# Research on the High Speed of Piston Pumps Based on Rapid Erecting of Launch Vehicles
## Methodology  
### _Erection Dynamics Model_  
The rapid erection of missiles is divided into three stages. 1. Before erection, the missile is placed on the frame at a horizontal or slight inclination angle and is restrained by various means to ensure the safety of the transportation process and prevent the missile from fleeing forward or tumbling horizontally in the case of braking deceleration and sharp turning. 2. During erection, the hydraulic system is injected with oil by a high-pressure axial piston pump. The piston rods of the multi-stage hydraulic cylinder are forced to lengthen sequentially, causing the missile to spin around the bottom hinge with a specific motion law and gradually transition to a vertical state. 3. After erection, the missile is in a vertical state. The travel switch is triggered, the control system issues a stop command, the hydraulic cylinder is locked, and the launch procedure can be carried out. As shown in Figure 2, the missile is subjected to the combined action of gravity. \(G(x,\,y)\), hydraulic cylinder support force \(F\), and the constraint force of each hinge point at time \(t\) during the erection, ignoring the friction force of the hydraulic rod when it is extended. The pivot point \(O\) is located at the junction of the launcher's tail and the frame. The point \(O_{1}\) is the rotating centre of the erecting cylinder around the hinge position in the middle of the frame. The point \(O_{2}\) is the rotating centre of the erecting cylinder around the launcher. Figure 2: Diagram of erection. The vertical distance between point \(O\) and point \(O_{1}\) is \(L_{3}\). \(OO_{1}=L_{1}\), \(OO_{2}=L_{2}\) and \(O_{1}O_{2}=L(t)\) increase with time. The movement law of missile erecting is indicated by the change in erecting angle \(\theta(t)=O_{2}Ox\). The force arm of the hydraulic cylinder support force \(F\) relative to the rotating centre point \(O\) is:  
\[l=L_{2}\sin(180^{\circ}-\angle O_{1}O_{2}O) \tag{1}\]  
In \(\Delta OO_{1}O_{2}\), and \(L_{2}\) are constants. According to the sine theorem, we can get:  
\[\frac{L_{1}}{\sin\angle O_{1}O_{2}O}=\frac{L(t)}{\sin(\theta_{0}+\theta(t))}= \frac{L_{2}}{\sin(\angle O_{2}O_{1}O)} \tag{2}\]  
According to the cosine theorem, it can be obtained:  
\[L(t)^{2}=L_{1}^{2}+L_{2}^{2}-2L_{1}L_{2}\cos(\theta_{0}+\theta(t))L_{1}M_{ \rm J} \tag{3}\]  
The expression of force arm \(l\) can be simplified as:  
\[l=\frac{L_{1}L_{2}\sin(\theta_{0}+\theta(t))}{L(t)} \tag{4}\]  
The moment of the erection mechanism about the centre of rotation \(O\), according to the moment of moment theorem, consists of the moment of weight \(M_{\rm G}\), the moment of inertia \(M_{\rm J}\) and the moment \(T\) provided by the hydraulic cylinder:  
\[T=M_{\rm G}+M_{\rm J} \tag{5}\]  
\[M_{\rm G}=G\sqrt{x^{2}+y^{2}}\cos(\psi+\theta(t)) \tag{6}\]  
\[M_{\rm J}=\tilde{J}\tilde{\theta}(t) \tag{7}\]  
where \(J\) is the rotational inertia of the missile with point \(O\) as the rotation centre. By combining Equations (5)-(7), the hydraulic cylinder support force \(F\) can be obtained:  
\[F=\frac{G\sqrt{x^{2}+y^{2}}\cos(\psi+\theta(t))+\tilde{J}\tilde{\theta}(t)}{L _{1}L_{2}\sin(\theta_{0}+\theta(t))}L(t) \tag{8}\]  
To solve the hydraulic cylinder support force, combine Equations (3) and (8) and add the hydraulic cylinder expansion length curve \(L(t)\). Missiles are typically built in a 150-second uniform acceleration-uniform speed-uniform deceleration sequence. In order to satisfy the needs of modern warfare's quickness, the \(L(t)\) curve is given to acquire the hydraulic bearing capacity \(F^{\prime}\) during the erection process based on the completion of the erection action in 30 s. (The hydraulic bearing capacity \(F^{\prime}\) and the hydraulic cylinder support force \(F\) are a pair of interaction forces). Figure 3 depicts the specific results. The hydraulic bearing capacity increases first and subsequently declines as the erecting angle increases.