outer tube, and the contact stiffnesses corresponding to the two contact forces are \(K_{1}\) and \(K_{2}\). The relative motion produces kinetic friction forces \(f_{1}\) and \(f_{2}\), which are simultaneously affected by elastic force \(T\) from the spring. The dynamic equation of the contact pin rotation can be simplified to the following formula:  
\[M_{1}=F_{N1}R_{1}\cos(\alpha+\theta) \tag{11}\]  
\[M_{f1}=\mu F_{N1}R_{1}\sin(\alpha+\theta) \tag{12}\]  
\[M_{2}=F_{N2}R_{2}\sin\Bigl{(}\theta+\beta+\gamma-\frac{\pi}{2}\Bigr{)} \tag{13}\]  
\[M_{f2}=\mu F_{N2}R_{2}\cos\Bigl{(}\theta+\beta+\gamma-\frac{\pi}{2}\Bigr{)} \tag{14}\]  
The geometric relationship designed by the mechanism and the expression of \(M_{2}\) above are easy to obtain. Only when \(\gamma+\beta\geq\frac{\pi}{2}\) does the direction of \(F_{N2}\) allow for \(M_{2}\) to act as the driving torque for the rotation of the contact pin during the rotation of the contact pin, namely,  
Figure 4:0  
When \(\gamma+\beta+\theta\leq\frac{\pi}{2}\), the direction of \(F_{N2}\) enables \(M_{2}\) to act as a resistance torque during the rotation of the contact pin, namely:  
Figure 4:1  
After the core barrel and the pipe shoe had been lifted and removed, the contact pin needs to be able to self-unlock to complete the unlocking. From a dynamic point of view, when \(\gamma+\beta\geq\frac{\pi}{2}\), \(M_{2}\) becomes the resistance torque for the contact pin to flip in the opposite  
Figure 4:2 Three-dimensional design diagram of trigger mechanism installed in corer. direction, which renders the contact pin incapable of self-unlocking, so it is required to be located between \(\gamma\), \(\beta\), and \(\theta\). The corresponding satisfaction relationship is:  
Figure 4:3  
The kinetic equation for unlocking the contact pin is:  
Figure 4:4  
Three different angle combinations were used to simulate the contact pin mechanism (\(\gamma=\theta=15^{\circ}\), scheme 1: \(\alpha=30^{\circ},\ \beta=60^{\circ}\); scheme 2: \(\alpha=45^{\circ},\ \beta=45^{\circ}\); scheme 3: \(\alpha=60^{\circ},\ \beta=30^{\circ}\)). According to the contact-pin self-unlocking condition described in Formula (17), on the basis of the optimal combination of \(\alpha\) and \(\beta\) angles, parameters \(\gamma\) and \(\theta\) are changed within a reasonable range, and optimal \(\alpha\) and \(\beta\) values are explored. On the basis of the combination scheme, the best combination of \(\gamma\) and \(\theta\) angles was obtained. In ADAMS, for dynamic differential equations, the backward differentiation formulation (BDF) rigid integration program with variable coefficients is used to solve the rigidity problem of mechanical system characteristics [37]. The smaller the simulation step size is, the greater the simulation accuracy. After comprehensive comparison and consideration, the GSTIFF/SI2 integration method was selected in this paper.